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