Question

The research shows that alkali metal clusters (with a certain number of atoms in the cluster) exhibit higher stability than others. An example is Natrium, where clusters containing x=2,8,18,20,58 are especially stable. The stability can be explained in different ways, and one of them is based on expanding the quantum mechanical harmonic oscillator from one dimension to three dimensions - as is usually is done for one dimensional wells into three-dimensional wells. Assume an isotropic, three dimensional harmonic oscillator to constuct a solution that can explain part of the cluster sizes described above. (a). What are the energy eigenvalues for the quantum system described above? (b). Find the degeneracy for the different energy states, and also find how many electrons that can occupy each state? (c). Which of the numbers above can be explained by this model?, and what other models could you propose that might explain the stabilities observed experimentally?

Answer 1

a of a dullahar for a Quantum harmonic oscillator energy levels are defined as: E = (Mr+ Ral/₂) twnt (My e/) hwy Deganeracy of na = 0, 1, 2... ny=0,1,2... n₂=0, 1, 2.. | . (nget ya) twz Since we have isshopic oscillator b) Degeneracy of different state is given by 2 2 1 . Wr= Wya Wz=0 E= Mat Bay & na +32) ho E = (n+3) ho where ns met hy ena A 1 -01 3- gr = 16+1)(n+3) n denotes State ire around state, n=o. Ist excited a n al 4 soon for

I c) This model is satisfied by the nos. 2,8, 20 only. since é is a funion By Pauli's exclusion principle each state can accomodate maximum of 2e s. . for any state haning defereracy (gn) No of es = 2 gn where In a degeneracy of Stake n.