QMPT 540

Quantum Theory of Many-Particle Systems, Phys. 540

• IPM?

• Atoms?

• Nuclei: more now

• Other questions about last class?

• Assignment for next week Wednesday --->

• Comments?

QMPT 540

Nuclear shell structure• Ground-state spins and parity of odd nuclei provide further

evidence of “magic numbers”

• Character of magic numbers may change far from stability (hot)

A. Ozawa et al., Phys. Rev. Lett. 84, 5493 (2000)

• N=20 may disappear and N=16 may appear

Odd-N even Z

Odd-N even Z

Tz=1/2

Tz=9/2

Tz=0

Tz=5

odd

QMPT 540

Empirical potential• Analogy to atoms suggests finding a sp potential ⇒ shells + IPM

• Difference(s) with atoms?

• Properties of empirical potential– overall?

– size?

– shape?

• Consider nuclear charge density

Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987)

QMPT 540

Nuclear density distribution• Central density (A/Z* charge density) about the same for nuclei

heavier than 16O, corresponding to 0.16 nucleons/fm3

• Important quantity

• Shape roughly represented by

•

• Potential similar shape

�ch(r) =�0

1 + exp�

r�cz

⇥

c � 1.07A13 fm

z � 0.55fm

QMPT 540

Empirical potential• BM1

• Central part roughly follows shape of density

• Woods-Saxon form

• Depth+ neutrons

- protons

• radius with

• diffuseness

U = V f(r) + V�s

�� · s

�2

⇥r201r

d

drf(r)

f(r) =⇤1 + exp

�r �R

a

⇥⌅�1

V =⇤�51± 33

�N � Z

A

⇥ ⌅MeV

R = r0 A1/3 r0 = 1.27 fm

a = 0.67 fm

QMPT 540

Analytically solvable alternative• Woods-Saxon (WS) generates finite number of bound states

• IPM: fill lowest levels ⇒ nuclear shells ⇒ magic numbers

• reasonably approximated by 3D harmonic oscillator

• Eigenstates in spherical basis0 1 2 3 4 5 6 7 8 9 10

r (fm)

!60

!50

!40

!30

!20

!10

0

10

20

V(r

) (M

eV

)

UHO(r) =12m�2r2 � V0

A=100

H0 =p2

2m+ UHO(r)

HHO |n⌅m⇤ms⇥ =���(2n + ⌅ + 3

2 )� V0

⇥|n⌅m⇤ms⇥

QMPT 540

Harmonic oscillator• Filling of oscillator shells

• # of quanta

# of particles “magic #” parity 0 0 0 2 2 + 1 0 1 6 8 - 2 1 0 2 + 2 0 2 10 20 + 3 1 1 6 - 3 0 3 14 40 - 4 2 0 2 + 4 1 2 10 +

4 0 4 18 70 +

N = 2n + ⇥

N n ⇥

QMPT 540

Need for another type of sp potential• 1949 Mayer and Jensen suggest the need of a spin-orbit term

• Requires a coupled basis

• Use to show that these are eigenstates

• For eigenvalue

• while for

• so SO splits these levels! and more so with larger

|n(�s)jmj� =�

m�ms

|n�m�ms� (� m� s ms| j mj)

� · s = 12 (j

2 � �2 � s2)

� · s

�2|n(�s)jmj⇤ = 1

2

�j(j + 1)� �(� + 1)� 1

2 ( 12 + 1)

⇥|n(�s)jmj⇤

�

j = � + 12

j = �� 12

12�

� 12 (� + 1)

QMPT 540

Inclusion of SO potential and magic numbers• Sign of SO?

• Consequence for

• Noticeably shifted

• Correct magic numbers!

V�s

�� · s

�2

⇥r201r

d

drf(r)

0f 72

0g 92

0h 112

0i 132

V�s = �0.44V

QMPT 540

208Pb for example• Empirical potential & sp energies

• A+1: “sp energies” directly from experiment

• A-1:

• also directly from

• Shell filling for nuclei near stability follows empirical potential

H0 a� |208Pbg.s.⇥ =�E(208Pbg.s.)� ��

⇥a� |208Pbg.s.⇥

H0 a†� |208Pbg.s.� =��� + E(208Pbg.s.)

⇥a†� |208Pbg.s.�

EA+1n � EA

0

EA0 � EA�1

n

QMPT 540

Comparison with experiment• Now how to explain this potential …

QMPT 540

Closed-shells and angular momentum• Atoms: consider one closed shell (argument the same for more)

• Expect?

