The Physics of Nuclei II: Nuclear Structure

NNPSS: July 9-11, 2007 1

The Physics of NucleiII: Nuclear Structure

Ian ThompsonNuclear Theory & Modelling Group

Lawrence Livermore National Laboratory.

University of California, Lawrence Livermore National Laboratory

University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. This work was supported in part from LDRD contract 00-ERD-028

UCRL-PRES-232487

NNPSS: July 9-11, 2007 2

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 3

Segré Chart of Isotopes

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

NNPSS: July 9-11, 2007 4

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Segré Chart of Isotopes

NNPSS: July 9-11, 2007 5

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

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• Let’s start with the semi-empirical mass formula, Bethe-Weizsäcker formual, or also the liquid-drop model.

– There are global Volume, Surface, Symmetry, and Coulomb terms

– And specific corrections for each nucleus due to pairing and shell structure

Nuclear masses: what nuclei exist?),,(),,( ANZBENmZmANZM nh −+=

€

BE(Z,N,A) = aV A − aSurf A2 3 − asym

Z − N( )2

A+

aCoulZ(Z −1)A−1 3 + ΔPair + δShell

One goal in theory is to accurately describe the

binding energy

MeV 71.0

MeV 21.23

MeV 34.18

MeV 85.15

=

=

==

Coul

Symm

Surf

V

a

a

aa

Values for the parameters,Values for the parameters,A.H. Wapstra and N.B. Gove, Nulc. Data A.H. Wapstra and N.B. Gove, Nulc. Data

Tables 9, 267 (1971) Tables 9, 267 (1971)

NNPSS: July 9-11, 2007 7

Nuclear masses, what nuclei exist?

€

BE(Z,N,A) = aV A − aSurf A2 3 − asym

Z − N( )2

A+ aCoulZ(Z −1)A−1 3

Liquid-drop isn’t too bad!There are notable problems though.

Can we do better and what about the microscopic structure?

Bethe-Weizsäcker formulaBethe-Weizsäcker formulaFrom A. Bohr and B.R. Mottleson,Nuclear Structure, vol. 1, p. 168

Benjamin, 1969, New York

NNPSS: July 9-11, 2007 8

• How about deformation?

• For each energy term, there are also shape factors dependent on the quadrupole deformation parameters and

Nuclear masses, what nuclei exist?

€

BSurf ≈1+2

5

5

4πβ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−2

21

5

4πβ

⎛

⎝ ⎜

⎞

⎠ ⎟

3

cos3γ +K

BCoul ≈1−1

5

5

4πβ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−1

105

5

4πβ

⎛

⎝ ⎜

⎞

⎠ ⎟

3

cos3γ +K

€

R θ,ϕ( ) = R0 1+ a20Y20 θ,ϕ( ) + a22 Y22 θ,ϕ( ) + Y2−2 θ,ϕ( )( )[ ]

a20 = β cosγ

a22 = β 2( )sinγ

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


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Shell structure - neutron separation energies

• So do neutron separation energies



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• Binding energies show preferred magic numbers– 2, 8, 20, 28, 50, 82, and 126

More evidence of shell structure

Ex

pe

rim

en

tal

Sin

gle

-pa

rtic

le E

ffe

ct

(Me

V)

From W.D. Myers and W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)

Deviation from average binding energy

NNPSS: July 9-11, 2007 12

• Goeppert-Mayer and Haxel, Jensen, and Suess proposed the independent-particle shell model to explain the magic numbers

Origin of the shell model

22

4040

2020

7070

112112

88

168168

M.G. Mayer and J.H.D. Jensen, Elementary Theory of Nuclear Shell Structure,p. 58, Wiley, New York, 1955

Harmonic oscillator with spin-orbit is a reasonable approximation to the nuclear mean field

NNPSS: July 9-11, 2007 13

• Anisotropic harmonic oscillator

Nilsson Hamiltonian - Poor man’s Hartree-Fock

€

p2

2m+

m

2ωx

2x 2 + ωy2y 2 + ωz

2z2( ) − 2hω0κ

r l ⋅

r s − hω0κμ

r l 2

hω0 = 41A1 3 MeV

ωx = ω0e5

4 πβ cos γ +2π 3( )

ωy = ω0e5

4 πβ cos γ −2π 3( )

