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Post Graduate NuclearExperimental Techniques
(4NET) Course Notes
By
Dr. Paddy Regan,
Department of Physics
University of Surrey
Guildford, GU2 7XH, UK
e-mail [email protected]
October 2003
Contents
1 Electromagnetic Probes of Nuclear Structure. 2
1.1 Gamma-Ray Decay Selection Rules. . . . . . . . . . . . . . . . . . 3
1.2 Internal Conversion Electrons. . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Electric Monopole Decays. . . . . . . . . . . . . . . . . . . 7
1.2.2 Magnetic Monopoles. . . . . . . . . . . . . . . . . . . . . . 9
2 Studies of Nuclear Structure at High Angular Momentum. 10
2.1 Fusion Evaporation Reactions. . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Beam Currents, Energies and Target Thicknesses. . . . . . 11
2.1.2 Compound Nucleus Excitation Energy and Maximum An-
gular Momentum. . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Compound Nucleus Decay. . . . . . . . . . . . . . . . . . . 16
2.1.4 Excitation Functions. . . . . . . . . . . . . . . . . . . . . . 17
2.1.5 Spin Assignments: Gamma-ray Angular Distributions. . . 19
2.1.6 Anisotropies and Gated Angular Distributions. . . . . . . . 20
2.1.7 DCO Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Determining the Intrinsic Structure of Rotational Bands. . . . . . 24
2.2.1 Rotational Frequency, Moments of Inertia and Alignments. 25
2.2.2 Particle-Core Coupling. . . . . . . . . . . . . . . . . . . . . 29
2.2.3 Branching Ratios and g-Factors. . . . . . . . . . . . . . . . 32
2.2.4 Two State Mixing. . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Selected Topics in High Spin Nuclear Structure. . . . . . . . . . . 39
2.3.1 Shape Coexistence and Superdeformation. . . . . . . . . . 39
2.3.2 Band Terminations. . . . . . . . . . . . . . . . . . . . . . . 41
2.3.3 High K-Isomers and Pairing Reduction. . . . . . . . . . . 41
2.3.4 Octupole Correlations. . . . . . . . . . . . . . . . . . . . . 44
i
2.4 Branching Ratios and g-Factors in High-K Bands. . . . . . . . . . 45
3 Experimental Gamma-ray Spectroscopy. 48
3.1 Germanium Semi-Conductor Detectors. . . . . . . . . . . . . . . . 48
3.2 Gamma-Ray Spectroscopy with Germanium Detectors. . . . . . . 49
3.2.1 Response Function of Germanium Spectra. . . . . . . . . . 49
3.2.2 Germanium Detector Efficiency. . . . . . . . . . . . . . . . 51
3.2.3 The Compton Suppressed Spectrometer (CSS). . . . . . . 52
3.3 Gamma-Ray Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Resolving Power and Total Photopeak Efficiency. . . . . . 54
3.3.2 Add-Backs from Clover/Cluster Detectors. . . . . . . . . . 56
3.3.3 Polarization Measurements. . . . . . . . . . . . . . . . . . 56
3.3.4 Gamma-ray Tracking. . . . . . . . . . . . . . . . . . . . . 56
4 Channel Selection In Fusion-Evaporation Reactions. 63
4.1 Inner Multiplicity Sum-Energy Balls. . . . . . . . . . . . . . . . . 63
4.2 Studies of Very Neutron Deficient Nuclei. . . . . . . . . . . . . . . 65
4.2.1 Charged Particle Balls. . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Kinematic Focussing and Conversion from Lab to COM
Energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.3 Silicon Detectors. . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.4 CsI(Tl) Balls Using Pulse Shape Discrimination. . . . . . . 78
4.3 Neutron Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4 Recoil Detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.1 Recoil Mass Separators. . . . . . . . . . . . . . . . . . . . 100
4.4.2 Gas Filled Separators. . . . . . . . . . . . . . . . . . . . . 107
4.4.3 Recoil Decay Tagging. . . . . . . . . . . . . . . . . . . . . 108
4.4.4 Recoil Filter Detectors. . . . . . . . . . . . . . . . . . . . . 109
5 Measurement of Lifetimes of Bound Nuclear States. 113
5.0.5 Weisskopf Single Particle Estimates. . . . . . . . . . . . . 114
5.0.6 Determining Nuclear Quadrupole Deformation from Life-
times of E2 Transitions. . . . . . . . . . . . . . . . . . . . 115
5.1 Electronic Timing Methods. . . . . . . . . . . . . . . . . . . . . . 116
5.1.1 Gamma-ray Spectroscopy Across Isomers . . . . . . . . . . 120
ii
5.2 The Recoil Distance Method. . . . . . . . . . . . . . . . . . . . . 124
5.2.1 Feeding Corrections and Gating From Above. . . . . . . . 128
5.2.2 The Differential Decay Curve Method. . . . . . . . . . . . 133
5.3 The Doppler Shift Attentuation Method. . . . . . . . . . . . . . . 136
5.3.1 Lineshape Analysis. . . . . . . . . . . . . . . . . . . . . . . 139
6 Measurement of Magnetic Moments. 142
6.1 Measurement of Nuclear Magnetic Dipole Moments. . . . . . . . . 144
6.1.1 Corrections in the Ion-Implantation Perturbed Angular Dis-
tribution Technique. . . . . . . . . . . . . . . . . . . . . . 146
6.1.2 Analysis of Precession Data with Limited Angles. . . . . . 148
6.1.3 Transient Field Measurements. . . . . . . . . . . . . . . . 149
6.1.4 Time Differential Perturbed Angular Distributions. . . . . 150
7 Spectroscopy of Neutron Rich Nuclei. 153
7.1 Using Fusion Evaporation Reactions . . . . . . . . . . . . . . . . 153
7.2 Incomplete Fusion/Massive Transfer Reactions. . . . . . . . . . . 154
7.3 Deep Inelastic Reactions. . . . . . . . . . . . . . . . . . . . . . . . 156
7.3.1 Maximum Angular Momentum in DIC. . . . . . . . . . . . 159
7.3.2 Useful Formulae for Binary Reaction Studies. . . . . . . . 160
7.3.3 Doppler Correction. . . . . . . . . . . . . . . . . . . . . . . 164
8 Spectroscopy With Radioactive Ions Beams. 170
8.1 Production of Radioactive Beams. . . . . . . . . . . . . . . . . . . 171
8.1.1 Projectile Fragmentation. . . . . . . . . . . . . . . . . . . 171
8.1.2 Particle Identification in Fragmentation. . . . . . . . . . . 173
8.1.3 Isomeric Ratios and Angular Momentum Population. . . . 184
8.1.4 Projectile Fission . . . . . . . . . . . . . . . . . . . . . . . 188
8.1.5 Intermediate Energy Coulex. . . . . . . . . . . . . . . . . . 189
8.1.6 Double Fragmentation and In-beam spectroscopy . . . . . 190
8.1.7 Beta-Decay Measurements. . . . . . . . . . . . . . . . . . . 190
8.2 ISOL Based Techniques. . . . . . . . . . . . . . . . . . . . . . . . 190
8.2.1 On-Line Mass Separators. . . . . . . . . . . . . . . . . . . 191
8.2.2 The 19Ne∗+40Ca Experiment at Louvain La Neuve. . . . . 198
1
Chapter 1
Electromagnetic Probes ofNuclear Structure.
The focus of this course is the study of nuclear structure by the measurement
of particle bound nuclear state decays. These decays proceed mainly via elec-
tromagnetic interactions and can be investigated by detecting the emitted elec-
tromagnetic radiation as the state de-excites, ie. through the emission of either
gamma-rays or conversion electrons. While the strong force is the dominant in-
teraction within nuclei, the EM interaction is an excellent probe since (a) it is well
understood and (b) its weak nature compared to the strong force means that it
does not petrurb the system very much. The course will be based on methods of
exciting nuclei via interactions between an energetic beam of ions and a metalic
foil. In particular, we will look at the study of the nucleus at high values of angu-
lar momentum and exotic proton to neutron ratios. We will investigate methods
of producing exotic nuclear matter ‘in-beam’ and different methods of selecting
specific nuclei for study from a large ‘background’ of other nuclear species which
may be produced.
We can measure basic nuclear properties of excited nuclear states such as
excitation energy, angular momentum (spin) and parity using conservation laws
and electromagnetic selection rules. In addition, as we shall see later, the deter-
mination of decay probablilities of nuclear states (ie. their lifetimes) gives direct
information on the make-up of the initial and final states and can reveal highly
collective, deformed structures within the nucleus.
2
1.1 Gamma-Ray Decay Selection Rules.
For a gamma-ray decay between states of initial spin Iπi and final state spin Iπ
f ,
the gamma-ray selection rules are that the decay can proceed by a photon of
multipole order L where,
|Ii + If | ≥ L ≥ |Ii − If | (1.1.1)
Note that because the intrinsic spin of the photon is 1h, gamma-ray transitions
from 0+ → 0+ states are forbidden. These transition decay by electron conversion
and/or internal pair formation (if the transition energy is above 1.022 MeV).
M3, E4, M5,E6, M7, E8
5
3
4
+
+
+
("mixed")
(‘pure’)E2
M1, E2("mixed")
M7, E8, M9M3, E4, M5, E6,
M1, E2,
M3, E4, M5, E6, M7
-3
4
2
+
+
E1, (M2)
E3, M4, E5, M6, E7E2
M3,....,E6
E1, (M2)E3, M4, E5
(b)
(a)
Figure 1.1: Schematic decay scheme showing the effect of gamma-ray selectionrules on allowed multipolarities.
The parity of a magnetic transition of multipole order L is given by π =
3
(−1)L+1, while that of an electric transition is given by π = (−1)L.
The transition probability for a state decaying from state Ji to state Jf ,
separated by energy Eγ, by a transition of multipole order L is given by [1, 7]
Tfi(λL) =8π(L+ 1)
hL ((2L+ 1)!!)2
(
Eγ
hc
)2L+1
B(λL : Ji → Jf ) (1.1.2)
where B(λL : Ji → Jf) is called the reduced matrix element.
As figure 1.1 shows schematically, typically, the lowest multipolarity transi-
tions dominate the decays. This is an effect of the differing transition probabilties
for different multipoles (see chapter on measuring nuclear lifetimes).
The transition probability for a mixed multipolarity transition (usually re-
stricted to M1/E2 decays for in-beam decays) can be calculated in terms of the
multipole mixing ratio, δ [1, 2]. The mixing ratio for ∆I=1, parity non-changing
transitions is given by the ratio of the reduced matrix elements for the E2 and
M1 components. This is related to their partial transition probabilities, T , by
the simple equation [2]
δ2E2/M1 =
T (E2 : J− > J − 1)
T (M1 : J− > J − 1)(1.1.3)
The experimentally measured branching ratio for competing ∆I = 2 (E2)
and ∆I = 1 and (M1/E2) transitions is related to the ratio of reduced transition
probabilities (B(M1) and B(E2)) by the expression [3, 4]
B(M1)
B(E2)= 0.697
E52
E31
1
1 + δ2E2/M1
Iγ(∆I = 1)
Iγ(∆I = 2)(1.1.4)
where Iγ are the experimentally measured gamma-ray intensities for the com-
peting transitions. (Note the ∆I = 1 intensity contains both M1 and E2 admxi-
tures).
1.2 Internal Conversion Electrons.
A competing process to gamma-ray emission in the decay of bound nuclear states
is internal conversion where an atomic electron is emitted. Here the EM field
of the nucleus interacts with an atomic electron and the energy released by the
nuclear decay is transferred to the electron causing it to be ejected from the
atom. The electron is released with a kinetic energy equal to the energy difference
4
between the nuclear states minus the atomic binding energy for the electron shell
from which it was emitted. Thus the kinetic energy of the conversion electron is
given by,
Ee− = Eγ − B.E. (1.2.5)
where Eγ is the energy of the competing gamma decay and B.E. is the electron
binding energy.
202Po
counts
transition energy (K) in Po [keV]
electrons
400 600 800 10000
2000
4000
385.7
442.7
571.2
526.2
676.8
912.1
gammas
2x10
0
400
800delayed (35 - 550 ns)
385.7
442.7
571.2
676.8
526.2
912.1
Figure 1.2: Experimental internal conversion and gamma-ray spectra for transi-tions in 202Po. The 386, 443, 571 and 677 are E2 decays, the 526 is an E1 andthe 912 is an E3.
Figure 1.2 shows a comparison of electron and gamma-ray spectra for decays
in 202Po [8]. Note that while the gamma-ray transitions are single lines, the
electrons come in groups. This reflects the fact that the electrons come out with
different energies depending on which shell they are emitted from. Electrons
from the 1s, or K-shell are most likely to be emitted, and are most bound, thus
causing them to have the lowest energy. Electrons from the L and M shells can
also be observed with higher energies (since their binding energies are less).
The total decay intensity from a particle bound state is given by I where,
5
Experimental ICC’s (T. Kibedi et al., Australian National University)
202Po
ICC [K shell]
Transition energy [keV]
200 400 600 800 1000 1200-310
-210
-110
010
E1
M1
E2
M2
E3
Figure 1.3: Experimental internal conversion coefficients for transitions in 202Po.These data were taken from the reaction 194Pt(12C,xn)202Po at a beam energy of76MeV [8].
I = Iγ +∑
i
Iec = Iγ
(
1 +∑
i
αi
)
(1.2.6)
where αi is the internal conversion coefficient for the ith electron shell. Inner
shell electrons (K,L,M) are more likely to be converted than outer lying ones, as
long as the energy of the transition is greater than the electron binding energy
for that shell.
Experimentally, electron conversion coefficients are very useful as they are
dependent on the multipolarity of the transition [10, 11] and can thus give infor-
mation on the spin and parity of nuclear states. Transition multipolarities can be
assigned by either measuring the absolute internal conversion coefficients αi = Γe
Γγ
or the ratio of the partial conversion coefficients (eg. αM
αL).
In the case of mixed multipolarity ∆I=1 transitions, the experimentally de-
termined value of the electron conversion coefficient directly gives the magntitude
of the E2/M1 mixing ratio. Figure 1.3 shows the experimentally determined in-
ternal conversion coefficients for 202Po determined from the spectra shown in
figure 1.2.
The size of the electron conversion coefficient increases with (a) decreasing
transition energy, (b) increasing Z of the nucleus and (c) increasing multipolarity.
The decays from isomeric (long lived) nuclear states are often accompanied by a
low energy transition and/or a large change in multipolarity. Such decays can be
well studied using pulsed beam techniques by observing the conversion electrons
6
emitted between the beam bursts (see later).
The emission of a conversion electron results in an electron vacancy being
filled and the subsequent emission of a characteristic X-ray which can be used to
identify the proton number of the nucleus of interest.
The internal conversion coefficients for electric (E) and magnetic (M) multi-
poles can be calculated using the following expressions [5],
α(EL) ≈ Z3
n3
(
L
L+ 1
)
(
e2
4πǫohc
)4 (2mec
2
Eγ
)L+ 5
2
(1.2.7)
α(ML) ≈ Z3
n3
(
e2
4πǫohc
)4 (2mec
2
Eγ
)L+ 3
2
(1.2.8)
Note that there has been a recent report of the first example of ‘bound state’
internal conversion, where the electron is not emitted from the atom, but rather
raised to a higher lying atomic bound state [6].
1.2.1 Electric Monopole Decays.
Electric monopole decays (E0) between two 0+ states decay only by electron
conversion (and/or internal pair formation for transition energies greater than
1.022 MeV).
The total transition probability for E0 decays is given by [12] is given by
Γ(E0) =1
τ= ρ2
(
0+i → 0+
f
)
∑
j
Ωj (Z, k) (1.2.9)
where τ is the (partial) lifetime for the E0 decay, ρ is the (dimensionless)
monopole strength parameter and Ωj are the electronic factors (analagous to
internal conversion coefficients) [13]. The electronic factors are tabulated in ref-
erence [13] and depend on the Z of the nucleus and the energy of the transition.
In most cases, K-conversion dominates. The nuclear structure information is
contained in the monopole strength parameter which is defined by [12]
ρ =< 0+
f |∑
j ejr2j |0+
i >
eR2=< 0+
f |m(E0)|0+i >
eR2(1.2.10)
where R is the nuclear radius (1.2A1
3 fm) and m(E0) is the electric monopole
operator.
7
Figure 1.4: Out of beam decay spectra for the reaction 144Sm(33S,p2n)174Ir at 153MeV. Non-yrast E0 decays are observed in 174Os which was populated via thebeta-decay of 174Ir. The beam was incident on a 1.3 mg/cm2 target and irradiatedfor 4 seconds followed by a 4 sec measuring cycles. The lines in 174Os correspondto the following decays: 546 keV, 0+ → 0+, pure E0 ; 532 keV, 2+ → 2+, mixedE2+M1+E0; and 555 kev 4+ → 4+, mixed E2+M1+E0. Note the absence of a532 keV line in the gamma-ray spectrum [16].
The single particle units for E0 decays are given by [17],
ρ2sp = 0.5A− 2
3 (1.2.11)
where A is then nuclear mass number. This gives a useful scaling of E0
strengths, independent of mass number.
The lifetime of E0 decays can be used to infer the degree of mixing and/or
the change in deformation between two 0+ configurations [12, 14, 15, 17] and is
a very useful tool in the study of shape coexistence in nuclei.
The E0 matrix element can also be used to measure the admixture of differ-
ent nuclear states with different radii (ie. different deformation). The electric
8
monopole operator can be expanded [18, 296] in terms of the quadrupole and
triaxial deformation parameters β and γ respectively such that
M(E0) =(
3Z
4π
)
[
4π
5+ β2 +
(
5(√
5)
21√π
)
β3cosγ
]
(1.2.12)
In the limit of simple two-state mixing between configurations with deforma-
tions γ1, β1 and γ2, β2, if a is the mixing amplitude between the configurations,
the resulting monopole strength is given by
ρ2(E0) =(
3Z
4π
)2
a2(
1 − a2)
[
(
β21 − β2
2
)
+
(
5√
5
21√π
)
(
β31 cos γ1 − β3
2 cos γ2
)
]2
(1.2.13)
Most observed 0+ → 0+ E0 decays are between states where at least one of the
states is predominantly spherical in nature [12, 14] and it is usual to keep terms
only up to order β2 in equations (2) and (3). However, it has been suggested [15]
that in the case of prolate oblate mixing, the second term may become important
since the first vanishes for equal deformations of opposite sign.
Note that E0 decays also occur between nuclear states with the same spin
and parity (ie. Jπ → Jπ), although these will also compete with higher mutipole
gamma-decays. A review of E0 decays can be found in [17].
1.2.2 Magnetic Monopoles.
In principle, decays from 0− to 0+ states could decay by magnetic monopole type
transitions [19]. Such decays are forbidden to decay by photon emission but it
has been proposed [20] that they may decay via mono-energetic electron emission
by a cascade of virtual E1 and M1 pairs. To date this decay mechanism has not
been observed, although several searches have been made, eg. [19].
9
Chapter 2
Studies of Nuclear Structure atHigh Angular Momentum.
In order to fully understand the physics of the nucleus, one needs to examine the
effect of extreme conditions, (such as high temperature/excitation energy, distor-
tions of the nuclear shape, exotic proton to neutron ratios and large rotational
stresses), on nuclear matter. This chapter will deal with the study of high values
of angular momentum on the nucleus. We shall examine the formation of high
spin states via the mechanics of fusion-evaporation reactions and look at some
of the ways of analysing the spectroscopic information gained in such pursuits to
characterise the different nuclear structures observed in the decay of high spin
states.
2.1 Fusion Evaporation Reactions.
In order to study high spin states, one requires a reaction which will impart the
largest possible angular momentum into the nucleus of interest. Figure 2.1 shows
schematically different types of nuclear reaction depending on the value of the
impact parameter, b.
Fusion-evaporation reactions are the best way experimentally of producing
high spin states with large cross-sections. In a fusion-evaporation reaction, the
kinetic energy of the collision in the centre of mass frame is converted into ex-
citation energy of the compound system. The amount of angular momentum
transferred into the compound nucleus is given by b× p where b is the impact
parameter and p is the linear momentum, mv, of the beam. The angular momen-
10
elastic (Rutherford)scattering
b
Rfusion
inelastic scattering(Coulex)
DIC
Figure 2.1: Various types of heavy-ion collisions as a function of impact param-eter.
tum transfered is simply l = mvb. Thus, the higher energy the beam particles,
the more angular momentum will be transfered into the compound system. Note,
however, that as figure 2.1 shows, fusion reactions only occur for small values of
impact parameter, with other nuclear reactions occuring at increased target-beam
distances.
2.1.1 Beam Currents, Energies and Target Thicknesses.
Due to the small size of the nucleus, most of the beam particles simply ‘miss’
the target nuclei. Total fusion cross-sections are usually of the order of 1 barn
(10−28 m−2) for beam energies around the Coulomb barrier (∼ 3 → 5 MeV/A).
(Note the geometrical area for the reaction of two nuclei will be approximately(
1.22(A1 + A2)1
3
)2fm2). This fusion cross-section drops dramatically for heavier
nuclei where fission begins to dominate over fusion-evaporation. Typical beam
currents for fusion-evaporation experiments are of the order of a few particle
nano-amps (∼1010 particles per second) and are used to bombard relatively thin
target foils of thicknesses of the order of 1 mg/cm2 (dividing by the target density
gives the physical thickness).
Higher beam currents can be used to study nuclei at energies below the
Coulomb barrier, but a limit is usually set by either (a) the production of the
machine supplying the beam or (b) the deadtime of the data acquisition elec-
11
tronics/detectors in the experiment. Targets often have a thicker gold or lead
backing to stop the recoiling nuclei within the view of the detector system. The
choice of these high-Z stoppers is due to their higher Coulomb barrier which re-
duces the likelihood of beam-induced fusion events in the backing. Also, higher Z
stoppers cause the nucleus to slow down faster. The beam and target assemblies
are housed in high vacuum of around 10−6→7 Torr. A beam stop, or Faraday cup,
is usually placed downstream, behind the target and acts as a monitor for the
beam current.
Figure 2.2 gives a schematic set-up of a typical, thick target fusion-evaporation
experiment.
radiation detectors(gammas, e-)
Pb/Au stopper
targetthin
~ E-7 Torr beam stop/Farady cup
vaccum vessel
beamI~1 pnA outgoing beam
Fusion ~ 1mb
Figure 2.2: Schematic of a typical ‘in-beam’ set-up for the study of high-spinstates using a fusion evaporation reaction.
2.1.2 Compound Nucleus Excitation Energy and Maxi-mum Angular Momentum.
For fusion to occur the beam nuclei must have sufficient kinetic energy to over-
come the Coulomb repulsion between the two positively charged nuclei. Fusion-
evaporation reactions require the formation of a compound nucleus. This de-
scribes a hot nuclear system which lives long enough (> 10−20 s) for thermody-
namic equilibrium to occur, during which time the compound system ‘loses its
12
memory’ of how it was formed in terms of the make up of the target and projectile
nuclei [1]. However, quantities such as total energy and angular momentum are
conserved. By conservation of energy, the compound nucleus will be formed at
an excitation energy which depends on the centre of mass kinetic energy of the
collision and the Q-value for compound nucleus formation such that
13
Eex = Ecm +Qfus (2.1.1)
Ecm is the kinetic energy of the collision which is transfered to the compound
system. It can be calculated by taking the kinetic energy of the beam, EB and
subtracting the kinetic energy of the recoiling compound system, ER. Thus
Ecm = EB − ER (2.1.2)
By conservation of momentum, for beam and target masses of MB and MT
respectively, the velocity of the recoiling compound, VR can be calculated using
MBVB = (MT +MB)VR (2.1.3)
and by conservation of energy,
Ecm = EB − 1
2(MT +MB)V 2
R (2.1.4)
substituting in for VR, and recalling that EB = 12MBV
2B, we obtain
Ecm = EB
(
1 − MB
MT +MB
)
(2.1.5)
The maximum angular momentum that can be transferred in a fusion-evaporation
reaction will occur when the two nuclei are just touching in a peripheral collision.
(This is the so-called sharp cut-off approximation which assumes the nuclei are
‘hard spheres’ without a diffuse surface). The fusion cross-section will be a sum
of partial waves (depending on the size of the impact parameter). In the sharp
cut off approximation, the assumption is that the transmission coefficient Tl for
nuclear penetration falls to zero for l > lmax and has a value of 1 for l ≤ lmax.
Thus, the total fusion reaction cross-section, σf can be written as a sum of partial
waves upto lmax such that [1]
σf = π
(
λ
2π
)2 lmax∑
l=0
(2l + 1)Tl ≈ π
(
λ
2π
)2
(lmax + 1)2 (2.1.6)
where λ is the wavelength of the entrance channel given by
λ =h
2√
2Ecmµ(2.1.7)
14
Ecm is the kinetic energy of the collision in the centre of mass and µ is the
reduced mass of the system such that µ = ABAT
AB+ATwhere AB and AT are the
masses of the beam and target nuclei respectively.
The value of lmax, calculated using the reduced mass of the system, µ =MT MB
MT +MB, and from conservation of energy and angular momentum is given by [1]
hlmax = µvR (2.1.8)
where the velocity v, can be calculated using conservation of energy in terms
of the kinetic energy of the collision in the centre of mass and the Coulomb barrier
(Vc), by the expression
1
2µv2 = Ecm − Vc (2.1.9)
Substituting in for v we obtain,
l2max =2µR2
h2 (Ecm − Vc) (2.1.10)
where R is the maximum nucleus-nucleus distance for which a reaction can
occur and is given (in fm) empirically by [1]
R = 1.36(
A1
3
B + A1
3
T
)
+ 0.5 (2.1.11)
The Coulomb barrier energy Vc (in MeV) is given by
Vc = 1.44ZBZT
R(2.1.12)
It is clear from equation 2.1.10 that those collisions which maximise the value
of the reduced mass (ie. symmetric reactions) will have the largest input angular
momentum for a given centre of mass energy.
The experimental data on compound nucleus reactions shows that at very
high bombarding energies, the angular momentum in the compound system is
somewhat less than given by lmax in equation 2.1.10. This is because for higher
energies, compound fusion formation can only occur for smaller impact parame-
ters (ie. not the peripheral collisions used to calculate lmax). The critical angular
momentum lcrit, is the maximum angular momentum for which fusion can occur
and can be estimated by the expression [1]
15
(
lcr +1
2
)2
=µ (RT +RB)3
h2
(
4πγRBRT
RT +RB
− ZBZT e2
(RT +RB)2
)
(2.1.13)
where γ ≈0.9 MeV fm−2 is the surface tension of the nucleus.
2.1.3 Compound Nucleus Decay.
It typically takes around 10−21→22 s for a beam nucleus to pass a target nucleus.
If the beam and target nuclei do interact and fuse together, thermodynamic
equilibrium occurs within about 10−20 second, after which the compound system
decays by either high energy gamma-ray emission (such as giant resonance de-
cays) and/or by nucleon evaporation, where neutrons, protons and α-particles are
emitted [1, 21]. Due to the effect of protons and alpha-particles having to tunnel
through the Coulomb barrier, charged particle emission is inhibited compared
to neutron evaporation for compound systems closer to stability. Once the com-
pound nucleus moves further to the neutron deficient side, the neutron separation
increases and the proton separation decreases allowing charged particle (proton
and alpha) emission to compete and often dominate over neutron evaporation.
nucleon separationenergy above yrast
yrast line(locus of yrast states)
gammas in residual nuclei
compound nucleus
Exc
itatio
n en
ergy
Angular momentum
4
3
2
1
E-20 secs
E-9 sec
E-15s
Figure 2.3: Schematic of the formation of high-spin residual nuclei from com-pound nucleus decay.
16
Due to the very high density of states in the highly excited compound system,
the evaporated particles have a statistical energy spectrum and reduce the exci-
tation energy of the compound system by around 5-8 MeV per nucleon, yet only
remove 1→2 h of angular momentum. Particle evaporation will continue until
the system reaches a state where the excitation energy is less than the particle
separation energy above the yrast line. The yrast state is the state of lowest
energy for a given value of angular momentum.
It takes around 10−15 seconds for the compound nucleus to decay into the
residual nucleus. Note that as figure 2.3 shows, the final nucleus created is de-
termined by the entry point (relative to the yrast line) in the excitation en-
ergy/angular momentum plane. Generally speaking, residual nuclei formed by
the emission of fewer evaporated particles have higher initial angular momenta
and excitation energy distributions than higher multiplicity evaporation channels.
As shown later, this effect can be used to experimentally select transitions from
specific evaporation channels using total energy/gamma-ray multiplicity detec-
tors.
The probability for a compound nucleus evaporating a particle (usually mean-
ing a proton, neutron or alpha particle) is proportional to the density of final
states and a barrier (Coulomb usually) transmission coefficient, given by [1]
T (li, Ep(i)) = exp
− −2h∆
(2mp (V − Ep))1
2
(2.1.14)
where V is the height of the barrier and ∆ is its width. Note that the shape of
the spectrum for the emitted particles is different for charged particles (protons
and alphas) compared to neutrons due to the effect of the Coulomb barrier. In
neutron deficient compound systems, the neutron separation energy is so high
(upto 15-20 MeV), that it is larger than the height of the Coulomb barrier at a
given excitation energy, so charged particle emission is then favoured.
