The theory of irradiation damage in graphite

Carbon, 1977,Vol. IS, pp. 117-127. PergamonPress. Printed in Great Britain

THE THEORY OF IRRADIATION DAMAGE IN GRAPHITE B. T. KELLY UKAEA Reactor Fuel Element Laboratories, Springfields, Salwick, Nr Preston, Lancashire, England (Received21 Seprember1976) Abstract-The theory of irradiation damage in graphite under fast neutron irradiation is reviewed and then modified to take account of: (1) The growth in size of dislocation loops in the course of the irradiation. (2) A limiting density of unit c-axis edge dislocations created by vacancy collapse parallel to the basal planes. (3) The diffusion of lattice vacancies parallel to the basal planes. (4) The finite crystallite size parallel to the basal planes. The theory is used to calculate the lattice defect concentrations as a function of dose over a wide range of doses and temperatures. The crystal dimensional changes due to these defects are compared with experimental data derived from experiments on reactor grade graphite and highly oriented pyrolytic graphite at temperatures from 300to 1400°C to doses 5 1 x 10z2n/cm* (EDN) and found to be in remarkably good agreement.

1.

INTRODUCTION

The first nuclear reactor

used artificial polycrystalline

graphite to moderate the neutrons emitted by the uranium oxide fuel [ 11.Since that time similar graphites have been used extensively in nuclear reactors and intensive scientific studies carried out on the effects of neutron irradiation on the dimensions, physical and mechanical properties [2-51. The low atomic weight of the carbon atom and the low spatial density of atoms in the crystal structure of graphite produce a remarkable simplification of the atomic displacement process by comparison with most solids [3,6]. It is found to a good approximation that a neutron flux 4 n/cm*/sec (measured on the Equivalent DID0 Nickel scale and hence proportional to the atomic displacement rate) produces G atomslatomlsec displacements from the crystal lattice which may be assumed to be randomly distributed-that is there are no significant “spike” effects[3]. The inherent simplicity of the displacement process has encouraged attempts to formulate a detailed theory of irradiation damage in graphite, which agree well with experiment over limited ranges of conditions [7]. However, as has been pointed out by Henson et al. [8] in particular, there is no satisfactory theory for doses in excess of -2x 10” n/cm’ (EDN), the only previous attempt breaking down at about this dose[9]. In this paper a model described by Kelly et al.[9] is modified to make it suitable for calculations for doses of
and physical property changes of polycrystalline reactor graphites [lo, 111, and direct measurements of crystal dimensional changes on highly oriented polycrystalline pyrolytic graphite of near theoretical density[l2-141. It is well known that graphite crystals grow parallel to the hexagonal axis under neutron irradiation and contract parallel to the basal planes[lO], with very small or insignificant changes in crystal lattice parameters in the conditions considered here [8]. In this work the strain measured parallel to the hexagonal axis of a crystal is denoted by e,, and the strains parallel to the basal planes by e,, and e,, in the usual elastic tensor notation[lS]. The lattice parameter changes are, as usual, denoted by AC/C, and Au/a, for in the interlayer and intra-layer spacings respectively[3]. Examples of the data available[8,14] are given in Figs. l-3. In the irradiation conditions considered it has been well established by transmission electron microscopy[l6,17] that the growth e,, is due to the nucleation and growth between the graphite layer planes of hexagonal or circular dislocation loops with Bmger’s vector c/2 parallel to the hexagonal axis. The concentration of interstitial atoms ci contained in these loops is related to the crystal strain by[18] co

c In (1 + e,,).

(1)

(where we have neglected any effect of loops of similar type formed from vacancies. In general vacancy loops do not appear to grow to a size where they can collapse [ 1I].) The contraction of the basal planes has been attributed[9] to the collapse of rather special vacancy structures parallel to the basal planes. The concentration of lattice vacancies in these structures is related to the crystal strain by

(2) 117

118

B. T. 0.15

KELLY

l1350°C / / / / / /

:

//

/ + 350°C

/

//

300°C ,450”C

0 1

2

3

4

I

1

5

6x10*’

Dose, n/cm’ (EDN) Fig. 1. Interstitial concentration in loops-pile grade A graphite (8).

IC

Pyrolytic data

300% 350°C 450°C 640°C

0 a 0 +

Theory IV, = 10” cm-*

d 5

Neutron dose, n/cm’ (EDN) 0

J

-5 Fig. 2. Crystal strains in pyrolytic graphite irradiated in Dido-piuto (after Kelly and Brocklehurst [141)comparison with theory (no correction for lattice parameter changes).

