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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 260 18 "Compton scattering" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 423 "We used \+
conservation laws for linear momentum and kinetic energy to solve one-
dimensional collision problems. In more dimensions the number of degre
es of freedom associated with the particle motion exceeds the number o
f constraints provided conservation laws. This means that we will not \+
be able to obtain complete solutions. Nevertheless, it is interesting \+
to investigate the constraints supplied by the conservation laws." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Suppose t
hat a particle of mass " }{TEXT 259 1 "m" }{TEXT -1 28 " is impinging \+
with velocity " }{TEXT 257 1 "v" }{TEXT -1 12 "0 along the " }{TEXT 
258 1 "x" }{TEXT -1 28 "-axis on a particle of mass " }{TEXT 256 1 "M
" }{TEXT -1 530 " that is initially at rest. The particles interact by
 a central force (force acts only along the mutual separation). To att
empt a solution based on conservation laws means to not take into acco
unt the details of the interaction. A complete dynamical approach invo
lves solving Newton's equations (or variants thereof, such as Lagrange
's, or Hamilton's forms), which takes into account the force law, and \+
which automatically obeys the conservation laws. An example of such a \+
complete solution was the problem of firing a cannonball." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 284 "1) The conservati
on of angular momentum for the motion of objects subject to central fo
rces implies that the motion is confined to a plane. This reduces the \+
three degrees of freedom in the position vectors of the masses to two.
 The number of degrees of freedom is reduced from 6 to 4." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "2) The conservatio
n of energy provides one scalar constraint, i.e., 3 degrees of freedom
 remain." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
86 "3) The conservation of linear momentum in the scattering plane pro
vides 2 constraints." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 282 "Thus, the problem cannot be completely determined from
 the conservation laws. This is illustrated in this worksheet, first i
n general, and then specifically for the scattering of photons off ato
mic electrons resulting in an energy transfer to the electron, i.e., C
ompton scattering." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 324 "We set up the equations for the constraints 2 and 3, i.e
., we begin in the scattering plane (having made use of angular moment
um conservation). We assume complete elasticity for the collision, alt
hough it is not difficult to incorporate in the energy balance a given
 amount of binding energy that has to be supplied to mass " }{TEXT 
264 1 "M" }{TEXT -1 37 " in order to free it from the target." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "En:=m/2*v0^2=m/2*v^2+M/2*V^2
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EnG/,$*&%\"mG\"\"\")%#v0G\"\"#
\"\"\"#F)F,,&*&F(F-)%\"vGF,F-F.*&%\"MGF))%\"VGF,F-F." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 99 "To specify the momentum components after the co
llision we introduce two angles with respect to the " }{TEXT 261 1 "x
" }{TEXT -1 42 " axis: theta for the scattering particle (" }{TEXT 
262 1 "m" }{TEXT -1 46 ") and phi for the particle initially at rest (
" }{TEXT 263 1 "M" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "Momx:=m*v0=m*v*cos(theta)+M*V*cos(phi);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%MomxG/*&%\"mG\"\"\"%#v0GF(,&*(F'\"\"\"%\"vGF(
-%$cosG6#%&thetaGF(F(*(%\"MGF(%\"VGF(-F/6#%$phiGF(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "Momy:=0=m*v*sin(theta)+M*V*sin(phi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MomyG/\"\"!,&*(%\"mG\"\"\"%\"vGF*-%
$sinG6#%&thetaGF*F**(%\"MGF*%\"VGF*-F-6#%$phiGF*F*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 26 "What are we interested in?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The masses of the particl
es and the initial speed " }{TEXT 265 1 "v" }{TEXT -1 20 "0 are usuall
y known." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
29 "Unknown are four quantities: " }{TEXT 266 1 "v" }{TEXT -1 2 ", " }
{TEXT 267 1 "V" }{TEXT -1 105 ", and theta and phi. How can we use the
 three constraints to learn as much as possible about the problem?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Before we
 jump into the general situation, let us first consider the equal-mass
 case:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "En1:=subs(M=m,En)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$En1G/,$*&%\"mG\"\"\")%#v0G\"\"
#\"\"\"#F)F,,&*&F(F-)%\"vGF,F-F.*&F(F-)%\"VGF,F-F." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "Momx1:=subs(M=m,Momx);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&Momx1G/*&%\"mG\"\"\"%#v0GF(,&*(F'\"\"\"%\"vGF(-%$cos
G6#%&thetaGF(F(*(F'F,%\"VGF(-F/6#%$phiGF(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "Momy1:=subs(M=m,Momy);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&Momy1G/\"\"!,&*(%\"mG\"\"\"%\"vGF*-%$sinG6#%&thetaGF
*F**(F)\"\"\"%\"VGF*-F-6#%$phiGF*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 81 "Given that we have three equations we can ask for a solution fo
r three variables:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sol:=
solve(\{En1,Momx1,Momy1\},\{v,V,theta\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$solG6&<%/%\"vG%#v0G/%\"VG\"\"!/%&thetaGF,<%/F(,$F)!
\"\"F*/F.%#PiG<%/F(*&F)\"\"\"-%$sinG6#%$phiGF8/F.,&F<F8F4#F2\"\"#/F+*&
-%$cosGF;F8F)\"\"\"<%/F.,&F<F8F4#F8F@FA/F(,$F7F2" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 8 "Maple's " }{TEXT 19 5 "solve" }{TEXT -1 77 " engine
 finds a set of four acceptable solutions. Let us see what they imply.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Set 1 \+
 " }{XPPEDIT 18 0 "\{v = v0, V = 0, theta = 0\};" "6#<%/%\"vG%#v0G/%\"
VG\"\"!/%&thetaGF)" }{TEXT -1 46 " describes the situation before the \+
collision." }}{PARA 0 "" 0 "" {TEXT -1 6 "Set 2 " }{XPPEDIT 18 0 "\{v \+
= -v0, V = 0, theta = Pi\};" "6#<%/%\"vG,$%#v0G!\"\"/%\"VG\"\"!/%&thet
aG%#PiG" }{TEXT -1 54 " describes the mirror image of the situation in
 set 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
43 "Sets 3 and 4 assign the same speed to mass " }{TEXT 268 1 "M" }
{TEXT -1 129 ". If we assume that theta and phi are in the range  0..P
i/2, and -Pi/2..0 respectively, i.e., both masses move to the right wi
th " }{TEXT 272 1 "m" }{TEXT -1 27 " deflected in the positive " }
{TEXT 271 1 "y" }{TEXT -1 15 " direction and " }{TEXT 270 1 "M" }
{TEXT -1 41 " moving downward, it is clear that set 4 " }{XPPEDIT 18 
0 "\{theta = phi+1/2*Pi, V = cos(phi)*v0, v = -v0*sin(phi)\};" "6#<%/%
&thetaG,&%$phiG\"\"\"%#PiG#F(\"\"#/%\"VG*&-%$cosG6#F'F(%#v0G\"\"\"/%\"
vG,$*&F2F(-%$sinGF1F(!\"\"" }{TEXT -1 36 " is the interesting case. Th
e speed " }{TEXT 269 1 "v" }{TEXT -1 20 " should be positive." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "An intere
sting conclusion can be reached:" }}{PARA 0 "" 0 "" {TEXT -1 307 "appa
rently the scattering angle phi is arbitrary (in the range -Pi/2..0), \+
and the projectile scattering angle theta is given in terms of phi. Th
e well-known textbook result for the equal-mass elastic scattering cas
e is obtained, namely that the two objects emerge at right angles with
 respect to each other." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 230 "Furthermore, given a choice of angle phi, the distr
ibution of kinetic energies between incident and hit particle is fixed
. For each scattering angle phi (or theta) the kinetic energy of both \+
the impinging and hit object are fixed." }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assign(sol[4]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "print(v,V);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$,$*&%#v0G\"\"\"-%$sinG6#%$phiGF&!\"\"*&-%$cosGF)F&
F%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "KEm:=m*v^2/2;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$KEmG,$*(%\"mG\"\"\")%#v0G\"\"#\"
\"\")-%$sinG6#%$phiGF+F,#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "KEM:=m*V^2/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$KEMG,$*(%\"m
G\"\"\")-%$cosG6#%$phiG\"\"#\"\"\")%#v0GF.F/#F(F." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "m:=1; v0:=10;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G\"#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot([KEm,KEM],phi=-Pi/2..0,
color=[blue,green],axes=BOXED);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Obviously the sum of the kinetic
 energies of the two masses (note " }{TEXT 274 1 "M" }{TEXT -1 1 "=" }
{TEXT 275 1 "m" }{TEXT -1 44 ") is independent of phi, as it has to eq
ual " }{TEXT 276 1 "m" }{TEXT -1 3 "/2*" }{TEXT 277 1 "v" }{TEXT -1 4 
"0^2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 278 57 "The extreme cases correspond to the following situations
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 17 "phi=
0, theta=Pi/2" }{TEXT -1 37 ": the incident mass remains at rest (" }
{TEXT 19 5 "KEm=0" }{TEXT -1 56 "), and the angle theta is irrelevant.
 The hit particle (" }{TEXT 273 1 "M" }{TEXT -1 151 ") moves in the fo
rward direction with maximum kinetic energy. This represents a head-on
 collision that can be calculated in a one-dimensional geometry." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 18 "phi=-Pi/2
, theta=0" }{TEXT -1 57 ": the incident mass transfers no energy to th
e hit mass (" }{TEXT 19 5 "KEM=0" }{TEXT -1 338 "), phi is irrelevant.
 This corresponds to the limit of a grazing collision when the particl
es interact only very weakly. More interesting is the observation that
 before the limit is reached (small-angle projectile scattering) the h
it particle recoils with an angle of almost 90 degrees, and picks up a
 small amount of kinetic energy only." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 19 21 "phi=-Pi/4, theta=Pi/4" }{TEXT -1 46 "
: equal sharing of energy after the collision." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We are now ready to consi
der the general case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "En:=m/2*v0^2=m/2*v
^2+M/2*V^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EnG/,$*&%\"mG\"\"\")
%#v0G\"\"#\"\"\"#F)F,,&*&F(F-)%\"vGF,F-F.*&%\"MGF))%\"VGF,F-F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Momx:=m*v0=m*v*cos(theta)+M*
V*cos(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MomxG/*&%\"mG\"\"\"%
#v0GF(,&*(F'\"\"\"%\"vGF(-%$cosG6#%&thetaGF(F(*(%\"MGF(%\"VGF(-F/6#%$p
hiGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Momy:=0=m*v*sin(
theta)+M*V*sin(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MomyG/\"\"!
,&*(%\"mG\"\"\"%\"vGF*-%$sinG6#%&thetaGF*F**(%\"MGF*%\"VGF*-F-6#%$phiG
F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol:=solve(\{En,Mom
x,Momy\},\{v,V,theta\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG6%<
%/%\"VG\"\"!/%&thetaGF)/%\"vG%#v0G<%F'/F-,$F.!\"\"/F+%#PiG<%/F(,$*&*(%
\"mG\"\"\"F.F;-%$cosG6#%$phiGF;\"\"\",&%\"MGF;F:F;!\"\"\"\"#/F-*&*&-%'
RootOfG6#,,*$)%#_ZGFDF@F;*()-%$sinGF>FDF@F:F@FBF;!\"%*&FBF@F:F@FD*$)F:
FDF@F2*$)FBFDF@F2F;F.F@F@FAFC/F+-%'arctanG6$,$*&*(FQF;FBF@F<F@F@FHFC!
