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Seniority. A really cool and amazing thing. 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?
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Seniority A really cool and amazing thing
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?
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)
Seniority Scheme – Odd Tensor Operators (e.g., magnetic dipole M1) Fundamental Theorem * 0 + even ≠ odd
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 Recall: 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 jnjv 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 injnconfiguration, let’s consider them explicitly: Starting from V0δαα΄ = 0 if v = 0 or 1 No 2-body interaction in zero or 1-body systems Hence, only second term: (n even, v = 0) (nodd, v = 1) These equations simply state that the ground state energiesin the respective systemsdepend 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 = Independent ofn!! 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 =
Foundation Theorem for Seniority For odd tensor interactions: < j2ν J′│Ok│j2J = 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 ΔE│ ≡ E(ν = 2, J) – E(ν = 2, J) =constant ν 8+ 6+ 4+ ν = 2 ν= 2 2+ ν = 0 ν = 0 0+ 2 4 6 8 n jnConfigurations
To summarize two key results: • For odd tensor operators, interactions • One-body matrix elements (e.g., dipole moments) are independent of nand 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. 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. Another example: Consider states of a j2 configuration: 0,2,4,6, …: The 2,4,6, … have seniority 2, so the B(E2: 4+ 2+) is an even tensor ( cause E2) seniority conserving transition, and hence follows the above rule. We will see an example of this next.
