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An Introduction to Nuclear Shell Model Prince Ahmad Ganai National Institute Of Technology Srinagar IUAC Dehli
Introduction
Empirical Facts ●
●
"Magic Numbers" in Nuclear Structure It is found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. In particular, there are "magic numbers" of neutrons and protons which seem to be particularly favored in terms of nuclear stability:
2, 8, 20, 28, 50, 82, 126 ●
Nuclei which have both neutron number and proton number equal to one of the magic numbers can be called "doubly magic", and are found to be particularly stable.
Empirical Facts ●
Calcium provides a good example of the exceptional stability of "doubly magic" nuclei since it has two of them.
●
The existence of several stable isotopes of calcium may have to to with the fact that Z=20, a magic number
●
Empirical Facts ●
●
Part of the motivation for the shell model of nuclear structure is the existence of "magic numbers" of neutrons and protons at which the nuclei have exceptional stability, implying some kind of "closed shell". One indication of this stability is the enhanced abundance of isotopes which have a magic number of neutrons or protons.
Empirical Facts
Empirical Facts ●
Further evidence of the uniqueness of these numbers is the fact that the end points of all four of the natural radioactive series are nuclei which have magic numbers of either N or Z.
●
The lead end products have 82 protons, a magic number, and the bismuth has 126 neutrons, also a magic number. The lead208 is doubly magic with Z=82, N=126.
●
Thorium > 232Th > 232Th > 228Ra + a 208Pb
●
Neptunium >237Np > 237Np > 233Pa + a 209Bi
●
Uranium >238U > 238U > 234Th + a >206Pb
●
Actinium > 235U >235U > 231Th + a > 207Pb
Empirical Facts ●
Evidence comes from absorption crosssections for neutrons.
The stability of those nuclei with magic numbers of neutrons makes them less likely to be excited by neutron bombardment. The probability for absorption of an incident is expressed as an effective crosssection which is presented by the target nucleus to those incoming neutrons.
Empirical Facts ●
●
Binding Energy for the Last Neutron as Evidence of Shell Structure. This dependence of the energy to remove the last neutron is strong evidence for a kind of shell structure. At the magic numbers, the shell is "closed" and it is hard to remove a neutron.
Shell Model of Nucleus ●
Densegas type models of nuclei with multiple collisions between particles didn't fit the data, and remarkable patterns like the "magic numbers" in the stability of nuclei suggested the seemingly improbable shell structure.
●
●
With the enormous strong force acting between them and with so many nucleons to collide with, how can nucleons possibly complete whole orbits without interacting? This has the marks of a Pauli exclusion principle process, where two fermions cannot occupy the same quantum state. If there are no nearby, unfilled quantum states that are in reach of the available energy for an interaction, then the interaction will not occur.
Shell Model of Nucleus ●
●
If there is not an available "hole" for a collision to knock a nucleon into, then the collision will not occur. There is no classical analog to this situation. The evidence for a kind of shell structure and a limited number of allowed energy states suggests that a nucleon moves in some kind of effective potential well created by the forces of all the other nucleons.
Physics behind the story ●
Kinetic energy of nucleon is roughly 40 MeV.
●
Rest mass energy is 938 MeV => Non relativistic approach.
●
●
●
●
Nuclear Size and DeBroglie wave length of nucleus are comparable Quantum mechanics at work Thus nuclear system to the first approximation can be treated as nonrelativistic quantum many body system. All we need to do is to solve Schroedinger equation for interacting nucleons.
Physics behind the story ●
Many body Hamiltonian
●
H=T+V(two body potential)
●
H=T+U+VU
●
VU=Residual Interaction
●
To first approximation one can choose VU =0
●
Now we need to find appropriate U.
Single Particle Model ●
Early development > Magic numbers
●
More refinements > Other nuclear properties.
●
Extreme single particle model
●
Nucleons pair to spin zero > Even – Even
●
Odd A nucleus > Unpaired Nucleon
●
To determine the how nucleon fills various quantum states, it is necessary to specify mean filed.
One Body Potentials ●
Square Well
●
Harmonic Oscillator
●
Woods Sexson
Harmonic Oscillator ●
General consideration of motion in mean field
●
Schroedinger equation > Separation of variables
●
Enl=(2n+l1/2)hw
●
Harmonic oscillator shells 2,8,20,40,112,168
●
1s>1p>2s1d>2p1f>3s2f1h>
●
Harmonic oscillator hw is related with A
Spin Orbit Coupling ●
Harmonic oscillator magic numbers are 2,8,20,40 ,70.
●
No single potential can alone explain the magic numbers.
●
One includes spinorbit splitting which means that radial Schroedinger equation not only depends on L but also j of single particle state.
●
This results in splitting of j=l(+)1/2
●
f(r)l.s > j=l+1/2 is depressed relative to j=11/2
●
The following level scheme is predicted.
Level scheme
Single particle Levels
Lecture II
Model Formalism
Shell Model Formalism
Formalism
Formalism
A self consistent potential should represent the combined action of all other nucleons in the system, one can not ignore the residual interaction which is treated as perturbation in the system of independent particle motion. Normally the antisymmetrized wave functions are employed for the description of many particle states. However, as the nuclear sates are independent of their orientation in space, the total angular momentum J is conserved and is thus a good quantum number.
