8. Models of the nucleus

8.1 Fermi-gas model • The theoretical concept of a Fermi-gas may be applied for systems of weakly interacting fermions, i.e. particles obeying Fermi-Dirac statistics leading to the Pauli exclusion principle. — The free electron gas is an example of solid state physics, where electrons move quasi-freely in a background of positively charged ions. — In nuclear physics, protons and neutrons are considered as moving quasi-freely within the nuclear volume. The binding potential is generated by all nucleons. • Simple picture of the nucleus: — Neutrons and protons are distinguishable fermions and are therefore situated in two separate potential wells. — In a first approximation, these nuclear potential wells are considered as rectangular. • Fermi momentum, Fermi energy and its relation to the nuclear binding energy: — Neglecting spin, the number of states of protons, respectively, neutrons within the nuclear volume V is given by dn =

4πp2 dp (2π~)

3

V

(8.1)

— At temperature T = 0, i.e. for the nucleus in its ground state, the lowest states will be filled up to a maximum momentum, called the Fermi momentum pF . The number of these states follow from integrating eq.(8.1) from 0 to pF : n=

V p3F 6π 2 ~3

(8.2)

Since each state can be filled with two nucleons of the same type, N=

V p3F,n , 3π2 ~3

Z=

V p3F,p 3π 2 ~3

(8.3)

where pF,n and pF,p are the Fermi momenta of the neutrons, respectively, the protons. Using our prior knowledge from electron scattering, i.e. (see section 6.6), V =

4 3 4 3 πR = πR0 A, 3 3

R0 = 1.21 fm

(8.4)

we obtain for Z = N = A/2 and equal radius for the two separate potential wells of the protons and the neutrons a Fermi momentum of

90

8. Models of the nucleus

pF = pF,n = pF,p

~ = R0

µ

9π 8

¶1/3

≈ 250 MeV/c

(8.5)

Hence, the nucleons move in the nucleus with large momentum. — The Fermi energy is the energy of the highest occupied nucleon level: EF =

p2F ≈ 33 MeV 2mN

(8.6)

mN denotes the nucleon mass (mn ≈ mp in the present context). The difference B 0 between the edge of the potential well and the Fermi level is rather constant for different nuclei and equals the average binding energy per nucleon, B/A ≈ 7 − 8 MeV (note that, since the potential well is created by the nucleons, removing a few of them changes the well depth in a way that B 0 stays approximately constant). Hence, the depth of the potential well, V0 , is approximately independent of A and given by V0 = EF + B 0 ≈ 40 MeV

(8.7)

Kinetic and potential energies of the nucleons are thus of the same order. In this sense, nucleons are rather weakly bound in the nucleus (similar to the case of electrons in a metal).

Fig. 8.1. Sketch of the proton and neutron potential wells and states in the Fermi gas model [B. Povh et al., Particles and Nuclei, Springer, 2002].

• Coulomb repulsion of the protons leads to an asymmetry in Z and N : — The neutron potential well is deeper than the proton one, since the former have no Coulomb interaction. On the other hand, for a stable nucleus, the Fermi levels of the protons and the neutrons have to be the same, otherwise it would decay to an energetically more favourable state through a β transition. As a result, there are more neutron states than proton states occupied, which explains the fact that N > Z for heavier stable nuclei. — The binding energy as a function of N − Z can be estimated using the Fermi-gas model: the mean kinetic energy per nucleon is R pF Ekin p2 dp 3 p2F ≈ 20 MeV (8.8) = hEkin i = 0 R pF 2 5 2mN p dp 0 The total kinetic energy of the nucleus is

Ekin (N, Z) = N hEkin,n i + Z hEkin,p i =

¢ 3 ¡ 2 N pF,n + Zp2F,p 10 mN

(8.9)

8.2 Shell model

Hence, using eq.(8.3) and eq.(8.4), µ ¶2/3 5/3 3 ~2 9π N + Z 5/3 Ekin (N, Z) = 10 mN R02 4 A2/3

91

(8.10)

where the radii of the proton and the neutron potential well have again been taken as equal. Expanding this expression in the difference n − Z leads to ! µ ¶2/3 Ã 3 ~2 9π 5 (N − Z)2 A+ + ... (8.11) Ekin (N, Z) = 10 mN R02 4 9 A For fixed A the kinetic energy has a minimum for N = Z. The first term corresponds to the volume energy in the Weizsäcker mass formula (see section 2.4), the second one to the symmetry (or asymmetry) energy. The asymmetry energy grows with the neutron (or proton) surplus, thereby reducing the binding energy (this consideration, although instructive, neglected the change of the nuclear potential connected to a change of N on cost of Z. This additional correction turns out to be as important as the change in kinetic energy).

