7 Nuclear Force and its Properties -3

Sanjay Kumar Chamoli

Learning Outcomes

From this module students may get to know about the following:

The role of scattering experiments in understanding nuclear interaction .
The importance of low energy neutron-proton scattering.
The basic properties of nuclear force in general.

1.2. Nucleon-nucleon interaction

One of the important limitations with deuteron is that it doesn’t have an excited state and therefore nuclear interaction can only be studies in the ground state (i.e. l = 0 state) and not in the excited state. So the deuteron case yields limited information about nucleon-nucleon interaction. To study the nuclear interaction in detail, the nucleon-nucleon scattering experiments are needed.

The scattering experiments in general can be performed in the following two ways ;

(a) An incident beam of nucleons is scattered from a target of nucleons.

In this method, the scattering of a single nucleon will include the complicated effects of multiple encounters and is very difficult to extract the properties of the interaction between individual nucleons.

(b) An incident beam of nucleons is scattered from a target of hydrogen.

In this method, the incident neutron is scattered by individual protons mainly. The multiple encounters are greatly reduced by large spatial separations between nucleons, so properties of nucleon-nucleon interactions can be deduced without complications.

In general, the nucleon-nucleon scattering resembles with optical diffraction.

Intensity of scattered nucleons varies with distance like 1/r2.
Intensity of scattered nucleons depend on the angular coordinates and . A detector placed at a distant point records both incident & scattered beam.
The n-p scattering is the simplest one as complications due to Coulomb forces are not present.
In (n-p) scattering neutron proton system is analyzed in the state of positive energy, i.e., in a situation when they are free.
Simplest target is hydrogen gas but nylon sheet & paraffin are also used.
The target protons are not free but are bound in molecules. The molecular binding energy is so small about 1 eV, therefore, for the impinging neutrons of energy greater than 1 eV, protons can be treated as free.
The presence of electrons also do not affect the process because they are too light to cause any appreciable trouble to incoming neutrons
In the interaction process, some neutrons get captured and form deuteron ( -rays are emitted), but the majority undergo elastic scattering.

1.1.2 scattering at low energy:

To solve n-p scattering problem using quantum mechanics we assume the nuclear interaction by a square-well potential, as assumed in case of deuteron. Contrary to the case of deuteron, here the incident particles are free (E > 0)

Fig. 1: The square-well potential for E>0.

Consider an incident neutron strikes the target on its surface. The impact parameter is b ≈ 1 fm. If the incident particle has velocity v, its angular momentum relative to the target will be mvR.

We have,

mvR = lh

Where l is the angular momentum.

If mvR << lh, then only l =0 interactions are likely to occur. The kinetic energy is given by

Where m is the reduced mass and it is given by

The effect of a scattering potential is to shift the phase of the scattered wave at points beyond the scattering regions, where the wave function is that of a free particle.

Figure 2 shows the variation of phase shift of scattered wave of a free particle at points beyond the scattering regions with different nature of scattering potential. The phase sift is positive for attractive potential and it is negative for repulsive potential.

Fig. 2: Phase shift of scattered wave of a free particle at points beyond the scattering regions for the different nature of scattering potential.

General theory of scattering gives differential cross section:

So, l = 0 phase shift is directly related to probability for scattering. δ0 can be evaluated from simple square-well model & compared with experimental cross section..

To compare with low-energy n-p scattering data, a realistic estimate of cross section is needed. This can be obtained by putting values of all related terms:

Example: Let, the incident energy, E ≦ 10 keV.

Also, as V0 = 35 MeV from the analysis of the deuteron bound state problem.

Figure 3 shows the experimental data of neutron-proton scattering cross sections. The experimental value of neutron-proton scattering cross sections is constant at low energy and decreases with E at large energy as predicted by theory. But the low-energy cross section, ~20 b, is not in agreement with our calculated value of 4.6 b. Thus the theoretical estimate is not equal to the experimental value.

