1' Fundamentals of Radiation Damage and Defect Production

1
1. Fundamentals of Radiation Damageand Defect
Production
NE220 Spring 2008 Brian D. Wirth and D. R.
Olander Nuclear Engineering Department
Ref Sect 17.2, D.O. text
2
 Energetic elementary particles

• neutrons interact with nuclei
• gamma rays interact with electrons
• - photo-electric effect (? atom ? e recoil
  Ee Ee I)
• - pair production (? ? e e Ee Ee E? -
  1.02 MeV
• e ? 2 ? E? 0.51 MeV annihilation
  rad.)
• - compton scattering (? e ? ?
  e E? E? Ee)
• electrons (accelerator- produced, beta
  particles Compton)
• fission fragments, recoils from alpha decay
• light decay products (alpha, proton)
• accelerator-produced heavy ions
• recoil atoms (ions) produced from scattering
  collisions  

3
Ion interactions with electrons of solid

•   Termed Electronic stopping - dominant at high
  ion energies
• Electrons of the medium are removed from the
  nucleus leading to ionization along the ion track
•  Characterized by the stopping power, or energy
  loss per unit path length
•  Most of the ionized electrons are thermalized
  and re-captured by nuclei - energy is degraded
  into heat
•  In insulators, some electrons are captured in
  defect sites, producing changes in electrical
  properties (e.g., resistivity)
• ion track is straight

4
Ion (electron) interactions with nuclei in solids

•  termed nuclear stopping - predominates at low
  ion energies
•  elastic scattering of moving ion and
• - bare nuclei at high ion energy (Coulomb
  forces only)
• - nuclei and bound electrons at low ion energy
• also characterized by a stopping power
•  produces permanent displacement of atoms of the
  solid
•  displacements are the cause of physical and
  mechanical
• property changes - radiation damage and
  radiation effects
• ion track irregular

5
Energy transfer in an Elastic Collision
Center-of-mass frame
(1)
Momentum conservation in CM Initial m1u10m2u20
(2) Final m1u1fm2u2f (3) Kinetic energy
conservation in CM (dropping the
1/2) Both are satisfied only if
u1f u10 (5a) and u2 f
u20 vCM (5b)
(4)
Ref Sect 17.2, D. O. text
6
Converting from CM to Lab for v2f in terms of v10
Combined diagram
Apply law of cosines to blue triangle use (5b)
Use (1)
Convert to KE
T E2f recoil energy
T 1/2?E(1 - cos?)
(6)
Tmax ?E
E E10 incident ion energy
(8)
(7)
7
Lab scattering angle of ion and recoil
law of sines to red triangle yields but
u1f u10, u10 v10 - vcm vcm m1/(m1m2)v10,
so law of cosines to red triangle
yields Combining Similarly, starting
from the law of sines for the blue triangle leads
to
(9a)
(9b)
8
Total elastic scattering cross section ?(E)

• N density of target particles
• ?(E) total scattering cross section ( 10-18
  nm2)

N?(E)dx probability of the collision of an
incident particle with a target particle in dx
collisions per unit path N ?(E) Reciprocal is
the average distance between collisions mean
free path
(10)
9
Differential angular cross section ? (E,q)d?dx

• ? (E,q)d?dx N probability of collision in dx
  which scatters the incident particle into a
  center of mass angle within the range (q, d?)
• d? Differential solid angle (steradians) in
  center- of-mass coordinates
• azimuthally symmetry
• no ? dependence of ?
• integrate over 0 ? ? ? 2?

Also
(11)
10

• Differential energy-transfer cross section

N?(E,T)dTdx probability of collision in
path-length dx which transfers energy
in the range of (T,dT) to the target
particle The differential energy and angular
cross sections are related by

The total and
differential cross sections are related by
(11a)
T 1/2?E(1 - cos?)
(12)
11
Stopping power
Energy lost by a moving particle in path dx
take average of T over all possible values
(note limits)
or
(13)
Stopping power has electronic and nuclear
components
12
Range
range is the average path length traversed by a
particle projected along the initial direction
before it comes to rest
(14)
13
Interatomic potentials
The interatomic potential function, V(r), between
atoms and ions separated by a distance r only
the repulsive portion of V(r) is important.
gt 103 eV - Electrostatic repulsion between the
positively charged nuclei (Coulomb potential)  
- tens of eV - electron clouds begin to overlap.
To avoid violating the Pauli exclusion principle,
electron(s) of one of the species must be
promoted to higher energy levels
14
Interatomic Potential Functions
(15)
(16)
(17)
(18)
(19)
How to get s(E,T) from V(r) ???
15
Binary collision dynamics

• Objectives
• Determine the orbit of two particles in an
  elastic collision
• 2) Relate the interaction potential V(r) to the
  differential cross section ?(E,?)
• Known
•  interaction potential, V(r)
•  m1 is initially moving with E m2 initially at
  rest

16
Conversion of kinetic energy to potential energy
during a collision
 Because of momentum conservation, KE of the CM
must be retained throughout the collision
cannot be converted to PE initial kinetic
energy in the CM system is the energy available
for conversion to potential energy
kinetic energy in center of mass system during
collision
(20)
(21)
17
Closest Approach in a Head-On Collision
 head- on collision in the CM system ro
distance of closest approach in a head-on
collision. attained when Ec 0
given, V(r), solve for ro Special case
m1 m2
(22)
(23)
18
Initial configuration
m1r1 m2r2 r r1 r2
(24)
rsin?
(25a)
(25b)

• Decompose the particle velocity vectors u10 and
  vcm
• into radial and tangential components
• angular momentum of a particle about CM

(i 1,2)

• p impact parameter distance between axes of
  approaching particles in CM

19
Total angular momentum
(1)
(25a)
(25b)
(24)
?
(26)
before collision

• p impact parameter distance between axes of
  approaching particles in CM

20

• ? is the angle between the bisector and the line
  joining the two particles (r) The function r(?)
  is called the orbit
• before collision ? ?/2 when r ?
• at distance of closest approach ? ? /2 when
  r ro

(27a)
(27b)
(28)
Bisector line ? r0 through CM
21
Limit of ? as r ? ?

• ? ABC ? (by symmetry of the orbit)
• ?? 2? ?
• ?ACB ?/2 - ? (definition of bisector)

5. ?BAC ? (parallel-line theorem) 6. ?in
?ABC ? ? (?/2 -?) ? (sum of the angles
of a triangle) 7. Eliminate ? between 2. and 6.
? ?/2
22

• Conservation of angular momentum
• using (26)
• substitute (25a) (25b), solve for
•  

(29)
Conservation of total energy
Use (25a) (25b)
and (29) in above Ec0
Use
23
Use (20) for Ec0 and
(30)
At the distance of closest approach, dr/d? 0,
and from (30)
(31)
Integrate (30) over ½ of the orbit (ro r ?
?/2 ? ?/2)
(32)
(32) is the CLASSICAL SCATTERING INTEGRAL
(gives p as a function of ?)
24

• An impact parameter that lies in the range p to
  pdp forms an annular ring of area 2ppdp.
• A particle moving in this ring is scattered into
  the cone 2ps(E,q)d(cosq).

Equating these two differentials and solving
(33)

• The classical scattering integral (32) gives p
  as a function of ?
• p(?) is substituted into (33) to give the
  differential cross section
• Except for a few forms of V(r), this process is
  numerical.

25
The Rutherford cross section
Insert the Coulomb potential (15) into (32) and
nondimensionalize
(34)
and (31) becomes
(35)
where
(36)
(34) can be integrated analytically
26
Using (35), the arguments of sin-1 are
and
?the argument of the first sin-1 term is unity,
and the argument of the second is ?/(2-?), so
Using (35) and (36), p2 can be expressed in terms
of ?, which can then be eliminated by the above
equation to yield
?? ? as p?0 ??0 as p??
(37)
note
(sin2(?/2) (1 - cos?)/2 has been used)
27
Substituting (37) into (33), replacing Ec0 with E
using (22) yields Rutherford cross section
Converting to the energy-transfer cross
section using (6) (11a) yields Note
e214.4 eV-Å Characteristics of the Rutherford
cross section  Decreases with increasing
particle energy  Strongly forward-peaked
(i.e., becomes large as q or T ? 0) The
total scattering cross section (defined by (12))
is infinite.
(38a)
(38b)
28
Hard-Sphere Approximation
For low energy collisions, V(r ) becomes very
steep and can be approximated by a step
function ro is the sum of the hard
sphere radii - it is also the
distance of closest
approach for any V(r) - 2ro is sometimes
called the collision diameter -
total cross section Scattering is isotropic
in the CM system all q are equally probable.
From (11a), The differential energy-transfer
cross section is
(39)
(40)
Differential cross section
(41)
(42)
29
Equivalent Hard Sphere Model

•   avoids having to calculate s(E,q) by the exact
  method of evaluating the classical scattering
  integral.
•  is applicable for relatively low ion energies,
  where the function V(r) is very steep
•  Allows ro (determined from (21)), and hence ?
  (by (42)), to be a function of E
• retains isotropic scattering of the true hard
  sphere model.

30
Example - Born - Mayer potential (19) (V(r)
Ae-r/? )
at distance of closest approach (m1 m2) use
(22)
(43)
(44)
Use in (42)
31
Example Inverse-square potential Cu on Cu

• Use (18) (VA/r2) with A 7.7 eV-nm2
• From (22)
• From (42)
• for a 10 keV Cu ion moving in copper
• (10 keV) ?x7.7 eV-nm2/104 eV 2.4x10-3 nm2
• The mean free path is calculated from (10) with
• N 9.4x1021 atoms/cm3 (molecular density)
• l (Ns)-1 (2.4x10-3x9.4x1021x10-21
  cm3/nm3)-1 44 nm

(44a)
32

• effective charge of a moving ion all bound
  electrons of the moving ion with orbital
  velocities less than v10 are stripped

(45)
(46)
Example Zr fission fragment E 100 MeV M1
91 Z1 40 
Substitute into (45) (Z1)eff 26
protons 5 MeV (Z1)eff 1 5 keV proton
(Z1)eff 0.6
33
Energy Loss to Electrons (ionization of the solid)
Energy levels of electrons on an atom
34
Energy Loss to Electrons (ionization of the solid)

• Electrons of the medium Rest mass, mo 1/1836
  amu charge 1
• depending on ion energy, one of two types of
  interaction with electrons occurs
• The ion category depends on the Fermi energy of
  the electrons (see slide 33), which is ?F 5 eV
  their maximum velocity, from (45), is
•   High-energy ions ( v10 gt 3vF) interact with
  the bound or inner-shell electrons, which
  exhibit an average ionization energy of
• low-energy ions interact with one or two
  outer-shell or valence (conduction) electrons

(46)
(47)
35
Electronic stopping v10 gt 3ve

• The solid provides a total electron density NZ,
  where N is the atom density and Z is the atomic
  number.
• The stopping power is calculated from (13)
  with
• The lower limit on the energy transferred is
  ( For T lt To),
• the electrons cannot be excited since the
  higher states are occupied)
• - For a 5 MeV proton in iron (Z 26), Tmax
  10.9 keV, Tmin 0.3 keV
•  The interaction is purely Coulombic, so the
  Rutherford cross section (38b) is used in (13)
• (49)
• The factor of 2 in the numerator is a quantum
  mechanical correction. The formula is the
  Bethe-Bohr stopping power

(48)
36
Energizing electrons (v10 lt ve)

• In a collision with a conduction electron, T
  must be sufficient to excite the electron to
  empty states above the Fermi energy.

v10
ve
vef
v10
vef- v10
v10ve
(see slide 33)
v10
ve
v10
vef

• Because the ion and electron are comparable
  velocities, the energy transferred in a collision
  depends on the initial directions

v10-ve
vefv10
ve
ve

• Three cases are shown on the right

v10
v10
ve
ve
v10
Note because m1gtgtmo, the CM is stationary and
located on the ion
vef-v10
37
1) head-on
2) ion and electron moving in the same direction
electron loses energy
3) v10 ? ve - electron speed in its initial
direction does not change the ion sees a
stationary electron and the energy transfer is
given by (6) and (7)
38
Electron stopping power (v10 lt ve)