\"#*&,(FBF2F:F;*&FBF@FPF@FDF@FHFC" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
164 "We are now used to the fact that the initial condition and its mi
rror image appear as solutions to the system of equations. Let us anal
yze the interesting solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "assign(sol[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "v;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%'RootOfG6#,,*$)%#_ZG\"\"#\"
\"\"\"\"\"*()-%$sinG6#%$phiGF,F-%\"mGF.%\"MGF.!\"%*&F6F-F5F-F,*$)F5F,F
-!\"\"*$)F6F,F-F;F.%#v0GF.F-,&F6F.F5F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 2 "V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*(%\"mG\"
\"\"%#v0GF'-%$cosG6#%$phiGF'\"\"\",&%\"MGF'F&F'!\"\"\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "theta;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'arctanG6$,$*&*(-%$sinG6#%$phiG\"\"\"%\"MG\"\"\"-%$cosGF+F-F/-
%'RootOfG6#,,*$)%#_ZG\"\"#F/F-*()F)F9F/%\"mGF-F.F-!\"%*&F.F/F<F/F9*$)F
<F9F/!\"\"*$)F.F9F/FA!\"\"!\"#*&,(F.FAF<F-*&F.F/F;F/F9F/F2FD" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 281 "We notice that the relationship b
etween theta and phi is more complicated. Let us first analyze the sim
ple part. The kinetic energy of the hit particle is still a function o
f its scattering angle. Note that the arctan function can take two arg
uments (look it up in the help pages)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "KEM:=M/2*V^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$K
EMG,$*&**%\"MG\"\"\")%\"mG\"\"#\"\"\")%#v0GF,F-)-%$cosG6#%$phiGF,F-F-*
$),&F(F)F+F)\"\"#F-!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Fo
r fixed scattering angle phi we can explore the dependence of this kin
etic energy on the mass ratio." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "phi:=-Pi/4; v0:=10; m:=1; M:=r*m;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$phiG,$%#PiG#!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#v0G\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG%\"rG" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 4 "KEM;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"rG\"
\"\"*$),&F%\"\"\"F*F*\"\"#F&!\"\"\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "KEtot:=m/2*v0^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&KEtotG\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "We perform a sem
ilog-plot 'by hand' and cover the range of mass ratios from 0.001 to 1
000 on the abscissa." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plo
t(subs(r=10^s,KEM)/KEtot,s=-3..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 641 
236 236 {PLOTDATA 2 "6%-%'CURVESG6$7eo7$$!\"$\"\"!$\"1*)*4?*f+'*>!#=7$
$!1+++vq@pG!#:$\"1b'4$GN]&p#F-7$$!1++D^NUbFF1$\"1?!RQN:,]$F-7$$!1++]K3
XFEF1$\"1W>e$\\&*Qp%F-7$$!1++]F)H')\\#F1$\"13fTbzZ/jF-7$$!1++D'3@/P#F1
$\"1d2x9q8^%)F-7$$!1++Dr^b^AF1$\"1P8p!R3#36!#<7$$!1++D,kZG@F1$\"1!f\"Q
Y\"RfY\"FM7$$!1++Dh\")=,?F1$\"1&3CJMT`&>FM7$$!1++DO\"3V(=F1$\"1=^kw9M,
EFM7$$!1+++NkzV<F1$\"1k?8&=35[$FM7$$!1++]d;%)G;F1$\"1!)e$HhTv[%FM7$$!1
+++0)H%*\\\"F1$\"1Cv,0x3]fFM7$$!1+++vl[p8F1$\"1<2y#*GHcyFM7$$!1+++&>iU
C\"F1$\"1G'[p<w+-\"!#;7$$!1++DhkaI6F1$\"1#3#z9Mk$G\"F`p7$$!1+++]XF`**F
`p$\"1ic&G\"3\\n;F`p7$$!1+++vL`!Q*F`p$\"1NJA$)HDa=F`p7$$!1++++Az2))F`p
$\"14Pq&y*Gb?F`p7$$!1++Dc!e:9)F`p$\"1EFL7aE1BF`p7$$!1++]7RKvuF`p$\"1wg
4?A(Qd#F`p7$$!1,+D1Qf&)oF`p$\"1xUL4$*HAGF`p7$$!1-+++P'eH'F`p$\"1,/7xtj
yIF`p7$$!1++D1j$)[cF`p$\"1=M\"HZMYO$F`p7$$!1****\\7*3=+&F`p$\"1$3RYtS)
\\OF`p7$$!1*****\\#eo&Q%F`p$\"1$*fjZ8K9RF`p7$$!1)***\\PFcpPF`p$\"1g:I1
Y.lTF`p7$$!1)**\\7$HqEJF`p$\"1)*3$oV7US%F`p7$$!1)****\\7VQ[#F`p$\"1@'z
AeDCh%F`p7$$!1)**\\P9(\\$*=F`p$\"1XD.8prpZF`p7$$!1)***\\i6:.8F`p$\"1@;
Id87*)[F`p7$$!1()*\\i:sw%)*FM$\"1L_8HnFO\\F`p7$$!1$***\\(oKQm'FM$\"1Na
cY^oq\\F`p7$$!1'*\\7`H\">2&FM$\"1hPpD-*H)\\F`p7$$!1***\\(=K**zMFM$\"1h
c'*)f#)>*\\F`p7$$!1-]P%[t!))=FM$\"1*GyR?Qw*\\F`p7$$!1b+++v`hHF-$\"1uVs
t=%***\\F`p7$$\"1&*\\7`MSd8FM$\"1XI8w!z()*\\F`p7$$\"1&**\\ilg4,$FM$\"1
@v!3^'*R*\\F`p7$$\"1&*\\Pfy^kYFM$\"1vr#>-3c)\\F`p7$$\"1&***\\i]2=jFM$
\"1]>\"Q%yjt\\F`p7$$\"1&**\\(o%*=D'*FM$\"1#4H\\N+\"R\\F`p7$$\"1++](QIK
H\"F`p$\"1ezH\\&y2*[F`p7$$\"1******\\4+p=F`p$\"1Z![Q%yXvZF`p7$$\"1****
\\7:xWCF`p$\"1/\\V\"z1Ri%F`p7$$\"1++v$\\>m1$F`p$\"1`(=F%)3^U%F`p7$$\"1
,++vuY)o$F`p$\"1.-M+Dj'>%F`p7$$\"1++](G(*3L%F`p$\"1Y2B5oFPRF`p7$$\"1)*
*****4FL(\\F`p$\"1i(\\:erAm$F`p7$$\"1)***\\P$>=g&F`p$\"1HJ.*=kaQ$F`p7$
$\"1)****\\d6.B'F`p$\"15Y'**3pu5$F`p7$$\"1***\\785%QoF`p$\"1$e]ccgD%GF
`p7$$\"1++](o3lW(F`p$\"1wG!*e4y&e#F`p7$$\"1***\\iX)p@\")F`p$\"1J>tnO,9
BF`p7$$\"1*****\\A))oz)F`p$\"1baZ-CDf?F`p7$$\"1(***\\iid.%*F`p$\"1u-E1
4YY=F`p7$$\"1+++Ik-,5F1$\"1Qgb[')p\\;F`p7$$\"1+++D-eI6F1$\"1=zL!QdNG\"
F`p7$$\"1++v=_(zC\"F1$\"1RK6Q6K75F`p7$$\"1+++b*=jP\"F1$\"1\"=?7,iNu(FM
7$$\"1++v3/3(\\\"F1$\"1)z$4d:P!)fFM7$$\"1++vB4JB;F1$\"16J#o^\"QUXFM7$$
\"1+++DVsY<F1$\"14gb&eU%eMFM7$$\"1++v=n#f(=F1$\"1;,Q\"o=>f#FM7$$\"1+++
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F1$\"1(4=152D5\"FM7$$\"1+++5jypBF1$\"1R(y\\Q$Rj%)F-7$$\"1++]Ujp-DF1$\"
1mE:l'*3YiF-7$$\"1+++gEd@EF1$\"1b[t(>avv%F-7$$\"1++v3'>$[FF1$\"1crY[Lj
dNF-7$$\"1++D6EjpGF1$\"1%>-\\jKHp#F-7$$\"\"$F*$\"1))*4?*f+'*>F--%'COLO
URG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"s6\"%!G-%%VIEWG6$;F(F_bl%(
DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "We learn from the graph that the \+
equal-mass case is the most effective for energy transfer. If the mass
 ratio " }{TEXT 279 2 "r " }{TEXT -1 2 "= " }{TEXT 280 1 "M" }{TEXT 
-1 1 "/" }{TEXT 281 1 "m" }{TEXT -1 91 " is either small or large, the
n the fraction of the kinetic energy transferred to the mass " }{TEXT 
282 1 "M" }{TEXT -1 57 " in collisions with phi = -45 degrees becomes \+
very small." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 153 "The other variables that were determined looked somewhat clums
y. A square root was not carried out to avoid case distinctions. With \+
the specification of " }{TEXT 283 1 "m" }{TEXT -1 5 " and " }{TEXT 
284 1 "v" }{TEXT -1 43 "0 they are, of course, somewhat simplified:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "v;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&-%'RootOfG6#,(*$)%#_ZG\"\"#\"\"\"\"\"\"!\"\"F.*$)%
\"rGF,F-F/F-,&F2F.F.F.!\"\"\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "simplify(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%'RootOfG
6#,(*$)%#_ZG\"\"#\"\"\"\"\"\"!\"\"F.*$)%\"rGF,F-F/F-,&F2F.F.F.!\"\"\"#
5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(subs(r=1,%));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+5y1rq!\"*" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 77 "Maple picks one of the roots, unfortunately the one wi
th the negative sign..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 90 "Given that a simple square root is at issue we conve
rt the RootOf to a radical expression:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "v:=convert(v,radical);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"vG,$*&*$-%%sqrtG6#,&\"\"\"F,*$)%\"rG\"\"#\"\"\"F,F1F1,&F/F,F
,F,!\"\"\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "To display the fr
action of the total kinetic energy remaining with mass " }{TEXT 302 1 
"m" }{TEXT -1 108 " after the collision for the special case of phi = \+
-45 degrees as a function of the mass ratio we calculate:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(subs(r=10^s,v)^2/v0^2,s=-3..3)
;" }}{PARA 13 "" 1 "" {GLPLOT2D 643 239 239 {PLOTDATA 2 "6%-%'CURVESG6
$7eo7$$!\"$\"\"!$\"1-*z+%*R+)**!#;7$$!1+++vq@pG!#:$\"1/prk\\/t**F-7$$!
