8/30/2013 PHY 711 Fall 2013 -- Lecture 2 1
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 2:
1. Comments on Maple software
2. Chapter 1 – scattering theorya) Rutherford scattering
b) Scattering for arbitrary potential
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 2
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 3
Some additional Maple examples
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 4
Scattering theory:
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 5
angleat detectorintoscatteredis that beamincident ofAreaareaunit per particlesincident ofNumber
particleper targetat particlesdetectedofNumber sectioncrossalDifferenti
dd
Figure from Marion & Thorton, Classical Dynamics
sinsin
ddbb
dddbbd
dd
bdbdb
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 6
Note: The following analysis will be carried out in the center of mass frame of reference.
In laboratory frame: In center-of-mass frame:
V1
m1
mtarget
v1m
origin vCM
r
Also note: We are assuming that the interaction between particle and target V(r) conserves energy and angular momentum.
1vr μ
mmmm
μ
target1
target1
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 7
)(2 2
2
rVr
m
E
rmin
In center of mass reference frame:
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 8
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 9
2
2
)()(:sy variable trajectoroftion Transforma
:momentumangular ofon Conservati
rddr
dtd
ddr
dtdr
rtr
dtdr
m
m
)(22
1
)(22
1
:framemassofcenter in theenergy ofon Conservati
2
22
2
2
22
rVrrd
dr
rVrdt
drE
mmm
mm
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 10
)(22
1
)(22
1
2
22
2
2
22
rVrrd
dr
rVrdt
drE
mmm
mm
)(2
2
/
)(2
2
for Solving
2
2
2
2
2
2
42
rVr
E
rdrd
rVr
Erddr
(r))r(
mm
mm
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 11
)(2
2
/
2
2
2
rVr
E
rdrd
mm
bE
vE
bv
m
m
m
221
:separationlargeat tion simplificaFurther
2
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 12
),(
)(1
/
)(2
2
/
:clearsdust When the
2
2
2
2
2
2
EbErV
rbrbdrd
rVr
E
rdrd
mm
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 13
0)(1
:where
)(1
/
min2
min
2
2
2
2
0 min
max
ErV
rb
ErV
rbrbdrd
r
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 14
Relationship between max and q:
22
)(2
max
max
q
q
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 15
EuVub
dub
ErV
rbrdrb
ErV
rbrbdr
r
r
r
)/1(1
12
)(1
/12
)(1
/22
22
/1
0
2
2
2
2
2
2
max
min
min
min
q
q
q
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 16
EuVub
dubr
)/1(1
12
:equationangleScattering
22
/1
0
min
q
1/2
1sin21
12
12b2b
11)(
:examplescatteringRutherford
2
1
22
/1
0
2
min
min
q
buubdub
brrErV
r
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 17
2/sin
2/cos2
1/2
1sin2
:continuedscatteringRutherford
2
1
q
b
b
2/sin1
16sin 4
2
q
d
dbbdd
8/30/2013 PHY 711 Fall 2013 -- Lecture 2 18
Hard sphere scattering
2cos
that showyou willhomework your For Db
4sin
2Dddbb
dd
