Seniority A 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] Short range attractive interaction – what are the energies? 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

Seniority Scheme – Odd Tensor Operators (e.g., magnetic dipole M1) Fundamental Theorem * 0 + even ≠ odd

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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. Recall: J  0  V0 lowest energy for occurs for smallest v, largest  largest lowering is for all particles coupled to J = 0  v = 0 lowest energy occurs for (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 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. So: g.s. of e – e nuclei have v = 0 J = 0+! Reduction formulas of ME’s jn  jv achieve a huge simplification n-particle systems  0, 2 particle systems

Since v = 0 ( e – e) or v = 1 ( o – e) states will lie lowest in jn configuration, let’s consider them explicitly: Starting from V0 δαα΄ = 0 if v = 0 or 1 No 2-body interaction in zero or 1-body systems Hence, only second term: (n even, v = 0) (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 E = E = = Independent of n !! Constant Spacings between v = 2 states in jn (J = 2, 4, … j – 1) 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  =

Odd Tensor Interactions Foundation Theorem for Seniority 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 Odd Tensor Interactions = + V0 Int. for J ≠ 0 No. pairs x pairing int. V0 < 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 g.s. of e – e nuclei are 0+!! ΔE│ ≡ E(ν = 2, J) – E(ν = 2, J) = constant ν 8+ 6+ 4+ 2+ ν = 2 ν = 0 0+ ν = 0 2 4 6 8 n 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)

So: Remarkable simplification if seniority is a good quantum number When is seniority a good quantum number? (let’s talk about 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.

Think of levels in Ind. Part 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 !!! 7/2

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+

SUMMARY