Fig. 3: The Neutron-proton scattering cross section at low energy (Phys. Rev. C3, 1886 (1970).

Implications of scattering:

Experimental cross section of n-p scattering is constant at low energy, and decreases with E at large energy as predicted theoretically.
Scattering cross sections depend strongly on the energy of incident neutron. For E <10 MeV, scattering is essentially due to neutrons with zero angular momentum (l=0) & hence in centre of mass system, angular distribution of scattered neutrons is isotropic.
Observed cross section (~ 20 b) is well above theoretical estimates (~ 5 b).
For the solution to this discrepancy, we must study the relative spins of the incident and scattered nucleons.

Summary :

The Deuteron case is not sufficient to full understand the nuclear interaction. Though it is a simple system but it has got various limitations. So for the wider understanding of the nuclear interaction the n-p scattering experiments are needed. At low energy, the huge disagreement between the measured and the theoretical cross sections (calculated using parameters obtained from the study of deuteron) strongly suggest the presence of spin dependent component in the nuclear interaction. The scattering cross section from singlet state is different from the scattering cross section from the triplet state.

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References:

Introduction to Nuclear Physics – by Keneth S Krane.
Introductory Nuclear Physics – by Samuel S M Wong.
Nuclear Physics – by R R Roy & B P Nigam.
Elementary Nuclear Theory by Hans A. Bethe and Phillip Morrison.
Introduction to Nuclear Physics, 2nd Edition, W.N.Cottingham & D.A. Greenwood.
Concept of Nuclear Physics by B L Cohen, McGraw Hill.
Nuclear Physics ; an Introduction by S.B. Patel.
The Origin of the Concept of Nuclear Force by L.M. Brown and Rechenberg.
Theoretical Nuclear Physics by John M. Blatt and Victor F. Weisskopf.
Experimental techniques in Nuclear Physics by Dorin N. Poenaru & Walter Greiner
Exotic Nuclear Excitation by S.C. Pancholi
Nuclear spectroscopy Part B, by Fay Ajzenberg- Selove
Theory and Problems of modern Physics (Schaum’s outline Series)
Basic Ideas & Concepts in Nuclear Physics – by K Heyde
The “Particles of Modern Physics” by J. D. Stranathan, Philadephia: Blakiston.
5. Nuclear Physics by Irving Kaplan, Narosa Publishing House.

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Did you know ?

The Deuteron is a simple system to understand the nature of nuclear force BUT it is not a perfect system.
As the deuteron doesn’t have any excited state so very limited properties of nuclear interaction can be studies.
For detailed study, nucleon-nucleon (neutron-proton, proton-proton and neutron –neutron) scattering experiments need to be performed.
In the nucleon-nucleon scattering the interactions does not change the incident particles, i.e., incoming and outgoing particles are the same. The change is in the path of incoming nucleons, i.e., they are deviated from their original paths
In the scattering process the outgoing particles may have same energy as that of incident particles (elastic scattering) or may have the changed energy value (inelastic scattering).
In the neutron-proton (n-p) scattering, the neutron proton system is analyzed in the state of positive energy, i.e., in a situation when they are free.
In the n-p scattering, the presence of electrons do not affect the process because they are too light to cause any appreciable trouble to incoming neutrons.
In the n-p scattering when the neutrons impinge on protons, some of them are captured to form deuteron and the balance of energy is radiated in the form of rays; but the great majority of neutrons undergo elastic scattering.
In n-p scattering process, the interactions between two nucleons is so strong that the neutrons changed their velocities in magnitude as well as in direction.
The proton – proton (p-p) scattering is due to the presence of coulomb repulsion between two protons.
The Coulomb repulsion increases the change of direction the account of which is made in estimating the nuclear forces.
The neutron neutron (n-n) scattering is not practically possible because of the non-availability of neutron target (because neutron decays into proton in a few minutes). However, their are evidences to support that if n-n forces are similar to p-p forces, a bound state for two neutrons cannot exist.

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