1++D^NUbFF1$\"13;YY))*\\'**F-7$$!1++]K3XFEF1$\"1#=k]/hI&**F-7$$!1++]F)
H')\\#F1$\"1TeW?_&p$**F-7$$!1++D'3@/P#F1$\"1$H_)H')[:**F-7$$!1++Dr^b^A
F1$\"1o3$4;z\"*))*F-7$$!1++D,kZG@F1$\"1T=O&31M&)*F-7$$!1++Dh\")=,?F1$
\"1$f(oleY/)*F-7$$!1++DO\"3V(=F1$\"1)[NB&e')R(*F-7$$!1+++NkzV<F1$\"1&z
'[\"=**=l*F-7$$!1++]d;%)G;F1$\"18kqQeC^&*F-7$$!1+++0)H%*\\\"F1$\"1Z#)
\\H7*\\S*F-7$$!1+++vl[p8F1$\"1I>s52P9#*F-7$$!1+++&>iUC\"F1$\"1t80BQ#*z
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+++vL`!Q*F-$\"1mox;quX\")F-7$$!1++++Az2))F-$\"1\"H'H9-rWzF-7$$!1++Dc!e
:9)F-$\"1vsm(eMPp(F-7$$!1++]7RKvuF-$\"1BR!*zx7EuF-7$$!1,+D1Qf&)oF-$\"1
Cdm!p+x<(F-7$$!1-+++P'eH'F-$\"1*fzGii8#pF-7$$!1++D1j$)[cF-$\"1\"e'3FbO
NmF-7$$!1****\\7*3=+&F-$\"1;4Ol#f,N'F-7$$!1*****\\#eo&Q%F-$\"11SO_'yc3
'F-7$$!1)***\\PFcpPF-$\"1Q%)p$Rl\\$eF-7$$!1)**\\7$HqEJF-$\"1+\"pJc(y&f
&F-7$$!1)****\\7VQ[#F-$\"1x.s<Wd(Q&F-7$$!1)**\\P9(\\$*=F-$\"1du'p3$GI_
F-7$$!1)***\\i6:.8F-$\"1!Q)pU'y36&F-7$$!1()*\\i:sw%)*!#<$\"1oZ'3FBP1&F
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**\\i]2=jFbu$\"1[!)=c@OE]F-7$$\"1&**\\(o%*=D'*Fbu$\"1042X'**31&F-7$$\"
1++](QIKH\"F-$\"1W?q]9A4^F-7$$\"1******\\4+p=F-$\"1`>:c@aC_F-7$$\"1***
*\\7:xWCF-$\"1'4l&3K4w`F-7$$\"1++v$\\>m1$F-$\"1Z7Gd6*[d&F-7$$\"1,++vuY
)o$F-$\"1'zf'*\\nL!eF-7$$\"1++](G(*3L%F-$\"1b#p(*=BF1'F-7$$\"1)******4
FL(\\F-$\"1Q-X=%GxL'F-7$$\"1)***\\P$>=g&F-$\"1po'4\"e`9mF-7$$\"1)****
\\d6.B'F-$\"1!RN+\"4`#*oF-7$$\"1***\\785%QoF-$\"1;%\\VVRu:(F-7$$\"1++]
(o3lW(F-$\"1Br4T!>UT(F-7$$\"1***\\iX)p@\")F-$\"1q!oAL')fo(F-7$$\"1****
*\\A))oz)F-$\"1YX_(fZ2%zF-7$$\"1(***\\iid.%*F-$\"1E(RP4RN:)F-7$$\"1+++
Ik-,5F1$\"1jRW^8I]$)F-7$$\"1+++D-eI6F1$\"1$3i'>EW;()F-7$$\"1++v=_(zC\"
F1$\"1hn)=')yw)*)F-7$$\"1+++b*=jP\"F1$\"1#)z())zVcA*F-7$$\"1++v3/3(\\
\"F1$\"1=1HWG'>S*F-7$$\"1++vB4JB;F1$\"1)o<$[=wX&*F-7$$\"1+++DVsY<F1$\"
1*RW9ubTl*F-7$$\"1++v=n#f(=F1$\"1!*>'=833u*F-7$$\"1+++!)RO+?F1$\"1p/x(
y,T!)*F-7$$\"1++]_!>w7#F1$\"1m.D#p?J&)*F-7$$\"1++v)Q?QD#F1$\"1!>Q**G\\
(*))*F-7$$\"1+++5jypBF1$\"18-:mgO:**F-7$$\"1++]Ujp-DF1$\"1t%[L5Rv$**F-
7$$\"1+++gEd@EF1$\"1_E-eWU_**F-7$$\"1++v3'>$[FF1$\"1F`^mOUk**F-7$$\"1+
+D6EjpGF1$\"1!)4lt12t**F-7$$\"\"$F*$\"1+*z+%*R+)**F--%'COLOURG6&%$RGBG
$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"s6\"%!G-%%VIEWG6$;F(F_bl%(DEFAULTG" 
1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 141 "Obviously, this is the complement to the previou
s graph. Let us observe the corresponding scattering angle for the inc
ident particle of mass " }{TEXT 285 1 "m" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "theta:=convert(theta,radical);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG-%'arctanG6$*&%\"rG\"\"\"*$-%
%sqrtG6#,&\"\"\"F0*$)F)\"\"#F*F0F*!\"\"*&F*F**$-F-6#F/F*F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(subs(r=10^s,theta),s=-3..3);" 
}}{PARA 13 "" 1 "" {GLPLOT2D 644 260 260 {PLOTDATA 2 "6%-%'CURVESG6$7S
7$$!\"$\"\"!$\"0nomm'******!#=7$$!1+++vq@pG!#:$\"1yN,)4'R^8F-7$$!1++D^
NUbFF1$\"1vPm1#3iv\"F-7$$!1++]K3XFEF1$\"1D-WoZ-eBF-7$$!1++]F)H')\\#F1$
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.cF-7$$!1++D,kZG@F1$\"1f;EAw,RuF-7$$!1++Dh\")=,?F1$\"1a#ysP[B(**F-7$$!
