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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 47 "One dimensional lattice \+
with two types of atom." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 775 "This Maple worksheet generates two animations. They
 illustrate the optical and acoustic modes of vibration for a one dime
nsional chain of two types of atom, alternating along the chain, with \+
neighbouring masses joined by springs. The curves are drawn with the m
asses in the ratio appropriate for sodium chloride: the lighter sodium
 atoms are at the odd numbered sites and appear as red circles in the \+
plots, while the heavier chorine atoms live at the even sites and are \+
denoted by blue circles. The background continuous curves of sin (k x \+
- omega(k) t) appear in yellow, one for each type of atom. Remember th
at the physical atomic displacements are only defined at the atomic si
tes where x = n a for some n (a is the separation between neighbouring
 masses, set to 1 here)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 531 "To see the plots, first open the section labelled
 \"Input Commands\" by clicking on the \"+\" symbol in the box at the \+
left margin. Then evaluate the long chain of commands (lines in red, b
eginning with \">\") by placing the cursor anywhere on the input and t
yping return. This is quite slow (takes about 30 seconds). When it's f
inished, Maple will type \"animations generated!\" in blue at the bott
om. Now you can see the plots by evaluating the two commands \"optical
_mode\" and \"acoustic_mode\" in the sections at the end of the worksh
eet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "J
MF 10 Jan 1996. Revised 22 Jan 1998." }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 14 "Input Commands" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
71 "omegaplus := (k,mr) -> sqrt(1 + mr + sqrt((1+mr)^2 - 4*mr*sin(k)^2
))/2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "omegaminus := (k,mr) -> sq
rt(1 + mr - sqrt((1+mr)^2 - 4*mr*sin(k)^2))/2:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "aplus := (k,mr) -> cos(k)/(1 - 2*omegaplus(k,mr)^2):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "aminus := (k,mr) -> cos(k)/(1 -
 2*omegaminus(k,mr)^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "pwaveplu
s := (k,x,t,mr) -> sin(k*x - omegaplus(k,mr)*t):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "pwaveminus := (k,x,t,mr) -> sin(k*x - omegaminus(k,mr
)*t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "nacl := 23/35:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 87 "Curves := animate(\{pwaveplus(Pi/6,x,t,nacl),aplus(Pi/6,nacl)*
pwaveplus(Pi/6,x,t,nacl)\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "x=0.
.11,t=0..2*Pi*19/(20*omegaplus(Pi/6,nacl)),frames=20,color=yellow):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PtList1 := proc(k,t,mr)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "  local i; evalf([seq([2*i,pwaveplus(k,2*
i,t,mr)],i=0..5)]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PtList2 := proc(k,t,mr)" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 76 "  local i; evalf([seq([2*i+1,aplus(k,mr)*pw
aveplus(k,2*i+1,t,mr)],i=0..5)]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 
"end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Points1 := [seq(plot(PtLis
t1(Pi/6,j*Pi/(10*omegaplus(Pi/6,nacl)),nacl)," }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "    style=point,symbol=circle,color=blue),j=0..19)]:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Points2 := [seq(plot(PtList2(Pi
/6,j*Pi/(10*omegaplus(Pi/6,nacl)),nacl)," }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "    style=point,symbol=circle,color=red),j=0..19)]:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "PtAnim1 := display(Points1,insequ
ence=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "PtAnim2 := display(P
oints2,insequence=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "PtAnim \+
:= display(\{PtAnim1,PtAnim2\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"optical_mode := display(\{Curves,PtAnim\}):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 90 "Curves := animate(\{pwaveminus(Pi/6,x,t,nacl),aminus(
Pi/6,nacl)*pwaveminus(Pi/6,x,t,nacl)\}," }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 72 "x=0..11,t=0..2*Pi*19/(20*omegaminus(Pi/6,nacl)),frames=20,colo
r=yellow):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PtList1 := proc(k,t,m
r)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "  local i; evalf([seq([2*i,pw
aveminus(k,2*i,t,mr)],i=0..5)]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "
end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PtList2 := proc(k,t,mr)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "  local i; evalf([seq([2*i+1,aminus
(k,mr)*pwaveminus(k,2*i+1,t,mr)],i=0..5)]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Points1 := \+
[seq(plot(PtList1(Pi/6,j*Pi/(10*omegaminus(Pi/6,nacl)),nacl)," }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "    style=point,symbol=circle,color
=blue),j=0..19)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Points2 := [se
q(plot(PtList2(Pi/6,j*Pi/(10*omegaminus(Pi/6,nacl)),nacl)," }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 51 "    style=point,symbol=circle,color=red),j=
0..19)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "PtAnim1 := display(Poin
ts1,insequence=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "PtAnim2 :=
 display(Points2,insequence=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "PtAnim := display(\{PtAnim1,PtAnim2\}):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "acoustic_mode := display(\{PtAnim,Curves\}):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "print(`animations generated!`);" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Optical Mode" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 461 "To see the optical mode animation, evaluate the \+
following command by placing the cursor anywhere on it and typing retu
rn. This will pop up a graph to show the animation. On the row of VCR-
like buttons that appear at the top of the screen, click on the right \+
button with the loop symbol. This will cause the animation to repeat g
iving the illusion of continuous motion. To play the animation click o
n the play button, a right-pointing triangle, second from left." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "optical_mode;" }}}}{SECT 1 {PARA 3 
"" 0 "" {TEXT -1 13 "Acoustic Mode" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
67 "To see the acoustic mode animation, evaluate the following command
." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "acoustic_mode;" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 19 "Dispersion Relation" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 175 "If you want a reminder, here is a plot of the dispe
rsion relation, omega(k), for a chain of two types of atom. It is plot
ted with a mass ratio appropriate for sodium chloride." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "plot(\{omegaplus(k,nacl),omegaminus(k,nacl)\},
k=-Pi/2..Pi/2);" }}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }