The true mystery of the world is the visible,
The true mystery of the world is the visible, not the invisible Oscar Wilde
K
Cu
Na
Outline of lectures (I)
• L1: Probing the nuclear fingerprint on the atom
1.1 Hyperfine structure
1.2 Isotope shifts
• L2: Doppler-free laser spectroscopy
2.1 Introduction to lasers and laser spectroscopy
2.2 The crossed beams methods and fast beams collinear laser spectroscopy
2.3 The structure of K isotopes & the shell model
2.4 The N=60 region and nuclear deformation
Outline of lectures (II)
• L3: Efficiency, sensitivity and precision
3.1 Selective Radioactive Ion Beam production 3.2 In-source resonance ionization spectroscopy - ISOL (hot cavity approach) - IGISOL (gas cell approach) 3.3 In-gas jet spectroscopy & CRIS 3.4 Towards the superheavy elements 3.5 229Th (a nuclear clock) Outlook
THEORY • H. Kopfermann, Nuclear Moments, Academic Press, New York (1958) • W.H. King, Isotope Shifts in Atomic Spectra, Plenum Press, New York
(1984) • G.K. Woodgate, Elementary Atomic Structure, McGraw-Hill (1970) • D.M. Brink and G.R. Satchler, Angular momentum, Oxford Science
Publications, third edition (1994)
LASERS • J. Wilson and J.F.B. Hawkes, Lasers Principles and Applications,
Prentice Hall Europe (1987)
LASER SPECTROSCOPY • W. Demtroder, Laser Spectroscopy Basic Concepts and Instrumentation, 3rd
Edition Springer-Verlag (2003)
RESONANCE IONIZATION • V.S. Letokhov, Laser Photoionization Spectroscopy, Ac. Press, (1987)
Between 1995 and 2010 (B. Cheal and K.T. Flanagan,
J. Phys. G 37 (2010) 113101)
Before 1995 (J. Billowes and P. Campbell,
J. Phys. G 21 (1995) 707) Unpublished
Submitted to PPNP (Campbell, Moore, and Pearson 2015)
Previous laser spectroscopic reviews
What are limits of nuclear existence?
Do new forms of nuclear matter exist?
Are there new forms of collective motion?
Does the ordering of quantum states change?
Key questions in nuclear science
Between 1995 and 2010 (B. Cheal and K.T. Flanagan,
J. Phys. G 37 (2010) 113101)
Before 1995 (J. Billowes and P. Campbell,
J. Phys. G 21 (1995) 707) Unpublished
Submitted to PPNP (Campbell, Moore, and Pearson 2015)
Previous laser spectroscopic reviews
Recent work mostly by
collinear laser spectroscopy
Recent work mostly by resonance
ionization spectroscopy (RIS)
N=4
N=2
N=3
N=1
Balmer Series
Lyman Series
N=∞
A historical note on atomic spectroscopy
• 1704: release of Newton’s ”Opticks”. Sun’s light can be dispersed into a ”spectrum”
L
S
J=L+S
• Increasing the resolution by a factor of ~5000 reveals
a fine structure splitting of hydrogen
Increasing in resolution
λ=656.279 nm (N=3→N=2)
F=J+I
2P3/2
2P1/2
2S1/2
Fj
Fi
• A further factor of 1000 in resolution reveals a finer
splitting due to the coupling of the nucleus with the
electronic orbital
→ Hyperfine structure
Hyperfine interaction = the interaction of nuclear magnetic and electric moments with electromagnetic fields (produced at the nucleus
by the orbiting electrons)
Lets consider the effect on an atomic orbit of spin J
J
Nuclear spin
I
Electron spin
Hyperfine interactions in free atoms
e
I
J
F
The atomic and nuclear spins
couple to form the total
angular momentum
F = I + J
Each state J has
several F-states:
States of the same I and J but coupled to different angular momenta F will have slightly different energies
Nuclear magnetic moment
Contributions from orbiting charge and intrinsic spin
Magnetic (dipole) moment
The magnetic dipole moment of a state of spin I = expectation value of the z-component of the dipole operator μ :
The magnetic moment (or g factor) therefore tells us about the valence nucleon orbits and couplings. What about filled shells?