1++DO\"3V(=F1$\"1U*>\"4\"obL\"!#<7$$!1+++NkzV<F1$\"1l0kVum.=Ffn7$$!1++
]d;%)G;F1$\"1x\"RM#p0]BFfn7$$!1+++0)H%*\\\"F1$\"1Tr)ffu`;$Ffn7$$!1+++v
l[p8F1$\"1K-5otCoUFfn7$$!1+++&>iUC\"F1$\"1qP[ei/#p&Ffn7$$!1++DhkaI6F1$
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*Fhp7$$\"1****\\7:xWCFhp$\"1$Q#pJk2`5F17$$\"1,++vuY)o$Fhp$\"1GeHA$Hm;
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\"1UX%=j`wc\"F17$$\"1+++gEd@EF1$\"1-ELphSo:F17$$\"1++v3'>$[FF1$\"19gNb
6,p:F17$$\"1++D6EjpGF1$\"1WH5BiWp:F17$$\"\"$F*$\"1I#GrK'zp:F1-%'COLOUR
G6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"s6\"%!G-%%VIEWG6$;F(Fgz%(DEF
AULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 21 "The equal-mass case (" }{TEXT 286 1 "s" }
{TEXT -1 4 "=0, " }{TEXT 287 1 "r" }{TEXT -1 38 "=1) results in theta=
Pi/4 as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "In the limit of " }{TEXT 291 2 "r " }{TEXT -1 2 "= " }
{TEXT 289 1 "M" }{TEXT -1 1 "/" }{TEXT 290 1 "m" }{TEXT -1 16 " going \+
to zero (" }{TEXT 288 1 "s" }{TEXT -1 184 " large, negative), i.e., a \+
massive incident particle as compared to the hit particle, the scatter
ing angle for the incident particle goes to zero (note that phi is fix
ed at phi=-Pi/4)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "In the limit of " }{TEXT 294 2 "r " }{TEXT -1 2 "= " }
{TEXT 292 1 "M" }{TEXT -1 1 "/" }{TEXT 293 1 "m" }{TEXT -1 20 " going \+
to infinity (" }{TEXT 295 1 "s" }{TEXT -1 323 " large, positive), i.e.
, a light incident particle as compared to the hit particle, the scatt
ering angle theta approaches Pi, i.e., reflection while the heavier ob
ject exits at phi=-45 degrees. This includes the limit of a head-on co
llision with perfect reflection, as the previous graphs showed that in
 the limit of large " }{TEXT 297 1 "r" }{TEXT -1 44 " no energy is tra
nsferred to the heavy mass " }{TEXT 296 1 "M" }{TEXT -1 41 " (its exit
 angle phi becomes irrelevant)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 145 "Another interesting case to investigate \+
involves fixing the energy of the hit particle and asking for the allo
wed range of scattering angles phi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Finally we are ready to discuss \+
" }{TEXT 298 18 "Compton scattering" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 428 "The discussion is slig
htly different to take into account the fact that photons are not clas
sical massive particles. It was observed in the passage of light throu
gh matter that some light (X rays in particular) was emitted with an i
ncreased wavelength, i.e., reduced energy. The change in wavelength de
pended on the scattering angle of the observed photons. Compton's expl
anation is based on a corpuscular model for the photons." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The problem can b
e set up as before, except that one needs expressions for photon energ
y and momentum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 55 "The photon momentum before the collision is labeled as " 
}{TEXT 303 1 "p" }{TEXT -1 88 "0 and given in terms of Planck's consta
nt h, the frequency nu, and the speed of light c:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "p0:=h*nu0/c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p0
G*&*&%\"hG\"\"\"%$nu0GF(\"\"\"%\"cG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 49 "The momentum conservation law can be expressed as" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Momx:=p0=h*nu/c*cos(theta)+P
*cos(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MomxG/*&*&%\"hG\"\"\"
%$nu0GF)\"\"\"%\"cG!\"\",&*&*(F(F+%#nuGF)-%$cosG6#%&thetaGF)F+F,F-F)*&
%\"PGF)-F36#%$phiGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M
omy:=0=h*nu/c*sin(theta)+P*sin(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%MomyG/\"\"!,&*&*(%\"hG\"\"\"%#nuGF+-%$sinG6#%&thetaGF+\"\"\"%\"cG
!\"\"F+*&%\"PGF+-F.6#%$phiGF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
211 "To formulate energy conservation we need to take into account the
 relativistic energy-momentum relationship (in the limit of high photo
n energies the electron can be picking up a large amount of kinetic en
ergy)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
153 "The kinetic energy before the collision is rest energy of the ele
ctron (it is assumed to be unbound, i.e., free, and at rest, we denote
 its rest mass as " }{TEXT 299 1 "M" }{TEXT -1 24 ") and the photon en
ergy:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "En:=h*nu0+M*c^2=h*
nu+Eel;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EnG/,&*&%\"hG\"\"\"%$nu0
GF)F)*&%\"MGF))%\"cG\"\"#\"\"\"F),&*&F(F0%#nuGF)F)%$EelGF)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 105 "Before we can solve we need to express t
he electron energy after the knock-out in terms of its rest mass " }
{TEXT 300 1 "M" }{TEXT -1 24 " and its final momentum " }{TEXT 301 1 "
P" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "En1:=s
ubs(Eel=sqrt(M^2*c^4+P^2*c^2),En);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$En1G/,&*&%\"hG\"\"\"%$nu0GF)F)*&%\"MGF))%\"cG\"\"#\"\"\"F),&*&F(F0%
#nuGF)F)*$-%%sqrtG6#,&*&)F,F/F0)F.\"\"%F0F)*&)%\"PGF/F0F-F0F)F0F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol:=solve(\{Momx,Momy,En1\}
,\{nu,P,phi\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG6\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Matters are more complicated obvio
usly..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
97 "We can eliminate the electron emission angle phi by squaring the m
omentum conservation equations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "ex1:=solve(Momx,P*cos(phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$ex1G*&*&%\"hG\"\"\",&%$nu0GF(*&%#nuGF(-%$cosG6#%&thetaGF(!\"\"F(
\"\"\"%\"cG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ex2:=so
lve(Momy,P*sin(phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ex2G,$*&*(
%\"hG\"\"\"%#nuGF)-%$sinG6#%&thetaGF)\"\"\"%\"cG!\"\"!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eq1:=ex1^2+ex2^2=P^2;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$eq1G/,&*&*&)%\"hG\"\"#\"\"\"),&%$nu0G\"\"\"*&
%#nuGF0-%$cosG6#%&thetaGF0!\"\"F+F,F,*$)%\"cG\"\"#F,!\"\"F0*&*(F)F,)F2
F+F,)-%$sinGF5F+F,F,*$)F:\"\"#F,F<F0*$)%\"PGF+F," }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "eq1:=simplify(eq1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq1G/*&*&)%\"hG\"\"#\"\"\",(*$)%$nu0GF*F+\"\"\"*(F/F
0%#nuGF0-%$cosG6#%&thetaGF0!\"#*$)F2F*F+F0F0F+*$)%\"cG\"\"#F+!\"\"*$)%
\"PGF*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sol:=solve(\{En
1,eq1\},\{P,nu\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<$/%#nuG,
$*&*(%$nu0G\"\"\"%\"MGF,)%\"cG\"\"#F,F,,(*(%\"hG\"\"\"F+F4-%$cosG6#%&t
hetaGF4F4*&F3F,F+F,!\"\"*&F-F4F.F,F:!\"\"F:/%\"PG*&*(-%'RootOfG6#,4*$)
%#_ZGF0F,F4*()F3F0F,)F+F0F,)F5F0F,F:*(FIF,FJF,F5F,F0*,F3F,F+F,F5F,F-F,
F.F,\"\"%*&FIF,FJF,F:**F3F,F+F,F-F,F.F,!\"#*&)F-F0F,)F/FNF,FQ*,F-F,F.F
,FKF,F3F,F+F,FQ*(FSF,FTF,F5F,F0F4F+F,F3F,F,*&F/\"\"\"F1\"\"\"F<" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "nu;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,$*&*(%$nu0G\"\"\"%\"MGF')%\"cG\"\"#F'F',(*(%\"hG\"\"\"F&F/-%$cosG6
#%&thetaGF/F/*&F.F'F&F'!\"\"*&F(F/F)F'F5!\"\"F5" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 226 "If we are just interested in the wavelength shift a
s a function of photon scattering angle, we do the following. First we
 make use of the relationship  lambda*nu=c to obtain an expression for
 the scattered photon's wavelength:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "eq2:=nu=c/lambda;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$eq2G/,$*&*(%$nu0G\"\"\"%\"MGF*)%\"cG\"\"#F*F*,(*(%\"hG\"\"\"F)F2-%$
cosG6#%&thetaGF2F2*&F1F*F)F*!\"\"*&F+F2F,F*F8!\"\"F8*&F-F*%'lambdaGF:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now we substitute the frequen
cy nu0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq3:=subs(nu0=c/
lambda0,eq2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,$*&*&)%\"cG
\"\"$\"\"\"%\"MGF,F,*&%(lambda0G\"\"\",(*&*(%\"hG\"\"\"F*F5-%$cosG6#%&
thetaGF5F,F/!\"\"F5*&*&F4F,F*F,F,F/F:!\"\"*&F-F5)F*\"\"#F,F=\"\"\"F:F=
*&F*F,%'lambdaGF:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq3:=s
implify(eq3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,$*&*&)%\"cG
\"\"#\"\"\"%\"MGF,F,,(*&%\"hG\"\"\"-%$cosG6#%&thetaGF1F1F0!\"\"*(F-F1F
*F1%(lambda0GF1F6!\"\"F6*&F*F,%'lambdaGF9" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 146 "The wavelength shift Delta is given by subtracting the o
riginal wavelength lambda0 from lambda; the latter is obtained by isol
ating lambda in eq3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Del
ta:=simplify(solve(eq3,lambda)-lambda0);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&DeltaG,$*&*&%\"hG\"\"\",&!\"\"F)-%$cosG6#%&thetaGF)F)\"\"\"*&
%\"MG\"\"\"%\"cG\"\"\"!\"\"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 331 
"The wavelength shift Delta does not depend on the original wavelength
 lambda0 and is given as a multiple of a natural length scale for the \+
electron, namely the Compton wavelength of the electron  h/(M*c). If w
e factor out this unit, the wavelength shift as a function of the phot
on scattering angle is given in the following graph:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "plot(1-cos(theta),theta=0..Pi);" }}{PARA 
13 "" 1 "" {GLPLOT2D 648 302 302 {PLOTDATA 2 "6%-%'CURVESG6$7S7$\"\"!F
(7$$\"1fD2LzxZo!#<$\"1jbD!3(oVB!#=7$$\"1Npx*G*f!G\"!#;$\"1%4#fcIZ)=)F/
7$$\"15@exGm]>F3$\"1'o-bp<l*=F,7$$\"199!=3o^i#F3$\"1`(3IV5gU$F,7$$\"1!