119

The theory of irradiation damage in graphite 40

7 a? + z = E d -2 L ci m z f 0

. +

30

.

0

20

0 390°C + 440°C 0 660°C

I

0

2x 1022 Neutron dose, n/cm2 (EDN) l

‘; a? + =.

Theory IV,= 10” 350 A 450 0 650 V

Data

10

0 0

-10

. 0

l

.

= -20

Fig. 3. Crystal strains in pyrolyticgraphiteirradiationsin the Dounreay fast reactor (after Kelly and Brocklehurst[l4]. The lattice also contains free vacancies in the form of

single vacancies, of concentration c.. The theory used to calculate the defect concentrations is given below and then related to the c,, C”,(c”)Eollapaed experimental crystal strains using eqns (1) and (2).

activation energy for motion of the interstitial group parallel to the basal planes ( = 1.17eV) [ 191. k, is Boltzmann’s constant and T is the temperature. Insertion of the appropriate numerical values leads to N, = 2.68 x 10” G”* exp

2. THEORY 2.1 Interstitial loop density The interstitial dislocation loops in irradiated graphite are nucleated in a relatively small dose range (~10’~ n/cm’ (EDN), after which they grow by capturing further displaced atoms diffusing in two dimensions between the layer planes [17]. The nucleation processes have been treated by various authors and three distinct expressions obtained for the density of nuclei N, cm-‘, depending upon the process assumed to be operating. (a) Homogeneous nucleation. According to Reynolds and Thrower [ 181nucleation of an interstitial dislocation loop occurs when two diffusing groups of interstitial atoms combine to form an immobile nucleus. The density of these nuclei increases until a newly created interstitial atom has a greater chance of joining a loop than meeting another interstitial. The theory of Greenwood et al. [ 191 appropriate to three dimensions, was readily modified for two dimensional diffusion by Reynolds and Thrower [ 191 to give

.

(b) Heterogeneous nucleation. Two separate processes have been considered for heterogeneous nucleation of interstitial loops, associated respectively with the presence of substitutional boron atoms [20-221 or unknown impurities [ 171 (which may be preferentially located in grain boundaries). According to the work of Mayer et al.[21] when the substitutional boron content exceeds about one part/million the interstitial loop density is given by: N, = 2.5

[

9 01“’C,,"' exp (E/2kT)

(3)

where K, is approximately constant. The various symbols in eqn (3) are: rO, the radius of the loop nucleus at formation, generally taken to be 2~ lo-* cm; Do, the diffusion constant for the interstitial group moving parallel to the basal planes (=4 cm’/sec)[l9]; a, the jump distance parallel to the basal planes, 1.42x lo-‘cm; d, (=c/2), the interlayer spacing (3.35 x lo-’ cm); E, is the

O-1

where: Cb, is the boron concentration; No, is the number of atoms/cm’ in graphite (1.13 X 10Z3cm-‘); Y,,, is an atomic vibration frequency (expected to be about lOI*set-‘; but -1@ see-’ from experiment[21]) and E, is an activation energy (= 1.25eV), in itself given by: E=Ef,+Eb

= KG’12exp [E/2kT]

(4)

(6)

where: Ei., is the activation energy for interstitial motion parallel to the basal planes and Eb, is the binding energy between an interstitial atom and a substitutional boron atom. Substitution of the numerical values of the parameters leads to N, = 6.3 x 10” G1’*C:”exp (7025/T)

(7)

remarkably similar to (4). In the second case, where the nature of the nucleating

120

B. T. KELLY

sites was uncertain, Reynolds and Thrower [17] found that NI = Mn/[n + (N - n) exp (-E&T)] G zexp

(EJkT)

(8)

where M is the number of mobile interstitials N is the number of interstitial atom sites n is the number of impurity or nucleation sites

all referred to unit volume

where R is the capture probability/collision and I& is the height of the energy barrier against interstitial vacancy recombination (II, = 0.24 eV) [25]. This equation is not suitable for use in high dose calculations because it makes no allowance for the increasing capture probability for interstitials by loops as the latter grow to finite size. Lidiard and Perrin[26] have formulated the full two dimensional diffusion theory of the growth of interstitial loops in graphite in these conditions and applied the results to comparatively low dose data. For the high doses with which this paper is concerned, their eqn (2.9) becomes, to a good approximation.

