Transcript Seniority

Seniority
• Enormous simplifications of shell model calculations,
reduction to 2-body matrix elements
• Energies in singly magic nuclei
• Behavior 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. The essential theorem
amounts to “odd + 0 doesn’t equal even” !!
How to simplify the calculations?
Note a key result for 2-particle systems
Tensor Operators
Don’t be afraid of the fancy name.
Ylm
e.g.,
Y20
Even, odd tensors:
Quadrupole Op.
k even, odd
To remember: (really important to know)!!
δ interaction is equivalent to an odd-tensor interaction
(explained in deShalit and Talmi)
Or, this:
O
Seniority Scheme – Odd Tensor Operators
(e.g., magnetic dipole M1)
Fundamental Theorem
*0
+ even ≠ odd
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.
J0



0
lowest energy for jn v J V jn v J
occurs for smallest v, largest n  v V0
2
largest lowering is for all particles coupled to J = 0
lowest energy occurs for
v =0
(any unpaired nucleons contribute less extra binding from the residual interaction.)
v = 0 state lowest for e – e nuclei
v = 1 state lowest for o – e nuclei
Generally, lower v states lie lower than high v
a)
g.s. of e – e nuclei have v = 0
b)
Reduction formulas of ME’s
jn  jv
achieve a huge simplification
J = 0+!
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.
n
jn v  J  Vik jn v  J 
ik
v
jv v  J  Vik jv v  J 
ik
nv
2
V0 δαα΄
= 0 if v = 0 or 1
No 2-body interaction in
zero or 1-body systems
n
jn J = 0  Vik jn J = 0  n2 V0
(n even, v = 0)
ik
n
j J = j  Vik jn J = j 
n
ik
n 1
2
V0
(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
j v = 2, J   E jn , v = 0, J= 0
 n,


E
Independent of n !!




j2 J V j2 J  n 2 2 V0 
=
j2 J V j2 J  V0
n
2
V0
Constant
Spacings between v = 2 states in jn (J = 2, 4, … j – 1)
E jn , v = 2, J   E jn , v = 2, J






n-2

  j2 J V j2 J 
V0 
2

 j2 J V j2 J 

n2
2
V0
= j2 J V j2 J  j2 J V j2 J
= E j2 , v  2, J - E j2 , v  2, J
All spacings constant !
Low lying levels of jn configurations (v = 0, 2) are independent of number of particles in orbit.
Can be generalized to  =  
i
i
jini
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 midshell.
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)
For odd tensor interactions:
< j2 ν J′│Ok│j2 J = 0 > = 0 for k odd, for all J′ including J′ = 0
Proof: even + even ≠ odd
n
j J Vik j J
n
n

j J Vik j  J

=
ik
+
ik
n
2
V0
No. pairs
x pairing int.
V0 < 0
Int. for J ≠ 0
ν = 0 states lie lowest
g.s. of e – e nuclei are 0+!!
ΔE ≡ E(ν = 2, J) – E(ν = 0, J = 0) = constant
ΔE│ ν ≡ E(ν = 2, J) – E(ν = 2, J) = constant
ν=2
8+
6+
4+
ν=2
2+
ν=0
n
0+
2
4
6
jn Configurations
8
ν=0
When is seniority a good quantum number?
(let’s talk about
jn J configurations)
•If, for a given n, there is only 1 state of a given J
Then nothing to mix with.

v is good.
•Interaction conserves seniority: odd-tensor interactions.
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.