1C

lass

ical

Pot

. Sca

tter

ing

Cla

ssic

al P

ot. S

catt

erin

gC

lass

ical

Pot

. Sca

tter

ing

W. Udo Schröder, 2008

Classical Potential Scattering Deflection Function

To infinity

Angular Momentum

p

L 1 2 Conservative potentialconserves energy E.

Central potential

V(r )

V(r ) V(r )

From infinity

Distance of closest approach

p

r

Central potentialconserves angular momentum L.

V(r ) V(r )

2

Coul Nucl 2

( 1)V(r , ) V (r ) V (r )

2 r

2C

lass

ical

Pot

. Sca

tter

ing

infinity

Scatter Angle Classical deflection function

maps radial dependence

Coul Nucl 2V(r , ) V (r ) V (r ) 2 r

Cla

ssic

al P

ot. S

catt

erin

g

Scatter Centerb = Impact Parameter

Symmetric trajectoryr = classical turning point ( E (r )=0 )

maps radial dependenceof potential in elastic scattering to an angular dependence of the

Cla

ssic

al P

ot. S

catt

erin

g

0r2 d dr dr

r0 = classical turning point ( Er(r0 )=0 ) dependence of the scattering cross section d d ( )

W. Udo Schröder, 2008

0r

Classical Potential Scattering Deflection Function

To infinity

Angular Momentum

p

L 1 2 2Coul Nucl 2

( 1)V(r , ) V (r ) V (r )

2 r

From infinity

Distance of closest approach

p

r2

r cm

2 2

drE ( ) E V(r , )

2 dt

d ( 1)E ( ) r

radial

tangential

3C

lass

ical

Pot

. Sca

tter

ing

infinity

Scatter Angle

t 2

2 2

t cm2

2 2 2

d ( 1)E ( ) r

2 dt 2 r

d 2 dr 2E ( ); E V(r , )

dt dtr

tangential

Cla

ssic

al P

ot. S

catt

erin

g

Scatter Centerb = Impact Parameter

Langer modification :r2 d dr dr

Symmetric trajectoryr0 = classical turning point ( E (r )=0 )

2 2 2t

2cm

E ( )d d drdt dtdr r E V(r , )

Cla

ssic

al P

ot. S

catt

erin

g

2( 1) ( 1 2)

Langer modificatio

d 1 ( 1 2)

n :0r

2 d dr dr 0point ( Er(r0 )=0 )

dr ( 1 2)2

W. Udo Schröder, 2008

2

cm

d 1 ( 1 2)dr r 2 E V(r , )0

2rcm

dr ( 1 2)2

r 2 E V(r , )

Classical Cross Section and Deflection Function

All projectiles impinging on area 2 bdb are scattered to solid angle element daround angle

dF=2 R’dR’

around angle

bd ( ) 2 b db

dF R dR

b db R’

dR’

R

2 R’dR’

4C

lass

ical

Pot

. Sca

tter

ing

2 2

1

b

dF R dRd 2 2 sin d

R R

d ( ) 2 b db b( ) dd 2 sin d sin db

2 bdb

dR’

R’=RsindR’=Rd

Cla

ssic

al P

ot. S

catt

erin

g d 2 sin d sin dbdR’=Rd

1

bd ( ) b( ) d1 2L

Langer : bp

Cla

ssic

al P

ot. S

catt

erin

g

b

d sin db

12 1d ( ) d

p

Strength of deflection of trajectories ( ) determines

W. Udo Schröder, 2008

12 1d ( ) dd sin d

trajectories ( ) determines differential cross section d /d

Coulomb Trajectories

2 2

2 21 2

2

V(r , ) :r r2 r 2 r

e ( 1 LA)Z Z

Hyperbolic trajectories in 1/r potential

0 cm

2cm

r r2 r 2 r

L b d 2E b const.

