J. Phys. Ch.
Solids, 1975,Vol. 36, pp. 13534363. Pexgmoo Press. Printed in Orrat Brifain
THERMAL CONDUCTIVITY OF yIRRADIATED LiF AT LOW TEMPERATURES KURT GUCKELSBERGER Centre d’Etudes NuclBaires de Grenoble, Dtpartement de Transfert et Conversion d’Energie, Service Basses TempCratures,CCdex85, GrenobleGare, France and
KARL NEUMAIER Zentratinstitut fiir Tieftemperaturforschung, der Bayrischen Akademie der Wissenschaften, Garching, Reaktorgekde, B.R.D. (Received 19 September 1974;accepted I April 1975)
AhstraetThe thermzd ~ndu~~ty
of LiF singIe crystals which where yirradiated in a Co”source at room temperature with doses ranging from 8.5 lo’ Rad to 3.6 10’Rad was measured in the temperature range from 60 mK to lOOK.?‘he most heavily irradiated specimen was also measured after annealing treatments at temperatures between 260°C and 400°C. From a numerical analysis of the thermal conductivity data we derive the following interpretation of the thermal resistivity due to the radiation damage. The defects created are threefold: (a) Fcenters which act as point defects, (b) small aggregates of point defects with a diameter of about 10A which are roughly thirty times less numerous than the Fcenters (c) iarge scale aggregates containing several thousands of lattice sites which appear at irradiation doses 2 10’Rad. Each of these defects acts on the thermal conductivity in a different tern~~~~ range and are identi~ as interstitial clusters &rough their characteristic behaviour during radiation andsubsequent annealing.
1. INTRODUCTION
2. EXPJWhfENT~
The coloration of alkalihalides by ionizing radiation and 2.1 Specimen The pure specimens used in this investigation were its influence on the low temperature thermal conductivity single crystals purchased from Harshaw in 1964,196s and was one of the first subjects investigated in q~ti~tive thermal conductivity work[l,2]. However, these early 1972. All contain essentially the same amount of residual attempts failed to give a clear answer, presumably impurities. As judged from ITC measurements (C. Laj, because available specimens were not pure enough and private communication) and from thermal conductivity maxima in the u&radiated, annealed state the impurity the introduced defects are rather complex. We have now excellent information on the intrinsic concentration is less than lo’ (lo‘mol% or 6.3 thermal condu~ti~ty of LiF on the basis of the Callaway 10” cm‘). The three ma~esium doped specimens of Fig. model[3,4] and rather pure material from recent (later 1 were prepared by C. Laj in Saciay. All specimens were irradiated first with 5 106Rad to than 1964) Harshaw production. A detailed model of Fcenter production by ionizing radiation has been insure proper cleaving. The dimensions are typically developed since[5,4] as well as models for phonon 0.4 x 0.4 x 4.0 cm3. Measurements on pure samples were scattering by precipitates [7,8] and by resonant modes 191. made after an annealing treatment for 24hr at 750°C The present ~ves~~tion was started to take advantage followed by a slow cooldown to room temperature with a of all this new notation and to look for possible cooling rate of less than 1 degree per minute and in au correlations between Fcenter production and thermal atmosphere of pure argon to prevent contamination with resistivity. The hypothesis was that Fcenters may OHions[ll]. The doped samples were irradiated in the introduce a localized mode and thus create resonant asreceived state. All samples were sandblasted prior to phonon scattering[lO]. It soon became apparent, how mounting to insure diffuse phonon scattering by the boundaries. Irradiation was carried out at room temperaever, that this is not the case. In Section 2 of this paper, expe~enta1 details are ture in a Co” ysource of approximately 5 1~Rad~~ given and results on the evolution of thermal conductivity activity. Owing to its great penetration depth, yduring both, yirradiation and subsequent annealing of the irradiation produces homogenous coloration, a prerequismost heavily irradiated specimen are displayed. In ite to good thermal conductivity measurements[12]. Section 3 we describe briefly the model of Farge 161on the Measurements were made with standard steady state creation of Fcenters and the Callaway model for thermal heat flow techniques in three different cryostats: a conductivjty which we will use to interpret our experi He3fHe4 dilution refrigerator for tem~ratures between mental results. In Section 4 two conflicting but equally 40 mK and l8 K, in a He3cryostate for the temperature acceptable models (resonant scattering of phonons and range from 0.5 to l8 K and a conventional cryostat for scattering by precipitates) to explain our data obtained temperatures between l5 and 1OOK. The details are after irradiation are given. Then, by comparing each with described elsewhere[13]. The error in the thermal the data obtained after annealing we propose a choice conductivity, k, is essentially *5% over the whole range between them which we then discuss in Section 5 in the and overlap between sections taken with different bath light of evidence from other experimental ~vestigations. temperat~es was always satisfactory. 1353
1354
K.
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5
I
0 /
/
l
A
TEMPERATURE
K.
NEUMAIER
A annealed firradiated
106
2.7107
240
,
B
5106 , //
80
: :
GUCKELSBERGER and
K
Fig. 1. Thermal conductivity of several specimens containing diierent amounts of magnesium and irradiated with 5 106Rad each. Note the characteristic change of slope around 15K in the case of the purest specimen.
2.2 Experimental results The experimental results of thermal conductivity as a function of temperature (k(T)curves) are given in Figs. 14. Since magnesium is the most common trace impurity in otherwise quite pure LiF, we show tirst the k(T)curves of irradiated specimens containing different amounts of magnesium (Fig. 1). The irradiation dose was in each case 5 1o”Rad but the thermal conductivity is increasingly reduced at increasing magnesium concentrations. To demonstrate this very clearly we have included a second k(T)curve obtained from a recent, high purity specimen (curve D of Fig. 2). Although the dose was in this case roughly 5 times higher than in the other specimens, the reduction in thermal conductivity is comparable to the one observed in a specimen containing 74ppm of magnesium. Furthermore, one notices in the pure irradiated specimen a distinct change of slope of k(T) between about 15 to 25 K which is absent in the magnesium containing irradiated specimens. This important feature will be discussed in detail later on. Concerning the magnesium doped specimen, it is now well known that magnesium present in such concentrations produces thermal resistivity in the same temperature range by clustering[9], its value depending critically on the thermal history of the specimen. Further investigation must clarify how much of the observed shift of the maximum and the additional reduction of thermal
Co 60
85 105Rad
C
3.6 lo6 Rad
D
2.7 10 7 Rad
E
7.0 10 7 Rad
F
1.7 10’
Rad
G
3,6 lo*
Rad
81
IO*
10l
loo
TEMPERATURE
10’
,
lo*
K
Fig. 2. Thermal conductivity of pure irradiated LiF. Curve (A) annealed Harshaw 119)crystal, curves (B) and (C) Harshaw1968 specimens; curves (D), (F) and (G) Harshaw 1964 specimen, curve (E) obtained from a Harshaw1972specimen. Solid lines are calculated k(T) curves with parameters given in Table 1.
in these specimens may be attributed to precipitates of magnesium rich phases. It is of note that the k( T)curves in Fig. 1 are strikingly similar to some of those published earlier[ 1,2] and we may infer that Cohen’s specimen contained about 200ppm and Pohl’s specimens about 1OOppm magnesium.+ In Fig. 2 we present results obtained from recent very pure specimens containing less than 1Oppm impurities. Most of the data were obtained from a Harshaw 1964 crystal but we have included two curves (B) and (C)from a 1968 specimen and another, curve (E) from a boule purchased in 1972 (the source activity was then about 2 ldRad/hr). The general agreement is quite good. The solid lines are the results of model calculations and should be ignored at present as those of Figs. 35. One sees two essentially separate phenomena: (1) At low doses, the k(T) curves are reduced essentially around the conductivity maximum, i.e. 15 K. There is a welldefined change of slope of k(T) at this temperature. (1) At irradiation doses higher than about 10’Rad, the thermal conductivity is also reduced at low temperatures (T 5 2 K) and in the case of curve (E), the boundary limited value of k is reached only at the lowest temperatures measured (T s 0.1 K). We have treated one specimen according to the +A typical residual magnesium content of 100ppm in 1954, prescription given by Farge et al.[lS] to create and about 10ppm for 1960 and about 1ppm for I%5 Harshaw material destroy the complex Ecenter. In Fig. 3 the results has been reported[14]. obtained after each step are displayed. The crystal had conductivity
Thermal conductivity of yirradiated LiF at low temperatures
LiF
pure
G y irradiated
P
1355
3.6 106 Rod
H
anneaied
260%
J
reirradialed
2hrs
7 106 Rod
, K
TEMPERATURE
Fig. 3. Search for the Ecenter in the thermalconductivityof the Harshaw1964specimen. The indicated treatments are known to
TEMPER~~RE
, K
create and destroy the Ecenter[lS]. Solid lines are calculated.
