Question

4. When an external magnetic field B is applied, a "spin-1" ion has 3 magnetic states with energies given Em=aBm, m=-1,0,1, where a is a constant of order a few times the Bohr magneton up = en/(2m). (Note: the notation here is quite different from that of Kittel & Kroemer who use "m" for the elementary magnetic moment which we have denoted a. In our terminology, m=ms is an integer quantum number: m=-1 labels the "spin down" state, m=0 labels the "spin middle" state and m=+1 labels the "spin-up" state). (a) In terms of a, B and T give an expression for the internal energy of the ion, assumed to be acted on by a heat bath. (b) Also give an expression for the mean value, <m>, of the spin projection quantum number m. (Note: both "< >" and a "bar over the top" are often used to denote a thermal average, so you could equally well have been asked for m.) (c) Show that U and <m> are proportional to 1/T at "high" temperatures, so that the Curie law of magnetism holds for spin-1 paramagnets as well as for the spin-1/2 paramagnet treated in the notes. What constitutes a "high" temperature in this context? (Hint: e 1 +x for x << 1.)

Answer 1

La When an apy When an external magnetic field applied to a spin i system the energies are given an Em = a Bm. m = 0,47 Where a is a const. of the order of a few torines the Botor magrelos Us Events then internal energy is given as- ABM uzoen 7 (shera 7 = B-k] - Baßm a poco Baßm =+ Žalmyė Ba ¿Bemas a tßaß - aß e tot aße Baß ßaß aB & Bar Site an tepaB -able U = 1 +26001 lektionen - 2a 8 Sin Á ( a) ) It 2 eoth (as I s 16) the mean value of opin co. m is written projection quantum my= afm = 12 +0+le

- 2 sinh letes ME Tt2 cash created In the high aB 441 temperature limil- then we use the fact that Teritx, ēr 1-se then sin helen- * L N 23 d a la -17 TRT 2 Coth (AR) enero e o 11 11 then we get U. - lala <my - - 2. de . 172 en 1+2 High temperature means we are not consider the quantum detail of the system