Lattice parameter changes in irradiated graphite

Car&n

1965, Vol. 3, pp. 277-287.

Pergamon Press Ltd.

Printed in Great Britain

LATTICE PARAMETER IN IRRADIATED

CHANGES

GRAPHITE

R. W. HENSON and W. N. REYNOLDS Atomic Energy Research Establishment,

Harwell, Berkshire, England

(Received 15 Jane 1965) Abstract-Changes in the X-ray lattice parameters of Grade A graphite have been studied in detail during isothermal irradiations to high doses in the temperature range 200-350°C. The results show several new features not hitherto discussed in the literature and have been analysed in the light of the crystalline elastic constants. The changes have been related to the corresponding changes in overall dimensions and of Young’s modulus and the rate of accumulation of stored energy. The estimated energy of formation of a vacancy is increased from a previous value of 5.5 f 1 eV to 7.3 f 1 eV and the activation enemv for its motion within a basal plane-is greater than 2.5 eV. Vacancies have an appreciable mutual at&&ion, and they recombine less readily with migrating interstitials when they have formed small groups. The basal plane contraction per vacancy is in reasonable agreement with recent theoretical estimates and un-nucleated interstitial atoms appear to occupy 2-4 atomic volumes.

1. INTRODUCTION

c-spacing and decreases in u-spacing were frequently reported in early X-ray studies of radiation damage effects at ambient temperature in graphite. Interest fell off when it was discovered that changes were much smaller at power reactor temperatures, particularly when it was shown(‘) that the lattice parameter changes then constituted only a minor part of the overall crystallite dimensional changes. Consequently, there has not hitherto been a systematic study of parameter changes as a function of dose under isothermal conditions at various temperatures. Some results(2) relating to 200” and 650°C were reported two years ago. Since then some improvements in technique have been perfected, and it is the purpose of this paper to present a range of results from irradiation at ZOO”, 250” and 350°C which can be correlated with known changes in other properties. CONSIDERABLE increases

in

2. EXPERIMENTAL RESULTS The measured changes in c and a-spacing are shown in Fig. 1 as calculated from displacements of both line peaks and the mean of the half-heights. The measurements were made at 20°C with a 19 cm Debye Scherrer powder camera, using copper 277

radiation by the same method as previously describedc2). Special precautions were taken to ensure that the film was maintained at the same temperature from exposure in the camera until measurement. The camera was calibrated by the direct measurement of the linear distance between the knife edges(3). It can be shown that this method is insensitive to errors in the alignment of the beam and the knife edges, even when there is differential shrinkage between the two films. The line position on the film was measured with a manual microphotometer, with a discrimination of 0.01 mm. All the measurements were made using pile grade A graphite, for which c,=6-720 A, a,=2.462

L%and p=O.ZO

At an irradiation temperature of 350°C the line shifts were small, and the line widths only slightly increased. However, the 12.0 and 00.6 lines were too diffuse to measure, leaving only four lines for the determination of the c-spacing, the 00.2 a, 00.2 /3, 00.4 CL,00.4 /3, and four lines for the aspacing, the 1010 a, 10.0 8, 11-O a and 11-O/I. An extrapolation method was used to remove the systematic errors. The estimated error in the final result was f 0.05 %.

R. W. HENSON

I

1

I

I

I

and W. N. REYNOLDS

I

I

I

I

1

I I

I I c(

0

LATTICE

I

I

PARAMETER

I

I

CHANGES

I

I

IN IRRADIATED

GRAPHITE

279

I

a a

280

R. W. HENSON and W. N. REYNOLDS

I

0

I

I

I

IO

20 NEUTRON

I

I

I

30 DOSE N.ChK2

I

40 x I020

FIG. l(c). 350°C.

At 200°C and 250°C the shifts were very large and the line widths so increased that a nickel filter had to be used to prevent overlapping of the lines. Only the 00.2 CIand the 10.0 c1were still measurable at high doses, so the spacings had to be determined from a single line. No corrections were applied, the systematic errors being minimized by calculating only the relative change in spacing. The unirradiated 00.2 a line was slightly asymmetrical, so the origin for all measurements was taken at the centre of half peak. As the value of Aala approaches 2% the 10.0 o! line becomes very close to the 10.1 cc This means that the value of Aa/a obtained is liable to be overestimated and a better value was obtained from the 11.0 a for moderate doses. The systematic error in AC/C was calculated from the data obtained with unirradiated material. This rises with increasing dose to 0.05 % when Ac/c=9%. The main source of error is due to the difficulty of measuring the position of the very broad and diffuse lines at high doses. This