• energy transferred only to electrons near the
  Fermi level collisions with electrons deep in
  the Fermi sea cannot transfer energy because the
  higher levels are occupied.
•  The density of electrons that can accept energy
  from the slow-moving ion is
• Ne density of conduction electrons (N)
• the flux (current) of electrons in the CM Ie
  neve
• cross section for ion - electron
  collisions

39
Lindhard model of electron stopping
or
(50)

• Detailed calculation by Lindhard gives

(51)
Solid N atoms/nm3 Z atomic number
40
Transition from electronic stopping to nuclear
stopping
lower limit on Te to promote it to the
ionization level
(see slide 33)

•  minimum ion energy needed to transfer Te,min
  from case 3 on slide 37
• EC from case 1 is higher than case 3 by (ve/v10)
  100 see (45) and (46)
• EC is approximately when nuclear stopping takes
  over from electronic stopping (see slide 11)
• - E gt Ec - all energy loss by ionization
  (Bethe-Bohr or Lindhard)
• - E lt Ec - all energy loss by nuclear stopping

(52)
All stopping powers calculated from (13) using
(38b), (44) or (44a)
41
The primary event
Primary Knockon Atoms (PKA) are created by
neutron/ion atom collisions.
- T gt Ec only electronic energy
loss - no
displacements - T lt Ec only
atomic displacements (nuclear
stopping) which produces
vacancies and interstitials
En initial neutron energy En neutron energy
following collision E energy of projectile
(PKA or secondary SKA) T energy transferred to
target (stationary atom)
42
Displacement cascade

• The PKA strikes another atom, creating a
  secondary knock-on (SKA)
• The SKA hits another atom, making another SKA
• At each collision, the projectile energy is
  partitioned between the target atom and the
  recoiling projectile
• collisions continue until the struck atoms
  receive energies less than Ed

V vacancy empty lattice site left behind
when atom leaves I self-interstitial atom
when displaced atoms finally stop
43
Minimum Energy of recoil to be displaced

• the displacement energy depends on PKA direction
• Ed represents an average over all possible
  directions
• in fcc crystal structure
•   Ed displacement energy minimum energy
  transferred to a lattice atom to displace it (
  25 50 eV)
• EFp energy to form Frenkel pair (V I)
  thermodynamically
• ( 5eV)

Displacement Energy (eV)
44
Number of displacements per PKA of energy E

• Displacements stop when average recoil energy is

final number of displaced atoms (the
cascade)
(53)
This is the Kinchin-Pease model
45
Effect of PKA energy Loss by electron stopping

• Approximate method
• If E gt Ec, all energy loss from E to Ec is by
  electron stopping - no displacements
• Lindhard method
• Continuous blending of electronic stopping and
  nuclear stopping (displacements)
• 
  (55)

(54)
(56)
(see (17))
46
Comparison of displacement models
Lindhard Damage efficiency Displacements in
Fe
EC - see (52)
Displacements
EC for Fe
47
Lattice Structure Effects

• Previous analysis assumed random atom locations
  in solid with mean separation, N-1/3 however,
  solids have structure
•   focusing - transfer of atoms along an atomic
  row in nearly head-on collisions (Replacement
  Collision Sequence)
• channeling - motion of PKA long distances along
  an open direction in the lattice
• These processes affect both ? and the cascade
• structure (spatial correlation of V-I)

48
Focusing
Within a range of PKA energy and direction with
respect to an atomic row, a series of
collisions transmits energy and one atom along
the line of atoms (see sketch below) The first
atom A0 (the PKA) dislodges A1 and falls into its
site. This is called a replacement
collision.  Subsequent collisions become more
head-on (?n1lt ?n), hence the name focusing.
(angles in the lab frame) The replacements
continue until the recoil energy is too small to
displace the next atom.
a0 atom spacing along the row
Initial direction of recoil A1
based on the equivalent hard-sphere model (slide
29)
Initial direction of recoil A2
Collision generating PKA
Initial direction of PKA
49

