Theories of fission gas behaviour

Annals of NuclearEnergy, Vol.3, pp. 41 to 53. PergamonPress 1976. Printedin Northern Ireland

THEORIES OF FISSION GAS BEHAVIOUR J. W. C. Du~s Companhia Brasileira De Tecnologia Nuclear and K. R. MERelC,X Exxon Nuclear Company, Inc. Richland, Washington 99352, U.S.A. (Received 18 August 1975)

NOMENCLATURE a a0 B b c D D' D, Dq D~ E F _F f fr fB F G g,g' J,~ k K L m N n nc P P" Po Q Q* R r s T t u V v x fl 7

spherical radius (grain or bubble) radius of a bubble under no strain Van der Waal's constant magnitude of a Burger's vector concentration of atoms in solution thermal diffusion coefficient irradiation enhanced diffusion coefficient surface self diffusion coefficient diffusion coefficient in gas volume self diffusion coefficient energy level dose or total fissions fission rate number of gas atoms per fission fraction of gas atoms in a bubble trapped per sputtering event fraction of gas atoms escaping per activation event bubble driving force free energy probability per second material flux Boltzmann constant thermal conductivity diffusion length concentration of trapped atoms number of atoms in each bubble concentration of matrix gas bubbles critical concentration of inert gas atoms for homogeneous bubble formation pressure in a gas bubble effective pressure for one species of gas external pressure the heat of activation for atom diffusion heat of transport per molecule gas constant radial coordinate bubble spacing absolute temperature time ratio gas atom knock out rate to number of gas atoms volume of a bubble bubble velocity linear dimension. number of bubbles nucleated per fission fragment gas production rate per unit volume surface energy of uranium dioxide

AG AH r/ 0 2 2, 20 # v v0 ~r ~ffl f~ .Q,

critical distance for activated gas atom to travel without an interaction free energy change heat of solution change fraction of atoms per second receiving energy greater than a certain minimum level number of collisions per fission fragment capable of causing gas atoms pair separation numerical factor for thermal gradients length of a fission track or range of fission fragment normal lattice spacing nearest neighbour spacing in the lattice shear modulus number of diffusing atoms per unit surface area jump frequency of an atom cross section line tension of a bowed dislocation fission fragment flux atomic column of matrix atoms atomic volume of a vacancy. INTRODUCTION

The behaviour of inert gases created during the irradiation of reactor fuel materials is of importance in the study of fuel swelling, of inert fission gas release and of the mechanical behaviour of reactor fuel elements. Although the phenomena stressed in this paper primarily deal with the steady state behaviour of fuel elements, they are also relevant to the understanding of transient behaviour. The movement of fission gas as single atoms in solution in the fuel matrix and their clustering in the form of bubbles, the resolution of bubbles into the matrix, and bubble migration and coalescence are phenomena that enter into the behaviour of the fission gases. This paper reviews the theoretical developments and experimental evidence that have played a significant role in the evolution of the current models used to describe the behaviour of the inert fission gases. Many of the initial investigations and theories attempted to explain gas release and swelling in 41

42

J. W. C. DL~S and K. R. MERCKX

terms of some controlling phenomena. The more recent models have combined phenomena and have either developed simulated stochastic models, Warner and Nichols (1970), or dealt with distribution functions of bubbles, Li et aL (1970). Such models are then used in conjunction with fuel performance codes (Duncombe et aL, 1970. Jankus and Weeks, 1970). Since prototypic reactor environments create conditions which make it difficult to evaluate phenomena individually, the use of composite models may well be required to explain observations of prototypic fuel materials. An understanding of the competing processes and proposed mechanisms is necessary to evaluate performance prediction codes and to interrelate the observations made of irradiated fuel materials. The evolution of different performance models may well be warranted to represent steady state behaviour, transient behaviour, cyclic behaviour, and power ramp behaviour. This review has concentrated on the phenomena in the belief that a general review would not be biased in favour of a proposed performance model and that a broad treatment of the subject is required to evaluate the significance of assumptions in performance models. GAS A T O M SOLUBILITY

