The Physics of Nuclei
II: Nuclear Structure
Ian Thompson
Nuclear Theory & Modelling Group
Lawrence Livermore National Laboratory.
University of California, Lawrence
Livermore National Laboratory
UCRL-PRES-232487
NNPSS: July 9-11, 2007
University of California, Lawrence Livermore National
Laboratory under contract No. W-7405-Eng-48. This
work was supported in part from LDRD contract 00ERD-028
1
Structure Synopsis
• Heavier Nuclei, so:
• Simpler Structure Theories:
– Liquid-Drop model
– Magic Numbers at Shells; Deformation;
Pairing
– Shell Model
– Hartree-Fock
(Mean-field; Energy Density Functional)
NNPSS: July 9-11, 2007
2
Segré Chart of Isotopes
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
NNPSS: July 9-11, 2007
3
Segré Chart of Isotopes
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
NNPSS: July 9-11, 2007
4
Nuclei with A≥16
• Start with Liquid-Drop model
– Parametrise binding energies
– Look for improvements
• Magic Numbers for Shells
• Shell Corrections
• Deformation
• Pairing
• Shell Model
• Mean field: Hartree-Fock
NNPSS: July 9-11, 2007
5
Nuclear masses: what nuclei exist?
M ( Z , N , A)  Zmh  Nmn  BE ( Z , N , A)
•
Let’s start with the semi-empirical mass
formula, Bethe-Weizsäcker formual, or also
the liquid-drop model.
Z  N 
2
BE (Z,N, A)  aV A  aSurf A 2 3  asym
A

aCoul Z(Z 1)A1 3   Pair  Shell
One goal in theory is 
to
accurately describe the
binding energy
– There are global Volume, Surface, Symmetry,
and Coulomb terms
– And specific corrections for each nucleus due
to pairing and shell structure
aV  15.85 MeV
aSurf  18.34 MeV
aSymm  23.21 MeV
Values for the parameters,
A.H. Wapstra and N.B. Gove, Nulc. Data
Tables 9, 267 (1971)
aCoul  0.71 MeV
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6
Nuclear masses, what nuclei exist?
Z  N

23
BE (Z,N, A)  aV A  aSurf A  asym
 aCoul Z(Z 1)A1 3
A
2
Liquid-drop isn’t too bad!
There are notable problems
though.

Can we do better and what
about the microscopic
structure?
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 168
Benjamin, 1969, New York
Bethe-Weizsäcker formula
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7

Nuclear masses, what nuclei exist?
•
•
How about deformation?
For each energy term, there are also shape factors dependent on the
quadrupole deformation parameters b and g
BSurf
2  5  2  5 
 1 
b   
b  cos 3g 
5  4   21  4  
BCoul
2
3




1
5
1
5
 1 
b  
b  cos 3g 

5  4   105  4  
2
3


R,   R0 1 a20Y20 ,   a22 Y22 ,   Y22 , 
a20  b cos g

a22  b

2 sin g
Note that the liquid drop always has a minimum for a spherical shape!
So, where does deformation come
from?

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Shell structure - evidence in atoms
•
Atomic ionization potentials show sharp discontinuities at shell
boundaries
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 191
Benjamin, 1969, New York
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9
Shell structure - neutron separation energies
• So do neutron separation energies
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 193
Benjamin, 1969, New York
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10
More evidence of shell structure
Binding energies show preferred magic numbers
– 2, 8, 20, 28, 50, 82, and 126
Deviation from average binding energy
Experimental Single-particle Effect (MeV)
•
From W.D. Myers and W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)
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11
Origin of the shell model
•
Goeppert-Mayer and Haxel,
Jensen, and Suess proposed the
independent-particle shell model
to explain the magic numbers
168
112
Harmonic
oscillator with
spin-orbit is a
reasonable
approximation
to the nuclear
mean field
70
40
20
8
2
M.G. Mayer and J.H.D. Jensen, Elementary Theory of Nuclear
Shell Structure,p. 58, Wiley, New York, 1955
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12
Nilsson Hamiltonian - Poor man’s Hartree-Fock
• Anisotropic harmonic oscillator
p2 m 2 2
  x x   y2 y 2   z2 z 2  2  0  s   0
2m 2
 0  41A1 3 MeV
 x   0e
 y   0e
 z   0e
5
b cosg 2  3
4
5
b cosg 2  3
4
5
b cosg
4