_ _G N, “2 A - i$ o-iF C”“2

ET is the interstitial-impurity binding energy. In their studies of single crystal graphite E, was found to be 0.22eV, while for Pile Grade A graphite E, was deduced to be 0.37 eV with ~~~~ - 1014cmM3. It is not in fact clear that there is any distinction between the observations as is discussed by Thrower[23], all of the nucleation may be boron controlled. However we are only concerned with estimating N{ for various conditions. It must also be noted that there is evidence that the nucleation density is increased compared to these expressions when the separation of crystallite boundaries perpendicular to the basal plane defining a crystallite size L,, is less’than the coplanar loop separations given by the nucleation processes above[23]. In the graphites considered here with ,5*- 2 x lo-’ cm this occurs at about 300°C (DID0 Equivalent temperature) and it is possible that N, is then relatively independent of temperature at - 10” cm-3. 2.2 Interstitial Imp growth The N; interstiti~ loops/unit volume now grow by capturing those interstitial atoms diffusing in two dimensions which evade annihilation by the background concentration of vacancies with concentration c,, assumed uniform. If P is the probability of an interstitial atom escaping annihilation when it encounters a vacancy then the concen~ation c, of interstitial atoms in the loops after a neutron dose y (measured on the DID0 nickel scale) n/cm* is191

where z is the number of new crystal lattice sites explored/jump. This equation was first derived by Reynolds[24] and subsequently employed by Kelly et al., in an earlier version[9] of this theory. According to Reynolds, P = 1 -exp(-~~/k~ =l-Rsay.

wo

(11)

(The Bessel function ratio K,(p&K,(p,J in Lidiard and Perrins equation tends to a value of 1.4, which is here absorbed into NJ where A is a constant and a is the area/atom in an interstitial loop. If y* 4 y is the dose at which nucleation ceases, eqn (11) can be written as:

and also as

2.3 Vacancy laces-~j~itation of density When the interstitial loop nucleation dose is complete, it is supposed, following the treatment of Kelly et al.191 that the overlap of the groups of displacements in the lattice produces steadily more complex vacancy structures in the basal planes which collapse paraiiel to the basal planes independent of the defect content of the neighboring layer planes. These structures are idealised into the concept of collapsed lines of vacancies (known as “tadpoles”) which are in fact edge dislocation dipoles localised in single basal planes. These collapsed lines continue to grow after their formation by capturing vacancies created close to their ends, athermally, over most of the tempe~t~e range. It is possible, as discussed for instance by Jenkins[27] that concentrations of these defects may exist in a graphite crystal prior to irradiation, some in the body of the crystal and some in grain boundaries-further consideration is given to this possibility later. The rate of growth of a collapsed vacancy line at temperatures where vacancies are immobile is193 (14) where a is the diameter of a vacancy (- 1.42X lo-’ cm) and x is the number of neighbours of the end of a line at which instantaneous collapse occurs. In the originai

121

The theory of irradiationdamagein graphite version of this theory the rate of creation of vacancy lines was assumed to be[9],

(19 This equation assumes that vacancies become divacancies by adjacent displacement, the di-vacancy then collapsing to such a degree that it no longer traps diffusing interstitials. This theory agrees remarkably well with experimental data on crystal dimensional changes up to doses of about 3 x 10” n/cm’ (EDN), but considerably underestimates the vacancy concentrations deduced from experiment[8], probably because of the inadequacy of eqn (9). Simmons has pointed out that the observed vacancy concentrations (deduced from lattice parameter and thermal conductivity changes) would lead to enormous vacancy line densities and basal shrinkages at high doses [7,8]. He proposes an alternative description in which the line ends are largely responsible for lattice parameter changes ha/a, and phonon scattering-it is however difficult to reconcile this with annealing data which show complete recovery of all properties except dimensional changes [28] at about 1500°C. Thrower[23] has considered other possible explanations in terms of nucleation in twist boundaries. A survey of high dose dimensional change data and analysis, using simple models, suggests strongly that the basal plane contraction rate tends to the same value at temperatures where vacancies are immobile, although the dose required to reach this rate increases with temperature [29]. According to Kelly et al. [ 121,the rate of basal plane contraction is:

de,,_de,,_ NLx ._G dr

dy

N,,

4

area grows from -+/2 for the di-vacancy to 4 at r, = 2(,/?r,“* might be more appropriate. However we assume as an approximation that for doses where the lines are fully developed this effect can be included by replacing eqn (15) with (17) which has the solution: N,=N,,,[l-exp[-$?l)c.(y)dy}]