2.1.4 Excitation Functions.
If an experiment is interested in studying a particular nucleus or set of nuclei, it
is usual to perform an excitation function to decide on the optimum beam energy
to maximise the cross-section and angular momentum input for the channel of
interest. Clearly, increasing the beam energy will both increase the maximum
17
50 60 70 80 90EBEAM (MeV)
101
102
103
104
105
106
σ (a
rb. u
nits
)
p4n (109
Ag)
p3n (110
Ag)
5n (109
Cd)
4n (110
Cd)
798 keV
249 keV
191 keV
125 keV
335 keV
523 keV
Figure 2.4: Excitation function for various products of the reaction 18O+96Zr[22].
input angular momentum, but will also increase the excitation energy of the
compound system, resulting in more particles being evaporated.
Figure 2.4 shows the relative intensity of various known gamma-ray transi-
tions from nuclei of interest in fusion of an 18O beam on a 96Zr target (forming
the compound nucleus 114Cd. Note how the relative intensity of the higher multi-
plicity channels such as the 5n to 109Cd and p4n to 109Ag increase with increasing
beam energy in the region of this excitation function, while the four particle out
channels to 110Cd and 110Ag peak at a beam energy of around 60 MeV, before
falling off. Note that the shape of the excitation function of all lines should be
similar for a given channel and thus, the variation of gamma-ray intensity of
a line with beam energy can be used to identify a transition with a particular
evaporation product.
18
2.1.5 Spin Assignments: Gamma-ray Angular Distribu-tions.
In order to determine the spins (and infer the parities) of excited nuclear states
formed in fusion-evaporation reactions, one can measure the angular distribution
of the gamma-ray transitions. In order to observe an anisotropic distribution
one needs to populate the nucleus in a way which gives rise to states of aligned
angular momentum with a specific orientation in space. This is achieved in fusion-
evaporation reactions, where the angular momentum vector (l = r × p) is in a
direction (to a good approximation) perpendicular to the beam direction (see
figure 2.5).
after evaporationsubstate alignment
beam direction
target
reaction plane
Figure 2.5: Schematic of initial orientation in fusion-evaporation reactions.
The orientation of the nucleus will be slightly attenuated by the emission of
evaporated particles (neutrons, protons and alpha-particles) and by the emission
of gamma-rays. The effect will be to provide a substate, or m-state alignment,
peaked symmetrically about the the m = 0 value corresponding to the reaction
plane.
The general formula for the angular distribution function is given by [1, 23]
19
W (θ) =∑
k
AkPk (cosθ) (2.1.15)
where W (θ) is the gamma-ray intensity measured at angle θ to the beam
direction; for parity conserving decays, such as gamma-ray emissions, k=even
numbers less than or equal to 2l where l is the angular momentum taken away
by the emitted photon; Pk (cosθ) are the standard Legendre polynomials; and Ak
is the angular distribution coefficient. The Ak value depends on the substate or
m-population distribution and the values of the initial and final state spins [23].
By measuring the intensity of a gamma-ray transition as a function of detec-
tor angle about the beam direction, a full angular distribution can be obtained
from which the values of A2 and A4 can be obtained by fitting the distribution to
equation 2.1.15. The Ak values can be used to experimentally distinguish between
transitions of different multipolarities [23, 24]. Similarly, fitting the experimen-
tally observed A2 and A4 coefficients for transitions of known multipolarity (and
possibly mixing ratios) gives a measure of the degree of substate alignment for
that spin. Generally, in fusion evaporation reactions, transition multipolarities
can be restricted to angular momentum values of 2 or less (ie. usually only E2,
M1 or E1 decays are observed).
For a pure dipole ∆I = 1 transition (E1), the angular distribution will be
given by
W (θ) = A0 1 + A2P2 (cosθ) (2.1.16)
where P2 (cosθ) = 12(3cos2θ − 1) and A0 is the ‘true’ intensity.
For a quadrupole (∆I=2) transition (E2), the distribution will have the form,
W (θ) = A0 1 + A2P2 (cosθ) + A4P4 (cosθ) (2.1.17)
where P4 (cosθ) = 18(35cos4θ − 30cos2θ + 3)
2.1.6 Anisotropies and Gated Angular Distributions.
The fixed (and often limited) angular granularity of modern gamma-ray ar-
rays coupled to the complexity of the singles spectra often observed in fusion-
evaporation reactions mean that a full angular distribution analysis may not be
viable. It is often enough to be able to tell the difference in the anisotropy between
20
∆I=1 and ∆I = 2 type transitions. In the case where only a few gamma-ray
detector angles are used, the A4 coefficient is set to a value of zero (eg. [25]).
Alternatively, coincidence data can be used to form ‘gated singles spectra’ at
specific detector angles, from which an anisotropy can be taken (eg [22]).
Pohl et al. defined one particular type of anistropy as
A = 2
(
W (37) −W (79)
W (37) +W (79)
)
(2.1.18)
where the angles correspond to detector positions in the Chalk River 8π
gamma-ray array.
-90 -60 -30 0 30 60 90arctan(δE2/M1)
-1.5
-1.0
-0.5
0.0
0.5
1.0
Ani
sotr
opy
∆ I = 0∆ I = -1∆ I = -2
Figure 2.6: Theoretical values for various transitions for the gamma-rayanisotropy as defined in reference [22].
Figure 2.6 shows the theoretically expected value for this anistropy for various
21
multipolarity decays (assuming A4 = 0). Note the different values of A2 for ∆=1
decays depending on the value of the E2/M1 mixing ratio.
Figure 2.7 shows the experimentally measured anisotropies for 108Pd and109Ag. The difference between pure ∆I = 2 (E2) and pure ∆I = 1 (E1) transi-
tions is clear. Mixed M1/E2 transitions are generally easy to identify for small
values of mixing ratio, but note that for large, positive values of the mixing ratio,
the anisotropy can not be distinguished from a pure E2 transition. The multi-
polarities for such transitions are assigned on the basis of other branches to the
same level in the decay scheme.
Figure 2.7: Experimentally measured value of anisotropy for 108Pd and 109Ag.Note the clear separation between ∆I = 2 and ∆I = 1 transitions [22]. Also,note the lower values of for stretched E2 transition in 108Pd resulting from a largerdestruction of reaction alignment associated with the emission of an α-particle.
22
2.1.7 DCO Ratios.
For coincidence measurements, the Directional Correlations from Oriented states
(DCO) method can be used to infer the spin differences between states observed
by the measurement of the gamma-decay between them [26, 27, 28]. Fusion-
evaporation reactions produce many gamma-ray transitions and consequently,
using singles data to obtain angular distribution/anisotropy data is usually only
useful for (a) strong transitions and/or (b) strongly populated reaction channels.
In obtaining high-spin data, it is more common to measure coincidence data,
using an electronic master gate of detecting at least two transitions. By gating
on a known transition in a given nucleus, much cleaner, angle gated spectra can be
obtained from which an anisotropy can be obtained. However, since the gamma-
ray detector angles are usually fixed in such experiments, the extra condition
that at least two gamma-rays must be measured introduces angular correlations
into the data, thus altering the observed angular distribution compared to true
singles data.
Since such reactions are still aligned, one can use these correlations to discrim-
inate between transitions of differing multipolarities. By measuring coincidence
data from detectors are different angles, once can construct sets of angle gated
gamma-gamma coincidence matrices, in offline sorting. These 2 dimensional spec-
tra correlate the intensities of coincidence transitions in a single cascades by the
angle at which the gamma-ray was measured. By setting a software gate on a
transition energy of known multipolarity (usually E2) on both axes and projecting
the spectra for the two angles, it is possible to distinguish transition multipolari-
ties by the intensity of the projected transition for given angles. Figure 2.8 shows
the gamma-ray spectra gated on a known E2 transition in 61Cu and projects
transitions of known E2 and E1 multipolarity. Note the difference in relative
intensity between the two spectra for the two types of decay.
By taking the ratio of the number of counts for a given transition in these two
spectra, a DCO ratio can be obtained, which can be used to tell the distinguish
between different multipolarities. Figure 2.9 shows the results of two different
types of DCO ratios (different detector angles used) for 106Cd, formed using two
different fusion-evaporation reactions. Note that in both cases, there is a clear
discrimination possible between the different types of transition.
23
Figure 2.8: Angle gated DCO correlation spectra showing the difference be-tween E2 and E1 transitions in 61Cu from the reaction 24Mg+40Ca→3p+61Cu(AYEBALL data [106]).
2.2 Determining the Intrinsic Structure of Ro-
tational Bands.
Using coincidence relationships and information from gamma-ray anisotropies
and DCO ratios, one can come up with a (complicated) decay scheme with many
different structures observed. Since it has been shown experimentally that yrast
or near yrast are preferentially populated in fusion evaporation reactions, it is
usual to assume that the spins of states increase with increasing excitation energy.
In order to understand the intrinsic make-up of the various structures ob-
served, the information contained in the decay scheme has to be used to construct
variables which one can compare with theoretical predictions.
A common treatment in the study of deformed nuclei is to take the exper-
imental information obtained on nuclear states (ie. their spins and excitation
24
Gamma-ray Energy (keV)0 200 400 600 800 1000 1200
0.5
1.0
1.5
∆J=1∆J=2 J->J
R’
R’’
∆J=1∆J=2 J->J
0.0
0.5
1.0
1.5
Figure 2.9: DCO ratios for transitions in 106Cd from (a) the CAESAR array and(b) the 8π array [3].
energies) and transform this into the intrinsic (rotational) frame of the nucleus.
2.2.1 Rotational Frequency, Moments of Inertia and Align-ments.
It can be intuitive to observe the effect of increasing/decreasing rotational fre-
quency on the structure of the nucleus.
Using a simple rotational model, the rotational frequency, ω, for a transition
between states of spin I + 1 and I − 1, with projections along the symetery and
rotation axes of K and Ix respectively, is given by the expression [30, 31, 32],
25
ω =dE(I)
dIx(I)≈ E(I + 1) −E(I − 1)
Ix(I + 1) − Ix(I − 1)(2.2.19)
The value of K is usually taken as being equal to the spin of the lowest energy
state in the band, known as the bandhead. The difference in the excitation energies
of the two states, E(I+1)−E(I−1), is simply the measured gamma-ray transition
energy, Eγ. The value of Ix can be calculated using Pythagoras theorem, such
that
Ix(I) =√
I(I + 1) −K2 ≈√
(
I +1
2
)2
−K2 (2.2.20)
substituting in to the expression for ω,
ω ≈ Eγ√
(I + 32)2 −K2 −
√
(I − 12)2 −K2
(2.2.21)
for decays from a state of spin I + 1 to one of I − 1. Note that at high spins
where I >> K, this expression simplifies to ω ≈ Eγ
2.
The quasi-particle aligned angular momentum, ix is the spin generated by the
breaking of the core, valence quasi-particles along an axis perpendicular to the
axis of symmetry. This is shown schematically in figure 2.10.
ωc
ix
i x
ω
R
K
I
j2
j1
j
Ix
R
jI
K
Ix
Figure 2.10: Schematic of components of angular momentum for a deformednucleus.
It can be approximated by taking the total angular momentum and subtract-
ing a fixed amount of spin due to the rotational motion of the ‘inert’, deformed
26
nuclear core. The quasi-particle aligned angular momentum ix, is a function of
rotational frequency and can be defined by
ix(ω) = Ix(ω) − Iref(ω) (2.2.22)
where Ix is the projection of the total angular momentum along the axis of
rotation and can be calculated using the expression,
Ix =√
I(I + 1) −K2 (2.2.23)
Iref is the reference angular momentum (of the core) which will be subtracted
and is calculated in terms of the Harris parameters [34], such that
Iref =(
I(0) + I(2)ω2)
ω (2.2.24)
The Harris parameters are usually fitted to states in the band to give a con-
stant alignment above the first band-crossing.
[h]
i//
h [MeV]// ω0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-2
0
2
4
6
8
10
12
14
16
18
106Cd band 1106Cd band 2106Cd band 3105Cd h band 11/2
106Cd band 4
Figure 2.11: Alignment plots for band structures in 106Cd and the νh 11
2
band in105Cd using Harris parameters of I0=7.0h2/MeV and I1=15.0h4/MeV3 [3].
27
Figure 2.11 shows the alignment plots as a function of rotational frequency for
structures in 106Cd and the rotational band built upon the neutron h 11
2
orbital in
the neighbouring 105Cd. At a rotational frequency of approximately 0.45 MeV/h,
band 1 in 106Cd shows a rapid decrease in rotational frequency and a large increase
of approximately 10h in aligned angular momentum. This effect is known as a
‘backbend’ [77, 78] and consitutes the crossing of a the ground state band with the
first two-quasi-particle band, which comes lower in excitation energy compared to
the simple ground state structure (ie becomes yrast) for spins above 10h. (This
is due to the two-quasi-particle band having a higher moment of inerta compared
to the ground state configuration due to a reduction in pairing correlations).
At higher spins still, there is another, smaller increase in alignment of approx-
imately 3 h in the same band. This is interpreted as the alignment of a pair of
neutrons occupying g 7
2
orbitals. If the structure of a band already contains on
of the ‘aligning orbitals’, the alignment can not proceed by the Pauli exclusion
principle and the bandcrossing is said to be blocked. Note that the h 11
2
band in105Cd already has a initial alignment of around 5h at the bandhead (from the
unpaired neutron), and thus the first (h 11
2
)2 alignment (or ‘crossing’) observed in106Cd is blocked in this structure.
The intrinic alignments are additive [31] and this is useful when quantifying
more complex structures. That is to say, at a given frequency
ix(12) = ix(1) + ix(2) (2.2.25)
where ix(12) is the alignment of the two-quasi-particle structure and ix(1) and
ix(2) are the intrinsic alignments of the single quasi-particle components, 1 and
2. Note that bands 2 and 3 in 106Cd, both the h 11
2
and g 7
2
alignments observed
in band 1 appear to be absent. This, together with the large, initial alignement
of these bands, may be taken as suggesting that bands 2 and 3 are two-quasi-
particle bands built consisting of both and unpaired h 11
2
and g 7
2
neutron coupled
together.
Moments of Inertia.
There are three types of moment of inertia used to describe high-spin rotational
structures, the static (I(0)), kinematic (I(1)) and dynamic (I(2)). The static
moment of inertia is defined by the simple relation,
28
Erot(I) =h2
2I(0)(I)I(I + 1) (2.2.26)
The kinematic moment of inertia is given by
I(1)(I) =I
ω(2.2.27)
while the dynamic moment is given by
I(2) =dI
dω≈ 4h
∆Eγ(2.2.28)
Note a calculation of the dynamic moment of inertia requires only the differ-
ence in transition energy between two decays and has no inherent spin depedence.
It is thus a useful quantity to use in cases where the decay out of a band is not
measured and the spin of the states not well established (for example in stud-
ies of super-deformed nuclei). The variation of the dynamic moment of inertia
with rotational frequency can also be used to determine the intrinsic structure of
rotational bands (see for example Fallon et al. [33]).
2.2.2 Particle-Core Coupling.
For axially symetric nuclei, rotational bands can be characterised by the single
particle excitations upon which the bandheads are built. The way that the the
odd-particle couples to the nuclear core gives rise to a number of effects (Coriolis
mixing and increased/decreased magnetic dipole moments) which show up in the
decay schemes observed for such structures [35].
29
Figure 2.12: Rotational properties of the yrast band in 106Cd, showing the back-bend due to the (11
2)2 neutron configuration crossing the yrast, ground state,
vacuum configuration, giving rise to a backbend. Note that the decrease in pair-ing correlations above the backbend gives an increased moment of inertia [3].
30
K+1
K+5
K+3
K+4
K+2
Khigh-j, low-KRotation alignedde-coupled bandLarge Coriolis mixing
Deformation alignedK bandheadInterband I=1 decaysStrongly coupled
K
j
j+5
j+3
j+1
j
j+2
j+4
(a) (b)
Figure 2.13: Schematic of decays schemes and descriptions of strong and weakcoupling of an odd-nucleon to a deformed nuclear core.
Figure 2.13 shows the two extremes of particle core coupling for deformed
nuclei. For equatorial orbits, the odd-particle has a large angular momentum
projection on the axis of rotation (ie. a large intrinsic alignment ix) and a corre-
spondingly small projection on the symmetry axis, or K value.
In the deformation aligned case, ix is small and K is large. The bandhead is
taken to have spin K and the spectra show both odd and even spins.
The strong coupling limit breaks down in the case where the Coriolis effects
are large. The Coriolis operator can be written by [36]
Hc = −2h2
I(
I.j− Ω2)
(2.2.29)
where j is the single particle angular momentum and Ω is its projection on the
nuclear axis of symmtery. In the case of weak coupling (coming from population
of high-j orbitals with small K projections), the rotation of the core is directly
coupled to the odd-particle, and indeed, the bands built on such structures have
energy spacings strikingly similar to the even-even neighbouring nuclei [35, 36].
The orbitals involved in decoupled bands are usually high-j, unnatural par-
ity states, which ‘intrude’ down from the next harmonic oscilator shell in the
31
spherical shell model as a consequence of the spin-orbit interaction. They are
thus known as intruder orbitals. In such, decoupled bands, the even-spins of the
core which coupled to the single particle are depressed in energy compared to the
odd-spins (corresponding to states where the odd-particle angular momentum is
not maximally aligned with the rotation axis). Since, these states will then be
non-yrast, they are usually not observed in fusion-evaporation reactions.
Magnetic Dipole Transition Strengths.
The strength of in-band B(M1) decays depends on the single particle structure
of the odd-particle. In the case where K 6= 12
and If = Ii − 1, the B(M1) decay
strength is given by [35, 235]
B(M1) =3
4πµ2
N (gK − gR)2K2 (I −K)(I +K)
I(2I + 1)(2.2.30)
where gK and gR are the single particle and collective g-factors respectively.
Thus, measuring the lifetime of a state which decays by a pure M1 decay allows
a measurement of the B(M1) and thus a deduction of (gK − gR) which will be
particle dependent.
These particle and core g-factors gK and gR, relate to the magnetic dipole
moment, µ, of a state of spin I with projection K by the expression [35, 88] by
µ = gRI + (gK − gR)K2
I + 1(2.2.31)
The value of gR, the g-factor of the core is approximated by the ratio of protons
to total nucleons in the nucleus (ZA). Thus, if the magnetic dipole moment can
be measured (see later) for a known spin, the K value of the state can be infered.
2.2.3 Branching Ratios and g-Factors.
The intrinsic structure of a rotational band can also be inferred in a model de-
pendent way by measuring the intensity on the ∆I=1 and ∆I=2 branches and
their energies [3]
δ2
1 + δ2=
2K2(2I − 1)
(I + 1)(I − 1 +K)(I − 1 −K)
(
E1
E2
)5 Iγ(∆I = 2)
Iγ(∆I = 1)(2.2.32)
32
where δ is the E2/M1 mixing ratio, K is the projection of the angular mo-
mentum on the axis of symmetry and I is the state spin. E1 and E2 are the
dipole and quadrupole transition energies in MeV and Iγ are the relative γ-ray
intensities.
The magnitude of the quantity |gK−gR|Q0
, (where Q0 is intrinsic quadrupole
moment, and gK and gR are the g-factors for the intrinsic state and collective
core respectively), is given by
|gK − gR|Q0
=0.93E1
δ√I2 − 1
(2.2.33)
The gK value can be approximated in the large deformation limit (ie. that
K remains a good quantum number) for a multi-quasi-particle state, using the
expression
KgK =∑
ΩgΩ =∑
ΛgΛ +∑
ΣgΣ (2.2.34)
where for Λ is the projection of the orbital angular momentum on the axis
of symmetry and Σ = ±12
is the single particle intrinsic spin projection. For
protons gΛ=1 and for neutrons gΛ=0. The gΣ values are simply the g-factors for
the single proton and neutron (+5.59 and –3.83 respectively), attenuated by a
factor of 0.6→0.8 to account for the fact that they exist in the nuclear medium
and are thus not ‘free’ [37].
Note that equation 2.2.33 only gives the magnitude of the value of gK − gR
and therefore there may be two different configurations which give branching
ratios consistent with the experimentally measured values. This anomaly can be
overcome, if one can obtain good angular distribution information on the ∆I = 1
interband transitions from which an A2 angular distribution coefficient can be
extracted and the sign of the mixing ratio inferred. The sign of the mixing ratio
is the same of the sign of the quantity gK−gR
Qo[39], ie.
sgn (δ) = sgn
(
gK − gR
Q9
)
(2.2.35)
Figure 2.14 shows a gK − gR analysis for the high-K band in 136Sm [38]. This
work assumed a value for the quadrupole moment of +4.5 eb (from the measured
deformation of the ground state band in this nucleus) and a negative value for the
mixing ratio, δ (from the measured angular distribution of the interband ∆I=1
33
Figure 2.14: Comparison of gK − gR values for the rotational band built on thehigh-K isomeric state in 136Sm. The observed branching ratios are consistentwith a Kπ=8− assignment and rule out the Kπ = 7− configuration [38].
transitions in the band). It demonstrates clearly that the Kπ = 8−, two-quasi-
neutron configuation is the only one consistent with the measured branching
ratios.
Donau and Frauendorf’s Geometrical Model.
Donau and Frauendorf [40, 41, 3] used a simplified geometrical model to estimate
the B(M1) values for a multi-quasi-particle band. For bands with no signature
splitting, the B(M1) value can be estimated (in units of µ2N) by,
B(M1) =3
8π
K2
I2
[
(g(1) − gR)(√
I2 −K2 − i(1)x
)
−(
g(2) − gR
)
i(2)x
]2(2.2.36)
where the g are g-factors for specific particles and ix is the contribution to
the aligned angular momentum from that particle. The superscripts (1) and (2)
correspond to deformation aligned and rotation aligned particles respectively.
In the rotational model, the B(E2 : I → I − 2) is given by
B(E2) =5
16πQ2
o
3(I −K)(I −K − 1)(I +K)(I +K − 1)
(2J − 2)(2J − 1)J(2J + 1)(2.2.37)
34
where Q0 is the intrinsic quadrupole moment. For an unstretched (J → J−1),
E2 transitions, the B(E2 : I → I − 1) is given by
B(E2) =5
16πQ2
o
3K2(J −K)(J +K)
(J − 1)(J)(2J + 1)(J + 1)(2.2.38)
106CdB(M1)/B(E2)
Spin
19 21 23 25 27 29 31
010
110
210
( µ/eb)2
(h ) (g d )ν ν11/2 7/2 5/22
(h ) (g d ) (g )ν ν π11/22
7/2 5/2 9/22
(h ) (g )ν π11/22
9/22
(h ) (g g ) ν π11/22
9/2 7/2
Figure 2.15: Branching ratio comparison for different configurations in the mutli-quasi-particle structure in 106Cd. [3].
The experimental B(M1)B(E2)
branching ratios (in units ofµ2
N
e2b2) are given by the
expression
B(M1)
B(E2)= 0.697
E52
E31
1
1 + δ2
Iγ(∆I = 1)
Iγ(∆I = 2)(2.2.39)
As figure 2.15 shows, by comparing the experimentally observed B(M1)B(E2)
values
with those extracted using the B(E2)s from the rotational model and the B(M1)s
from the geometrical model, a specific assignment of a mutli-quasi-particle state
can be made.
2.2.4 Two State Mixing.
Often two nuclear configurations of the same spin and parity can not be described
in terms of exactly pure configurations but rather linear combinations of two
different configurations which wavefunctions φ1 and φ2 [164].
We can write the wavefunction for the final state in terms of admixtures of
the two basis states φ1 and φ2, ie
35
unperturbed basisstates
φ1
φ 2
αφ + βφ1 2Ψ =
Ψ = −βφ + αφ1 2
experimentally observed,mixed states
E1
E2
h1
h2
> 2V
Figure 2.16: Effect of mixing two basis states to form a mixed initial state.
Ψ = αφ1 + βφ2 (2.2.40)
The general way this is written is in the matrix form of the Schrodinger
equation such that
HΨ = EΨ (2.2.41)
In matrix form this may be written in terms of the unperturbed energies of
the basis states (h1 and h2) and the interaction which mixes the two states, (such
as pairing or Coriolis effects) v. Then
(
h1 v12
v12 h2
)(
αβ
)
= E
(
αβ
)
h1 and h2 are the unperturbed energies that the states would have if there was
no interaction between the two configurations, ie. the energies of the theoretical
states corresponding to the pure wavefunctions φ1 and φ2. The eigen values, E are
the experimentally observed energies of the two states and v12 is the interaction
matrix element which corresponds to the strength of the interaction between the
two configurations. α and β are the eigenvectors.
(
h1 − E v12
v12 h2 − E
)(
αβ
)
= 0
36
Since α and β are not zero, the determinant of the first matrix must equal
zero. In the case of simple two state mixing, the energy of the observed states (the
eigenvalues) are related to the unperturbed energies and the interaction matrix
element by solving the determinant
[
h1 −E v12
v12 h2 − E
]
= 0
which gives
(h1 − E) (h2 − E) − v212 = 0 (2.2.42)
This is a quadratic equation in E which has two roots (which are the two
energy eigen-values) given by
E =
(
h1 + h2
2
)
±
√
√
√
√
(
h1 − h2
2
)2
+ v212 (2.2.43)
The wavefunctions of the experimentally observed states can then be written
in terms of a linear combination of the basis states φ1 and φ2. The strengths
of the two final states in terms of the basis states weighting, α and β can be
calculated in terms of the h1, h2, E1, E2, and v12 by solving the expressions
α(h1 − E) + v12β = 0 (2.2.44)
for the higher energy state and
α(h2 − E) + v12β = 0 (2.2.45)
for the lower one and normalising with the condition that
α2 + β2 = 1 (2.2.46)
Gathering the terms and summarising, the two mixed states with energies
given by equation 2.2.43 have wavefunctions which can be written in terms of
Ψ =
√
(
1 ∓ Y
2
)
φ1 ±√
(
1 ± Y
2
)
φ2 (2.2.47)
where
37
Y =X
√
(1 +X2)(2.2.48)
and
X =(h2 − h1)
2v12(2.2.49)
Alignment (h)
Rotational frequency (keV)0 100 200 300 400
0
2
4
6
8
10 180W|V|=141.5 keV
Figure 2.17: Alignment plots for the yrast and Kπ=8+ band in 180W showing theunperturbed bands [42].
From the experimentally observed (perturbed) energies of two states, an es-
timate of the minimum value of the mixing matrix element can be obtained,
and the unperturbed energies of the basis states obtained. These can then be
used to calculate the unperturbed alignments for the two configurations which
are thought to mix at the band crossing point. Figure 2.17 shows the bandcross-
ing of the yrast states in 180W (decay scheme shown in figure 2.18) which has
been attributed to an interaction between the K=0 ground state band and the
Kπ = 8+, t-band [42]. Note the interaction between the even spin members of
the two bands, while the odd-spin members of the Kπ=8+ remain unperturbed.
This approach can be expanded for more than two bands. For interactions
between three bands, the unperturbed energies can be obtained by diagonlising
the three dimensional determinant [43],
38
Figure 2.18: Partial decay scheme for 180W showing the ground state and Kπ =8+ band. The alignment in the yrast band is interpreted as a t-band crossing[42].
h1 −E v12 v13
v12 h2 − E v23
v13 v23 h3 − E
= 0
2.3 Selected Topics in High Spin Nuclear Struc-
ture.
In this short section, we will briefly discuss a few ‘hot topics’ in the high spin
study of nuclei. Interested parties are encouraged to look up the references for
the topics of interest.
2.3.1 Shape Coexistence and Superdeformation.
Shape coexistence occurs when two competeing minima exist in the nuclear poten-
tial energy surface corresponding to different shapes of the nuclear mean field[44].
Such effects often occur in nuclei with oe near magic numbers of protons and
39
neutrons. Here, a spherical configuration competes with a deformed shape corre-
sponding to particle-hole excitations across the shell gaps. This deformed config-
uration populates deformation driving orbitals (those which drop down rapidly
with energy for increasing deformation). Good examples of studies of spherical-
prolate shape coexistence in heavy nuclei for very neutron deficient nuclei around
the Z=82 magic number can be found in references [46, 45, 47, 48].
The study of very elongated or superdeformed nuclei is an extreme form of
shape coexistence [49, 50, 51]. Superdeformed decays are characterised by long
cascades of regularly spaced E2 transitions, with a large I(2) moment of inertia
very large, in-band B(E2) values, corresponding to highly collective decays. Using
the rotational model, these B(E) correspond to nuclear quadrupole deformations
of β ∼ 0.35 → 0.6 or major to minor axis ratios of between 3:2 [53, 54, 55, 56]
and 2:1 [57, 58]. The regions where superdeformed nuclei have been observed,
are close to those nuclei with proton and neutron numbers which correspond to
large shell gaps (regions of low level density) at large prolate deformations in
the deformed shall model potential. For example, the superdeformed nuclei in
the A∼150 region are centred about 15266 Dy86, where both proton and neutron
numbers correspond to superdeformed magic numbers [52]. Similar regions of
superdeformation have been identified around 13258 Ce74 [65, 59, 64], 192
80 Hg112 [50]
and 8238Sr44 [61].
