Irradiation induced dimensional changes in graphite: The influence of sample size

Journal of Nuclear Materials 420 (2012) 241–251

Contents lists available at SciVerse ScienceDirect

Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat

Irradiation induced dimensional changes in graphite: The influence of sample size M.V. Arjakov a, A.V. Subbotin a, S.V. Panyukov b,⇑, O.V. Ivanov b, A.S. Pokrovskii c, D.V. Kharkov c a

Scientific and Production Complex Atomtechnoprom, Moscow 119180, Russia P.N. Lebedev Physics Institute, Russian Academy of Sciences, Moscow 117924, Russia c State Scientific Center of the Russian Federation – Research Institute for Nuclear Reactors, Dimitrovgrad 433510, Russia b

a r t i c l e

i n f o

Article history: Received 15 April 2011 Accepted 7 October 2011 Available online 12 October 2011

a b s t r a c t Dimensional changes in irradiated anisotropic polycrystalline GR-280 graphite samples as measured in the parallel and perpendicular directions of extrusion revealed a mismatch between volume changes measured directly and those calculated using the generally accepted methodology based on length change measurements only. To explain this observation a model is proposed based on polycrystalline substructural elements – domains. Those domains are anisotropic, have different amplitudes of shapechanges with respect to the sample as a whole and are randomly orientated relative to the sample axes of symmetry. This domain model can explain the mismatch observed in experimental data. It is shown that the disoriented domain structure leads to the development of irradiation-induced stresses and to the dependence of the dimensional changes on the sizes of graphite samples chosen for the irradiation experiment. The authors derive the relationship between shape-changes in the finite size samples and the actual shape-changes observable on the macro-scale in irradiated graphite. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Reactor-grade graphites used in nuclear engineering as neutron moderators and structural materials, are subject to damage due to high dose irradiation. Irradiation-induced physical, mechanical and dimensional effects significantly affect performance characteristics of graphite. Detailed analyses of irradiation effects such as shape-changes, irradiation dose evolution of elastic moduli, thermal expansion coefficient, temperature and electrical conductivities, irradiation creep, etc., point to an essential role of graphite morphology in all irradiation effects [1–5]. The morphology of graphite is formed by two main components, viz., a filler and a binder, as well as microcracks and technological pores [1,4,5], which are to a large extent determined by the technology of graphite fabrication. The binder has a fine crystalline structure whereas the structure of the filler is hierarchically based on microcrystallites which are more or less perfect hexagonal crystal lattices of two types – ABAB. . . and ABCABC, the fraction of which may differ [6]. Between the crystallite having the hexagonal structure and the macrographite as a whole, there is a succession of spatial structural levels of organization of crystallites, binder, and ensembles of microcracks that form the morphology of graphite [5,7]. These multiscale structures determine the symmetry of elastic properties of reactorgrade graphite and its irradiation-induced shape changes. ⇑ Corresponding author. E-mail addresses: [email protected] (A.V. Subbotin), panyukov@ lpi.ru (S.V. Panyukov), [email protected] (O.V. Ivanov). 0022-3115/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnucmat.2011.10.009

The elasticity of the hexagonal crystallographic lattice is described by five independent constants c11, c12, c13, c33 and c44 [8]. It has been established that reactor-grade graphites both extruded and pressed have properties of a transversely isotropic medium [1,9], and to describe their elasticity five independent variables are also needed, e.g., E\, Ek, Gk, m\, mk – Young’s and shear moduli as well as Poisson’s ratio in the isotropy plane and normal to it, respectively. The irradiation-induced shape-change of graphite has similar symmetry, allowing the introduction of the diagonal gradient tensor of an irradiation-induced shape-change (see below). For extruded graphites the axis of the axial symmetry is dictated by the direction of extrusion (k direction) whereas the directions perpendicular to it (\ directions) define the plane of isotropy [1]. The well-known reactor-grade graphite GR-280 [5] belongs to this group of graphites. In the present paper we discuss the experimental results obtained by studying irradiation-induced shape-changes of graphite GR-280. The notations used in this paper are collected in Appendix A. 2. Description of experiment Several hundred cylindrically shaped graphite samples 6 and 8 mm in diameters and 50 mm long with parallel (k) and perpendicular (\) extrusion directions were cut out of the bulk material. These samples were irradiated at temperatures of 460 ± 25 °C, 550 ± 25 °C, 640 ± 25 °C in the research fast-breeder reactor BOR-60 up to the dose levels of 3  1026 neutrons/m2 at the energies En > 0.18 MeV. Measurements were carried out at the dose

242

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

intervals 0.6  1026 neutrons/m6. The samples were measured both in the longitudinal and transverse directions at the 4.5 mm step along the cylinder axes. The diameters between 16 equidistant pairs of points over the circle were measured and used to characterize the cross-sections of the samples. All the data are presented as functions of neutron doses in the spectrum of reactor BOR-60. The dose conversion to the neutron spectra of thermal reactors is performed in Ref. [10]. The conversion of BOR-60 neutron fluxes into equivalent neutron fluxes of thermal reactors requires a special consideration and will be given in a separate paper. Numerous measurements of shape changes of samples were made and the two observations were discovered that require an additional explanation:  The initial circular cross-section of a sample under the irradiation acquires an increasingly pronounced elliptical shape and orientations of ellipses vary randomly along the sample on the scale of 6 mm.  Results of the calculations of the relative volume changes using the traditionally accepted methodology [1–3]

ðDV=VÞT ¼ ðDL=LÞk þ 2ðDL=LÞ? ; k

ð1Þ

\

(where (DL/L) and (DL/L) are relative length changes of the samples with parallel and perpendicular extrusion directions) increasingly diverge with irradiation dose from the results of the direct measurements of the relative volume changes. As we show below, the reason for such a mismatch is that expression (1) only accounts for the relative elongations of samples to relative volume change while neglecting changes in cross-sectional areas. We present the interpretation of the above facts in the framework of a model evolved by authors that takes into account the hierarchic morphology of graphite [11]. Elastic properties of graphite are determined by hierarchy of irradiationinduced affine (with constant deformation gradient tensor) shape-changing regions of different spatial scales. We introduce the concept of the ‘‘domain’’ as an affinely deformed region of the intermediate size (5 mm) between the grains of graphite and macro-graphite. The domain has the same symmetry (but different values) as the irradiation-induced shape change of macrographite, but nevertheless, its symmetry axis does not coincide with the axis of extrusion. We use this concept to develop a formalism that allows combinding all the above facts. 3. Experimental data processing In this section we describe the geometry of graphite samples before and after irradiation and introduce the main physical concepts to describe irradiation-induced changes of their shapes. Using these concepts we deduce the relation between the local characteristics of a domain structure and experimentally observed irradiation-induced shape-changes of graphite samples. 3.1. Domains Graphite is a polycrystal and like each polycrystal it consists of small microcrystallites. The most important distinction of graphite from common polycrystals is the presence of mesoscale threedimensional objects having the size n J 0.5 cm (for brevity called domains, see Fig. 1). Each domain consists of many microcrystallites, that posses the properties of a transversely isotropic medium and can be characterized by individual orientation of the symmetry axis. In the preparation of graphite the local orientations of anisotropy axes of different domains deviate from their average

(a)

(b)

Fig. 1. Graphite is a polycrystal consisting of domains (shown schematically by hexahedrons) of typical size n. Each domain can be characterized by the local direction of anisotropy (shown by small arrows). (a) In case of a regular structure all domains have the same orientations. (b) In case of an irregular structure these directions are random vectors with preferred orientation (shown by large arrow), determined by extrusion direction of the sample. Radiation-induced shape-changes of randomly oriented domains lead to a rise in cross-domain stresses with irradiation dose.

orientation due to preparation technology. The changes in the shapes of domains under irradiation are restricted by their local environment since theses changes are constrained by neighboring domains of different orientations. This leads to the development of internal stresses in the bulk material, see Fig. 1b. These stresses result in an additional dimensional effect in graphite. The knowledge of the physics of a dimensional effect is extremely important for understanding the behavior of bulk graphite materials whereas for the most part the experimental data are obtained for finite size graphite samples. Because of the transversal isotropy of a domain, its deformation can be described by two principal strains: along the principal axis of axial symmetry, (Dl/l)k, and perpendicular to it, (Dl/l)\. These strains determine the irradiation-induced gradient tensor b F 0, which is diagonal in domain coordinate system and has axes that are directed along the main axes of deformation:

0 B b F0 ¼ @

1 þ ðDl=lÞ?

0

0

1

0

1 þ ðDl=lÞ?

0

0

0

1 þ ðDl=lÞk

C A:

ð2Þ

3.1.1. Regular structure Before irradiation a sample has the shape of a cylinder of length L and diameter d. After irradiation the shape of the cylinder will change due to irradiation-induced shape-changes of randomly oriented domains. We shall first consider the case when the orientations of axial symmetry axes of all the domains in the sample coincide with the extrusion direction. Thus, each sample may be characterized by the orientation angle x between the direction of the axial symmetry and the axis of the sample, see Fig. 1a and Fig. 2a. Although the approximation of the regular texture can be only approved for a negligible disorder in the domain orientation, it is usually used in literature. Relative deformations of samples along i = x, y, z-directions in the coordinate system related to the sample are described by the deformation gradient tensor b F ideal ; ðDLi =Li Þideal ¼ ð b F ideal Þii  1. In case of the regular texture the deformation tensor has the form of:

b b 1 ðxÞ b 1 ð xÞ b F ideal ¼ R F0R 1

ð3Þ

b 1 ðxÞ is a tensor of rotation by the orientation angle x: where R

0

1

0

0

0

sin x

cos x

1

C b 1 ð xÞ ¼ B R @ 0 cos x  sin x A:

ð4Þ

We conclude that after irradiation the sample becomes an oblique elliptic cylinder.

243

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

(a)

(b)

d

ez

ez

L0

ey

ex

2

ey

d( )

dmax =d ¼ 1 þ ðDl=lÞ? ; dmin =d ¼ 1 þ ðDl=lÞ? cos h þ ðDl=lÞk sin h

L1

ð5Þ

b u ; hÞ ¼ R b 2 ðu Þ R b 1 ðhÞ and the tensor of rotation in the plane Here Rð 0 0 b 2 ðu Þ is defined by expression of the cross-section of the sample, R 0

 sin u0 cos u0 0

0

ð8Þ 2

2

dmin dmax

ð7Þ

L2

3.1.2. Structure with random domain orientations The orientation of a domain axial axis in a cylindrical sample can be characterized by two angles: the azimuth angle u0 in the crosssectional plane (ex, ey) and the polar angle h between the domain axial direction e0z and the cylinder axis ez, see Fig. 2b. These angles  0 and x ¼  vary randomly in space, and their average values, u h depend on how the sample was cut out of a bulk graphite array. Below we shall consider the cases of both parallel (with the average axial direction along the cylinder axis, x = 0°, superscript k) and perpendicular (with the average axial direction perpendicular to the cylinder axis, x = 90°, superscript \) extrusion directions. The macroscopic deformation of a sample is described by the deformation gradient tensor b F defined in the coordinate system of the sample. Deformations of an elastic body are usually described by Lagrangian strain tensor b E which is related to the gradie ^ ent tensor as b E ¼ ðb Fb F  1Þ=2 where f   means transposition. For the axial axis rotated by angles (u0, h) relative to the axis of the cylinder, the gradient tensor takes the form of

cos u0 B B b 2 ðu Þ ¼ B sin u R 0 0 @ 0

2

Thus, the initially circular cross-section of the diameter d after irradiation takes an elliptical shape, see Fig. 2c. The maximal dmax and minimal dmin diameters of the ellipse are given by equations:

Fig. 2. (a). Before irradiation the sample is cylindrically shaped and has circular cross-sections. (b) Two different coordinate systems are introduced:   (ex, ey, ez) is attached to the sample with axis ez along cylinder axis, e0x ; e0y ; e0z relate to a domain with axis e0z along the direction of the domain axial symmetry. (c) After irradiation the sample takes a ‘‘crankshaft’’ shape with elliptical cross-sections, characterized by maximal dmax and minimal (dmin)k diameters measured along the main axes rotated to random angles (u0)k in the cross-sectional planes, where k is the index of the cross-section.

0

2

L3

ex

b b u ; hÞ b b 1 ðu ; hÞ F0R F ¼ Rð 0 0

2

d ðuÞ ¼ dmax cos2 ðu  u0 Þ þ dmin sin ðu  u0 Þ

(c)

ð9Þ

According to Eq. (8) the maximal diameter dmax is identical for all segments of the sample while the minimal diameter dmin varies randomly in different segments. 3.3. Relative elongation We use L0k to denote the distance between segments before irradiation. After irradiation the length Lk of the kth segment of the crankshaft cylinder can be found as the projection of the vector b F ðL0k ez Þ on the direction ez of unit vector along the axis of the cylinder: 2

Lk ¼ L0k ½1 þ ðDl=lÞ? sin h þ ðDl=lÞk cos2 h

ð10Þ

The total length of the sample is found as the sum of the lengths of all its segments

L¼

N X

Lk

ð11Þ

k¼1

From Eq. (11) we derive the relative change of the length 2

ðDL=LÞ ¼ ðDl=lÞ? sin h þ ðDl=lÞk cos2 h ¼ ðDl=lÞk þ e

ð12Þ

Here (DL/L) is measured as a relative elongation of the whole length of the sample, while (Dl/l)\ and (D l/l)k are corresponding components of the principal strain, see Eq. (2). We use the notation    for averaging over all the cross-sections of the sample:

e ¼

"   # N 1X Dl Dl 2 ek ¼  sin h; N k¼1 l ? l k

ð13Þ

and ek is the flattening factor of kth cross-section:

ek ¼

dmax  ðdmin Þk Ddmax  ðDdmin Þk ¼ d d

ð14Þ

1

C C 0 C: A 1

3.4. Volume change

ð6Þ

The orientation of the gradient tensor b F of a domain randomly varies in space on the length of about the domain size n and remains constant at that length scale. As a result of such variations in b F the sample is deformed after irradiation taking a ‘‘crankshaft’’ shape, see Fig. 2c. The bending of the cylindrical sample at the domain junctions is described by the effects of higher order in angle deviations and will be neglected below, although this effect is observed experimentally. Consider the shape of the crankshaft cylinder in more detail: 3.2. Transverse dimensions Before irradiation the cross-section of the sample has a circular shape with angular dependence of the diameter d0(u) = (dcosu)ex + (dsinu)ey. After irradiation, the shape d(u) of the cross-section is determined by the projection of the vector b F d0 ðuÞ on the cross-sectional plane (ex, ey). By calculating the square of the cross-sectional diameter in the deformed sample we find its dependence on the azimuth angle u:

The volume of the sample can be found as the sum of the volumes of all its segments, see Fig. 2c:

V¼

N X k¼1

p 4

dmax ðdmin Þk Lk ’

p 4

dmax dmin L

ð15Þ

Here p4 dmax ðdmin Þk is the area of the kth cross-section. Using this equation we receive the relative change in the volume of such a ‘‘free’’ sample

ðDV=VÞF ¼ ðDdmax =dÞ þ ðDdmin =dÞ þ DL=L ¼ 2ðDdmax =dÞ þ DL=L  e

ð16Þ

According to Eqs. (8) and (12) the volume change can be expressed via the principal strains of the domain:

^ ¼ 2ðDl=lÞ þ ðDl=lÞ ðDV=VÞF ¼ Trð b F 0  1Þ ? k

ð17Þ

where Tr(  ) is the sum of diagonal elements. Eq. (17) demonstrates the self-consistency of our definition of principal strains since it reproduces the experimentally measurable volume change.

244

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N  1 X 2 ak þ bk þ c2k ; N k¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   4 2 1 2 2 2 bk þ c2k  dmax ; dmin ¼ ak  k 3 3 3 1 ck ðu0 Þk ¼ arctan ; 2 bk

3.5. Experimental data processing

2

dmax ¼

Below we shall calculate two principal strains (Dl/l)k and (Dl/l)\ through the irradiation-induced change of experimentally measurable maximal diameter and length of a sample:

ðDl=lÞk ¼ DL=L  e; ðDl=lÞ? ¼ Ddmax =d

ð18Þ

Here we used Eqs. (12) and (8). In our experiments the set of the diameters (di)k of the samples is measured along N equally separated cross-sections k for M equidistant angles ui = 2pi/M. Each of those sections can be characterized by the minimal diameter of the ellipsis (dmin)k and the angle (u0)k of its rotation in the crosssectional plane, while the maximum diameter dmax of the ellipse is the same for the whole sample. The values of those parameters are found from the experimental set of the diameters {(di)k} via minimizing the mean squared deviation

r2

N X M h i2  1 X 2 2 ¼ di  dk ðui Þ k NM k¼1 i¼1

ð20Þ ð21Þ ð22Þ

where ak are the average squared diameters of the samples and bk,ck are their cosine and sine Fourier transforms for kth cross-section:

ak ¼

M   1 X 2 di ; k M i¼1

ð23Þ

bk ¼

M   2 X 2 di cosð2ui Þ; k M i¼1

ð24Þ

ck ¼

M   2 X 2 di sinð2ui Þ: k M i¼1

ð25Þ

ð19Þ 4. Interpretation of data

relative to dmax and the set of N individual diameters (dmin)k and 2 angles (u0)k for each of N sections. Here dk ðuÞ is given by Eq. (7) with the respective parameters (dmin)k and (u0)k. The solution of those minimum conditions is quite cumbersome and rendered in Appendix B. Here we show the result

(a) 10

In this section we present the observed dependences of geometrical sizes of samples on fluence U. Below we shall show only the curves averaged over a series (a = A, B, C) of samples with the same extrusion direction (k and \), diameter (6 and 8 mm) and the

(b)

( L/L)||

8

8

6

6

4

4

2 x 10

0

1

2 C6C8

-2

A8

2

B8 A6 3

0

B6

26

B6 A6B8

( L/L)

10

C6 C8

1

A8

3

2 x 1026

-2

Fig. 3. Neutron fluence U dependence of relative length changes (in %) averaged over all samples (adx) of assembly a = A, B, C with diameter d (6 mm – thin lines, 8 mm – thick lines) of parallel (k) (a) and perpendicular (\) (b) extrusion directions x (in neutrons/m2, at En > 0.18 MeV).

(a)

(b) o

25

A (460 C) 6 mm 15

V/V

6 mm

25

A

20

F

10

T

10

0

5 1

x 1026

0

20

A

10

C

0 1

2 x 10 2

C

5

3

2

B

8 mm

15

15

5

-5

(c) B

26

3

-5

1

2

3 x 1026

Fig. 4. (a) Relative volume change DV/V (in %) as a function of fluence U (in neutrons/m ) in assembly A at temperature T = 460 ± 25 °C. Two solid lines (F) show our model predicted expession (Dl/l)k + 2(Dl/l)\, two dash-dotted lines denote experimentally measured DL/L + 2(Dd/d) in 6 mm samples of both parallel (k) and perpendicular (\) extrusion directions, single dotted line (T) shows the traditionally used combined expression (DL/L)k + 2(DL/L)\ calculated using the experimental data for those two extrusion directions. Since similar curves are obtained for samples from assemblies A, B, C of all diameters d, in figures (b) and (c) we shall not differentiate between different extrusion directions in samples having diameters d = 6 mm and 8 mm, respectively. We only show the single dependence of DV/V = DL/L + 2(D d/d) on U for assemblies A,B and C (solid lines, at the average temperatures of 460, 550 and 640 °C, respectively), and also plot predicted in [11] asymptotic dependence of these curves K[(Dl/l)\  (Dl/l)k]2 (dashed lines, K = 7 for all curves) at high fluence U.

245

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

average irradiation temperature T. Samples of assembly A were exposed to irradiation in reactor Bor-60 at the temperature T = 460 ± 25 °C, samples of assemblies B and C were irradiated at the temperatures of 550 ± 25 °C and 640 ± 25 °C, respectively.

Using Eq. (12) we find the following expression for combined formula (1) traditionally used to describe the volume change DVT under irradiation:

4.1. Length changes

where e and e are flattening factors for the two extrusion directions of the samples. We observe that Eq. (28) gives results that differ (DVT < DVF) from the actual relative change in the sample volume given by Eq. (26) (see the dotted line in Fig. 4) due to the ellipticity, e > 0, of sample cross-sections. We also conclude that the traditional expression (28) cannot be used to calculate the actual volume changes in graphite samples. The replacement of the actual volume change DVF by DVT can be safely applied only to the case of the regular texture of graphite, see Fig. 1, as well as to macro-scale samples having disordered structures. In finite size samples the disorder in the domain orientations (Fig. 1b) leads to a significant difference between the commonly used (Eq. (28)) and actual (Eq. (26)) volumes of samples.

ðDV=VÞT ¼ ðDV=VÞF  ð2e?  ek Þ < ðDV=VÞF ; k

Most of the information on the irradiation-induced shapechanges of graphite is usually generated using the data describing changes in the lengths L of the samples with a neutron fluence U. For example, the volume change DV is usually estimated by formula (1) using the data on length variations in samples of parallel, (DL)k, and perpendicular, (DL)\, extrusion directions. The dependence of relative elongations (DL/L)k and (D L/L)\ of the studied samples on fluence U is shown in Fig. 3. As can be seen, the change in sample size strongly depends on its shape because of different constraints imposed on domains changing their shapes in the samples having different diameters d and extrusion directions x. The irradiation-induced shape-changes of domains are weakly restricted in samples with perpendicular extrusion direction having the least diameter d = 6 mm while the domains experience the strongest restrictions in samples having parallel extrusion direction and the diameter d = 8 mm.

ð28Þ

?

4.3. Parameters of domains

Fig. 4 shows the relative volume change vs fluence U. The evolution of a sample volume vs the fluence U might be calculated in two different ways: viz., using variations in macroscopic sample sizes (Eq. (16)) or changes in sizes of domains (Eq. (17)). As one can see in Fig. 4a, both the ways give essentially the same result:

The orientation of the domain anisotropy axis is determined by two angles: the azimuthal angle u0 and the polar angle h, see Fig. 2b. Below we estimate the parameters of the domains basing on the knowledge of the statistics of those angles in the samples under consideration. The azimuthal angle u0 determines the orientation of the axial axis in the cross-sectional plane. The angles (u0)k are random variables of the cross-section number k that correlate at a distance of about the domain size n. In general, the amplitude of angle fluctuations might differ in different samples. In order to exclude the effect of such amplitude variations consider the ratio

ðDV=VÞF ¼ DL=L þ ðDdmax þ Ddmin Þ=d ¼ ðDl=lÞk þ 2ðDl=lÞ?

r¼

4.2. Volume changes

ð26Þ

We also observe that the sample volume slightly depends on extrusion direction x. Therefore, the values of principal strains (Dl/l)k and (Dl/l)\, by which the volume change is expressed, can be considered to be an inherent characteristic of graphite, weakly depending on the sample extrusion direction x. In contrast to the volume change (see Fig. 4), the length L and the diameter d of the sample (see Fig. 3) strongly depend on the sample size. These values depend not only on principal strains but also on the flattening factor e, because of different constraints imposed on domains changing their shapes under irradiation in small size samples having different extrusion directions. We observe volume variations to be depressed in samples of larger diameters d = 8 mm because of stronger constraints imposed on shape-changing domains in larger samples. The dependence of a volume change on fluence U is highly nonmonotonous: at low fluences U < Umin graphite shrinks at dose U, and dilates at high fluences U > Umin. The dotted lines in Fig. 4a and b show the dependence

K½ðDl=lÞ?  ðDl=lÞk 2

ð27Þ

at the fit coefficient K = 7. As one might see this dependence describes adequately well at high U > Umin an asymptotic behavior of volume changes in graphite independent of temperatures, diameters and extrusion directions. Although such an asymptotic behavior was theoretically predicted in Ref. [11], the origination of the universality of the coefficient K in Eq. (27) remains unclear. The knowledge of the asymptotic behavior of irradiation-induced volume changes is important both to understand the deformation mechanisms undergoing in graphite under irradiation and to extrapolate experimental data in regions of higher irradiation doses unattainable in the current experiments.

2rg Nr n

ð29Þ

of two sums for each sample

rg ¼

N N X X

½ðu0 Þk  ðu0 Þj 2 ;

rn ¼

k¼1 j¼kþ1

N 1 X

½ðu0 Þk  ðu0 Þkþ1 2

ð30Þ

k¼1

The first sum includes all pairs of cross-sections (an analog of a gyration radius), while the second one refers only to the nearest pairs (an analog of end-to-end distance of a polymer chain). Since each of those terms has a quadratic form of angles (u0)k, their ratio (29) does not depend on the amplitude of fluctuations and only encrypts the information on a domain size. If there are no correlations between angles uk this ratio tends to 1 in the limit of large N ? 1 when we can substitute [(u0)k  (u0)j]2 in Eq. (30) for its average, ½ðu0 Þk  ðu0 Þj 2 ¼ 2u20 . In order to determine the dependence of the parameter r on the domain size n, consider the case when the group of l consecutive angles unl+j, j = 1, . . ., l has the same value ^ n for integer n,l, whereas the values of u ^ n for each group n of u are independent random variables. At N ? 1 substituting [(u0)k  (u0)j]2 in Eq. (30) by its average (0 for k and j belonging to the same group and 2u20 otherwise) we find that the parameter r equals the length l of the group. Therefore, we can identify

Table 1 Average domain size n = rL0k in samples having parallel (k) and perpendicular (\) extrusion directions. Assembly (°C)

A (460)

Diameter (mm)