Fig. 5. The contribution of individual relaxation rates to the total relaxation rate. Curves (N) and (J) as in Fig. 4. Curve (1) calculated with the totaI relaxation rate for curve (w plus the point defect term Ao’. Curve (2): calculated with relaxation rate of curve (1) plus the dislocation term GO. Curve (3): calculated with relaxation rate of curve (2) pIus enhanced frequency independent term C. Curve (f): calculated with the totai relaxation rate including the cluster term. All parameters used are given in Table 1, line 9. 2 K) and decreases between 4 K and 30 K. Thus the change of slope of k(T) is shifted by the annealing ~eatment towards a lower temperature. Subsequent reirradiation with a dose of 7 10”Radwhich is known to destroy the newly created Ecenterreduces the conductivity slightly at the lowest tempera~es (curve J) and leaves the high temperature conductivity essentially unchanged. Thus the Ecenter does not produce a well defined change in k(T) possibly because its relative abundance is rather low [5]. We consider this specimen as heavily irradiated and annealed at 260°C and proceed with the annealing treatment. The resutts for subsequent isochronal 1 hour anneals at 300 (K), 350 (L) and 400 (M)“Care displayed in Fig. 4. Qne notes a rapid recovery of k(T) at the highest and lowest temperatures (below 2 K and around 25 K) after the treatments at 350 and 400°C. (2’ 
LiF
pure
l_irradiatrd J
amealed
260 “C 2hrs
I twvxJia~ thm
75 lO%od
anrtt~txl
f
[email protected]
36 lOBRad
300%
1 hr
II

35o*c
lhi
M
u

400%
1 hr
N

II 
mml 10l
760°C 12hrs
10’ TEMPERATURE
,
K
Fig. 4. [email protected] b&aviour of the Harshaw 1964specimen. Solid lines are calculated.
received a total gradation dose of 1.3 x la” Rad (curve G which is curve G in Fig. 2). Anneding the crystal for two hours at 260°Cproduces the Ecenter and gives curve H. The conductivity increases slightly at Iow temperatures
3. h4ODRE.S FORTEEOCTAGON OF DATA 3.1 Creation of Fcenters The relevant features of Fcenter creation by ionizing radiation as discussed by Farge[6] and which are needed in the present context are as follows: (1) At room temperature, the radiation energy transferred to the lattice creates a number of Frenkel defects, i.e. a fluorine
1356
K.
GUCKELSBERGER
vacancy (acenter)interstitial pair. Some of these pairs recombine immediately whereas some of the highly mobile interstitials become trapped, thus leaving behind a vacancy which captures an electron to become an Fcenter. (2) The nature of the interstitialtraps is not known but there must be two kinds: one which is saturable to account for the rapid stage I coloration and another type of trap which has an unlimited capacity to trap interstiti~s but whose near must be smaller than the free Fcenter number in stage II coloration. The increase of Fcenter concentration in very pure LiF is proportional to the square root of the product of the intensity and irradiation time (It)“’ in stage II coloration. This law was observed for Fcenter concentrations between about 5 10”cm” an 10’9cm3 (which corresponds roughly to 10”Rad and 5 10”Rad respectively; see also Fig. 9). Experimentally no Fcenter clustering is observed during irradiation at room temperature [S]. From experiments[5,24] it is further known that the Fcenter concentration remains unchanged during annealingto up to 200°C and that it decreases rapidly during isochronai one hour anneals between 200 and 400°C. At this temperature they have virtually disappeared as well as the complex Fcenter aggregates, i.e. IV, M and Rcenters. 3.2 Model calculations of thermal conductivity We have carried out a quantitative analysis of our thermal conductivity data within the framework of the Caliaway model which is now well established and has been used in many investigations[3,8]. Here we will discuss briefly the limits imposed on establishing invididual phonon scattering crosssections through the overlap of different contributions. In the Callaway formalism one uses a total relaxation rate to describe phonon scattering
and
K. NEUMAIER
ing cross section of point defects can be calculated from Klemens’ formula[20].
t,!,. = Aw ‘; A = Fr I =
= 1.22 iO'@r(set’)
7 cis:; s*=St2+ (St + s&)
VO= 16 lOmucm’ and Y= 5 10’cm see’ in LiFf8], ci is the fractional concentration of the ith impurity SC contains three contributions due to the relative maSs change AM/M, force constant change AFIF and ionic radius change AR/R of the perturbed unit ceil. Berman and Brock[3] have also shown that this holds for isotopes which scatter phonons through their mass diierence alone. Neumaier [8] calculates the scattering cross section of a Li’vacancy in Mg’doped LiF as given below and finds good agreement with experiment. Here we include a calculation of the scattering cross section of a Fvacancy again using molecular units:
G' F
0.074 0.20
0.43 0.43
0.19 0.42
0.4 0.7
(4) Phonon scattering by dislocations: It is equally well established that phonon scattering by dislocations can be described by a relaxation rate proportional to the phonon frequency 7& = Gw ; G = I&,(f.
However, the scattering cross section rD given by Kiemens[20] To = 6.04 lo‘b*y* where b is the Burgers vector and y the Griineisen constant, overestimates the Some of the terms which must be included are dislocation density ND by a factor of 100 to 1000 if one compares the dislocation density ND of deformed wellknown and we consider them in turn. (1) Intrinsic phononphonon scattering was derived for specimens derived from thermal conductivity measureLiF by Berman and Brock [3], and we use the coefficients ments with that determined by etch pit counts [21]. Recent modeis[221 have reduced in some cases the discrepancy and functional forms given by these authors. (2) At low enough temperatures (T s4K in Lie, the to a factor of 2 to 3. The point defect and dislocation relaxation times have thermal conductivity is limited by the size of the specimen insulators. According to in common that they are slowly varying functions of in pure, mon~~s~ine frequency. Thus they affect the k( Tjcurves over most of Casimirfll], diffuse phonon scattering is then independent of frequency and can be calculated from the width of the accessible temperature range. In Fig. 5 we have drawn calculated k(T) curves which the specimen: show the successive contributions of the different parameters deduced for this particular curve (see Table 1). In this figure one sees clearly the overlap of the where L is the width of the specimen and v the velocity different contributions. In the absence of other scattering of sound. For specimens of square cross section the mechanisms or by going to sufficiently low temperatures, equivalent width L = 1*12(1,x L)f may be used where 11 the Casimir term may be established with confidence as and 12are the lateral dimensions. f is a correction factor well as the Rayleigh term in the high temperature region taking account of the finite length[l8]. Several inves where the intrinsic conductivity is approached. This is not tigators iind that this relation holds to within 15% in true for the dislocation term whose infIuence on k(T) is most pronounced in the inte~~iate region. Often, one LiFU, 191.(see however the discussion in Ref. 141). (3) Scattering from point defects: The phonon scatter must use the trend of several curves to establish its
Harshaw 68”’
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The dose rate was (1) 1  4 lo’ Rad/hr (2) 1  6 10’Rad/hr, (3) 1  2 lcr’Radlhr, (4) This was the first ~iation dose for Harshaw 64 specimen. The data are only displayed in Fig. 6 since they are similar to curve (C). (5) Treatments from this line on are subsequent to a total irradiation dose of 3.6 lo” Rad. (6) This specimen was only measured up to 2 K.
Harshaw 72
2
1
1
Treatment
Specimen
Interstitial cluster
Table 1. Crystal origin, treatment and model parameters used to obtain the calculated thermal conductivity curves in Figs. 24 and 6
1358
K.
GUCKJUBERGERand
K.
NEUKUER
contribution in the cases where no combination of boundary and point defect scattering can represent the results adequately. By inspection it is immediately clear that no combination of slowly varying relaxation times may represent our results in the intermediate temperature range (about 15 K in curve (D) or around 0.5 K in the case of curve (E) Fig. 2). We must thus introduce an additional relaxation time which is a rapidly varying function of frequency. There are two phonon scattering processes with the required behaviour: resonant scattering and scattering from extended defects. The corresponding expressions are given in the next section. 4. ANALYSIS OF DATA
In the case of the annealed, unirradiated specimen of pure, LiF an excellent fit was obtained by using the coefficients listed in Table 1, lines 1 and 13. The boundary scattering term corresponds to the calculated Casimirvalue to within lo%, the dislocation term is consistent with Berman and Brock’s results on similar specimens and the experimental point defect scattering coefficient corresponds to an equivalent residual magnesium concentration of 5 10ppm when the isotope contribution of the Li6 defects is subtracted. This value should be considered with care since phonon scattering from point defects is sensitive to most defects and not only to divalent cationic impurities as for example ITCmeasurements are. These latter indicate a smaller concentration (about 1 ppm) of divalent cationic impurities. However 1 ppm of silver [14] would already contribute through its mass defect alone about 50% of the derived figure. It is thus reasonable to assume that a few ppm of impurities are present in the pure crystal. 4.1 Resonance scattering Walker [ lo] concludes that evidence for resonance scattering by Fcenters may be seen in several irradiated alkali halides. A fist analysis of our data on an irradiated specimen (Harshaw 1964) was carried out on the following basis: we kept the intrinsic phononphonon scattering relaxation rates constant and adjusted the parameters C, G and A of the total relaxation rate. In addition, we introduced a resonance relaxation rate of the form
fB
N annealed
TEMPERATURE
,
K
6. Thermal conductivity of specimen Harshaw 1964 after successive irradiations. Solid lines are calculated with the resonance relaxation rate. Parameters are given in Table 1 columns 10to 14. Fig.
constant resonance frequency of o. = 9.2 10” rad set’ = 49 cm‘. From similar calculations, Walker [lo] finds 9 10” rad see’ in irradiated KC1 and 1.4 10” rad set’ in irradiated NaF. In irradiated LiF, the coefficient D is found to increase proportionalIy to the square root of dose or, since all irradiations were carried out in the same source within a few months, proportionally to (irradiation time)“*. Pohl[23] has found an empirical relationship between the coefficient D and the resonance frequency for a great number of impurities D = new,’
where 00 is the resonance frequency appropriate to a transition between two energy levels separated by an energy IroO. w is the phonon frequency and D is the scattering coefficient containing as a factor the concentration of resonance scatters, n. This empirical formula was used successfully by Pohl and his coworkers to explain phonon scattering by localized modes due to molecular impurities like NO* or atomic impurities like Li’ in several alkali halides (see Fig. 41 in Ref. 191). The results of this calculation are displayed in Fig. 6 where the solid lines are k(T) curves computed from parameters given in Table 1. All fits were made with a
where the constant c is about lo“rad set’ cm’ to within an order of magnitude. The concentration n of Fcenters in the case of 5 106Rad was determined optically by M. Cagnon of the group of Mme. Lambert to be n =
8 10” cm‘.
This value is consistent with Cagncms[24] results on specimens coming from the same boule and which we shall quote in the following. Another check was made on
1359
Thermalconductivityof yirradiatedLiF at lowtemperatures specimen Harshaw 68 (36 1O”Rad) by C. Hoentzsch (University Stuttgart) which was equally consistent. From this concentration n, the corresponding coefficient D and the resonance frequency we compute c = 5 lo” rad set’ cm’ in very good agreement with the value used by Pohl[23]. Since D increases in the same way as n this value of c remains obviously constant for all irradiation doses applied. Now let us turn to the annealing behaviour of this specimen (Figs. 3 and 4). Annealing for two hours at 260°C reduces the Fcenter concentration by at least a factor of two, as is well known from experimental results [5,24]. In the k(T) curves the change of slope shifts towards lower temperatures after this treatment. This may be due to two processes: (1) The annealing has influenced the defects which are summarily taken into account by the coefficients C, A and Gwhich will be discussed laterand the net result is just such a shift. In this case we can keep the resonance frequency constant and adjust all other parameters. We get a still reasonable fit with the parameters given in Table 1 columns 9 to 12 line 8. The coefficient D has to be increased by a factor of two and the point defect scattering term decreased by about 30%. This eliminates the Fcenter as a resonance scatterer. (2) The resonance frequency has changed during annealing. Then again, this resonance frequency cannot be an intrinsic feature of the Fcenter. Other possible defects produced during yirradiation of LiF have either another production rate (dose dependence) than Fcenters or are unstable at room temperature like the isolated interstitial fluorine [25]. The whole set of data can, however, be described consistently if we assume that the major contribution to thermal resistivity is due to extended defects. 4.2 Phonon scattering by extended defects As has been shown earlier[7,8,26,27], additional thermal resistivity may be attributed to the formation of extended defects in the lattice. These may be formed by inclusions of foreign matter as for example MgFz precipitates in LiP[8]. Two limiting cases are well defined: if the size r of the extended defect is much smaller than the phonon wavelength 27ru/w where v is the velocity of sound, then they act as “Rayleigh’‘scatterers and the effective scattering cross section is proportional to 0’. In the other limit where r is much greater than the wavelength, the effective scattering cross section will be independent of the phonon frequency and proportional to the geometrical cross section. Ying and Truell[28] find for both cases:
r;;e:cc= NW;
u =
(4/9)nr2. (m/v)‘; pr 9 2.
m/v 6 1 m/v B 1
One notes the strong dependence on size which goes in the long wavelength limit as r6. At low temperatures only phonons of long wavelengths are excited so that the Rayleighlimit will be observed at low temperatures and the geometrical limit at high temperatures. This effect is clearly demonstrated in Fig. 7. Crossover will occur where the wavelength of the majority of phonons is about JPCS Vol. 36 No. 12D
TEMPERATURE
,
K
Fig. 7. Typical case for phonon scattering from extended defects. The high and low frequency limits are drawn in separately, together with the composite relaxation rate to demonstrate the generation of the characteristic change of slope in the k(T) curves due to extended defects.
equal to the dimension of the defect. Since the characteristic changeofslope in k(T) must occur in the temperature range accessible to measurement and interpretation (0.1 to 15 K[31]) the size of the clusters observable may vary from 8 A to 1000A. A shift in the change of slope of k(T) towards lower temperatures signals an increase in size of the extended defects. If their size is not uniform, k( T)curves of the type measured on magnesium doped, irradiated crystals (Fig. 1) are observed [291. For a numerical analysis of our results, we have used the scattering relaxation time developed earlier [8].
