Interazioni nucleari forti e forza nucleare

V

Coulomb Term

This term represents the electrostatic repulsion between protons, and is directly

derived from the potential electrostatic energy shown in Equation 2.8 . Indeed, the

value of can be approximated using Equation (2.8), with = (Ze) and the

2 2

a q

C

radius as = 1/3

R r A

0

3 1 3 1 (Ze) 3e 3e 1) 1)

2 2 2

2 2 − −

q Z Z(Z Z(Z

= = =

= ≈ a

E C C

5 4πε 5 4πε 20πε 20πε

1/3 1/3 1/3 1/3

R r A r A r A A

0 0 0 0 0 0 0 (2.12)

in which become 1) to reflect that electrostatic repulsion could only exists

2 −

Z Z(Z

when nucleus contains more than one proton. Via this method, the Coulomb term

constant is found to be 3e 2

= = 0.714M (2.13)

a eV

C 20πε r

0 0

Asymmetry Term

This term is sometimes referred to as the ”Pauli term”, as it is derived from the

which states that no two can occupy the same

Pauli exclusion principle, fermions

20

quantum state.

The asymmetry term accounts for the charge-symmetric nature of the binding force

between nucleons. In fact, in the hypothetical absence of Coulomb repulsion, the

stable nuclei would have = for every value of Despite the fact that ≈

N Z A. N Z

for 20, as shown in Figure 2.4, beyond this value a larger number of neutrons is

A <

required to balance and reduce the repulsion between protons. Therefore, this term

drops to 0 when calculated on a symmetric nucleus.

Neutrons and protons are different types of particles and therefore occupy differ-

ent quantum states. Thus, in a nucleus that has a significantly higher number of

neutrons than protons, neutrons are forced by the exclusion principle to occupy a

higher energy levels, decreasing the total binding energy. More generally, any asym-

metry between and reflects itself into a higher overall energy of the nucleus. To

N Z

visualize the principle, we can picture two different pools of states, where neutrons

and protons occupy one pool each, as represented in Figure 2.8; each horizontal line

of the pool shows a different level of energy state, and each level can contains two

quantum states at most. It becomes clear that the difference consistently

−

N Z

increases the level of energy. In fact, the term shows a quadratic behavior in relation

to the difference. Furthermore, the denominator in the term shows that the effect

of the asymmetry becomes less relevant as increases.

A

Figure 2.8: The image represents two different distribution of equal number of nucle-

ons, which corresponds to two different energy level. The difference between and

N

is the numerator of the equation; note that if the difference is 0, the asymmetry

Z

term is lead to 0.[29] 21

Pairing Term

The pairing term takes its name from its role in capturing the spin-coupling effect,

and it depends on the parity of and Indeed, for each proton and each neutron

N Z.

, and, again according to the exclusion principle,

there are two possible spin states 1

± 2

for any type of particle and on any given energy level only a pair of spins

opposite

(+ ) is admissible. As a consequence, nuclei with all spins paired this way have

12 12

−

,

lower overall energy (i.e., higher binding energy), than nuclei with some unpaired

spin. Thus, fixing to zero the contribution to the binding energy of the configuration

with one unpaired spin, either among the protons or among the neutrons (i.e. the

case of odd we have a positive contribution to when is even and both

A), B A N

and are even, and a negative contribution when is even and both and are

Z A N Z

odd. Summarizing, the pairing term is :

δ(N, Z)

• = 0 for odd

δ(N, Z) A;

• = +δ for even and with both and even ;

δ(N, Z) A N Z

0

• = for even and with both and odd .

δ(N, Z) δ A N Z

0

Empirical observations have shown that is 1000 , and it decreases as

≃ A

δ keV

0

increases.

The dependence on the mass number is usually written as:

12 (2.14)

= =

kP ±

δ a A

P

0 1/2

A

where is determined from experimental data. Historically, it was considered

kP

similar to , but recent findings suggest that its value is better approximated to

34

−

. In the next image it is plot the pairing term against the mass number.

12

− 22

Figure 2.9: Magnitude of pairing term. [29]

2.3 Forces that show saturation

As discussed in Sections 2.2, the binding energy shows saturation, which means

that each nucleon can attract a finite number of neighbors, and this also explains

the short radius of the nuclear force. Such a feature suggests that the force among

the nucleons is not a fundamental interaction, but rather a residual interaction bind-

ing together globally neutral objects. Indeed, saturation and short-range are also

exhibited by forces between molecules, such as forces, which are

Van der Waals

residual weak electrostatic forces with a significantly short range potential; in fact,

the force rapidly decreases with distance, more so than the fundamental electrostatic

interaction.