100.0 300.0 500.0 700.0 900.0Energy (keV)
0.0
1000.0
2000.0
3000.0
Cou
nts
192Hg Band 2
194Hg Band 3
200 400 600 800Egamma (keV)
0.6
0.8
1.0
1.2
∆Spi
n (h
)
90
100
110
120
130
J(2) (
Mev
/h2 )
10
20
30
40
50
Sp
in (
h)
Figure 2.19: Identical superdeformed bands in 192Hg and 194Hg [66].
A partilcularly interesting and puzzling aspect of the study of superdeformed
40
bands is the presence of rotational bands with almost identical energy spacing in
SD bands of neighboring nuclei, the so called identical bands effect [62, 63, 64, 65].
This is surpring in light of the expected drop in pairing corrlations caused by the
presence of an odd, unpaired nucleon which in normal deformed nuclei, causes a
reduction in the I(2) moment of inertia by about 10-15%.
Much work has been invested over the past 10 years in identifying the direct
decays out of SD bands into the lower spin yrast states. This information is very
important as without it, the spin and excitation energy of the SD band memebers
can not be exerimentally determined with any certainty. There have been recent
ports of direct links from SD bands in the A∼190 region [67, 68], however such
links in the A∼80 and 150 regions currently remain elusive.
2.3.2 Band Terminations.
Since the nucleus is a finite system, the collective angular momentum observed in
a nucleus must have a microscopic basis. It is generated by the individual valence
particles. Thus, the maximum spin available will be determined by the maximal
coupling of the single particle spins for those particles outside the inert core. In
order to generate higher angular momentum valuesthe core must be broken and
the collective band structure will be ’terminated’ This can occur rapidly, over
one or two transitions [69, 70, 71] or more gradually over a number of transitions
[72, 73, 74]. The latter effect is termed ’soft’ of ’smooth’ band termination [75]
and is characterised by an increase in gamma-ray transition energy (corrsponding
to a reduction in the moment of inertia and collectivity) with increasing spin.
2.3.3 High K-Isomers and Pairing Reduction.
For deformed nuclei with valence nucleons with large angular momentum pro-
jections, Ω, on the nuclear axis of symmtery, we can assign a total projection of
the angular momentum on the symmtery axis, K, where K =∑
j Ωj . The K-
selection rule for gamma-decays is that for K remaining a good quantum number,
∆L ≥ ∆K, where ∆L is the multipolarity of the transition linking two bands
differering by ∆K. Thus, if ∆K is (≥ 2), then unusually high order multipoles
are required for such decays to occur, and an isomeric state results. For nuclei
in the vicinity of Z∼72 and N∼106, both the proton and neutron Fermi sur-
faces lie in the vicinity of high-Ω orbitals. (K-isomeric states have also been
41
Figure 2.20: Smooth band termination in 110Sb [76].
observed in N=74 isotones [90, 91, 92, 93, 94]). By breaking pairs of nucle-
ons, very high-Ω states can be obtained relatively cheaply in energy and these
compete energtically with the yrast states and a large number of K-isomeric
states and their associated rotational bands have been observed in this region
[37, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]. The study of the rotational bands
built in these states is of particular interest since, these configurations are built
on configuration consisting of upto eight unpaired nucleons [80, 9] and the effect
this has on the overall pairing correlation can be invetsigated in a configuration
dependent manner by obtaining the moment of inertia for the various seniority
band structures and noting the increase with number of broken pairs [80].
The reduction in the moment of inertia of a nucleus with pairing correlations,
compared to the rigid body value can be estimated using the Migdal formula [89],
Ip,n = Irig
(
1 − x+√
1 + x2
x√
1 + x2
)
(2.3.50)
where
x = δhω0
2∆p,n(2.3.51)
42
where ∆ is the pair gap parameter, hω0 is the harmonic oscilator frequency
and I is the moment of inertia.
Figure 2.21: Partial decay scheme for 178W showing 0, 2, 4, 6 and 8 quasi-particlestructures [9].
By investigating the moments of inertia of mutli-quasi-particle states (where
various numbers of valence pairs are blocked), the effect on pairing on the mo-
ment on inertia (and visa versa) can be investigated for various specific orbital
configurations. As an example, figure 2.21 shows the partial level scheme for178W, which shows a large number of different rotational bands built on 0, 2, 4, 6
and eight quasi-particle states. The moments of inertia of the various bands can
be extracted from the gamma-ray transition energies and these are plotted in fig-
ure 2.22. The increase in moment in inertia with increased blocking (ie. reduced
pairing correlations) is clear. However, note that even for the eight quasi-particle
structure, the value if still some way from the rigid body value.
43
0 5 10 15 20 25 30Bandhead Spin (h)
0.0
20.0
40.0
60.0
80.0
Mom
ent o
f in
ertia
(h
2 MeV
-1)
Exact calc.BCS calc.Expt.Rigid body
gsb
2-qp4-qp
6-qp8-qp
Figure 2.22: Experimental and calculated moments of inertia for various multi-quasi-particle bands in 178W [9].
2.3.4 Octupole Correlations.
The study of reflection assymetric, octupole deformed nuclei at high spins [95]
is centred around regions of the nuclear chart where there are sets of nuclear
orbitals which differ in both total spin J and orbital angular momentum l, by
three units, ie ∆J = ∆L = 3, such as the following pairs of spherical orbitals,
(j 15
2
, g 9
2
), (i 13
2
, f 7
2
), (h 11
2
, d 5
2
) and (g 9
2
, p 3
2
). Regions where octupole correlations are
particularly favoured to occur where both protons and neutron occupy regions of
the Fermi surface where these combinations of such orbitals lie closeby in energy.
For example, in the actinide region around Z∼88 (i 13
2
, f 7
2
) and N∼134 (j 15
2
, g 9
2
)
[95, 96, 97] and in the neutron rich rare earth region around Z ∼56 (h 11
2
, d 5
2
),
N ∼88 (i 13
2
, f 7
2
) [95, 99, 100].
Nuclei with static octupole deformations (β3) are characterised rotational
bands of opposing parity interleaved by enhanced electric dipoles (E1) (see fig-
ure 2.23). Typical E1 strengths in heavy nuclei range between 10−4 and 10−7
Weisskopf units. For nuclei which exhibit octupole deformation, this increases to
44
0+2+
4+
(6+)
(8+)
(10+)
(12+)
(14+)
(16+)
(18+)
(20+)
1– 242.13– 317.3
(5– )
(7– )
(9– )
(11– )
(13– )
(15– )
(17– )
(19– )
0+84.5 2+4+
6+
8+
10+
12+
(14+)
(16+)
(18+)
(20+)
(22+)
(24+)
(26+)
(28+)
1– 215.93–5–
7–
9–
11–
(13– )
(15– )
(17– )
(19– )
(21– )
(23– )
(25– )
(27– )
0+2+ 67.7
4+
6+
8+
10+
12+
14+
16+
18+
(20+)
(22+)
(24+)
(26+)
(28+)
1– 253.73– 321.55– 447.0
7–
9–
11–
13–
15–
17–
(19– )
(21– )
(23– )
(25– )
(27– )
111.2
190.7
248.4
292.9
330.1
363.9
396.0
425.5
452.3
476.7
229.3
289.0
338.3
379.6
415.0
444.8
470.8
166.5
228.6
276.1
313.7
345.3
373.8
400.2
424.4
447.1
467.1
485.1
500.8
518.2
142.6
207.9
265.9
314.8
356.7
391.1
419.4
442.9
462.8
479.8
496.8
511.6
144.0
205.0
252.9
290.7
321.3
347.3
369.8
391.1
411.3
431.6
452.9
472.6
492.5
180.2
231.0
275.3
314.5
348.5
378.4
404.4
427.4
448.2
467.0
484.0
153.1
149.3
157.4
173.1
192.1
211.4
172.2
140.1
180.9
206.2
206.0
182.2
160.5
151.2
210.5
188.6
173.3
166.5
176.5
189.5
167.6
148.3
180.4
202.1
214.9
221.8
222
88Ra
224
88Ra
226
88Ra
0+2+ 63.8
4+
(6+)
(8+)
(10+)
(12+)
(14+)
(16+)
(18+)
(20+)
(22+)
(1– ) 474.1(3– ) 537.6(5– ) 656.0
(7– )
(9– )
(11– )
(13– )
(15– )
(17– )
(19– )
0+ (2+) 57.4(4+)
(6+)
(8+)
(10+)
(12+)
(14+)
(16+)
(18+)
(1– ) 710.9(3– ) 785.8(5– ) 879.8
140.8
206.9
262.6
309.0
347.8
378.9
403.8
422.4
436.1
446.8
225.0
272.0
312.3
348.3
380.1
408.6
129.4
192.3
247.3
293.6
332.2
364.1
390
412
418.4
380.8
343.6
308.3
277.5
254.6
228
88Ra
230
88Ra
Figure 2.23: Decays schemes for octupole deformed Radium (Z=88) nuclei show-ing enhanced E1 decays between rotational structures of differing parities [97, 97].
between 10−3 and 10−2 Wu. The E1/E2 branching ratio can be used to estimate
the intrinsic electric dipole moment of the nucleus, D0, which is related to the oc-
tupole deformation. The electric dipole moment is related to the B(E1 : Ii → If)
by the expression [95]
B(E1 : Ii → If ) =3D2
0
4π< Ii010|If0 >2=
3D20
4π
(I −K)(I +K)
I(2I + 1)(2.3.52)
2.4 Branching Ratios and g-Factors in High-K
Bands.
The γ-ray branching ratio λ, for a given state in a rotational band, is given by,
λ =T2(E2)
T1(E2) + T1(M1)=
P2(E2)
P1(E2) + P1(M1)(2.4.53)
45
where T1 and T2 are the γ-ray intensities for spin changes of 1 and 2 respectively.
P1 and P2 are the corresponding transition probabilities (∝ intensity). Note
that the ratio T1(M1)T1(E2)
= 1δ2 , where δ is the quadrupole/dipole mixing ratio. The
transition probabilities1 are related to the reduced transition probabilities (B(XL))
by equations 2.4.54.
B(E2; Ii → If) =1
1.225 × 109E5γ
P (E2; Ii → If)
B(M1; Ii → If) =1
1.758 × 1013E3γ
P (M1; Ii → If) (2.4.54)
where E1 and E2 are the M1 and E2 transition energies, in MeV, respectively.
Substituting for the transition probabilities in equation 2.4.53 using equations 2.4.54
yields,
λ =(
E2
E1
)5 B2(E2) × 1.225 × 109
B1(E2) × 1.225 × 109 +B1(M1)E−21 × 1.758 × 1013
(2.4.55)
The reduced in-band transition probabilities1 are given by,
B(E2; IiK → IfK) =5
16πe2Q2
o| < Ii2K0|IfK > |2
B(E2; IiK → IfK) =5
16πe2Q2
o| < Ii1K0|IfK > |2 (2.4.56)
B(M1; IiK → IfK) =3
4πe2| < Ii1K0|IfK > |2(gK − gR)2K2
where Qo is the intrinsic quadrupole moment and gK and gR are the intrinsic
and rotational gyromagnetic ratios respectively. The relevant Clebsch-Gordon
coefficients2 are given below.
E2(∆I = 2) =
[
3(I −K)(I −K − 1)(I +K)(I +K − 1)
(2I − 2)(2I − 1)I(2I + 1)
]1/2
E2(∆I = 1) = −K[
3(I −K)(I +K)
(I − 1)I(2I + 1)(I + 1)
]1/2
(2.4.57)
M1(∆I = 1) = −[
(I −K)(I +K)
I(2I + 1)
]1/2
1K.E.G. Lobner in, The Electromagnetic Interaction in Nuclear Spectroscopy, W.D. Hamil-ton (Ed), North-Holland (1975) Chapter 5
2The Theory of Atomic Spectra, Condon and Shortley (1935) reprinted (1963) p76-77
46
where I is the angular momentum of the initial state. K is assumed to be a
good quantum number throughout. Substituting equations 2.4.57 and 2.4.58
into equation 2.4.55 results in λ =,
(
E2
E1
)5
516πe2Q2
o
[
3(I−K)(I−K−1)(I+K)(I+K−1)(2I−2)(2I−1)I(2I+1)
]
× 1.225 × 109
516πe2Q2
o
[
3(I−K)(I+K)(I−1)I(2I+1)(I+1)
]
× 1.225 × 109 + 34π
[
e2 (gK−gR)2K2
E2
1
(I−K)(I+K)I(2I+1)
]
× 1.758 × 1013
(2.4.58)
Rearranging these terms gives equation 2.4.59
λ =(
E2
E1
)5
516
[
3(I−K−1)(I+K−1)(I+1)(2I−1)2K2
]
516
[
3(I−1)(I+1)
]
+ 34
[
(gK−gR)2K2
Q2oE2
1
(
1.758×1013
1.225×109
)]
(2.4.59)
This leads to the following expression for the branching ratio.
λ =(
E2
E1
)5
[
(I−K−1)(I+K−1)(I+1)(2I−1)2K2
]
1 + 45
[
(gK−gR)2
Q2oE2
1
(I2 + 1)(1.435 × 104)]
(2.4.60)
This equation is identical to that quoted by Alexander et al.3 To convert Qo from
units of e·fm2 to e·b, there is an additional factor required, namely, Qo → Qo
100.
Making this substitution gives equation 2.4.61
(gK − gR)
Qo=
√
5
4(0.8347)
E1√
(I2 − 1)
(
1
λ
(
E2
E1
)5[
(I −K − 1)(I +K − 1)(I + 1)
2K2(2I − 1)
]
− 1
)1/2
(2.4.61)
This result can also be expressed in terms of the quadrupole/dipole mixing ratio
δ,(gK − gR)
Qo= 0.933
E1
δ√
(I2 − 1)(2.4.62)
where δ is related to the quadrupole admixture q, by,
q =δ2
1 + δ2=
2K2(2I − 1)
(I −K − 1)(I +K − 1)(I + 1)
E51
E52
λ (2.4.63)
The factor (gK−gR)Qo
should be a constant for a given rotational band.
3P. Alexander, F. Boehm, and E. Kankeleit, Phys. Rev. 133 (1964) B284
47
Chapter 3
Experimental Gamma-raySpectroscopy.
3.1 Germanium Semi-Conductor Detectors.
The basic idea behind using semiconductor materials to detect radiation is that
through interactions with the radiation, it is possible to excite electrons from
the valence band into the conduction band (assuming that the energy of the
radiation is larger than the semiconductor band gap). This leaves a hole behind
in the valence band, thus a electron-hole pair has been created. The liberated
charge can be then swept away by an applied voltage.
For temperatures greater than absolute zero, thermal energy is shared by the
electrons in the crystal lattice and thus it is possible to for an electron to be
thermally excited across the band gap into the conduction band. The probabilty
per unit time of thermally creating an electron-hole pair, P (T ), as a function of
absolute temperature, T , is given by a Boltzmann function,
P (T ) = CT3
2exp(
− Eg
2kT
)
(3.1.1)
where Eg is the band gap, k is Boltzmann’s constant and C is a constant
which is material dependent.
Thus for small value of a band gap (as is the case for semiconductors), there is
a large probability of thermal excitation, which in a detector would be a source of
unwanted noise. In order to reduce this thermal noise, semiconductor detectors
should be operated at low temperature (usually 77 K, liquid nitrogen tempera-
ture).
48
If radiation enters the electrically neutral depletion layer, electron-hole pairs
can be created. The electrons will flow towards the positive potential and the
holes to the negative. This charge can be collected and converted to an output
voltage by a pre-amplifier. The number of electron-hole pairs created, and thus
the size of he output voltage, is proportional to the energy of the radiation.
3.2 Gamma-Ray Spectroscopy with Germanium
Detectors.
Gamma-ray spectroscopy is used to (a) identify the quantum levels in a nucleus
to probe the physics of nuclear structure and (b) identify radioactive substances
by measuring their characteristic decay gamma-rays (eg 662 keV line in 137Cs).
General considerations for a good gamma-ray spectrometer device are that (a)
it must have excellent energy resolution, (b) a good photopeak efficiency and (c)
good timing properties.
While sodium iodide has a better efficiency than germanium, the excellent
energy resolution of germanium (better than 0.2 % at 1.333 MeV) makes it the
gamma detector of choice for high resolution studies. The main problems with
germanium are
• The most probable interaction for most gamma-rays is Compton scattering
and a sizeable portion of gamma-rays that enter the detector will scatter
out of the detector before the full energy has been absorbed in the crystal.
This gives rise to a large Compton background.
• Germanium detectors must be kept at liquid nitrogen temperature for good
resolution. This means that bulky liquid nitrogen dewars must be included
in the detector apparatus.
3.2.1 Response Function of Germanium Spectra.
(Knoll p289-293, p301.)
The typical germanium spectrum is made up from a number of different com-
ponents. These include,
1. The Full Energy Peak. The peak corresponding to where all the incident
radiation’s energy has been collected by photoelectric absorption (some
49
fraction will have been Compton scattered before p-e absorption). In a
perfect, idealised detector, all the counts would be in this peak.
2. The Compton Background. The background of counts with energies
less than the full energy peak where some of the incident radiation has
been Compton scattered out of the detector.
3. The Compton Edge. In a Compton scattering event, the energy removed
by the electron is given by ∆E, where ∆E = hν−hν ′ = hν( hν
mec2(1−cosθ)
1+ hν
mec2(1−cosθ)
)
.
Therefore, the minimum energy removed by scattering an electron (when
θ=180) is given by ∆E= Eγ
1+2Eγ
mec2
, where mec2 = rest mass of the elec-
tron= 511 keV. This minimum amount of energy being lost in a Compton
scattering, gives rise to the Compton background being essentially cut off
at an energy ∆E below the full energy peak. Note for Eγ >> 511 keV,
∆E → mec2
2≈ 250 keV.
4. Escape Peak(s). For photon energies greater than twice the electron
rest mass energy (511 keV ×2 = 1.022 MeV) there is a probabilty of pair
production where an electron-positron pair is created. The positron may
then recombine with an atomic electron in the detector and decay back to
2×511 keV gamma-rays. One or both of these 511 keV gammas may then
escape from the detector with no further interaction. If the initial gamma-
ray enegy has an energy Eγ , then the escape peaks lie at energies Eγ–511
keV and Eγ–1.022 MeV
5. Backscatter Peak. These correspond to gamma-rays which have been
scattered backwards in material surrounding the detector. The energy for
back scattered gamma-rays is approximately equal for all incident energies
at between 200 and 250 keV. The energy of the backscatter peak corre-
sponds to the energy of the photon after it is scattered. This is simply the
energy difference between the full energy peak and the Compton edge.
6. Annihilation Peak. One observes a peak at 511 keV due to annihila-
tion radiation from pair-production caused by the initial radiation in the
surrounding material (assuming Eγ > 1.022 MeV). The 511 keV is then
measured in the detector.
50
7. X-ray Escape Peaks. A characteristic X-ray is emitted by the material in
the photoelectric process. This is usually reabsorbed, but occasionally it can
escape the detector. Thus a peak with an energy equal to the photo-peak
energy minus the X-ray energy can appear. This is only really a problem for
(a) low gamma-ray energies and (b) detectors with large surface to volume
ratios.
3.2.2 Germanium Detector Efficiency.
(See Knoll p427.)
The efficiency of a germaium detector is usually given relative to that of a
3in × 3in NaI(Tl) detector for the 1333 keV gamma-ray in 60Ni (from a 60Co β-
source). Typical efficiencies range from 30 % to 70 % for hyperpure germanium
detectors.
The efficiency response varies as a function of energy and is usually empirically
deduced using a variety of standard calibration sources such as 152Eu and 133Ba
(see figure 3.1.)
Figure 3.1: Gamma-ray spectra for (a) 152Eu and (b) 133Ba efficiency calibrationsources.
51
3.2.3 The Compton Suppressed Spectrometer (CSS).
(Knoll p421.)
In order to reduce the (unwanted) Compton continuum events in a germanium
gamma-ray spectrum, and thus increase the signal to noise ratio, the germanium
detector can be surrounded by a high efficiency gamma-ray scintillator (usually
Bi4Ge3O12 or BGO). This shield acts as veto for Compton events which scatter
out of the germanium detector. The peak to total for the 1173+1333 keV lines
from the 60Co source for an unsuppressed detector is only about 20 %. This rises
to 50–60 % for a suppressed detector with almost no loss in the number of counts
in the full energy peak.
Liquid
nitrogen
‘cold finger’
incoming
gamma-ray
photomultiplier
tubes
GeBG
O
BG
O
NaI NaI
Figure 3.2: A Compton suppressed germanium detector
The improvement in the peak to total achieved by using anti-Compton shields
is even more crucial when using more than one germanium detector at a time to
measure more mulitiple gamma-rays emitted in a cascade. This is known as co-
incidence spectroscopy. For an event where two gamma-rays from a cascade are
measured (a γ − γ or γ2 event), for two unsupressed detectors, the probability
that both gamma-rays (for Eγ ∼1 MeV) will be in the full energy peak is
approximately 0.2×0.2=0.04. If however, both detectors are surrounded by a
suppression shield this rises to 0.6×0.6=0.36, almost an order of magntitude
improvement. The increase is even larger for higher multiplicity events.
3.3 Gamma-Ray Arrays.
Much nuclear spectroscopy is carried out by measuring the gamma-rays emitted
following a fusion-evaporation reaction. This tends to form the nucleus of in-
52
Energy (keV)
Nu
mb
er o
f C
ou
nts
(a) Eurogam I
Figure 3.3: Effect of higher fold gating on incresing the signal to noise [174].
terest in a high spin state (typically around 30-50 h) which decays by emitting
a cascade of gamma-rays. The ideal tool for gamma-ray spectroscopy would be
able to measure all of the individual gamma-rays in a cascade with 100% effi-
ciency. In practice this is not possible, however, over the last 10 years there have
been significant advances in the development of gamma-ray spectrometers con-
sisting of large numbers of germanium detectors (or ‘arrays’) such as TESSA3
[101], the 8π spectrometer [103], EUROGAM [174, 173] and GAMMASPHERE
[104]. These devices have revolutionised the field of gamma-ray spectroscopy by
allowing hitherto unthinkable sensitivity to weak transitions. The following sub-
sections will describe the important factors in array design and how the maximum
performance can be achieved.
53
3.3.1 Resolving Power and Total Photopeak Efficiency.
The resolving power [174] of a gamma-ray spectrometer is given by R where
R =
(
SEγ
∆Efinalγ
)
PT (3.3.2)
SEγ is the ‘average’ energy spacing consecutive γ-ray transitions in a ‘typical’
cascade. This is clearly reaction dependent and can not be ‘improved’ upon.
Therefore, in order to improve the resolving power on the array improvements
must be made on either the energy resolution, ∆Efinalγ , and/or the peak to total
fraction, PT .
∆Efinalγ is the full width at half maximum resolution of γ-rays obtained in
the detector spectrum. This is made up from a number of effects including the
intrinsic, statistical resolution of the detector, the effect of Doppler broadening,
the spread in the recoil cone and recoil velocity.
The total resolution of a given γ-ray transition in a fusion-evaporation reaction
is given by [174]
∆Efinalγ =
(
∆E2int + ∆E2
open + ∆E2ang + ∆E2
vel
) 1
2 (3.3.3)
Doppler Shifts and Doppler Broadening.
If the nucleus of interest recoils out of the target with a velocity v, its energy
is Doppler shifted when measured in a detector at angle θ to the recoil velocity
direction to a value Es, where, if E0 is the ‘unshifted’ energy,
Es(θ) = Eo
√
1 − (vc)2
1 − vccosθ
≈ E0
(
1 +v
ccosθ
)
(3.3.4)
Since the geramnium detector measuring the gamma-ray has a finite opening
angle, there will be a spread in the Doppler shifted energy across the face of the
detector (see figure 3.4).
If the opening angle of the detector is ∆θ, then differentiating equation 3.3.4
gives,
dEs
dθ≈ ∆Es
∆θ≈ Eo
v
csinθ (3.3.5)
54
germanium detector
E = Eo (1 + v/c cos )θs
θ
beam direction, velocity, v
∆θ∆θ
γ
∆Ε = Εοv/c sin θs ∆θ
Figure 3.4: Schematic of the effect of finite opening angle for a detector causingDoppler broadening of the gamma-ray lineshape.
The full expression, including broadening due to velocity spread (see fig. 3.5),
can be approximated by
∆Es ≈ Eocosθ∆v
c− Eo
v
csinθ∆θ (3.3.6)
Clovers and Segmented Clover Detectors.
One method which can be used to improve the energy resolution deteriation from
Doppler broadening is to segment the germanium detector, thereby reducing the
effective solid angle of its face. Clover detectors use four 25% germanium crystals
housed in the same can. The detectors are smaller than the typical 70% germaium
crystals and thus have a smaller intrinsic efficiency, however, this limitation can be
overcome by adding together gamma-rays measured in two neighboring segments
(from events which Compton scatter from one crystal to the next) [176, 177].
55
Figure 3.6 shows the effect of clover detectors at 90 in the EUROBALL
array compared to the same reaction using 70% detectors. The improvement in
resolution, and thus sensitivity is clear. In some more modern cases, the clover
elements are segmented electrically to give an even better defined localisation
[178]. This is particularly useful for fast moving recoils and has been implemented
for example in the EXOGAM array for use with radioactive beams [105].
3.3.2 Add-Backs from Clover/Cluster Detectors.
Improvements in the peak to total can be achieved by using larger volume germa-
nium detector, by improving and maximising the Compton suppression and/or
the use of add-back such as in the cluster detector [179] (see figure 3.7).
A further advantage of using a clover type detector [176, 178] comes when try-
ing to look at higher energy gamma-rays. As figure addback shows, the majority
of counts with energies less than 1 MeV are detected in a single element. How-
ever, for energies above this value, the probability of Compton scattering from
one crystal into a neighboring one (spectrum (c)) is large enough that there is a
significant increase in the full energy peaks obtained using the detector in add-
back mode. This increases the sensitivity of the array by improving the overall
signal to noise (or peak to total) of the device.
3.3.3 Polarization Measurements.
Clover detectors can also be used to remove the anomaly between electric and
magnetic dipole transition by measuring the polarization of the gamma-rays
[186, 185, 187]. This type of analysis uses the fact that the direction of Compton
scatters is different for electric and magnetic type transitions. Thus, by com-
paring the efficiency corrected intensities of ‘added-back’ lines from horizontal
and vertical scatters, the electric or magnetic (or mixed) nature of the transi-
tion can be eludidated. This is highlighted for experimental data taken from the
EUROGAM 2 array in figures 3.8 and 3.9.
3.3.4 Gamma-ray Tracking.
Work is currently underway on investigating the possibilities of using algorthyms
to track Compton scatters through a highly segmented germanium shell to make
56
the next generation of gamma-ray spectrometers (eg. the GRETA project [180,
181, 182]). The idea uses the Compton formula to determine whether two events
in a germanium shell are two separate gamma-rays or, two ‘tracks’ of a single
events which has Compton scattered. Using an algorythm (see eg. [183]) a
decision can be made offline, as to whether to ‘add-back’ or ‘reject’ Compton
scatters. Such algorythms depend on a good knowledge of the interaction points
of the gamma-rays in the crystals, which can be further improved by looking at
the charge collection times of the signals and their pulse shapes [184].
57
Figure 3.5: Gamma-ray spectra taken from the reaction 28Si on a 40Ca targetwith a v
cof 1.8%. Note the doppler broadening is larger away from 90 in this
case due to the large spread in recoil energy associated with the alpha-particleevaporation [106] .
58
1400 1450 1500
400
800
1200
1600
1400 1450 1500
200
400
600
800
γ-ray energy (keV)
Num
ber
of c
ount
s
Phase I
Phase IIFWHM = 4.8 keV
FWHM = 6.1 keV
Figure 3.6: Effect of using clover detectors in improving energy resolution inEUROGAM phase II [174].
59
(f) (e) (h)
(a) (b) (c)
(g)
(d)
Figure 3.7: Spectra gated on add-backs from cluster detectors at PEX using thereaction 28Si+40Ca [107].
60
0 5 10 15 20 25 30
–0.2
0.0
0.2
0.4
0.6
0.8
Band 3 (M1)
Band 12 (E2)
Spin (–h)
Fig. 5
γ–ra
y po
lari
satio
n P
/home/esp/text/figs/i117/pol3.psk Paul et al PRC
Figure 3.8: Polarization measurements showing the difference between magneticand electric type decays in 132Ce [186, 185].
61
0.0 0.5 1.0 1.5
–10
0
10
0.0
0.5
1.0
1.5
Eγ (MeV) Fig. 4
Cou
nts
(104 )
/home/esp/text/figs/i117/polspec.psk Paul et al NPA
(a) 117
I gated
(b) Ungated
↓ Magnetic
↑ Electric
↓ Magnetic
↑ Electric
Figure 3.9: Difference spectra for horizontal and vertical scatters for transitions in117I showing how a polarization measurement can be use to differentiate betweenelectric and magnetic transitions (see [186, 185].
62
Chapter 4
Channel Selection InFusion-Evaporation Reactions.
The statistical nature of nucleon emission from the hot, compound system means
that fusion-evaporation reactions tend to populate more than one residual nu-
cleus. In order to associate gamma-rays with specific nuclei, some form of extra
channel selection is required. There are two main ways to determine the nucleus
which emitted the gamma-rays, either by identifying the nucleus itself (recoil sep-
arator, gas filter techniques) or by measuring the particles evaporated from the
compound system (charged particle balls, neutron detectors). The former gen-
erally have the advantage of being ‘cleaner’ and less susceptible to contaminant
nuclei (from for example isotopic impurities in the target) while the latter tend
to have a larger detection efficiency.
In this section we will discuss four main types of channel selection; (a) inner
sum-energy balls [101]; (b) light charged particle detection arrays [109, 110, 111,
112, 113, 114, 124]; (c) neutron detectors [136, 25] and (d) recoil detectors and
separators [143, 144, 146, 147].
4.1 Inner Multiplicity Sum-Energy Balls.
The entry point for fusion evaporation products in the excitation energy versus
total angular momentum plane is generally different for different reaction prod-
ucts and (typically) depends on the number of evaporated particles. Usually,
residual nuclei formed by the evaporation of fewer particles leave the final nu-
cleus in a state of higher spin and excitation energy than channels with higher
63
particle emission multiplicities. If a high efficiency, high granularity γ-ray detec-
tor (calorimeter) can be placed around the target position, this effect can be used
to give a degree of channel selection between different residual species [101] (see
figures 4.1 and 4.2). This form of channel selection is particularly effective when
the majority of the fusion cross-section goes into evaporation channels consisting
of only neutrons (ie. 3n,4n,5n,...etc).
Figure 4.1: Fold and Sum Energy spectra from the inner ball of the Chalk River8π spectrometer for the reaction 76Ge(34Sn,xn)110−xCd. Spectra (a) and (b) are
gated by the 540 keV 152
− → 112
−and 633 keV 2+ → 0+ transitions in 105Cd and
106Cd respectively. Note the lower values of fold and sum energy observed for thehigher neutron multiplicity evaporation channel [3].
For example the 8π gamma-ray spectrometer at Chalk River contains a 71
element inner ball made up of BGO elements [112]. This can be used to measure
the total energy emitted in a decay cascade (related to the inital excitation en-
ergy of the nucleus), with the number of detectors firing, giving an estimate of
the total gamma-ray multiplicity of the cascade (which is related to the initial
angular momentum of the system). As figures 4.1 and 4.2 clearly demonstrate,
the entry point of a nucleus in the spin/excitation place depends on the num-
ber of evaporated neutrons for the 76Ge+36S reaction. By setting software gates
64
Fold (~ multiplicity ~ total input angular momentum)
Sum
Ene
rgy
(ent
ranc
e ex
cita
tion
ener
gy)
5n + 105Cd4n + 106Cd
Figure 4.2: Two dimensional Fold and Sum Energy spectra from the inner ballof the Chalk River 8π spectrometer for the reaction 76Ge(34Sn,xn)110−xCd gated
by the 540 keV 152
− → 112
−and 633 keV 2+ → 0+ transitions in 105Cd and 106Cd
respectively.
in offline sorting on various fold/sum-energy conditions, the relative intensity of
transitions from specific evaporation residues can be enhanced.
4.2 Studies of Very Neutron Deficient Nuclei.
For very neutron deficient compound nuclear systems, the reduced proton sepa-
ration energies compared to compound nuclei closer to stability mean that it is
easier for protons and α-particles to tunnel through the Coulomb barrier. This
allows charged particle evaporation to compete with and finally dominate over
neutron emission, and the population of between 10 and 20 different nuclei in a
single reaction is possible. The effectiveness of the particular method of channel
selection used depends on the final nucleus of interest. For example, in studies
of A∼80 nuclei along the N=Z line, these nuclei are accessible via a 2-neutron
evaporation channel [138, 139, 140, 141, 142] for which a charged particle detec-
tor array would not be suitable, whereas a recoil-separator and split-anode gas
ionisation-chamber, or neutron-detection system might yield useful information.
65
4.2.1 Charged Particle Balls.
The main job of charged particle detector arrays is to detect and disciminate
between the light charged particles evaporated from a compound nucleus. The
types of detectors are usually either some form of fast-slow scintilator sandwich
known as a phoswich detector [110, 111, 109], a thin silicon ∆E detector [134,
113, 114] or some form of scintilator such as CsI(Tl) which uses pulse shape
discrimination [112, 124].
66
Figure 4.3: (a) Total projection of the 28Si+40Ca reaction from PEX. (b) Samespectrum gated on the 2p condition. (c) 2p gated spectrum with 3p componentsubtracted. (d) 2p gated spectrum with both 3p and 4p components subtracted[106].
67
Figure 4.4: Particle gated identification spectra gated numbers of emitted protonsfrom the ANU particle detector ball with higher multiplicities subtracted [38].
68
The ideal charged particle detector has almost 4π coverage around the target
position (with gaps for the beam to come and possibly go out), good timing
properties, good granularity (to reduce the probability of two charged particles
hitting the same detector) and good discrimination between different types of
particles. In practice however, there are gaps and dead regions between the
detectors, and this coupled with kinematic focussing effects means that such
arrays are never 100% efficient. The result is for example a γ ray from a 3
proton evaporation channel will be present in the 3p, 2p 1p and 0p gated spectra.
Careful, normalised subtractions of higher multiplicity charged particle events
must be applied to achieve clean spectra (see figure 4.3). However, as shown in
figure 4.4, once these contaminatnts from higher multiplicity channels have been
subtracted, very clean identification spectra can be obtained
4.2.2 Kinematic Focussing and Conversion from Lab to
COM Energies.
Fusion evaporation reactions evaporate light particles isotropically in the rest
frame of the compound nucleus. However, as shown schematically in figure 4.5,
the fact that the recoil is moving forward in the laboratory frame means that
the angular distribution of emitted of charged particles in a fusion evaporation is
forward focussed in the laboratory frame see figure 4.5.
θPlab
Vcm
V P lab
V R lab
Vpcm
VRcm
Figure 4.5: Schematic of the kinematics of particle evaporation following a fusion-evaporation reaction. Note how the recoil direction is altered by the particleevaporation.
If the beam energy is EB and EB = 12MBv
2B, where MB is the mass of the
beam nuclei and vB is the beam velocity in the lab. frame, then by conservation
69
of momentum, the velocity of the centre of mass vcm is given by
70
MBvB = (MB +MT ) vcm (4.2.1)
and by substitution
vcm =MB
(MB +MT )
√
2EB
MB
(4.2.2)
where MT is the mass of the target nuclei. The velocity of an evaporated
proton in the rest frame of the compound nucleus can be calculated (by the
cosine rule) such that
v2pcm
= v2cm + v2
plab− 2vcmvplab
cosθlab (4.2.3)
where vplabis the emitted proton velocity in the laboratory frame and θlab is
the lab angle between the beam direction and the direction of the emitted proton.
By substitution
v2pcm
=MB
(MB +MT )22EB + v2plab
− 2MB
(MB +MT )
√
2EB
MBcosθlab (4.2.4)
The proton energy is the centre of mass (intrinsic frame of the compound
system) is given by
Epcm=
1
2Mpv
2pcm
(4.2.5)
Multiplying both sides of equation 4.2.4 by 12Mp gives
1
2mpv
2pcm
= Epcm=
1
2Mp
MB
(MB +MT )22EB+1
2Mpv
2plab
−1
2
Mp2MB
(MB +MT )
√
2EB
MBvplab
cosθlab
(4.2.6)
Given that vplab=√
Eplab
Mp, substituting in and collecting the terms of equa-
tion 4.2.6, we obtain the following expression for the energy of the evaporated
proton in the centre of mass frame in terms of the measured proton lab energy
(Eplab), the angle of emission in the lab frame (θlab), the beam energy (EB) and
the masses of the beam (MB) and target (MT ) nuclei respectively,
Epcm=
MpMB
(MB +MT )2EB + Eplab− 2
(MB +MT )
√
MPMBEBEplabcosθlab (4.2.7)
71
A similar expression can be obtained for α-particle emission.
Note one problem with using charged particle balls for channel selection are
the presence of target impurities. In particular, relatively light ions such as car-
bon and oxygen are common target contaminants. However, the lighter mass of
these nuclei, means that in general, the effect of kinematic focussing for particles
evaporated from beam induced reactions on these nuclei is greater than for the
(typically) heavier target nuclei. Thus, the angular distributions of the emitted
charged particles for such contaminants are usually more forward focussed (in the
laboratory frame) compared to the ‘true’ reaction products [132] and thus they
can be distinguished.
Scintilators.
The basic premise of a scintilator detector is that the incoming radiation interacts
with the electrons in the scintilator, exciting them into higher energy states. As
these states decay, they emit X-rays or light, which is converted into a charge by
means of a light-pipe, photo-cathode and photomultiplier tube assembly.
prom
pt
fluo
resc
ence delayed
fluorescence
(phosphorescence)
abso
rptio
n
S0
S3
S2
S1
SINGLET STATES
T2
T3
TRIPLET STATES
T1
Figure 4.6: Schematic of the decay of prompt and delayed light from an organicscintilator.
Often, depending on which quantum levels the electron decays through, this
72
light is emitted with two characteristic decay lifetimes, a fast one known as
prompt fluorescence and a slower type known as delayed fluorescence (see fig-
ure 4.6). The relative intensities of these two types of light depends on the type
of radiation which excited the scintilator in the first place and as we shall see
later can be used to distinguish between different types of radiation.
Phoswich Detectors.
Phoswich detectors are combinations of two detectors, in optical contact. The
front detector consists of a thin (∼ 100 µm) ∆E scintilator detector, which has
a fast (∼ 5ns) decay component to its light output, which is particle dependent.
Typically, the heavier the detector particle, the larger the fast signal. The second
detector (thickness ∼5 mm), placed behind the thin ∆E detector, is used to stop
the particle and usually gives out light with a much longer decay time (∼ 1µs)
which is particle independent. The total signal from the two detectors is taken via
a light pipe, into a photomultiplier tube. The anode signal from the PMT is then
integrated in two different time regions by setting two separate time discriminator
gates on the short (∆E) and long (E) components of the signal (see figure 4.7).
By making a 2-dimensional array of ∆E versus E, different evaporation particles
can be identified. In order to stop signals from scattered beam entering the
phoswich detector, thin metal absorber foils (Al or Au) are placed on the front
face of the detector. These foils are thick enough to stop the (higher Z) beam
particles, but thin enough to allow through evaporated protons and α-particles.
Examples of phoswich detectors are the ANU particle detector ball [38] and
the ERICIUS array [111] which use a combination of fast and slow plastic scin-
tilators and the Penn 4π array [110, 132, 287] which uses fast plastic as the ∆E
detector with CaF2(Eu) as the stopping detector.
One problem with phoswich detectors is that due to kinematic focussing,
charged particles emitted at backward angles in the lab frame tend to have much
reduced energies compared to forward angles. If the lab energy is too small, and
the ∆E detector is too thick, the particles will stop in the ∆E fast scintilator
detector. The thickness of the ∆E detector is then a compromise between particle
identification resolution and the low energy threshold. Also, the light pipes and
photomultiplier tubes associated with converting the light output from a phoswich
detector to an electrical pulse constrict the geometery and available space which
73
for example germanium gamma-ray detectors may take up.
74
NE102A3 mmCaF2 (Eu)
0.1 mm
"DE""E"
light
out
put /
vol
tage
time
slowfast
α
p
0-40 ns 60 - 250 ns
slow (E)
fast
(D
E)
p
α
Figure 4.7: Schematic of the operation of a phoswich detector for discriminationbetween protons and alpha particles.
4.2.3 Silicon Detectors.
Si Energy Loss Detectors.
Thin (∼150–200 µm) silicon detectors have also been used to give channel se-
lection between protons and α-particles emitted in fusion evaporation reactions
[113, 114, 134, 345]. Particle identification is achieved due to the differential en-
ergy loss of protons and α-particles. An advantage of such detectors is that they
can be made to be very compact and do not require the photo-multiplier tubes
that phoswich detector systems do. They can easily be made into 4π geometry
and the thin nature of the detector means that attenuation of reaction gamma-
rays in the ball material is kept to a minimum. This is an important factor in
keeping the low-energy gamma-ray detection efficiency of such experiments at a
75
premium.
76
Figure 4.8 shows a schematic of the NBI silicon inner ball used in the PEX
array [114]. Note the extra granularity of the forward angle detectors. This is to
take into account the effect of kinematic focussing of the reaction products, and
to attempt to keep the count rate for each individual element at a reasonable
level. (If the count rate in an individual element is too high, the dead time for
the whole system is increased and the beam current and thus total reaction rate
in the experiment has to be reduced).
Due to the lower Coloumb barrier, evaporated protons tends to have lower
energies than evaporated α-particles. Thus to first order, the energy deposited
in the detector can be used to distinguish protons and α-particles. The problem
is the high energy tail of the proton distribution leaks into the α-particle energy
regime. This is compensated for in the Si ball type detector by making the
detector thin enough that higher energy protons are not stopped in the detector
and thus leave only a fraction of their energy. As figure 4.9 shows, a good degree
of discrimination is available using such a device.
The main disadvantages of such detectors are that they are rather sensitive
to radiation damage and also, in general, they do not retain good information on
particle energies.
PSD with Si Detectors.
It has long been established that pulse shape discrimination between electron
and hole associated rise times can be used in silicon detectors to provide some
form of disrcimination between different types of radiation [115, 116]. There have
been a number of recent developments which use pulse shape discrimination for
with planar silicon detectors to discriminate between different heavy ions, from
protons, alphas and heavier ions, including the RoSiB detector [118, 119, 117],
which uses a zero cross-over taken from a bipolar amplifier output, verses the
total energy deposited in the detector, to differentiate between different types of
ions.
(E,∆E) Discrimination with Silicon Balls.
The Bethe equation as used in phoswhcih style detectors can also be employed for
silicon detector telesccope to provide good proton/alpha discrimination in fusion-
evaporation reactions. The EUCLIDES and ISIS arrays [120] are constructed in
77
TARGET
(a)
(c)
1611
20(b)
OUT
IN
(d)
5
31
OUT IN1
23
4
7,8,9
6,10
11,15
16,20
12,13,14
17,18,1923,24
21,22,25
26,27,30
28,29
IN
31
28
22 27
26
21
23
29
24
30
25
1 5
42
3
13 138
12
1712
7
11 6 10
15
15
1914
914
18
Figure 4.8: Schematic of the PEX Si inner ball geometry for selection of lightcharged particles [107, 114].
a ‘football’ geometry of interlocking pentagonal and hexagonal silicon detector
telescopes made up of an energy loss, ∆E, and residual energy E detectors.
Note the separation of the protons and alpha particles in figure 4.10 and also
the locii corresponding to events where more than one evaporated particle has hit
a single telescope (such as αp. Note that due to the kinematics of typical fusion-
evaporation reactions, α particles are often not observed at backward angles in
the laboratory frame, ie. they are more forward focussed than protons.
4.2.4 CsI(Tl) Balls Using Pulse Shape Discrimination.
Some of the best inner balls for channel selection are made of elements consisting
of single crystals of CsI(Tl) scintillator [112, 124]. The light output from CsI(Tl)
has two parts, a fast component (τ = 0.4 →1.0 µs), the amplitude of which
78
Figure 4.9: Energy spectra for charged particles from the PEX inner Si ball forthe reaction 28Si+40Ca showing the effect of kinematic focussing [106].
is radiation type dependent and a slower (τ = 7 µs), radiation independent
component. Particle identification can be achieved using a similar pulse shape
discrimination technique as outlined above for the phoswich type detectors. High
efficiency, high granularity devices based on this method, such as the ALF-ball
[112] and the MICROBALL [124] have been used very successfully inside the 8π
and GAMMASPHERE gamma-ray spectrometer arrays, to give high degrees of
channel selection. In both cases, to save space and reduce the material between
the target position and surrounding germaniun detectors, the light is converted
into charge using silicon PIN photodiodes.
Figure 4.13 demonstrates the proton/alpha-particle particle discrimination
afforded for an element of the ALF ball. The two axes in figure 4.13 are total
light output (which is related to the total energy deposited in the detector) verses
a zero-cross-over time. This corresponds to the time difference between a T=0
79
reference time (such as beam burst of prompt-γ-ray signal) and response time of
the CsI(Tl) detector (which is radiation type dependent). This response time can
be obtained by taking an amplitude invariant zero-crossing point (obtained by
taking the preamplified output pulse from the detector through a bipolar shaping
amplifier [112]).
80
Figure 4.10: Particle hit pattern rom the ISIS silicon E,∆E array for differentangles in the lab frame for the reaction 40Ca on 40Ca. Note the general forwardfocussing of the particles and the absence of alpha particles at backward angles[121]. The lower detector numbers correspond to detectors aty forward angles.
Channel Selection Using Proton/Alpha COM Energies.
While a charged particle detection system can be used to separate evaporation
channels consisting of purely charged particles, another method must be used
to distinguish between channels containing evaporated neutrons as well as the
charged particles. For example a spectrum gated on the condition that 2 pro-
tons are detected will contain counts from 2p,2pn,3p,3pn,4p,α2p,...etc channels.
The transitions from higher charged particle multiplicities will come into such a
spectrum (due to the fact that such devices are never 100% efficient) but can be
identified by their presence in higher multiplicity gated spectra (eg. 3p gated will
contain 3p,3pn,4p,...etc. but should not contain the 2p and 2pn channels).
81
φ = 108.0°
10 20 30
E (MeV)
2
4
6
8
10
12
14 ∆
E (
MeV
)
φ = 148.3°
10 20 30
E (MeV)
2
4
6
8
10
12
14
2p
p p
Figure 4.11: Particle identification spectra from the ISIS silicon E,∆E array fordifferent angles in the lab frame for the reaction 40Ca on 40Ca for backward ISISangles in the lab frame. Note the general forward focussing of the particles andthe absence of alpha particles at backward angles [120, 121, 122].
φ = 31.7°
10 20 30
E (MeV)
2
4
6
8
10
12
14
∆E
(M
eV)
φ = 69.9°
10 20 30
E (MeV)
2
4
6
8
10
12
14
α p
α
2p 2p
α
p p
Figure 4.12: Particle identification spectra from the ISIS silicon E,∆E array forforward angles in the lab frame for the reaction 40Ca on 40Ca [120, 121].
82
Tim
e re
spon
se o
f C
sI(T
l)
Light Output
p
α
Figure 4.13: Time verses total energy spectrum for one element of the chalk RiverALF ball for the reaction 76Ge+34S. Note the protons and alpha particle locii areclearly separated.
One method of distinguishing between for example 2p and 2pn events is to
compare the average evaporated proton energy (in the centre of mass frame).
The 2p proton channel will tend to have a higher average proton energy than
the 2pn channel (since some of the available excitation energy is removed by the
neutron).
Another way of thinking of this is to realise that the total excitation energy
of the compound nucleus is fixed (by the kinetic energy of the reaction and the
Q-value for the reaction). By measuring the proton energies in the centre of mass
frames and subtracting them from the total compound excitation energy, one is
left with the ‘residual’ excitation energy of the system. This can be removed by
either the emission of an (unobserved) neutron or by gamma-ray emission (if the
excitation energy of the residual nucleus is below the particle evaporation thresh-
old). The probability of emitting a further particle depends on the excitation
energy the system is left in after a particle emission. For example, if the nucleus
emits two protons with relatively high energies, the residual system will be left
in a lower energy state than for cases where the two protons remove less energy.
Since there is more residual energy left in the latter case, it is more likely that
83
such a scenario will result in the emission of a further nucleon (neutron) to bring
the final nucleus below the particle evaporation threshold. This is shown to be
true experimentally in the work by Balamuth et al. and Pohl et al. [132, 135].
Similarly, the average energy of protons emitted in the 2p channel is larger
than the average energy of protons emitted in the 3p channel and so on [110, 123,
135, 112]. Thus, even using a single charged particle detector, and setting off-
line gates on measured average proton energy will yield some selection between
different proton evaporation channels.
Selection of High Spin Cascades by Particle Evaporation Spectra
Pohl et al. [135] showed that there is a correlation between the proton emission
spectra (and thus the total entrance excitation energy of the residual system)
and the population of high spin states. This is a simple effect to understand, the
less excitation energy taken away by the emitted particles, the more is available
for gamma decay in the residual nucleus. The experimental data taken from the58Fe(27Al,3p)82Sr reaction [135] highlighting this effect is shown in figure 4.15.
The Total Energy Plane Method.
As pointed out previously, a problem with using charged particle balls to iden-
tify weak, low-multiplicity channels in very neutron deficient systems, such as
the 2p channel, is that the raw 2 proton gated spectrum will be ‘contaminated’
with lines from higher charged particle multiplicity evaporation channels (eg,
3p,4p,α2p,α2pn...) If the low-multiplicity channel of interest is populated weakly
in the reaction, the breakthrough lines from the higher multiplicity channels will
dominate the spectra. A clever method has been suggested by Svensson et al.
[133] to overcome this problem.
In a fusion-evaporation reaction, the total energy in the centre of mass frame
for a given exit channel, ECM , is given by
ECM = TCM +Q (4.2.8)
where TCM is the centre of mass kinetic energy of the beam-target collision
(ie. beam energy in the centre of mass frame) and Q is the reaction Q-value for
the different (specific) exit channels.
84
This total excitation energy bought into the compound system is then emitted
in the form of γ-ray decays and emitted nucleon kinetic energies such that
85
Figure 4.14: Centre of mass proton energy spectra for 27Al+58Fe reaction at abeam energy of 92 MeV, gated on transitions in 82Sr and 81Sr from the 3p and3pn channels respectively [135]. The lower average proton energy for the 3pnchannel is clearly demonstrated.
86
Figure 4.15: Entry-point excitation energy centroids in coincidence with discretegamma-rays depopulating states at excitation energies between 0 and 8 MeV in82Sr. Note that the higher excitation energy states, typically come from nucleiformed with higher energy entry points (ie. lower proton evaporation energies)[135].
87
ECM = Hγ + Tpart (4.2.9)
where Hγ is the total energy emitted as gamma-rays and Tpart is the total
emitted particle kinetic energy in the centre of mass frame. Note that ECM is
constant for a specific channel. Therefore if Hγ and Tpart can be measured event
by event and plotted in a 2 dimensional array (called the total energy plane
or TEP). Since the total available ECM is constant for a given channel, events
corresponding to a channel where all the evaporated particles have been measured
should lie on a straight line in this TEP (see figure 4.16).
Tot
al P
artic
le K
inet
ic E
nerg
y
Total gamma-ray energy
3p
Ecm
Ecm
a3p 3pn
4p
etc
Figure 4.16: Schematic Total Energy Plane for a 3p out channel showing a con-stant line for 3p events and clear separation from the higher multiplicity chargedparticle events [133].
Events from higher multiplicity channels where some of the Tpart has not
been measured will lie below this line on the TEP plane. Using a 4π charged
88
particle detector (to obtain the centre of mass energies of the emitted particles)
in conjunction with a 4π gamma-ray inner ball (to obtain an estimate for the
Hγ) one can increment such a plane event by event. By setting software gates
on various regions of the TEP plane, excellent channel selection between particle
multiplicities can be obtained.
Kinematic Corrections for Doppler Effects.
Figure 4.17: Spectra showing the SD band in 85Nb using the reaction58Ni(36Ar,α2p)85Nb at an energy of 180 MeV. The improvement in signal tonoise due to particle gating and kinematic corrections are clear [166].
By conservation of momentum, the evaporation of particles from the com-
pound system may have the effect of causing the recoiling nucleus to alter its
direction and (velocity) with respect to the beam direction. In the case of thin
target experiments, where the recoiling nucleus is not stopped in view of the
89
detectors, this spread in velocities and increase of the recoil cone can cause a
deteriation of the gamma-ray energy resolution [174].
This is particularly a problem for light nuclei (≤ 100) which are formed by the
emission of α-particles. However, by measuring the energy of the emitted charged
particles and converting back to the centre of mass energy, the momentum of
the evaporated particles can be obtained. By conservation of momentum, the
momentum of the recoiling nucleus (in the centre of mass frame) can then be
reconstructed on an event-by-event basis. By converting this back into the lab
frame, the change in direction and velocity of the recoil due to the charged particle
emission can be calculated and used to improve the gamma-ray energy resolution
of the emitted gamma-rays [131].
The measured gamma-ray energy, Eγ will be shifted from its correct energy,
E0, by (a) the Doppler shift and (b) the change in direction/velocity due to the
emission of light charged particles. In general, for i emitted particles of mass mi,
this can be written as
Eγ = E0
(
1 + ∆E0 −n∑
i=1
∆Ei
)
(4.2.10)
where
∆E0 =MCN
MR
vCN.dγ
c(4.2.11)
and
∆Ei =Mi
MR
vi.dγ
c(4.2.12)
where vCN.dγ is the vector dot product between the velocity vector of the
recoil and the unit vector of the direction of gamma-ray emission (ie. VCN times
the cosine of the angle between the recoil and gamma-ray directions in the lab
frame).
Figure 4.17 shows the improvement in energy resolution (and thus peak to
total) obtained for the superdeformed band in 85Nb by kinematically correcting
the spectra for charged particle evaporation.
90
4.3 Neutron Detection.
The detection of neutrons evaporated from compound nucleus reactions is usu-
ally achieved using organic scintilators containing a large amount of hyrdrogen.
A common type of detector consists of the liquid scintilator NE213 [136]. Dif-
ferentiation between neutron and gamma-ray induced events is usually achieved
by one of two methods: (a) time of flight and/or (b) pulse shape disrimination.
Figure 4.18 shows a time spectrum for events in a neutron detector with respect
to the RF of the beam pulse for the reaction 40Ca+24Mg from the AYEBALL
array at Argonne National laboratory. The gamma-ray induced events all have
a common time profile (since all photons travel at the same speed), while the
neutrons are slower and arrive later. Note the time spectrum can also be thought
of as a neutron energy spectrum.
Figure 4.18: Neutron time of flight spectra showing clear separation betweengamma-ray and neutron event. From the reaction 40Ca+24Mg on AYEBALL[106].
In general, the larger the distance between the neutron detectors and the
target position, the better the TOF discrimination between gamma-rays and
neutrons. However, by placing the detectors further back, one loses detection
efficiency. In order to maximise the neutron detection efficiency, the detec-
tors must be placed closer to the target position and pulse shape discrimination
[136, 127, 125] is used to separate gamma and neutron events (see figure 4.19).
As with the phoswich detector, the prompt component of the scintilator light
is particle dependent (neutron induced, proton scattering events have a larger
prompt component than gamma-rays events). (Note that the neutron detectors
91
can also be used as a gamma-ray multiplicity detector [136]).
gammasneutronsN
eutr
on T
ime
of F
light
Zero-crossing time
Figure 4.19: Time of flight verses z component for neutrons measured in the28Si+40Ca reaction at PEX [106].
In the studies of very neutron deficient nuclei, it is often desirable to combine
both charged particle and neutron channel selection devices to obtain clean, high-
resolution gamma-ray spectra for a specific, weakly populated channel. In modern
arrays, different forms of channel selection are often used in tandem to give a
maximum effect. A good example of this is the use of both charged particle
inner balls and neutron detectors to study very neutron deficient nuclei around
N∼Z∼100 at the Niels Bohr Institute, Denmark [126, 127, 128, 129]. The PEX
array (see figure 4.21) is an extension of this project, as is the EUROBALL
neutron wall [130].
Figure 4.20 shows the effect of using both charged particle and neutron se-
lection in identifying transitions in the neutron deficient nucleus 6231Ga from the
reaction 40Ca+28Si, via the αpn evaporation channel. It is clear that the signal to
92
noise for the lines associated with the nucleus of interest improves dramatically
with each extra level of channel selection.
Once gamma-rays from such a channel have been identified, gamma-gamma
coincidence data such as that shown in figure 4.22 can be used to work out the
decay sequence of the transitions to form a decay scheme.
93
Figure 4.20: Particle and neutron identification spectrum showing different resid-ual channels in the reaction 28Si+40Ca from PEX [106].
94
Figure 4.21: Schematic of the PEX array, used in conjunction with a small, siliconinner ball and a wall of neutron detectors.
95
Figure 4.22: Alpha-neutron gated γ − γ coindicence gates showing transitions in62Ga from the reaction 28Si+40Ca taken from PEX [106]. The matrix was gatedon the condition that one α-particle and either one proton and/or one neutronwere also detected.
96
Determining Charged Particle and Neutron Multiplicity.
Figure 4.23: Efficiency corrected spectra used to identify states in 61Co. (a) Thepre-scaled γ singles data, (b) proton–γ (c) proton–neutron–γ and (d) the α − γcoincidence spectra from the 16O+48Ca reaction. [25].
