Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics

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14.2 General Properties of Nuclei 423

0 123456 78910

Nuclear radius (fm)

Density x 10

(kg m

-3

)

Fig. 14.4 Distribution of matter density as a function of the nuclear radius for some nuclei. The

curves represent the parameterization of the experimental data with the Wood–Saxon function

(14.8)

The differential cross-section for the elastic scattering of a neutron on a

nucleus (n-nucleus) shows, as a function of the scattering angle, peaks and valleys

typical of a diffraction pattern. Indeed, the interaction can be interpreted in a

quantum-mechanical treatment (optical model of the nucleus) which describes the

perturbation of the nucleus on the incident neutron in terms of a potential well

and an absorption term. The problem may be analyzed by solving the Schr

odinger

equation with a potential well which is described by a complex function. This

corresponds to the optical analogue of a semitransparent sphere, which diffuses and

absorbs light. The results for the nucleus radius can be summarized in a formula

similar to (14.8) with the following parameters:

R D 1:2A

1=3

fm;tD 0:75 fm:

There is a surprising agreement between the electromagnetic form of the nucleus

and the shape of nuclear potential, which depends on the distribution of nuclear

material in the nucleus and on the nuclear force range. Figure 14.4 shows the

distribution of matter density .r/ for different nuclei.

All measurement methods, even different from those mentioned, give consistent

results. From these measurements, the followings conclusions can be drawn:

• The nuclear matter distribution is approximately equal to the electric charge

distribution.

• To a good approximation, the distributions have spherical symmetry.

• Charge and matter distributions are approximately uniform in a sphere of

radius R.

A potential well is the region surrounding a local minimum of the potential energy.

424 14 Fundamental Aspects of Nucleon Interactions

• The mean square radius distributions are proportional to R D R

1=3

, with R

1:2  10

13

cm.

•ThevalueofR

depends only slightly on the measurement method.

•Thenuclear volume is proportional to the number of nucleons, R

/ A.

• The density of nuclear matter is high: the mass of the nucleon is equal to

1:7 10

27

kg, and the nuclear density is 

D 2:410

kg m

3

, i.e., 2:410

times that of water.

14.2.4 Electromagnetic Properties of the Nuclei

The electromagnetic properties of the nuclei are described by the charge density

.rIt/ and the current density j.rIt/, in the region with radius R. The nuclei

are subject to the action of electric and magnetic ﬁelds produced by charges and

currents of the atomic electrons as well as to the action of external ﬁelds. The

charge and current densities have the nucleus spin direction (I) as symmetry axis.

The electromagnetic ﬁeld produced by atomic electrons has the direction of the total

angular momentum as its symmetry axis.

The proton and the neutron have spin s D 1=2 and magnetic moment 

s and



s respectively, as described in Sect. 7.14.4. It is useful to deﬁne the gyromagnetic

factor g  =

as the ratio between the intensity of the magnetic moment and

the nuclear magneton 

jej„

D 3:15  10

8

eV/T. The magnetic moment

of the nucleus is induced by the magnetic moments of individual nucleons and

by the orbital motion of protons. Also in this case, it is parallel to the nuclear

spin I. The gyromagnetic factor of the nucleus, g

, can be positive or negative,

that is,



D g



I: (14.9)

Different methods are used to measure the magnetic moment of the nuclei. They

are based on the interaction of the nuclear magnetic dipole moment with the atomic

magnetic ﬁelds or with artiﬁcial magnetic ﬁelds.

14.3 Nuclear Models

Unlike the atomic model, there is no a single model able to explain all nuclear

properties. The reasons are mainly due to the fact that a massive central body

representing the center of attraction is not present; in addition, the analytical form

of the potential is not known. In this section, we shall describe some of the different,

but complementary, nuclear models.

14.3 Nuclear Models 425

BE/A

dn / dE

Fig. 14.5 Potential well model for protons and neutrons in a nucleus and evolution of the nucleon

energy as a function of the state density dn/dE

14.3.1 Fermi Gas Model

The Fermi model is a statistical model based on the following assumptions:

1. The nucleons are spin 1/2 fermions (in a nucleus, Z protons and N D A  Z

neutrons) which move freely, like a gas, within the nucleus.