Protons: gl = +1 gs = +5.586
Neutrons: gl = 0 gs = -3.826
E = - m . Be = - m Be cos q
The interaction energy depends on angle θ
J
Nuclear spin
I
Electron spin
Magnetic dipole interaction
Since and
then the interaction Hamiltonian can be expressed as
m
The different energy shifts of the different F-states are then
where
Be can be calibrated by measuring the energy shifts for an isotope of a known magnetic moment.
Spectroscopic quadrupole moment
The electric quadrupole moment provides a measure of the deviation of charge distribution from sphericity:
I
q
m
Z
this assumes a well-defined deformation axis
Using angular momentum algebra, we get
Experiments measure the maximum ”projection” of the intrinsic quadrupole moment along the quantization axis
Note for nuclear spin I=0 and I=1/2 the spectroscopic quadrupole moment vanishes even if the intrinsic shape is deformed. The intrinsic moment can in turn be related to the quadrupole deformation parameter β2
Oblate Spherical
Prolate
In a uniform electric field the energy of an electric quadrupole moment is independent of angle and there is no quadrupole interaction. However, there is an angle-dependence in an electric field gradient.
Quadrupole deformation
β2 -ve +ve 0
Electric quadrupole interaction
E = ¼ e Q0 VJJ P2(cos q) J
Nuclear spin
I
Electron spin
Electric field gradient along J-direction due to atomic electrons.
Energy shifts of the F-states are then
where
The hyperfine factor “B” is measured by experiment
The electric field gradient VJJ may be calibrated with an isotope with known Qs
The original fine structure level E(J) is perturbed so that the final energy due to hyperfine effects:
Summarizing the hyperfine splitting
E(J)
5/2
3/2
1/2
5/2 A
3/2 A
-B
+5/4 B
1/4 B
5/2 A + 5/4 B
3/2 A - 9/4 B
)1()1()1( where
)12)(12(2
)1()1()1(
2)( 4
3
JJIIFFC
JIJI
JJIICCBC
AFE
201Hg
Nuclear spin I=3/2
Atomic spin J=1 nucleusat field magnetic)0(
,)0(
e
eI
B
JI
BA
m
Access to nuclear spin I (number of HF components) and µI
The original fine structure level E(J) is perturbed so that the final energy due to hyperfine effects:
Summarizing the hyperfine splitting
E(J)
5/2
3/2
1/2
5/2 A
3/2 A
-B
+5/4 B
1/4 B
5/2 A + 5/4 B
3/2 A - 9/4 B
)1()1()1( where
)12)(12(2
)1()1()1(
2)( 4
3
JJIIFFC
JIJI
JJIICCBC
AFE
201Hg
Nuclear spin I=3/2
Atomic spin J=1
nucleusat field magnetic)0(
,)0(
e
eI
B
JI
BA
m
Access to nuclear spin I (number of HF components) and µI
nucleusat
gradient field electric)0(
),0(
JJ
JJs
V
VeQB
Access to Qs
But...
All you actually need is,
82
But...
All you actually need is,
82 Page
JU FU I FL JL 1
(2FL+1) (2JL+1) (2Fu+1) { } Intensities:
Splittings:
Only 3 transitions allowed (no F=0 → F=0)
In practise: example of K
• The nuclear spin can be unambiguously determined from the number of HF components (relative spacings and intensities)
• Selection rules obeyed, ΔF=0, ±1, F=0 → F=0 forbidden
49K I=1/2
51K I=3/2
J. Papuga et al., PRL 110 (2013) 172503
Frequency (MHz)
102
101
100
99
98
96
94
93
92
95
97
(Pauli 1924!!!)
B Cheal et al, PLB 645 (2007) 133
= Frequency difference in an electronic transition between two isotopes
Isotope 1
Isotope 2
Isotope shifts of electronic transitions
The Nobel Prize in Chemistry 1934 was awarded to Harold C. Urey "for his discovery of heavy hydrogen".
H. Urey (1932): Splitting in the Balmer series in natural hydrogen
→ ”heavy” hydrogen (deuterium!)