rises from an initial value of f 0.05 % to f 0.3 % when Ac/c=9%. Some of the high dose points at 200°C were not measured with the photometer. These are marked with a different symbol in Fig. 1, and are subject to a larger error. Some line profiles for irradiation at 250°C are shown in Fig. 2. The films were not individually calibrated, so the amplitudes are not strictly comparable. However, the exposures were nominally the same, and the deviation from linearity of the density exposure scale was only 4 % at a density of 2.5. Line widths and asymmetries are shown in Fig. 3. 3. THEORY A detailed survey was recently completed of the loops revealed by transmission electron microscopy of irradiated graphite, and their quantitative relation to observed overall dimensional changesc4). It was shown that the c-axis growth of a crystal is chiefly due to two effects, viz. the forcing apart of neighbouring planes by a population of small inter-

LA’M’ICE

PARAMETER

SAMPLE

DOSE

CHANGES

IN IRRADIATED

GRAPI-IITE

281

X 10-20N.CMT2

0 I.70 7.39 16.24 2426

61 (4) 61 2.c

:: ;; z w 0

1.c

0

I

I

I

I

f

I

I

IO

I

8

6

4

2

0

-2

-4

+ FIG. 2. 00&t line profiles after irradiation at 2W’C.

p

The dashed lines are asymmetrical

stitial clusters of atoms, and the formation of fra~en~ new basal planes by the growth of some of these interstitial dusters into large loops. In Fig. 1 are therefore included the experimental values of the total c-axis crystallite growth e,, and the total basal plane contraction ers corresponding to each irradiation condition. The values are those given for pyrolytic material by SIMMONS et QZ$~), and it can be seen that after a short initial period, es9 exceeds AC/Cas the interstitial loops grow. However, Fig. 1 also shows that after a somewhat longer initial period, the basal plane shrinkage esu exceeds that of Aala. This might suggest that vacancy loops are forming and collapsing on pyramidal planes, but consideration of the weakness of the interaction between neighbouring basal planes has led various workers, e.g. KELLY(~), to the idea that groups of vacancies in the same basal plane may collapse in more or less linear structures restricted to their own plane.

in opposite senses.

It is therefore proposed to analyse the results given in Section 2 in terms of the following model : the interstitials and vacancies produced by the irradiation are present partly in the form of collapsed lines and loops, which cause direct changes in exx and exz, and partly in solution either singly or in small uncollapsed groups. The vacancies and interstitials in solution aBeet e,, and erx via the lattice parameters. They may be regarded aa creating anisotropic internal pressures proportional to their respective concentrations, n, and ni and related to the parameter changes via the elastic moduli. This idea is analogous to that used in the study of thermal expansion by NELSON and RILEY”). The relationship may be expressed by the equation : $=S33

:

(Pti+Pbw)fzS13

(P*i+Pav)

=(~11+~12)(~,i+~~)+~l3(~ci+~,?)

(1)

R. W. HENSON and W. N. REYNOLDS

282

z

0.6

ii! -W-4

E

0

0

I

I

I

0

Fm. 3(a). Line asymmetries ($)

where Ki ni=Sss K2

K2

n,+or P&-2&3

~=(S11+&2)

Pd;

Pa,+&3

(&1+&2) =1=

(2) Kl tzi

pa+%3

Sg3 P,-+-2Si3 Pk

Pm; pci

’

1 30

DOSE N.CW2

repk-($)

where the P’s are the internal pressures, the subscripts c and a referring to those directions in the crystal, and i and v to interstitials and vacancies, and the S’s are the compliance moduli. In terms of the defect concentration, the equations become : %+ -#” Q2

I

20

NEUTRON

AC ,=K1 Aa ;=K2

I

IO

x 1020

aplr_hekht meMand (b) widths

(&1+&z)

and a’=

Pcv

pa,+st3

S33 P,+2Si3

P,

Following the calculations of COULSON et al.(“) of the change of x-bonding by the introduction of vacancies, KELLY recently showed that (Sl l + &2) pal- 0.14 n, and also that &a Pci lies in the range l-5%. This naturally suggests that K1 g Ka so that in a case where ni=nV, A = AC -- Aa -=-cl. As pointed out by KELLY(~), such a l c a case occurs in the results obtained after irradiation in liquid nitrogen by PLUCHERY(~), who found A=O.O86. This is close to the value of -ol = of 0.075 found from SPENCE’S@) -&3/S33 values of the compliance moduli if it is assumed

LATTICE PARAMETER

CHANGES

IN IRRADIATED

GRAPHITE

283

15 -

25

IO -

0

X IO -

250°c

0

0

NV (VACANCIES)

x

~~ (INTERSTITIALS)

:: S-

O0 00xX

O

9xX

5-

6 35ooc

@@

QQQ~B;~‘@ O--f?@

IO

20

I

NEUTRON DOSE N’.CtK2

I 30

I

4

x 1020

FIG. 4. Calculated total densities of vacancies N, and interstitials Ni.

that P,+< Pe.Pb may therefore be neglected and Kr tli=Sss Pk. It follows that lcra[, which is >,lo,l, gives the maximum value of A. KELLY assumed that Pcv=O, so that -cr~=-(Sr1+S1a)/2S1s=0.215. However, the A values obtained from the data of Fig. 1 almost all lie between 0.25 and O-47, so that P, cannot be neglected. If the maximum observed value of A is assumed to correspond to the condi-

tion where the concentration of interstitials is so low that they make a negligible contribution to the parameter changes compared to that due to the vacancies, we may write a, = -0.56. This corresponds to writing S’s3 Pm= - 0*42n, and Sr3 Pm= 0.03 1 fl”, .*.K2 = - 0.11, assuming KELLY's value of (&1+&z)

pa,.

From the data, an upper limit for [Kal may be

R. W. HENSON

and W. N. REYNOLDS

X'

b-

/

KI ni

0

UNCOLLAPSED

X

4-

VACANCIE!

NON-NUCLEATED INTERSTITIALS

2oo”c 2-

0 : F d

K2nv 2

: s z v

2-

X X---_--_-_X

KI ni

9 5 J 2

2soec

IK2nV I),

X

350%

0.2 I 0

I IO

I

I 20

NEUTRON

interstitials and vacancies,

KIni and K

of vacancies in solution

must be greater than NO=(ez8-$)

+2(ezz-$?),

and 14

0

I 30

DOSE N.CM-2

FIG. 5. Calculated relative densities of un-nucleated

found. Thus the number

I

< I$1

This inequality must be reinforced by any Poisson contribution from interstitials. The lowest value

Of Is1 Obtained from the data Of Fig* good agreement with the ideas expn Then, solving equation (2), we find: &+&5

s 0.13, in d above.

LATTICE

K2 %=

PARAMETER

O-865

CHANGES IN IRRADIATED GRAPHITE

(3b)

Consequently, the total number of inters&i& present is : Ni=e=*-t+R and that of vacancies is

so that these quantities may be calculated if K, and K2 are known. The value of K, is not known, but stored energy considerations have recently shown that it is greater than 2. A calculation by AGRANOVICH and SEMENO~('~) suggests that for single interstitials K, =5.3. For the interstitials in small groups with which we are concerned, we may take K,=3. In general, for 5.3 > Kl > 2, the actual value of K, has little effect on Ni. Values of Ni and N, are illustrated for the three temperatures involved in Fig. 4. In each case the value of Nv increases in direct proportion to the dose. The total interstitial density, after lagging somewhat at lower doses, later rises to a value not significantly different from Nv .There are many possible explanations for this. It may be that in the early stages, inter&it& may migrate to positions of low potential energy in the polycrystalline structure, and after these positions are filled the accumulation rates of vacancies and interstitials are equal. On the other hand, systematic errors are possible in the experimental determinations of, e.g. ezz or Si a or the use of the theory expressed in equation (1) with fixed constants may not be justifiable at all damage levels. Two annealing experiments have been carried out which confirm, at least qualitatively, the conclusions of this section. In the first, described by GOGGIN et aZ.c2),a specimen irradiated to 5.5 x 102’ n cmm2 at 200°C was annealed at approximately 300°C intervals up to 1840°C. The recovery of the c-spacing showed two distinct but approximately equal maxima in rate, at 650°C and at 1500°C. The recovery of the a-spacing showed a maximum at 1500°C. If the second peak in the c-spacing recovery rate may therefore be taken as character-