• Collisions follow the Born-Mayer potential (43)
  for which the equivalent hard sphere radius is
• r0i rln(2A/Ei-1)
•  The collisions are characterized by a series of
  triangles formed by the initial positions of Ai
  and Ai1 and the off-axis position of Ai at
  collision.
• Applying the law of sines to this triangle

(57)

• given E0, ?0 and ao, all subsequent angles are
  determined from (57)

50
Critical angle for focusing For each initial
recoil energy E0, there corresponds a critical
angle above which focusing does not occur
 using this condition in (57) gives
(58)

• Obtaining r01 from the Born-Mayer potential (43)

Critical Energy for Focusing When the recoil
energy becomes large enough that q0f 0, a
maximum energy for focusing to occur is
obtained
51

• Example The Born-Mayer parameters for copper are
  A 20 keV and r 0.02 nm
• the atom-atom distance in the lt100gt direction is
  a0 0.36 nm. Using these values gives 5
  eV, a value too low to initiate a focusing event
  ( lt Ed).
• In the lt110gt direction, the atom spacing is
  a0/v2 0.26 nm, and 66 eV. If the PKA
  recoil energy is E0 40 eV, the critical
  injection angle of q0f 37.
•  The net result of the focused replacement
  process is a vacancy where atom 0 used to be and
  atom i is in an interstitial position. Atoms 0 ?
  i-1 have replaced the atom initially in front of
  them.

52
Channeling
  The mechanism of channeling is a process by
which an ion move along channels bounded by
the close-packed atomic rows.  The sketch
below shows a lt110gt channel in the fcc lattice.
The channel geometry is characterized by the
distance between host atoms in the bounding rows
(lch) and the effective radius of the channel Rch
( defined by pRch2 actual open area)
53
Channeling

•  For the lt110gt channel in the fcc lattice, these
  quantities are
• lch and
• where a0 is the lattice constant.
• The recoil atom enters the channel with energy E0
  and
• direction q0 (in the lab frame). The axial and
  radial velocity
• components are

54
The Channel Potential

•  The motion of the moving atom is similar to
  that of a ball rolling along a horizontal gully
  it oscillates in the lateral (r) direction and
  slowly decelerates due to friction in the axial
  (z) direction
• Laterally, the trajectory can be approximated as
  simple harmonic motion with a parabolic channel
  potential given by
• where k is the force constant (due to the
  potential interaction between the moving atom and
  the enclosing atom rows). ? above is for
  the Born-Mayer potential function (19)

55
The period and wavelength of the radial
oscillation are  The initial amplitude
of the transverse oscillation, rmax0, is
determined by equating the initial radial kinetic
energy of the recoil atom to its potential energy
at rmax0 For an initial recoil
energy E0, the maximum injection angle based on
the criterion that rmax0 Rch the
critical channeling angle decreases with
increasing E0. However, there is no maximum
energy at which q0ch 0, as there is for
focusing.
56
Minimum energy for channeling, Ech

• Ech is dictated by the condition that the
  wavelength l be at least two atom spacings along
  the channel
• Example For the lt110gt channel in copper,
• - Rch 0.12 nm (slide 53).
• - the force constant (slide 54) is k 5.6x104
  eV/nm2.
• Ech 365 eV
• For E0 1 keV, 0.91 radians
• Causes of Channeling Termination
•  As the ion moves down the channel, electronic
  stopping reduces its energy and the wavelength
  decreases (slide 55).
• The parabolic energy law (slide 54) fails when
  l 2-3 x lch
•  Vibrational motion of atoms in rows enclosing
  channel reduces Rch, resulting in a wide angle
  collision

57
Features of Channeling Focusing
Termination of channeling
 relation between the critical energies for the
focusing, replacement and channeling
58
Neutron scattering from atoms of the solid

• In the UO2 fuel, the main source of radiation
  damage is the fission-fragment flux
• In the structural metals, neutrons, and to a
  lesser extent gamma rays, are the principal
  sources of damage
• Collision of a neutron of energy En and a
  lattice atom produces a PKA energy E
• ?n(En,E) differential energy transfer
  scattering cross section (elastic and inelastic)
• For elastic isotropic scattering