For evaporation channels containing more than one type of the same particle,
the probability of measuring at least one of these particles increases with the
particle multiplicity. For example, the probability of measuring a proton from
a 2p channel is approximately twice as large as measuring a proton from a 1p
channel, simply because in the former case there are twice as many ‘chances’ to
measure the proton. This can be useful when identifying gamma-rays with specific
evaporation channels. By comparing the intensities of gamma-ray transitions in
the raw, ungated spectra with the same spectrum gated by the condition of
measuring one type of particle, the various evaporation channels fall into discrete
groups depending on the evaporated particle multiplicity of that channel [126,
25, 127, 128].
Figure 4.23 shows identification spectra for the reaction 16O+48Ca gated on
charged particles and neutrons taken from the U. Penn 4π array [110, 25]. The
main channels of interest were the pxn channels to the 62Co and 61Co channels
97
via the pn and p2n channels respectively. The ratio of counts for lines between
the 1p and 1p1n gated spectra are shown in figure 4.24 and the lines from the
different neutron multiplicities are clearly separated.
Figure 4.24: Plot of ratio of relative intensities of states in proton–neutron–γ andproton–γ spectra, identified in (a) the 16O+48Ca and (b) the 18O+48Ca reactions.Note how the neutron multiplicities, used to identify the residual decay productsare clearly separated and the effect of differing Q-values on the neutron detectionefficiency in the two reactions [25].
Figure 4.24 also shows the same plot for the evaporation channels for the
same target but using an 18O reaction. Note that the apparent neutron efficiency
is different for the 16O induced reaction. This is a kinematic focussing effect
due to the different Q-values for the two reactions. Thus the recoil cones of the
neutrons in the lab frame for the two reactions are different. In the 18O case,
the evaporated neutrons typically have higher energies than those in the 16O
98
reaction. Thus the neutrons from the 18O reaction are less forward focussed in
the laboratory frame. Since the neutron detector was placed at forward angles,
their detection efficiency is reduced relative to the 16O data.
99
4.4 Recoil Detectors.
As shown above, the use of ancillary detectors to measure evaporation products
from compound nucleus reactions results in a dramatic improvent in the signal to
noise and allows the identification of transitions in channels produced with low
cross-sections (≤ 10µb). However, one problem with such a method is that one
is not actually measuring the nucleus of interest but rather inferring it from the
evaporated particles. Isotopic impurities in the target at the level of only one per
cent of so can can cause problems with identification of very weak channels due
to the large differences in for example, neutron evaporation probability between
isotopic compound nuclei differing by only a few neutrons.
A more direct way of determining which nucleus specific gamma-ray tran-
sitions come from is to detect the recoiling nucleus itself. This is the premise
behind the use of recoil separator type devices. Their main job is to collect the
recoils and separate them from the (much more intense) flux of beam particles. In
addition, there may be other contaminant nuclei which one would like to remove
such as products from fusion-induced fission or deep-inelastic type reactions. The
rejection of non-evaporation type events is usually achieved by passing the par-
ticles through some form of electric and magnetic fields which separate the ions
by their mass over charge state (AQ
). There are usally two electro-magnetic fields,
one to do the main job of ‘dumping’ the beam and the second one to give disper-
sion in the focal plane with the AQ
of the evaporation residues. There is usually a
position sensitive detector (such as a PPAC or microchannel plate) at the focal
point and the position of the recoil on this detector yields information on its AQ
ratio.
4.4.1 Recoil Mass Separators.
The Daresbury recoil mass separator [143] used two Wien filters (crossed magnetic
and electric fields) to separate the beam from the evaporation residues. Crossed
E and B fields are velocity filters and will only allow through particles with a
certain velocity. Since the recoils are slower than the beam particles, setting the
fields so that the residues passed through the Wien filter means that the faster
beam particles are deflected. The final AQ
separation was achieved using a dipole
magnet which dispersed the recoils along a focal plane where they are detected
100
using a microchannel plate.
Three more modern devices are the FMA or Argonne Fragment Mass Analyser
[144] (see figure 4.25), the Oak Ridge RMS [145] and CAMEL [146] at the INFN,
Legnaro, Italy. They work on the basis of two electric dipole fields which give
the beam separation with a magnetic quadrupole magnet in the middle to give
steering and focussing.
DipoleBeamStop
ElectricField
MagneticField
ElectricField
NeutronDetectors
Target
Ion-Chamber
Dipole
F.M.A
Chamber
Ge Detectors
AYEBALL
LinacTandem
ATLAS
P.P.A.C
Figure 4.25: Schematic of the AYEBALL array used in conjunction with an arrayof neutron scintilator detectors, the Argonne Fragment Mass Analyser and a splitanode ionisation chamber [29].
One problem with recoil separator devices is that the recoiling ions have a
distribution of charge states when they leave their thin production target. The
dispersion allowed by the final dipole magnet and the limit of the size of the
postion detector (micro-channel plate or PPAC) typically limits to twothe number
of charge states which can be focussed through such a device and identified at the
final focus. It is usual in experiments using recoil separators to initially perform
a charge state sweep, where the field settings of the electro-magnetic devices
are set to the charge state which will maximise the yield of a specific isobar.
Figure 4.26 shows the results of a charge state sweep for A=87 recoils for the
reaction 36Ar+54Fe from the FMA.
Clearly, there will be a susbantial loss in detection efficiency due to the fact
that not all ionic charge states can be measured.
However, such devices do allow very clean isotopic identification of recoils with
mass resolutions of upto one part in 300. Figure 4.27 shows a two-dimensional
spectrum taking from the FMA of the x-position on the PPAC verses the energy
loss in the PPAC for the reaction 24Mg+40Ca. Clear identification of different
isobaric species is allowed. Note the presence of two charge states on the focal
plane.
By setting gates around these mass locii in offline sorting, one can associate
gamma-ray transitions with specific recoils. Figure 4.28 shows the results of these
101
Figure 4.26: Results of a charge state sweep for A=87 recoils for the reaction36Ar+54Fe [29].
mass gates for the 24Mg+40Ca reaction and there is a clear difference between
the spectra.
However, due to the fact that there is some overlap between the mass locii in
the focal plane, there are some bleed throughs from one mass gate to another.
This is a problem when the contaminant nucleus is produced much more readily
than the one of interest (eg. the production cross-section for the A=61 recoils
is an order of magnitude greater than for the A=62). If there are sufficient
statistics this problem can be overcome by subtracting normalised portions of
higher mass spectra from the spectrum of interest to obtain ‘clean’ or isobarically
pure gamma-ray identification spectra as shown in figure 4.29
Z-Separation Using Split Anode Ionisation Chambers.
The discussion above shows how one can obtain the mass of a nucleus using
a separator. However, for full identication, the atomic number must also be
obtained. In the case of high-Z nuclei, where the main contamination comes
from the large fission background, it is often possible to determine the Z of the
specific recoils by the energies of the co-incident X-rays (associated with electron
conversion decays) observed in the gamma-ray identification spectra [48, 45, 46].
However, for lower-Z recoils, where the X-ray energies are much decreased, a
different way of acertaining the Z of the recoil must be used.
One successul method has been to put a split anode ionisation chamber behind
the position focus of a separator. The energy loss in the first part of this chamber
102
∆Ε
A/Q
5762
Q=11
61
60
59 58 59
Q=10
Figure 4.27: Argonne FMA two-dimensional spectrum of X-position versus energyloss in the PPAC for the reaction 40Ca+24Mg showing clear separation of thevarious isobars. Note the presence of two charge states on the focal plane [29].
(∆E) can then be plotted against the total energy deposited in the chamber by
the recoil E. If the velocity of the ions is high enough, it has been shown that
recoils of the same mass but different Z will deposit different amounts of energy
in the first part of the ion-chamber [154, 155]. The ion chamber associated with
the Daresbury recoil separator [143] was used very successfully to identify the
first excited states in very neutron deficient N=Z nuclei from 64Ge upto 84Mo
[139, 140, 141, 142] by detecting the very weakly populated 2-neutron evaporation
channel. The ion-chamber was used to separate out events from the same mass
but from the (much larger) 2p and pn channels.
Figure 4.30 shows the ion-chamber ∆E signal for A=58 gated recoils from the
reaction 40Ca+24Mg from the AYEBALL array with the FMA. There is a clear
separation between the ∆E signals associated with the 58Ni and 64Zn (6411
∼ 5810
recoils from the α2p ands 2p2n+44Ca channels resepctively.
103
Figure 4.28: Raw, mass gated spectra from the reaction 40Ca+24Mg [29].
By gating on these different portions of the ion-chamber ∆E spectrum in
offline sorting and performing normalised subtractions of contaminants to decon-
volute the different overlapping channels, one can achieve the isotopically ‘pure’
identification spectra shown for 58Ni in figure 4.31.
104
Figure 4.29: A=58 gated spectra from the reaction 40Ca+24Mg with and withoutsubtractions from higher mass contaminants [29].
Figure 4.30: Rotated ∆E ion chamber signals gated on known transitions in 58Niand 64Zn. The separation by the Z of the recoil is clearly demonstrated [29].
105
Figure 4.31: Gamma-ray spectra from the AYEBALL array for the reaction40Ca+24Mg showing the effect of FMA and ion-chamber gating. The transitionfrom the A
Q= 5.8 recoils 58Ni and 64Zn are clearly separated [29].
106
4.4.2 Gas Filled Separators.
The main problem with recoil separator type devices is that since they trans-
mit the recoils in more than one charge state, they have a limited transmission
efficiency (≤ 10 → 15 %). In cases where the main aim of the experiment is
to simply identify the fusion-evaporation events and separate them from beam
particles and possibly fission background, the transmission efficiency can be dra-
matically improved by using a gas filled separator such as RITU (see figure 4.32
[147]).
Figure 4.32: Design drawing of the RITU gas filled separator [147].
RITU is a QDQQ type device (ie. a quadrupole magnet for collection and
focussing, followed by a dipole magnet for the separation, followed by two further
quadrupole magnets for final focussing. In vaccum mode, the mass resolution of
this device is around a part in 100.
The main idea behind such a device is to use an dipole magnet to separate the
fusion-evaporation products from the background, as in the usual vaccum-mode
recoil separators, but to fill this field region with a dilute gas such as helium.
Collisions between the reaction products and the gas atoms lead to a charge state
107
focussing effect. Ions then follow trajectories approximately determined by the
average charge state of the ion through the gas (independent of the initial charge
state of the ions as they exit the thin production target).
The magnetic rigidity of the dipole magnet for a gas filled separator is given
approximately by the expression [147, 148]
Bρ =mv
eqav=
mv(
evv0
) 1
3
= 0.0227A
Z1
3
Tm (4.4.13)
where mv is the momentum of the reaction product and vo is the Bohr velocity
(=2.19×106 ms−1). Typical gas pressures of about 1mb of helium are used.
4.4.3 Recoil Decay Tagging.
If the neutron deficient nucleus of interest has a substantial decay width by either
α-decay or direct proton emission, this can be used as an experimental tag with
which to identify gamma-ray transitions from excited decays in that nucleus.
This technique is known as recoil decay tagging [156, 157, 158, 159, 160, 161].
The basic idea is shown schematically in figure 4.33.
beam
EDMD
ED
8.2m 0.4m
implantedrecoils
PPACFMA
target
AYEBALL
DSSD
Figure 4.33: Schematic of using the FMA for RDT experiments [159].
Fusion residues are selected using online mass separation such as the FMA or
RITU. A pixellated charged particle detector is placed at the final focus of the
separator a charged particle detector, which gives good position resolution for the
incoming recoils. Gamma-rays emitted from fusion-evaporation reactions can be
correlated with an evaporation residue detected at the focus of the separator.
The pixel that the recoil hit is recorded and if a second event is measured in the
same pixel (but without a recoil signal through the separator) it is assumed that
this second event is the α (or proton) decay of the recoil.
108
Figure 4.34: Proton and alpha-particle energy spectra from the decays of theproducts from the 92Mo+58Ni reaction at a beam energy of 260 MeV [161].
Since α and direct proton decays have discrete energies, these are definitive
signatures of a particular nucleus and can be correlated with specific recoils by
their position in the DSSD detector. The limiting factor in such experiments is
that the average time between 2 recoils hitting the same pixel should be large
compared to the lifetime of the α or direct proton decay.
Using this method, in-beam spectroscopy of neutron deficient nuclei with
fusion-evaporation cross-sections as low as 10 µb can be achieved [158].
Figure 4.36 shows the comparison between the recoils mass gated and RDT
gated γ − γ coincidence spectra for very neutron deficient Rn isotopes. The
extra degree of channel selection afforded by gating on the alpha-decay of the
compound nucleus allows this weak channel to be identified.
4.4.4 Recoil Filter Detectors.
Another way of improving the signal to noise and measured resolution of gamma-
ray lines emitted in fusion-evaporation reactions using a thin target is to detect
the recoils directly using an array of thin scintilators [162, 163] to detect those
recoils which are scattered in the target out of the beam direction and into the
recoil cone.
Figure 4.37 shows schematically how such a detector works. Using a pulsed
beam, one can measure the time of flight for the recoils (and other reactions
products such as scattered beam, fission products, etc) to reach the filter detector.
109
Figure 4.35: (a) Gamma-ray spectra gated on A=147 residues by the FMA, (b)Gamma-ray spectra gated by the tagged proton decaying from the h 11
2
of 147Tm
(c) the proton d 3
2
isomeric state in 147Tm [161].
The (slower moving) evaporation residues can be selected in off-line anaylysis by
gating on this time of flight signal. Note that if the target/detector distance
is known, this time of flight gives a direct measure of the recoil velocity. This
coupled with the angular information obtained by knowing which recoil filter
detector element was hit, allows excellent correction for Doppler broadening due
to the spread in recoil cone [163].
This type of detector has also proved immensely useful in identifying low
cross-section evaporation products in heavy nuclei where the background from
fission products is a large problem [47]. In these cases, fission can make up 99%
or more of the total fusion yield.
110
0
2
4
6
8
100 200 300 400 500 600 700 800gamma energy (keV)
0
100
200
300
400
500
coun
tsco
unts
selected usingrecoil’s alpha decay
selected usingrecoil mass only(for comparison)
433 keV
504 keV
Hf Coulex176
(4 2 )
(2 0 )+ +
→
→
+→ +
Figure 4.36: Comparison of FMA-γ − γ projections for (a) no recoil conditionand (b) using RDT using the reaction 176Hf+28Si at a beam energy of 142 MeV[159].
111
RFD
evap.residues.
pulsedbeam
gamma-rayarray
γ
beam
scat
tere
d be
am
evap
.re
sidu
es
coul
exfi
ssio
n
Time of Flight (ns)
0 100 200
Figure 4.37: Schematic of the design and operation of the recoil filter detectorwhich uses time of flight to distinguish between evaporation residues and un-wanted background events.
112
Chapter 5
Measurement of Lifetimes ofBound Nuclear States.
In this chapter we will investigate various methods for the measurement of nuclear
lifetimes. (The review by Nolan and Sharpey-Schafer [202] is an excellent source
of information on these topics). The deduction of lifetimes is important for a
number of reasons. The lifetime of the nuclear state τ is related to its intrinsic
width by the Heisenberg uncertainty principle such that
Γτ ≥ h (5.0.1)
The probability of decay is proportional to the intrinsic energy width Γ and
depends soley on the matrix element between the initial and final states and the
operator which governs the decay between them, such that [202]
Γ α | < φf |M |φi > |2 (5.0.2)
where M is the operator for the decay and φf and φi are the wavefunctions
of the initial and final states respectively. The lifetimes of the decay thus re-
veal information on the nature of the states. The lifetime can be compared with
Weisskopf single particle estimates to help deduce the spin difference between
the initial and final states, or if this is already known (from for example an-
gular distribution/correlation data) can be used to deduce other effects such as
an enhancement in E2 strength consistent with a highly collective (deformed)
structure.
By measuring the lifetime of a nuclear state, one is really measuring the decay
probability from one quantum state to another. For electromagnetic decays, the
113
transitions probability from a state Ji to a state Jf (summed over all possible
magnetic substates) by a transition of energy Eγ is given by [7]
Tfi(λL) =8π(L+ 1)
hL ((2L+ 1)!!)2
(
Eγ
hc
)2L+1
B(λL : Ji → Jf ) (5.0.3)
where B(λL : Ji → Jf) is called the reduced matrix element.
Measuring the lifetime (decay probability) of a nuclear state thus gives a value
for the B(λL : Ji → Jf).
For lifetimes, τ in units of seconds where the transition probability per unit
second, T = 1τ, (Eγ in MeV),
T (E1) = 1.587 × 1015E3γB(E1) (5.0.4)
T (E2) = 1.223 × 109E5γB(E2) (5.0.5)
T (E3) = 5.698 × 102E7γB(E3) (5.0.6)
T (M1) = 1.779 × 1013E3γB(M1) (5.0.7)
T (M2) = 1.371 × 107E5γB(M2) (5.0.8)
T (M3) = 6.387 × E7γB(M3) (5.0.9)
The units ofB(Eλ) are e2fm2λ and the units ofB(Mλ) are (eh2Mc)2 (fm)2λ−2.
Due to the long range involved in the lifetimes of nuclear states, different
techniques must be employed in order to measure nuclear states. In this chapter
we will deal with methods to deduce nuclear lifetimes in the region from 10−15 →10−3 seconds.
5.0.5 Weisskopf Single Particle Estimates.
Lifetimes of nuclear states are sometimes described in terms of of Weisskopf units
(W.u.), which give a yardstick as to the lifetime range expected for a typical decay
of a fixed multipolarity. The Weisskopf single particle estimates are based on a
114
single proton in a spherical orbit. The expressions for the single particle estimates
for the reduced transition probabilities are given in table 5.1. To convert these
into lifetimes, simply substitute in the single particle B(Mλ) values into the
equations for the transition rates given in the previous section.
The general equations for the single particle estimates for the reduced prob-
ability matrix element are [35],
B(Wu :EL) =1.22L
4π
(
3
L+ 3
)2
A2L3 e2fm2L (5.0.10)
for electric transitions and
B(Wu :ML) =10
π1.22L−2
(
3
L+ 3
)2
A2L−22
(
eh
2Mc
)2
fm2L−2 (5.0.11)
for magnetic ones. M is the single nucleon mass and A is the atomic mass
number.
Transition Multipolarity T 1
2
(1 spu) (seconds)
E1 6.76 × 10−6E−3γ A− 2
3
E2 9.52 × 106E−5γ A− 4
3
E3 2.04 × 1019E−7γ A−2
E4 6.50 × 1031E−9γ A− 8
3
M1 2.20 × 10−5E−3γ
M2 3.10 × 107E−5γ A− 2
3
M3 6.66 × 1019E−7γ A− 4
3
M4 2.12 × 1032E−9γ A−2
Table 5.1: Weisskopf single particle estimates for transition half-lives [190, 191].Note, Eγ is in keV and A is the mass number.
Typcal recommended upper limits rate for the different types of transitions
can be found for the different mass regions in references [192, 193, 194, 195].
5.0.6 Determining Nuclear Quadrupole Deformation fromLifetimes of E2 Transitions.
For deformed nuclei, the deformation, or deviation from sphericity, is related to
the intrinsic quadrupole moment of the nucleus, Qo, which is in turn related to
the B(E2) of the collective, E2 transitions in the system by the expression,
115
B(E2) =5
16πQ2
o| < JiK20|JfK > |2 (5.0.12)
where < JiK20|JfK > is a Clebsh-Gordon coefficient for the transition de-
caying from a state of spin I to one of I − 2 is given by
< JiK20|JfK >=
√
√
√
√
3(J −K)(J −K − 1)(J +K)(J +K − 1)
(2J − 2)(2J − 1)J(2J + 1)(5.0.13)
Thus by susbtitution, the lifetime is related to the quadrupole moment by the
expression [53],
1
τ= 1.223E5
γ
5
16Q2
o| < JiK20|JfK > |2 (5.0.14)
The quadrupole moment can be related to the quadrupole deformation pa-
rameter β2 by the expression [108],
Q0 =3√5πZR2β2
1 +1
8
√
5
πβ2....
(5.0.15)
Therefore, assuming the rotational model, measuring the intrinsic lifetime of
a state in a stretched E2 cascade (rotational band) can allow the deduction (in a
model dependent way) of the nuclear deformation.
5.1 Electronic Timing Methods.
The law of radioactive decay states that at time t, the number of nuclei left in a
particular state of lifetime τ (given that there were N0 at t = 0), is given by
N(t) = N0exp(
− t
τ
)
(5.1.16)
If the lifetime of the nuclear state we wish to measure is long compared to
the intrinsic timing properties of the (germanium) detector, one can simply use
pulsed beam techniques to determine the number of decays of a state as a function
of time.
The general experimental idea is summarised in figure 5.1. A thick or backed
target is irradiated by a beam to form the nucleus of interest for a period of time
which is short compared to the lifetime of the isomer we wish to measure. (Note
116
the recoil must stop in the view of the gamma-ray detectors). The beam-on period
is also usually used to start a clock, ie. to define a population at time t = 0. The
beam is then switched off for a period and gamma-rays decaying from the isomer
(called delayed, or out-of-beam decays) are measured in the germanium detectors.
The relative time difference between when the target was irradiated and when
the gamma-ray is measured (usually done with a time to amplitude converter
or TAC, or a time to digital conver, TDC). Over a period of time, a full time
(decay) spectrum for the state can be obtained. For a single, long lived state,
with no long lived feeding transitions, the time spectrum will be an exponential
(ie. a straight line on a log scale), which can be simply fitted to the radioactive
decay law to obtain the lifetime of the state.
prompt, in beam gammas
isomer, T1/2 > few ns
(in delayed portion of TDC/TAC)delayed, out of beam gammas
(in prompt peak)
time
Log
N
time
Log
N
beam off period(delayed gammas)
next beam pulse,
beam
inte
nsity
stop clock
beam pulse(prompt gammas)
start clockreset clock
t=0
Figure 5.1: Schematic using pulsed beams to measure nuclear lifetimes electron-ically.
Electronic timing methods have been extensively used to measure lifetimes of
K-isomers in the mass 130 [93, 94, 38] and 180 regions [83, 86, 87, 84, 9].
Figure 5.2 shows the time spectra gated by gamma-ray lines below the T 1
2
=6µs,
Kπ = 8− isomer in 138Gd [93]. Figure 5.3 shows the gamma-gamma coincidence
117
data for 138Gd lines observed out of beam. The first 4 decays in the yrast band
are observed, together with the 583 keV transition which decays out of the iso-
mer. Note that the 10+ → 8+ 616 keV decay previously observed in prompt
decay studies [189] is not present.
138Gd
counts
time [ s]µ
0 10 20 30010
110
210
Figure 5.2: Decay curve following the decay of the T 1
2
=6 µs, Kπ = 8− isomer in138Gd [93].
118
138Gd
counts
energy [keV]0 200 400 600 800 1000
0
100
200
221
<---- 384 (gate)
489
556
583 0.0 0 +
220.8 2 +
605.2 4 +
1094.3 6 +
1649.9 8 +
2233.1 (8 ) -
220.8
384.4
489.1
555.6
583.2
T = 6 1/2 s
Figure 5.3: Out of beam, gamma-ray coincidence spectrum showing transitionsin coincidence with the 384 kev 4+ → 2+ transition in 138Gd [93].
119
Other examples of such work are to be found in studies of high spin, yrast
trap decays in trans-lead nuclei by electric octupole (E3) decays [196, 197, 246,
198, 199, 200, 201]. A particularly good example of this technique at high spins is
the proposed 8.5 MeV, spin 34+ state in 212Fr, which decays by an E3 transition
with a meanlife of 34±3 µs [196].
5.1.1 Gamma-ray Spectroscopy Across Isomers
In order to establish prompt decays into isomeric states one needs to be able to
correlate coincidences between transitions above and below isomeric states. For
lifetimes below around 20 µs, this is possible using a pulsed beam (with the beam
pulses separated by the order of 2 µs) by allowing each gamma-ray detector to
have its own individual time signal with respect to the beam pulse (or other fixed
time reference such as an RF signal).
Figure 5.4: Single TDC and TDC difference spectra for reaction 11B+176Yb usinga pulsed beam of width 1 ns separated by 1400 ns [292]. Note the clear separationbetween in-beam and out-of-beam event and the well defined regions correlatingto co-incidences above and below isomeric states.
120
The upper portion of figure 5.4 shows a typical TDC time spectrum for
gamma-rays detected in a pulsed beam measurement. Note that the majority
of counts occur during the ‘in beam’ period. The bump to the right of the
prompt, in-beam peak is due to low energy gamma-rays (or X-rays) which occur
in the beam pulse but have poor timing properties (due to poor charge collection
times in the detector). These thus ‘walk’ out of the prompt time gate. This effect
can be corrected for in off-line software analysis.
Figure 5.5: (a) Spectrum of ‘earlies’ obtained from a sum of transitions belowthe 12+ isomer in 106Cd. (b) Out-of-beam,prompt-γ − γ spectrum gated on the633 keV 2+ → 0+ transition in 106Cd [3].
One can software gate on the various portions of this spectrum to define those
gamma-rays which occur ‘in-beam’ or ‘out-of-beam’. It is thus possible to create
a gamma-gamma coincidence matrix of gamma-rays which occur in the in-beam
period correlated by delayed coincidences with gamma-rays which are detected
in the out-of-beam period. By gating on in-beam or out-of-beam portions of
the TDC singles spectra, one can determine whether a gamma-ray was measured
in-beam (early) or out-of-beam (delayed) (see figure 5.5).
121
0 0 +
633 2 +
1494 4 +
6 + 24922504 6 +
8 + 3044
3367 8 +3354 7 +
8 + 3788
4121 9 +
4816 10 +
5419 12 +
6227 14 +
7119 16 +
8100 18 +
9250 20 +
10561 22 +
12049 24 +
13726 (26 ) +
15584 (28 ) +
21052331 5 +
7 + 3084
3409 7 - 9 - 3678
4324 11 -
5214 13 -
6264 15 -
7518 17 -
8884 19 -
10350 21 -
11941 23 -
5 - 2629
3319 6 -3507 8 -
4106 10 -
4967 12 -
5976 14 -
7121 16 -
8411 18 -
9877 20 -
4436 10 +4660 12 +
5253 (13 ) +
5573
5987 5912
55585770
6101
6858
7480
1716 2 +
2371 3 -
4183 8 +
9319 + X (18 ) +
9722 + X (19 ) +
10161 + X (20 ) +
10664 + X (21 ) +
11168 + X (22 ) +
11740 + X (23 ) +
12311 + X (24 ) +
12951 + X (25 ) +
13614 + X (26 ) +
14322 + X (27 ) +
15065 + X (28 ) +
15861 + X (29 ) +
632.8
861.3
997.7
874.7
1009.4
540.9 552.8
1284.5
1076.7
1028.4695.3
703.3
1471.5
602.4
807.9
892.3
980.8
1150.6
1310.6
1487.6
1677.6
1857.4
433.4
862.31295.9
610.6
225.8
836.4
592.5 754.2
171.1311.6633.9
269.1
906.0 917.6 780.6
1134.8
524.2
690.5422.
8 187.6
827.4
598.3
861.3
1008.6
1145.3
1290.1
1466.0
645.6
889.8
1050.4
1253.6
1366.5
1465.8
1591.4
315.0335.7223.6
592.8
659.7319.9
414.2
304.9
542.6
517.7
330.5
757.7
621.9
403.6
1622.6
438.5
502.6
504.4
572.5
571.0
639.7
663.3
718.8
733.0
795.6
842.4
1007.0
1143.8
1302.6
1452.1
941.5
1077.2
1211.4
1382.3
1529.0
= 90 ns τ
Cd106 48 58
14
3
2
Figure 5.6: Decay scheme for 106Cd [3].
122
By taking the time difference between two detectors, one can create the type
of time difference spectrum shown in the lower portion of figure 5.4. If for example
one wanted to look at gamma-rays in prompt coincidence with each other, but
delayed with respect to the beam pulse (to look at decay cascades below an
isomer), one would gate on the prompt-prompt region of the TDC difference
spectrum (thus ensuring that the gamma-rays were in prompt coincidence with
respect to each other) but with the extra condition on the individual TDC spectra
such that the transitions were measured out of beam.
Similarly, one can put the condition on the TDC difference spectrum to look
for coincidences between early and delayed gamma-rays. Figure 5.5 shows the
gamma-ray spectra for transitions above and below the 12+ isomer in 106Cd [3].
counts
time [ns]0 500 1000 1500
110
210
310
410
510
610
1950(150) ns
145,170,194,217,236 and 710 keVb)
counts
time [ns]0 200 400 600 800
010
110
210
310
410
222(8) ns
145, 170 or 194 keV (start)131 keV (stop)
a)
Figure 5.7: Time spectra for isomeric decays in 175Ta (a) shows the lifetime of
the 92
−[514] isomer in 175Ta (TDC difference spectrum) (b) shows the lifetime of
the 212
−isomer in 175Ta (pulsed beam TAC singles spectrum) [87].