(18)

neglecting any inherent collapsed line defects. The alternative mechanism leading to a saturation of the line density is the mutual annihilation of line end dislocations of opposite sign. The loss of vacancy line ends is now given by d$ = -[number of new lattice sites explored/unit dose] x[fraction of lattice sites occupied by line ends] ~[fraction of ends capable of mutual annihilation] (19) leading to I,2

N; = ;xc.N, [

-

I

10z2cme3.

The simultaneous operation of both mechanisms, that is combination of eqns (17) and (19) has the solution at saturation

(16)

which leads to a value of NL+5 X 10” cm-’ using G/I$ = 0.55 x lo-*’ at/at/n cm* (EDN)[29], x = 3, etc. It is contended therefore that the original model of creation of collapsed vacancy lines requires modification to produce a saturation of the line density NL. There are two obvious modifications to the theory which can produce an effect of this kind. Heald and Speight [30] in considering the theory of void growth in irradiated metals have pointed out that there must be a limiting dislocation density in a crystal lattice which corresponds to the overlap of the dislocation cores-that is a mean separation of about 10 Burger’s vectors. The density of edge dislocations in a single basal plane corresponding to NL = 5 X 10” crnm3gives rise to a separation of - 19 x lo-* cm, that is between the values expected for partial and total in-plane dislocations (14x IO-*cm and 24x 10-*cm respectively) in good agreement with this concept. It is proposed to include this effect in the theory by assuming that for a fully developed collapsed line of vacancies, no new line can be nucleated in an area $ about each end. A fully developed line must clearly have a length L > 2 ($/-llr)“*,for shorter lines the areas overlap and an approximation in which the excluded

= N, to a good approximation,

(21)

thus (18) is adequate for inclusion in the theory. 2.4 Vacancy mobility At high irradiation temperatures the vacancies become mobile and diffuse in the basal planes. It has been shown in previous work[31] that vacancies which reach crystal boundaries collapse there, contracting the basal plane. The same occurs with vacancies diffusing to the ends of collapsed lines. Using the methods of the rate theory of irradiation damage described by Brailsford and Bullough[32] modified to take the two dimensional defect diffusion into account gives for these two vacancy loss rates [33] 82D ,,“eE= - e

(dc.> F

NLcu

(22)

and

boundaries

(23)

122

B. T. KELLY

where 0, = 4. exp (- E,,JkT) is the vacancy diffusion coefficient paraliel to the basal planes and z, is a small nume~cal factor (order unity) to allow for defect-sink forces. The vacancies which take part in both these processes collapse, and thereby contract the basal planes[31]. Equation (16) must now be replaced by:

Collecting all of the vacancy loss terms together and using eqn (13) as the vacancy source strength leads to ~=~($~N,[~c~““dYIc~“]-XC,-2x~} 2” -u80 Id, jQ--JN~t~(~)YV}c”. (

N, = 5 x l@’ cmm3, No = 1.13x 1g3cm-‘, a = 1.42x 10m8 cm,

d = 3.35 X lo-’ cm, R = exp (- 2786/I),

2” = 1 and Lt, = 1.0exp [ - (52,240/T)] corresponding to an activation energy for vacancy motion in the basal phures of 4.5 eV (see the discussion). A computer programme was prepared by Mr M. Toes of the UKAEA Central Technical Services (Risley) which for a given set of values of 4, G, N,, L, and T outputs values of c,, Ci(=e,,), NLt e, and cZvat doses of y = 0.5,1,2,3,4,5,8,10

x l(t2’n/cm’ (EDN).