From J.M. Eisenberg and W. Greiner, Nuclear Models,
p 542, North Holland, Amsterdam, 1987
Oblate, g=60˚
Prolate, g=0˚
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2
Nuclear masses
Shell corrections to the liquid drop
•
Shell correction
– In general, the liquid drop does a good job on the bulk properties
• The oscillator doesn’t!
– But we need to put in corrections due to shell structure
• Strutinsky averaging; difference between the energy of the discrete spectrum and the
averaged, smoothed spectrum
Discrete spectrum
Mean-field single-particle spectrum
Smoothed
Discrete
i
15
F
 gd
g(E)
i
N
g~    
20
g     i 
N
10
F
5

0
 gd
   i 
    i 
f 
 g 
~F
~
 g  d


E Discrete 
Smoothed spectrum
f x   L x e  i 
2
is a modified Gaussian
g 
~F
EStrutinsky   g~  d

0
20
40
60
80

E (MeV)
Shell  E Discrete  E Strutinsky
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g2
14
2 g
Nilsson-Strutinsky and deformation
• Energy surfaces as a function of deformation
From C.G. Andersson,et al., Nucl. Phys. A268, 205 (1976)
Nilsson-Strutinsky is a mean-field type approach that allows for
a comprehensive study of nuclear deformation under rotation
and at high temperature
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Nuclear masses
Pairing corrections to the liquid drop
• Pairing:
 Pair Z,N, A  12A1 2 ;
= 0;
even - even(odd - odd)
odd - even(even - odd)

 n  14 {BE N  2,Z   3BE N 1,Z 
+ 3BE N,Z   BE N  1,Z }
 p  14 {BE N,Z  2  3BE N,Z 1
+ 3BE N,Z   BE N,Z  1}
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 170
Benjamin, 1969, New York
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16
The Shell Model
• More microscopic theory
– Include all nucleons
– Fully antisymmetrised wave functions
– Include all correlations in a ‘shell’.
– Harmonic Oscillator basis states.
• Effective Hamiltonian
– Still needed!
– Find or fit potential matrix elements,
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17
Many-body Hamiltonian
•
Start with the many-body Hamiltonian
pi2
H 
 VNN ri  rj 
i 2m
i j
•
Introduce a mean-field U to yield basis
 pi2

 H    U ri  VNN ri  rj  U ri 
2m
 i j
i 
i
0f1p N=4
0d1s N=2
1p N=1
0s N=0
Residual interaction
•
•
The mean field determines the shell structure
In effect, nuclear-structure calculations rely on perturbation theory
The success of any nuclear structure calculation depends on
the choice of the mean-field basis and the residual interaction!
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Single-particle wave functions
•
•
With the mean-field, we have the basis for building many-body states
This starts with the single-particle, radial wave functions, defined by the
radial quantum number n, orbital angular momentum l, and z-projection m
 nlm r   Rnl rYlm rˆ 
– Now include the spin wave function:
•
 Ss
1
2 z
Two choices, jj-coupling or ls-coupling

– Ls-coupling
nlms r   nlms r  Ss  Rnl rYlm r  Ss
z
1
2 z
z
1
2 z
– jj-coupling is very convenient when we have a spin-orbit (ls) force


 nljj r   Rnl r Yl rˆ   
z

Yl rˆ    1
S
2

jjz
S
1
2

jjz
  lm 12 sz | jjz Ylm rˆ  1Ssz
2
msz
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Multiple-particle wave functions
•
•
 1 sz
Total angular momentum, and isospin;
2
Anti-symmetrized, two particle, jj-coupled wave function
T