(59)
and
(60)
59
Fast neutron flux
f(En)dEn flux of neutrons with energies between
En and En dEn (n/cm2-s-eV) the flux
spectrum
(61a)
Note the lower limit of 0.1 MeV neutrons of
lower energy cause little displacement damage
Fast fluence
(61b)
60
Displacement cross section
Fast-reactor neutrons
Fusion neutrons (14 MeV) ?d(En)
displacement cross section Integral of ?n(En,E)
and ?, the number of displacements produced by
the PKA of energy E. The simplest model
(59) hard sphere model
(53) Kinchin-Pease model
61
Case 1 ?En lt EC
(62a)
Case 2 ?En gt EC
(62b)
For 1 MeV neutrons on iron M 56 EC 56 keV
? 0.069 ??En 69 keV Ed 40 eV ?En/Ed
1725 EC /Ed 1400
Avg. no. of displacements by a collision with a
neutron of energy En (63)
?s 3.5 barns? ?d 1460 barns
62
Displacement cross-sections vs. neutron energy
m 56 Ed 25 eV x 0.8 ssel 3.5 b
(62a) (62b)
63
Displacement Rate/Dose Rate
(64)
(atoms/cm3)
Dose Rate
(65)
Evaluate ?d at average neutron energy (gt
0.1 MeV).
Approximation
(65a)
64
The end of the cascade the displacement spike

• At low PKA energies, the atom-atom interaction
  cross section becomes large
•  For projectile and target atoms of equal mass (
  (L 1), the cross section for displacing
  collisions is
• For the Born-Mayer equivalent-hard-sphere model
  ((44)), the mean distance ((10)), between
  displacing collisions is)
•  when E is 50 - 300 eV, l ? ao, the
  lattice constant,
• so every atom in the path of the
  moving atom is hit and displaced a
  displacement spike

Copper
(66)
total cross-section
Displacing cross-section
l (Å)
s (Å2)
(65)
65
Displacement spike Early Qualitative Ideas
Later, but still qualitative, version of the
displacement spike (also called depleted zone).
Original model of the displacement spike, J.A.
Brinkman, (1956)
66
Computer Simulation of Neutron Damage
 Follow the atom trajectories in a crystallite
of 5000 atoms, when one atom is energized by a
neutron scattering event  The force on each
atom is the sum of forces from all neighboring
atoms in the crystal Newtons 2nd law (mass m)
position of ith atom rij
separation between atoms i and j V(rij)
potential energy between i-j pair  The system
of 3N (coupled) differential equations is solved
numerically (the displacement energy, Ed,
need not be specified)
(67)
67
15o from (010) 22.5o from (010)
40 eV PKA in the (100) plane of Cu
For high PKA energies, the Kinchin-Pease model
overestimates the number and types of
displacements  No account is taken of -
vacancy-interstitial (V-I) recombination V I
null (restores perfect lattice) -
Clustering of like defects nV
Vn mI Im - Spatial configuration of damage
68
Damage Due to Ion Bombardment
 Electronic stopping
Bethe-Bohr (49) Lindhard, (50)  Nuclear
stopping use appropriate V(r) (slide 14) in (13)
or use ?(Ei,E) ion-lattice atom differential
cross-section, e.g., Rutherford (38b)
69

• Ion energy vs penetration distance, Ei(x)
• Displacement rate vs. x
• or
• (69) is the ion analog of (65) for fast neutrons.

(67)
(68)
(69)
70
Comparison of ion and fast neutron damage

• Examples
• 20 MeV C I 1014 cm-2sec-1
• gives 4x10-3 dpa/s in a thin layer
• dpa varies with penetration depth
• neutrons give 6x10-8 dpa/s
• dpa distribution is uniform
• - Since energy transfer for ions
• is governed by an atomic cross section (10-17
  cm2) while that
• for neutrons is a nuclear cross section (10-24
  cm2),
• (?d)ions gtgt (?d)neutrons

71
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