Using the gamma-gamma-time coincidence techniques, one can gate on any
123
two gamma-ray energies in a gamma-gamma-time-difference cube (3D-coincidence
matrix) and project the time difference between any two gamma-rays. Figure 5.7
shows the time difference spectra gated by transitions above and below the 92
−
isomer in 175Ta [87]. The upper portion of figure 5.7 is a TDC difference spec-
trum gated on transitions above and below the 228 ns 92
−isomer, while the lower
spectrum is a singles TAC spectrum showing the lifetime of the 212
−, 1950 ns
isomer [87]. For reference, the decay scheme of 175Ta is shown in figure 5.8.
619 13/2 +
0 7/2 +130 9/2 +
284 11/2 +
461 13/2 +
658 15/2 +
872 17/2 +
1101 19/2 +
1341 21/2 +
1592 23/2 +
1850 25/2 +
2118 27/2 +
2394 29/2 +
2681 31/2 +
2979 33/2 +
3287 35/2 +
3611 37/2 +
3947 39/2 +
4297 41/2 +
4661(43/2 ) +
5037(45/2 ) +
9/2 - 5/2 -
131 9/2 -
276 11/2 -
446 13/2 -
640 15/2 -
856 17/2 -
1093 19/2 -
1350 21/2 -
1621 23/2 -
1909 25/2 -
2205 27/2 -
2515 29/2 -
2825 31/2 -
3143 33/2 -
3457 35/2 -
1552 17/2 +1651 19/2 +
1793 21/2 +
1969 23/2 +
2173 25/2 +
2402 27/2 +
2656 29/2 +
2931 31/2 +
3225 33/2 +
1566 21/2 -
1877 23/2 -
2202 25/2 -
2537 27/2 -
2879 29/2 -
3231(31/2 ) -
1729 21/2 +
1895 23/2 +
2086 25/2 +
2298 27/2 +
2531 29/2 +
2782 31/2 +
3052 33/2 +
3338 35/2 +
3640 37/2 +
3957 39/2 +
4282 41/2 +
4619(43/2 ) +
4966(45/2 ) +
1279 (15/2)
3216(31/2 ) -
3526(33/2 ) -
3762
4041
4329
4635
129.6
154.3
176.9
196.9
214.4
228.8
240.5
250.1
258.8
284.3
373.9
443.0
490.6
526.0
563.1
606.7
659.2
714.0
331.6
411.3
469.3
509.0
543.7
584.5
632.3
686.2
740.0
144.6
170.0
193.6
216.6
236.4
256.9
271.1
288.7
295.7
309.6
310.4
318.3
314.5
410.4
493.4
559.9
605.3
627.9
363.9
453.1
528.3
584.6
619.0
632.0
131.471.9
833.4
98.8
142.7
175.5
204.1
229.4
253.6
275.1
293.7
379.8
482.9
569.1
318.6
433.7
528.2912.1
695.7458.9
680.3
894.2
1090.6
932.4
311.2
325.2
334.5
342.5
351.9
635.9
677.2
659.5
694.0
216.5
473.3
709.6
165.7
191.0
212.4
232.3
251.3
269.5
286.6
302.1
316.3
325.5
337.4
346.0
403.7
483.3
555.9
618.7
662.9
357.1
444.4
520.4
588.8
641.1
684.0
163.4
235.7
279.1
288.0
305.8
646.8
336.4
678.8
310.4
Ta175 73 102
7/2 [404] +
5/2 [402] +
9/2 [514] - 222ns
5ns 1950ns0.9ns
( 0.5ns)
( 2ns)(35/2,37/2)
(37/2,39/2)
(39/2,41/2)
(41/2,43/2)
1/2 [541] - 123.451.5+X
( 1ns)
Figure 5.8: Partial decay scheme for 175Ta [87].
5.2 The Recoil Distance Method.
In order to measure lifetimes in the region between 10−8 → 10−12 s the most
commonly used method is the Recoil distance or ‘plunger’ method [202]. The
nuclei are formed in a thin (∼500 µg/cm2) target usually by a fusion-evaporation
reaction. The recoils then fly out of the thin target towards a thick, stopper or
‘plunger’ where they are stopped.
124
The Recoil Distance Doppler-Shift Method(RDM/RDDS)
Target Stopper
Eu E = E (1+ v/c cos( ))us θ
θ
Detector
Decay Curve
v ~ 1-2 % cv
u: unshifteds: shifted
d
Figure 5.9: Schematic of the Recoil Distance Method.
The idea behind this technique is shown graphically in figure 5.9. The premise
of the RDM is to measure the difference in the intensity of gamma-rays decaying
either in flight of when stopped in the plunger as a function of target-stopper
distance. Gamma-rays which are observed in a detector at angle θ to the recoil
direction will have their energies shifted by the Doppler effect such that to first
order,
Es(θ, t) = Eo
√
1 − vc
1 − vccosθ
≈ Eo
(
1 +v
ccosθ
)
(5.2.17)
125
where Eo is the unshifted transition energy and v is the recoil velocity.
The gamma-rays which are emitted between the target and the plunger will
decay from a recoil with velocity v, while those that decay while stopped in the
plunger will have their unshifted energy E0. The gamma-ray lineshape will be
split into two parts, the shifted and stopped component. The intensity in the
shifted peak Is is given by
Is = N0
(
1 − exp
(
− d
vτ
))
(5.2.18)
where N0 is the total number of decays (total number in the shifted and
stopped peaks) d is the recoil distance (ie. target–stopper distance) and τ is the
measured, apparent lifetime of the state. The nuclei which decay in the stopper
(emitting the gamma-ray at the stopped energy, E0) will give rise to a line with
intensity,
Io = Noexp
(
− d
vτ
)
(5.2.19)
Since v and d are known, the lifetime can be determined using R where,
R =Io
Io + Is= exp
(
− d
vτ
)
(5.2.20)
Descriptions of various plungers can be found in references [222, 202]. The
target-stopper distance can be measured using a micrometer screw [202] or using
the capacitance between target and stopper (in effect parallel plates).
126
Figure 5.10: Schematic of the NORDBALL plunger taken from [228] .
127
The velocity of the recoil can be calculated directly from the difference in
energy between the shifted and unshifted peak values at a given angle using
equation 5.2.17. Good examples of RDM experiments can be found in references
[217, 226, 227, 228, 229, 230]
5.2.1 Feeding Corrections and Gating From Above.
The observed lifetime of a nuclear state depends on the lifetimes of the states
which feed it. The relationship between the intrinsic lifetime of state and the
observed or apparent lifetime is given by the Bateman equations of radioactive
decay [203], such that
dNi(t)
dt=
N∑
j=1+1
Njλij −Ni
i−1∑
j=1
Niλi (5.2.21)
where Ni(t) is the population of level i at time t.
In order to deduced the intrinsic lifetime of a state from the measured or
‘apparent’ lifetime, one must first correct for the lifetimes of the states which
feed into the level of interest. In the case of fusion-evaporation reactions, often
much of the side-feeding intensity comes from unresolved continuum states of
which the lifetime is not known and this can cause problem with the fitting of
RDM data.
In singles measurements, the decay properties and intensities of all the feeding
transitions for the state of interest must be known in order to solve the Bate-
man equations. In practice, this is very difficult to achieve and can give rise to
erroneous values for the lifetimes. The advent of high-efficiency gamma-ray ar-
rays has somewhat alleviated this problem. If a number of discrete gamma-rays
are in a connected cascade, one can determine the feeding completely by setting
a gamma-gamma coincident gate on the shifted component of a line above (ie.
higher spin) than the state of interest. The coincidence requirement also has
the large advantage of cleaning the spectra up, thus reducing the possibility of
contaminant transitions giving false values for the shifted or unshifted intensities.
Figure 5.11 gives a good example of the use of the coincidence technique
in the RDM for high spin states in 110Cd [229]. the spectra are gated on the
shifted component of the 335 keV 10+ → 8+ yrast transition. Note the change
in the relative intensities of the stopped and shifted components for the various
128
lines with stopper distance The decay scheme for this nucleus and the fits to the
lifetime data are shown for reference in figures 5.12 and 5.13.
129
Figure 5.11: Coincident plunger spectra gated on the 10+ → 8+, 335 keV transi-tion in 110Cd. Taken from Piiparinen et al. [229].
130
0 +0
2 +658
4 +1542
6 +2480
8 +32758 +3440
10 +3611
12 +4172
14 +5026
16 +6100
18 +7325
6 –28968 –3056
10 –3824
12 –4931
8 –3428
10 –4182
509312 –
7 –2879
9 –3346
11 –4173
524913 –
15 –6181
5 – 25405 –2660
10 +4078
12 +4889
14 +5857
14 +5676
8 +3187
10 +4620
0 +1473
2 +1783
4 +2251
6 +2877
658
885
938
795
171
561
854
1075
1224
159
768
1107
755
911
467
827
1076
932
811
968
1433
467
626
1126
70813
3539
7
399563
802
637
707
424
787
1504
335164
960
265
1155
998
1117
399339219236356
177
531 548290
478
9.2 ps
< 3 ps
< 3 ps
< 4 ps
< 4 psτ = 670 ps
80 ps
1.0 ps
2.0 ps
12.0 ps
2.0 ps
< 1.5 ps
750 ps
71 ps
3.0 ps
< 2.0 ps
8.6 ps
1.5 ps
4.7 ps
5.0 ps
A B
C
S
G
E
F
H
Cd6248110
Figure 5.12: Decay scheme for 110Cd by Piiparinen et al. [229].
131
10+(3611) → 8+Gate: 561
τ = 670(35) ps
2+(658) → 0+Gate: 795 + 885 + 811
τ = 9.2(6) ps
12+(4172) → 10+Gate: 795 + 335
τ = 12.0(6) ps
9-(3345) → 7-Gate: 658 + 885
+ 938 + 399
τ = 71(4) ps
100Mo(13C,3n)110Cd E = 44 MeV Θ = 143°
Shifted peakUnshifted peakFit
Distance µm
Num
ber
of C
ount
s
10 100 1000 10 100 1000
102
103
104
103
104
102
103
104
102
103
104
Figure 5.13: Fits to RDM data for 110Cd from Piiparinen et al. [229].
132
5.2.2 The Differential Decay Curve Method.
The usual method of analysis of high-spin data in RDM experiments is to fit the
lifetimes of state including the effect of feeding from high spin states using the
Bateman equations. The individual lifetimes in a cascade are generally calculated
using a χ2 minimization program. This works well when the nature and intensity
of the levels feeding the state of interest are well known, however, if this informa-
tion is less well known, systematic errors can be introduced given erroneous values
for the calculated lifetimes. The Differential Decay Curve Method [231, 232, 218]
allows one to eliminate the effects of the feeding lifetimes for the transition by
gating from transitions above the transition of interest (see figure 5.14).
(gating transition)(detemines feeding)
side
feed
ing
"direct feeding transition"
level, i
decay from level of interest
Z
X
Y
XsXu
Zs
Zu
YuYs
Figure 5.14: Schematic of the gating in a cascade for use with the differentialdecay curve method.
The basis of the DDCM is that the Bateman equations can be reduced to the
expression [232, 218],
τi =−Ni(t) +
∑
h bhiNh(t)d(Ni(t))
dt
(5.2.22)
where τi is the intrinsic lifetime of the state of interest, Ni is the population
of that state, Nh are the populations of those states which feed the state i and
bhi are the branching ratios for the feeding transitions. d(Ni(t))dt
is the differential
of Ni with respect to time at time t.
133
If there are three transitions in cascade, Y ,Z and X, (see figure 5.14) where
X is the transition out of the state on interest (level i), Z is the transition
which directly feeds state i and Y is a transition in the cascade above Z and
X, then the lifetime of the state i can be calculated directly, (ie accounting for
feeding) by setting a gamma-ray co-incidence gate on the shifted component of
the higher lying, transition Y and measuring the intensities of the moving and
shifted components of the transitions X (decaying from the state of interest) and
Y (directly feeding the state of interest).
µ31 m
300
0
200
0
300
0
200 240 250 290 320290
cou
nts
Energy [keV]
213 keV 256 keV 298 keV
0 20 40 60 80 100
50
0
150
100
0
100
200
300
400
d[ m]
co
un
ts
µ
Lifetimes in the SD band of Pb194
Decay out ~ 15%
decay curves (shifted)
µ22 m
µ97 m
298
256
213
R. Krucken et al. PRL 73, 3359 (94)
:
Figure 5.15: Decay curves for the decay of the SD band in 194Pb [219].
This is given by [232, 218]
τ(ti) =IXs(tk) − αIZu(tk)
d(IXs(tk)dt
(5.2.23)
where the I is the intensity measured in the gate on the shifted component of
the higher lying transition Y and the subscripts u and s correspond the unshifted
134
(stopped) and shifted components of the lineshape respectively.
α =IXu + IXs
IZu + IZs(5.2.24)
135
and
d(IXs(tk))
dt=IXs(tk+1) − IXs(tk−1)
tk+1 − tk−1(5.2.25)
and t = dv.
This tecnhique has the advantages [232] that (a) only directly measured in-
tensities are involved in the analysis, no lifetimes or feeding intensities have to
be known; (b) only flight-time differences are important in the analysis, the ab-
solute target-stopper distance is not involved, thus a systematic error in this
measurement does not affect the results
5.3 The Doppler Shift Attentuation Method.
If the lifetime of the state is of the same order of the slowing down time in a
target/backing (∼ 10−12 sec) then the RDM can no longer be used to determine
the lifetime (as the target–stopper distance has to be made too short). However,
if the experimenter has a knowledge of the slowing down process of the recoil in
the target (and target backing), one can use the lineshape of the transition of
interest as a function of detector angle to determine the nuclear lifetime. This
method is known as the Doppler Shift Attentuation Method.
Recalling that the observed gamma-ray energy for a transition of energy Eo,
emitted from a recoil of velocity v at an angle θ to the detector is given by Es
where
Es ≈ Eo
(
1 +v
ccosθ
)
(5.3.26)
The gamma-ray energy spectrum will now have a lineshape depending on what
velocity (between vo and zero) the recoil had when it emitted the gamma-ray.
For example, on average, decays from faster (∼ 10−15s) levels will have hardly
slowed down in the target/backing material when the gamma-ray is emitted,
while decays from states with slower lifetimes (∼ 10−12s) maybe almost totally
stopped. Measuring the centroid of the total lineshape gives a measure of the
average velocity at which the gamma-ray was emitted. The centroid results are
usually expressed in terms of a Doppler Shift Attenuation Factor F (τ), where
F (τ) =vav
vo=
1
v0τ
∫ ∞
0v(t)exp
(
− t
τ
)
dt (5.3.27)
136
where v0 is the initial recoil velocity and vav is the average recoil velocity when
the gamma-ray is emitted.
The slowing down of the recoil with as a function of time (v(t)) is usually
taken from tables of stopping powers such as those by Braune [204], Northcliffe
and Schilling or Ziegler [205].
The stopping power is usually separated into two effects, electronic and nuclear
stopping. In electronic stopping, the recoil slows down due to interactions with
atomic electrons in the stopping material. Since the mass of the recoil is much
larger than the mass of the electrons, many collisions are required to remove all
the energy of the recoil.
In the nuclear stopping process, the recoils lose energy in a small number of
discrete steps due to nuclear collisions. These can cause the nucleus to scatter
and alter its direction, which must be accounted for (since a change in direction
will alter the measured, Doppler shifted energy). This is usually accounted for
using either the Blaugrund formulism [206] or using Monte Carlo simulations of
the recoil velocity profiles [207, 208, 209].
137
700 900 1100 1300 1500 1700 1900 2100
Energy (keV)
-200
200
600
1000
1400Cou
nts
-200
200
600
1000
1400
Ce band 1132
Ce band 1132
31.7 + 37.4o o
o o142.6 + 148.3
Figure 5.16: DSAM spectra for forward and backward angles for the yrast su-perderformed band in 132Ce taken from GAMMASPHERE [56].
138
0.7 1.2 1.7Energy (MeV)
0.4
0.6
0.8
1.0
F(τ
)
132Ce 1
131Ce 1
0.7 1.2 1.7Energy (MeV)
132Ce 2
132Ce 3
0.7 1.2 1.7 Energy (MeV)
131Ce 2
a) b) c)
Figure 5.17: Measured and calculated fractional Doppler shifts for hte highlydeformed bands in 131,132Ce [56].
5.3.1 Lineshape Analysis.
Note that the lifetime of the state can also be gained directly from the gamma-ray
lineshape (dNdE
) since this gives directly the velocity distribution of the recoils as
they emit the gamma-ray.
dN(t) =1
τexp
(
− t
τ
)
dt (5.3.28)
A knowledge of the slowing down process or stopping power of the recoil, dvdt
,
thus allows the lineshape to be calculated for a given value of τ . (Note that since
Es and v have a linear relationship, it is simple to translate from the measureddNdE
to the calculated dNdv
, the dNdt
= dNdv.dvdt
).
DSAM with Thin Targets for Very Fast Decays.
In the case of very fast transitions (∼10−15s) such as those observed in the su-
perdeformed bands in the A∼80 region the effect of the recoil slowing down in
139
395.0 405.0 415.00.0
500.0
1000.00.0
200.0
400.0
coun
ts
0.0
500.0
1000.0
435.0 445.0 455.0Energy [keV]
0.0
200.0
400.0
600.0
0.0
100.0
200.0
300.00.0
200.0
400.0
600.0
455.0 465.0 475.00.0
200.0
400.0
600.0
0.0
100.0
200.0
300.00.0
200.0
400.0
600.0
197Pb (1)
403 keV 446 keV 467 keV
35o+50
o
90o
130o+145
o
Figure 5.18: DSAM lineshape fits to transitions in the M1 band of 197Pb [216].
the thin target can be used to obtain a Doppler shift between the forward and
backward angles of a large array, from which a fractional Doppler shift can be
obtained [213, 214, 167]. As in the usual DSAM, the difference in gamma-ray
energy measured between detectors at forward and backward angles can be used
to infer the velocity at which the recoil was moving when the gamma-ray was
emitted.
Figure 5.19 shows the angle gated spectra for the SD band in 87Nb from
GAMMASPHERE [167]. For the highest spin transitions there is a clear differ-
ence in centroid for the forward and backward angle detectors. From this shift,
an F (τ) value can be deduced from which an estimate of the lifetime (and thus
quadrupole moment) can be obtained.
140
(a)
(b)
Figure 5.19: (a) DSAM spectra for thin target data on the SD band in 87Nb. (b)F(τ) curves for the superdeformed and normal deformation bands in 87Nb [167].
141
Chapter 6
Measurement of MagneticMoments.
The magnetic dipole moment, µ, is defined classically as the vector cross-product
of a current I and area about which the charge circulates, A. In the nuclear
case, the measurement of a magnetic dipole moment gives extremely sensitive
information on the nature of particles causing the current (ie. whether protons or
neutrons) and the single particle orbital(s) which the active nucleon(s) occupy. As
such they are extremely sensitive probes of nuclear wavefunctions. One can draw
the semi-classical analogy of higher angular momentum orbitals being further on
average from the centre of the nucleus and thus sweeping out larger areas, giving
rise to larger magnetic dipole moments.
The g-factor relates the magnetic dipole moment, µ, and the spin of the state
by the expression [1, 235],
g =µ
I(6.0.1)
In the case of odd-A nuclei, the Schmidt model [235] gives an estimate of the
values of g-factors for pure, single nucleon states moving in shell model orbits,
independent of the nuclear core.
For a single, independent nucleon, the single particle magnetic dipole moment,
µ can be calculated using the expression [233],
µ =gl[l(l + 1) + j(j + 1) − s(s+ 1)] + gs[s(s+ 1) + j(j + 1) − l(l + 1)]
2(j + 1)(6.0.2)
Substituing j = l ± s and s = 12, the Schmidt values for g-factors for pure
142
protons and neutron orbitals can be calculated. For spherical shell model orbitals
where j = l + 12
(eg. p 3
2
, d 5
2
...)
g =1
j
1
2gs +
(
j − 1
2
)
gl
(6.0.3)
and for orbitals with j = l − 12
(p 1
2
, d 3
2
, g 7
2
...),
g =1
j + 1
−1
2gs +
(
j +3
2
)
gl
(6.0.4)
Where gl=1 and gs=+5.587 for protons and gl=0, gs=-3.826 for neutrons
repsectively. The emprical data suggested that the gs values need to be attenuated
by a factor of approximately 0.7 to account for fact that the odd nucleon is not
‘free’ but included in the nuclear medium.
As table 6.1 shows the emprically g-factors for single particle shell model
states.
Particle Orbital g-factorNeutrons h 11
2
–0.21
g 9
2
–0.24
g 7
2
+0.21
d 5
2
–0.33
d 3
2
+0.44
s 1
2
–1.76
Protons h 11
2
+1.17
g 9
2
+1.27
g 7
2
+0.72
f 5
2
+0.54
d 5
2
+1.38
d 3
2
+1.33
p 1
2
–0.23
s 1
2
+2.90
Table 6.1: g-factors for intrinsic spherical orbitals [234].
In reality, most nuclear states are not pure single particle states but super-
positions of a number of different configurations which are mixed together. A
measurement of the magnetic dipole moment can be used to infer the underlying
single particle structure and purity of such a state.
143
6.1 Measurement of Nuclear Magnetic Dipole
Moments.
If a magnetic field is applied to a nucleus with magnetic dipole moment, µ, a
torque, T , will occur which will cause the nucleus to twist around or ‘precess’.
This torque can be calculated using the vector cross product of the magnetic
dipole moment and the applied magnetic field. The rotational frequency of this
precession, is given by the Larmor frequency, ωL where [1]
ωL =µB
hI= −gµNB
h(6.1.5)
110Cd
angular distribution
335 keV ext field up
0 10 20 30 40 50 60 70 80 90
0.8
1.0
1.2
= -107 24 mrad + -ωτ
θ
335 keV ext field down
0.8
1.0
1.2
a = +0.31 2
a = -0.08 4
Figure 6.1: Angular distributions used to measure the g-factor of the yrast 10+
state in 110Cd (E2 decay) [236].
144
110Cdangular distribution
399 keV ext field up
0 10 20 30 40 50 60 70 80 90
0.8
1.0
1.2
θ
= -302 37 mrad + -ωτ
399 keV ext field down
0.8
1.0
1.2
a = -0.23 2
a = 0.00 4
Figure 6.2: Angular distributions used to measure the g-factor of the yrast 7−
state in 110Cd (E1 decay) [236].
For a state with lifetime τ , the effect of this torque will be to shift the measured
gamma-ray angular distribution by an angle ∆θ (essentially because the nucleus
twists around relative to the reaction plane, before it decays) where
∆θ = ωLτ = −gτ BµN
h(6.1.6)
Thus, if the lifetime of a nuclear state is known, by measuring the shift in
the angular distribution in the presence of a known magnetic field, compared to
the usual in-beam distribution, the g-factor can be deduced. Figures 6.1 and 6.2
show the shifts in the observed angular distributions for both field up and down
directions for two different configurations in 110Cd.
The size of the applied magnetic field, B required to cause the nucleus to
145
precess sufficiently so that a rotation of angle ∆θ can be measured, for a state
decaying with lifetime τ , can thus be estimated using the expression [1],
Bτ =∆θh
gµN(6.1.7)
Thus, the shorter the nuclear lifetime, the greater the required magnetic field
to observe a shift in the angular distribution function. A rule of thumb from
equation 6.1.7, is that for a shift of 100 mrads, a Bτ value of approximately
2×10−9 Ts is needed. Thus for liftimes of the order of a nanosecond, external
fields of around 1 Tesla are required, which is at the limit of what can be provided
using an external magnet. For excited states with lifetimes of ∼ 10−12 s, required
larger fields than can be obtained using such magnets, and the very large internal
(or transient) fields of the ion moving through a ferromagnetic crystal have to be
used.
In the case where the applied magnetic field is perpendicular to the reaction
plane, the angular distribution, for a nucleus precessing with the Larmor fre-
quency for time t, will have a, perturbed angular distribution, given by [1, 235]
W (θ, t, B) =∑
k
AkPk (cosθ − ωLt) (6.1.8)
If the lifetime of the state of interest τ is small compared to the electronic
time resolution of the system, the intergrated precession angle, (∆θ = ωLτ) over
the lifetime of the state is measured and the angular distribution will have the
form, [235]
W (θ, t, B) =∑
k
∫ ∞
0
1
τexp
(
− t
τ
)
AkPk (cosθ − ωLt) (6.1.9)
6.1.1 Corrections in the Ion-Implantation Perturbed An-gular Distribution Technique.
The ion-implantation, integral perturbed-angular-distribution (IMPAD) technique
[235, 236, 237], assumes that following a heavy-ion reaction, for an ion coming to
rest in a magnetic host, the total precession of the angular distribution, ∆θ, is
given by,
∆θ = ωLτ + ∆θtr + ∆θfeed, (6.1.10)
146
external static fieldof ~ 0.1 ->1 Tesla to provide B x T.
productiontarget
S
Nthin ferro-mag. layer(gives rise to large transient B)
external mag fieldto provide directionalpolarisation of transientfield
N
S
stopper
beam beam
(a) Static field, for t > 1ns (b) Transient field, for t~10 ps
Figure 6.3: Schematic methods of the (a) static field and (b) transient fieldmethods of measuring g-factors of exited nuclear states.
where ∆θtr is the precession due to the transient field, which acts on the nucleus
as it slows down in the ferromagnetic medium and ∆θfeed is the average static-
field precession accumulated in states which feed the state of interest. The static-
field precession of the level of interest is given by ωLτ , where
ωL = −gµN
hBst (6.1.11)
and Bst is the static hyperfine magnetic field, while g and τ are the the mean
life of the level and its gyromagnetic ratio, respectively.
The magnitude of the transient field contribution may be estimated using the
expression [236],
∆θtr = −〈g〉µN
h
∫ Ts
0Btr(t)dt (6.1.12)
where Btr(t) denotes the transient magnetic field, Ts is the time the ion takes to
come to rest in the ferromagnetic layer, and 〈g〉 denotes the average g-factor of the
(continuum) states that are populated while the nuclei experience the transient
field.
The precession induced in the state of interest due to precessions induced
in higher-lying feeding transitions can be be deduced by measuring the shift in
147
the angular distributions of the direct feeding transitions directly in the same
experiment.
6.1.2 Analysis of Precession Data with Limited Angles.
It is often not necessary to measure the entire shift in the angular distribution to
determine the g-factor. In the case where there are a limited number of gamma-
ray detectors, one can measure the ‘up-down’ intensity ratio by placing detectors
at angles symmetric to the beam direction and measuring the ratio of counts
in the precession shifted decay with the field in both up and down directions
[236, 237, 238, 239, 240]. To determine the field up-down counting asymmetry
and reduce possible systematic errors, the double ratio, ρ, is defined for a ±θdetector pair by
ρ =
√
√
√
√
N+θ(↑) ×N−θ(↓)N+θ(↓) ×N−θ(↑)
(6.1.13)
where N+θ(↑) denotes the number of counts in the θ = +θ detector for field up
direction, etc. For small precessions, ∆θ ≤ 100 mrad, the precession angle may
be obtained from
∆θ =ǫ
S(6.1.14)
where
ǫ =1 − ρ
1 + ρ(6.1.15)
and S, the logarithmic derivative of the angular distribution, is given by
S =1
W (θ)
dW
dθ(6.1.16)
The angular distribution has the usual form:
W (θ) = A0 1 + a2P2(cosθ) + a4P4(cosθ) (6.1.17)
One explicit form for evaluation of Eq. 6.1.16 is [236]
S =−8 sin(2θ)[12a2 + 5a4 + 35a4 cos(2θ)]
64 + a2[16 + 48 cos(2θ)] + a4[9 + 20 cos(2θ) + 35 cos(4θ)](6.1.18)
148
Therefore, if the angular distribution coefficients of the decaying transition
are known, one can calculate the precession angle (and thus the g-factor is the
lifetime of the state in known) directly from equation 6.1.14.
6.1.3 Transient Field Measurements.
For nuclear lifetimes of excited nuclear states of the order or 10−11→12 s, the
product of the lifetime and static field strength is not usually enough to cause
a measureable precession. In order to obtain a measureable shift in the angular
correlation/distribution, the nucleus has to be caused to experience a much larger
‘transient’ field caused by its movement through a ferromagentic medium. The
basis of the transient field technique [235, 241, 242] is shown schematically in
figure 6.3b. The nucleus is formed in a thin production target and then passes
through a thin ferromagnetic layer (usually iron or gadolinium), where it experi-
ences a strong transient field of the order of 1000 T. A non-magnetic stopper is
then placed behind the ferromagnetic layer to stop the recoil, where it emits the
gamma-rays from which a shift in the angular distribution/correlation function
can be deduced.
One problem with this technique is that it is rather hard to correct for the
shifts in the angular correlations caused by interactions with states above the state
of interest. One method proposed to remove this problem is to use a ‘plunger’
set-up, as used to measure lifetimes in the recoil distance method [241].
The size of the transient field experienced by the nucleus as it passes through
the polarised ferro-magnetic layer can be hard to measure and where possible, a
calibration of the field strength is performed using the measurement of a state
with a known g-factor. The size of the transient field depends on the ratio of the
recoil velocity as it passes through the ferromagnetic layer and velocity of the 1s
orbital electons in the medium. An estimate of the transient field as a function
of the recoil of proton number Z and recoil velocity, v, can be given by [242, 243],
BTR(v) = αZ(
v
v0
)
exp(
−β vv0
)
(6.1.19)
where v0 = c137
is the Bohr velocity and α and β are constants obtained fits
to experimentally determined transient fields [244].
149
Magnetic Moment Measurements Using A Plunger and the TransientField Technique.
A problem with deducing magnetic moments from the transient field technique
is correcting for the effect of the precessions of states which feed the state of
interest. This problem can be solved using a plunger [241]. The basic idea is
that the recoils come out of the thin production target with velocity v and are
stopped in a backing target placed a distance d behind the production target.
Those gamma-rays which are emitted while the nucleus is in flight between the
production target and the stopper, are, as observed in a fixed angle detector,
shifted from their true energies by the Doppler effect.
Gamma-gamma coincidence data is taken and a gate is set on the shifted
component of transition above the state of interest. This ensures that the nucleus
was moving when this state decayed and thus had not yet entered the stopper
(and thus had not yet experienced the transient field). The angular correlation
of the stopped component of the transition of interest is then measured and a
precession frequency extracted, from which a g-factor can be deduced.
6.1.4 Time Differential Perturbed Angular Distributions.
For nuclear state lifetimes of between 1 ns and 1 µs, it is possible to measure
the Larmor frequency directly using an in-beam method. The basis of the Time
Differential Perturbed Angular Distribution (also known as the ‘spin rotation’)
technique [235, 234, 247, 246, 248], is to use a pulsed beam to irradiate the target
and form the isomeric state, which is implanted into a ferromagnetic medium,
where it is subjected to an applied static magnetic field, Bst whose value is known.
The decay of the transition(s) out of the state are measured as a function of time
delay after the target was irradiated. The time spectra for such experiments show
the typical exponential decay associated with a nuclear lifetime, but a smooth
oscilation, with the periodicity of the Larmor frequency, is superimposed upon
this decay spectrum. This effect is caused by the nucleus precessing around in
the presence of the magnetic field, which causes the angular distribution (and
thus measured intensity at a fixed detector angle) to oscillate with time. The ob-
served intensity of an observed γ-ray, measured at angle θ to the beam direction,
decaying from a state of lifetime, τ at time t is then given by [235]
150
I(θ, t, B) = Ioexp(
− t
τ
)
W (θ, t, B) (6.1.20)
where W (θ, t, B) is the angular distribution function for the state given in
equation 6.1.8.
A good example of the use of the TDPAD technique to measure the g-factors
of high-spin isomeric states is shown in figure 6.4. Note that the Larmor frequency
can be shown up to a larger extent by measuring the gamma-ray intensity as a
function of time using two detectors at symmetric angles ±θ to the beam direction
and plotting the intensity ratio, R, defined by R = I(+θ)−I(−θ)I(+θ)+I(−θ)
151
214Fr
R(t)
time (ns)
472 keV ratio
0 100 200 300 400 500 600
-0.4
-0.2
0.0
0.2
0.4
214Fr
counts
472 keV (135 deg)
310
410
510
472 keV (-135 deg)
310
410
510
Figure 6.4: Time differential perturbed angular distribution spectra showing theoscillation in the time spectra associated with the Larmor frequency [246].
152
Chapter 7
Spectroscopy of Neutron RichNuclei.
The structure of very neutron deficient nuclei extends to nuclei at and in some
cases even beyond the proton drip line. Using stable beam/target combinations,
it is possible to use fusion-evaporation reactions to form very neutron deficient
compound systems. Unfortunately, this very fact, means that the study of very
neutron-rich systems at high spins using this method is not possible. However,
there are a number of in-beam techniques for the study of nuclei with a neutron
excess which will be discussed in this chapter.
7.1 Using Fusion Evaporation Reactions
It is possible to study a few specific cases of β-stable and a few nucleons to the
neutron rich side of the valley of stability, using fusion-evaporation reactions.
The basic idea is to form as neutron-rich a compound system as possible and use
a charged particle detector to detect any evaporated charged particles such as
protons and α-particles [249, 22, 25]. Such events are inhibited by the Coulomb
barrier, however, the clean selection allowed by modern day charged particle de-
tectors means that very clean charged particle gated, γ-ray identification spectra
can be obtained.
Figures 7.1 highlights the effectiveness of this technique for identifying the
yrast states of the neutron rich nucleus 63Co [25] (note that the β-stable neutron
rich isotope is 59Co). The gamma-ray transitions associated with cobalt isotopes
are identified by having the gating condition that an evaporated proton must be
153
(a)
(b)
Figure 7.1: (a) Singles identification spectra and (b) proton gated γ −γ coincidence data on the neutron rich nucleus 63
27Co33 populated via the18O(48Ca,p2n)63Co reaction [25].
measured. The isotope is then identified by means of either, the average proton
energy, an excitation function or a measurement of neutron multiplicity.
The same method can also be used to identify high-spin states in the most
neutron rich stable nuclei [22, 251]. These are often nuclei where many of the non-
yrast structures are known from neutron capture and pick-up/transfer reaction
studies. The addition of knowledge of high-spin states allows a full spectroscopic
study of such nuclei.
7.2 Incomplete Fusion/Massive Transfer Reac-
tions.
Another way of studying the near yrast states of slightly-neutron to medium
spins is to use a light projectile which partially fuses with a heavy target nucleus,
154
close to the nucleus of interest [80]. This method, known as massive transfer
or incomplete fusion has been used to to study quasi-particle alilgnments in the
A∼100 region [250] and more recently to study high-K structures in stable A∼180
nuclei [81].
0
5
10
15
0
4
8
0 200 400 600 8000
1
2
Eγ(keV)
R55
(F/M
B)
R60
(F/M
B)
RF(
60/5
5)(a)
(b)
(c)
179Hf
178Hf
177Hf
178Hf
177Hf
177Hf
178Hf
Figure 2.
Figure 7.2: Angular distributions of α-particles observed in the reaction176Yb+9Be. Note that the incomplete fusion reactions to populate higher spinsin 178Hf have a more forward focussed distribution compared to the pure fusionevaporation products [81].
While charged particles emitted in fusion-evaporation reactions have an isotropic
distribution in the centre of mass frame, the break up products from incomplete
fusion are more forward focussed. In general, the higher the spin state popu-
lated in an incomplete fusion reaction, the more peripheral the collision and the
more foward focussed the outgoing particle distribution. Figure 7.2 shows the
α-particle anisotropies for various channels observed in the 176Yb+94Be5 reaction
[81]. Note that the higher the gamma-ray energy corresponding in general to the
decay of higher spin states in the rotational bands of 178Hf), the more forward
focussed the α-particle anisotropy. Note that the lower spin states are probably a
combination of fusion-evaporation and massive transfer, so the anisotropy effect
155
is more pronounced for higher spins. Figure 7.2 shows α-particle gated γ-ray
spectra obtained for the reaction 176Yb+94Be5, which is proposed to the band
built on top of the τ=31 year, Kπ = 16+ isomer in 178Hf, identified as high-spin
states by the α-particle anisotropy.
7.3 Deep Inelastic Reactions.
For neutron-rich nuclei, it is difficult to observe the yrast sequence to high spins
(such as through the backbend) due to the preferential population of neutron-
deficient species in fusion-evaporation reactions and the low angular momentum
involved in the fission process. The use of deep-inelastic reactions to populate
near yrast states in slightly neutron-rich nuclei is now well established [252, 254,
255, 256, 257, 258, 259, 96, 97, 98, 264, 265, 266, 267, 268, 269, 270, 271, 272] and
provides an efficient way of studying the yrast states of stable and slightly neutron
rich nuclei. Figure 7.3 shows schematically the use of deep inelastic reactions to
populate neutron rich nuclei.
136
Xe
Neutronsevaporated
232
Th
Newbeam like
nuclei
Newtargetlike
nuclei
Target
Beam
N:Z equilibration
Nucleons exchangedEb ~ 10-15%above Ec
Figure 7.3: Schematic of the use of deep inelastic collisions to populate yraststates of neutron rich nuclei.
It has been shown experimentally that the binary system equilibrates into
systems with approximately equal N:Z ratios [260, 261, 262, 263]. Thus, the
extra neutron excess of heavy, stable targets, means that bombarding these with
lighter beams usually results in an overall flow of neutrons onto the lighter, beam
like fragments. Since these are neutron rich nuclei, charged particle evaporation
is strongly hindered by the Coulomb barrier and thus the total Z of the compound
system (=Zbeam+Ztarget) is usually conserved in the break up. Typically, between
2 and 6 neutrons will be evaporated from the two hot binary fragments and thus
a specific nucleus will be accompanied by a number of binary partner nuclei,
comprising of between 2 and 4 isotopes of the same element. If transitions in
156
these binary partners are known, the relative intensity of the various binary
partner products can be used to identify the isotope of other, possibly unknown
neutron rich fragment.
A major problem with using deep inelastic reactions to study beam like prod-
ucts is their large recoil velocity. One approach to solve this problem is to use a
thick or backed target which stops the beam-like product within a few ps. Thus,
transitions from decays with apparent lifetimes greater than this stopping time
have no Doppler shift and can be clearly resolved. Figure 7.4 shows the region
of the Segre chart populated with the binary reaction 86Pd+110Pd [259].
66
64
Cd
Pd
Ru
Zr
Ag
Rh
Tc
Y
Kr
Br
Rb
Sr
Nb
Mo
50 52 54 56 58 60 6248
Se
= STABLE NUCLEUS
100
104Ru
108Pd
Mo
86Kr
110Pd
= BEAM / TARGET NUCLEI
Figure 7.4: Region of the Segre chart poplated by binary collisions using thereaction 110Pd+86Kr [259].
Figure 7.5: Spectra from the 86Kr+110Pd binary reaction. Comparison of align-ments between odd and even-N Mo, Pd and Ru nuclei populated in this reaction.
Figure 7.5 shows the spectra for 104Ru studied using the binary reaction86Pd+110Pd [259]. The 104Ru lines of interest can be clearly selected, as can
157
the presence of the Sr binary partner lines. The data in figure 7.5 extends the
yrast band of 104Ru upto spin 14+. As figure 7.5 shows, thus is high enough to
obtain useful insights into the alignment processes in these nuclei.
In heavier deformed nuclei, relatively high spins can be obtained using binary
reactions with thick/backed targets and a high efficiency gamma-ray arrays [98,
271, 272]. Figure 7.6 shows the high spin cascade in 234Th and 230Th observed
using GAMMASPHERE obtained from the binary reaction 136Xe+232Th [97].
Note that the higher multiplicity gamma-ray gating afforded by using the higher
efficiency arrays gives rise to extremely clean spectra, with only the nucleus of
interest (and possibly a few binary partners) observed,
0
1
2
3
x 10
4 C
ount
s
50 150 250 350 450 550 650Energy (keV)
0
1
2
3
x 10
4 C
ount
s
182.
5T
h
Th
Th
237.
4
285.
7
327.
9
365.
339
7.9
426.
845
2.3
Xe
483
.4
Th
493.
747
6.3
427.
639
6.8
361.
8322.
2
277.
8
228.
2
173.
4
Th
X-r
ays
X-r
ays
120.
8T
h
(a)
(b)
454.
1
Xe
589
.0
Figure 7.6: Gamma-ray spectra produced for the binary reaction 136Xe+232That beam energy of 833 MeV. (a) double gated spectrum showing ground state-band transitions in 234Th. (b) spectrum showing ground state-band transitionsin 230Th. Transitions labelled ‘Xe’ are cross-coincident gamma-rays attributedto 138Xe, the projectile-like partner of 230Th. Gamma-ray peaks marked ‘Th’ inboth spectra are contaminant transitions in the yrast band in Coulomb-excited232Th [96, 97].
158
7.3.1 Maximum Angular Momentum in DIC.
Spin-wise, the use of deep inelastic reactions are not particularly efficient in gen-
erating very high spin states. (Recent studies of (discrete-line) spin-input us-
ing thin target deep-inelastic reactions can be found in reference [268]). Unlike
fusion-evaporation reactions, not all of the input angular momentum of the reac-
tion goes into the intrinsic angular momentum of the final products. Most of the
angular momentum goes into the relative spin of the two fragments with respect
to each other. It can be shown by a semi-classical argument that in the pres-
ence of strong friction between two rolling spheres (ie. the target and beam like
nuclei), known as the rolling limit, 57
of the maximum initial maximum angular
momentum (for a grazing collision), Lmax goes into the relative motion of the two
fragments while only 27
goes into the intrinsic spins of the beam and target like
fragments respectively [253].
The maximum input spin into the beam and target like fragments of masses
AB and AT respectively can be estimated using the following semi-classical ex-
pressions [253, 254, 280]. Figure 7.7 shows how increasing the input beam energy,
increases the angular momentum input of the target and beam nuclei. Note also,
that these fold distributions are ‘two-peaks’ corresponding to low-fold reactions
such as Coulex and higher fold reactions such as deep-inelastic collisions. This
is highlighted further in figure 7.8 which shows the fold distribution for states
gated on the yrast band in 100Mo for increasing spin. The lower spin states are
populated mostly by direct Coulomb excitation, while the higher spin states are
populated via transfer followed by neutron emission, and have a considerably
higher entry spin.
LBLF =2
7
1
1 +(
AT
AB
) 1
3
Lmax (7.3.1)
LTLF =2
7
1
1 +(
AB
AT
) 1
3
Lmax (7.3.2)
159
7.3.2 Useful Formulae for Binary Reaction Studies.
The following section is a summary of useful formula for studies of transfer/binary
style reactions and was compiled by Prof. P.A. Butler of Liverpool University.
Elastic scattering
Projectile mass M1 on target M2 with lab. energy E0.
M1 scatters at lab angle ψ, c.m. angle θ,
M2 scatters at lab angle ζ , c.m. angle φ = π − θ = 2ζ
Lab. energy of scattered nucleus M1 is
E1
E0
= 1 − 2M1M2
(M1 +M2)2(1 − cosθ) = (
M1
M1 +M2
)2cosψ ± [(M2
M1
)2 − sin2ψ]1
22
Use only plus sign unless M1 > M2, in which case ψmax = sin−1(M2
M1
)
Lab. energy of recoil nucleus M2 is
E2
E0
= 1 − E1
E0
=4M1M2
(M1 +M2)2cos2ζ
where ζ ≤ π2
Angles
sinζ = (M1E1
M2E2
)1
2 sinψ
tanψ =sin2ζ
M1
M2
− cos2ζ=
sinθM1
M2
+ cosθ
θ = ψ + sin−1(M1
M2)sinψ = π − 2ζ
cosθ = 1 − 2cos2ζ
ψ = π/2 − ζ if M1 = M2
160
Rutherford Scattering Cross Sections (in mb/sr).
dσ
dΩcm= 1.3(
Z1Z2
Elab
M1 +M2
M2)2 1
sin4 θcm
2
(Elab is in MeV)
dσ
dΩscat,lab=
dσ
dΩcm
sin2θ
sin2ψcos(θ − ψ)
dσ
dΩrecoil,lab=
dσ
dΩcm4sin
θ
2
Inelastic scattering.
Same notation as for elastic scattering except that lab energies of scattered beam
and recoil are respectively E3 and E4. Define the following terms:
Q = (M1 +M2 −M3 −M4)c2
ET = E0 +Q = E3 + E4
A =M1M4
(M1 +M2)(M3 +M4)
E0
ET
B =M1M3
(M1 +M2)(M3 +M4)
E0
ET
C =M2M3
(M1 +M2)(M3 +M4)(1 +
M1
M2
Q
ET) =
E4′ET
D =M2M4
(M1 +M2)(M3 +M4)(1 +
M1
M2
Q
ET) =
E3′ET
The total scattered energy in the c.m. is
E′ = E3′ + E4′ =M2
M1 +M2(ET +
M1
M2Q) = Ecm +Q
161
Lab energy of light product
E3
ET
= B +D + 2(AC)1
2 cosθ
Lab energy of heavy product
E4
ET= A + C + 2(AC)
1
2 cosφ
Angles:
sinζ = (M3E3
M4E4)
1
2 sinψ
sinθ = (E3/ET
D)
1
2 sinψ
Deep Inelastic or Fission
with initial kinetic energy, put M2 = 0 and
Q ≈ 0.107Z2
A1
3
+ 22.2
(A is in u and Q in MeV) so that
E4 =M4E3
M3 +M4[cosζ ±
√
M3Q
M4E3− sin2ζ ]2
Use only + sign unless M3QM4E3
< 1, when
ζmax = sin−1(M3Q
M4E3
)1
2
Compound Nucleus
Projectile M1 with lab energy E0, target M2 as above.
Lab energy of compound nucleus
ECN =M1
M1 +M2
E0
Excitation energy of compound nucleus
Ex = (E0 − ECN) +Q =M2
M1 +M2E0 +Q = Ecm +Q
162
Coulomb Barrier
Interaction radius is
R = 1.16(A1
3
1 + A1
3
2 + 2)
where A is in u and R is in fm, then
Vcm = 1.44Z1Z2
R
Vlab = (1 +M1
M2
)Vcm , Elab = (1 +M1
M2
)Ecm
Maximum Angular Momentum.
ℓmax = 0.219R[µ(Ecm − Vcm)]1
2
where
µ =M1M2
M1 +M2
Grazing Angle.
The grazing angle [274], where the cross-section for peripheral, deep-inelastic
reactions will be maximised, is defined as the angle at which the distance of
closest approach, d, given by [5, 273],
d = [Z1Z2e
2
4πǫ0Ek].[1 + cosec(
θ
2)] (7.3.3)
(where Z1 and Z2 are the atomic numbers of the two nuclei involved and E
is their kinetic energy) is equal to the sum of the nuclear radii (ie. when the two
nuclei are just touching), given by,
d = 1.25(A1
3
1 + A1
3
2 )fm (7.3.4)
where A1 and A2 are the nuclear mass numbers.
A quick estimate of the grazing angle can be obtained using,
1
2(1 + cosec
θc.m.
2) =
E
V
163
Reaction Cross Section (in barns).
σR ≈ 0.67
µEcmℓ2max
Velocity in % c.
v = 4.634
√
E
A
(E in MeV, A in u)
For transfer reactions the two nucleon transfers are strongly dependent on the
Q-value,
Q = Qgg −Q(∆Z)
where,
Q(∆Z) =[(
Z3Z4
Z1Z2
)
− 1]
Vcm
Z1 and Z2 correspond to the projectile and target before the reaction. Z3 and
Z4 correspond to the projectile-like and target-like fragments after the reaction.
The more positive the Q-value the higher the population. Note that Q(∆Z)=0
for only neutron transfers. Qgg is the mass difference between initial and final
products.
7.3.3 Doppler Correction.
By stopping the beam like products in the target, transitions with lifetimes less
than a few ps are Doppler shifted and thus can not be resolved. In order to observe
the highest spins of neutron rich products populated in deep inelastic collisions,
a thin target is required, which allows the beam-like products to decay in flight.
In this case, a method is required to obtain the recoil velocity and emission angle
of the beam like fragment so an appropriate Doppler correction can be applied
to reduce the overall Doppler broadening of the gamma-ray transitions [?].
The HIPS detector, shows in figure 7.9 gives a position sensitive signal which
gives the recoil direction and uses also uses the time of flight to give a degree
of mass resolution for beam-like products in binary reactions. Figure 7.10 shows
the improvement in the gamma-ray energy resolution of beam-like products in
164
the binary reaction using a 86Kr beam on a 116Sn target by detecting the beam
like fragment in the HIPS detector and correcting for the Doppler effect.
Similarly, a recoil filter detector [47, 163] can be used to obtain similar infor-
mation and Doppler corrections.
The CHICO detector [276] is a large solid angle set of PPACS which uses
time of flight to get some mass infomation on the fragments and allows excellent
Doppler correct when used with the GAMMASPHERE array. In addition to in-
beam experiments, it has been used to extend the decays schemes of a number
of neutron-rich fission products around A ∼100 upto discrete spins of 20h using
a thin, fission source [277, 278, 279]
Determination of Angular Corrections in DIC.
If the recoiling beam (or target like fragment) is detected at angles θ1 and φ1 to
the beam axis, and the (doppler shifted) gamma-ray emitted from the recoiling
fragment is detected in a germanium detector and angles θ2, φ2 to the same axis,
in order for the Doppler correction to be applied, the angle between the emitted
gamma-ray and charged particle fragment directions (θdop) must be known. This
can be calculated using simple geometry assuming two unit vectors, v1 and v2,
where the (x, y, z) components of these vectors are given by x1 = sin(θ1)cos(φ1),
y1 = sin(θ1)sin(φ1), z1 = cos(θ1) and x2 = sin(θ1)cos(φ2), y2 = sin(θ2)sin(φ2),
z1 = cos(θ2) respectively. The angle between these two vectors can be calculated
using the vector dot product, ie. v1.v2 = v1v2cos(θdop). Therefore
cos(θdop) = sin(θ1)cos(φ1)sin(θ2)cos(φ2)+sin(θ1)sin(φ1)sin(θ2)sin(φ2)+cos(θ1)cos(θ2)
(7.3.5)
collecting the terms and recalling the trigonometric identity cos(A − B) =
cosAcosB + sinAsinB, the above expresssion reduces to
cos(θdop) = sin(θ1)sin(θ2)[cos(φ1 − φ2)] + cos(θ1)cos(θ2) (7.3.6)
165
Figure 7.7: Fold distributions gated on the low-lying 601 keV transition in 100Mo,following the binary reaction 100Mo +136Xe at beam energies of 650, 700 and750 MeV respectively. Note that the overall spin of the fragments increasesdramatically with beam energy [281]. Data taken from the 8π array at Berkeley.
166
Figure 7.8: Fold distributions gated on the low-lying yrast transitions in 100Mo,following the binary reaction 100Mo +136Xe at a beam energy of 700 MeV. Notethat low-lying states are predominantly populated vua Coulomb excitation, whilethe higher ones have a higher overall multiplcity distribution, associated with adeep-inelastic collision, probably followed by neutron emission [281]. Data takenfrom the 8π array at Berkeley.
167
ygri
d
cato
de
dela
y-ch
ip
xgri
d
10MΩ
10MΩ
10MΩ
10MΩ
10MΩ
10MΩ
xgri
d
ygri
d
cato
de
dela
y-ch
ip
housing
Stop
Detector
Ionisation
Chamber
Foils &Support Grids
Field Rings
FrischGrid
Anodes
10
Gate-valve
Anode HT
Signal
Signal
Signal
Signal
Signal
Grid HT
0 10 20 30 40 50cm
Start-Detector
S.Schwebel et.al.Manchester University
Figure 7.9: Scale drawing of the HIPS vessel for detecting recoils from deep-inelastic reactions [275].
168
0 500 1000 1500 2000 2500 3000 3500Channel Number
0
50
100Doppler corrected spectrum
0
50
100
150
HIPS-Ge coincidence γ-ray spectra86
Kr + 116
Sn
Raw Ge spectrum
Cou
nts
per
chan
nel
Figure 7.10: HIPS-γ coincidence spectra for the binary reaction 86Kr+116Sn high-lighting the effect of Doppler correction on the detected lineshape [275].
169
Chapter 8
Spectroscopy With RadioactiveIons Beams.
The restriction of using beams of ions which are stable against radioactive decay
places constraints on those nuclei which can be produced in the laboratory for
study. Figure 8.1 shows the Segre chart and shows the predicted proton and
neutron drip lines. There around predicted to be around 7,000 nuclei which lie
within these driplines, of which less than 300 are stable against radioative decay.
N=Z
0 20 40 60 80 100 120 140
Neutron number N
0
20
40
60
80
100
Ato
mic
num
ber
Z
Figure 8.1: Segre chart showing the stable isotopes predicted proton and neutrondriplines [282].
Figure 8.2 shows those compound nuclei which can be formed using stable
170
beam target combinations. Note, virtually all of these systems are neutron defi-
cient with respect to the stable isotopes.
Proton Drip Line
Neutron Drip Line
0 20 40 60 80 100 120 140
Neutron number N
0
20
40
60
80
100
Ato
mic
num
ber
Z
Figure 8.2: Segre chart showing the compound nuclei which can be formed infusion-evaporation reactions using stable beam/target combinations [282].
In order to study the nuclear proporties of the entire nuclear chart, the ability
to induce nuclear reactions with radioactively unstable nuclear beams is impor-
tant. This chapter will look at methods of producing beams of radioactive ions
of the desired energy and intensity to be useful in nuclear structure studies.
8.1 Production of Radioactive Beams.
There are two main methods of producing radioactive nuclear beams [283], pro-
jectile fragmentation [284] and isotope separation on-line or ISOL [285, 286]. As
table 8.1 shows, both techniques have advantages and disadvantages depending
of the energy regime and beam intensity requirements of the experiment.
8.1.1 Projectile Fragmentation.
In a projectile fragmentation reaction, a high energy, heavy ion beam bomards a
target at energies of 30 MeV/A and above [294]. The result of a nuclear collision
at these energies is often that the beam loses a number of protons and neutrons
171
Method Advantages DisadvantagesProjectile Fast delivery times (∼ µs) low beam intensityFragmentation No chemical contraints Final beam deceleration difficult
Reliable operation Limited target thicknessHigh collection efficiency Large energy spreadSimple target design Moderate isobaric purity
ISOL Thick target Decay losses due to slow releaseHigh beam purity Needs post accelerationLow energy spread Radiation contamination in target‘Useful’ beam energies Complex target design
Yield depends on beam/target chem.
Table 8.1: Comparison of the ISOL and projectile fragmentation methods ofradioactive beam production.
but a beam like fragment carries on with a velocity similar to the iniital beam.
In this way, many radioactive species can be created using a single beam/target
combination. The beam-like fragment is then separated from the primary beam
using a set of electro-magnetic focussing and steering devices provided by for
example the LISE3 spectrometer [293] at the GANIL facility, France.
Figure 8.3 shows the identification spectrum (time of flight verses energy
loss) for the fragmentation of a 92Mo beam on a nickel target at an energy of
60 MeV per nucleon using the LISE3 spectrometer at GANIL. Note, the particle
identification afforded in fragmentation reactions using the time of flight (AQ
)
and energy loss (Z) technique means that each nuclear species produced can be
individually identified.
This technique of radioactive beam production has the advantages of high
beam purities and high collection efficiency (all the products are very forward
focussed in the lab frame due to the large beam velocity), but has disadvantages
for use in providing beams for fusion-evaporation reactions in that (a) the beam
intensities are generally smaller than are useful for evaporation reactions and
(b) a set of degraders have to be introduced to slow down the beam to energies
around the Coulomb barrier ∼ 4 − 5 MeV/A causes a spread in beam velocity.
However, this method is useful in experiments (as discussed below) where the
radioactive beam itself is the nucleus of interest.
172
Zr80
66As
__qA = 2
T Z
/23/ 15/2 012 2
B = 1.9068ρ Tm
Mo 42Zr 40Sr 38
states of beamdifferent charge
"beam"
Z
Kr 36
Figure 8.3: Particle identification spectrum from the fragmentation of 92Mo atGANIL with no degrader or Wien filter selection [287].
8.1.2 Particle Identification in Fragmentation.
One of the initial benefits of projectile fragmentation is that it enables the identi-
fication of nuclei on an event-by-event basis, and also can be used to show where
the drip-lines occur, by the absence of any nuclei in specific places on the particle
identification plot. For example, the nuclei 81Nb and 85Tc are thought not to be
bound against direct proton emission due to their absence in the fragmentation
of 92Mo [321].
The parameters used to identify the beam-like fragments are: (a) the mag-
netic rigidity of the dipoles which select the ions, Bρ; (b) the time-of-flight,
TOF,assuming that the distance, L, between the production target and the fo-
cal point, where the ions are implanted, is constant for every fragment; (c) the
173
energy-loss, ∆E and (d) the total kinetic energy, K, of the beam-like fragments.