2. A single nucleon is subject to the collective action of all other nucleons. This

action is represented by a spherically symmetric potential well U.r/ in a region

of size R D R

1=3

3. The nucleons gas is degenerate, i.e., the kinetic energy is much larger than the

thermal energy kT . The nucleons are in the state of lowest accessible energy for

the Pauli exclusion principle.

Based on these simple assumptions, the Fermi model provides information on the

state density (Fig. 14.5) and on the nucleon kinetic energy for nuclei with A large

enough .A > 12/ to use statistical criteria. The number of states for a spin s D 1=2

fermion is calculated from the number density:

dn D .2s C1/

dVd˝p

Integrating over the solid angle (

d˝ D 4)andvolume(V D 4=3R

A), and

recalling that h D 2„ and .2s C 1/ D 2, one has

dn D

2  4V

8

„

dp D

3



„



dp: (14.10)

The maximum of the momentum is called the Fermi momentum, and it is determined

by requiring that the integral of (14.10) matches the number of protons (or neutrons)

in the volume V :

9



„



D ZI

9



„



D .AZ/; (14.11)

426 14 Fundamental Aspects of Nucleon Interactions

and

c D



9



1=3

„c





1=3

I p

c D



9



1=3

„c



2.A  Z/



1=3

: (14.12)

The numerical values of p

and p

can be derived assuming that 2Z=A ' 2

.A  Z/=A ' 1 and R

D 1:25 fm. One ﬁnds

D p

' 240 MeV/c: (14.13)

The corresponding kinetic energy is called the Fermi energy = E

D p

=2M

30 MeV (a comparable value is obtained for the neutron). The nonrelativistic

approximation used here is sufﬁciently accurate. For heavy nuclei, the Fermi energy

of neutrons is slightly larger than that of protons since 2.AZ/=A >1. For example,

for uranium .Z D 92; A D 238/, the proton and neutron Fermi energies are

respectively E

D 28 MeV and E

D 32 MeV. The depth of the potential well is

the sum of the Fermi energy and the binding energy per nucleon, that is,

U D E

C BE=AI BE=A ' 8 MeV/nucleon ! U D .35  40/ MeV:

For protons, the potential well is deformed by the electrostatic energy that produces

a potential barrier that has a dependence  1=r for r>R(Fig. 14.5).

The average kinetic energy per nucleon

can be calculated (adding and

mediating the contributions of p and n) from (14.10)

=2m/dn

' 20 



1 C



A  2Z





MeV: (14.14)

slightly depends on the asymmetry term  D Œ.A2Z/=A

.When2Z ' A,as

for light nuclei,  D 0. For heavy nuclei, the average kinetic energy has a minimum

for Z D A=2; it increases when an excess of neutrons is present. The correction

factor  contributes only by 3% to the average kinetic energy per nucleon for the

case of

238

U .

14.3.2 Nuclear Drop Model

The nuclear drop model is a collective model of the nucleus which describes the

nuclear binding energy with a few parameters. The model uses analogies with a

liquid droplet and is based on the following assumptions:

• The interaction energy between two nucleons is independent on the nucleon type.

• The interaction is attractive at a short-range, R

int

(as in the case of a liquid droplet

in which the molecules have dipole–dipole interactions).

14.3 Nuclear Models 427

• The interaction is repulsive at distances r  R

int

• The binding energy of the nucleus is proportional to the number of nucleons.

From these considerations, a formula for the binding energy may be obtained. It

takes into account a main volume term and some correction factors.

Volume term. Each nucleon is strongly related to only the few nearby nucleons.