ωz = ω0e5

4 πβ cosγ

Prolate, Prolate, =0˚=0˚Oblate, Oblate, =60˚=60˚

From J.M. Eisenberg and W. Greiner, Nuclear Models, p 542, North Holland, Amsterdam, 1987

NNPSS: July 9-11, 2007 14

• 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

Nuclear massesShell corrections to the liquid drop

€

g ε( ) = δ ε −ε i( )i

∑

N = g ε( )dε−∞

ε F

∫

EDiscrete = εg ε( )dε−∞

ε F

∫

Discrete spectrumDiscrete spectrum

( ) ( )

( )

( )∫

∫

∑

∞−

∞−

=

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

F

F

dgE

dgN

fgi

ii

ε

ε

εεε

εε

εεδγ

εεε

~

Strutinsky

~

~

~

~

Smoothed spectrumSmoothed spectrum

€

δShell = EDiscrete − EStrutinsky

€

f x( ) = L x( )e− ε −ε i( )2 γ 2

2 π γ

is a modified Gaussian

γ ≈ hΩ

Mean-field single-particle spectrumMean-field single-particle spectrum

0 20 40 60 800

5

10

15

20

g(E)

E (MeV)

Smoothed Discrete

NNPSS: July 9-11, 2007 15

• Energy surfaces as a function of deformation

Nilsson-Strutinsky and deformation

Nilsson-Strutinsky is a mean-field type approach that allows for a comprehensive study of nuclear deformation under rotation and at high temperature

From C.G. Andersson,et al., Nucl. Phys. A268, 205 (1976)

NNPSS: July 9-11, 2007 16

• Pairing:

Nuclear massesPairing corrections to the liquid drop

€

ΔPair Z,N, A( ) ≈ − +( )12A−1 2; even - even(odd - odd)

= 0; odd - even(even - odd)



€

Δ n = 14 {BE N − 2,Z( ) − 3BE N −1,Z( )

+ 3BE N,Z( ) − BE N +1,Z( )}

Δ p = 14 {BE N,Z − 2( ) − 3BE N,Z −1( )

+ 3BE N,Z( ) − BE N,Z +1( )}

NNPSS: July 9-11, 2007 17

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,

NNPSS: July 9-11, 2007 18

Many-body Hamiltonian• Start with the many-body Hamiltonian

• Introduce a mean-field U to yield basis

• The mean field determines the shell structure

• In effect, nuclear-structure calculations rely on perturbation theory

€

H =r p i

2

2mi

∑ + VNN

r r i −

r r j( )

i< j

∑

0s N=0

0d1s N=2

0f1p N=4

1p N=1

€

H =r p i

2

2m+ U ri( )

⎛

⎝ ⎜

⎞

⎠ ⎟

i

∑ + VNN

r r i −

r r j( )

i< j

∑ − U ri( )i

∑

Residual interactionResidual interaction

The success of any nuclear structure calculation depends on the choice of the mean-field basis and the residual interaction!

NNPSS: July 9-11, 2007 19

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

– Now include the spin wave function:

• Two choices, jj-coupling or ls-coupling– Ls-coupling

– jj-coupling is very convenient when we have a spin-orbit (ls) force

€

ϕ nlm

r r ( ) = Rnl r( )Ylm

ˆ r ( )Ssz2

1χ

€

ϕ nlmsz

r r ( ) = ϕ nlmsz

r r ( )χ 1

2 sz

S = Rnl r( )Ylm

) r ( )χ 1

2 sz

S

€

ϕ nljjz

r r ( ) = Rnl r( ) Yl

ˆ r ( )⊗ χ 12

S[ ]

jjz

Ylˆ r ( )⊗ χ 1

2

S[ ]

jjz

= lm 12 sz | jjz( )Ylm

ˆ r ( )χ 12 sz

S

msz

∑

NNPSS: July 9-11, 2007 20

Multiple-particle wave functions

• Total angular momentum, and isospin;

• Anti-symmetrized, two particle, jj-coupled wave function

– Note J+T=odd if the particles occupy the same orbits

• Anti-symmetrized, two particle, LS-coupled wave function

€

ψJMTTz

j1 j2 = ϕ n1l1 j1

r r 1( )⊗ϕ n2l2 j2

r r 2( )[ ]

