Phenomenological properties of nuclei

1) Introduction - nucleon structure of nucleus

2) Sizes of nuclei

3) Masses and bounding energies of nuclei

4) Energy states of nuclei

5) Spins

6) Magnetic and electric moments

7) Stability and instability of nuclei

8) Exotic nuclei

9) Nature of nuclear forces

Introduction – nucleon structure of nuclei. Atomic nucleus consists of nucleons (protons and neutrons).

Number of protons (atomic number) – Z. Total number nucleons (nucleon number) – A.

Number of neutrons – N = A-Z. NAZ Pr

Different nuclei with the same number of protons – isotopes.

Different nuclei with the same number of neutrons – isotones.

Different nuclei with the same number of nucleons – isobars.

Different nuclei – nuclides.Nuclei with N1 = Z2 and N2 = Z1 – mirror nuclei

Neutral atoms have the same number of electrons inside atomic electron shell as protons inside nucleus.Proton number gives also charge of nucleus: Qj = Z·e

(Direct confirmation of charge value in scattering experiments – from Rutherford equation for scattering (dσ/dΩ) = f(Z2))

Atomic nucleus can be relatively stable in ground state or in excited state to higher energy – isomers (τ > 10-9s).

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Stable nuclei have A and Z which fulfill approximately empirical equation:

Reliably known nuclei are up to Z=112 in present time (discovery of nuclei with Z=114, 116 (Dubna) needs confirmation).Nuclei up to Z=83 (Bi) have at least one stable isotope. Po (Z=84) has not stable isotope. Th , U a Pu have T1/2 comparable with age of Earth.

Maximal number of stable isotopes is for Sn (Z=50) - 10 (A =112, 114, 115, 116, 117, 118, 119, 120, 122, 124).

Total number of known isotopes of one element is till 38. Number of known nuclides: 3104 (r. 2011).

Sizes of nucleiDistribution of mass or charge in nucleus are determined.

We use mainly scattering of charged or neutral particles on nuclei.

Density ρ of matter and charge is constant inside nucleus and we observe fast density decrease on the nucleus boundary. The density distribution can be described very well for spherical nuclei by relation (Woods-Saxon):

where α is diffusion coefficient. Nucleus radius R is distance from the center, where density is half of maximal value. Approximate relation R = f(A) can be derived from measurements: R = r0A1/3

where we obtained from measurement r0 = 1,2(1) 10-15 m = 1,2(2) fm (α = 1,8 fm-1). This shows on

permanency of nuclear density. Using Avogardo constant

or using proton mass: 315

27

30

p

3

p

m102.134

kg1067.1

r34

m

R34

Am

we obtain 1017 kg/m3.

High energy electron scattering (charge distribution) smaller r0.Neutron scattering (mass distribution) larger r0.

Distribution of mass density connected with charge ρ = f(r) measured by electron scattering with energy 1 GeV

Larger volume of neutron matter is done by larger number of neutrons at nuclei (in the other case the volume of protons should be larger because Coulomb repulsion).

)Rr(0

e1)r(

Deformed nuclei – all nuclei are not spherical, together with smaller values of deformation of some nuclei in ground state the superdeformation (2:1 3:1) was observed for highly excited states. They are determined by measurements of electric quadruple moments and electromagnetic transitions between excited states of nuclei.

Neutron and proton halo – light nuclei with relatively large excess of neutrons or protons → weakly bounded neutrons and protons form halo around central part of nucleus.

Experimental determination of nuclei sizes:

1) Scattering of different particles on nuclei: Sufficient energy of incident particles is necessary for studies of sizes r = 10-14m. De Broglie wave length λ = h/p < r:

Neutrons: mnc2 >> EKIN → → EKIN > 16 MeV

Electrons: mec2 << EKIN → λ = hc/EKIN → EKIN > 100 MeV

2) Measurement of roentgen spectra of mion atoms: They have mions in place of electrons (mμ = 207 me): μ,e – interact with nucleus only by electromagnetic interaction. Mions are ~200 nearer to nucleus → „feel“ size of nucleus (K-shell radius is for mion at Pb 3 fm ~ size of nucleus)

3) Isotopic shift of spectral lines: The splitting of spectral lines is observed in hyperfine structure of spectra of atoms with different isotopes – depends on charge distribution – nuclear radius.

4) Coulomb energy of nucleus: Reduction of Coulomb energy EC and the same reduction off binding energy of nucleus (energy of uniformly charged sphere)

2

C

2

C ZR

e

5

3E

Mirror nuclei – same nuclear binding energy, different Coulomb energy. Difference of binding energy is given by EC difference.

5) Study of α decay: The nuclear radius can be determined using relation between probability of α particle production and its kinetic energy.

KIN2mEh/

Masses of nuclei

Nucleus has Z protons and N=A-Z neutrons. Naive conception of nuclear masses:

M(A,Z) = Zmp+(A-Z)mn

where mp is proton mass (mp 938.27 MeV/c2) and mn is neutron mass (mn 939.56 MeV/c2)

where MeV/c2 = 1.78210-30 kg, we use also mass unit: mu = u = 931.49 MeV/c2 = 1.66010-27 kg. Then mass of nucleus is given by relative atomic mass Ar=M(A,Z)/mu.

Real masses are smaller – nucleus is stable against decay because of energy conservation law. Mass defect ΔM: ΔM(A,Z) = M(A,Z) – (Zmp + (A-Z)mn)

It is equivalent to energy released by connection of single nucleons to nucleus - binding energy B(A,Z) = - ΔM(A,Z) c2

Binding energy per one nucleon B/A:

Maximal is for nucleus 56Fe (Z=26, B/A=8.79 MeV).

Possible energy sources:

1) Fusion of light nuclei2) Fission of heavy nuclei

8.79 MeV/nucleon 1.4·10-13 J/1,66·10-27 kg = 8.7·1013 J/kg

(gasoline burning: 4.7·107 J/kg) Binding energy per one nucleon for stable nuclei

Measurement of masses and binding energies:

Mass spectroscopy:

Mass spectrographs and spectrometers use particle motion in electric and magnetic fields:

Mass m=p2/2EKIN can be determined by comparison of momentum and kinetic energy. We use passage of ions with charge Q through “energy filter” and “momentum filter”, which are realized using electric and magnetic fields:

EQFE

and then F = QE BvQFB

for is FB = QvBvB

The study of Audi and Wapstra from 1993 (systematic review of nuclear masses) names 2650 different isotopes. Mass is determined for 1825 of them.

Frequency of revolution in magnetic field of ion storage ring is used. Momenta are equilibrated by electron cooling → for different masses → different velocity and frequency.

Electron cooling of storage ring ESR at GSI Darmstadt

Comparison of frequencies (masses) of ground and isomer states of 52Mn. Measured at GSI Darmstadt

Velocity filtr: v = E/B

At GSI Darmstadt fragment separator (FSR) makes possible to produce different isotopes and storage ring (ESR) makes possible to measure big number of nuclear masses. Accuracy is ΔM = 0,1 MeV/c2, that means relative accuracy ΔM/M ~ 10-6. Possibility to measure also very short isotopes τ > 30 s (with electron cooling), τ ≈ μs (without electron cooling).

Similar device is at CERN (ISOLDE)

Exploitation of reaction energy balance:

Useful also in the case where mass spectroscopy is not working (neutral particles).

Determination of neutron mass as example:

1) We measure energy of γ quantum essential for deuteron split: BHnd 11

10

21

2) Deuteron mass is: md = mn + mH - B

3) Masses of hydrogen and deuteron are measured by spectroscopy.

4) Neutron mass is: mn = (md - mH) + B.

Masses of other instable particles and nuclei can be determined by this method (ΔM/M ~ 10-8).

Are nucleons localized at nuclei? B/A 8 MeV /A Energy necessary for nucleon separation 8 MeV

De Broglie wave length = h/p binding state condition 2r = n (n natural number) /2 shows typical size. 8 MeV << 939 MeV → nonrelativistic approximation

are Agree with nuclear sizes.

Can be electrons localized at nuclei? Electron with EKIN = 8 MeV is relativistic even ultrarelativistic:

can not

fm10MeV89402

fm197MeV3,142

Ec2m

c2

E2m

h

p

h

KIN2

0KIN0

155fm8MeV

fm197MeV3,142

E

c2

p

h

KIN

Excited energy statesNucleus can be both in ground state and in state with higher energy – excited state

Every excited state – corresponding energy→ energy level

Energy level structure of 66Cu nucleus (measured at GANIL – France, experiment E243)

Deexcitation of excited nucleus from higher level to lower one by photon irradiation (gamma ray) or direct transfer of energy to electron from electron cloud of atom – irradiation of conversion electron. Nucleus is not changed. Or by decay (particle emission). Nucleus is changed.

Scheme of energy levels:

Quantum physics → discrete values of possible energies

Three types of nuclear excited states:

1) Particle – nucleons at excited state EPART

2) Vibrational – vibration of nuclei EVIB

3) Rotational – rotation of deformed nuclei EROT (quantum spherical object can not have rotational energy)

it is valid: EPART >> EVIB >> EROT

Obtaining of excited state of nuclei:

1) Beta or alpha decays2) Inelastic scattering of charged particles or nuclei – Coulomb excitation3) Nuclear reactions

The big number of different isotopes can be produced using the fragment separators and radioactive beams make possible.

Experiment E243(LISE-GANIL-France)

Measurement of properties of transitions between excited states:

1) Energy spectra and angular distribution of gamma rays2) Energy spectra of conversion electrons

Measurement of excited state properties:

Energy spectra and angular distribution of particles from scattering or reactions Gamma ray spectrum of deexcitation of

70Ni levels (experiment E243)

Isotope identification obtained by device LISE (GANIL-France)

Spins of nucleiProtons and neutrons have spin 1/2. Vector sum of spins and orbital angular momenta is total angular momentum of nucleus I which is named as spin of nucleus

Orbital angular momenta of nucleons have integral values → nuclei with even A – integral spin nuclei with odd A – half-integral spin

Classically angular momentum is define as . At quantum physic by appropriate operator, which fulfill commutating relations:

prl

lˆ

ilˆ

lˆ

There are valid such rules:

2) From commutation relations it results, that vector components can not be observed individually. Simultaneously and only one component – for example Iz can be observed .2I

3) Components (spin projections) can take values Iz = Iħ, (I-1)ħ, (I-2)ħ, … -(I-1)ħ, -Iħ together 2I+1 values.

4) Angular momentum is given by number I = max(Iz). Spin corresponding to orbital angular momentum of nucleons is only integral: I ≡ l = 0, 1, 2, 3, 4, 5, … (s, p, d, f, g, h, …), intrinsic spin of nucleon is I ≡ s = 1/2.

5) Superposition for single nucleon leads to j = l 1/2. Superposition for system of more particles is diverse. Extreme cases:

slˆ

jˆ

i

ii

i sS,lˆ

L,SLI

LS-coupling, where i

ijˆ

Ijj-coupling, where

1) Eigenvalues are , where number I = 0, 1/2, 1, 3/2, 2, 5/2 … angular momentum magnitude is |I| = ħ [I(I+1)]1/2

2I

22 1)I(II

Magnetic and electric momenta

Ig j

Ig j

Magnetic dipole moment μ is connected to existence of spin I and charge Ze. It is given by relation:

where g is g-factor (sometimes named also as gyromagnetic ratio) and μj is nuclear magneton:

114

pj MeVT1015.3

c2m

e

Bohr magneton: 111

eB MeVT1079.5

c2m

e

For point like particle g = 2 (for electron agreement μe = 1.0011596 μB). For nucleons μp = 2.79 μj and μn = -1.91 μj – anomalous magnetic moments show complicated structure of these particles.

Magnetic moments of nuclei are only μ = -3 μj 10 μj, even-even nuclei μ = I = 0 → confirmation of small spins, strong pairing and absence of electrons at nuclei.

Electric momenta:

Electric dipole momentum: is connected with charge polarization of system. Assumption: nuclear charge in the ground state is distributed uniformly → electric dipole momentum is zero. Agree with experiment.

)aZ(c5

2Q 22

Electric quadruple moment Q: gives difference of charge distribution from spherical. Assumption: Nucleus is rotational ellipsoid with uniformly distributed charge Ze:(c,a are main split axles of ellipsoid) deformation δ = (c-a)/R = ΔR/R

Results of measurements:

1) Most of nuclei have Q = 10-29 10-30 m2 → δ ≤ 0.1

2) In the region A ~ 150 180 and A ≥ 250 large values are measured: Q ~ 10-27 m2. They are larger than nucleus area. → δ ~ 0.2 0.3 → deformed nuclei.

Generally apply to:

1) All odd electric multiple moments disappeared

2) All even magnetic multiple moments disappeared

3) For state with total angular momentum I, mean value of all moments, which order of multiple L > 2I disappeared. Nuclei with I = 0,1/2 has not electric quadruple moment.

Measurement of magnetic moments

Magnetic dipole moments of nucleus are measured by their interaction with magnetic field. Energy of magnetic dipole in magnetic field is:B

BEmag

A) Magnetic moments of nuclei can be obtained from splitting hyperfine structure (interaction between electron cloud and nucleus).

B) On the base of motion of magnetic dipole through magnetic fields:

1) Beam of neutral atoms come through inhomogeneous magnetic field force : F = ZBZ/z acts on magnetic moment, oriented it and focused beam to the point C. (Axe z is in the direction of magnetic field changes)

2) Homogeneous magnetic field of magnet C not created force. In this place orientation of magnetic dipole is changed by high frequency field (induced by dipole transitions) with frequency = ΔEmag /ħ obtained by induction coil.

3) Inhomogeneous magnetic field B focused on detector only atoms with changed orientation. Atoms with not changed orientation are loosed.

C) Measurement of magnetic resonance: Sample is placed to homogeneous magnetic field. Energy difference corresponding to different projections of angular momentum IZ : ΔEmag = gμΔIZB. For dipole transitions ΔIZ = ±1 : ΔEmag = ħ L = gμB → L = (1/ħ) gμB where L is Larmor frequency. Resonance is observed by energy absorption at induction coil.

Source HF

Detector

Stability and instability of nucleiStable nuclei: for small A (<40) is valid Z = N, for heavier nuclei N 1,7 Z. This dependence can be express more accurately by empirical relation:

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For stable heavy nuclei excess of neutrons → charge density and destabilizing influence of Coulomb repulsion is smaller for larger number of neutrons.

Even-even nuclei are more stable → existence of pairing N Z  number of stable nucleieven even 156 even odd 48odd even 50odd odd 5

Magic numbers – observed values of N and Z with increased stability.

At 1896 H. Becquerel observed first sign of instability of nuclei – radioactivity. Instable nuclei irradiate:

Alpha decay → nucleus transformation by 4He irradiationBeta decay → nucleus transformation by e-, e+ irradiation or capture of electron from atomic cloudGamma decay → nucleus is not changed, only deexcitation by photon or converse electron irradiationSpontaneous fission → fission of very heavy nuclei to two nucleiProton emission → nucleus transformation by proton emission

Nuclei with livetime in the ns region are studied in present time. They are bordered by:

proton stability border during leave from stability curve to proton excess (separative energy of proton decreases to 0) and neutron stability border – the same for neutrons. Energy level width Γ of excited nuclear state and its decay time τ are connected together by relation τΓ ≈ h. Boundery for decay time Γ < ΔE (ΔE – distance of levels) ΔE~ 1 MeV→ τ >> 6·10-22s.

Exotic nucleiNuclei far away from stability curve: 1) with large excess of neutrons

2) with large deficit of neutrons (excess of protons)

Effort to study all isotopes between boundaries of proton and neutron stability.

Double magic nuclei: 100Sn is such nucleus with maximal numbers of neutrons and protons

Firstly observed at GSI Darmstadt at Germany and at GANIL Caen at France

Cases of observation of nucleus 100Sn at GSI Darmstadt

1) with very high energy2) with very high spin3) with large deformation → quadruple moments (superdeformed till hyperdeformed)

Highly excited states:

Device for exotic nuclei studies at GSI Darmstadt

Superheavy nuclei: for large A and Z stability is increasing – existence of magic numbers → existence of stability island. Nuclei up to Z = 112 and also 114 and 116 (mainly GSI Darmstadt, JINR Dubna and Berkeley) were confirmed and have names, discovery of element 113 have been confirmed just now. Elements with Z = 115, 117 and 118 (Dubna and Berkeley) need confirmation.

Table of isotopes in the region of superheavy elements (situation in 2000)

Hypernuclei: One or more neutrons are changed by neutral hyperon Λ. ΛH3, ΛHe5, ΛLi9,

ΛO16, ΛFe56, ΛBi209, ΛΛHe6, ΛΛBe8). Other hyperons (Σ, Ξ, Ω) interact strongly with nucleons and they decay quickly to Λ (reactions with strangeness conversation) → hypernucleus is not produced. First discoveries (1952) during cosmic rays studies. Today more than 33 hypernuclei are known. Production by intensive meson beams. Decay time τ ≈ τΛ ≈10-10s.

They make possible to study influence of strangeness on nuclear force properties – demonstrate existence of attractive forces between Λ and nucleons (BΛp < Bnp).

Exotic atoms: 1) mion atoms – electron is changed by mion 2) positronium – bound system consists of electron and positron 3) antiprotonic helium atoms – bound system consists of nucleus and antiproton

Halo nuclei: consist of strongly bounded coreoften stable isotope and very weak bounded neutrons or protons around

Borromean nuclei: weakly bound system, everyits part is not bounded alone

Antinuclei: antiproton, antineutron, antilambda, pozitron and other antiparticles are produced. Possible existence of antinuclei. Up to now only the lights: antideuteron, antihelium 3

Antiatoms: First antiatom (antihydrogen) at CERN (1996) – creation of electron and positron pair during antiproton movement through electromagnetic field of nuclei was used (it resolves problem of positron capture by antiproton).

Antiproton decelerator at CERN makes possible production of thousands antihydrogens, capture of antiprotons tomagnetic trap, mixture with positrons → creation ofantihydrogen – detection by anihilation

One case of antihydrogen anihilation – production of 4 mesons (p + anti-p) and 2 (e + e+)

Nature of nuclear forces

The forces inside nuclei are electromagnetic interaction (Coulomb repulsion), weak (beta decay) but mainly strong nuclear interaction (it bonds nucleus together).

For Coulomb interaction binding energy is B Z (Z-1) B/Z Z for large Z non saturated forces with long range.

For nuclear force binding energy is B/A const – done by short range and saturation of nuclear forces. Maximal range ~1.7 fm

Nuclear forces are attractive (bond nucleus together), for very short distances (~0.4 fm) they are repulsive (nucleus does not collapse). More accurate form of nuclear force potential can be obtained by scattering of nucleons on nucleons or nuclei.

Charge independency – cross sections of nucleon scattering are not dependent on their electric charge. → For nuclear forces neutron and proton are two different states of single particle - nucleon. New quantity isospin T is define for their description. Nucleon has than isospin T = 1/2 with two possible orientation TZ = +1/2 (proton) and TZ = -1/2 (neutron). Formally we work with isospin as with spin.

In the case of scattering centrifugal barier given by angular momentum of incident particle acts in addition.

Expect strong interaction electric force influences also. Nucleus has positive charge and for positive charged particle this force produces Coulomb barrier (range of electric force is larger then this of strong force). Appropriate potential has form V(r) ~ Q/r.

Spin dependence – explains existence of stable deuteron (it exists only at triplet state – s = 1 and no at singlet - s = 0) and absence of di-neutron. This property is studied by scattering experiments using oriented beams and targets.

Tenzor charakter – interaction between two nucleons depends on angle between spin directions and direction of join of particles.

Intermediate particles with similar masses were discovered and named as π mesons. Attractive and repulsive nuclear force is than intermediated exchange of charged and neutral mesons:

p + π - → n, n + π + → p, p + π0 → p, n + π0 → n

Protons and neutrons emit and absorb mesons. Why their masses are not changed?

Uncertainty principle: ΔEΔt ≥ ћ → violation of energy conservation is allowed if it is shorter then ћ/ΔE. Maximal range of nuclear forces is R = 1.7 fm. Then the smallest time of nucleon transit is: Δt = R/c. Value of violation of energy conservation is during emission of meson with mass mπ: ΔE = mπc2. If time of violation will be Δt we obtain for maximal possible energy violation (meson mass): mπc2 = ћc/R (the same as earlier shown)

Further mesons (η, ρ, φ …) were found, also two-meson exchange.

MeV120fm 1.7

MeVfm197

R

ccmc2

Exchange nature of nuclear forces:

r

eV(r)

mcr

short range → nonzero rest mass of intermediate particles. H. Yukawa proposed corresponding potential

where m is mass of intermediate particle and ћ /mc is its Compton wave length. We put Compton length equal to range R of nuclear forces and we determine mass of intermediate particle: