Question

1 Conmder an isolated syatem consistng of a large mumber N of very weakly interactung localized particles of spin 1/2 Each particle has an intrinmc magnetic momnent which can pont ether parallel or antparallel to an appled field H The energy E of the syatem is then gven by E--uH (n,-n), where n, is the number of spins algned parallel to H and ng the mumber of spuss aligned antiparallel to H. (a) Write down an expresnon for the denmty of states w (E) of this system Explan (5) (b) Show that the number of microstates between E and E+8E is pven by your reasonung leading to your expreson for wE) (5) N &E a(E)- N/2-E/ (2pH)" (N/2+ E/(2pH) 2pH (e) By makung approprate appraxamations, ahow that 2 (B) the form of a Gausan, can also be wntten in E n (E)- Cexp V2NuH where C is a constant Clearly explan any approxcumationa and/or assumptions you make un obtain the result (15) (25]

Answer 1

Let $n_{1}$ be the number of particles aligned parallel to H with energy $E_{1}=-\mu Hn_{1}$ and $n_{2}$ be the number of particles aligned anti parallel to H with energy $E_{2}=+\mu Hn_{2}$ . Total energy .

EE

Total number of particles N is given by .

N=n1n2

Since density of states is given by
w(E)dE dE
, We will find
(E)
first.

(b)

E= -HH(n-n,) E:-en-NHH ACE) RCni) SE (n) NI ni! CN-ni SSince dE2H SE nCE) ani n (N-n, MH 2-E MM SE ACE):N! 2pH ):n ;(nk N tE E 2

(c)

taking In nCE) -In 2NH In NI In Ln ACE) 2 2 ac to stirlings approximaton In n nlnn -n In nCE) N In N -In2M N+E 2HM 2 2 2 Now t E HH In N Ln 2 2HH (DA ) ln N + In In 2 2 MH in AlE) afier Subshituton gves Ln In nCE)= N In 2 X 2N

Un nCE) In 2MH E 2NLHH) N In 2 ACE) Cep E 2. where 2N HH C = 2N

(a)

wCE) dA (E) n CE)= C ( fom part (c)) -2EC erf EN CHM) dennity of stat es wLE) F 2 2N ( Where wLE) is -EC 2 wCE)= NH 2NCH C 2 2E 2N CHSS w CE) ,2N(MH)