Computational coding on nuclear models
Abstract
INTRODUCTION
Atom and its Nucleus
An Atom is the basic building block of matter and it consists of sub atomic particles electrons (ve), protons (+ve) and neutrons (no charge). Protons and neutrons are collectively bound by nuclear forces to form ‘nucleus’ and this nucleus is located at the center of the atom. Electrons revolve around the nucleus. Nuclear forces are stronger than the coulomb force which makes protons and neutrons stay together. Atomic general notation is given as
A: mass number.
Z: atomic number or proton number.
Mass number is given as A= Z+N.
N: nutron number.
Nuclear Chart
Each point plotted on the graph thus represents the nuclide of chemical element. The elements which have the same atomic number and different neutron number are called “Isotopes”. The elements which have same neutron number but a different proton number are called “Isotones”. The elements which have same mass number and different proton and neutron number are called “Isobars”. Nuclides with the same difference between neutrons and protons (N−Z) are called “Isodiaphers”. “Nuclear drip line” is the boundary delimiting the zone beyond which atomic nuclei decay by the emission of proton or neutron, nuclei decay by emitting protons is called “Proton drip line” or nuclei decay by emitting neutrons is called “neutron drip line”. From the graph it is clear that for a stable nucleus, N>Z.
NUCLEAR MODELS
Nuclear models focus on nucleons in the nucleus that which configuration it follows while filling in orbitals, effect of charge on nucleus means quadrapole and magnetic momentum and configuration in high spin nucleus. There are several nuclear models are available some of examing below.
Liqid Drop Model
According to this model atomic nucleus behaves like the molecules in a drop of liquid. In the ground state nucleus is spherical, if the specific kinetic energy is added, this spherical nucleus may be distorted into a dumbbell shape and then may be split into two fragments.
Formula for binding energy is given as [1]
Volume term
The first term is the volume term a_{v} that describes how the binding energy is mostly proportional to mass number A. The constant of proportionality is a fitting parameter that is found experimentally to be a_{v} = 15.5MeV [1].
Surface term
The surface term, a_{ s} also based on the strong force, is a correction to the volume term. We explained the volume term as arising from the fact that each nucleon interacts with a constant number of nucleons, independent of A. While this is valid for nucleons deep within the nucleus, those nucleons on the surface of the nucleus have fewer nearest neighbors. This term is similar to surface forces that arise for example in droplets of liquids, a mechanism that creates surface tension in liquids.
Coulomb term
The third term a c derives from the Coulomb interaction among protons and of course is proportional to Z. This term is subtracted from the volume term since the Coulomb repulsion makes a nucleus containing many protons less favorable. Constant a_{ c} can be estimated as 0.691 MeV [1].
Symmetry term
The Coulomb term seems to indicated that it would be favorable to have less protons in a nucleus and more neutrons. However, this is not the case and we have to invoke something beyond the liquiddrop model in order to explain the fact that we have roughly the same number of neutrons and protons in stable nuclei. The shape of the symmetry term is (A−2Z). It can be more easily understood by considering the fact that this term A goes to zero for A = 2Z and its effect is smaller for larger A. From above it is clear that for smaller nuclei the symmetry effect is more important. The coefficient is a_{ sym} = 23MeV[1].
Pairing term
The final term is linked to the physical evidence that likenucleons tend to pair off. Then it means that the binding energy is greater (δ > 0) if we have an eveneven nucleus, where all the neutrons and all the protons are pairedoff. If we have a nucleus with both an odd number of neutrons and of protons, it is thus favorable to convert one of the protons into a neutrons or viceversa. Thus, with all other factor constant, we have to subtract (δ < 0) a term from the binding energy for oddodd configurations. Finally, for evenodd configurations we do not expect any influence from this pairing energy (δ = 0). The pairing term is then with a p ≈ 34MeV[1].
Calculating binding energy of carbon 12
Shell Model
According to this model the motion of each nucleon is given by the average attractive force of all the other nucleons. As nucleons are added to the nucleus, they drop into the lowest energy shell permitted by the Pauli principle which requires that each nucleon have a unique set of quantum numbers to describe its motion. When all the protons or neutrons in a nucleus are in filled shells, the number of neutrons or number ofprotons is called “Magic numbers”. Some of the Magic numbers are 2,8,20,28,50,82,126. Like ^{40}Ca . elements have both proton and neutron Magic numbers are called doubly Magic numbers.
The nuclear spin represents the total angular momentum of the nucleus. The nucleus is although composed of neutrons and protons but it acts as if it is a single entity which has intrinsic angular momentum (I=l+1/2 or l1/2). Parity is the behaviour of wave function under the reflection of space coordinates through the origin determine the parity of the system.
Spin and parity of 26 Fe with N=29.
