NEEP 541 – Damage and Displacements

Fall 2003Jake Blanchard

Outline Damage and Displacements

Definitions Models for displacements Damage Efficiency

Definitions Displacement=lattice atom knocked from its

lattice site Displacement per atom (dpa)=average number of

displacements per lattice atom Primary knock on (pka)=lattice atom displaced by

incident particle Secondary knock on=lattice atom displaced by

pka Displacement rate (Rd)=displacements per unit

volume per unit time Displacement energy (Ed)=energy needed to

displace a lattice atom

Formal model To first order, an incident particle

with energy E can displace E/Ed lattice atoms (either itself or through knock-ons)

Details change picture Let (E)=number of displaced

atoms produced by a pka

Formal Model

m

d

m

d

m

d

T

Ed

dd

T

Ed

T

Ed

dTTET

dEEENR

dEdTTETENR

dTdETEENTR

),()(

)()(

),()()(

),()()(

0

0

0

What is (E) For T<Ed there are no displacements For Ed <T<2Ed there is one

displacement Beyond that, assume energy is shared

equally in each collision because =1 so average energy transfer is half of the incident energy

Schematic

pka

skatka

displacements

1 2 4 2N

Energy per atom

E E/2 E/4 E/2N

Displacement model Process stops when energy per atom

drops below 2Ed (because no more net displacements can be produced)

So

d

N

dN

ETE

or

ET

22)(

22

Kinchin-Pease model

T

Ed 2Ed Ec

More Rigorous Approach Assume binary collisions No displacements for T>Ec No electronic stopping for T<Ec Hard sphere potentials Amorphous lattice Isotropic displacement energy Neglect Ed in collision dynamics

Kinchin-Pease revisited

E

EE

EE

dTTTEE

E

dTTTEEEdTE

EE

spherehardEETE

dTTTETEdTETE

TTEE

0

00

00

)()(1)(

)()()()()(

;1;)(),(

)()(),()(),(

)()()(

Kinchin-Pease revisited

E

E

d

E

E

E

E

E

E

EE

E

E

d

d

d

d

d

dTTEE

EE

dTTE

dTE

dTE

E

dTTE

E

dTTdTTE

TEwdwwdTTE

2

2

2

0

0

00

0

0

)(22)(

)(2202)(

)(2)(

)()(

;)()(

Kinchin-Pease revisited Solution is:

For power law potential, result is:dEEE2

)(

d

d

s

d

EEE

EEE

s

EsEE

257.0)(3

252.0)(2

122

)( 11

Electronic Stopping Repeat with stopping included Hard sphere potentials

dd

e

ENk

EEE

EkdxdE

241

2)(

Hard sphere collision cross section (independent of E)

Don’t need cutoff energy

any more

Comprehensive Model Include all effects (real potential,

electronic stopping) Define damage efficiency:

3/1

22

4/36/1

88.0/2

4.04.313.011)(

2)()(

Zaa

aeZE

E

EEEE

B

d

Damage Efficiency