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