8.2 Shell model • Magic numbers: Nuclides with certain proton and/or neutron numbers are found to be exceptionally stable. These so-called magic numbers are 2, 8, 20, 28, 50, 82, 126

(8.12)

— Nuclei with magic proton or neutron number possess an unusually large number of stable or long lived nuclides. — A nucleus with a magic neutron (proton) number requires a lot of energy to separate a neutron (proton) from it. — A nucleus with one more neutron (proton) than a magic number is very easy to separate. — The following doubly magic nuclei are stable: 4 2 He2 ,

16 8 O8 ,

40 20 Ca20 ,

48 20 Ca28 ,

208 82 Pb126

(8.13)

• Eigenstates of the nuclear potential : — The wavefunctions of a spherically symmetric potential can be separated into a product of a radial part Rn (r) and an angular part Y m (θ, ϕ). The quantum number n denotes the number of nodes +1 of Rn (r) at r > 0, and l = s, p, d, f, g, h, i, ... is the quantum number of orbital angular momentum. The energy is independent of the "magnetic quantum number" m. The levels belonging to n are 2 (2 + 1) degenerate, the first factor 2 due to the two possible orientations of the nucleon spin. — The nuclear force is very short-ranged ⇒ the form of the potential corresponds to the density distribution of the nucleons within the nucleus. For very light nuclei (A . 7), the nucleon distribution has Gaussian form (corresponding to a harmonic oscillator potential), for heavier nuclei it is may be parameterised by a Fermi distribution (see section 6.6). The latter corresponds to the Woods-Saxon potential : Vcentral (r) =

−V0 1 + e(r−R)/a

(8.14)

92

8. Models of the nucleus

Table 8.1. Hierarchy of the energy eigenstates corresponding to the central Woods-Saxon potential. The first three magic numbers 2, 8 and 20 can be understood as nucleon numbers for filled shells, but the larger magic numbers cannot be obtained that way. N = 2 (n − 1) + l nl

0

1

2

2

3

3

4

4

...

1s

1p

2s

1d

1f

2p

1g

2d

...

degeneracy

2

6

2

10

14

6

18

10

...

States with E ≤ Enl

2

8

10

20

34

40

58

68

...

• Spin-orbit coupling: this is the essential ingredient into the nuclear shell model. In atomic physics, this effect causes only a small effect, the fine sturcture of atomic levels. In nuclear physics, it leads to considerable changes in the hierarchy of the energy levels (see fig. 8.2). Formally, spin-orbit coupling may be introduced as a term proportional to s: V (r) = Vcentral (r) + V s (r)

h si ~2

(8.15)

The expectation values are h si 1 = (j (j + 1) − ( + 1) − s (s + 1)) = ~2 2

½

/2 − ( + 1) /2

for j = + 1/2 for j = − 1/2

(8.16)

The spin-orbit energy thus leads to an energy splitting linearly rising with orbital angular momentum: ∆E

s

=

2 +1 · hV s (r)i 2

(8.17)

Experimentally, V s < 0, hence states with j = + 1/2 are always below states with j = − 1/2 (in contra-distinction to the effect of electron spin-orbit coupling in atomic physics). The nuclear levels are quoted as n j,

e.g. 1f7/2 and 1f5/2

(8.18)

• Nuclear magnetic moments in the shell model: — Magnetic moment operator : A

µnucl = µN ·

1X (g ~ i=1

with (see sec. 2.6.1) ½ 1 for protons g = 0 for neutrons

i

+ gs si )

and

(8.19)

gs =

½

+5.58 for protons −3.83 for neutrons

(8.20)

— We consider now the special situation with only a single nucleon in addition to closed shells. The angular momenta of all the nucleons filling the closed shells couple to zero, such that the expectation value of the magnetic moment is determined by µnucl =

µN hψ nucl |g ~

+ gs s| ψ nucl i

(8.21)

8.2 Shell model

93

Fig. 8.2. Single particle energy levels calculated in the shell model. The spin-orbit coupling lifts the degeneracy of the levels denoted by quantum numbers n . Magic numbers appear when the gaps between levels are particularly large [B. Povh et al., Particles and Nuclei, Springer, 2002].

— Wigner-Eckart theorem: The expectation value of any vector operator of a system is equal to the projection onto its total angular momentum. Here, the latter is the nuclear spin J (here J is defined by the j of the single nucleon): µnucl = gnucl µN

hJi ~

(8.22)

with gnucl =

hJMJ |g · J + gs s · J| JMJ i hJMJ |J2 | JMJ i

(8.23)

— Since 2 · j = j2 +

2

− s2

and

2s · j = j2 + s2 −

2

(8.24)

we obtain gnucl =

g {j (j + 1) + ( + 1) − s (s + 1)} + gs {j (j + 1) + s (s + 1) − ( + 1)} (8.25) 2j (j + 1)

94

8. Models of the nucleus

The magnetic moment of the nucleus is defined as the value measured when the nuclear spin is maximally aligned, i.e. |MJ | = J. Then, hJi = J~ such that µ ¶ µnucl gs − g 1 = gnucl J = g ± J for J = j = ± (8.26) µN 2 +1 2 An excellent agreement with this theoretical prediction is found for the nuclei quoted in the table, which are doubly magic up to a single nucleon or a single hole (a nulceon missing to fill a shell). Table 8.2. Comparison of magnetic moments in the shell model with experimental values. The nuclides shown correspond to single proton or neutron states (A = 17) or single holes in a proton or neutron shell (A = 15). The latter are denoted with a superscript −1. Nucleus

state

JP

µnu c l /µN (model)

µnu cl /µN (experiment)

15

N

1/2−

15

O

p-1p−1 1/2 n-1p−1 1/2

−0.264

−0.283

17

O

n-1d5/2

5/2+

p-1d5/2

+

17

F

1/2− 5/2

+0.638

+0.719

−1.913

−1.894

+4.722

+4.793