6

8

B (550) 6

8

C (640) 6

8

rk r\

1.2 1.6

1.2 1.4

1.3 1.4

1.1 1.3

1.1 1.3

1.4 1.5

246

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

n = rL0k as an average domain size along the direction of the cylinder axis, L0k is a distance between different sections of the sample, see Fig. 2. In Table 1 we present the results of calculating the parameter r for the samples of parallel (rk) and perpendicular (r\) extrusion directions averaged over all such samples. Since the distance between different cross-sections of a sample is L0k = 4.5 mm, we estimate n ’ 6 mm as a typical domain size. As is shown in Table 1, the samples with the perpendicular extrusion direction have r higher than samples having the parallel extrusion direction (r\ > rk). The larger longitudinal size of domains with the perpendicular extrusion direction relates to a less restricted environment of such domains that more weakly constrains the irradiation-induced evolution of a domain shape. The orientation of the domain axial axis relative to the global axis of the cylinder is described by the polar angle h (see Fig. 2c). Combining Eqs. (13) and (18) we find the average value 2

sin h ¼

e Ddmax =d  DL=L þ e

ð31Þ

via analyzing the experimentally acquired data we find (see Fig. 5) 2 the dependence of sin h on fluence U for samples having different diameters d = 6, 8 mm and extrusion directions x (k and \). The question is whether the axial axis might rotate as a result of irradiation-induced shape-changes of domains. Inspecting Fig. 5 we come to the conclusion that this is not the case, and the orientation angle h does not vary with the fluence U. The statistics of this angle only weakly depend on temperature T during irradiation, and thus are mainly determined by fabrication conditions. Special instances of importance are demonstrated by 6 mm samples extruded in the perpendicular direction. They are highly inhomogeneous and contain areas where irradiation-induced shape-changes are slightly restricted by their environment. This heterogeneity of a sample is 2 responsible for an apparent rise of sin h above 1 with an irradiation dose (when the samples become more heterogeneous) in Fig. 5. This effect is not available in samples of 8 mm diameter which supports our previous estimation of the domain size to be n ’ 6 mm. 4.4. Principal strains The neutron fluence U dependences of principal strains (Dl/l)k and (Dl/l)\ in all the studied samples are illustrated in Fig. 6. Below we use those data to discuss the dependence of the strains on the shapes and sizes of samples while the dose dependence will be studied later. The dimensional effect is most pronounced in case of a parallel principal strain (Dl/l)k, see Fig. 6a. At a low fluence U, the domain always shrinks in the parallel direction. The shrinking amplitude (Dl/ l)k is maximal in samples extruded in the perpendicular direction

(a)

2

(b)

since the domains in those samples are less restricted in growing as dose U increases with respect to samples having a parallel extrusion direction. This effect is strongly dependent on a sample size. While the strain (Dl/l)k dramatically changes with the extrusion direction x in 6 mm samples, the value (Dl/l)k changes rather weakly in 8 mm samples. This conclusion is in agreement with the above estimated domain size n ’ 6 mm; and we expect the dimension effect to be low only in samples having the diameter d > 8 mm. We assume that the variation in (Dl/l)k with the fluence U is related to the corresponding growth of microcracks because of the radiation induced shape changes in microcrystallites that are hampered by internal stresses in graphite. Such a constrained effect of internal stresses is minimal in samples extruded in the perpendicular direction. A similar behavior is also observable in case of perpendicular principal strain (Dl/l)\. At low U the value of (Dl/l)\ weakly depends on fluence U: A domain shrinks in the perpendicular direction at low temperatures T [ 600 °C and extends at high temperatures of > 600 °C. The principal strain (Dl/l)\ grossly varies with the extrusion direction x only in 6 mm samples, and does not depend on x in 8 mm samples. The amplitude of the (Dl/l)\ variation with the neutron fluence U is maximal in samples least constrained in the perpendicular principal direction that are subjected to parallel extrusion and minimal in samples extruded in the perpendicular direction. The amplitude of the anisotropy in principal strains increases with a rise in the irradiation temperature T. We expect this effect to be related to the evolution of an ensemble of anisotropic microcracks and a rise in microcrack volumes with temperature T. An increase in microcrack volumes with a temperature rise is responsible for the corresponding increase in the sample volume that is clearly observable in Fig. 4. At the high temperature (T = 640 °C) the contribution made by microcracks is so large, that at low doses it prevents the shrinkage in the sample length in the parallel principal direction, (Dl/l)k. Although there is always a region of the initial volume shrinking since at low U the domain shrinks in the perpendicular principal direction, (Dl/l)\. Principal strains (Dl/l)k and (Dl/l)\ demonstrate a wide variation of non-monotonous dependences on fluence U, see Fig. 6. To understand the reason for this behavior we can present those strains in the form of:

"     # Dl 1 DV 2 Dl Dl ;  ¼  l k 3 V 3 l ? l k " #       Dl 1 DV 1 Dl Dl þ ¼  l ? 3 V 3 l ? l k

ð32Þ ð33Þ

The non-monotonous dependence of these strains on U is related to the respective non-monotonous dependence of the volume, see

(c)

Fig. 5. Dependence of sin h on fluence U (in neutrons/m2) for samples of assemblies a = A, B, C having different average temperatures. Symbols dx correspond to samples of diameters d = 6 and 8 mm and extrusion direction x (k and \).

247

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251 B6

l/l

2

0

1

B6||

15

A6 II

C6

x 1026

2

3

B8|| B8

l/l

A6|| A8||

B6

10

A8 A6

C8

-2

C8||

C6||

-4

C8||

B6||

B8|| A6||

C6||

5

C6

B8 A8||

C8

-6

x 1026

0

A8

2

1

3

Fig. 6. Dependence of principal strains (Dl/l) k and (Dl/l)\ (in %) on neutron fluence U (in neutrons/m2) in samples (adx) of assemblies a = A, B, C with diameter d (6 mm – thin lines, 8 mm – thick lines) and extrusion direction x (k – solid lines, \ – dash lines). The designations are the same as in Fig. 3.

Fig. 4. The dependence of the difference in principal strains, (Dl/l)\  (Dl/l)k, on fluence U is monotonous and can be extrapolated by a quadratic function (as it was first predicted in [11]), see Fig. 7a: 2

ðDl=lÞ?  ðDl=lÞk ’ bl U

Table 2 Coefficients bl and be in Eqs. (34) and (35) for different extrusion directions x, diameters d and assemblies a.

ð34Þ

With an increase in the neutron fluence U the shapes of cylindrical samples become more and more crankshaft-like. The amplitude of this effect is characterized by the average flattening factor e of the elliptic cross-sections of the sample, Eq. (13). via inspecting the experimental data we find that e grows quadratically with fluence U: 2

eðUÞ ’ be U

ð35Þ

Dependence (35) is illustrated in Fig. 7b where we plot e for different assemblies, diameters and extrusion directions in samples as a function of U2. Coefficients bl and be for different extrusion directions x, diameters d and assemblies a are given in Table 2. The factor bl monotonously grows with temperature T under irradiation (assemblies A to C). The difference in bl in samples extruded in parallel, (bl)k and perpendicular, (bl)\ extrusion directions decreases with an increase in the sample diameter, however, a gap between these values exists even for the maximal diameter d = 8 mm. Therefore, we conclude that in our experiment even the thickest samples have elastic properties that differ from the properties of bulk graphite. The coefficients be increase with a rise in the temperature T. According to Eq. (13) the two coefficients bl and be are proportional to each other: 2

be ¼ sin hbl

ð36Þ

B6||

( l/l) - ( l/l)||

16

B8 B8||

B

C

A

B

C

(bl)k  1054 (bl)\  1054 (be)k  1054 (be)\  1054

1.8 0.7 0.4 0.9

2.5 1.0 0.7 1.2

2.5 1.2 0.5 1.5

1.7 2.0 0.7 1.4

2.0 2.2 0.4 1.6

2.3 2.8 0.4 2.0

C8

12 C8|| C6|| B6 A6

C6

(b)

B8

12 A8

C8

B6

8

A6 C6

A8||

B6||

A6|| B8||

4

4

C6|| x

0

8 mm

Higher be values of samples extruded in the perpendicular direction 2 (x = p/2) are related to higher values of sin h with respect to the case of parallel extrusion direction (x = 0). We expect that for bulk graphite there should not be any difference between the two ways of measurements of the volume changes, DVT = DVF, Eqs. (26) and (28). At a first glance the simple quadratic dependence (34) looks very surprising, taking into account the complex non-monotonous dependence of volume changes on the fluence U, see Fig. 4: at a low dose U < Umin a sample shrinks while at high doses it dilates. It is commonly assumed that appearance at high fluences U > Umin of growing branches in volume and elongation dependences is related to the formation of a new set of microcracks in graphite at moderate fluences U > Umin. The simple dependence (34) demonstrates that actually nothing special happens to graphite above the cross-over fluence Umin. In Ref. [11] we propose an alternative explanation of such a non-monotonous dose dependences to be the result of the interplay of two constantly acting physical phenomena: irradiation-induced changes in a shape and variations in elastic moduli of crystallites in polycrystal graphite. At low doses,

A6|| A8||

A8

8

6 mm A

= (dmax- dmin )/d

(a)

Diameter d Assembly a

0

2

4

6

1052

8

C8||

0

0

2

4

x

6

10

52

8

Fig. 7. Dependence (in %) (a) of principal strain difference (Dl/l)\  (Dl/l)k and (b) of the flattening factor  e on squared fluence U in all samples. The notations are as in Fig. 3.

248

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

because of domain disorientation (see Fig. 1b) following the method developed in Ref. [12]. Let us analyze the physical implication of differences in the volume changes (DV/V) obtained in Eqs. (17) and (1). Expression (17) corresponds as it has already been mentioned, to the irradiation-induced ‘‘free dilatation’’. Now let us turn to Eq. (1). The longitudinal elongation in this expression (for example, (DL/L)k in the parallel extrusion direction) describes the actual strain in a sample taking into account the random orientation of its domains. Transverse elongation ((D L/L)\ in the parallel extrusion direction) in expression (1) corresponds to the totally suppressed deformations of the adjacent distinctly oriented domains. Eq. (1) gives a relative volume change in a simply connected macro-graphite assembled of samples of say, parallel extrusion direction when all the discrepancies between the adjacent shape-changing domains are straightened by stresses along the domain boundaries. The difference in values (17) and (1) determines an excess volume fraction of such overlapping irradiation-induced regions:

U < Umin the change in the shape of domains prevails leading to the initial shrinking of graphite samples. At high doses, U > Umin the main contribution to graphite deformation is made by variations in elastic moduli leading to an expansion of graphite described by the asymptotic dependence (27). 4.5. Relative contributions Fig. 8 shows the relative contributions made by four important geometrical characteristics of assembly A samples: an irradiationinduced change in length L, a change in diameter d and two principal strains (Dl/l)k and (Dl/l)\ as functions of fluence U. According to Eq. (26) there is a linear dependence between these variables. We observe that the measured elongations L and d are found between the principal elongations (Dl/l)k and (Dl/l)\ due to the constraints of the domain shape within the volume of graphite. Similar dependences (not shown here) were derived for assemblies B and C. 5. Theory: internal stresses of graphite

d0

Intuitively, when all the domains are strictly oriented (see Fig. 1a), the irradiation-induced changes in their shapes do not result in any internal stresses. According to the terminology of Ref. [12] such domains are in the state of a free dilatation. In this state the relative volume changes of an arbitrarily shaped sample and of macro-graphite are the same. This idealized case corresponds to expression (1) and to the upper curves in Fig. 9 in the sense that they describe changes in the domain shapes almost without stress, despite the apparent disorientation of the domains. In this case internal stresses are absent because the size of a domain is comparable to the minimal size (the diameter) of the sample. In reality the situation is fundamentally different for macro-graphite, since all the dimensions are much larger than the domain size. Below, we consider this case and describe the emerging internal stresses

A

   F  T DV DV DV   ¼ 2e?  ek V V V

ð37Þ

After the equilibrium is reached at the internal boundaries between the domains the excess volume fraction d0(DV/V) partially relaxes. Considering macro-graphite to be an elastic body containing an ensemble of equilibrium dilatating objects whose dimensions are much smaller than the size of a body with free boundaries, one can find a dilatation component of the total strain. According to Ref. [12] we get in the leading order in (DV/V):

^Þ ’ Trðu

2 1  2m ð2e?  ek Þ 3 1m

ð38Þ

Since in the reference state the relative volume change is (DV/V)T we find the following expression for the relative change in the volume of macro-graphite:

15

6 mm,

A 8 mm,

Elongation changes

15 10 d

d

10

5 5 0

x 10

1

2

26

3

L ||

-5

A

0

1

L 3

2

||

-5

15

6 mm,

x 1026

A 8 mm,

Elongation changes

15 10 L

10

5 5

d 26

0

L d

x 10

1

2

3

0

||

x 1026

1

2

3

-5 -5

||

Fig. 8. Dependence of relative elongations (in %) of samples irradiated in reactor BOR-60: DL/L (dashed thin lines, symbol L), Dd/d (solid thin lines, symbol d), (D l/l)\ (solid thick lines, \) and (Dl/l)k (dashed thick lines, k) on fluence U of assembly A samples irradiated at average temperature T = 460 °C. The notations are as in Fig. 3.

