Received 22 August 1963. Oen and Robinson l) have suggested that the large disagreement between radiation damage measure- ments in metals and defect ...
Volume 6, number 3
EFFECT
PHYSICS
OF
CHANNELLING
Research
ON
LETTERS
15 September 1963
DISPLACEMENT
P. SIGMUND Atomic Energy Commission Establishment Rise, Roskilde,
CASCADE
THEORY
Denmark
Received 22 August 1963
Oen and Robinson l) have suggested that the large disagreement between radiation damage measuretheory might be ments in metals and defect cascad reduced by taking the channelling 27 effect into account. An energetic atom moving in certain almost force-free crystal directions may loose a large part of its energy in sub-threshold collisions instead of creating defects as postulated by cascade theory for a random lattice. Oen and Robinson have made the following assumptions : 1. There exists a probability P for a particle to be channelled after a displacement collision, independent of a) its energy and b) its history. 2. A channelled particle creates exactly one defect, i.e., it remains as an interstitial in the lattice. 3. The collision process may be described by a hard core cross section. 4. Ionisation effects may be neglected. These authors have used the Kinchin-Pease 3, model for the displacement mechanism with a threshold energy Ed and have obtained the following expression for the average number v(E) of defects in a cascade of knock-on energy E. l-2P p v(E) = G(gd) (1) --.l-2P’ E==d. In order to get agreement between experimental results and eq. (1) Oen and Robinson need a channelling robabilit P = 0.07. Machine calculations on Cu 2Pand Fe 41 however, suggest that P is only of the order of 0.0;. The purpose of this letter is to test the above assumptions. It will turn out that v(E) is, contrary to ordinary cascade theory 5), strongly dependent on the cross section, and further that the fluctuations of v(E) are larger and ionisation effects may not be neglected. la. The energy dependence of the channelling probability is not known in detail. However, as P varies only slowly compared to v(E) in the range covered by machine calculations 2, (l-10 keV), it may be taken as a constant as a first approximation. A somewhat better approach would be P(E)
=Ofor
EC
El,
P(E)=PforE>E1,(2)
where El is some limit for channelling collisions supposed to be smaller than 1 keV. The choice of El has considerable influence on v(E) but, as we shall see, this is a special feature of the hard core cross section. lb. The assumption of P being independent of the path of the particle before a displacement collision seems reasonable for cross sections which are not too strongly forward peaked. If, however, the average deflection angle gets much smaller than half the angular distance of the channels, it seems necessary to take the angular dependence of the channelling effect explicitly into account. In order to get isotropic damage one has to deal with polycrystalline material. The average diameter of the micro-crystals should not be smaller than the range of the channelled particles. 2. Recent machine calculations 4) on iron indicate that the separation between channelling collisions (v = 1) and normal displacement collisions (v >> 1) is rather sharp at an angle of about 5o from any main channelling direction. Hence, assumption 2 seems justified at least for a lattice without defects. However, if there is a large defect concentration the channelling effect will be destroyed. One might expect a more rapid increase of damage rate at doses which, because of the large ranges of channelled particles, might be appreciably smaller than the saturation dose. As such an increase has never been observed in neutron damage experiments one might conclude that channelling does not play an important role in cascade theory. However, as pointed out by R. Sizmann (private communication) the defect clusters created by neutron irradiation may be considered to be saturated with defects, so that a channelled particle meeting a cluster will be slowed down rapidly without creating further defects. On the other hand Beevers 10) has observed an increase in damage rate by irradiating Al-foils with protons. Though channelling might be possible at much lower energies in Al than in Cu, the average primary energy in this experiment is so small that one cannot be sure to deal with this kind of saturation effect. 3. The total probability for a particle to be chan251 .
Volume 6, number 3
PHYSICS
LETTERS
-__
_____._~~
~~~~ ~~~
nelled during the slowing down process is P times the number of displacement collisions it makes before coming to rest. If the cross section favours small energy transfers this c creases. Hence, the use of a sonable) forward peaked cros petted to give a further reduction of Y(E) : We derive v(E) from the integral equation
15 September 1963
P*O.rn, 4 .OD
P.o.05,,Ed/Ec=0,1J: P.001,
Ed/Ec=O.l
Pr0.01,
e7~aO.2
P.o.05, .y~*O,l
EK GdTu(Z-)+j
u(E)=P+(l-P)[s
P405,
Ey~dl.2
Ed if E 3 2&j; u(E) = 1 if E S 2Ed. For P = 0 this is the well-known
cascade equation for the Kinchin-Pease model 3,9). K(E, 2’) is the differential cross section and o(E) = j’EK(E, T) dT. If P + 0, we get v(E) = 1 with probability P or a displacement collision with probability 1-P. It is easilvE/25, ’ verified that (1) is a .solution of eq.- (3) for K/a = l/E Fig. 1 Solid curves : Average number of defects in a (hard core interaction). cascade of energy E for different channelling probabilities P and parameters EC. For considering the qualitative influence of K on Dashed curves : Hard core results ‘). v(E) one may use the expression 5) fore, that the shape of the cross section plays only K exP (+E - T)/Ec a secondary role in cascade theory. However, if a = 2E, sinh E/2 EC ’ P f 0, the slope of the cross section determines which is perhaps less realistic than other analytical the height of the asymptote forms of forward-peaked cross sections, but has V(=‘) w Ec/2PEd. (8) the advantage of calculational ease and contains, furthermore, a parameter EC scaling the slope of K. As the average energy transfer per displacement collision is approximately equal to EC, the total channelling probability becomes 1 for particles with energies higher than E. = EC/P. The average defect number is expected to approach an asymptotic value at this energy: Inserting (4) into eq. (3), multiplying by sinh (E/2E,), and taking the second derivative, one gets a differential equation with the solution 1-P v(E) = 1 + 2~,
sinh 2p- ‘(Ed/EC)
Ec/PE
li
Xs
dE'
sinh-2p (E’/2E,)
.