E 2

Distance of closest approach d(b) r0

5C

lass

ical

Pot

. Sca

tter

ing

0

0 cm

Distance of closest approach d(b) r

Central collision: b 0d d(b 0) A E

Cla

ssic

al P

ot. S

catt

erin

g

2

0 cm r 020 0

22

A Lr r E E (r ) 0

r 2 r

A Lr r 0

Cla

ssic

al P

ot. S

catt

erin

g

20 0

cm cm

2 2

0cm cm cm

r r 0E 2 E

A A Lr

2E 2E 2 E

W. Udo Schröder, 2008

cm cm cm

2 20 0 0

2E 2E 2 E

d(b) r d 2 d 2 b

Coulomb Trajectories

Distance of closest approach2 2

0Lcm cm cm

Distance of closest approach

A A Lr

2E 2E 2 E

1 22cm2EL 2A Ldr

cm cm cm

2 20 0

2E 2E 2 E

d d 2 d 2 b

0

6C

lass

ical

Pot

. Sca

tter

ing

0L

cm2 2 2r

2

2EL 2A Ldr

rr r

variable transform

x : L r dx dr L r

Cla

ssic

al P

ot. S

catt

erin

g

0L

1 20 2cm

x

0

2E 2Adx x x

L

x A L

x : L r dx dr L r

Cla

ssic

al P

ot. S

catt

erin

g

0

2cm x

0

x A Larc sin

2E (A L)

x L A 1

W. Udo Schröder, 2008

0

20 x

x L A 1arc sin

1 (2b d )

Coulomb Deflection Functions

0

0

20 x

x L A 1arc sin

1 (2b d )

2 2

0

1 1cos

2E L2b 11Ad0

7C

lass

ical

Pot

. Sca

tter

ing

0

0

0

2b d tan (r )2 2A

2b d cot cot2 2E 2

Cla

ssic

al P

ot. S

catt

erin

g

0cm

2b d cot cot2 2E 2

21 2e Z Z 1b 2 arctan

2E b

Cla

ssic

al P

ot. S

catt

erin

g

cm

b 2 arctan2E b

2 21 2 1 2

Sommerfeld Parameter

e Z Z e Z Z2 arctan

W. Udo Schröder, 2008

1 2 1 2

2cm

e Z Z e Z Z

2E2 arctan

1 2

The Rutherford Formula

2 1d ( ) dd sin d

2 2

2

1 1 dtan

2 1 2 2 dcos ( 2) ( 1 2)d

2 cos ( 2)

8C

lass

ical

Pot

. Sca

tter

ing

22

2 2 2

d2 cos ( 2)

d ( 1 2)2 2

tan cos ( 2) sin ( 2)2

Cla

ssic

al P

ot. S

catt

erin

g

2

2

1d ( ) 1d 2 sin sin ( 2)

22d ( )

Rutherford Cross Section

Cla

ssic

al P

ot. S

catt

erin

g

2 2

2

2 2

2

cos( 2) 12 sin sin( 2) sin ( 2)

cos( 2) 14 sin( 2)cos( 2) sin( 2) sin ( 2)

4d sin ( 2)

221 2d ( ) e Z Z 1

W. Udo Schröder, 2008

24sin( 2)cos( 2) sin( 2) sin ( 2)1 24

cm

d ( ) e Z Z 1d 4E sin ( 2)

Finite Size Coulomb Potentials

Coulomb potential between point Point Nucleus

1 cr r R

Coulomb potential between point charge and homogenous sharp sphere

Finite Size Nucleus

221 2

1

( ) 13 ,

2

c

hsCoul

cc c

r r R

V r e Z Z rr R

R R

9C

lass

ical

Pot

. Sca

tter

ing

More realistic (if non-adiabatic)

Cla

ssic

al P

ot. S

catt

erin

g

More realistic (if non-adiabatic) Coulomb potential:2 overlapping diffuse charge distributions (Fermi)

Cla

ssic

al P

ot. S

catt

erin

g

VCoul: similar at large distances, large differences for large projectile/target overlaps

W. Udo Schröder, 2008

overlaps

Coulomb Deflection Function

Monotonic for monotonic dependence V(r)

Smaller b (l) larger

Non-monotonic for non-monotonic potentials

10C

lass

ical

Pot

. Sca

tter

ing

monotonic potentials

Cla

ssic

al P

ot. S

catt

erin

gC

lass

ical

Pot

. Sca

tter

ing

1

n

n

b ( )d ( ) dd sin db

Multi-valued deflection functions:Different b same

W. Udo Schröder, 2008

nn bd sin dbDifferent b same

“Rainbow” Scattering

d ( ) b( ) dbd sin d

Large d /db small cross sections L Large d /db small cross sections

Peak cross section for d /db ˜0

2( ) r rb b b

L

11C

lass

ical

Pot

. Sca

tter

ing

( )

( )

1

r r

r r

b b b

b b

db

Orb

itin

g

Use semi-

Cla

ssic

al P

ot. S

catt

erin

g

2b rrd

d ( )d

Coulomb Rainbow

Orb

itin

g

Use semi-classical WKB

Cla

ssic

al P

ot. S

catt

erin

g

L

Molecular Rainbow Scattering

12 6

Lennard-Jones (6-12) Potential

12 6( ) 4V r r r

Differential Cross Section E/ =2

12C

lass

ical

Pot

. Sca

tter

ing

Differential Cross Section E/ =2

Cla

ssic

al P

ot. S

catt

erin

gC

lass

ical

Pot

. Sca

tter

ing

b

W. Udo Schröder, 2008

Oscillations due to quantum wave effects

Effective Potentials & Elastic Deflection FunctionsDifferent relative magnitudes of interaction potentials Vcoul+VN+VDifferent relative magnitudes of interaction potentials Vcoul+VN+V

13C

lass

ical

Pot

. Sca

tter

ing

Very heavy ions: No capture, no fusion “nuclear rainbow”

Cla

ssic

al P

ot. S

catt

erin

g

Asymmetric heavy (or light) system

Symmetric heavy system does not

Very heavy ions: No capture, no fusion “nuclear rainbow”

Cla

ssic

al P

ot. S

catt

erin

g

(or light) system shows orbiting:Perfect balance of Coulomb, centrifugal and nuclear forces

system does notshow orbiting:Perfect balance of Coulomb, centrifugal and nuclear forces

W. Udo Schröder, 2008

and nuclear forces (at o= 50)

and nuclear forces impossible

Fluctuations in Scattering

Nucleus-nucleus interactions = superposition of nucleonic interactions fluctuating forces

1P , exp

interactions fluctuating forces probability distribution in

14C

lass

ical

Pot

. Sca

tter

ing

22

1P , exp

2 ( )2

Vanishing fluctuations :

lim P , d d

Cla

ssic

al P

ot. S

catt

erin

g

0lim P , d d

Washes out artificial singularitiesFolded cross section:

Cla

ssic

al P

ot. S

catt

erin

g

22

22

d 1exp d

d sin 2 ( )2

Folded cross section:

W. Udo Schröder, 2008

d sin 2 ( )2

Riedel et al., 1979

Mapping the Potential

16 208O Pb16 208O Pb107 MeV

Ecm=200MeV

15C

lass

ical

Pot

. Sca

tter

ing

Ecm=107MeV

Cla

ssic

al P

ot. S

catt

erin

gC

lass

ical

Pot

. Sca

tter

ing

Elastic scattering: Increasing bombarding energy

“pocket” in effective potential disappears

W. Udo Schröder, 2008

“pocket” in effective potential disappears orbiting and nuclear rainbow disappear peaks in cross section disappear

Reactive Scattering Deflection FunctionsConcept of deflection function extended to inelastic (dissipative) scattering, E

Partial Wave Sum

2(2 1)

dd sin d

d1 2L

Langer : b 1 2p

17

2

Langer : b 1 2p

db d d 2 bdb 2 1

Cla

ssic

al P

ot. S

catt

erin

g

l

0 lmax

max2 2

max max0

max

(2 1) ( 1)

1:

Cla

ssic

al P

ot. S

catt

erin

g

2 2 2max max max( 1)Geometric Cross Section

Cla

ssic

al P

ot. S

catt

erin

g

max max max( 1)

kR R22 2 2 2

max max max

2maxR

Geometric Cross Section

W. Udo Schröder, 2008

Partial (l) Wave Decomposition of

l-windows2 2 2

max max max( 1)

2 2R R

l-windows2(2 1)

2 2

2 1 1

ij j i

j j i i

R R

l

2312

01

18C

lass

ical

Pot

. Sca

tter

ing

l

l0 l1 l2 lmax

???

, ?

,

is measureable for a given event class

how to infer

ambiguousE*(l)

M (l)

Excitation energyParticle

Cla

ssic

al P

ot. S

catt

erin

g ??? ,

,

ambiguous

Need independentMonotonic functi

observable fon of

Mn(l)

Zmax(l)

Particle MultiplicityZ of largest product

Different possible functions f(l )

2 2

( )

j i

Monoton function f

df

ic

Cla

ssic

al P

ot. S

catt

erin

g

2

1( )

21

f ff

Experimental tool to estimate l»1

Different possible functions f(l )

22j i

ij

dff d

W. Udo Schröder, 2008

exp2

1( ) ( )

2estf

Conservation of Flux/Cross Section Sum Rule

d ( ) Only long-range d ( )d

PNPoint NuclRuthd

Only long-range interaction: Coulomb point-nucleus beyond short range of nuclear forcePoint NuclRuth

Coul

dd

nuclear force

PNtot Ruth el R

19C

lass

ical

Pot

. Sca

tter

ing

Measured elastic el

el Rmax max 180

000

No elastic events measured R

tot Ruth el R

observableTheoretical sum observable

Reaction cross sectionR

R2 2

max

Cla

ssic

al P

ot. S

catt

erin

g

dd

Ela

stic

Sca

tter

ing

PN exptRuth el

R0

d d2 d sin

d d

Cla

ssic

al P

ot. S

catt

erin

g

Reactions Tran

sfer

Ela

stic

Sca

tter

ing

Determine reaction cross section from the value for the quarter-point angle

expteld 1

expte

NR

lPuthd

d dd

1Elastic

Rea

ctio

ns

W. Udo Schröder, 2008

0 F R

Reactions/Fusion

Tran

sfer

Ela

stic

Sca

tter

ing

1 4

elPNRuth

d 14d

0

14

R1 4 max

Rea

ctio

ns

Energy and Angular Distributions

Determine elastic energy = f( ), fit standard line shape, determine elastic cross section el( )

20 exptel

PNRuth

d 14d

Cla

ssic

al P

ot. S

catt

erin

gC

lass

ical

Pot

. Sca

tter

ing

Cla

ssic

al P

ot. S

catt

erin

g

1 4

W. Udo Schröder, 2008

Plot ratio elastic/Rutherford cross section = f( ), determine quarter-point

reaction total integrated cross section R.

84 209 036 83 1 4

max max R

Kr(600MeV ) Bi : 66.7

268.5 R 14.17fm 1.9belmax 1 4

1 4

Distance of Strong Absorption RSAe Z Z2

cm

e Z Zd E b e Z Z

E

b d

221 2

1 2

0

( ) 1 1 2 ( )2

1( ) cot

2 216O 2 2

1 3 1 3SA Int 1 2R R 1.7 A A

16O Beam

Strong Surface Separation

21

Cla

ssic

al P

ot. S

catt

erin

g

Strong Surface Separation

Cla

ssic

al P

ot. S

catt

erin

g

12C Beam

Cla

ssic

al P

ot. S

catt

erin

g

SA 1 2s R C C

Different parameterizations

W. Udo Schröder, 2008

SA P T P T

1R C C 4.49 C C

6.35

Different parameterizationslarge uncertainties (Ecm , A/Z dependencies?)

22

Cla

ssic

al P

ot. S

catt

erin

gC

lass

ical

Pot

. Sca

tter

ing

Cla

ssic

al P

ot. S

catt

erin

g

W. Udo Schröder, 2008