j1 j 2
JMTTz

 1   2
  n1 l1 j1 r1    n 2 l2 j 2 r2   1
T
T
1
2
1
2
TTz
JM
j1  j 2 J T

r1   n l j r2 
JM
n 2 l2 j 2
1 1 1

21 12 
– Note J+T=odd if the particles occupy the same orbits
•

Anti-symmetrized, two particle, LS-coupled wave function
LS
JMTTz


S JM
L
L 

l1 l 2 L S T
S
S
  n1 l1 r1    n 2 l2 r2   1
 n 2 l2 r1    n1 l1 r2    1 1   1 2 



2
2


 1   2
T
T
1
2
1
2
TTz
21 12 
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Two-particle wave functions
•
Of course, the two pictures describe the same physics, so there is a way to
connect them
( ˆj  2 j 1)
– Recoupling coefficients
j j
 JMTT
1 2
•

z
l1

ˆ
ˆ
ˆ
ˆ

  j1 j1LSl2
L
LS

1
2
1
2
S
j1 
 LS
j 2  JMTTz
J 

Note that the wave functions have been defined in terms of r1 and
often we need them in terms of the relative coordinate r  r1  r2
r2 , but
– We can do this in two ways
• Transform the operator
1 Fl r1,r2  Y
l
ˆ  is
ˆ 
V rQuadrupole,
 Ylarge

1  r2   4   l=2, component
l r1
l r2 
2l  1 
l
and very important
2l  1
Fl r1,r2  
Pl cos  V r1  r2 sin 12d12

2 0
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00
21
Two-particle wave functions in relative coordinate
•
Use Harmonic-oscillator wave functions and decompose in terms of the
relative and center-of-mass coordinates, i.e.,
 
r  r1  r2 ;
•
•
 
R  r1  r2 2
Harmonic oscillator wave functions are a very good approximation to the
single-particle wave functions
We have the useful transformation
 r    r 
L M 
n1 l1
1
n1 l1
1

 M nlNL;n l n l  r    R
L M 
11 2 2
nl
NL
nlNL

–
–
–
–
2n1+l2+2n2+l2=2n+l+2N+L
Where the M(nlNL;n1l1n2l2) is known as the Moshinksy bracket
Note this is where we use the jj to LS coupling transformation
For some detailed applications look in Theory of the Nuclear Shell Model, R.D.
Lawson, (Clarendon Press, Oxford, 1980)
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Many-particle wave function
•
•
To add more particles, we just continue along the same lines
To build states with good angular momentum, we can bootstrap up from the
two-particle case, being careful to denote the distinct states
– This method uses Coefficients of Fractional Parentage (CFP)


j NJM   j N 1J jJ j NJ
J 
•

j N 1J   j

JM
Or we can make a many-body Slater determinant that has only a specified Jz
and Tz and project J and T

n l
1 n l
i i j i j zi
1,2, , A 
j j
j j jz j
A!
n l
l l j l j zl
r1 n l j j r2 
r1 n l j j r2 
n l
n l
r1 n l j j r2 
n l
i i i zi
j j
j zj
l l l zl
i i j i j zi
j j
j j jz j
l l j l j zl
rA 
rA 
rA 
In general Slater determinants are more convenient

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Second Quantization
•
•
Second quantization is one of the most useful representations in many-body
theory
Creation and annihilation operators
–
–
–
–
Denote |0 as the state with no particles (the vacuum)
ai+ creates a particle in state i;
ai 0  i , ai i  0
ai i  0 ;
ai 0  0
ai annihilates a particle in state i;
Anti-commutation relations:
ai ,aj  ai,a 
j  0



a
,a

a
,a
 i j   j i  ij
•
Many-body Slater determinant


 i r1  i r2 
1  j r1   j r2 
 i rA 
 j rA 
A!
 l r1  l r2 
 l rA 
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 al
a j ai 0
l  j  i
24
Second Quantization
•
Operators in second-quantization formalism
– Take any one-body operator O, say quadrupole E2 transition operator er2Y2,
the operator is represented as:
O   j O i aj ai
ij
where j|O|i is the single-particle matrix element of the operator O
– The same formalism exists for any n-body operator, e.g., for the NNinteraction

V
1
ij V kl A ai a j al ak   ij V kl A ai a j al ak

4 ijkl
i j,kl
ij V kl
A
 ij V kl  ij V lk
• Here, I’ve written the two-body matrix element with an implicit assumption that it is
anti-symmetrized, i.e.,
r    i r ai ai