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

249

(a)

(b)

(c) Fig. 9. Fluence U dependence of relative volume changes (in %) in samples of diameters d = 6 and 8 mm for assemblies a = A, B, C irradiated at temperatures T = 460, 550 and 640 °C: upper dash-dotted lines F give actual values, (DV/V) F, Eq. (16), lower dotted lines U correspond to traditionally used expression, (DV/V)T, Eq. (1), and middle solid lines M are predicted by us for bulk graphite, (DV/V)M, Eq. (39).

^Þ ðDV=VÞM ¼ ðDV=VÞT þ Trðu  k  ? DL DL 2 1  2m þ2 þ ð2e?  ek Þ ’ L L 3 1m

6. Discussion

ð39Þ

The relation between different definitions of a relative volume change, (DV/V)F, (DV/V)T and (DV/V)M is shown in Fig. 9 with account for the value of Poisson’s ratio m = 0.2 for graphite GR-280. The obtained result is in agreement with a frequently encountered in literature experimental fact – large size samples of extruded graphite change their shapes more intensively than thinner samples. For example, the paper [13] presents the results of measuring dimensional changes in irradiated 10 cm  10 cm  61 cm samples as well as in cylindric ones having the diameter of 1.09 cm and the length of 10 cm for the case of extrusion in the parallel and perpendicular directions. The observed values of the volume changes (shrinkage) in larger samples are approximately twice as large. Using expressions (37) and (39) the discussed relation between macro-graphite and thin graphite samples can be written down as

 M  F DV DV 1 1þm ¼  ð2e?  ek Þ V V 3 1m

ð40Þ

In this work we discuss the data on both the longitudinal and transverse measurements of irradiation-induced shape-changes in cylindrically shaped graphite samples. These data evidence the following:  Under irradiation the cross-sections of the samples change their shapes from the initially circular to elliptical ones, and the orientations of the main ellipse axes randomly vary along a sample.  The relative volume changes obtained by direct measurements of (DV/V)F, as well as using the commonly accepted expression (DV/V)T in terms of the relative length variations of samples with parallel and perpendicular extrusion directions, differ essentially (see Fig. 9). To explain those facts, the authors proposed a new model of the extruded graphite morphology, considering it to be an ensemble of dilated domains, that are randomly oriented in relation to the global axis of the symmetry (direction of extrusion) of macro-graphite. In this model the symmetry properties of both macrographite and domains are the same, i.e., the transversal isotropy.

250

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

For each cross-section of cylindrical samples 6 and 8 mm in diameters with parallel and perpendicular extrusion directions we present the results of measuring the values (DL/L)k, (DL/L)\,dmax and dmin. The authors derived a closed system of equations for 2 parameters (Dl/l)k, (Dl/l)\ and sin h characterizing the deformation of domains and their orientations as well as calculated those values. The authors predict that the domains of samples with transverse dimensions on the order of a domain size have to experience a free (non-constrained) irradiation-induced dilation. In this case, 2 the values (Dl/l)k, (Dl/l)\ and sin h should match the ones of samples having parallel and perpendicular extrusion directions. This condition is met by the samples 6 mm and 8 mm in the diameters (see Fig. 8). The data on such freely-dilating domains were used in the subsequent analysis and evaluations. The main results might be summarized as follows:

a. In macro-graphite the irradiation-induced dilation of randomly 2

oriented (sin h > 0, see Fig. 1b) domains leads to a development of cross-domain stresses. Of particular interest is the dilatation component of a stress leading to a difference between the real irradiation-induced volume change (DV/V)F and the traditionally used expression (DV/V)T for these volume variations. Note that in case of regularly ordered domains (see Fig. 1a) the internal stresses disappear and all the expressions for the volume change are the same,

 F  T  M DV DV DV ¼ ¼ V V V

ð41Þ

b. It is shown that the shrinkage curves derived by both direct measurements and by calculations of relative size changes do not describe the genuine shrinkage curve of macro-graphite. We derived the relation

 M  k  ? DV DL DL 2 1  2m ¼ þ2 þ ð2e?  ek Þ V L L 3 1m

ð42Þ

that allows the shrinkage curve of macro-graphite to be obtained by direct measurements of strains in samples having small sizes. c. Our analysis leads to an important conclusion that there are restrictions for sizes of experimental samples. In case when the minimal sample size Lmin is much larger than that of the domain n, Lmin n, the elongation measurements directly match the behavior of macroscopic graphite. However, such a condition is very hard to attain in reactor experiments. The simplest realizable condition is to fit the ratio Lmin ’ n, followed by the subsequent conversion of obtained data using the relation (42). In our experiments, as is seen from Fig. 8, the condition Lmin ’ n is realized for a set of samples 6 mm and 8 mm in diameters. This criterion is in an adequate agreement with the constancy of the principal elongations (Dl/l)k and (Dl/l)\ of dilatating domains having different orientations. Acknowledgements The authors consider it to be their pleasant duty to thank B.N. Bennesch, A.N. Maltsev and A.V. Nechitailo who made a significant contribution into the experiments. Appendix A. Glossary of symbols    averaging over all cross-sections of the sample, see Eq. (13). k and \: superscripts (k and \) show extrusion direction x of samples; subscripts (k and \) describe corresponding components of principal strains of domains, Eq. (2).