AAAA ''
Nuclear radius: few × 10-15 m Atomic radius: few × 10-10 m
Point nuclear charge: Coulomb potential (-1/r)
Potential is slightly deeper for the smaller isotope: s-electrons more
tightly bound
The nuclear volume effect (field shift)
The finite spatial extent (volume) of the nucleus gives an electrostatic potential difference to that of the Coulomb potential
– this perturbs the electron wavefunction Ψe(r)
V(r
)
r
V(r)
r
Isotope shift of atomic level
The nuclear volume effect II
rdrrVreE eefs
3
0)()()(
The difference in this energy shift between two isotopes can be expressed as
where δV(r) is the change in Coulomb potential between two isotopes. We assume constant electron density in the region of the
nucleus and arrive at the change in frequency of a transition:
',22
0
2', )0(
6
AA
e
AA
fs rh
Ze
Change in electron density at the nucleus between the two atomic levels comprising the
transition (electronic part)
Difference in the mean
square nuclear charge radius
(nuclear part)
The Finite Nuclear Mass Effect
i e
inkinetic
m
p
M
PE
22
22Kinetic energy
(nucleus + electrons)
In the Centre of Mass frame, the nucleus has a momentum equal and opposite to that of the
sum of the electrons
i
in pP
ji
ji
i
inuclear ppM
pM
E ˆˆ1
ˆ2
1 2This leads to a nuclear kinetic energy:
recoil correlations
”Unfortunately” this is an effect which we must take account of, even though it has no ”nuclear physics” interest…
E
One can finally solve the Schrodinger equation to calculate the energy change between two isotopes A and A’ (and the transition
frequency)
Normal and specific mass shifts
ji
ji
i
i
u
ms pppAA
AA
mE ))ˆˆ(2ˆ(
'
'
2
1 2
Normal mass shift • due to the finite
nuclear mass • easy to calculate
Specific mass shift • due to electron correlations • no analytical expression
• calcuable only for 2- and 3- electron systems
'
')(
AA
AAKK SMSNMSSMSNMSms
1/A2 A>>1
Isotope shifts in practise
IS = FS
2δ r
MS +
|e(0)|2 6hε0
THEORY EXPERIMENT
Ze2
To evaluate IS data:
- mass data from Atomic Mass Evaluation (2012) - SMS either calculated (ab-initio, MBPT, coupled cluster) or evaluated
via non-optical data (elastic e scattering, muonic atom X-rays) - Field shift factor from non-optical, semi-empirical, atomic theory
(accurate to 15%)
Cheal, Cocolios and Fritzsche, PRA 86 (2012) 042501
To summarize the effects so far…
Isotope (A-1)
Isotope A
Isotope (A-1)
Isotope A
F=5/2
F=3/2
F=1/2
Point nucleus + Finite size + Magnetic + Electric dipole quadrupole
Example: J=1, I=3/2
+ higher order multipoles
(too small to consider in laser measurements)
These energy shifts of may be only a few parts per million of the energy of an optical atomic transition. Optical techniques provide the sensitivity and precision required to measure these effects.
qm coseB )(cos4
120 qPVeQ JJMass shift + Field shift
The important message(s) from Lecture 1
Sizes
Spins
Qs
μ
Model Independent
(measured)
Ground-state spin assignments underpin all excited levels. We measure them directly!
F. Charlwood et al, Hyp. Int. 196 (2010) 143
1974
Mn58 3+,4+
65 s
Mn58 2+,3+
65 s
Mn58 3+ (0+)
3.0 s 65 s
b – 3.8, .. g 810.8, 1323.1, 459.2,…
E 6.25
Mn58 3+ (0)+
3.0 s 65 s
b – 6.1, .. g 1447-, 2227
b – 3.8, .. g 810.8, 1323.1, 459.2,…
E 6.25
Mn58 1+ (4)+
3.0 s 65 s
b – 6.1, ..
g 1446.5,
2433.1,
…
b – 3.8, ..
g 810.8,
1323.1,
459.2,
…
IT 71.78
E 6.25
Mn58 (1)+ (4)+
3.0 s 65 s
b – 6.1, ..
g 1446.5,
2433.1,
…
b – 3.8, ..
g 810.8,
1323.1,
459.2,
…
IT 71.78
E 6.25
1980 1988
1996 2002 2010
End of Lecture 1