2
i

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Second Quantization
•
Matrix elements for Slater determinants
(all aceklm different)
acekl ac am aeklm  0 al ak ae ac aa ac am am al ak ae aa 0
 0 al ak ae ac aa ac al ak ae aa 0  acekl ac aekl
  0 al ak ae ac aa al ak ae ac aa 0   acekl acekl
 1
Second quantization makes the computation of expectation
values for the many-body system simpler
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Second Quantization
•
Angular momentum tensors
– Creation operators rotate as tensors of rank j
– Not so for annihilation operators
a˜ jm  1
•
j  jz
a jm
Anti-symmetrized, two-body state


1
ja j b : JM,TTz  
a j a 1  a j b 1
2
2
1 ab

JM
0
TTz

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Shell-model mean field
•
One place to start for the mean field is the harmonic oscillator
– Specifically, we add the center-of-mass potential
HCM
2
1
1
m2
2 2
2
 Am R   mri  
ri  rj 
2
2
2A
i
i j
– The Good:

• Provides a convenient basis to build the many-body Slater determinants
• Does not affect the intrinsic motion
• Exact separation between intrinsic and center-of-mass motion
– The Bad:
• Radial behavior is not right for large r
• Provides a confining potential, so all states are effectively bound
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Low-lying structure –
The interacting Shell Model
•
•
The interacting shell model is one of the most powerful tools available
too us to describe the low-lying structure of light nuclei
We start at the usual place:
 pi2

H    U ri  VNN ri  rj  U ri 
2m
 i j
i 
i
0f1p N=4
0d1s N=2
•
Construct many-body states |i so that
i   Cin n

•
1p N=1
0s N=0
n
Calculate Hamiltonian matrix Hij=j|H|i
– Diagonalize to obtain eigenvalues
  H11
 H 21
 

H
 N1

 i r1  i r2 
1  j r1   j r2 
A!
 l r1   l r2 
H12  H1N 

H 22




 H NN 
 i rA 
 j rA 
 al
 l rA 
a j ai 0

We want an accurate description of low-lying states
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Shell model applications
•
The practical Shell Model
1. Choose a model space to be used for a range of nuclei
•
E.g., the 0d and 1s orbits (sd-shell) for 16O to 40Ca or the 0f and 1p orbits for 40Ca
to 120Nd
2.
3.
4.
5.
We start from the premise that the effective interaction exists
We use effective interaction theory to make a first approximation (G-matrix)
Then tune specific matrix elements to reproduce known experimental levels
With this empirical interaction, then extrapolate to all nuclei within the
chosen model space
6. Note that radial wave functions are explicitly not included, so we add them
in later
The empirical shell model works well!
But be careful to know the limitations!
NNPSS: July 9-11, 2007
30
Simple application of the shell model
•
A=18, two-particle problem with
16O
core
– Two protons: 18Ne (T=1)
– One Proton and one neutron: 18F (T=0 and T=1)
– Two neutrons: 18O (T=1)
Example: 18O
Question # 1?
•
How many states for each Jz? How many states of each J?
– There are 14 states with Jz=0
•
•
•
•
•
N(J=0)=3
N(J=1)=2
N(J=2)=5
N(J=3)=2
N(J=4)=2
NNPSS: July 9-11, 2007
31
Simple application of the shell model, cont.
Example:
Question #2
•
What are the energies of the three 0+ states in 18O?
– Use the Universal SD-shell interaction (Wildenthal)
 0d
 3.94780
0d 5 / 2 0d 5 / 2 ; J  0, T  1 V 0d 5 / 2 0d 5 / 2 ; J  0, T  1  2.8197
1s  3.16354
0d 5 / 2 0d 5 / 2 ; J  0, T  1 V 0d 3 / 2 0d 3 / 2 ; J  0, T  1  3.1856
5/ 2
1/ 2
 0d
3/ 2
 1.64658
Measured relative to 16O core
Note 0d3/2 is unbound
0d 5 / 2 0d 5 / 2 ; J  0, T  1 V 1s1/ 21s1/ 2 ; J  0, T  1  1.0835
1s1/ 21s1/ 2 ; J  0, T  1 V 1s1/ 21s1/ 2 ; J  0, T  1  2.1246
1s1/ 21s1/ 2 ; J  0, T  1 V 0d 3 / 2 0d 3 / 2 ; J  0, T  1  1.3247
0d 3 / 2 0d 3 / 2 ; J  0, T  1 V 0d 3 / 2 0d 3 / 2 ; J  0, T  1  2.1845
NNPSS: July 9-11, 2007
32
Simple application of the shell model, cont.
Example:
Finding the eigenvalues
•
3.522
3.522
1.964
1.964
-0.830
-1.243
-1.348
-1.616
-2.706
-3.421
11.341
10.928
10.823
10.555
9.465
8.750
2+
-6.445
-7.732
-7.851
-8.389
-9.991
5.726
4.440
4.320
3.781
2.180
0+
-12.171
0.000
Set up the Hamiltonian matrix
– We can use all 14 Jz=0 states, and
we’ll recover all 14 J-states
– But for this example, we’ll use the
two-particle J=0 states
|(0d5/2)2J=0
|(1s1/2)2J=0
1+
|(0d3/2)2J=0
10.7153 1.0835 3.1856


H   1.0835 8.4517 1.3247


 3.1856 1.3247 1.1087
NNPSS: July 9-11, 2007
3+
4+
33
What about heavier nuclei?
• Above A ~ 60 or so the number of configurations just gets to be
too large ~ 1010!
• Here, we need to think of more approximate methods
• The easiest place to start is the mean-field of Hartree-Fock
– But, once again we have the problem of the interaction
• Repulsive core causes us no end of grief!!
• So still need effective interactions!
• At some point use fitted effective interactions like the Skyrme force
NNPSS: July 9-11, 2007
34
Hartree-Fock
•
•
There are many choices for the mean field, and Hartree-Fock is one
optimal choice
We want to find the best single Slater determinant 0 so that
0 H 0
 minimum
0 0
0 H 0  0 H 0  0
A
0   ai 0
i1
• Thouless’ theorem
– Any other Slater determinant  not orthogonal to 0 may be
written as



  exp Cmiam ai 0
mi

– Where i is a state occupied in 0 and m is unoccupied
– Then

0  Cmiam ai 0
and 0 0  Cmi 0 am ai 0  0
im
im
NNPSS: July 9-11, 2007
35
Hartree-Fock


pi2 1
1
H    V ri  r j  T  Vij
2 ij
2 ij
i m
•
•
Let i,j,k,l denote occupied states and m,n,o,p unoccupied states
After substituting back we get

occupied
mTi 

jm V ji
A
0
j
•
This leads directly to the Hartree-Fock single-particle Hamiltonian h with
matrix elements between any two states  and b

hb  Tb 
occupied

j V jb
A
j
 Tb  Ub

NNPSS: July 9-11, 2007
36
Hartree-Fock
•
We now have a mechanism for defining a mean-field
– It does depend on the occupied states
– Also the matrix elements with unoccupied states are zero, so the first order
1p-1h corrections do not contribute
occupied

mTi 
jm V ji
A
 m hi 0
j
•
We obtain an eigenvalue equation (more on this later)

h i  i i
E  0 H 0   i T i 
i
1
ij V ij

2 ij
A

1
i T i  i 


2 i
• Energies of A+1 and A-1 nuclei relative to A

E A1  E A   m
E A1  E A   i
NNPSS: July 9-11, 2007
37
Hartree-Fock – Eigenvalue equation
•
Two ways to approach the eigenvalue problem
– Coordinate space where we solve a Schrödinger-like equation
– Expand in terms of a basis, e.g., harmonic-oscillator wave function
•
Expansion
– Denote basis states by Greek letters, e.g., 
i   Ci 