Formalism ●
Slater determinant wave functions are not suitable as basis.
●
Contain states of different angular momenta.
●
Construct many particle states of definite angular momentum.
●
Angular momentum coupling
●
C.G coefficients
Formalism ●
●
●
Pauli principle requires that these states are antisymmetric in the coordinates of identical particles. This combination of rotational and permutational anti symmetry introduces CFP's With these coefficients it is possible to construct anti symmetric many particle wave functions of definite angular momentum.
Formalism ●
●
●
●
Another method in use > mscheme All rotational symmetry is discarded and the many particle wave functions are not coupled to well defines angular momentum. One specifies whether a single state |nljm> is occupied or not. Number of dimensions increases with subshells and particle number.
Formalism ●
Configuration space
●
Basis wave functions
●
Diagonalize the residual interaction
●
Complete space > Matrices of infinite dimensions
●
Truncation of Configuration space is required.
Formalism ●
Truncation is dictated by the capacity of computer.
●
Truncation will effect the results.
●
●
Effect of truncation , one has to make distinction between effective and true operators. For results to be acceptable one imposes the condition
Second Quantization ●
●
●
Description of many nucleon system one can advantage of properties of fermion creation and inhalation operators to generate a complete set of many body wave functions. Direct relation between the algebra of anticommuting creation and inhalation operators and the required anti symmetry of the many nucleon wave function All operators can be expressed in terms of creation and inhalation operators.
Second Quantization.
Second Quantization
●
Multi particle State operators
Second Quantization
Second Quantization
Second Quantization
Second Quantization
Computational Procedure
Overview of our SSM program ➢
Works in J scheme
➢
Calculate basis for each shell followed by multibasis construction.
➢
Calculate CFP's
➢
Workout all single J shell matrix elements
➢
Calculate Hamiltonian for identical particles
➢
Calculation of NP Hamiltonian
➢
Carry out diagonalization> ( Most of the memory and processor speed is required at this point)
Summary
Nuclear Physics strives to arrive at a fundamental understanding of strong interaction and nature of the nuclei. CEBAF and TJNAF probe the inner structure of nuclear constituents to test the predictions of QCD.
RHIL at BNL examines the nature and phases of quark gluon matter.
Near future RIA will study the physics in unexplored regime , in the regions of nuclear chart involving short lived nuclei near the limits of stability as defined by extream proton – neutron rich differences.
Exotic nuclei > Astrophysical phenomena
References
Thanks
Introduction
Empirical Facts ●
●
"Magic Numbers" in Nuclear Structure It is found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. In particular, there are "magic numbers" of neutrons and protons which seem to be particularly favored in terms of nuclear stability:
2, 8, 20, 28, 50, 82, 126 ●
Nuclei which have both neutron number and proton number equal to one of the magic numbers can be called "doubly magic", and are found to be particularly stable.
Empirical Facts ●
Calcium provides a good example of the exceptional stability of "doubly magic" nuclei since it has two of them.
●
The existence of several stable isotopes of calcium may have to to with the fact that Z=20, a magic number
●
Empirical Facts ●
●
Part of the motivation for the shell model of nuclear structure is the existence of "magic numbers" of neutrons and protons at which the nuclei have exceptional stability, implying some kind of "closed shell". One indication of this stability is the enhanced abundance of isotopes which have a magic number of neutrons or protons.
Empirical Facts
Empirical Facts ●
Further evidence of the uniqueness of these numbers is the fact that the end points of all four of the natural radioactive series are nuclei which have magic numbers of either N or Z.
●
The lead end products have 82 protons, a magic number, and the bismuth has 126 neutrons, also a magic number. The lead208 is doubly magic with Z=82, N=126.
●
Thorium > 232Th > 232Th > 228Ra + a 208Pb
●
Neptunium >237Np > 237Np > 233Pa + a 209Bi
●
Uranium >238U > 238U > 234Th + a >206Pb
●
Actinium > 235U >235U > 231Th + a > 207Pb
Empirical Facts ●
Evidence comes from absorption crosssections for neutrons.
The stability of those nuclei with magic numbers of neutrons makes them less likely to be excited by neutron bombardment. The probability for absorption of an incident is expressed as an effective crosssection which is presented by the target nucleus to those incoming neutrons.
Empirical Facts ●
●
Binding Energy for the Last Neutron as Evidence of Shell Structure. This dependence of the energy to remove the last neutron is strong evidence for a kind of shell structure. At the magic numbers, the shell is "closed" and it is hard to remove a neutron.
Shell Model of Nucleus ●
Densegas type models of nuclei with multiple collisions between particles didn't fit the data, and remarkable patterns like the "magic numbers" in the stability of nuclei suggested the seemingly improbable shell structure.
●
●
With the enormous strong force acting between them and with so many nucleons to collide with, how can nucleons possibly complete whole orbits without interacting? This has the marks of a Pauli exclusion principle process, where two fermions cannot occupy the same quantum state. If there are no nearby, unfilled quantum states that are in reach of the available energy for an interaction, then the interaction will not occur.