285

istic of a vacancy-annealing process, and the first as due to interstitials, the annealing properties are very much what would be expected from equation (3). In a second experiment, a specimen irradiated at 350°C to a dose of 346 x 102’ n cmq2 was annealed at 1100°C. In this case no recovery could be measured, again in accordance with the implication of Fig. 5 that in this case almost all the c-spacing increase is due to vacancies. The analysis of this section shows that under isothermal irradiation conditions the interstitial concentration ci rapidly reaches a dynamical equilibrium. This equilibrium is subject to quite sudden increases at certain critical doses which AC are reflected in the steps in the - curves. They C

occur at: 2OO”C*6 x 10” n cm” 25O“C~ 8 and 19x 10” n cms2 350°C; 10x 10” n cms2 The step at 350°C is less marked than those at the lower temperatures, as illustrated in Fig. 5. The Klni values show close correlation with the line widths of Fig. 3. It is at once clear that the step at 200X! and the first step at 250°C coincide with the doses at which exx begins to exceed @, i.e. where vacancy coma plexes begin to collapse in the basal plane. For this reason, it is suggested that these steps are the result of a sudden consequential decrease in the vacancy/interstitial recombination probability. This idea is supported by the acceleration in e,, which is also observed at about the same points on the dose scale. At both 200 and 250°C these steps are associated with a vacancy concentration of about 3 %. At 350°C exX exceeds *

even at the smallest

doses, when the concentrzion is less than 0.1%. It is suggested that this is essentially due to a certain limited amount of vacancy mobility within a basal plane at this temperature, which permits some rearrangement of vacancies into small groups with some collapse. If we assume a single vacancy may diffuse about ten atomic spacings in one month at 35O”C, the corresponding activation energy of the motion must be close to 2.5 eV. This must be taken as a lower limit for free diffusion,

R.,W: HENS0.N

286

and W. N. REYNOLDS

however, as it includes effects due to vacancyvacancy attraction. It is close to the lower limit obtained by BAKER and KELLY(I 3, from quenching experiments. 4. STOREZD ENERGY AND MEfxANKAL PROPERTIES

The conclusions of Section 3 may be used to analyse the stored energy data earlier reported for irradiations under similar conditions(14). Thus all the excess stored energy which is released by the complete oxidation of an irradiated sample may be taken as due to the vacancies and interstitials in solution, aa by comparison the energy stored by nucleated loops and lines of defects must be negligible. An attempt was recentlyt4) made to establish the energy of a vacancy, E, by writing the stored energy Q as

where A is the stored energy per unit change in c-spacing due to interstitials in solution together with their corresponding vacancies and B is the energy per unit c-axis growth due to nucleated inter&k& plus that of their vacancies. Since the nucleated interstitials occupy one atomic volume for the graphite lattice and have negligible energy in themselves, B=E,. Graphical analysis of the data in Table 1 leads to the conclusion that Ev lies in the range 5.5 f 1 eV. TABLE 1. INITXALDEFECT ACCUMULATION RATESPER ~O*@N.CM-8 Irradiation Temperature

Stored KIni F$$$

AC

(%I

(%I

ezz (%)

&

4

o-13

44

200

97

O-58

O-36

l-4

l-3

250

38

0.09

0.12

0%58

0.29

350

14

0.03

0.028

o-15

O-065

150°C

183

&tt”

4.2

An obvious step now is to write

Q=&

& nl+B1

&

nv

(5)

The data of Table 1 lead to the conclusion in this case that E,=B, K,=1*04 eV, if it is assumed that K,=O.14. The difficulty of quenching vacancies into graphite is well known, so that such a value is

unacceptably low. This may have arisen partly because Kz is greater than 0~14 at doses less than 10” n cmW2 which, as mentioned in the previous section, is onk possible explanation of the apparent excess of total vacancy population N, over total interstitial population Ni at low doses. We may, however, still use the data of Table 1

as a measure of the population of &collap&h vacancies associated with nucleated interstitials. Then we have:

This

amounts to correcting the procedure of and THROWER~~’to allow for that part of c-spacing increase due to vacancies. The conclusion is that B2 =E, lies in the range 7.3 + 1 eV. This is still lower than the theoretical value of 11 eV computed by COULSONet al.(“), but is reasonably consistent with the value of E, found when the vacancies are in di-vacancy form, by COULSON and POOLE. The decrease in ES on formation of di-vacancies is consistent with the vacancyvacancy attraction mentioned in Section 3. Equation (6) may also be applied to ranges of somewhat higher dose where Klni has a stationary value. Consideration of the dose ranges 2-4x 10” n cmW2 at 200°C and 2-6x 10” n crnm2 at 250°C in Fig. 5 leads to values of E, of 2 eV in regions where the vacancy concentrations are about 2-3 %. This suggests that even quite widely separated vacancies have strong interactions, as would indeed be expected from the suggestion of KELLY that the introduction of vacancies causes an overall change in the n-bonding. ~YNOLDS