C C   
*
i
j
 C  C b  b
*
i
ij

i
i
– From the variational principle, we obtain the eigenvalue equation

occupied

 T b   j V bj
b 
j
or

Cib  iCi
A


  h b Cb C
i
i
i
b
NNPSS: July 9-11, 2007
38
Hartree-Fock –
Solving the eigenvalue equation
•
•
As I have written the eigenvalue equation, it doesn’t look to useful
because we need to know what states are occupied
We use three steps
1. Make an initial guess of the occupied states & the expansion coefficients Ci
•
For example the lowest Harmonic-oscillator states, or a Woods-Saxon and Ci=i
2. With this ansatz, set up the eigenvalue equations and solve them
3. Use the eigenstates |i from step 2 to make the Slater determinant 0, go
back to step 2 until the coefficients Ci are unchanged
The Hartree-Fock equations are solved self-consistently
NNPSS: July 9-11, 2007
39
Hartree-Fock – Coordinate space
•
Here, we denote the single-particle wave functions as i(r)
occupied
occupied

*
3

  r1   
 j r2 V r1  r2  j r2 d r2 
 i r1     *j r2 V r1  r2  j r1  i r2 d 3 r2  i i r1




2m
 j

j
2
2
1 i
Direct or Hartree term: UH
•
Exchange or Fock term: UF
These equations are solved the same way as the matrix eigenvalue
problem before
1. Make a guess for the wave functions i(r) and Slater determinant 0
2. Solve the Hartree-Fock differential equation to obtain new states i(r)
3. With these go back to step 2 and repeat until i(r) are unchanged
Again the Hartree-Fock equations are solved self-consistently
NNPSS: July 9-11, 2007
40
Hartree-Fock
Hard homework problem:
•
•
M. Moshinsky, Am. J. Phys. 36, 52 (1968). Erratum, Am. J. Phys. 36, 763
(1968).
Two identical spin-1/2 particles in a spin singlet interact via the Hamiltonian
H  12 p12  r12  12 p22  r22  
•

  
Use the coordinates r  r1  r2 

1
2
r1  r2  
2
  
2 and R  r1  r2 
2 to show the exact
energy and wave function are
E
3
2
1

2  1
 2  1 
 2   1 
1
2
r,R 
exp  R 2 
r 2 
 exp
34
2
 
 



34
•
Note that since the spin wave function (S=0) is anti-symmetric, the spatial
wave function is symmetric

NNPSS: July 9-11, 2007
41
Hartree-Fock
Hard homework problem:
•
The Hartree-Fock solution for the spatial part is the same as the Hartree
solution for the S-state. Show the Hartree energy and radial wave function
are:
EH  3
 1
   1 3 2
   1 2     1 2 
r1,r2   
r1 exp
r2 
 exp

2
2



 

20
E/h
15

10
Eexact
EH
5
0
0
5
10
15

20
25
30
NNPSS: July 9-11, 2007
42
Hartree-Fock with the Skyrme interaction
•
•
In general, there are serious problems trying to apply Hartree-Fock with
realistic NN-interactions (for one the saturation of nuclear matter is
incorrect)
Use an effective interaction, in particular a force proposed by Skyrme


1
1
1
v12  t 0 1 x 0 Ps  r1  r2   t11 x1Ps  r1  r2  12  22  12  22  r1  r2 


2
2i
2i
1
1
1
1
t 2 1 x 2 Ps  1  2   r1  r2  1  2  W 0 s1  s2   1  2   r1  r2  1  2 
2i
2i
2i
2i
t 3 r1  r2  r2  r3 


–
•



 





Ps is the spin-exchange operator
The three-nucleon interaction is actually a density dependent two-body,
so replace with a more general form, where  determines the
incompressibility of nuclear matter
1
t3 1 x 3Ps r1  r2  r1  r2  2
6
NNPSS: July 9-11, 2007
43
Hartree-Fock with the Skyrme interaction
•
•
One of the first references: D. Vautherin and D.M. Brink, PRC 5, 626
(1972)
Solve a Schrödinger-like equation
2


i
i



U
r

iW
r


s

r









r 



i

*
z
z
z
z
2m z r 






z labels protons
or neutrons
– Note the effective mass m*
2

2m*z r 

2
2m

1
1
t1  t 2 r   t 2  t1 z r 
4
8
– Typically, m* < m, although it doesn’t have to, and is determined by the
parameters t1 and t2
•
•