T& = Nvrr2
(l+R, .exp(b/3))(ro/v)4; (1 +R1. exp (ro/3v));
m/v d b r/v > b
r the defect radius, N the concentration of extended defects per cm”; b and Rr
Here, o is the phonon frequency,
depend on each other so that we have a threeparameter relaxation tune. According to calculations made by Walton and Lee [30]
the form of the scattering cross section in the region where ro  bv depends essentially on the ratio of the densities p (cluster)/p(matrix) for a wide range of acoustic mismatch v (cluster)/v (matrix). We can simulate Walton and Lee’s curves [30] with our phenomenological factor (1 t RI exp (rolfv) quite well considering the fact that thermal conductivity is not sensitive to details in the structure of the scattering crosssection which corresponds to a small band of frequenciesL71. The factor l/3 in the exponent gives a transition region for bv < rw < 1Obv (Fig. 8) as required. For a cluster density higher than the matrix density we have RI >O and for a cluster density
1360
K.
GUCKELSBER~ER and K. NEUMAIER
(l*R,exp(l13Mr~v~
w/v4
(1 lR,exp(w/k))
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’ ’ ’ 5““J
2
lo2
1 lo6 III b loa Dose(Rad)
aI1 10'
Fig. 9. Variation of the parameters deduced from curve fitting as a function of total irradiation dose. For comparison, the results of Cagnon[24] on the Fcenter concentration of similar specimens are given together with his results on the elastic limit.
rw Iv
Fig. 8. Scattering crosssection for extended defects as a function of the reduced frequency.
lower than the matrix density we have exp(b/3)s Rt < 0. The value RI = 2.1 would then correspond to a cluster density somewhat higher than the matrix density which seems reasonable in the light of the interpretation given in the next section. We note that R, is reasonably well defined from the curvature of the changeofslope in k(T). A more delicate problem is that the derived number density N depends on the particular choice of the pair of parameters b and RI. Mode1calculations which we do not show here indicate that N should perhaps be defined within a factor of two or three. In any case the mode1 of Ying and Truell[28] is already a strong simplification as discussed in Refs. [7,30]. A more detailed investigation of the properties of the cluster scattering crosssection is outside the scope of the present work and we shall claim only a consistent description of our results. In Table 1 all of the coefficients used to obtain the fits (solid lines in Figs. 2,3 and 4) are given. For convenience we display in Fig. 9 these coeficients as a function of irradiation dose and in Fig. 10 the coefficients N and G obtained from the annealing of the 3.6 10”Rad irradiated specimen. We have drawn in Fig. 9 the Fcenter concentration as obtained by Cagnon[24] and in Fig. 10 the evolution of the elastic limit with 1 hr annealing treatment as measured by the same author on a specimen irradiated with 1O”Rad. For the thermal resistivity, induced by yirradiation into LiF we observe the following prh&al points: (1) The point defect scattering term ([email protected]) increases proportional to the square root of the irradiation dose as does the Fcenter concentratioa!S]. Here, AOis the point defect scattering term due to the isotopes and .residual impurities as determined from the annealed, unirradiated
0
Drnsity
N
cluster
G
Dislocation
T
l=/ow Stress
100
mo
0
Densify
A

300
600
500
htmr~~
Fig. 10. The cluster density N and dislocation density which is proportional to parameter G as a function of the annealing temperature. For comparison, the results of Cagnon[24] on the elastic limit of a similar specimen irradiated with [email protected] Rad is given.