Nuclear magnetic momentum
Nuclear magnetic moment is the magnetic moment of an atomic nucleus and arises from the spin of the protons and neutrons. The spinorbit interaction, which causes the observed fine structure of spectral lines, comes about because of the electromagnetic interaction of the electron's magnetic moment with the magnetic field generated by its motion about the nucleus. The effects are typically very small. So it has little effect on nucleus. Shell model gives a reasonable result with observed nuclear properties in the case of magnetic dipole moments[2]. The magnetic moment is computed from the mean value of the magnetic moment operator in the state with maximum z projection of angular momentum. Magnetic moments given as [2]
Magnetic moment of 3 Li with N=4.
Electrical quadrapole moment(Q)
The nuclear electric quadrapole moment is a parameter which describes the effective shape of the ellipsoid of nuclear charge distribution. The electric quadrapole moment is calculated by evaluating the electric quadrapole operator given as (3z^{2} r^{2})[2], in the state in which the total angular momentum of the odd particle has its maximum projection along the z axis. If Q=0 nucleus is sperical, Q>0 nucleus is prolate or Q<0 nucleus is oblet. The single particle quadrupole moment of an odd proton in a shellmodel state j given as [2]
Quadrapole moment of 9 F with N=10.
Shell model with appearing magic numbers
Nilsson Model
The nuclear shell model is generally used to describe spherical nuclei but it fails to explain the properties of deformed nuclei (nuclei with N and Z far from the closed shells) and rapidly rotating nuclei. To understand the basic properties of nucleons moving in a deformed nucleus, a model was introduced by S.G. Nilsson in 1955, named as the Nilsson model. First experimental examples were found of rotational bands in nuclei, with their energy levels following the same J (J+1) pattern of energies as in rotating molecules. Quantum mechanically, it is impossible to have a collective rotation of a sphere, so this implied that the shape of these nuclei was non spherical. The Nilsson model the modified oscillator potential is used to describe the motion of the nucleons. For an axial symmetric deformation, the Nilsson Hamiltonian takes the form[3]
Here ω_{0 }is the harmonic oscillator frequency in the spherical limit while ω _{z }and ω_{⊥} are the frequencies of the anisotropic oscillator in the zdirection and the directions perpendicular to the zdirection, respectively. The first term provides the kinetic energy of the nucleons and the second term defines the quadrupole deformation dependence describing spheroidal shapes. The third term is the spinorbit coupling and was introduced to reproduce the empirical shell gaps for different N and Z values. The last term is introduced to simulate the surface diffuseness depth, which leads to a proper singleparticle ordering by lowering the energies of the large l orbitals within an Nshell.
Cranking model
Because the potential is not spherically symmetric, the singleparticle states are not states of good angular momentum J. However, a Lagrange multiplier, known as a "cranking" term, can be added to the Hamiltonian. Usually the angular frequency vector ω is taken to be perpendicular to the symmetry axis, although tiltedaxis cranking can also be considered. Filling the singleparticle states up to the Fermi level then produces states whose expected angular momentum along the cranking axis has the desired value set by the Lagrange multiplier. To describe the singleparticle motion in a rotating nucleus, the cranking model approximation is frequently used, which was suggested by Inglis in 1954[3]. The basic idea of the cranking model is the following classical assumption: The nucleus with an angular momentum I ≠ 0 is rotating with a fixed frequency, ω, around a principle axis. The cranking Hamiltonian is then given by
h^{ ω} = h − ωj_{ x} [3].
where h^{ ω} is the Hamiltonian in the bodyfixed rotating system and h is the singleparticle Hamiltonian in the laboratory system. The x component of the singleparticle angular momentum is denoted by j _{x} , j_{ x} = l _{x} +s_{ x} . The term  ωj _{x} is analogous to the Coriolis and centrifugal forces in classical mechanics. The eigenvalues of h ^{ω} are the singleparticle energies in the rotating system, which are generally referred to as Routhians, e ^{ω}_{i} . The total singleparticle energy is calculated as the sum of the expectation values of the ”singleparticle energies” in the laboratory system[2].
The deformed harmonicoscillator Hamiltonian h_{osc} , with different oscillator frequencies ω_{ x} ,ω_{ y }and ω_{z }is expressed as[3]
Generally these three frequencies are expressed in terms of quadrupole deformation coordinates ε_{2} and γ , corresponding to the different ellipsoidal shapes.[3]
The parameter ε_{2 }gives the degree of the deformation of the nucleus, while γ gives its degree of axial asymmetry. Figure shows the values of ε_{2} and γ, which are used todescribe all possible ellipsoidal shapes with rotation around the three principal axes.[3]
ε_{2} = 0 corresponds to a spherical nucleus. The γ = 0◦ axis refers to prolate shape, where as the γ = −60◦ axis refers to oblate shape. These two shapes are axially symmetric with collective rotation around the perpendicular axis. At the border line, the nucleus rotates around the oblate (γ = 60◦ ) or prolate (γ = −120◦ ) symmetry axis corresponding to the noncollective limit. Away from the axes, (0 ◦ < γ < 60◦ ), (0◦ < γ < −60◦ ) and (−60◦ < γ < −120◦ ), the nuclear shape is triaxial, with rotation about each of the three different principal axis.