a = A, B, C – assemblies of samples irradiated at different temperatures, viz.: 460 ± 25 °C, 550 ± 25 °C and 640 ± 25 °C, assemblies A, B and C, respectively. ak, bk, ck – Fourier transforms of angular dependences of squared diameters of samples, Eqs. (23)–(25). d – diameter of a sample before irradiation (6 and 8 mm). (di)k – experimentally measured diameters of a sample for the angle ui in kth cross-section plane. d(u) – theoretical angular dependence of the diameter of a sample after irradiation, Eq. (7). dmax and dmin – maximal and minimal diameters of a crosssection, Eqs. (8) and (9). b F – gradient tensor of irradiation-induced deformation in coordinate system related to sample, Eq. (5). b F 0 – gradient tensor of irradiation-induced deformation in domain coordinate system, Eq. (2). b F ideal – gradient tensor of irradiation-induced deformation of regular structure, Eq. (3). L – total length of sample, Eq. (11). L0k – length of kth segment before irradiation. Lk – length of kth segment after irradiation, Eq. (10). DL/L – relative change in measured sample length, Eq. (12). DLi/Li – relative change in measured sample size along the axis i = x,y,z. (Dl/l)k, (Dl/l)\ – principal strains: along domain axis of axial anisotropy and perpendicular to it. N – number of equally separated cross-sections in sample. M – number of equidistant angles ui in a cross-section of sample. b 2 ðu Þ R b 1 ðhÞ – tensor of rotation in the plane of crossb u ; hÞ ¼ R Rð 0 0 section of sample. b 1 ðhÞ – tensor of rotation by polar angle h, Eq. (4). R b 2 ðu Þ – tensor of rotation by azimuth angle u0, Eq. (6). R 0 r = n/L0k – average dimensionless size of a domain, Eq. (29). T – temperature during irradiation. Tr(  ) – sum of diagonal elements. V – volume of sample, Eq. (15). bl and be – proportionality coefficients in dependences (34) and (35) on squared fluence U2. (DV/V) – relative change of sample volume under irradiation. (DV/V)F – relative change in volume of a ‘‘free’’ sample, Eq. (16). (DV/V)M – predicted relative change in volume of macro-graphite, Eqs. (39) and (40). (DV/V)T – traditionally used equation for the relative change of sample volume, Eq. (1). d0(DV/V) – excess volume fraction, Eq. (37). m – Poisson’s ratio. n – average domain size, see Fig. 1. x ¼ h – orientation angle between the direction of axial symmetry and the axis of sample (x = 0°, superscript k and x = 90°, superscript \), see Fig. 2. e – average flattening factor of sample, Eq. (13). ek – flattening factor of kth cross-section of sample, Eq. (14). h – polar angle between domain axial direction and cylinder axis, see Fig. 2b. r2 – mean square deviation, Eq. (19). U – neutron fluence (neutrons/m2). Umin – cross-over value of the neutron fluence at which shrinkage is replaced by expansion. (u0)k – azimuthal angle characterizing rotation of domain structure in the plane of kth cross-section, see Fig. 2b. ui = 2pi/M(i = 1, . . . ,M) – angles, in which the cylindrical sample is rotated upon measuring diameters of its cross-sections, (di)k.

251

M.V. Arjakov et al. / Journal of Nuclear Materials 420 (2012) 241–251

Appendix B. Solution of minimum conditions

The inspection of the above equations shows that bk and ck can be written down in the form of:

The angular dependence of the diameter (7) of a deformed cylinder having initially a circular cross-section can be rewritten down as

bk ¼

2

d ðuÞ ¼ ak þ bk cosð2uÞ þ ck sinð2uÞ

ðA:1Þ

 i 1h 2 2 ; d þ dmin k 2 max h   i 1 2 2 bk ¼ dmax  dmin cos½2ðu0 Þk ; k 2 h   i 1 ck ¼ d2max  d2min sin½2ðu0 Þk  k 2

ðA:2Þ

cos2 ð2ui Þ ¼

M X i¼1

M sin ð2ui Þ ¼ 2 2

ðA:3Þ

i¼1

M X

cosð2ui Þ ¼

i¼1

M X

sinð2ui Þ ¼ 0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ðak  ak Þ  b2k þ c2k þ bk þ c2k ¼ 0;  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi N P 2 2ðak  ak Þ þ b2k þ c2k  bk þ c2k ¼ 0

ðA:9Þ

Combining Eqs. (A.2) and (A.7) we receive

h



ak ¼ 12 d2max þ d2min

i

; k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h  i 2 2 b2k þ c2k ¼ 12 dmax  dmin

ðA:10Þ

ðA:4Þ References

i¼1

N  2  1X ¼ 2ak þ b2k þ c2k  4ak ak  2bk bk  2ck ck N k¼1

þ

ðA:8Þ

At the last step of our derivation we substitute Eq. (A.10) into Eq. (A.9) and find diameters dmax and (dmin) k, see Eqs. (20) and (21).

After performing some algebra we get

r2

where the azimuthal angle (u0)k is the solution of equation

k

and orthogonality relations

cosð2ui Þ sinð2ui Þ ¼

ðA:7Þ

k¼1

To find the coefficients ak,bk and ck from the experimental data we substitute Eq. (A.1) into Eq. (7). The result of this substitution can be simplified using normalization conditions

M X

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2k þ c2k sin½2ðu0 Þk 

Solving this equation with respect to (u0)k we find Eq. (22). Substituting Eqs. (A.7) and (A.8) into Eq. (A.6) we get

ak ¼

i¼1

ck ¼

bk sin½2ðu0 Þk  ¼ ck cos½2ðu0 Þk 

where

M X

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2k þ c2k cos½2ðu0 Þk ;

N X M h  i2 1 X 2 dk ; i NM k¼1 i¼1

ðA:5Þ

where ak, bk and ck are defined in Eqs. (20)–(22). via minimizing expression (A.5) with respect to (dmin)k, dmax and (u0)k we obtain equations for those variables

2ðak  ak Þ  ðbk  bk Þ cos½2ðu0 Þk   ðck  ck Þ sin½2ðu0 Þk  ¼ 0; N

P 2ðak  ak Þ þ ðbk  bk Þ cos½2ðu0 Þk  þ ðck  ck Þ sin½2ðu0 Þk  ¼ 0;

k¼1

ðbk  bk Þ sin½2ðu0 Þk  þ ðck  ck Þ cos½2ðu0 Þk  ¼ 0 ðA:6Þ

[1] John H.W. Simmons, Radiation Damage in Graphite, Pergamon Press, Oxford, 1965. [2] B. Kelly, Prog. Nucl. Energy 2 (1978) 219–269. [3] Irradiation damage in graphite due to fast neutrons in fission and fusion systems, IAEA-TECDOC-1154, September 2000. [4] G. Haag, Properties of ATR-2E graphite and property changes due to fast neutron irradiation, Forschungszentrum, Jülich, 2005. [5] Ya.I. Shtrombakh, B.A. Gurovich, P.A. Platonov, V.M. Alekseev, J. Nucl. Mater. 225 (1995) 273–301. [6] A.R. Ubbelohde, F.A. Lewis, Graphite and Its Crystal Compounds, Clarendon Press, Oxford, 1960. [7] W. Windes, T. Burchell, R. Bratton, Graphite technology development plan, INL/ EXT-07-13165, 2007. [8] J.F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1964. [9] A.E.H. Love, Mathematical theory of elasticity, London, 1952. [10] A.V. Subbotin, O.V. Ivanov, I.M. Dremin, V.P. Shevel’ko, V.A. Nechitailo, Atom. Energy 100 (2006) 199–209. [11] S.V. Panyukov, A.V. Subbotin, Atom. Energy 105 (2008) 25–32. [12] J.D. Eshelby, Solid State Physics, vol. 3, Academic Press, New York, 1956. pp. 79–144. [13] R. Nightingale, E. Woodruff, Nucl. Sci. Eng. 19 (1964) 390–395.