The binding energy of the nucleus is not proportional to the sum of the interaction

energy between all nucleon pairs (which is proportional to A.A  1/ ' A

), but to

the sum on only nearby nucleon pairs. Each pair is contained within an interaction

volume, V

int

, smaller than the total volume of the nucleus. This fact is explained

by the short-range hypothesis of nuclear force. Under these conditions, the binding

energy is / A, BE DCa

Surface term. The binding energy is reduced by a surface effect since the nucleons

on the nuclear surface have fewer neighboring nuclei with respect to the inner

ones and are therefore less bound. The surface of the nuclear volume is propor-

tional to A

2=3

; thus, the corresponding negative correction factor can be written

as a

2=3

Coulomb repulsion. The Coulomb repulsion between protons contributes to

reducing the binding energy. Measurements of electromagnetic form factors show

that the nuclei have approximately a uniform charge distribution. The electrostatic

energy of a sphere of radius R with uniform charge density is proportional to the

charge contained in the sphere, and inversely proportional to R  A

1=3

. Thus, the

correction term for the binding energy associated with the Coulomb repulsion is

a

1=3

Term due to the exclusion principle (or asymmetry term). The binding energy

is further reduced due to the contribution of the kinetic energy of nucleons: the

higher the kinetic energy, the lower the binding energy. The estimate based on the

Fermi gas model can be used. It takes into account the effect of fermion statistics

and the Pauli exclusion principle that favors nuclear conﬁgurations with equal

numbers of protons and neutrons. The average kinetic energy per nucleon is given

by (14.14); the total kinetic energy of the nucleus with A nucleons is a constant,

plus a correction term proportional to Œ.A 2Z/

=A.ForA=2 D Z, the correction

term is equal to zero; it becomes increasingly important as we move away from the

symmetry condition Z D N . The binding energy contribution due to this factor is

a

Œ.A  2Z/

=A.

Term due to conﬁgurations. It is experimentally observed that there is a sys-

tematic difference between conﬁgurations of nuclei with odd or even number of

protons and neutrons (Table 14.1). It is necessary to introduce a correction term in

the formula, like a

1=2

, to take this effect into account. The value and sign of a

are such that

428 14 Fundamental Aspects of Nucleon Interactions

AZND A  Z ˙a

(MeV)

(More stable) Even Even Even C12:6

(Intermediate) Odd 0

(Less stable) Even Odd Odd 12:6

Volume term

Surface term

Coulomb repulsion

Asymmetry term

Mass number A

100

200 250150

Binding energy per nucleon (MeV)

Fig. 14.6 Contribution of various terms of the Weizsacker formula for the binding energy per

nucleon as a function of A. The surface, Coulomb and symmetry terms are subtracted to the volume

term. The ﬁgure does not consider the conﬁguration term

Table 14.2 Values [14w3] of the constants in the

Weizsacker formula. a

takes into account the fact

that the nucleus is even-even, even-odd or odd-odd

(MeV) a

15.74 17.61 0.71 23.42 ˙12.6, 0

The ﬁnal result of all considered terms gives the Weizsacker formula of the

binding energy as a function of A and Z:

BE D a

A  a

2=3

 a

1=3

 a

.A  2Z/

1=2

. (14.15)

The value of the ﬁve parameters a

:::; a

are obtained from a ﬁt to the experimental

data (see Fig. 14.6). The parameters have the dimension of energy and are measured

in MeV. The values are given in Table 14.2.

14.3 Nuclear Models 429

From the binding energies, the masses of the nuclei can be obtained using

(14.5) (Bethe and Weizsacker formula). This formula gives the nucleus masses

as a function of A; Z and of the parameters a

:::; a

. The values calculated with

(14.15) show relatively large deviations from experimental data for small A.

Fewer discrepancies are observed for large A because the nuclear binding is

particularly strong for certain values of Z and N . These values are denoted as

magic numbers: since neither the Fermi gas model nor the nuclear drop model can

explain them, it is necessary to introduce an additional nuclear model, the shell

model.

14.3.3 Shell Model

The liquid drop model gives a fairly good description of the nuclear binding energy.

It also offers a qualitative explanation of nuclear ﬁssion, as discussed in Sect. 14.9.