(25)

The vacancy concentration c2, in the form of di-vacancies is given approximately by:

(A small fictitious value of c, at y = y* = lOI n/cm* (EDN) was used to start the integration routine.) Most of the UKAEA data has been obtained in either the DI~~PLUTO Materials Testing Reactors or the Dounreay Fast Reactor in core locations for which the following G and # apply]291

DIDOlPLUTO MTR.G = 1.89x lo-’ atlatlsec and qi = 3.4 x 10” n/cm*/sec (EDN). Dounreay Fast Reactor G = 47 x W atlatfsec and # = 8.5 x 1014n~cm*~sec(EDN). ~~C~~~UA~~~F~~Y The polycrysta~ine graphite and pyrolytic graphite has a The complete set of equations which constitute the crystallite size of La - 2 x 10-j cm and this was used for model are therefore: all calculations. The complete set of results was obtained for the two reactor locations at T = 573, 623, 723, 923, 1023, 1173, 1273,1373 and 1473°Kand for the values of Nt = IO’*,10” and 10’6cm-3 which cover most of the range of interest (obtained from eqns (3), (5) and (8)). The results follow a similar pattern in each case and it is simpler to present one set (N, = 10” cm-‘) in detail. Figure 4 shows the crystal strains at different temperatures as a function of dose up to 1~znfcm2 (EDN)-the &Zvalues can be compared qualitatively with the values for ci given by Henson et al.]181 and shown in Fig. 1. The similarity is obvious. A more detailed N,=N,,,[l-exp(-$?xlc.dy)] (281 comparison is shown in Fig. 5, where the rates of crystal dimensional change de,Jdy obtained from reactor grade and pyrolytic graphites irradiated in the Dounreay Fast dc,.=2.f_Gx c -5 Reactor and DI~~PLUTO are compared with those [ calculated for Ni = 10’7cm-3, and lO’“cm-“. The results (30) agree well with the curve for N, = 10” cm-’ up to -800°C. At higher temperatures, as previously described[291, a flux level effect is obtained. The higher temperature data e~~=-$j$-$%-;j~~ are then closer to the results for N, = 10’6cm-3. The theoretical results for crystal strain as a function of dose are shown in Fig. 2, assuming N! = lO”~rn-~. Figure 6 (31) [ ($)hx + ~)~~~~~~~ dy* compares the vacancy concentrations deduced from lattice parameter changes (8) in Pile Grade A graphite A number of parameters are used unchanged throughout with the total uncollapsed vacancy concentration C* the calculations calculated from where x’ c x neglecting formation by thermal diffusion. This completes the theory required for the calculations.

m 1 1 =,r”. (;j-&[-Q1i2)dy]

Gj# = 0.55 x 10W2’ at/at~n - cm-‘1291; A = 3.5 x lo-“, x=3,

c* = ci - 2&$.

(32)

This parameter C* therefore includes single vacancies,

123

in graphite

The theory of irradiation damage

N = 10”cm

’ i =34x 10’3n/cm2/sec L.=2x105cm

I

I Neutron

dose,

(EDN)

I

10 n/cm’

15X102’ (EDN)

300-65O”C

\

\

\

\ \

0

\,

1200°C

Fig. 4. Crystal

strains

as a function

di-vacancies and vacancy groups which are prevented from collapsing by the presence of the collapsed vacancy lines. 4. DISCUSSION

The modifications to the theory of irradiation damage in graphite represented by the proper inclusion of two dimensional loop growth, the limitation of the density of collapsed vacancy lines and vacancy mobility have produced a remarkable agreement with experiments in the following respect. (1) The basal plane shrinkage rate increases steadily to the limiting rate at temperatures where vacancies are immobile and then remains constant and independent of temperature. (2) The crystal growth parallel to the hexagonal axis agrees remarkably well with experiment over the range 300-8OO”C assuming an interstitial-vacancy repulsive barrier Es = 0.24 eV and a constant density of loop nuclei of 5 x lOI cmm3(=N,/2). At temperatures above 800°C it is necessary to assume a loop density of -5 x lOI cme3 for the explanation of the crystallite size effects but this is very sensitive to the value chosen for E,,,,. In Fig. 7 the values of interstitial loop density deduced in various ways are shown as a function of temperature of

900°C

of

dose

at various

temperatures.

irradiation. Reynolds and Thrower[l6] have deduced a nucleation density for Pile Grade A graphite of 10” cme3 at 350°Cand approximately lOI cmm3at 650°C. The effects of crystallite boundaries are not evident for irradiation temperatures below 250°C in well crystallised materials, compatible with a nucleation density of 10” cm-’ [29] at higher temperatures. The observation of a flux level effect on thermal conductivity (associated with vacancies) shows that h$ cannot be constant but must decrease with increasing temperature but this is beyond the sensitivity of the data. (3) The total uncollapsed vacancy concentration is comparable with that deduced from lattice parameter measurements but differs in detail, particularly at 300 and 350°C. Once the vacancies are mobile there is no significant experimental concentration as predicted theoretically. The major difficulty with the theory as applied to Pile Grade A graphite and pyrolytic graphite lies in the comparison of calculated and experimental vacancy concentrations. Examination of Fig. 6 shows that up to a dose of 5 x 10” n/cm* (EDN) the total vacancy concentration which has not collapsed in lines is in good agreement at 450 and 650°C but the curve shapes at low dose magnitudes are very different at 300 and 350°C. As

B

T. KELLY i 0.25

Low flux

I

300

I 1500

I

I

500 Temperature,

“C

1000

Fig. 5. Variation of crystal strain parallel to hexagonal axis at 5 X IO*’n/cm2 (EDN) comparison experiment.

noted above C* contains single vacancies (which are allowed to trap interstitials), di-vacancies (which are not allowed to trap interstitials) and various multiple vacancy groups which are not allowed to collapse (or trap interstitials). The experimental curves are deduced by assuming a constant relationship between lattice parameter change Aa/a the vacancy concentration. In fact each type of uncollapsed vacancy will have a different importance factor and to compare theory and experiment these factors (which are not accurately known) would have to be included and could change the curve shape in the required direction if simpler groups were more effective. It is more likely that a part of the discrepancy (also obvious at small strains in Fig. 2) is due to the inaccuracy of eqn (11) at small doses. Lidiard and Perrin[l6] have shown that their treatment gives a very rapid initial loop growth and hence initial vacancy concentration increase. This would also shorten the dose at which the basal shrinkage rate becomes constant. A further possible source of this discrepancy could be the presence at 300 and 350°C of a concentration of the so-called “sub-microscopic” interstitial clusters and their vacancies. Thus the discrepancies obvious in Fig. 6 may not be very important up to doses of 102’n/cm*. However as the dose builds up a large concentration of the uncollapsed groups of vacancies will build up-whereas it is clear experimentally that saturation of such defects occurs [8,14]. It is necessary clearly to introduce into the theory a very small probability for these defects to trap interstitials. There is no obvious way of introducing this into the theory at present, but it will improve agreement with the vacancy concentration at high doses and reduce the high dose growth rate along the c-axis as required (Fig. 3)[35].

of theory and

There are a number of less obvious objections to the theory (A) In less well graphitised materials higher rates of basal plane shrinkage are observed than would be allowed by a limiting density of collapsed vacancy lines N, = 5 X loZ”cm-‘. (B) In graphite of the same type irradiated at temperatures below 300°C basal shrinkage rates previously attributed to collapsed vacancy lines with a density NL of 3 x l@’ cm-’ occur. In boron doped graphites the basal shrinkage rates at these temperatures can be increased by a factor of about 3 [34] at least up to doses of 10” n/cm* at 575°C which would imply NL = 1.5 x 102lcm-‘. The first of these objections can be overcome by postulating that crystal boundaries contain edge dislocations at which instantaneous vacancy collapse can occur as well as collapse of vacancies reaching boun-, daries by diffusion. The number of these collapse points is approximately

&=2

La dp

(33)

where p is the separation of the points in the boundary. This can be comparable with N, for L. < 1OOOA.The parameter p is related to the tilt angle at boundaries. This possibility was first suggested by Harrison[36] and Jenkins[27] has also postulated the presence of such dislocations in the body of poorly graphitised crystallites. However these would presumably not be able to exceed -5 x 10”’cm-‘. It has also been shown that the smaller the crystallite size (or higher nucleation density) the lower the temperature at which vacancy diffusion becomes important [29].

125

The theory of irradiation damage in graphite

‘I 2 3 4 Neutron dose, nfcm* (EDN)

5x102’

(a) x10-2

5r

X l

I

I

I

4 2 3 Neutron dose, n/cm’ (EDN)

1 5x10*

(W Fig.

6.

Comparison of experimental[8] and theoretical vacancy concentrations. (a) Experimental vacancy concentration[8]; (b) total uncollapsed vacancy concentration C, - 2exx vs dose.