specimen. It decreases during annealing and at 400°C its value is only 15% of its initial value after an irradiation with 3.6 lo” Rad. (2) The small defectcluster terms (r and N): The effect on k(T) around 15 K is due to point defect aggregates of a welldefined size of about 10 A diameter which already exists at low irradiation doses (8.5 10’Rad). Their number increases also with the square root of the total irradiation dose but they are roughly 30 times less numerous than Fcenters (Fig. 9). At high doses (a5 1O’Rad) their concentration seems to saturate in the vicinity of 2 10’7cm3. Upon annealing they increase in size to about 16 A diameter and diminish in number. (3) The dislocation term (G). The coefficient G increases with the fourth root of total irradiation dose up to about 1dRad as does the flow stress as measured by Cagnon[24] (Fig. 9). During annealing, its evolution with temperature is again very similar to that of the flow stress
Thermalconductivity of yirradiatedLiF at low temperatures obtained by Cagnon on a specimen irradiated with lo” Rad. (4) The boundary scattering term (C). At low irradiation doses, its value is identical to the one derived from the crystal dimensions. Its increase for heavy doses (2 5 IO’Rad) may be attributed to the presence of large point defect clusters where the characteristic change of slope is outside the range of measurements. This point is discussed in detail in the next section. S. DISCUSSION
A parameter which is established with a fair amount of confidence is the increase in the point defect scattering term (AAo) during irradiation. It increases with the square root of the total irradiation dose and we ascribe it therefore to the Fcenters. Taking its value obtained at 5 106Rad where the overlap with the small defect cluster term is still small, as well as the measured concentration of Fcenters at this dose we find S*(Fcenter) = 0.8 2 20%. This is about 18 times the value calculated from the massdifference alone (S,’ = 4.4 IO*) and approximately equal to the fluorinevacancy scattering crosssection derived earlier. We note that Walker [ IO] concludes that Rayleigh scattering of an Fcenter in KC1 is about 30 times stronger than predicted from the mass difference alone. In this figure of S* the scattering by N, M and Rcenters which are essentially closely spaced Fcenters, is included. In a recent investigation of precipitation dynamics of magnesium doped LiF[3 l] it was concluded that three vacancies grouped as nextnearest neighbours in a (11l)plane, i.e. an arrangement similar to the Fcenter complexes, scatter phonons as if they were still isolated point defects. It is tempting to identify our clusters with agglomerates of interstitial fluorine ions. There is no doubt that some form of interstitial clustering takes place during irradiation at room temperature. The alternative, Fcenter clustering to complexes containing more than four components is quite unlikely[5]. Experimental conditions quite similar to ours were used in the investigation of Peisl et al. [32]. These authors find regions within the crystal of about 14A in diameter within which diffuse Xray scattering cannot be treated by elastic continuum theory. A more detailed investigation with the same technique by Spalt [33] leads to the conclusion that these regions are interstitial clusters. This last paper includes a study of the annealing behaviour of the clusters which gives a result quite similar to ours: a rapid disappearance at annealing temperatures of 350 to 400°C. It seems reasonable to suppose that thermal conductivity, i.e. diffuse phonon scattering, and diffuse Xray scattering ‘sees’ the same defect. The large scale clusters appearing at low temperatures and high irradiation doses should be similar to those reported by Hobbs et a1.[34]. These authors observe by direct electron microscopy that elongated planar interstitial clusters of large size (a few 100A wide and a few 1OOOA long) form during irradiation. We note that the general agreement for both cluster sizes is quite good.
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Next we turn to the number density N of the clusters. This is clearly the leastknown parameter since it depends on the choice for the parameter b. We have set this latter in agreement with the results of Walton and Lee [301equal to 0.75. We can then obtain certain informations from the relative change of the number density N: (I) The apparent saturation at high irradiation doses suggests a limited number of nucleation sites for the small clusters. (The absolute number of N would then be consistent with the idea that these sites are residual impurities since 2 10”cm’ corresponds to about 3 ppm lattice sites.) (2) Taking the results of Cagnon[24] on the flow stress measurements during irradiation and subsequent annealing one may infer that the clusters act as strong pinning points or as sources of dislocations which in turn determine the elastic behaviour. A similar observation was recently made in magnesium doped LiF[31] where the number of precipitates of magnesium rich phases grow in parallel during annealing. Both ideas should be now tested with simultaneous measurements on very high purity specimens but of exactly the same boule since it appears that even very low concentrations of residual impurities may still determine the physical properties of irradiated LiF. The absolute value of the number density N and the number of lattice sites contained in both large and small clusters is less well known but we may use it to gain some insight into the mechanism of cluster formation. The following is, to some extent, an educated