Van der Waals forces originate from the residual electrostatic interaction between un-

charged molecules; indeed, they arise from fluctuations in the electric dipole moment

between uncharged molecules or atoms, which lead to an attractive force between

them. They are considered as a combination of different electrostatic interactions :

• between permanent dipoles

Keesom force,

• between permanent dipole which causes instantaneous induced

Debye force,

dipole

• between instantaneous induced dipole

London dispersion forces, 23

Figure 2.10: (a) Permanent Dipole-Permanent Dipole or Keesom forces. They exist

only between polar molecules, being stronger than London forces for molecules of

equivalent size; (b) Permanent Dipole-Induced Dipole or Debye force. It arises from

the distortion of the charge cloud induced by a polar molecule nearby, a non-

i.e.,

polar molecule will be temporarily polarized in the vicinity of a polar molecule, and

the induced and permanent dipoles will be mutually attracted; (c) Instantaneous

Dipole-Induced Dipole or London forces. They result from electrostatic attraction

between temporary dipoles and induced dipoles caused by movement of electrons;

these are attraction forces that operate between all molecules and among isolated

atoms in noble gases. The strength of the forces is related to the number of electrons

present and hence to the size of the molecule (or isolated atom); Interactions

(d)

between molecules–temporary and permanent dipoles. [6]

The first contribution, termed Keesom forces, originates from the electrostatic

interaction between permanent dipoles, quadrupoles, and permanent multipoles,

and it is also temperature dependent; as a matter of fact it decreases with increasing

temperature.

Permanent dipoles occur when the atoms contained in a molecule have different

electronegativity, which means that the molecule has separation of charge, with one

part having a slight negative charge , and the other part having a slight positive

δ−

charge Thus, the charge creates an attraction force between two opposite poles

δ+.

of different molecules, as shown in 2.10.

The second contribution is termed Debye forces or Unlike the

polarization forces.

Keesom force, the Debye force arises between a permanent dipole and an induced

dipole; in fact, even if the molecule does not possess a permanent dipole moment, it

is influenced by the electrical field of neighboring particle that possesses a permanent

24

dipole moment, which creates a momentary polarization of the non-polar molecule,

driven by the shift of the electron cloud, which results in an attraction between

the two particles. In contrast to the Keesom interactions, the polarization force is

not temperature dependent because the induced dipole is free to move around the

permanent pole.

The last and most common contribution arises from the London force, or also

termed This is due to the random and temporary fluctuations of

dispersion force.

the electrons within molecules and atoms, differently from the Keesom and Debye

interactions that are present only on molecules. Every atom and molecule, includ-

ing non-polar molecules, have instantaneous non-zero dipole moment, because the

movement of the electrons in their orbitals temporarily produces a higher density

zone, which is richer in electrons and thus possesses a charge and a lower density

δ−,

zone, which contains fewer electrons and therefore has a charge This instanta-

δ+.

neous polarization, which could be driven either from a permanent polar molecule or

by repulsion of negatively charged cloud of electrons on non-polar molecules, leads

to the creation of attraction forces between the opposite poles of different particles.

It is possible to describe the effect of van der Waals forces by means of the Lennard-

potential, which is an empirical potential used to describe intermolecular

Jones

interactions, #

" 12 6

σ

σ (2.15)

(r) = 4ε −

·

V r r

in which is the distance between two interacting particles, is the depth of the

r ε

potential wall and is the distance at which the particle-particle potential energy

σ

is zero. The first term ( ) represents the repulsion potential due to the overlap

σr 12

V

of electrons orbitals, while the second term ( ) represents the attraction potential

σr 6

of Van der Waals forces. The potential clearly has a minimum for = = 2

1/6

r r σ,

min

where = = as shown in Fig. 2.11.

−ε,

V V

min 25

Figure 2.11: Graph of the Lennard-Jones potential function: Intermolecular poten-

tial energy as a function of the distance of a pair of particles. The potential

V

L

minimum is at = = 2 . [13]

1/6

r rmin σ

As expected, due to its nature, the Lennard-Jones potential shows a faster decay

with the distance compared with the well-known behaviour of the electrostatic

−1

r

interaction among charged objects. Indeed, the Van der Waals interactions are

among the weakest chemical forces. In fact, taking as an example the interactions

between two different atoms of hydrogen in different molecules of , the inter-

H

2

action energy is equal to 0.6 meV, far from the value of the binding energy of the

hydrogen, corresponding to 1.3 shown in figure 2.1.

≃ M eV

Thus, in the sense above described, Van der Waals forces are a force,

residual

a sort of remnant of the electromagnetic interactions, binding together electrically

neutr