The Fermi gas model is required to justify some numerical values such as the

asymmetry term Œ.A 2Z/

=A in (14.15). The shell model potential is chosen as a

simple three-dimensional square well, different for protons and neutrons. It explains

further experimental facts such as the existence of particularly stable nuclei. In this

model, the nucleons can move freely within the nucleus on quantum orbits. This is

consistent with the idea that they are subject to a global effective potential created

by the sum of the contributions of other nucleons.

Once again, the electromagnetic case can be used as a prototype. Indeed,

the atomic model (based on a Coulomb potential with radial symmetry, angu-

lar momentum quantization and the Pauli principle) successfully reproduces the

phenomenology of atoms: energy levels and valence. In addition, some elements

(noble gases, as helium (Z D2), neon (Z D10), argon (Z D18), krypton (Z D36),

:::) are characterized by a total angular momentum J D0, high electron binding

energies and low chemical reactivity.

Particularly stable nuclei are observed when the number of protons, Z,orthe

number of neutrons, N D A  Z is equal to 2; 8; 20; 28; 50; 82; 126 (magic

numbers). Nuclei with magic numbers have special features, for example, (1) many

isobar nuclei exist; (2) nuclear spin I D 0, and null magnetic dipole and electric

quadrupole moments; (3) high nuclear binding energy; (4) small nuclear cross-

section. The last two properties are more evident in doubly magic nuclei such as

He,

C a:::

208

Pb.

The goal of the shell model is to reproduce the magic numbers by solving

the Schr

odinger equation with a particular potential. The problem has a number

of difﬁculties because the shape of the nuclear potential is not exactly known.

Moreover, if a potential with radial symmetry is assumed, the center of sym-

metry is not well deﬁned since all the nucleons are the source of the nuclear

ﬁeld. Finally, it is not obvious how to extend to the nucleons in the nucleus

430 14 Fundamental Aspects of Nucleon Interactions

the concept of orbital. The Pauli principle and the success of the Fermi gas

model can help: if the nucleons gas is strongly degenerate, each nucleon is

in a given quantum state and does not interact with another nucleon but with

an exchange mechanism. This leads to an equation of motion for the single

nucleon, independent of what happens to the other nucleons (independent particle

model).

The eigenstates for a particle of mass m in a spherically symmetric poten-

tial U.r/ are .r;;/ D R

.r/Y

.; / where Y

.; / are the spheri-

cal harmonic functions. They are obtained by solving the radial Schr

odinger

equation,





„

C U.r/ C

„

`.` C 1/

2mr



u.r/ D Eu.r/; (14.16)

with u.r/ D u

.r/ D rR

.r/. One possible choice for U.r/ is the Woods–Saxon

potential:

.r/ D

1 C e

.rR/=t

; (14.17)

which reproduces the matter distribution in the nucleus (14.8). This choice allows

one to numerically solve the equation; it determines a sequence of particularly stable

states with the sequence 2, 8, 20, 40, 70, 112:::.

Another important progress was made by Mary Meyer and Hans Jensen (Nobel

laureate in 1963) by introducing in the potential a term for the spin-orbit interaction,

that is,

D U

.r/ C U

` s: (14.18)

This term is suggested by the observation that the interaction between nucleons has

a strong dependence on the spin state. Unlike the atomic interaction, the spin-orbit

term in nuclei has no origin from the magnetic dipole moment interaction with the

magnetic ﬁeld produced by the motion of charges. Such a term would move the

energy levels by smaller quantities than those observed.

Figure 14.7 shows the energy levels obtained by solving (14.16) with the two

potentials mentioned above, in addition to the simplest case of a harmonic oscillator

potential type.

The shell model which assumes independent particles (Independent Particle

Shell Model), besides the magic numbers, can make predictions on the spin, parity,

magnetic dipole and electric quadrupole moments of the nuclei. Given the simplicity

of the model, these predictions are not very accurate, but provide a useful base for

further improvements.