• Example: He

• Consider nuclear closed shell

|n⇤m� = ⇤ms = 12 , n⇤m� = ⇤ms = � 1

2 , ...n⇤m� = �⇤ms = 12 , n⇤m� = �⇤ms = � 1

2 ⇥

|(1s)2⌃ =1�2{|1s ⇥ 1s ⇤)� |1s ⇤ 1s ⇥)}

= |(1s)2 ;L = 0S = 0⌃

|�0⇥ = |n(⇤ 12 )jmj = j, n(⇤ 1

2 )jmj = j � 1, ..., n(⇤ 12 )mj = �j⇥

QMPT 540

Angular momentum and second quantization• z-component of total angular momentum

• Action on single closed shell

• Also

• So total angular momentum

• Closed shell atoms

Jz =�

n�jm

�

n���j�m�

�n⇥jm| jz |n�⇥�j�m�⇥ a†n�jman���j�m�

=�

n�jm

�m a†n�jman�jm

Jz |n⌅j;m = �j,�j + 1, ..., j⇤ =�

m

�m a†n�jman�jm |n⌅j;m = �j,�j + 1, ...,= j⇤

=⇥ j�

m=�j

�m⇤

|n⌅j;m = �j,�j + 1, ..., j⇤

= 0⇥ |n⌅j;m = �j,�j + 1, ..., j⇤

J± |n⌅j;m = �j,�j + 1, ..., j⇥ = 0

J = 0L = 0S = 0

QMPT 540

Nucleon-nucleon interaction• Shell structure in nuclei and lots more to be explained on the

basis of how nucleons interact with each other in free space

• QCD

• Lattice calculations

• Effective field theory

• Exchange of lowest bosonic states

• Phenomenology

• Realistic NN interactions: describe NN scattering data up to pion production threshold plus deuteron properties

• Note: extra energy scale from confinement of nucleons

QMPT 540

Nuclear Matter• Nuclear masses near stability

• Data

• Each A most stable

N,Z pair

• Where fission?

• Where fusion?

M(N, Z) =E(N, Z)

c2= N mn + Z mp �

B(N, Z)c2

0 50 100 150 200 250A

7.5

8

8.5

9

B/A

(MeV

)

QMPT 540

Nuclear Matter• Smooth curve

• volume

• surface

• symmetry

• Coulomb

Great interest in limit: N=Z; no Coulomb; A ⇒∞

Two most important numbers in Nuclear Physics

B = bvolA� bsurfA2/3 � 12bsym

(N � Z)2

A� 3

5Z2e2

Rc

bvol = 15.56 MeVbsurf = 17.23 MeVbsym = 46.57 MeV

Rc = 1.24A1/3 fm

B

A� 16 MeV �0 � 0.16 fm3

QMPT 540

Saturation problem of nuclear matterGiven VNN ⇒ predict correct minimum of E/A in nuclear matter as a function of density inside empirical box

Describe the infinite system of neutrons

⇒ properties of neutron stars

1 1.2 1.4 1.6 1.8 2 2.2kF (fmï1)

ï25

ï20

ï15

ï10

ï5En

ergy

/A (

MeV

)HJ

BJRSC

AV14 ParHM1 C

B

A

Sch

UNG

Par

AV14B

7%6.6%

6.5%5.8%

6.1%

5%

4.4%

QMPT 540

Isospin• Shell closures for N and Z the same!!

• Also 939.56 MeV vs. 938.27 MeV

• So strong interaction Hamiltonian (QCD) invariant for p ⇔ n

• But weak and electromagnetic interactions are not

• Strong interaction dominates ⇒ consequences

• Notation (for now) adds proton

adds neutron

• Anticommutation relations

• All others 0

mnc2 � mpc2

{p†�, p⇥} = ��,⇥

{n†�, n⇥} = ��,⇥

p†�

n†�

QMPT 540

Isospin• Z proton & N neutron state

• Exchange all p with n

• and vice versa

• Expect

• Consider

• will also commute with

|�1�2...�Z ;⇥1⇥2...⇥N � = p†�1p†�2

...p†�Zn†

⇥1n†

⇥2...n†

⇥N|0�

T+ =�

�

p†�n�

T� =�

�

n†�p�

[HS , T±] = 0

T3 =12[T+, T�] =

12

⇤

�⇥

�p†�n�n†

⇥p⇥ � n†⇥p⇥p†�n�

⇥

=12

⇤

�⇥

�p†�p⇥��,⇥ � n†

⇥n���,⇥

⇥=

12

⇤

�

�p†�p� � n†

�n�

⇥

HS

QMPT 540

Isospin• Check

• Then operators

obey the same algebra as

so spectrum identical and simultaneously diagonal !

proton

neutron

For this doublet

and

States with total isospin constructed as for angular momentum

[T3, T±] = ±T±

T1 = 12

�T+ + T�

⇥

T2 = 12i

�T+ � T�

⇥

T3

Jx, Jy, Jz

HS , T 2, T3

|rms�p = |rmsmt = 12 �

|rms⇥n = |rmsmt = � 12 ⇥

T 2 |rmsmt� = 12 ( 1

2 + 1) |rmsmt�T3 |rmsmt� = mt |rmsmt�

QMPT 540

Two-particle states and interactions• Pauli principle has important effect on possible states

• Free particles ⇒ plane waves

• Eigenstates of notation (isospin)

• Use box normalization (should be familiar)

• Nucleons

• Electrons, 3He atoms

• Bosons with zero spin (4He atoms)

• Use successive basis transformations for two-nucleon states to survey angular momentum restrictions

• Total spin & isospin; CM and relative momentum; orbital angular momentum relative motion; total angular momentum

|p s = 12 ms t = 1

2 mt⇥ � |pmsmt⇥

|p�

T =p2

2m

|p s = 12 ms⇥ � |pms⇥

( 12 mt2

12 mt1 |T MT ) = (�1)

12+ 1

2�T ( 12 mt1

12 mt2 |T MT )

QMPT 540

Antisymmetric two-nucleon states• Start with

• then

• and use

• as well as

|p1ms1mt1 ;p2ms2mt2⌅ =1⌃2{|p1ms1mt1⌅ |p2ms2mt2⌅ � |p2ms2mt2⌅ |p1ms1mt1⌅}

=1⌃2

�

SMS

�

TMT

{( 12 ms1

12 ms2 |S MS) ( 1

2 mt112 mt2 |T MT ) |p1 p2 S MS T MT )

� ( 12 ms2

12 ms1 |S MS) ( 1

2 mt212 mt1 |T MT ) |p2 p1 S MS T MT )}

P = p1 + p2

p = 12 (p1 � p2)

|p⇥ =�

LML

|pLML⇥ �LML|p⇥ =�

LML

|pLML⇥Y �LML

(p)

|�p⇤ =�

LML

|pLML⇤ ⇥LML| ⇥�p⇤ =�

LML

|pLML⇤ (�1)LY �LML

(p)

Y �LML

( ��p) = Y �LML

(⇥ � �p, ⇤p + ⇥) = (�1)LY �LML

(p)

( 12 ms2

12 ms1 |S MS) = (�1)

12+ 1

2�S ( 12 ms1

12 ms2 |S MS)

QMPT 540

Antisymmetry constraints for two nucleons• Summarize

• must be odd!– Notation T=0 T=1

|p1ms1mt1 ;p2ms2mt2⇤ =1⇧2

⇤

SMST MT LML

( 12 ms1

12 ms2 |S MS) ( 1

2 mt112 mt2 |T MT )Y �

LML(p)

⇥�1� (�1)L+S+T

⇥|P p LMLSMS TMT )

=1⇧2

⇤

SMST MT LMLJMJ

( 12 ms1

12 ms2 |S MS) ( 1

2 mt112 mt2 |T MT ) Y �

LML(p)

⇥ (L ML S MS |J MJ)�1� (�1)L+S+T

⇥|P p (LS)JMJ TMT )

L + S + T

1S0

3P0

3P1

3P2 �3 F2

1D2

3S1 �3 D1

1P1

3D2

...

QMPT 540

Two electrons and two spinless bosons• Remove isospin

• even!

• Two spinless bosons

• even!

L + S

|p1;p2⇥ =1⌅2

⇤

LML

Y �LML

(p)�1 + (�1)L

⇥|P p LML )

L

|p1ms1 ;p2ms2⇤ =1⇧2

⇤

SMSLML

( 12 ms1

12 ms2 |S MS) Y �

LML(p)

⇥�1 + (�1)L+S

⇥|P p LMLSMS )

QMPT 540

Nuclei• Different shells only Clebsch-Gordan constraint

• Uncoupled states in the same shell

• Coupling

• Only even total angular momentum

• With isospin

• J+T odd!

|�jm,jm�� = a†jma†jm� |�0�

|�jj,JM ⇥ =�

mm�

(j m j m⇥ |J M) |�jm,jm�⇥ =�

mm�

(j m⇥ j m |J M) |�jm�,jm⇥

=�

mm�

(�1)2j�J (j m j m⇥ |J M) (�1) |�jm,jm�⇥

= (�1)J�

mm�

(j m j m⇥ |J M) |�jm,jm�⇥

= (�1)J |�jj,JM ⇥

|�jj,JM,TMT ⇥ =�

mm�mtm�t

(j m j m� |J M) ( 12 mt 1

2 m�t |T MT ) |�jmmt,jm�m�

t⇥

= (�1)J+T+1 |�jj,JM,TMT ⇥

QMPT 540

40Ca + two nucleons• Spectrum

QMPT 540

Two fermions outside closed shells• Atoms: two different orbits ⇒ usual Clebsch-Gordan coupling

and no other restrictions

• Two electrons in the same orbit

• Coupling

• even # of states notation

• Carbon L=0 ⇒ S=0 1 1S

• L=1 ⇒ S=1 9 3P g.s.

• L=2 ⇒ S=0 5 1D

|�⇥m�ms,⇥m��m�

s� = a†⇥m�ms

a†⇥m��m�

s|�0�

|�⇧,LML,SMS ⇥ =�

m⇤m�⇤msm�

s

(⌅ m⇧ ⌅ m�⇧ |L ML) ( 1

2 ms 12 m�

s |S MS) |�⇧m⇤ms,⇧m�⇤m�

s⇥

= (�1)L+S |�⇧⇧,LML,SMS ⇥L + S

(2p)2

QMPT 540

Carbon atom• Why S=1 g.s.?

• D below S?