SeniorityA really cool and amazing thing

that is far more powerful than the casual way it is often invoked.

It is the foundation, for example, for:

• Enormous simplifications of shell model calculations, reduction to 2- body matrix elements• Energies in singly magic nuclei• Behaviour of g factors• Parabolic systematics of intra-band B(E2) values and peaking near

mid-shell• Preponderance of prolate shapes at beginnings of shells and of oblate shapes near shell ends

The concept is extremely simple, yet often clothed in enormously complicated math(s). The essential theorem amounts to “odd + 0 doesn’t equal even” !!

Seniority

• First, what is it? Invented by Racah in 1942. • Secondly, what do we learn from it?• Thirdly, why do we care – that is, why not just do full shell

model calculations and forget we ever heard of seniority? • Almost completely forgotten nowadays because big fast

computers lessen the need for it. However, understanding it can greatly deepen your understanding of structure and how it evolves.

• Start with shell structure and 2- particle spectra – they give the essential clue.

Magic nuclei – single j configurations: jn [ e.g., (h11/2)2 J]

0 + states lie lowest, by far. Pairs of

identical nucleons prefer to couple of

angular momentum zero

Tensor Operators

Don’t be afraid of the fancy name.

Ylm

e.g., Y20 Quadrupole Op.

Even, odd tensors: k even, odd

To remember: (really important to know)!!

δ interaction is equivalent to an odd-tensor interaction

(explained in deShalit and Talmi)

You can have 200 pages of this….

Or, this:

O

Fundamental Theorem

* 0 + even ≠ odd

Seniority Scheme – Odd Tensor Operators(e.g., magnetic dipole M1)

Yaaaay !!!

Now, use this to determine what v values lie lowest in energy.For any pair of particles, the lowest energy occurs if they are coupled to J = 0.

0

J 0

n nv vj J V j J 2n v

v = 0

lowest energy for occurs for smallest v, largest V0

largest lowering is for all particles coupled to J = 0

lowest energy occurs for

(any unpaired nucleons contribute less extra binding from the residual interaction.)

v = 0 state lowest for e – e nucleiv = 1 state lowest for o – e nucleiGenerally, lower v states lie lower than high v

a)g.s. of e – e nuclei have v = 0 J = 0+!

b)Reduction formulas of ME’s jn jv

achieve a huge simplification

n-particle systems 0, 2 particle systems

THIS is exactly the reason seniority is so

useful. Low lying states have low seniority so all

those reduction formulas simplify the

treatment of those states enormously.

nn n

iki k

v v

J Jj V jv

v vik

i k

v v

J Jj V j n v2

V0 δαα΄

= 0 if v = 0 or 1No 2-body interaction in zero or 1-body systems

2

nn n

iki k

n

0

J = 0 J = 0j V j V (n even, v = 0)

n nn nik

i k

1

2 0J = j J = jj V j V (n odd, v = 1)

These equations simply state that the ground state energies in the respective systems depend solely on the numbers of pairs of particles coupled to J = 0.

Odd particle is “spectator”

Further implications

Energies of v = 2 states of jn

n n, v , v

j =2,J E j =0,J=0 n n 2 20 0

22 2

j J V j J V V

2 20

j J V j J V

=

E

Independent of n !! Constant

Spacings between v = 2 states in jn (J = 2, 4, … j – 1)

n n n, v , v

2 2

0- 2

2j J V j J Vj =2,J E j =2,J

n 2 20

22

j J V j J V

2 2 2 2j J V j J j J V j J

, ,v , - v , 2 2E Ej 2 J j 2 J

=

=

E

All spacings constant !

Low lying levels of jn configurations (v = 0, 2) are independent of number of particles in orbit.

Can be generalized to = ii

niij

< j2 ν J │O′ k│j2 J = 0 > = 0 for k odd, for all J including J = 0′ ′ Proof: even + even ≠ odd

nn n

ik

i k

j J V j J

= ik

i k

j J V j J

+2

n V0

Int. for J ≠ 0 No. pairs x pairing int. V0 < 0

ΔE│ ≡ E(ν = 2, J) – E(ν = 2, J) = constantν

ν = 0 states lie lowestg.s. of e – e nuclei are 0+!!ΔE ≡ E(ν = 2, J) – E(ν = 0, J = 0) = constant

8+

6+

4+

2+

ν = 2ν = 2

0+

nν = 0 ν = 084 62

jn Configurations

To summarize two key results:

For odd tensor operators, interactions

• One-body matrix elements (e.g., dipole moments) are independent of n and therefore constant across a j shell

• Two-body interactions are linear in the number of paired particles, (n – v)/2, peaking at mid- shell.

The second leads to the v = 0, 2 results and is, in fact, the main reason that the Shell Model has such broad applicability (beyond n = 2)

v is good.

•Interaction conserves seniority: odd-tensor interactions.

nj J

When is seniority a good quantum number?When is seniority a good quantum number?

(let(let’’s talk about configurations) s talk about configurations)

•If, for a given n, there is only 1 state of a given J

Then nothing to mix with.

7/2

Think of levels in Ind. Part. Model: First level with j > 7/2 is g9/2 which fills from 40- 50.

So, seniority should be useful all the way up to A~ 80

and sometimes beyond that !!!

This is why nuclei are prolate at the beginning of a shell and (sometimes) oblate at the end. OK, it’s a bit more subtle than that but this is the main

reason.

A beautiful example

• Consider states of a jn configuration: 0,2,4,6, …

• The J=0 state has seniority zero, the J = 2,4,6, … have seniority 2

• Hence the B(E2: 2+ 0+) is an even tensor (because E2) seniority changing transition.

• The B(E2: 4+ 2+) is an even tensor seniority conserving transition.

• They therefore follow different rules as a function of n

2+