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PHY 711 Fall 2013 -- Lecture 3 19/2/2013
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 3:
Chapter 1 – scattering theory continued; center of mass versus laboratory reference frame.
PHY 711 Fall 2013 -- Lecture 3 29/2/2013
PHY 711 Fall 2013 -- Lecture 3 39/2/2013
Scattering geometry:
sin
d
dbb
d
d
PHY 711 Fall 2013 -- Lecture 3 49/2/2013
Relationship between scattering angle and impact parameter b for interaction potential V(r):
ErV
rb
rdrb
r )(1
/12
2
2
2
min
sin
d
dbb
d
d
0)(
1
:where
min2
min
2
E
rV
r
b
PHY 711 Fall 2013 -- Lecture 3 59/2/2013
2/sin
2/cos
1/
1sin2
)(for :scattering Rutherford
2
1
b
b
rE
rV
2/sin
1
4sin 4
2
d
dbb
d
d
Example of cross section analysis
PHY 711 Fall 2013 -- Lecture 3 69/2/2013
Hard sphere scattering:
2cos
that showedyou homework your For
Db
4sin
2D
d
dbb
d
d
Example of cross section analysis
PHY 711 Fall 2013 -- Lecture 3 79/2/2013
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 2013 -- Lecture 3 89/2/2013
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 2013 -- Lecture 3 99/2/2013
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 2013 -- Lecture 3 109/2/2013
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 2013 -- Lecture 3 119/2/2013
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 2013 -- Lecture 3 12
v1
V1
VCM
y
9/2/2013
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 2013 -- Lecture 3 139/2/2013
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 2013 -- Lecture 3 149/2/2013
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 2013 -- Lecture 3 159/2/2013
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
yy
yyycos4
2
CM
CM
LAB
LAB
d
d
d
d
PHY 711 Fall 2013 -- Lecture 3 169/2/2013
energy) mass ofcenter is that (Note
)( :scattering Rutherford2
21
Er
eZZ
r
ErV
2/sin
1
4 4
2
CM
d
d
Example of cross section analysis – CM versus lab frame
yyyyy
42
21
sin
coscos4
2
For
CM
CM
LAB
LAB
d
d
d
d
mm