Supersymmetry in Nuclei - American Chemical Society

1 Supersymmetry in Nuclei Recent Developments

Downloaded by KAOHSIUNG MEDICAL UNIV on June 6, 2018 | https://pubs.acs.org Publication Date: November 14, 1986 | doi: 10.1021/bk-1986-0324.ch001

F. Iachello A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, CT 06511

Recent developments in applications of supersymmetry to the study of complex spectra are discussed. In particular, extensions of supersymmetry to nuclei with an odd number of protons and neutrons are presented. 1.Introduction Since its introduction in nuclear physics in 1980 [IAC80], supersymmetry has been extensively used to describe properties of complex nuclei [BAL81a,BAL81b,BAL83,SUN83a,SUN83b,CAS84,IAC85]. In its original formulation, supersymmetry was used to link properties of nuclei with an even number of protons and neutrons (even-even nuclei) with those of nuclei with an odd number of protons and an even number of neutrons (odd-even nuclei) or viceversa (even-odd nuclei). This formulation was based on a description of properties of nuclei in terms of a set of collec tive variables (bosonic in nature) and a set of single particle variables (fermionic in nature). In treating the collective vari ables, no distinction was made between protons and neutrons (interacting boson model-1). It has been recognized since, that the proton-neutron degree of freedom plays an important role in the description of some properties of nuclei and thus i t must be explicitly introduced (interacting boson model-2). In addition to providing a better understanding of the properties previously described, the introduction of the proton-neutron degree of freedom leads to the possibility of describing properties of nuclei with an odd number of protons and neutrons (odd-odd nuclei). In this lecture, I w i l l discuss some properties of oddodd nuclei as obtained from supersymmetry. 0097-6156/ 86/ 0324-0002S06.00/ 0 © 1986 American Chemical Society

Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

Supersymmetry in Nuclei

IA C H E L L O

2.The

model In

the

states

interacting

of nuclei

[ARI77,OTS78 ] , and

neutron

have

I t

(μ=±2,±1,0)

and

and

(ν) degrees

annihilation s

^ν,μ^'


proton

bosons

neutron

v^*

i s

o

t

,

r

s

a

tors

by t >

tion

operators

able

ρ i s called neutron

neutrons. number

the (

)

When

(

(s-boson)

introduce

creation

operators.

of freedom assume

an

a r e added,

extra

label

I shall

bosons

The s i x p r o t o n

When

a n d J=2 ( d ^ *, s * ) proton

(π)

the creation

and

(π,ν),

denote

d^

s_*,

t h e 12

opera

ρ=ττ,ν. T h e c o r r e s p o n d i n g

will

be d e n o t e d

by b p . T h e t w o - d i m e n s i o n a l

F-spin.

correspond

)

nucleus, (neutron)

i s built

B

V

noting

to correlated

collective

annihila vari

a

I t i s worthwhile

f o r each

only

U

to

dynamical

bosons. J=0

(dy,s)

simplicity,

of proton

ν

(6) ®

f

momentum

12

collective

α = 1,...,6;

Hamiltonian Β

π

o

of

lying

where

Thus,

π

e

low

i n terms

angular

convenient

bosons

Ν (Ν )

shells.

ϋ

a p

k

model-2,

and s i xn e u t r o n

annihilation

operators

F

boson

are described

s i x

(d-boson).

and

3

their pairs

degrees

that

pairs

number outside

of protons

and

i s fixed

to the

t h e major

closed

of freedom

o u t o f t h e 36+36

the proton

are considered,

generators

of t h e group

(6),

(B) 1 u

G a„ ρ , · α ρ- ~b " a p b„ a ',p_ J

and

n

Λ

1ft

i s given

by

(Β) Η

( 1 )

u

(B) = Ε

+ 0

Χ L

(ρ)

(Β)

ε

α,α',ρ

G α,α'

αρ,α'ρ

(ρ,τ)

2

In U

this (

V

B

)

u α , α ' ,β,β ' ,ρ, τ

equation,

B

E^ ^

0

G α,α',β,Β'

i s assumed

(Β)

(Β)

G αρ,βρ

t o be i n v a r i a n t

(2) α'τ,β'τ

under

(6).


Β

υ" ^ ^(6)< π

4

N U C L E I O F F T H E LINE O F STABILITY

In single

addition

ticles.

The

ticle

levels.

The

momentum

several

single

^ (2j +1).

is

p a r t i c l e s are

fermions

is

(2j+1).

for of

protons

to

is

label

introduce

the

an

extra isospin

The

fermions ( π

Π

can

( Ω ) β (F)

be

υ

(

Γ

)

ν

G

=

written

in

terms

with

the

Ω

occupy is

a

.

Ω and

If

the

creation

and

a

label,

Hamiltonian

of

level

label

2

υ

par

creation

(ί=1,...,Ω),

i

single

degeneracy

i s the

neutrons.

also par

p a r t i c l e can

considered,

acquire

has

unpaired

particle

total

a^

one

the

occupying

single

here

freedom

are

unpaired

the

also

extra

from

a the

fermions,

operators

( i = 1 ,...,Ω ; r = 1 , 2 ) . T h e

of If

levels,

convenient

degree

annihilation

freedom,

. These

degeneracy

j

of

freedom

operators

proton-neutron

tinguishes

of

particle

It

i

annihilation


c o l l e c t i v e degrees

unpaired

angular

i

to

p a r t i c l e degrees

^ i r

which

dis

describing

the

generators

of

2 +

Ω

(Ω), a

f

a

i r i

ir

a,

i

1f n

(3)

r

ΙΓ,ΓΓ

and

i s given

by

(F)

(F)

H

=

Ε

Λ +

X

0

1

2

Q

U

v

the

' ( Ω ) .

States

in

number

denoted

by

of

unpaired for

This

one

ir , i' r

to

nucleus

fermions

protons has

interaction

(F) G ir,kr

1

assumed

each

M^

Finally, neutrons.

1

is

(r) G

(r,t) ν ii'kk

i ,i ,k ,k ,r ,t

E

The quantity ( F)

(r) ^ε i ,i' ,r ii'

be can

they for

thus

an

interaction

is written

be

contain.

M

(ii)

1

invariant

and

v

(F) G i· t,k t

under

;

char a c t er i zaed This

number

will

( Ω ) ® by be

neutrons. between

protons

as


and

1.

(BF)

Υ

V

(ρ,r) w αα'ϋ'

L

=

1

1

α ,α ,i , i , ρ , r

The

algebraic

Η


is

.

structure

H< > B

then

5

Supersymmetry in Nuclei

IACHELLO

H< >

that

+

F

+

of

υ

( Β ) π

of

V

Supersymmetry in Nuclei - American Chemical Society

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