Computer calculation flow chart
In the inhorot program, one can define a mesh in the (ε_{2} , γ) deformation plane by introducing a Cartesian (x,y) coordinate system with the yaxis along the γ = −120◦/60◦ axis. The number of points in x and ydirections, and the spacing between the points is selected. Each point in the mesh represents a specific value of ε_{2} and γ. In addition, the number of points in ε_{4 }and their values are specified. The inhorot program creates an input file for horot. The horot program is used to calculate the singleparticle energies in the mesh points. For every deformation point in the mesh. Finally, to plot the singleparticle energies as function of rotational frequency at a fixed deformation, and to plot singleparticle energies e_{i} at a rotational symmetric deformation vs m_{i} , the programs called cross, routhplot and eimi need to be performed.
Cranked NilssonStrutinsky (CNS) plots
In above graph the energies are calculated at a constant deformation of ε_{2} = 0.120, γ =  120^{0} and ε_{4} =0.58. The arrow on the left hand side indicates the Fermi level for N = 32 at ω = 0. Particle numbers at some energy gaps are encircled. The orbitals are labelled by the group to which they belong and the ordering in the group at rotational frequency, ω = 0. The orbitals which are involved in a theoretical description of ^{62}Zn include the N = 3 highj 1f_{ 7/2} shell, the upper lowj fp shells 1f_{5/2} , 2p_{3/2} , and 2p_{1/2} , and finally the N = 4 shell 1g_{9/2} . The jshells are pure only if the shape is spherical. In the deformed rotating potential, these jshells will mix. i.e., in the present approximation the wave functions of the singleparticle orbitals will have amplitudes in all the jshells of a specific N rot shell. However, it turns out that if the deformation is not too large, these orbitals can be classified as having their main amplitudes in either the highj intruder shell or in the other shells with smaller jvalues . The orbitals are labelled according to which jshell or group of jshells they belong and the ordering in the group at rotational frequency ω = 0. The (fp) group indicates the mixing of the 1f_{5/2} and 2p_{3/2 }subshells. It is not really possible to make a distinction between the 1f_{5/2} and 2p_{3/2 }shells. The mixing probability of the 2p_{1/2} subshell is low for these orbitals and does not contribute significantly to the wavefunctions of the upper fp shell orbits which are active in ^{62}Zn. The large signature splitting of the lowest highj 1g_{9/2} intruder orbital means that for configurations with one 1g_{9/2} neutron (or proton), one expects to observe only the favoured α = +1/2 signature.
By comparing Fermi levels of neutron and proton orbital plots it is clear that Fermi energy of neutron is greater than Fermi energy of proton, because of Coulomb forces among protons and between proton electron. Because of neutrons are having nutral charge so there are no Coulomb effects. In nucleus these Coulomb effects are low, that is the resaon both orbitals are nearly same.
Let us observe point p on the graph located at normalised rotational frequency 0.15. For rotational frequency less than 0.15 the (fp) state (blue dotted) has lowyer energy than black solid line state and for rotational frequency greater than 0.15 the black solid state has lowyer energy than the (fp) state (blue dotted), it means rotational frequency less than 0.15 the nucleon enters in (fp) orbital and greater than 0.15 it enters in solid line orbital.
CONCLUSION
In this summer research fellowship maximum amount of work finished satisfactorily. Majorly focused on nuclear models, studied the concept and implemented using different programming languages. Liquid drop model studied and calculated binding energy of nucleons which is useful in nuclear reactions. Shell model studied it provides information about magic numbers , magnetic moment and electric moments . Generated computer programs to calculate them and plotted Shell model. Next focused on deformation of nucleus studied Nilsson model and Nilsson cranking models and generated programs for plot and these plots are analyzed in different aspects.
REFERENCES
1. Introduction to Applied Nuclear Physics, Prof. Paola Cappellaro.
2. Introductury Nuclear Physics, Kenneth S. Krane, Oregon State University,ISBN 0471 80553X, year 1955.
3. Comprehensive Gammaray Spectroscopy Studies of ^{62}Zn , Licentiate Thesis, Jnaneswari Gellanki, Lund University, year 2011.
4. Shapes And Shells In Nuclear Structre, Sven Gosta Nilsson and Ingemar Ragnarsson, ISBN 0 521 37377 8, 1995.
5.Shell Structure, Crancking and Magnetic Phenomena In Nuclei, Ashok K. Jain, P. Arumugam.
6.Shape deformations in atomic nuclei ,Prof. Ikuko Hamamoto, Prof. Ben Mottelson.Prof. Ben Mottelson.
ACKNOWLEDGEMENTS
I would like to express my gratitude towards my project guide Prof. Samit Kr. Mandal for giving me this opportunity to work under supervision of him and inspired me as a personolity . I would like to express my special gratitude and thanks to Dr. Jnaneswari Gellanki for providing necessary information regarding the project , also for her support in completing the project and giving me such attention and time. My special thanks to cointerns and Phd students in lab for helping me in different aspects.
Source

Fig 2: https://en.wikipedia.org/wiki/Table_of_nuclides

Fig 6: introduction to nuclear physics by Krane

Fig 8: introductory nuclear physics by Krane
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