8/31/2015phy 711 fall 2015 -- lecture 31 phy 711 classical mechanics and mathematical methods...
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PHY 711 Fall 2015 -- Lecture 3 18/31/2015
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 3:
1. Scattering theory – some more details
2. Rutherford scattering
3. Transformation between lab and center of mass reference frame.
PHY 711 Fall 2015 -- Lecture 3 28/31/2015
Scattering theory:
PHY 711 Fall 2015 -- Lecture 3 38/31/2015
angleat
detector into scattered is that beamincident of Area
areaunit per particlesincident ofNumber
particle per target at particles detected ofNumber
section cross alDifferenti
d
d
Figure from Marion & Thorton, Classical Dynamics
sin sin
d
dbb
dd
dbbd
d
d
bdbdb
PHY 711 Fall 2015 -- Lecture 3 48/31/2015
r min
r( )f
f
2 2
2
2 2
2 2
In center of mass frame:
1( )
2 2
1 ( )
2 2
drE V r
dt r
drV r
d r r
2
2
2
/
2 ( )2
rd dr
E V rr
PHY 711 Fall 2015 -- Lecture 3 58/31/2015
2
2
Some details: conservation of angular momentum:
Transformation of trajectory variables:
( ) ( )
dr
dt
r t r
dr dr d dr
dt d dt d r
)(22
1
)(22
1
:frame mass ofcenter in theenergy ofon Conservati
2
22
2
2
22
rVrrd
dr
rVrdt
drE
PHY 711 Fall 2015 -- Lecture 3 68/31/2015
)(22
1
)(22
1
2
22
2
2
22
rVrrd
dr
rVrdt
drE
)(2
2
/
)(2
2
for Solving
2
2
2
2
2
2
42
rVr
E
rdrd
rVr
Er
d
dr
(r))r(
PHY 711 Fall 2015 -- Lecture 3 78/31/2015
)(2
2
/
2
2
2
rVr
E
rdrd
bE
vE
bv
2
2
1
:separation largeat tion simplificaFurther
2
PHY 711 Fall 2015 -- Lecture 3 88/31/2015
2
2
2
2
2
2
max min
When the dust clears:
/
2 ( )2
/
( )1
( , )
rd dr
E V rr
b rd dr
b V rr E
b E r r r
PHY 711 Fall 2015 -- Lecture 3 98/31/2015
0)(
1
:where
)(1
/
min2
min
2
2
2
2
0 min
max
E
rV
r
b
ErV
rb
rbdrd
r
PHY 711 Fall 2015 -- Lecture 3 108/31/2015
Relationship between max and :
max
max
2( )
2 2
PHY 711 Fall 2015 -- Lecture 3 118/31/2015
min
min
min
2
max 2
2
2
2
2
1/
2 20
/
2 2 ( )1
1 /2
( )1
12
(1 / )1
r
r
r
b rdr
b V r
r E
rb dr
b V r
r E
b duV u
b uE
PHY 711 Fall 2015 -- Lecture 3 128/31/2015
min1/
2 20
Scattering angle equation:
12
(1 / )1
r
b duV u
b uE
min
2minmin
2
min
1/1
2 2 2
2
0
Rutherford scattering example:
( ) 1 =0
1 1 1
2b 2b
1 12 2sin
1 2 / 1
r
V r b
E r r r
r b
b dub u u b
2min
2min
where:
( ) 1 0
b V r
r E
PHY 711 Fall 2015 -- Lecture 3 138/31/2015
2/sin
2/cos2
1/2
1sin2
:continued scattering Rutherford
2
1
b
b
2/sin
1
16sin 4
2
d
dbb
d
d
PHY 711 Fall 2015 -- Lecture 3 148/31/2015
2/sin
1
16sin 4
2
d
dbb
d
d
From webpage: http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/rutsca2.html#c3
What happens as 0?
PHY 711 Fall 2015 -- Lecture 3 158/31/2015
The results above were derived in the center of mass reference frame; relationship between normal laboratory reference and center of mass:
Laboratory reference frame: Before After
u1 u2=0 v1
v2
yz
m1 m2
Center of mass reference frame: Before After
U1 U2
V1
V2
m1 m2
PHY 711 Fall 2015 -- Lecture 3 168/31/2015
Relationship between center of mass and laboratory frames of reference
2211212211
212211
212211
mass ofcenter of Definition
vvVuu
Rrr
Rrr
R
mmmmmm
mmmm
mmmm
CM
CM
CM
CM
21
22111
21
1
:caseour In
mm
mm
mm
mCM
vvuV
V1
VCM
v1yCMVUu 11 CMVVv 11
U1
u1
VCM
PHY 711 Fall 2015 -- Lecture 3 178/31/2015
Relationship between center of mass and laboratory frames of reference -- continued
CMCM
CMCMCM
mm
m
m
m
mm
m
mm
m
m
VuUVUu
VuUVUuuV
121
1222
1
21
21
21111
21
1
2
:restat initially is Since
CM
CM
VVv
VVv
22
11
PHY 711 Fall 2015 -- Lecture 3 188/31/2015
V1
VCM
v1y
Relationship between center of mass and laboratory frames of reference
211
11
11
11
/cos
sin
/cos
sintan
coscos
sinsin
mmVV
VVv
Vv
CM
CM
CM
y
yy
VVv
For elastic scattering
PHY 711 Fall 2015 -- Lecture 3 198/31/2015
Digression – elastic scattering
2
21212
22212
1121
2212
12222
12112
1
CM
CM
VmmVmVm
VmmUmUm
Also note:
0 0
21
21
22112211
CMCMm
m
mmmm
VUVU
VVUU
/mm/UV/VV
mm
CMCM 2111
21
: thatSo
: thatnote Also
and
21
CM2211
UU
VVUVU
PHY 711 Fall 2015 -- Lecture 3 20
v1
V1
VCM
y
8/31/2015
Relationship between center of mass and laboratory frames of reference – continued (elastic scattering)
211
11
11
11
/cos
sin
/cos
sintan
coscos
sinsin
mmVV
VVv
Vv
CM
CM
CM
y
yy
VVv
22121
21
/cos/21
/coscos :Also
mmmm
mm
y
PHY 711 Fall 2015 -- Lecture 3 218/31/2015
Differential cross sections in different reference frames
y
y
y
y
cos
cos
sin
sin
d
d
d
d
d
d
d
d
d
d
d
d
LAB
CM
LAB
CM
CM
CM
LAB
LAB
2/322121
21
22121
21
/cos/21
1cos/
cos
cos
/cos/21
/coscos
:Using
mmmm
mm
d
d
mmmm
mm
y
y
PHY 711 Fall 2015 -- Lecture 3 228/31/2015
Differential cross sections in different reference frames – continued:
yy
cos
cos
d
d
d
d
d
d
CM
CM
LAB
LAB
1cos/
/cos/21
21
2/322121
ymm
mmmm
d
d
d
d
CM
CM
LAB
LAB
21 /cos
sintan :where
mm
y
PHY 711 Fall 2015 -- Lecture 3 238/31/2015
Example: suppose m1 = m2
1cos/
/cos/21
21
2/322121
ymm
mmmm
d
d
d
d
CM
CM
LAB
LAB
21 /cos
sintan :where
mm
y
20 that note
2
1cos
sintan :case In this
y
y
y
yyycos4
2
CM
CM
LAB
LAB
d
d
d
d
PHY 711 Fall 2015 -- Lecture 3 248/31/2015
energy) mass ofcenter is that (Note
)( :scattering Rutherford2
21
Er
eZZ
r
ErV
2
4
1
16 sin / 2
CM
d
d
Example of cross section analysis – CM versus lab frame
1 2
2
4
For
2 cos4cos
4 sinLAB CM
LAB CM
m m
d d
d d
y y yyy