In graphite samples irradiated at low temperat~es (<3OO”C)very large vacancy concentrations are created at doses too small for lines to grow to lengths where they can be regarded as fully developed. (The line length is roughly given by the dose/W’ in terms of number of collapsed vacancies[l2].) Thus until the lines have overlapped it is possible to have a larger concen~tion than 5 X W cme3, but this condition must eventually be reached. It is interesting to note that in irradiations at 30, 150, 200 and 250°C where this could occur, the basal shrinkage rate at high doses attributed to the collapse of vacancies in lines does decrease[I2]. There is insufficient evidence to indicate whether it falls to the limiting rate due to N,-but the reduction in rate is real and is followed by a long period of constant shrinkage rate[35]. The effect of substitutional boron is more difIicult to explain since theoretically the only effect is an increase in N,. In the same way, unless the boron effectively reduces the crystallite size L,, the theory predicts that a substant~1 CAR Vol

15. No. Z-F

decrease in shrinkage rate of the polycrys~lline graphite will occur for doses of -3-5 x 102’cmm3; certainly increasing N, can have the observed effect for doses below 3 x 102’n/cm* (EDN), which exceeds the doses achieved in such experiments at present. However this appears to be a definite theoretical prediction which is being tested expe~ent~ly using pyrolytic graphites to avoid pore generation effects. Two related topics remain to be discussed, the flux level effect and the use of an activation energy for vacancy motion parallel to the basal planes of 4.5 eV rather than the approximately 3 eV usually employed[23]. The value used here is chosen to produce vacancy diffusion effects for a crystallite size of 2000 A and densities of nuclei in the range considered. It is also, in the authors view, more compatible with thermal annealing data[28]. (Figure 5 indicates that it may be higher than 4.5 eV.) In reactor irradiations at diierent flux levels there will generally be smaller effects at high temperature than calculated here

B. T. KELLY S This work Pile Grade A graphite

x 10” 3

-Single cfystals[l5]

I

+

Reynolds and Thrower[l7]

Pile Grade A graphite

x

Brocklehurst

pyrolytic

and Kelly[35)

graphite

00 Irradiation

temperature,

“C

Fig. 7. Density of interstitial loops in irradiated graphite.

for constant N,, because all of the nucleation theories (except that which leads to a constant A$ due to crystal boundary effects) predict an increase of N, with G”’ which will offset the shift to higher temperature with increasing damage rate. At the lower temperatures an increase in N, by about a factor of five associated with the flux difference from DFR to the DIDO/PLUTO reactors gives an increase in e,, at a dose of 5 x 10” n/cm’ (EDN) of about 1.8-2 at each temperature. The data does not indicate an effect of this magnitude but perhaps a somewhat smaller one (Fig. 5) of about 1.3. As already noted the existence of a flux level effect in thermal conductivity data, which we attribute to the vacancies shows that N, = constant cannot be completely correct [37]. It should be noted that there must be an upper limit to the number of nuclei which can be formed in each theory, which is not reflected in the simple formulae for homogeneous and heterogeneous nucleation. As this low temperature limit is approached the flux level effect should diminish, and for temperatures below 300°C (DID0 Equ.) it is swamped by the large densities of sub-microscopic interstitial clusters. In conclusion, this work has considerably improved agreement between theory and experiment for crystal dimensional changes in graphite particularly by the introduction of a limiting in-plane dislocation density. If the original concept[9] of the creation and collapse of vacancy lines is correct a mechanism of this type must exist. Basically the argument is that even heavily dosed graphite under these conditions is little different from low dosed material in which line creation is occurring-thus the process must be self-limiting. In order to extend the theory to doses of say 5 x loZzn/cm* (EDN) for which data now exist[35], it will be necessary to take account of the following effects in addition to those described above associated with the capture of interstitials at multiple vacancies.

((u) The interstitial loops become comparable in size with their initial separation. The loop radius r, is given by rr,= (ci/lr dN,)“* which for N, = 10” cm-’ gives values of 308 A and 616 A for ci = 0.10 and 0.40 respectively. The flow of interstitials from inside the loop is now comparable with that from outside the loop and eqn (12) is no longer adequate. (0) The basal shrinkage decreases the effective crystal size ~5..and the distance between loops in the basal plane. (S) Diffusion of vacancies in loops will become important at slightly lower temperatures (rO< LJ2) than diffusion of vacancies to boundaries. This might be expected to produce some “structure” in the crystal dimensional change curves at -900°C. Acknowledgements--I acknowledge a substantial debt to Mr. M. Toes of the Central Technical Services of the UKAEA Risley for programming these and many other calculations. Mr. J. E. Brocklehurst was instrumental in pointing out a numerical error. Mr. R. V. Moore, UKAEA Member for reactors gave permission for this paper to be published.

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