14.4 Properties of Nucleon-Nucleon Interaction 431

1d 2s

1f 2p

1g 2d 3s

1h 2f 3p

1i 2g 3d 4s

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

2p3/2

1f5/2

2p1/2

1g9/2

1g7/2

2d5/2

2d3/2

3s1/2

1h11/2

1h9/2

2f7/2

2f5/2

3p3/2

3p1/2

1i13/2

100

106

110

112

126

112

Harmonic

oscillator

Woods-Saxon

potential

Spin-orbit

coupling

168

2 (2l + 1)

2j + 1

Fig. 14.7 Energy levels and magic number sequences obtained by solving (14.16)usinga

harmonic oscillator potential, the Woods–Saxon potential of (14.17) and the potential with a spin-

orbit coupling term (14.18). The last one reproduces the observed sequence of magic numbers

14.4 Properties of Nucleon-Nucleon Interaction

In the previous section, we described some approximations which explain most of

the features and physical properties of nuclei. Similarly, there are phenomenological

formulations of the nucleon-nucleon potential; however, they do not allow an ana-

lytical solution of the problem. The interactions between nucleons depend on many

factors: distance between nucleons, relative speed, spin, angular momentum:::.

A single simple formula (analogous to the Coulomb potential) from which the

characteristics of the interactions between nucleons can be derived does not

exist. The reason is due to the fact that the strong force acts (see Sect. 11.9)

via gluon exchange between quarks. The residual interaction between nucleons

432 14 Fundamental Aspects of Nucleon Interactions

is analogous to the Van der Waals electromagnetic force between two neutral

atoms or molecules. Indeed, in a ﬁrst approximation, a nucleon is neutral (without

color) for the strong interaction as an atom is neutral for the electromagnetic

interaction.

Some properties of nucleon-nucleon interactions can be obtained from the

analysis of the nucleus binding energy, from the characteristics of the deuteron

and from the nucleon-nucleon elastic scattering at low energy. The deuteron is the

simplest nuclear state and it represents the analogous of the hydrogen atom for the

electromagnetic interaction for the nuclear interaction. The deuteron binding energy

(BE D 2:225 MeV) is however too low to form excited states. The main properties

of interactions between nucleon pairs are:

1. The interaction is attractive, short-range and can be described by a central

potential U.r/. The shape of the potential is not known a priori; different choices,

for example, the square well, the Woods–Saxon potential or the harmonic

oscillator potential, lead to similar conclusions for some observables using as

input parameters: radius of the potential R<2fm, depth of the potential

' 40 MeV;

2. The interaction is symmetric and independent with respect to the electric charge.

The study of binding energies and of energy levels of isobaric mirror nuclei

show that the proton–proton, neutron–neutron and neutron–proton interactions

are similar. The same conclusion is reached by comparing the p  p, n  n,

p n elastic scattering at low energy. This property involves the conservation of

isospin in nuclear interaction;

3. The interaction is invariant under parity and time reversal transformations. As a

result (Table 6.3), the nuclei have neither electric dipole moment, nor a magnetic

or quadrupole moment;

4. The interaction depends on the spin. The nucleon-nucleon state with spin I D 0

(singlet) has different properties from that with spin I D 1 (triplet), suggesting a

spin-dependent interaction:

.r/ D U

.r/s

 s

 U

.r/s

 s

(14.19)

which is attractive in the triplet .t/ and repulsive in the singlet state .s/;

5. The interaction also has a noncentral potential term. To account for the magnetic

dipole moment and the electric quadrupole moment of the deuteron, it is assumed

that it is a mixed state, i.e., a superposition of states of even angular momentum

L. However, a potential with radial symmetry does not produce stationary

degenerate eigenstates with different values of L. Therefore, the nucleon-nucleon

interaction must also include a nonradial term (tensor potential), U

.r/.Since

the only deﬁned direction is that of the spin, the tensor potential can be built with

combinations which depend on spin and distance, for example, .s  r/ or .s r/,

which are invariant under parity and time reversal transformations;

6. The interaction is repulsive at small distances, r R. Nuclei have energy

and volume proportional to the number of nucleons: the nucleus cannot be