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

ddbb

dd

PHY 711 Fall 2013 -- Lecture 3 49/2/2013

Relationship between scattering angle and impact parameter b for interaction potential V(r):

ErV

rb

rdrbr )(1

/12

2

2

2

min

sin

ddbb

dd

0)(1

:where

min2

min

2

ErV

rb

PHY 711 Fall 2013 -- Lecture 3 59/2/2013

2/sin

2/cos

1/

1sin2

)(for :scattering Rutherford

2

1

b

b

rErV

2/sin1

4sin 4

2

ddbb

dd

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

2Dddbb

dd

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

vvVuuRrr

RrrR

mmmmmmmmmm

mmmm

CM

CM

CM

CM

21

22111

21

1

:caseour In

mmmm

mmm

CM

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

mmm

mm

mmm

mmm

m

VuUVUu

VuUVUuuV

121

1222

1

21

21

21111

21

1

2

:restat initially is Since

CM

CM

VVvVVv

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

/cossin

/cossintan

coscossinsin

mmVV

VVvVv

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

CMCMmm

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

v1V1

VCM

y

9/2/2013

Relationship between center of mass and laboratory frames of reference – continued (elastic scattering)

211

11

11

11

/cossin

/cossintan

coscossinsin

mmVV

VVvVv

CM

CM

CM

y

yy

VVv

22121

21

/cos/21

/coscos :Alsommmm

mm

y

PHY 711 Fall 2013 -- Lecture 3 139/2/2013

Differential cross sections in different reference frames

y

y

y

y

coscos

sinsin

dd

dd

dd

dd

dd

dd

LAB

CM

LAB

CM

CM

CM

LAB

LAB

2/322121

21

22121

21

/cos/21

1cos/coscos

/cos/21

/coscos

:Using

mmmm

mmdd

mmmm

mm

y

y

PHY 711 Fall 2013 -- Lecture 3 149/2/2013

Differential cross sections in different reference frames – continued:

yy

coscos

dd

dd

dd

CM

CM

LAB

LAB

1cos/

/cos/21

21

2/322121

ymm

mmmmd

dd

d

CM

CM

LAB

LAB

21 /cossintan :where

mm

y

PHY 711 Fall 2013 -- Lecture 3 159/2/2013

Example: suppose m1 = m2

1cos/

/cos/21

21

2/322121

ymm

mmmmd

dd

d

CM

CM

LAB

LAB

21 /cossintan :where

mm

y

20 that note

2

1cossintan :case In this

y

yy

yyy cos42

CM

CM

LAB

LAB

dd

dd

PHY 711 Fall 2013 -- Lecture 3 169/2/2013

energy) mass ofcenter is that (Note

)( :scattering Rutherford2

21

Er

eZZrErV

2/sin1

4 4

2

CMdd

Example of cross section analysis – CM versus lab frame

yyyyy

42

21

sincoscos42

For

CM

CM

LAB

LAB

dd

dd

mm