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{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\fs28  \tab }{
\fs28\ul STATISTICAL PHYSICS COMPUTER LAB #4\par \tab \tab MAGNETIC COOLING}{
\fs28 \par \par }{\fs24 Magnetic cooling is used to lower temperatures 
below about 1K.  The system used is a paramagnetic salt such as Cr(NO}{
\cf2\fs24\dn 3}{\fs24 )}{\cf2\fs24\dn 3}{\fs24 *9H}{\cf2\fs24\dn 2}{
\fs24 O or FeNH}{\cf2\fs24\dn 4}{\fs24 (SO}{\cf2\fs24\dn 4}{\fs24 )}{
\cf2\fs24\dn 2}{\fs24 *12H}{\cf2\fs24\dn 2}{\fs24 O in which the 
magnetic ions (Fe}{\cf2\fs24\up 3+}{\cf2\fs24 ) }{\fs24 are separated 
by many other atoms and thus interact very weakly.  For simplicity 
will assume the spin to be 1/2, so we can describe this system of 
spins in a magnetic field as a two-state model.   For simplicity will 
use a system of units with }{\object{\*\objclass \eqn}\rsltpict{
\*\objdata 
.EQN 21 26 21 0 0
{0:k.B}NAME:1
}}{\fs24  and }{\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 21 34 23 0 0
{0:g\m.B}NAME:1
}}{\fs24 .  }{\fs24 Will start from the free energy per spin }{\object{
\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 26 1 17 0 0
{0:f}NAME({0:T}NAME,{0:H}NAME):-{0:k.B}NAME*{0:T}NAME*{0:ln}NAME(2*{0:cosh}NAME(({0:g\m.B}NAME)/({0:k.B}NAME)*({0:H}NAME)/({0:T}NAME)))
}}{\fs24 .  Then we calculate the entropy by using s = -df/dT.}{\fs24         
 }{\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 32 1 31 0 0
{0:s}NAME({0:T}NAME,{0:H}NAME):{0:k.B}NAME*{0:ln}NAME(2*{0:cosh}NAME(({0:g\m.B}NAME)/({0:k.B}NAME)*({0:H}NAME)/({0:T}NAME)))-{0:k.B}NAME*{0:tanh}NAME(({0:g\m.B}NAME)/({0:k.B}NAME)*({0:H}NAME)/({0:T}NAME))*({0:g\m.B}NAME)/({0:k.B}NAME)*({0:H}NAME)/(
{0:T}NAME)
}}{\fs24 \tab              Note that an adiabat, i.e. constant s, 
corresponds to H/T = constant.\par The magnetic cooling proceeds in 
two steps: (i) start with the system in contact with a helium bath 
(fixed T) and increase the magnetic field.  By doing this the spins 
get alligned and thus the disorder and s are reduced. (ii) Decrease 
fast the magnetic field.  This process is adiabatic. Since H/T = 
constant and since we diminish H then T is reduced.}}
.EQN 49 6 16 0 0
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&&(_n_u_l_l_&_n_u_l_l_)&{0:s}NAME(1,{0:H}NAME),{0:s}NAME(.5,{0:H}NAME),{0:s}NAME(.1,{0:H}NAME)@&&(_n_u_l_l_&_n_u_l_l_)&{0:H}NAME
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{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN 1997}}