JM+ −1( )

j1 + j2 +J +Tϕ n2l2 j2

r r 1( )⊗ϕ n1l1 j1

r r 2( )[ ]

JM

{ }

χ 12

T 1( )⊗ χ 12

T 2( )[ ]TTz

2 1+ δ12( )

Tsz2

1χ

€

ψJMTTz

LS = ϕ n1l1

r r 1( )⊗ϕ n2l2

r r 2( )[ ]

L− −1( )

l1 +l2 +L +S +Tϕ n2l2

r r 1( )⊗ϕ n1l1

r r 2( )[ ]

L ⎛ ⎝ ⎜ ⎞

⎠ ⎟⊗ χ 1

2

S 1( )⊗ χ 12

S 2( )[ ]S ⎧

⎨ ⎩

⎫ ⎬ ⎭

JM

χ 12

T 1( )⊗ χ 12

T 2( )[ ]TTz

2 1+ δ12( )

NNPSS: July 9-11, 2007 21

Two-particle wave functions

• Of course, the two pictures describe the same physics, so there is a way to connect them

– Recoupling coefficients

• Note that the wave functions have been defined in terms of and , but often we need them in terms of the relative coordinate

– We can do this in two ways• Transform the operator

€

ψJMTTz

j1 j2 = ˆ j 1 ˆ j 1 ˆ L ̂ S

l112 j1

l212 j2

L S J

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪LS

∑ ψ JMTTz

LS€

( ˆ j = 2 j +1)

€

rr 1

€

rr 2

€

r =r r 1 −

r r 2

€

Vr r 1 −

r r 2( ) = 4π

−1( )lFl r1,r2( )

2l +1Yl

ˆ r 1( )⊗Ylˆ r 2( )[ ]

00

l

∑

Fl r1,r2( ) =2l +1

2Pl cosθ( )V

r r 1 −

r r 2( )sinθ12dθ12

0

π

∫

Quadrupole, l=2, component is large and very important

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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.,

• Harmonic oscillator wave functions are a very good approximation to the single-particle wave functions

• We have the useful transformation

– 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)

2rr ;rr 2121

rrrr+=−= Rr

€

φn1l1

r r 1( )⊗φn1l1

r r 1( )[ ]

′ L ′ M = M nlNL;n1l1n2l2( ) φnl

r r ( )⊗φNL

r R ( )[ ]

′ L ′ M

nlNL

∑

NNPSS: July 9-11, 2007 23

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)

• Or we can make a many-body Slater determinant that has only a specified Jz and Tz and project J and T

€

Φ 1,2,K , A( ) =1

A!

φn i li j i jzir1( ) φn i li j i jzi

r2( ) K φn i li ji jzirA( )

φn j l j j j jz jr1( ) φn j l j j j jz j

r2( ) φn j l j j j jz jrA( )

M O M

φn l ll j l jzlr1( ) φn l ll j l jzl

r2( ) K φn l ll jl jzlrA( )

€

jNαJM = jN−1 ′ α ′ J jJ} jNαJ[ ] jN−1 ′ α ′ J ⊗ j[ ]JM

′ α ′ J

∑

In general Slater determinants are more convenient

NNPSS: July 9-11, 2007 24

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 annihilates a particle in state i;

– Anti-commutation relations:

• Many-body Slater determinant

€

ai+,a j

+{ } = ai,a j{ } = 0

ai,a j+

{ } = a j+,ai{ } = δij

€

Φ=1

A!

φi r1( ) φi r2( ) K φi rA( )

φ j r1( ) φ j r2( ) φ j rA( )

M O M

φl r1( ) φl r2( ) K φl rA( )

= al+K a j

+ai+ 0

ijl >>>K

€

ai i = 0 ; ai 0 = 0

€

ai+ 0 = i , ai

+ i = 0

NNPSS: July 9-11, 2007 25

Second Quantization

• Operators in second-quantization formalism– Take any one-body operator O, say quadrupole E2 transition operator er2Y2,

the operator is represented as:

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 NN-

interaction

• Here, I’ve written the two-body matrix element with an implicit assumption that it is anti-symmetrized, i.e.,