Because of their large atomic size and inertness, the noble fission gases, krypton and xenon, are almost insoluble in crystalline materials. Irradiation rapidly leads to super-saturation whereupon precipitation of the krypton and xenon becomes thermodynamically favourable. The equilibrium concentration of gas occurs when the free energy of a constant pressure and temperature system is stable. For this condition the changes in free energy of the system, ~G, when 6n atoms are dissolved is 6n AG = ~n A H + ~nkT(ln c - In Co) = 0 (1) or

c = co exp ( - - A H / k T ) where AH is the heat of solution and c is the concentration. At low irradiation temperatures the gas diffusivity in the fuel is low and the entire gas content is held in supersaturated solution, Speight (1968). Higher irradiation temperatures promote the diffusion of gas atoms and their trapping in existing fission gas bubbles as well as on structural defects such as grain boundaries, dislocations, dislocation loops and precipitates, MacEwan and Morel (1966). These defects act as sites for temporary immobilization until resolution of the gas takes place. As a result of this interplay between gas precipitation and

resolution, a pseudo-gas kinetic solubility emerges which is much higher than its thermodynamic counterpart. Hurst (1962)developed a model for a gas precipitation process based upon diffusion theory and firstorder rate expressions for both gas immobilization (by traps) and thermally activated decomposition of the immobile species. The rates of change of concentration of atoms in solution, c, and of the concentration of trapped atoms, m, are described by the following equations: dc/dt -- D V2c - (D[L~)c + g ' m (2) and dm/dt = (D/L2)c -- g ' m

where D V2c is the rate of inward diffusion of gas atoms, D is the diffusion coefficient of the atom species, (D/L~)e is the rate of trapping, g ' m is the rate of return from traps, and g ' is the probability per unit time for an atom to leave the trap. The steady-state distribution between gas in solid solution and gas in precipitates is obtained when dm/dt = 0 or with a solubility limit of c[m = g'/(D/L~). The solubility limit obtained with the Hurst model must be modified during irradiation to account for gas generation and to reinterpret the coefficients to take into account dispersion of gas precipitates (redissolved) by the interaction of passing fission fragment recoils and the gas trapping by gas bubbles. Speight (1969) used the following modification of equation (2) to calculate release of fission gas from fuel grains: dc/dt = fl + D V~c - - g c + g ' m

(3)

and dm/dt = g c - g ' m ,

where fl is the rate at which gas is produced per unit volume, g is the probability per second of a gas atom in solution being captured by a bubble and g ' is the probability per second of a gas atom within a bubble being re-dissolved. The finite transform series solution for equations (3) with boundary conditions of spherical grain boundary at radius a acting as a perfect sink and with the assumption that the trapping sites (bubbles) are effectively immobile is: c = - ~ ( -- 1)~(2fla3/D ~r3j3r) sin (j~rr/a)/r 5=1

× {1 - exp [ - D g T Z ~ r 2 t / a 2 ( g ' + g)])

(4)

and oO

m = -- ~, ( -- 1)J(2gfla3/D ,~g[13r) sin (j~rr/a)/r

× {1 -- exp [ D g T ~ r t / a 2 ( g ' + g)]}

Theories of fission gas behaviour The steady-state kinetic solubility of krypton and xenon in UOe has been estimated by Nelson (1969), Marlowe (1970), and Pati (1971). When all the gas not in solution is contained within randomly distributed bubbles of equal size and no gross sinks collect the gas, including the influence of grain boundaries, equation (3) is reduced to: [3 + g ' m

(5)

= @c

That is, the gas generation rate together with the re-solution rate equals the capture rate by bubbles. Also, at time t the amount of gas in solution per unit volume is equal to the total amount of gas generated minus that in bubbles, i.e. c =

dr -

m =

fo

fldr

-(gc

-- f l ) / g '

(6)

The gas generation rate # is given by # = f/6, where is the fission rate and f is the number of gas atoms produced per fission with 0.3 atoms of krypton and xenon per atom fissioned. The probability per second, g, of a gas atom in solution being captured by n bubbles of radius a per unit volume, as obtained from Ham (1958), is g = 4~rDan. Hence, for temperatures below the bubble migration range, the gas solubility is given by: (7)

e = f F t -- (4~rDanc - - f F ) / g '

Assuming that the inert fission gases in bubbles obey Van de Waals' gas law and are in equilibrium with the surface tension, Pati (1971), calculated the amount of gas that should be in the bubbles if there was total retention, and compared the results with the experimental data, Speight (1968), (see Table 1). Ronchi and Matzke (1971) modified the solution of Speight (1969) to include grain boundary effects. This model has been successfully applied to oxide fuel for temperatures up to 1800°C. Because of the bubble growth on the grain boundaries, the gas concentration in the intragranular bubbles does not continually increase but actually reaches a maximum and subsequently decreases. Figure 1 shows the temperature dependence of time to Table 1. Experimental data of gas retention* Temperature Flux Burnup (°C) (Fiss/cm s) (Atom ~) 800 1200 1500 * Speight (1968).