From these quantities, the calculation of the fragment atomic mass number, A,
atomic number, Z, and charge, Q, is possible. Charged particles are deviated
by a magnetic field, B. This deviation is characterized by the bending radius,
ρ. This depends on the linear momentum, p, and the charge, Q = qe. From the
Lorentz law, one can derive that:
Bρ =p
qe(8.1.1)
Substituting for the linear momentum, this equation becomes
Bρ =Mβγc
qe, (8.1.2)
where M is the mass of the fragment, c is the speed of light in vacuum and
γ2 = 11−β2 with β = v
c, where v is the fragment velocity. If the mass is expressed
in atomic mass units (a.m.u.), M = Au where u is the atomic mass unit, and Bρ
is expressed in Tesla-meters (Tm), then:
Bρ = 3.105 × Aβγ
q. (8.1.3)
The measurement of the total kinetic energy of an implanted fragment, K, is
expressed in relativistic mechanics as:
K = Mc2(γ − 1). (8.1.4)
Therefore, it is possible to directly obtain the atomic mass number, A, for the
implanted fragment:
A =K
931.5 × (γ − 1). (8.1.5)
where K is expressed in MeV and A is in atomic mass units. Similarly, the
charge state, q is deduced from equations (8.1.3) and (8.1.5):
q =3.105 × βγK
931.5 × (γ − 1)Bρ. (8.1.6)
The total kinetic energy for each fragment, K, can be measured by adding
the energy deposited in the silicon ∆E and stopping detectors. The time-of-flight
from the production target to the final focus can be measured by taking the time
174
difference between a fast signal (rise time of a few nanoseconds) extracted from
the ∆E detector and the cyclotron radiofrequency (RF = 11 MHz). Calibrations
of the energy-loss in the silicon detectors and of the time-of-flight are performed by
simulating the velocity of the fragments at the LISE3 focal point and analyzing
the energy deposited by each fragment with a code based on Bethe’s formula
[320]. The atomic number, Z, is calculated using the energy-loss in a ∆E-silicon
detector. Relativistic corrections to Bethe’s formula lead to the expression:
∆E = a1Z2
β2[ln (a2β
2γ2) + a3β2 + a4] + a5 (8.1.7)
where the an are constants that can be obtained through calibration. The
atomic number, Z, can be expressed after integration of equation ([320]) as:
Z = c1
√
∆E
Y+ c2 + c3
∆E
Y+ c4β (8.1.8)
where:
Y =ln (5930 × β2γ2)
β2− 1 (8.1.9)
and c1, c2, c3 and c4 are constants which can be fitted.
The complete reconstruction of A, q and Z is therefore possible from the
measurement of the time-of-flight, the energy loss, and the total kinetic energy.
Figure 8.4 shows a calibrated particle identification spectrum following the frag-
mentation of a 60 MeV per nucleon 92Mo beam at GANIL [320].
Gamma-Ray Spectroscopy Using Projectile Fragmentation.
The products from projectile fragmentation reactions may populate long lived
excited states. If the lifetime of these states is long compared to the flight path
of the separator, their decay can be measured at the end of the separator by an
array of germanium detectors [287, 290].
Figure 8.5 shows a more selective region of the identification data from fig-
ure 8.3, with and without the condition that a delayed gamma-ray must be mea-
sured with 100 µ seconds of he fragment stopping in a silicon telescope detector.
Those fragments which are transmitted in an isomeric excited state will decay in
the silicon telescope and their gamma-rays detected in the surrounding germa-
nium detector array. By gating on the nuclear species of interest in the particle
175
1 2 4 8 16 32 64 128 256 512 1024 2048 4096
1.950 1.975 2.000 2.025 2.050 2.075 2.100 2.125
A/q
26
28
30
32
34
36
38
40
42
44
46
48
Z
N=Z
Nb
Tc
Y
Rb
Figure 8.4: Calibrated particle identfication spectra from the fragmemtation of92Mo [320].
identification spectra one can project the gamma-ray energy and time spectrum
for the isomeric decay.
176
(a) all (b) gammas
Kr (36)Rb (37)
Ge (32)As (33)
Se (34)Br (35)
Tz =3/2 1 1/2
time of flight (A/Q)
ener
gy lo
ss
Kr-73
Se-69
‘GRZYWACZ PLOT’ TO ID ISOMERS
92-Mo @ 60 MeV/A on Ni target at GANIL
Figure 8.5: Particle identification spectrum from the fragmentation of 92Mo atGANIL. Note the extra condition of observing a delayed gamma-ray highlightsthose nuclei in which isomeric decays have been observed.
177
Figure 8.6 shows a two-dimensional plot of time verses gamma-ray energy
taken for the fragmentation of a 92Mo beam from GANIL [291]. Note the lines
extending out corresponding to gamma-rays from isomeric decays. Note also,
the presence of the line corresponding to ‘prompt’ radiation, which serves as a
useful ‘time-zero’ calibration. The spectrum on the right of figure 8.6 has been
corrected for low-energy time walk [291].
0 500
Time Difference (ns)
0
500
1000
1500
2000
2500
3000
3500
4000
Gam
ma–
ray
ener
gy (
keV
)
γ
0 500
Time Difference (ns)
0
500
1000
1500
2000
2500
3000
3500
4000
Gam
ma–
ray
ener
gy (
keV
)
E = 734 keV
a) b)
Figure 8.6: Two-dimensional gamma-ray energy verses time spectra showing thedecays of the isomeric states following the fragmentation of a 60 MeV/A beamof 92
42Mo. [291]. Note the correction for ‘time-walk’ for lower energy gamma-raysin the right hand spectrum.
Figure 8.7 shows the gamma-ray and time spectra associated with the decay
of the 42 ns isomer in 74Kr obtained by gating on the identification spectra shown
in figure 8.5.
This is a very powerful technique in identifying isomeric decays in exotic nuclei
and can be used for both neutron deficient [287, 288, 290, 291] and neutron rich
systems [308, 313, 314, 315, 316]. Figures 8.8 and 8.9 show the gamma-ray energy
and time spectra for decays associated with isomeric states in the neutron rich
nucleus 6828Ni40, produced by the fragmentation of a 86Kr beam. Note the presence
of the 511 keV gamma-ray line, corresponding to the 1770 keV, 0+ → 0+ decay
by internal pair formation.
It has been observed [288] that the population of yrast and non-yrast iso-
178
τ = 42 ± 8 ns
150 200 250 300 350
time difference (ns)
100
101
coun
ts
Figure 8.7: Gamma-ray energy and associated time spectra showing the decay ofthe isomeric state in 74
36Kr from the fragmentation of a 60 MeV/A beam of 9242Mo.
[296, 291].
meric states in intermediate energy projectile fragmentation reactions varies sig-
nificantly, with yrast states being favoured. Values of the isomeric ratio for nuclei
produced using fragmentation reactions have been found to range dramatically
from case to case [288]. Indeed, the production of nuclei in their isomeric state
has been found to be dependent on the reaction mechanism and the velocity of
the fragment compared to that of the beam [297].
Note, the effective lifetimes of the nuclei through a separator are also extended
due to relativistic effects, such that through the separator, the effective lifetime
is
τrmeff = τo1√
1 − β2(8.1.10)
where β is vc.
Fragmentation can also be used at relativistic energies corresponding to pri-
mary beams of hundred of MeV per nucleon [309, 310, 308, 312]. This is particu-
larly useful for the identification of heavy, exotic fragments [308, 312, 316]. The
high beam velocity, as provided using the SIS accelerator at GSI, allows both a
larger production of the most exotic nuclei (since thicker targets can be used).
179
T1/2=290 ns (exp)
T (us)
68m2Ni
Figure 8.8: Time spectrum gated on 511 keV decays in 68Ni from the fragmenta-tion of 86Kr at GANIL. The lifetime suggests an E0 decay proceeding by internalpair formation [295].
The FRagment Separator (FRS) at GSI can be used to separate and identify
on an event by event basis the nuclei transmitted [307] using time of flight and
energy loss measurements [307, 309, 311]. In addition, the higher primary beam
velocity means that the ions have a higher probability of being fully stripped of
electrons through the spectrometer. A knowledge of the charge state is important
in order to resolve charge state ambiguities in the time of flight signal (∼ AQ
, see
fig. 8.11.) Figure 8.11 shows how the charge states of the transitted nuclei can
be obtained by measuring their relative position through the spectrometer as a
function of their time of flight. Fully stripped ions do not change their position
through the spectrometer, whereas hydrogen and helium like one do, causing a
longer effective time of flight (see figure 8.11).
High spin spectroscopy can be achieved following isomeric decays in these
fragments. Figures 8.12 shows the gamma-ray spectrum gated on (a) 179W show-
ing the decay of the Kπ = 352
−isomer. A prescription of how to estimate the
much angular momentum transfered to nucleus in a projectile fragmentation re-
action can be found in reference [317]. Figure 8.14 shows the predicted average
spins of the fragments for a Pb on Pb collision at 1 GeV per nucleon.
180
68m2Ni
Figure 8.9: Gamma-ray gate on 68Ni from the fragmentation of 86Kr showing the511 keV lines associated with internal pair formation [295].
181
Auc
IC:time
IC:dE,Q
γ
MW:x,y
βeρB
Q
Segmented Clover
=
+
Degrader
SCI
TOF
Super Cloversmall Clover
catcher
SCI
primary beam:
Pb @ 1 GeV
production target
Degrader
dipole : Bρmiddle focus S2
end focus S4
Array
Figu
re8.10:
Sch
ematic
ofth
eset-u
pusin
gth
eG
SIFragm
ent
Sep
aratorto
inves-
tigatedecay
sfrom
isomeric
states[315].182
2.56 2.58 2.60 2.62 2.64
A/Q
–40
–20
0
SC
21x
(mm
)
fully–stripped hydrogen–like helium–like
2.56 2.58
A/Z
–60
–40
–20
0
20
40
60
80
SC
41x
(mm
)
187
Ta 188
Ta
189
W 190
W
191
Re
192
Re 193
Re
194
Os
195Os
196Os
2.60 2.62
A/(Z–1)
192
Re 193
Re
194
Os 195
Os 196
Os
197
Ir 198
Ir 199
Ir
202
Pt 200
Pt 201
Pt
203
Au
2.64
A/(Z–2)
200
Pt 201
Pt
203
Au 204
Au
205
Hg 206
Hg
209
Tl
Figure 8.11: Fragment identification spectra following the fragmentation of a1 GeV/u 208Pb, highlighting the different charge states of the ions for an FRSsetting centred on 191W ions [316].
183
Figure 8.15 shows background subtracted gamma-ray spectra associated with
the decay of isomers populated in a variety of neutron-rich nuclei around A ∼190,
for different charge states. Note in particular, the 15 ns isomer in 200Pt, which
is observed due to the effective lengthening of the decay half-life through the
spectrometer by the ‘switching off’ of the electron-conversion component of the
decay [291, 316].
Figure 8.12: Gamma-ray spectra associated with the fragmentation of a 1 GeV/u208Pb, for fully stripped ions around 177Ta [313].
Elemental identification in relativistic fragmentation is achieved by a com-
bination of energy loss (usually measured in gas detector, known as a MUSIC
chamber, see fig 8.16) or by position on a variety of scintilation detectors through
the spectrometer.
8.1.3 Isomeric Ratios and Angular Momentum Popula-
tion.
The isomeric ratio, R, is defined as the probability that in the reaction a nucleus
is produced in an isomeric state [322, 323, 324].
These can be determined in the following way.
184
Firstly, the observed decay yield is calculated :
Y =Nγ (1 + αtot)
ǫeff bγ, (8.1.11)
where Nγ is the number of counts in the gamma line depopulating the isomer
of interest, αtot is the total conversion coefficient for this transition, ǫeff is the
effective efficiency and bγ is the probability that the decay proceeds through this
transition (i.e. the absolute branching ratio). The isomeric ratio is then given by
:
R =Y
Nimp F G, (8.1.12)
where Nimp is the number of implanted heavy ions, F is a correction factor for
the in-flight isomer decay losses and the factor G corrects for the finite detection
time of gamma radiation. The factor F is calculated from :
F = exp
[
−(
λq1TOF1
γ1+ λq2
TOF2
γ2
)]
, (8.1.13)
where TOF1 (TOF2) is the time of flight through the first (second) stage of
the spectrometer (eg. the FRS), γ1 (γ2) is the corresponding Lorentz factor and
λq1 (λq2) is the decay constant for the ion in the charge state q1 (q2). In most of
the experiment using relativistic ions, the in-flight ions were highly charged (in
most of the cases studied, the ions are fully stripped), and the decay constants
λq can differ considerably from the value for an electrically neutral atom, λ.
For the fully stripped ion, λ0 can be calculated from :
λ0 = λ∑
i
bγi
1 + αitot
, (8.1.14)
where the summation is over all the decay branches depopulating the isomer.
Finally, the correction factor G is calculated using:
G = exp(−λ ti) − exp(−λ tf), (8.1.15)
where ti and tf are the gamma delay-time limits set in the off-line analysis to
produce the delayed gamma spectrum.
When more than one gamma-ray line was observed to depopulate an isomer,
the isomeric ratio was calculated separately for the strongest lines and then av-
eraged.
185
In some cases, more than one isomer in the same nucleus is populated in the
reaction and a lower lying isomer may be partly fed by the delayed decay of a
higher lying metastable state. We adopt here the definition of isomeric ratio as
the probability that a state is populated promptly after production of the nucleus
in the reaction. Then, it can be shown that in a case where the upper state decays
with the probability (branching) bUL to the lower one, the isomeric ratio for the
latter can be calculated by :
RL =YL
Nimp FLGL
− bULRU
FLGL
[
λLGU − λUGL
λL − λU
FU +λ0
U
λ0L − λ0
U
GL (FU − FL)
]
,
(8.1.16)
where the indexes ’L’ and ’U ’ refer to the lower and the upper states, respec-
tively, and the second term on the right side represents the correction due to
feeding from the upper state.
The Sharp Cut-Off Model
To describe the population of an isomeric state in a fragmentation reaction we
separate the process into two steps. In the ablation phase of the reaction, highly
excited prefragments evaporate nucleons until the final fragment is formed with
an excitation energy below the particle emission threshold. Subsequently, a sta-
tistical gamma cascade proceeds down to the yrast line and then along this line
to the ground state. If a long-lived state lies on this decay path, part of the
cascade may be hindered or stopped depending on the lifetime of the isomer.
The isomeric ratio is equal to the probability that gamma decay from the initial
excited fragments proceeds via this isomeric state.
The crucial aspect of the first step is the distribution of the angular momentum
in the ensemble of the excited fragments just prior to the gamma de-excitation
step. This problem was addressed by de Jong, Ignatyuk and Schmidt [298] who
applied the statistical abrasion-ablation model [299] of fragmentation. Assum-
ing that any angular momentum taken away by evaporating particles is small
and can be neglected, they calculated the angular momentum distribution of
the final fragment as the superposition of the angular momenta of all prefrag-
ments contributing to the final fragment of interest using the ABRABLA code
[299]. Furthermore, they have shown that for a large mass difference between the
186
projectile and the fragment this distribution can be approximated by a simple
analytical formula :
PJ =2J + 1
2σ2f
exp
[
−J(J + 1)
2σ2f
]
, (8.1.17)
where σf , the so called spin-cutoff parameter of the final fragments, is given
by :
σ2f = 〈j2
z 〉(Ap − Af ) (νAp + Af )
(ν + 1)2(Ap − 1). (8.1.18)
Ap and Af denote the projectile and fragment mass numbers respectively, ν
is the mean number of evaporated nucleons per abraded mass unit and 〈j2z 〉 is the
average square of the angular-momentum projection of a nucleon in the nucleus.
It is generally assumed that the abrasion of one nucleon induces an excitation
energy of about 27 MeV [301], whereas the evaporation of a nucleon decreases
the energy by about 13 MeV, hence the parameter ν = 2 is taken. Values of 〈j2z 〉,
estimated on the basis of a semi-classical consideration of the angular-momentum
distribution in the Saxon-Woods potential [300, 298], are written as :
〈j2z 〉 = 0.16A2/3
p (1 − 2
3β), (8.1.19)
where β is the quadrupole deformation parameter.
Given the angular momentum distribution of the final fragment, one can con-
sider the probability that gamma decays will lead to an isomeric state of spin
Jm. First, we assume that the initial excitation energies are well above the exci-
tation energy of the isomer. One can make the extreme simplifying assumption
that all states with J ≥ Jm, and only those, decay to the isomer. A similar
approach, known in the literature as the ‘sharp cut-off model’, has been used
in studies of angular momentum distributions in compound nuclei [302] and in
fission fragments [303]. From Eq. 10 it follows :
Rth =∫ ∞
Jm
PJ dJ = exp
[
−Jm (Jm + 1)
2σ2f
]
. (8.1.20)
Substituting ν = 2 and introducing ∆A = Ap−Af , the above equations yield:
σ2f = 0.0178 (1 − 2
3β)A2/3
p
∆A (3Ap − ∆A)
Ap − 1. (8.1.21)
187
8.1.4 Projectile Fission
A similar method of producing high energy radioactive beams to projectile frag-
mentation is that of projectile fission. Here a heavy beam, such as 238U bombards
a light target (such as 1H). The heavier element usually fissions in such a reac-
tion, giving rise to a large number of neutron rich fragments moving forward in
the lab frame. These fragments are subsequently collected and identified using a
mass separator such as the FRS at GSI, Darmstadt, Germany [304]. Using this
technique, many new neutron rich isotopes have been identified for the first time
[318].
188
Figure 8.13: Predicted spin distribution for 179W fragments in the fragmentationof a 1 GeV per nucleon 208Pb on a 9Be target using the abrasion-ablation model[310], taken from ref [315].
8.1.5 Intermediate Energy Coulex.
Another technique which can employed to study the collective states of very ex-
otic nuclei produced in projectile fragmentation reactions is to Coulomb excite
the products on a heavy stopper (such as Pb) and measure the detected gamma-
rays. The large fragment velocity means that Doppler broadening effects can be
large and rather degrade the energy resolution. However, since the number of
lines is generally quite small, and gamma-ray detection efficiency is at a premium
due to the relatively small number of events, a higher efficiency, lower resolution
detetctor, such as NaI(Tl) is used instead of germanium. Work using this tech-
nique has can give direct measurements of both the energy of the lowest 2+ state
and the associated B(E2) for it decay to the ground state in even-even nuclei very
far from stability. Recent work using this technique on the neutron rich N=28
nuclei has suggested a break down of this ‘magic number’ for very neutron rich
species, such as 4416S [305, 306].
189
Neutron Number
Prot
on N
umbe
r
Figure 8.14: Predicted average fragments spins using the abrasion-ablation model[310] for 208Pb on 208Pb at 1 GeV per nucleon.
8.1.6 Double Fragmentation and In-beam spectroscopy
For double fragmentation at RIKEN in 34Mg see [326]. In-beam spectroscopy at
GSI see [327]. For in-beam fragmentation gamma-ray spectroscopy at GANIL,
see [328].
8.1.7 Beta-Decay Measurements.
The same technique as used for isomer decays can also be applied to measurement
of fast β-decay lifetimes [319, 320] (see figures 8.17 and 8.18).
8.2 ISOL Based Techniques.
The isotope separation on-line method uses a primary reaction to create the ra-
dioactive nuclei of interest (such as a fusion-evaporation reaction or the spallation
of a heavy nuclei by protons or beam induced fission). The radioactive nuclei must
then diffuse out of their production target where they are ionised and selected
by mass (using a dipole magnet). The ions must then be accelerated to useful
experimental energies and thus some form of post-accelerator is required.
For many years, ISOL based experiments have been performed to investigate
the ground state and beta-decaying properties of exotic nuclei, using instruments
such as the GSI On-Line Mass Separator [333, 334, 335, 336, 337, 338, 339, 340,
341, 342, 343, 344].
190
Figure 8.15: Gamma-ray spectra associated with isomers populated following thefragmentation of a 1 GeV/u 208Pb beam, for fully stripped, hydrogen line andhelium-like ions around 191W [316].
8.2.1 On-Line Mass Separators.
A typical on-line mass separator (see ref. [330] for a review) is effectively a dipole
magnet which separates ions emitted from a thermal (or similar type) ion-source
[331, 332]. The ions are usually singly ionized and extracted by a voltage, V .
Thus they have energies QeV , where Q is usually 1.
Following the Lorentz equation, these ions are bent in a radius ρ, via a mag-
netic field according to the equation,
Bρ =Auv
Qe(8.2.22)
where A is the nuclear mass number, u is the atomic mass units (approx1.6×10−27 Kg), e is the electron charge, Q is the ionic charge state and v is the ion
191
Figure 8.16: Energy loss signals in the music chamber at the end of the FragmentRecoil Separator, which allows elemental identification of the fragments. Thesespectra are all for the fragmentation of a 208Pb beam at 1 MeV/a, but withdifferent settings of the spectrometer to transmit different nuclei the final focus[313].
velocity. The ion velocity, v can be obtained since the total kinetic energy, E, of
the ions can be deduced from the expression,
E =1
2Auv2 (8.2.23)
substituting in to equation 8.2.22, we obtain
Bρ =Au√
2EAu
Qe=
√2EAu
Qe(8.2.24)
where Q is usually equal to unity.
Figure 8.19 shows gamma-ray spectra for mass separated nuclei with A=176,177
and 178 from the GSI on-line mass separator, formed using identical reaction,
ion-source and tape cycle time conditions [329].
Channel Selection, Grow-in Curves and Tape Systems.
In addition to the mass selection provided by the on-line mass separator, other
channel selection can be provided by, (1) the choice of reaction to form the nuclei
192
1
2
4
8
16
32
64
128
256
512
1024
1.960 1.970 1.980 1.990 2.000 2.010 2.020
A/q
34
36
38
40
42
44
46
ZN=Z
Tc
Nb
Y
Rb
Figure 8.17: Particle id-spectra as in figure 8.4, but with hardware beam-offcondition for N=Z nuclei [320].
of interest; (2) the use of various types of ions sources such as FEBIAD and
TIS [331, 332]; and (3) the use of different tape cycle speeds to select decays of
different lifetimes (typically the furthest from stability are the shortest lived).
The extracted ions are often sent to a tape counting station, where the gamma-
rays from the radioactivity (with liofetimes usually longer than hundreds of mil-
liseconds) is collected. In a tape drive system, the tape can then be caused to
move with different frequencies, which means that the detection system becomes
sensitive to different rates of decays. For example, if the tape moves the ra-
dioactivity away quickly, only the fast decaying products will be observed in the
detection system (since the longer lived decays will be moved away from the sight
of the detectors before they decay).
One Component Grow-ins.
For a single component decay, the grow-in curve, which represents the decay rate
of a specific activity is given by assuming a constant implantation rate, I0, then,
193
01020304050607080
0 200 400 600 800 1000
time (ms)
counts 74Rb
72(18) ms
0
50
100
150
200
250
300
350
0 200 400 600 800 1000
time (ms)
counts
82Nb52(6) ms
0
10
20
30
40
50
0 200 400 600 800 1000
time (ms)
counts
86Tc45(12) ms
020406080
100120140160180
0 200 400 600 800 1000
time (ms)
counts 78Y
50(8) ms
Figure 8.18: Time spectra for delayed β+ decay in the same strip as the detectedN=Z recoil, with the beam-off condition provided by figure 8.17 [320].
dN
dt= I0 − λN (8.2.25)
where λN is the decay rate, with a decay constant, λ.
Trying a solution of the form
N = a(1 − e−λt) (8.2.26)
where a is a constantto be determined, then differentiating equation 8.2.26,
we obtain
dN
dt= λae−λt = I0 − λa(1 − e−λt) (8.2.27)
therefore,
I0 = λa (8.2.28)
194
which gives us the general solution for a one component grow-in curve of
N =I0λ
(1 − exp(−λt)) (8.2.29)
Thus the count rate, λN = I0(1 − exp(−λt)).
Two component Grow-ins.
In the case of a two component decay curve, where for example a β decay feeds
into a long lived isomeric state, a two component fit for the grow-in curve must
be performed.
If I0 is the production/implantation rate as above, the higher lying state has
a decay rate of λ1N1 and the lower lying state (ie. the one fed by state 1) has an
intrinsic decay rate of λ2N2, then the total decay rate as measured for the two
states is given by the decay rate of state 1 (to produce state 2), minus the decay
rate of state 2, ie.
dN2
dt= λ1N1 − λ2N2 (8.2.30)
The solution to equation 8.2.30 is given by trying a solution of the form,
N2 = a(1 − e−λ1t) + be−λ2t − c (8.2.31)
with the boundary conditions that at t = 0, N2 = 0 = b− c.
therefore, differentiating equation 8.2.31, we obtain,
dN2
dt= aλ1e
−λ1t − bλ2e−λ2t = λ1N1 − λ2N2 (8.2.32)
and therefore
dN2
dt= I0(1 − e−λ1t) − λ2a(1 − e−λ1t) − λ2be
−λ2t + λ2c (8.2.33)
re-arranging and collecting the terms
aλ1e−λ1t − bλ2e
−λ2t = I0 − λ2(a− c) − (I0 − λ2a)e−λ1t − λ2be
−λ2t (8.2.34)
Equation 8.2.34 requires that at t = ∞, I0 = λ2(a − c) and comparing the
terms for the e−λ1t term, aλ1 = (I0 − λ2a). Rearranging, this gives that
195
a =I0
λ2 − λ1
(8.2.35)
also, collecting and comparing the non-exponential terms in equation 8.2.34
gives
I0 = λ2(I0
λ2 − λ1
− c) (8.2.36)
re-arranging this gives
c = I0(1
λ2 − λ1− 1
λ2) = b (8.2.37)
196
Figure 8.19: Gamma-ray spectra for different masses taken from the GSI on-linemass separator, following the binary collision of a 11.4 MeV/u 136Xe beam witha Ta target. In each case, the tape cycle time was taken to be 8 seconds. [329].
197
Substituting the results of equations 8.2.35 and 8.2.37 into equation 8.2.33
gives that the decay rate of level 2, is given by
λ2N2 =λ2I0
λ2 − λ1(1 − e−λ1t) − I0(
λ2
λ2 − λ1− 1)(1 − e−λ2t) (8.2.38)
Note that the limits of equation 8.2.38 give when λ2t >> 0 and λ2 >> λ1 (ie.
upper level is much longer lived than lower one,
λ2N2 ≈ I0(1 − e−λ1t) (8.2.39)
ie. the decay rate depends on the lifetime of the upper state.
Similarly, when λ1t >> 0 and λ1 >> λ2
λ2N2 ≈ I0(1 − e−λ2t) (8.2.40)
If we substitute r = λ2
λ2−λ1, into equation 8.2.38, it reduces to
count rate = const.× r(1 − e−λ1t) − (r − 1)(1 − e−λ2t) (8.2.41)
Figure 8.20 shows the grow-in curves for the decays from the ground state
and excited states of 177Lu into states in 177Hf, following a binary reaction at GSI
[329].
Figure 8.21 shows the effect of the different tape cycle times for A=177 nuclei
following the binary reaction of 136Xe on a natural Ta target at the GSI on-line
mass separator. Note the presence of longer lived decays in the spectra with the
extended cycle times.
8.2.2 The 19Ne∗+40Ca Experiment at Louvain La Neuve.
To date there is one report of the use of a (short lived) radiaoctive ion beam
used to induce a fusion evaporation reaction [345]. This used the cyclotrons at
Louvain La Neuve to initially produce radioactive 19Ne (T 1
2
=17 s) by accelerating
protons onto a 19F target and using the charge exchange reaction 19F(p,n)19Ne.
The 19Ne beam was then injected into a second cyclotron where it was acceler-
ated to Coulomb barrier energies and used to bombard a target of 40Ca. The
target position was surrounded with germanium detectors to measure discrete
gamma-rays and a thin silicon charged particle detector (LEDA) to identify any
198
evaporated α-particle and protons from fusion-evaporation events. Typical 19Ne
beam currents were around 0.1 pnA.
Figure 8.22 highlights the problems associated with doing in beam gamma-
ray spectroscopy with intense radiaoctive beams. The raw gamma-ray spectra are
completely dominated by transitions associated with the decay of the radioactive
beam (in this case 511 keV annihilation gamma-rays coming from the β+ decay
of 19Ne).
However, as figure 8.22 shows if the beam is pulsed and the time structure
of the measured gamma-rays can be recorded with respect to the beam pulses,
most of this unwanted ‘radioactive’ background can be subtracted, leaving a
‘pure’ spectrum of transitions associated with fusion-evaporation events.
As figure 8.24 shows, the effect of charged particle gating is less dramatic
than performing a subtraction using the time spectra gating using the out-of-
beam spectra, however, a significant improvement in the signal to noise is clearly
observed.
Figure 8.25 shows a comparison of the relative yields of the various evaporation
residues for the 19Ne induced reaction compared to a 19F (stable beam) reaction.
While, it is apparent that the cross-section of the most neutron deficient nuclei in-
creases by using a more neutron deficient (radioactive) beam (19Ne), the increase
is negated by the decrease in beam intensity. It is clear that future radioactive
beam facilities which will desire to use such beams for fusion-evaporation reac-
tions will require both more exotic beams (more than one nucleon from stability)
coupled to intensities of a least 1 pnA.
199
Figure 8.20: Grow-in curves for the beta-decay of the high-spin isomer in 177KLu(upper panel) and ground state of 177Lu into 177Hf. The upper panel populates asecond isomeric state in 177Hf and therefore requires a two component fit, whilethe lower panel is a simple one component fit with a half-life of 1.9 hours [329].
200
Figure 8.21: Effect of the tape cycle time in selecting different decay half-lifes.A=177 spectra with identical reaction and source conditions, but different tapecycle times [329].
201
Figure 8.22: Differents types of channel selection to pick out transitions fromfusion products for the 19Ne+40Ca reaction [345].
202
Figure 8.23: TDC time spectra from the (a) 19Ne and (c) 19F beams at LLN. Notethe increase in counts for the ”beam on” period corresponding to gamma-raysfrom beam-induced reactions.
203
Figure 8.24: Channel selection afforded by gating on evaporated charged particlesin the 19Ne+40Ca experiment [345].
204
Figure 8.25: Comparison of yields from various residual channels using bothstable (19F) and radioactive 19Ne beams on a 40Ca target [345].
205
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