\\D0[nkH$F3$\"1]=+nYK%Q&F,7$$\"1\"=Za&z%)=RF3$\"1Sou**f\"4e(F,7$$\"1eX
a()oGjXF3$\"1o3&=tOK-\"F37$$\"1%3**Hbm(H_F3$\"1#>pn.PmL\"F37$$\"1PRr4)
3T*eF3$\"1c1ZAqJ(o\"F37$$\"1#[)yykYxlF3$\"1JmUr?G'3#F37$$\"1s'ocGo$zrF
3$\"1a'=I[a$oCF37$$\"1Q0;gr'p&yF3$\"1H)\\!*eV5$HF37$$\"1GMt?$[t`)F3$\"
1yO#yct#GMF37$$\"1#p30k@I>*F3$\"1Me*\\\"zCORF37$$\"1S7\\HeV)y*F3$\"1,4
\"eCu,U%F37$$\"1$[))*)3W'\\5!#:$\"1qZ^\"e-7-&F37$$\"1'[@XS@'46Fdp$\"10
chwS**\\bF37$$\"19w$3G*Qz6Fdp$\"1I0(4s0^='F37$$\"1%3*3tc9T7Fdp$\"1*H)e
x[(Gw'F37$$\"1y0DBA!*38Fdp$\"1L9uwY*3T(F37$$\"1kE.#[AMP\"Fdp$\"1:P.$e^
!R!)F37$$\"1a@X.EuS9Fdp$\"1\"Rza(e7.()F37$$\"1WF)**[jD]\"Fdp$\"14jhj4?
=$*F37$$\"1hw\"ymX#p:Fdp$\"1B?W;M\\%)**F37$$\"1xM!*4(4&Q;Fdp$\"1nf]\\;
mn5Fdp7$$\"1DL:iU!))p\"Fdp$\"14GKJ'ew7\"Fdp7$$\"1)R1/.CRw\"Fdp$\"1VLS%
QH>>\"Fdp7$$\"1Id)H7*>J=Fdp$\"1%pG#*ypuD\"Fdp7$$\"1fb7wY,(*=Fdp$\"1?E@
=JY?8Fdp7$$\"1N5'zg%pg>Fdp$\"1va[KV4!Q\"Fdp7$$\"1pJ8:.SJ?Fdp$\"1V]@J#*
[W9Fdp7$$\"1+2zPD$\\4#Fdp$\"1EbZg`Y+:Fdp7$$\"1=!fhumF;#Fdp$\"1^xSVs*zb
\"Fdp7$$\"1bk3@YBCAFdp$\"1^)4N**>zg\"Fdp7$$\"1/b<W_V\"H#Fdp$\"1!\\+\"z
[')f;Fdp7$$\"1!GNMzlYN#Fdp$\"1O1a.]-1<Fdp7$$\"1njYO*f2U#Fdp$\"1^$RwQc7
v\"Fdp7$$\"11()=U!z`[#Fdp$\"1g!f;13Bz\"Fdp7$$\"1kHHd#HIb#Fdp$\"1n4n\"4
R<$=Fdp7$$\"1_r&[X%==EFdp$\"1o)*[w17m=Fdp7$$\"1K-%\\0:[o#Fdp$\"12my@%y
u*=Fdp7$$\"1gN,?R*3v#Fdp$\"1lWo$=VY#>Fdp7$$\"1/0LMNh6GFdp$\"1-1e4\"\\g
%>Fdp7$$\"1$oUX107)GFdp$\"1tF%RE!Hm>Fdp7$$\"146xe&[M%HFdp$\"18w?,OV!)>
Fdp7$$\"1.6)e58)4IFdp$\"1D8mL'H8*>Fdp7$$\"1t2RXCLtIFdp$\"1hKXw6n(*>Fdp
7$$\"1++X]EfTJFdp$\"\"#F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELS
G6$Q&theta6\"%!G-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 192 "The wavelength shift acquires a maximum in the backward direct
ion. The maximum inelasticity equals twice the Compton wavelength of t
he particle that has been hit (in our case a free electron)." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "In the forward \+
direction the photons pass through without energy transfer." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "It is useful to
 obtain an idea about the length scale of the Compton wavelength for a
n electron:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lambda[C]:=h
/(M*c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'lambdaG6#%\"CG*&%\"hG\"
\"\"*&%\"MG\"\"\"%\"cG\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "h:=6.63*10^(-34)*_N*_m*_s; #Planck's constant in SI u
nits" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$*(%#_NG\"\"\"%#_mGF(%#
_sGF($\"++++Im!#V" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "c:=3.*
10^8*_m/_s; # speed of light in SI units" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"cG,$*&%#_mG\"\"\"%#_sG!\"\"$\"*++++$\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "M:=9.11*10^(-31)*_kg; # electron mass in \+
SI units" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG,$%$_kgG$\"++++5\"*!
#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lambda[C];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&*&%#_NG\"\"\")%#_sG\"\"#\"\"\"F+%$_kgG!
\"\"$\"+)f0fU#!#@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(_
N=_kg*_m/_s^2,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#_mG$\"+)f0fU#
!#@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 409 "Thus the maximum waveleng
th shift that a free electron can provide equals approximately 5*10^(-
12) m, or 0.05 Angstroms, or 5 picometers. To put this in relation wit
h the photon wavelengths we note that hard X rays have wavelengths tha
t are thousands of times shorter than those of visible light (0.1 nm v
s 450-700 nm, 1 Angstrom = 0.1 nm). For hard X rays the wavelength shi
ft thus reaches the percent range." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 417 "We conclude that soft photons can backsc
atter from electrons without appreciable energy loss, while hard photo
ns can transfer a sizable amount of energy in a backscattering process
. Of course, matters are in reality more complicated: soft photons can
 excite electrons to undergo transitions between bound states. It is j
ust the consideration of the kinematics of elastic scattering that lea
ds to the above conclusion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{MARK "85 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }