Structure of nuclei

3224Nuclear and Particle Physics

Ruben SaakyanUCL

Fermi gas model. Assumptions• The potential that an individual nucleon feels is the

superposition of the potentials of other nucleons. This potential has the shape of a sphere of radius R=R0A1/3 fm, equivalent to a 3-D square potential well with radius R

• Nucleons move freely (like gas) inside the nucleus, i.e. inside the sphere of radius R.

• Nucleons fill energy levels in the well up to the “Fermi energy” EF

• Potential wells for protons and neutrons can be different– If the Fermi energy were different for protons and neutrons, the

nucleus would undergo decay into an energetically more favourable state

– Generally stable heavy nuclei have a surplus of neutrons– Therefore the well for the neutron gas has to be deeper than for the

proton gas– Protons are therefore on average less strongly bound than neutrons

(Coulomb repulsion)

• 2 protons/2 neutrons per energy level, since spins can be

Fermi momentum and Fermi energy

• The number of possible states available to a nucleon inside a volume V and a momentum region dp is

23

4( )

(2 )

Vn p dp dn p dp density of states factor

• In the nuclear ground state all states up to a maximum momentum, the Fermi momentum pF, will be occupied. Integration leads to the following number of states n. Since every state can contain two fermions, the number of protons Z and neutrons N are also given:

3 3 3

2 3 2 3 2 3

( ) ( ) ( )

6 3 3

p nF F FV p V p V p

n Z N

The nuclear volume V is given as 3 30 0

4 41.21

3 3V R R A R fm

from electron scattering

Fermi momentum and Fermi energy

• Assuming the depths of the neutron and proton wells are the same and Z = N = A/2, the Fermi momentum

1/3

0

9250 /

8n p

F F Fp p p MeV cR

• The energy of the highest occupied state, the Fermi energy is

2

332

is the nucleon mass

FF

pE MeV

MM

Fermi gas model. Potential

• The difference between the Fermi energy and the top of the potential well is the binding energy B’ = 7-8 MeV/nucleon that we already know from the liquid drop model

• The depth of the potential well V0 is to a good extent independent of the mass number A:

0 ' 40FV E B MeV

Derivation of symmetry term

Derivation of symmetry term (ctd)

Nuclear Models• The liquid drop model allows reasonably good

descriptions of the binding energy. It also gives a qualitative explanation for spontaneous fission.

• The Fermi gas model, assuming a simple 3D well potential (different for protons and neutrons) explained the terms in SEMF that were not derived from the liquid drop model.

• Nucleons can move freely inside the nucleus. This agrees with the idea that they experience an overall effective potential created by the sum of the other nucleons

• There are things which the Fermi gas model can not explain. This will lead us to the Shell Model.

The Shell Model

Basics

• The Shell Model is based very closely on the ideas from atomic physics: orbital structure of atomic electrons

• Atomic energy levels n = 1, 2, 3,… In nuclear physics we are not dealing with the same simple Coulomb potential: radial node quantum number n

• Atomic Physics: for any n there are energy-degenerate levels with orbital angular momentum l = 0,1,2,…,(n-1)

• For any l there are (2l+1) sub-states with different values of the projection of l along any chosen axis ml = -l, -l+1,…,0,1,…,l-1,l – magnetic quantum number Due to rotational symmetry of Coulomb potential these sub-states will be degenerate in energy

Basics• Since electrons have spin-1/2, each of the

states above can be occupied by 2 electrons with , corresponding to the spin-projection number ms=1/2. Again both states will have the same energy.

• Summarizing, any energy eigenstate in, say, H2 atom has quantum numbers (n, l, ml , ms ) and for any n there will be nd degenerate states1

2

0

2 (2 1) 2n

dl

n l n

Basics• This degeneracy can be broken if there is a

preferred direction in space (magnetic field). Recall spin-orbit coupling and fine structure.

• Going beyond H2 atom one has to introduce electron-electron Coulomb interaction.

• This introduces splitting to any level n according to l. The degeneracies in ml and ms are unchanged.

• If a shell or sub-shell is filled, then 0 and 0s lm m

In this case Pauli principle implies L = S = 0 and J = L + S = 0Such atoms (with paired off electrons) are chemically inertZ = 2, 10, 18, 36, 54

Nuclear Shell Structure Evidence Neutrons

Magic numbers

Binding energy curve revisited

Infinite Spherical Well

Spherical Harmonics

Shell structureInfinite Well/Harmonic oscillator

Shell Model Potential

Spin-Orbit Potential

Shell Model – Energy Levels

Shell Model – Energy Levels

Observed magicnumbers

28

20285082

126…

Spins in the Shell Model

• Shell model can be used to make predictions about the spins of ground states

• A filled sub-shell must have J=0• This means that, since magic number

nuclides have closed sub-shells, the contribution to the nuclear spin from protons/neutrons with magic number must be zero

• Hence doubly magic nuclei are predicted to have zero nuclear spin (observed experimentally)

Spins in the Shell Model

• All even-even nuclei have zero nuclear spin• Pairing hypothesis: For ground state nuclei,

pairs of n and p in a given sub-shell always couple to give a combined angular momentum of zero, even when the sub-shell is not filled.

• Last neutron/proton determines the net nuclear spin. – In odd-A there is only one unpaired nucleon. Net spin

can be determined precisely– In even-A odd-Z/odd-N nuclides we have an unpaired p

and an unpaired n. Hence the nuclear spin will lie in the range |jp-jn| to (jp+jn)

Parities in the Shell Model

• The parity of a single-particle quantum state depends exclusively on l with P = (-1)l

• P = Pi . A pair of particles with the same l will always have P = +1

• From pairing hypothesis we have:

• Pnucleus = Plast_p Plast_n

• The parity of any nuclide (including odd-odd) can be predicted (confirmed by experiment)

Magnetic moments in the Shell Model

= gj j N, N – nuclear magneton, gj – Lande g-factor

• For odd-odd nuclei we have to consider an unpaired n and an unpaired p

• For even-odd nuclei one has to “only” find out orbital and intrinsic components of magnetic moment of the single unpaired nucleon

2Np

e

M

Magnetic moments in the Shell Model

We need to combine gs s and gl l

( 1) ( 1) ( 1) ( 1) ( 1) ( 1)

2 ( 1) 2 ( 1)

IF 1/ 2

/ 2 for 1/ 2

1 11 for 1/ 2

2 1 2 1

j l s

j l s

j l s

j j l l s s j j l l s sg g g

j j j j

j l

jg g l g j l

jg g j g j j ll l

Magnetic moments in the Shell Model

Unlike spin and parities the Shell Model does not predictmagnetic moments very well

Since gl = 1 for p and gl = 0 for n, gs +5.6 for p and gs -3.8 for n

15.6 2.8 for 1/ 2

21 1 2.3

1 5.6 1 for 1/ 22 1 2 1 1

13.8 1.9 for 1/ 2

21 1.9

3.8 2 1 1

proton

proton

neutron

neutron

jg l j j l

jg j j j ll l j

jg j l

jjg j

l j

for 1/ 2j l

For a given j the measured moments lie between j = l -1/2 and j = l+1/2but beyond that the model does not predict the moments accurately

Excited states in the Shell Model

• First one or two excited states can be predicted relatively easily

• Consider 178O

protons: (1s1/2)2 (1p3/2)4 (1p1/2)2

neutrons: (1s1/2)2 (1p3/2)4 (1p1/2)2 (1d5/2)1 • 3 possibilities for 1st excited state

– One of the 1p1/2 protons to 1d5/2, giving (1p1/2)-1 (1d5/2)1

– One of the 1p1/2 neutrons to 1d5/2, giving (1p1/2)-1 (1d5/2)2

– 1d5/2 neutron to next level, 2s1/2 or 1d3/2 giving (2s1/2)1 or (1d3/2)1

• The 3d possibility corresponds to the smallest energy shift and therefore it is favourable

Excited states in the Shell Model

• Comparing the above predictions with experimental results it was found that the expected excited states do exist but not always in precisely the order anticipated

• Higher excited states calculation is much more complicated

• Collective model is an attempt to bring together shell and liquid drop models

• Recent encouraging developments in nuclear calculations due to progress in computing power

-decay. Fermi theory

• W, Z, quarks were not known. Theory based on general principles and analogy with QED

• Fermi’s Second Golden Rule

- transition rate, |M| - matrix element, n(E) – density of states (phase space determined by the decay’s kinematics)

1A AZ Z eX Y e

22( )M n E

-decay. Fermi theory

• g – dimensionless coupling constant, O - five basic classes of Lorentz invariant interaction operators– scalar S, pseudo-scalar P, vector V, axial-

vector A, tensor T– The main difference is the effect on the spin

states of the particles

• Fermi guessed that O should be of vector type (EM interaction transmitted by photon with spin-1)

* ( )f iM gO dV

Fermi coupling constant

• If we do not consider particle spins matrix element can be thought in terms of a classical weak interaction potential, like the Yukawa potential

• Point-like interaction. Matrix element in this case is just a constant M = GF/V

• GF – Fermi coupling constant• Can be applied to any weak process provided

the energy is not too great• Extracted from muon decay GF = 90 eV fm3

• Often quoted as 3 5 2/( ) 1.166 10FG c GeV

-decay. Electron momentum distribution

2

23

2

3 2

2( ) ( )

Recall that ( ) 4 and the same for 2

by changing variables using / /

4( ) and similar for ( )

(2 )

Since /

e

e e e e

e e e e

e e e e e

F

E E E

d M n E E n E dE

Vn p dp p dp n

dp dE E pc

Vn E dE p E dE n E

c

M G V

d

dE

2

22 2 4 2 43 7 4

where 2

Fe e e

e

Gp E p E p c E m c E E m c

c

-decay. Electron momentum distribution

22

3 7 2

2 2 2 2 2 2 2 2 2

3 7 3 7 3 3 7 3

2

If 0 then / and

( )

2 2 2

e Fe

E e e

F e F e F e e

e

dE Gd dp p E

dp dp dE c

m p E c

G p p G p E G p E Ed

dp c c c

Fermi screening factors F(Z, Ee)

Possible changes of nuclear spins are not taken into accountIf the change is > 1, the decay is suppressed

Spectra shifted for + w.r.t. -!

Kurie plots and the neutrino mass

• Studying -spectrum around the end point can be used to measure me

• Kurie plots are the most obvious

22 2

3 7 3

( , )

2

1( )

( , )

e F e e

e

e ee e e

F Z E G p E Ed

dp c

dH E E E

dp p K Z p

F(Z, Ee) and constants are here

Q

3H spectrum and the neutrino mass

3 3eH He e

3H most suitable isotope• Low Q, Q E0 = 18.6 keV• Simple atom

World’s best result (Mainz, Troitsk)me < 2.2 eV/c2

Future experiment: KATRINSensitivity: me ~ 0.2 eV/c2 Probably the lowest possible limit for this technique

Katrin detector transportation

from Deggendorf to Karlsruhe (400km away) but had to make a detour of…9000 km