Post on 13-Jan-2016
description
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%