From equation (6) we aIso find j!$ =2 eV where

Ei is the energy of an interstitial in solution. Ei cannot be less than E,, but it might not be very much greater, as the non-nucleated interstitials are in small groups. Thus AZ12 N 3, as suggested above. This value of K1 is also consistent with the observed changes in Young’s modulus E. Measurements at 200°C and 250°C cannot be used to determine ni since at these temperatures the interstitial populations soon reach the value required to

LATTICE

PARAMETER CHANGES

pin all dislocations and raise the shear modulus to the single crystal value (cf. REYNOLDS). At 35O”C, the

relative

increase

of modulus

$--

1

saturates at 75 %, in a region where K1 Q=O*O!~%. GOGGIN~'~) has recently shown by means of low temperature electron irradiations that a value of j!$- 1 of 35% is reached 0

when ni=O*Ol %. The

increase with dose of f-

1 is roughly linear up to n 100% and it therefore follows that Kim4. Although this value is in satisfactory agreement, it is less conclusive since the influence of the state of aggregation of the interstitials is unknown. 5. SUMMARY AND CONCLUSIONS

It is concluded that the c-spacing change per non-nucleated interstitial is in the range Z-1 times the concentration, and the a-spacing change per vacancy is broadly consistent with the value -0.14 times the concentration suggested by KELLY, but there is some indication that this figure is too small at low doses. The overall volume increase he suggested is not, however, observed, apparently because vacancies cause a c-spacing contraction in their own right of 0.42 times the vacancy concentration. The value given in an earlier paper for the energy of a vacancy of 5.5 4 1 eV becomes 7-3 + 1 eV when the c-spacing is corrected for a vacancy contribution. The c-spacing change per vacancy is an upper lit obtained from the data, so that the correction to E, may be overestimated. The lattice parameter changes, elastic modulus changes and build-up of stored energy thus appear to arise from unequal populations of un-nucleated

IN IRRADIATED

GRAPHITE

287

interstitials and vacancies. The form of these defects required to produce the observed X-ray line shapes has not yet been found. Finally, the activation energy of motion of a single vacancy within a given basal plane has a lower limit of 2.5 eV, but this must be corrected for a considerable vacancy-vacancy attraction. REFERENCES 1. SIMMONS J. H. W. and REYNOLDS W. N., Proc. Inst. Metals Symposium on Uranium and Graphite, p. 75 (1962). 2. GOGGIN P. R., HENSON R. W., PERKSA. J. and REYNOLDS W. N., Carbon 1, 189 (1964). 3. LIPSON H. and WILSON A. J. C., J. 5%. In&. 18, 144 (1941). 4. REYNOLDS W. N. and THROWER P. A., Phil. Mag.

12, 573 (1965). 5.

SIMMONS J. H. W., KELLY B. T., NETTLEY P. T. and REYNOLDS W. N., 3rd U.N. Conf. on the Peaceful

Uses of Atomic Energy, Paper 163 (1964). 6. KELLY B. T., Second S.C.I. Conference on Carbon and Graphite (1965). 7. NELSON D. P. and RILEY H. L., PYOC.Phys. Sm.

8.

(London) 57,477 (1945). SPENCE G. B., Proceedings of the Fifth Carbon Conference, Vol. 2, p. 531. Pergamon Press, Oxford

(1963). 9. ~LUCHERY M., edition Damage in Reactor Materials, p. 523 I.A.E.A., Vienna (1963). 10. KELLY B. T., Nahtre 207,257 (1965). 11. COULSON C. A., HERRAEZ ti. A.;Lyk M., SwTos E. and SJZNENTS.. Proc. Rev. Sot. A274. 461 (1963). AGRANOVICH$. M. andSEMmov L: P., j. N&lea* Energy (.4/B) 18,141 (1964). 13. BAKER C. and KELLY A., Nature 193,235 (1962). 14. BELL J. C., BRIDGEH., COTTRELLA. H., GFZEENOIJCZH G. B., REYNOLDS W. N. and SIMMO~ J. H. W.,

12.

Phil. Trans. Rov. Sot.. London 254. 361 (19621. 15. COUISONC. A. and kooul M. ~6, Carbon k 275 (1964). 16. bOLDS W. N., Phil. Mug. 11, 357 (1965). 17. GOGGIN P. R., Second S.C.I. Conference on Carbon and Graphite (1965).