.MCAD 304020000 1 74 90 0
.CMD PLOTFORMAT
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0 0 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
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.CMD FORMAT  rd=d ct=10 im=i et=3 zt=15 pr=5 mass length time charge temperature tr=0 vm=0
.CMD SET ORIGIN 0
.CMD SET TOL 0.001000000000000
.CMD SET PRNCOLWIDTH 8
.CMD SET PRNPRECISION 4
.CMD PRINT_SETUP 1.200000 1.218750 1.200000 1.200000 0
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.CMD DEFINE_FONTSTYLE_NAME fontID=10 name=User^7
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.CMD UNITS U=1
.CMD DIMENSIONS_ANALYSIS 0 0
.CMD COLORTAB_ENTRY 0 0 0
.CMD COLORTAB_ENTRY 128 0 0
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.TXT 2 1 7 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}{\f1\fcharset0\fnil Century Gothic;}{
\f2\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard {\cf2\fs20 COPYRIGHT 
MIRON KAUFMAN, 1997}{\cf2 \par \par            }{\cf2\f1\fs32\b  }{\cf2
\f1\fs36\b Statistical Physics Computer Lab #2}{\cf2\f1\fs32 \par }{
\cf2\fs28 \par }{\cf2 We consider the Einstein crystal. Start from the 
formula for the energy: U=3Nh}{\cf2\f2 w}{\cf2 /2 + 3Nh}{\cf2\f2 w}{
\cf2 /(exp(h}{\cf2\f2 w}{\cf2 /kT) - 1).   Then we determine the heat 
capacity:\tab C = dU/dT by using the symbolic processor of MATHCAD.}}
.TXT 21 0 40 0 0
Cg a73.000000,73.000000,108
{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain
\cf1\fs24 \pard {\cf2  I start from the energy per atom u expressed in 
units of k}{\cf2\dn B}{\cf2 .  The Einstein temperature T}{\cf2\dn E}{
\cf2  stands for h}{\cf2\f1 w}{\cf2 /k}{\cf2\dn B}{\cf2 .}}
.EQN 6 0 49 0 0
{0:T.E}NAME:90
.TXT 1 24 64 0 0
Cg a60.000000,60.000000,39
{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\cf2 This 
the Einstein temperature for lead.}}
.EQN 4 -24 41 0 0
{0:u}NAME({0:T}NAME):(3*{0:T.E}NAME)/(2)+(3*{0:T.E}NAME)/(({0:e}NAME)^(({0:T.E}NAME)/({0:T}NAME))-1)
.TXT 0 24 58 0 0
Cg a49.000000,49.000000,73
{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\cf2 Next I 
differentiate u with respect to T by using the symbolic processor.}}
.EQN 11 -24 52 0 0
{0:c}NAME({0:T}NAME):3*(({0:T.E}NAME)^(2))/(((({0:exp}NAME(({0:T.E}NAME)/({0:T}NAME))-1))^(2)*({0:T}NAME)^(2)))*{0:exp}NAME(({0:T.E}NAME)/({0:T}NAME))
.TXT 0 31 59 0 0
Cg a42.000000,42.000000,50
{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\cf2 This 
is the heat capacity per atom in units of k}{\cf2\dn B}{\cf2 .}}
.EQN 11 -30 55 0 0
{0:T}NAME:.3,.35;300
.EQN 3 -1 54 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:c}NAME({0:T}NAME)@300&&(_n_u_l_l_&_n_u_l_l_)&{0:T}NAME
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.TXT 43 0 56 0 0
Cg b73.000000,73.000000,29
{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\cf2\fs20 
COPYRIGHT MIRON KAUFMAN, 1997}}
.TXT 7 1 78 0 0
Cg a72.000000,72.000000,58
{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The fundamental equation 
s(u) for the Einstein crystal is:}
.EQN 7 0 80 0 0
{0:s}NAME({0:T}NAME):(3+({0:u}NAME({0:T}NAME)-(3*{0:T.E}NAME)/(2))/({0:T.E}NAME))*{0:ln}NAME(3+({0:u}NAME({0:T}NAME)-(3*{0:T.E}NAME)/(2))/({0:T.E}NAME))-3*{0:ln}NAME(3)-({0:u}NAME({0:T}NAME)-(3*{0:T.E}NAME)/(2))/({0:T.E}NAME)*{0:ln}NAME(({0:u}NAME(
{0:T}NAME)-(3*{0:T.E}NAME)/(2))/({0:T.E}NAME))
.TXT 9 0 81 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;\red0\green0\blue0;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard where s is 
entropy per atom and u is energy per atom, both expressed in units of k{
\cf2\dn B}.}
.EQN 4 -1 82 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:s}NAME({0:T}NAME)@300&&(_n_u_l_l_&_n_u_l_l_)&{0:T}NAME
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0 1 1 1 1 0 0 1 1 
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0 2 1 0 1 1 NO-TRACE-STRING
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0 4 3 0 1 1 NO-TRACE-STRING
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0 2 5 0 1 1 NO-TRACE-STRING
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0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 62 22 10 0 3 
.TXT 30 0 88 0 0
Cg a73.000000,73.000000,212
{\rtf\ansi \deff0{\colortbl;\red0\green64\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Next we graph s vs. u . 
Note that s(u) is a {\b concave} function as required by thermodynamic 
stability (2'nd law).  Note that at the lowest energy s = 0 and the 
slope of s(u) is infinite, i.e. 1/T=infinity or T = 0.}
.EQN 6 0 87 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:s}NAME({0:T}NAME)@900&&((3*{0:T.E}NAME)/(2)&_n_u_l_l_)&{0:u}NAME({0:T}NAME)
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0 1 1 0 1 1 NO-TRACE-STRING
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0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 62 29 10 0 3 
.TXT 41 0 84 0 0
Cg b73.000000,73.000000,29
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\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\cf2\fs20 
COPYRIGHT MIRON KAUFMAN, 1997}}