Shell Model of Nucleus ●
●
If there is not an available "hole" for a collision to knock a nucleon into, then the collision will not occur. There is no classical analog to this situation. The evidence for a kind of shell structure and a limited number of allowed energy states suggests that a nucleon moves in some kind of effective potential well created by the forces of all the other nucleons.
Physics behind the story ●
Kinetic energy of nucleon is roughly 40 MeV.
●
Rest mass energy is 938 MeV => Non relativistic approach.
●
●
●
●
Nuclear Size and DeBroglie wave length of nucleus are comparable Quantum mechanics at work Thus nuclear system to the first approximation can be treated as nonrelativistic quantum many body system. All we need to do is to solve Schroedinger equation for interacting nucleons.
Physics behind the story ●
Many body Hamiltonian
●
H=T+V(two body potential)
●
H=T+U+VU
●
VU=Residual Interaction
●
To first approximation one can choose VU =0
●
Now we need to find appropriate U.
Single Particle Model ●
Early development > Magic numbers
●
More refinements > Other nuclear properties.
●
Extreme single particle model
●
Nucleons pair to spin zero > Even – Even
●
Odd A nucleus > Unpaired Nucleon
●
To determine the how nucleon fills various quantum states, it is necessary to specify mean filed.
One Body Potentials ●
Square Well
●
Harmonic Oscillator
●
Woods Sexson
Harmonic Oscillator ●
General consideration of motion in mean field
●
Schroedinger equation > Separation of variables
●
Enl=(2n+l1/2)hw
●
Harmonic oscillator shells 2,8,20,40,112,168
●
1s>1p>2s1d>2p1f>3s2f1h>
●
Harmonic oscillator hw is related with A
Spin Orbit Coupling ●
Harmonic oscillator magic numbers are 2,8,20,40 ,70.
●
No single potential can alone explain the magic numbers.
●
One includes spinorbit splitting which means that radial Schroedinger equation not only depends on L but also j of single particle state.
●
This results in splitting of j=l(+)1/2
●
f(r)l.s > j=l+1/2 is depressed relative to j=11/2
●
The following level scheme is predicted.
Level scheme
Single particle Levels
Lecture II
Model Formalism
Shell Model Formalism
Formalism
Formalism
A self consistent potential should represent the combined action of all other nucleons in the system, one can not ignore the residual interaction which is treated as perturbation in the system of independent particle motion. Normally the antisymmetrized wave functions are employed for the description of many particle states. However, as the nuclear sates are independent of their orientation in space, the total angular momentum J is conserved and is thus a good quantum number.
Formalism ●
Slater determinant wave functions are not suitable as basis.
●
Contain states of different angular momenta.
●
Construct many particle states of definite angular momentum.
●
Angular momentum coupling
●
C.G coefficients
Formalism ●
●
●
Pauli principle requires that these states are antisymmetric in the coordinates of identical particles. This combination of rotational and permutational anti symmetry introduces CFP's With these coefficients it is possible to construct anti symmetric many particle wave functions of definite angular momentum.
Formalism ●
●
●
●
Another method in use > mscheme All rotational symmetry is discarded and the many particle wave functions are not coupled to well defines angular momentum. One specifies whether a single state |nljm> is occupied or not. Number of dimensions increases with subshells and particle number.
Formalism ●
Configuration space
●
Basis wave functions
●
Diagonalize the residual interaction
●
Complete space > Matrices of infinite dimensions
●
Truncation of Configuration space is required.
Formalism ●
Truncation is dictated by the capacity of computer.
●
Truncation will effect the results.
●
●
Effect of truncation , one has to make distinction between effective and true operators. For results to be acceptable one imposes the condition
Second Quantization ●
●
●
Description of many nucleon system one can advantage of properties of fermion creation and inhalation operators to generate a complete set of many body wave functions. Direct relation between the algebra of anticommuting creation and inhalation operators and the required anti symmetry of the many nucleon wave function All operators can be expressed in terms of creation and inhalation operators.
Second Quantization.
Second Quantization
●
Multi particle State operators
Second Quantization
Second Quantization
Second Quantization
Second Quantization
Computational Procedure
Overview of our SSM program ➢
Works in J scheme
➢
Calculate basis for each shell followed by multibasis construction.
➢
Calculate CFP's
➢
Workout all single J shell matrix elements
➢
Calculate Hamiltonian for identical particles
➢
Calculation of NP Hamiltonian
➢
Carry out diagonalization> ( Most of the memory and processor speed is required at this point)
Summary
Nuclear Physics strives to arrive at a fundamental understanding of strong interaction and nature of the nuclei. CEBAF and TJNAF probe the inner structure of nuclear constituents to test the predictions of QCD.
RHIL at BNL examines the nature and phases of quark gluon matter.
Near future RIA will study the physics in unexplored regime , in the regions of nuclear chart involving short lived nuclei near the limits of stability as defined by extream proton – neutron rich differences.
Exotic nuclei > Astrophysical phenomena
References
Thanks