€

O = j O i a j+ai

ij

∑

€

V =1

4ij V kl

Aai

+a j+alak

ijkl

∑ = ij V klA

ai+a j

+alak

i< j,k<l

∑

ij V klA

= ij V kl − ij V lk

€

ρ r( ) = ϕ i r( )2ai

+ai

i

∑

NNPSS: July 9-11, 2007 26

Second Quantization

• Matrix elements for Slater determinants (all aceklm different)

€

acekl ac+am aeklm = 0 alakaeacaaac

+amam+ al

+ak+ae

+aa+ 0

aeklaaceklaaaaaaaaaa caeklcacekl++++++ == 00

1

00

−=

−=−= +++++ aceklaceklaaaaaaaaaa aceklacekl

Second quantization makes the computation of expectation values for the many-body system simpler

NNPSS: July 9-11, 2007 27

Second Quantization

• Angular momentum tensors– Creation operators rotate as tensors of rank j– Not so for annihilation operators

• Anti-symmetrized, two-body state

€

˜ a jm = −1( )j + jz a j−m

€

ja jb : JM,TTz = −1

1+ δab

aja

12

+ ⊗ ajb

12

+[ ]

TTz

JM

0

NNPSS: July 9-11, 2007 28

Shell-model mean field• One place to start for the mean field is the harmonic oscillator

– Specifically, we add the center-of-mass potential

– 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

€

HCM =1

2AmΩ2

r R 2 =

1

2mΩ

r r i

2

i

∑ −mΩ2

2A

r r i −

r r j( )

2

i< j

∑

NNPSS: July 9-11, 2007 29

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:

• Construct many-body states |ϕi so that

• Calculate Hamiltonian matrix Hij=ϕj|H|ϕi– Diagonalize to obtain eigenvalues

€

H =r p i

2

2m+ U ri( )

⎛

⎝ ⎜

⎞

⎠ ⎟

i

∑ + VNN

r r i −

r r j( )

i< j

∑ − U ri( )i

∑

0s N=0

0d1s N=2

0f1p N=4

1p N=1

€

Ψi = Cinφn

n

∑

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

NNN

N

HH

HH

HHH

K

OM

L

1

2221

11211

We want an accurate description of low-lying states

€

φ= 1

A!

φi r1( ) φi r2( ) K φi rA( )

φ j r1( ) φ j r2( ) φ j rA( )

M O M

φl r1( ) φl r2( ) K φl rA( )

= al+K a j

+ai+ 0

NNPSS: July 9-11, 2007 30

Shell model applications

• The practical Shell Model1. 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. We start from the premise that the effective interaction exists

3. We use effective interaction theory to make a first approximation (G-matrix)

4. Then tune specific matrix elements to reproduce known experimental levels

5. 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 31

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

• 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

Question # 1?

NNPSS: July 9-11, 2007 32

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)

1845.21,0;001,0;00

3247.11,0;001,0;11

1246.21,0;111,0;11

0835.11,0;111,0;00

1856.31,0;001,0;00

8197.21,0;001,0;00

2/32/32/32/3

2/32/32/12/1

2/12/12/12/1

2/12/12/52/5

2/32/32/52/5

2/52/52/52/5

−=====

−=====

−=====

−=====

−=====

−=====

TJddVTJdd

TJddVTJss

TJssVTJss

TJssVTJdd

TJddVTJdd

TJddVTJdd

64658.1

16354.3

94780.3

2/3

2/1

2/5

0

1

0

=

−=

−=

d

s

d

ε

ε

ε

Measured relative to Measured relative to 1616O coreO coreNote 0dNote 0d3/23/2 is unbound is unbound

NNPSS: July 9-11, 2007 33

Simple application of the shell model, cont.

Example:

• 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

Finding the eigenvalues

€

H =

−10.7153 −1.0835 −3.1856

−1.0835 −8.4517 −1.3247

−3.1856 −1.3247 1.1087

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

00++ -12.171-12.171 0.0000.000

-7.851-7.851 4.3204.320

1.9641.964 1.9641.964

-1.616-1.616 10.55510.555

-6.445-6.445 5.7265.72633++

-8.389-8.389 3.7813.781

-3.421-3.421 8.7508.750

44++

-9.991-9.991 2.1802.180

-7.732-7.732 4.4404.440

-1.243-1.243 10.92810.928

3.5223.522 3.5223.522

-2.706-2.706 9.4659.465

22++

-1.348-1.348 10.82310.823

-0.830-0.830 11.34111.341

11++

|(0d5/2)2J=0 |(1s1/2)2J=0 |(0d3/2)2J=0

NNPSS: July 9-11, 2007 34

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 35

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 Φ0 so that

• Thouless’ theorem– Any other Slater determinant Φ not orthogonal to Φ0 may be written

as

– Where i is a state occupied in Φ0 and m is unoccupied

– Then

€

Φ0 H Φ0

Φ0 Φ0

= minimum δΦ0 H Φ0 + Φ0 H δΦ0 = 0

Φ0 = ai+ 0

i=1

A

∏

€

Φ=exp Cmiam+ ai

mi

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥Φ0

€

δΦ0 = Cmiam+ ai Φ0

im

∑ and Φ0 δΦ0 = Cmi Φ0 am+ ai Φ0 = 0

im

∑

NNPSS: July 9-11, 2007 36

Hartree-Fock

• Let i,j,k,l denote occupied states and m,n,o,p unoccupied states

• After substituting back we get

• This leads directly to the Hartree-Fock single-particle Hamiltonian h with matrix elements between any two states and

€

H =pi

2

mi

∑ +1

2V ri − r j( )

ij

∑ = T +1

2Vij

ij

∑

€

h β = α T β + jα V jβA

j

occupied

∑

= α T β + α U β

€

m T i + jm V jiA

j

occupied

∑ = 0

NNPSS: July 9-11, 2007 37

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

• We obtain an eigenvalue equation (more on this later)

• Energies of A+1 and A-1 nuclei relative to A

€

m T i + jm V jiA

j

occupied

∑ = m h i = 0

€

h i = ε i i

E = Φ0 H Φ0 = i T i +1

2ij V ij

Aij

∑i

∑ =1

2i T i + ε i[ ]

i

∑

iAAmAA EEEE εε −=−=− −+ 11

NNPSS: July 9-11, 2007 38

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.,

– From the variational principle, we obtain the eigenvalue equation

€

i = Ciα αα

∑

Ciα* C jα

α

∑ = δij Ciα* Ciβ

i

∑ = δαβ

€

T β + αj V βjA

j

occupied

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥Ciβ

β

∑ = ε iCiα

or

α h β Ciβ

β

∑ = ε iCiα

NNPSS: July 9-11, 2007 39

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 steps1. 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=δi

• With this ansatz, set up the eigenvalue equations and solve them

• Use the eigenstates |i from step 2 to make the Slater determinant Φ0, go back to step 2 until the coefficients Ci are unchanged

The Hartree-Fock equations are solved self-consistently

NNPSS: July 9-11, 2007 40

Hartree-Fock – Coordinate space

• Here, we denote the single-particle wave functions as ϕi(r)

• These equations are solved the same way as the matrix eigenvalue problem before

— Make a guess for the wave functions ϕi(r) and Slater determinant Φ0

— Solve the Hartree-Fock differential equation to obtain new states ϕi(r)

— With these go back to step 2 and repeat until ϕi(r) are unchanged

€

−h2

2m∇1

2φi r1( ) + φ j* r2( )V r1 − r2( )φ j r2( )d3r2∫

j

occupied

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟φi r1( ) − φ j

* r2( )V r1 − r2( )φ j r1( )∫j

occupied

∑ φi r2( )d3r2 = ε iφi r1( )

Direct or Hartree term: UDirect or Hartree term: UHHExchange or Fock term: UExchange or Fock term: UFF

Again the Hartree-Fock equations are solved self-consistently

NNPSS: July 9-11, 2007 41

Hartree-Fock

• 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

• Use the coordinates and to show the exact energy and wave function are

• Note that since the spin wave function (S=0) is anti-symmetric, the spatial wave function is symmetric

Hard homework problem:

€

H = 12 p1

2 + r12

( ) + 12 p2

2 + r22

( ) + χ 12

r r 1 −

r r 2( )

2

[ ]

( ) 2rrr 21

rrr−= ( ) 2rrR 21

rrr+=

€

E = 32 h 1+ 2χ +1( )

Ψ r,R( ) =1

πh( )3 4 exp −R2 2h( )

2χ +1

πh

⎛

⎝ ⎜

⎞

⎠ ⎟

3 4

exp −2χ +1

2hr2

⎛

⎝ ⎜

⎞

⎠ ⎟

NNPSS: July 9-11, 2007 42

Hartree-Fock

• 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:

Hard homework problem:

€

E H = 3h χ +1

Ψ r1,r2( ) =χ +1

πh

⎛

⎝ ⎜

⎞

⎠ ⎟

3 2

exp −χ +1

2hr1

2 ⎛

⎝ ⎜

⎞

⎠ ⎟exp −

χ +1

2hr2

2 ⎛

⎝ ⎜

⎞

⎠ ⎟

0 5 10 15 20 25 300

5

10

15

20

E/h

χ

Eexact

EH

NNPSS: July 9-11, 2007 43

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

– P 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

€

v12 = t0 1+ x0Pσ( )δr r 1 −

r r 2( ) +

1

2t1 1+ x1Pσ( ) δ

r r 1 −

r r 2( )

1

2i

r ∇12 −

r ∇ 22

( ) +1

2i

s ∇12 −

s ∇ 22

( )δr r 1 −

r r 2( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥+

t2 1+ x2Pσ( )1

2i

s ∇1 −

s ∇ 2( ) ⋅δ

r r 1 −

r r 2( )

1

2i

r ∇1 −

r ∇ 2( ) + W0

r σ 1 +

r σ 2( ) ⋅

1

2i

s ∇1 −

s ∇ 2( ) ×δ

r r 1 −

r r 2( )

1

2i

r ∇1 −

r ∇ 2( ) +

t3δr r 1 −

r r 2( )δ

r r 2 −

r r 3( )

€

1

6t3 1+ x3Pσ( )δ

r r 1 −

r r 2( )ρ α r

r 1 +r r 2( ) 2( )

NNPSS: July 9-11, 2007 44

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

– Note the effective mass m*

– 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

€

− r

∇ ⋅h2

2mτ z

* r r ( )

r ∇ + Uτ z

r r ( ) − i

r W τ z

r r ( ) ⋅

r ∇ ×

r σ ( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥φτ z

i r r ( ) = ε iφτ z

i r r ( ) zz labels protons or labels protons or

neutronsneutrons

€

h2

2mτ z

* r r ( )

=h2

2m+

1

4t1 + t2( )ρ

r r ( ) +

1

8t2 − t1( )ρ τ z

r r ( )

NNPSS: July 9-11, 2007 45

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 46

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:

€

H =r p i

2

2mi

∑ + VNN

r r i −

r r j( )

i< j

∑

€

H =r p i

2

2m+ U ri( )

⎛

⎝ ⎜

⎞

⎠ ⎟

i

∑ + VNN

r r i −

r r j( )

i< j

∑ − U ri( )i

∑

Residual interactionResidual interaction

Perturbation TheoryPerturbation Theory

NNPSS: July 9-11, 2007 47

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 48

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)H. Sagawa, PRC65, 064314 (2002) The tail of the radial density The tail of the radial density depends on the separation depends on the separation energyenergyS. Mizutori et al. PRC61, 044326 (2000)S. Mizutori et al. PRC61, 044326 (2000)

Drip-line studiesDrip-line studiesJ. Dobaczewski et al., PRC53, 2809 (1996)

NNPSS: July 9-11, 2007 49

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

J. Dobaczewski et al., PRC53, 2809 (1996)

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!

NNPSS: July 9-11, 2007 50

Beyond mean field

• Hartee-Fock is a good starting approximation– There are no particle-hole corrections to the HF ground state

– The first correction is

• 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 higher-

order 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)

0==+ ∑ ihmjiVjmiTmoccupied

jA

∑ −−+ijmn nmji

AAijVmnmnVij

εεεε4

1

You should look these up!You should look these up!A Shell Model Description of Light Nuclei, I.S. TownerA Shell Model Description of Light Nuclei, I.S. TownerThe Nuclear Many-Body Problem, Ring & SchuckThe Nuclear Many-Body Problem, Ring & Schuck

NNPSS: July 9-11, 2007 51

Nuclear structure in the future

Ab initio methods

Standard shell modelMean-fie

ld, Hartre

e-Fock

Monte Carlo Shell Model

With newer methods and powerful computers, the future of nuclear structure theory is bright!

The Physics of Nuclei II: Nuclear Structure

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