2Ed If P is small (.S 0.01) we may replace the integrand by exp (- PE ‘/EC), which holds over the whole energy range except for E’ cc EC. However, if we assume that EC is not too large (EC c 100 Ed) this region does not occur in eq. (5). Neglecting P with respect to 1 we get (fig. 1) v(E) Particularly,
x3
l+
1- exp[-[(E-2Ed)/EclPl_ 2P sinhEd/E,
(5)
if P = 0, then v(E)
= 1+
E-2Ed 2E,
sinh Ed/E,
(7)
’
If EC * Ed, this expression approaches the hard core
252
kiW
V
= E/2Ed.
One has, therefore, to be very careful in choosing a cross section in cascade calculations including channelling . Bearing in mind that quantitative conclusions may be dubious we try to fit our parameter EC to experiments of Coltman et al. 7), also discussed in 1). These authors compared the damage v(E1) of a low energy event in Cu (El = 382 eV) to v(E2) of a high energy event (E2 = 24 300 eV) and got the result u(E2)/@1) = 34. Taking v(E1) = E1/2Ed and v(E2) = Y(W) we get the condition
Lehmann 5) concluded,
there-
1 = 34
(9)
for EC/P. Hence, for P= 0.01 2,4), EC should be about 130 eV. As Ed = 30 eV for cu 8) our approximations appear to be rather well justified. As the average angular deflection is 9’ at 24 keV and larger at lower energies, assumption lb should still hold approximately. Assumption la is well satisfied for this cross section, as a high energy particle makes very many displacement collisions before coming down to energies where P is eventually zero. 4. As v(m) lies an order of magnitude below the commonly used ionisation limit (fig. 2) one might suppose that ionisation effects may be completely ignored. This conclusion does not seem to be correct, as we shall see that one has to expect large fluctuations of v(E) : The average square defect number per cascade
Volume 6, number 3
PHYSICS
LETTERS
15 September 1963
rp(E) is derived from the integral equation 9) v(E)
=P+
(1-P)
[.f
EK adTcp(T)
Ed + f-Ed
fdQ(E-T)+2
JGEdtdTv(T)v(E-
0
Ed
The solution for P = 0 and hard core cross ~(E)=(E/ZEd)2+0.15E/2Ed,
sectiong)
E 2,4Ed
is assumed to be valid at energies E G< EC/P, as suggested by the behaviour of v(E) (fig. 1). Athigher energies we may neglect Ed with respect to E, insert (4) and (6) into eq. (10) and solve the resulting second order differential equation. We apply the same approximation (P small) * as in eq. (5) and choose the integration constants so that r&E) approaches (11) at low energies (or P = 0). The resultant mean square fluctuation n(E)
= p(E) - v2(E)
= 0.15 &
d (I- e-EP/Ec)
02) sd)2(1_2E-Pe-P@‘Ec + (2PE C
approaches
the asymptotic
n(m) = 0.15 &
d
_ ,-2EP/Ec)
1 V’
.’ ld
ld
to’
10’
I xi
E/cYFig. 2. Comparison of different cascade models for Cu (Ede3OeV): viJ : Hard core result without channellinx. _. ionisation threshold assumed. uL : Lindhard’s curve 6), including ionisation effects. uch: Present model eq. (6), P = 0.01, E, = 130 ev, not including ionisation effects. “ch =*/n(E) = fluctuation of uch from eq. (12).
value (fig. 2)
+ (&
d)
References
2 m I?(m) .
1) O.S. Oen and M. T.Robinson, Appl. Phys. Letters 2
As
the relative fluctuation of v(E) is 1 at energies E 2 E, there is a considerable part of high energy cascades in which no channelling occurs. The defect number in these cascades is strongly limited by ionisation and self-annealing. It may be expected that a detailed study of the simultaneous effect of channelling, ionisation and self-annealing would lead to a further reduction of u(E) in high energy cascades. It is a pleasure for me to acknowledge the hospitality of the Danish AEC and valuable discussions with Professor 0. Kofoed-Hansen, civ. ing. H. S&ensen and, especially, civ. ing. H. H. Andersen. I am grateful to Professor G. Leibfried and Dr. R. Sizmann for important information concerning sect. 2. I wish to thank the NATO for a fellowship.
(1963) 83. 2) M. T.Robinson and O.S.Oen, Appl. Phys. Letters 2 (1963) 30. 3) G.H.KinchinandR.S.Pease, Repts.Progr.Phys.18 (1955) 1. 4) D.G.Besco and J.R.Beeler, General Electric Report T M 63-5-3. 5) C. Lehmann, Nukleonik 3 (1961) 1. 6) J. Lindhard et al. , Dan. Mat.fys.medd.33 (1963) No. 10. 7) R.R.Coltman et al., J.Appl.Phys.33 (1962) 3509. 8) R. v. Jan and A. Seeger, Phys. Stat.Sol. 3 (1963) 465. 9) G. Leibfried, Nukleonik 1 (1958) 57. 10) C. J.Beevers; Phil.Mag. 8 (1963) 1189.
* Repeated application of this approximation may restrict its validity to P-values still smaller than in the cal-
culation of u(E). The qualitative argument should be unaffected.
*****
253