The effective mass influences the spacing of the single-particle states
The bias in the past was for m*/m ~ 0.7 because of earlier calculations with realistic
interactions
NNPSS: July 9-11, 2007
44
Hartree-Fock calculations
•
The nice thing about the Skyrme interaction is that it leads to a
computationally tractable problem
– Spherical (one-dimension)
– Deformed
•
•
•
Axial symmetry (two-dimensions)
No symmetries (full three-dimensional)
There are also many different choices for the Skyrme parameters
– They all do some things right, and some things wrong, and to a large degree
it depends on what you want to do with them
– Some of the leading (or modern) choices are:
•
•
•
•
M*, M. Bartel et al., NPA386, 79 (1982)
SkP [includes pairing], J. Dobaczewski and H. Flocard, NPA422, 103 (1984)
SkX, B.A. Brown, W.A. Richter, and R. Lindsay, PLB483, 49 (2000)
Apologies to those not mentioned!
– There is also a finite-range potential based on Gaussians due to D. Gogny,
D1S, J. Dechargé and D. Gogny, PRC21, 1568 (1980).
•
Take a look at J. Dobaczewski et al., PRC53, 2809 (1996) for a nice
study near the neutron drip-line and the effects of unbound states
NNPSS: July 9-11, 2007
45
Nuclear structure
•
Remember what our goal is:
– To obtain a quantitative description of all nuclei within a microscopic frame
work
– Namely, to solve the many-body Hamiltonian:
pi2
H 
 VNN ri  rj 
i 2m
i j
 pi2

H    U ri  VNN ri  rj  U ri 
2m
 i j
i 
i
Residual interaction
Perturbation Theory

NNPSS: July 9-11, 2007
46
Nuclear structure
•
Hartree-Fock is the optimal choice for the mean-field potential U(r)!
– The Skyrme interaction is an “effective” interaction that permits a wide range
of studies, e.g., masses, halo-nuclei, etc.
– Traditionally the Skyrme parameters are fitted to binding energies of doubly
magic nuclei, rms charge-radii, the incompressibility, and a few spin-orbit
splittings
•
One goal would be to calculate masses for all nuclei
–
By fixing the Skyrme force to known nuclei, maybe we can get 500 keV
accuracy that CAN be extrapolated into the unknown region
•
This will require some input about neutron densities – parity-violating electron
scattering can determine <r2>p-<r2>n.
– This could have an important impact
NNPSS: July 9-11, 2007
47
Hartree-Fock calculations
•
Permits a study of a wide-range of nuclei, in particular, those far from
stability and with exotic properties, halo nuclei
H. Sagawa, PRC65, 064314 (2002)
The tail of the radial density
depends on the separation
energy
S. Mizutori et al. PRC61, 044326 (2000)
NNPSS: July 9-11, 2007
Drip-line studies
J. Dobaczewski et al., PRC53, 2809
(1996)
48
What can Hartree-Fock calculations tell
us about shell structure?
•
Shell structure
– Because of the self-consistency, the shell structure can change from nucleus
to nucleus
As we add neutrons, traditional
shell closures are changed, and
may even disappear!
This is THE challenge in trying to
predict the structure of nuclei at
the drip lines!
J. Dobaczewski et al., PRC53, 2809 (1996)
NNPSS: July 9-11, 2007
49
Beyond mean field
•
Hartee-Fock is a good starting approximation
– There are no particle-hole corrections to the HF ground state
mT i 
occupied

jm V ji
A
 mhi 0
j
– The first correction is
ij V mn A mn V ij
1

4 ijmn  i   j   m   n
•
A
However, this doesn’t make a lot of sense for Skyrme potentials
– They are fit to closed-shell nuclei, so they effectively have all these higherorder corrections in them!
•
We can try to estimate the excitation spectrum of one-particle-one-hole
states – Giant resonances
– Tamm-Dancoff approximation (TDA)
– Random-Phase approximation (RPA)
NNPSS: July 9-11, 2007
You should look these up!
A Shell Model Description of Light Nuclei, I.S. Towner
The Nuclear Many-Body Problem, Ring & Schuck
50
Nuclear structure in the future
Monte Carlo
Shell Model
Standard shell model
Ab initio methods
With newer methods and
powerful computers, the future
of nuclear structure theory is
bright!
NNPSS: July 9-11, 2007
51