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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 257 31 "One-dimensional Crystal \+
Lattice" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
827 "This Maple worksheet sets up an animation of the displacements of
 the atoms in a one-dimensional lattice. If you haven't used Maple bef
ore, all you need to know is that the input lines (in red with a > at \+
the beginning) denote commands which Maple can evaluate. If an input l
ine ends with a semi-colon, Maple will show the output in blue below i
t. Commands ending with a colon do not display their output. To use th
e worksheet, place the cursor on the first input line and press \"retu
rn\" to evaluate it. You can then work your way down, evaluating the c
ommands as you go. Maple will leave the cursor positioned on the next \+
input line each time by default, so you can just keep hitting \"return
\". Alternatively, under the \"Edit\" menu option, you can select \"Ex
ecute\" and then \"Worksheet\" to cause all the commands to be execute
d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "JMF
 10 Jan 1996. Revised 22 Jan 1998." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "The dispersion relation for long
itudinal waves in a one-dimensional line of masses connected by spring
s is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " \+
         omega^2 = 4 (C/M) sin^2(ka/2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 323 "where the masses are M and they are \+
spaced a apart, linked by springs with spring constant C. For transver
se oscillations of beads on a light elastic string under tension T you
 get the same dispersion relation with T/Ma replacing C/M. In the foll
owing I'll use units where 4 C / m = 1, so the dispersion relation bec
omes..." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "omega := k -> abs(sin(k/
2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "... and here is a plot o
f it. The Brillouin zone runs over -Pi to Pi, or -Pi/a to Pi/a in conv
entional units." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(omega(k),k=
-Pi..Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 316 "We will write down \+
a plane wave solution and show that for any k in the Brillouin zone, t
he shifted wavevector k' = k + 2 Pi ( or k + 2 Pi/a in conventional un
its, where a is the spacing between adjacent masses) gives the same ph
ysical motion for the masses. Here is a plane wave using the dispersio
n relation above." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Pwave := (k,x,
t) -> sin(k*x - omega(k)*t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "
We need to load some extra Maple commands for the animations, which is
 done by loading the \"plots\" package..." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 
"Now set up an animation of the plane waves for k = Pi/6 and k' = Pi/6
 + 2 Pi (Note: this takes a few seconds to evaluate). You will not see
 the animation yet!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Curves := an
imate(\{Pwave(Pi/6,x,t),Pwave(Pi/6+2*Pi,x,t)\}," }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "x=0..5,t=0..2*Pi*19/(20*omega(Pi/6)),frames=20,color=
yellow,numpoints=100):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Define \+
a procedure to plot the positions of the masses m." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "PtList := proc(k,t)" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "  local i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  [seq([i,Pwa
ve(k,i,t)],i=0..5)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Set up a sequence of sets of posi
tions of the masses  (Note: this takes a few seconds to evaluate)..." 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "Points := [seq(plot(PtList(Pi/6,
j*Pi/(10*omega(Pi/6))),style=point,symbol=circle,color=blue),j=0..19)]
:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "... and animate them. Again,
 you will not see the animation yet!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "PtAnim := display(Points,insequence=true):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 270 "Finally display the animation. What you will see \+
is background yellow curves showing the plane waves for the two values
 of k, together with the positions of the masses given by blue dots. N
ote how the positions of the masses are the same for both background p
lane waves." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 411 "When you activate the following command a graph should appear \+
to display the animation. When you click on this a set of VCR-like but
tons should appear at the top of the screen. Click on the right hand b
utton with the loop symbol: this will cause the animation to be played
 over and over again to give the illusion of continuous motion. Then h
it the \"play\" button (second from left) to see the animation. Enjoy.
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{Curves,PtAnim\});" }
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