NEEP 541 – Focusing and Channeling

Fall 2002Jake Blanchard

Outline Focusing

Definitions Modeling Criteria

Channeling Definitions Modeling Criteria

Definitions focusing=energy transfer in near head

on collisions along a row of close-packed atoms Larger distances covered Less displacements produced (more

heating) Only seen at low energies

Schematic

Angles must be small enough to keep collisions in a line

r0D

Criterion for Focusing We must have n+1<n

12

;2

2

sinsin

0

1

0

1

0

1

r

D

rD

rD

n

nnnn

nnn

n n+

1

Difference Equation

0

10

10

01

4

,22

12

12

rDso

r

Dfor

r

D

r

D

nn

n

n

nn

For focusing to take place

Potentials The sphere radius is energy dependent Use Born-Mayer potential Use equivalent hard sphere model

r

r

E

Ar

rrp

r

p

E

rV

rArV

2ln

2

2;0

0)(

1

exp)(

0

0

2

2

Hard sphere radius is equal to distance of closest approach in head on collision for Born-Mayer potential

Criteria for Focusing Focusing occurs if:

For fcc metals, close packed direction is <110>

2exp2

2ln24 0

DAE

E

ArD

r

r

Example: Copper For fcc metals, close packed

direction is <110>

eVE

eVE

AD

Aa

eVA

A

aD

r

140

70

55.2

61.3

102

2.0

2

10

0

4

0

Displacement Effects Let c=probability of either channeling or

focusing effect (independent of energy)

c

c

E

E

c

cE

dTTE

c

E

EccE

E

dTTccE

c

d

E

E

d

E

E

d

d

21221

1)(

)()1(2

)1(2)(

)()1(2)(

21

2

Displacement Effects Consider 10 keV PKA in Fe c=0 gives 200 displacements c=0.07 gives 100 displacements

What is the probability (c)? Assume Copper c will increase with decreasing energy

E (eV) 140 100 60c 0 0.028 0.065 2.8 1.951 1.185

E/2Ed 2.8 2 1.2

Channeling Long distance displacement of

energetic particles down open direction in lattice

Direction is same as for focusing Channeling occurs between close-

packed rows Channel radius for fcc is

approximately 0.3a0

Schematic

Diagram

x

y

pN

D

N atoms per period in lattice

Scattering Each atom passed contributes a

momentum change Use impulse approximation

N

ii

N

ii

N

ii

p

nn

pIvM

PvM

v

pI

pr

pdr

dr

dV

vP

vM

P

1222

1221

222

2

22

)(2

1

)(22

Scattering

N

ii

N

ii

N

ii

ypID

vMyFyM

vyvy

ypIvDMD

v

vDtdt

d

ypIvM

1222

22

122

2

2

1222

)(2

)(

;

)(2

)(2

Scattering

22/

///

00

2)(

2112)(

)()(2

)()(

yyeR

D

AyU

ep

yep

yep

D

AyU

dyypIypID

dyyFyU

chRch

bybyp

y

ii

y

2-D channel2 rows of

atomspi=p=Rch

Born-Mayer

Scattering

xE

yy

E

M

EMvT

MT

M

yy

yyF

sin

22

22

22

2

2)(

0

2

2

2

2

Period of oscillation

Criterion

E

RR

E

xy

Ey

Ey

yUEMvMv

chch

y

2

00

0

00

20

20

002

022

02

max,;max,

2sin

2

)(sinsin2

1

2

1

Critical angle

Channeling Notes Simulations indicate 1-10% of PKAs

channel There is no upper limit on energy When wavelength is about 2D,

large angle collision is likely

22

min 1.02

KDKEch

300 eV for Cu

End of Channeling Event

Summary Kinchin-Pease is used for

displacements Realistic cross sections reduce by

up to 50% Electronic losses reduce

displacements continuously Focusing and channeling reduce

by up to 50%