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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

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Seniority

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  1. 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” !!

  2. How to simplify the calculations? Note a key result for 2-particle systems

  3. 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)

  4. You can have 200 pages of this….

  5. Or, this: O

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

  7. Yaaaay !!!

  8. 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 jnjv achieve a huge simplification n-particle systems 0, 2 particle systems

  9. 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”

  10. 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  =

  11. 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)

  12. 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

  13. 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.

  14. 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

  15. 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.

  16. SUMMARY

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