speculation. If the clusters are assumed to be spherical, the number of unit cells contained in the clusters is defined as Nca = N. (4~/3)r’/ V,; V0= 16 lOmucm’. Up to about lo” Rad this turns out to be almost exactly the number of Fcenters per cm’ (Table 2, third column) if we use again Cagnon’s[24] results for the Fcenter density. The radius of the large clusters was in the case of 8 10’Rad irradiation about 300 A. For the higher doses, where the thermal conductivity data do not extend to suthciently low temperatures to determine the radius, we may estimate from the coefficient C and assuming the same radius, a number density N,r (large) through the short wavelength approximation (see Section 4.2): 7 ’ = C = N,&rge)t&. as was already used by Slack[35] and Worlock[l6]. The results are listed in Table 2, fourth column. The estimated number of unit cells contained in these large clusters rises rapidly above any reasonable value of the Fcenter concentration created by the highest irradiation doses. Since it is unreasonable to assume that the cluster size decreases with increasing irradiation dose, we conclude that the initial assumption of spherical, large clusters is not valid. This is not too surprising since Hobbs et 01.134)observe interstitial clusters of roughly this size but explain them as planar clusters of interstitials. Assuming a constant radius of 3OOAand a constant thickness of 15A for the platelets, the total number of lattice sites contained in both large and small clusters
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K. GUCKELSBERGER and K. NEUMAIJJR
Table 2. Comparison of Fcenter concentration nF with the number of perturbed unit cells N,, contained in the interstitial clusters as deduced from thermal conductivity measurements of yirradiated LiF dose (Rad) 0.85 10” 5 10” 2.7 10’ 8.0 10’ 1.7 lo” 3.6 lo”
~~~~~~~~~~
(cm‘) 3.2 10” 8.0 10” 2.0 10” 5.0 10” 8.0 10”
N&small) (cm‘)
N,&rge)t (cm‘)
NL$ (cm‘)
3.7 10” 7.5 10” 1.9 10”


4.010” 5.0 lo’*
10l’&O1~) l.41012 1~610”‘(l.10’~) 2.2 10” 4 lOL9(2 10’8) 5.5 lOI
tin parenthesisthe numberof unit cellsis ‘ven whichwould be contained in a platelet 15A thick and 300R. m radius. SN, is the number of large interstitial clusters as deduced from the adjusted parameter C, assuming a constant cluster radius of 300A as in the case of the 8*O.lO’Radirradiated specimen. come out as essentially equal to the number of Fcenters created at these irradiation doses as extrapolated from Cagnon’s results. This should be taken as an order of magnitude agreement only since the problem of an appropriate scattering cross section for nonspherical clusters is not completely solved at present.t To a lesser degree, the same holds for the small clusters. Spah[33] reports that the defect regions are nonspherical. However, the quoted diameters indicate a typical length to width ratio of perhaps 2: 1. Such deviations from sphericity are discussed by Seyfert 1291in connection with thermal conductivity work on neutron irradiated silicon. The conclusion is that the spherical approximation is still good to describe the size and the number of perturbed unit cells in the case of small clusters. As in other cases, thermal conductivity is not sensitive to small details. 6.CONCLUSIONS
We have for the fust time analysed an extensive set of experimental data for heavily irradiated and annealed specimens which covers the whole range between the boundary limited conductivity and the conductivity limited by intrinsic processes. All the deduced parameters follow a pattern which is consistent with results on radiation damage obtained from a number of different investigations. On the basis of the present analysis we suggest that the low temperature thermal conductivity of pure LiF during irradiation and subsequent annealing is dominated by phonon scattering from extended interstitial clusters of fluorine atoms and the dislocations which they create. The characteristic temperature dependence of thermal contRecently, Turk and Klemens[36] have published a relaxation rate for phonon scattering from thin platelets. They find in the limit of or/u 9 1 a relaxation rate
where h is the thickness and r the radius of the platelet. This relaxation rate predicts for curve (E) (Fig. 2 and 7) a thermal conductivity proportional to T at the high temperature side of the .. . . _I orp whereas we observe I( a y’.
ductivity at 15 K is caused by interstitial clusters of approximately 10 or 16A diameter respectively. Their concentration seems to saturate in the vicinity of 2 10” cm’ which is consistent with a nucleation mechanism by residual impurities. At high doses (ZlO’Rad) large clusters containing several la’ lattice sites appear. These findings are consistent with the conclusions drawn from diffuse Xray scattering data and direct observation by electron microscopy obtained on similar material. With respect to the interpretation of results of thermal conductivity in irradiated alkali halides many of the earlier results may be understood in terms of the present model. The initial assumption that Fcenters scatter phonons resonantly can be excluded. AcknowledgementsThe authors would like to thank Dr. A. M. deGoer for many discussions on the subject and Dr. M. Locatelli as well as Mr. D. Arnaud for performing the measurements with the dilution refrigerator. One of us (K.G.) would like to express his gratitude towards the Deutsche Forschungsgemeninschaft for a fellowship which made the present work possible.
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Thermal conductMy of yimdiated LIF at tow temperatures
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