-10TM 1014

-1.4 13

Percentage retained 100 75 10

43

.'2600

I

t

I

2200 -

t.) *

1800

t.1400

~ooo 3

I

I

I

4

5

6

log

(time),

7

sec

Fig. 1. Time to reach maximum gas concentration into intergranular bubbles. reach maximum gas concentration for this model and indicates that for temperatures above ll00°C the neglect of grain boundary effects is not applicable to prototypic reactor fuel materials. GAS

ATOM

DIFFUSIVITY

In a solid crystal, the mean time of stay of an atom at any one lattice site is assumed to be finite. The atomic process responsible for the effects of diffusion is generally accepted to be the random motion which each atom performs as it jumps from one site to the neighbouring site. In a substitutional solid solution, which is the case of xenon and krypton in oxide fuel, the jumps are made via vacant lattice sites (vacancy diffusion) and a particular atom succeeds in making a jump only when a vacant site becomes adjacent to it as a result of the movement of the other atoms. The diffusion coefficient D is related to these random motions, Chandrasekhar (1943), by the product of the square of the jump distance, x, the jump frequency of an atom, the probability of finding a vacant lattice site next to it times the probability of success in overcoming the energy barrier of the jump. At equilibrium, Brophy et aL (1964), these probabilities are exponentially related to Qf, the energy for vacancy formation and O,~, the energy of motion of a vacancy. Thus, the diffusion coefficient, Greenwood and Speight (1963), becomes D = D1~0x~exp [ - ( Q ! + a ~ ) / k T ] = D Oexp (--Q/RT) (8) where Q is the heat of activation for atom diffusion per mole and D o is the pre-~xponential constant. MacEwan and Stevens (1969) determined D by irradiating a sample and following the release

J. W. C. DIAS and K. R. MP.RCKX

44

of a radioactive isotope during an isothermal anneal. Booth (1957) derived the released fraction of an inert fission gas isotope by calculating the rate at which the gas diffused to the boundary of a "spherical" sample of radius a during the anneal without taking into account gas trapping and return (re-solution) from the traps. Davies and Long (1963), used this model on data from UO2 compacts over temperatures in the range 800 to 1600°C and found the values 7.8 x 10-6 cm~/sec and 71,000cal/mole for Do and Q, respectively. The theory for the diffusion of fission gas from a sphere taking into account trapping and return from traps was used to analyse experiments (Cornell, 1969, MacEwan and Morel, 1966) carried out on the release of xenon and krypton from an irradiated uranium dioxide sample during a post-irradiation anneal. An activation energy of 87,000 cal/mole was calculated. Cornell (1969) determined the diffusion coefficients of atomic xenon and krypton with bubble growth data in thin foils of UO~ preirradiated at temperatures below 900°C when isothermally annealed between 1300°C and 1600°C. He used the following expression derived by Speight for growth of bubbles of radius a after t seconds: 1

~/1 + 2yB/ka

x loge

+ 2~,B/KatT af

k'-~l + 2yB]KaIT

,/1 24Da1T s3

,

(9)

where a I is the final bubble radius after complete precipitation of gas, and s is the bubble nucleation spacing. The resultant diffusion coefficients were expressed by: D = (2.1 4- 1.1)10-4 exp [-(91,000 4- 14,000)/RT] (10) Bubble coalescence was not included in this interpretation; thus, growth by coalescence and gas pickup by sweeping could introduce errors in estimating the diffusion coefficients. The activation energies reported by Cornell (1969) and Childs (1963) using the methods described above have been found to vary between 24 and 128 kcal/mole, a variation that cannot be explained solely by experimental error. Many unexplained anomalies

exist within and between the results obtained, suggesting that some important variables were not recognized and controlled. For instance, incorrect characterization of the fuel, errors in burnup estimation and non-stoichiometry effects have been identified as possible sources of variations by MacEwan and Stevens (1964). The previous investigations for estimating the diffusion of xenon and krypton were for post irradiation conditions in the absence of neutron irradiation. Under a normal thermodynamical regime, diffusion is limited by an activation energy which is the sum of a formation energy of vacancies and an activation energy for migration, equation (8). In an irradiation environment, the dynamic vacancy concentration is sufficiently high that the large inert gas atoms combine with vacancies and limit the long range diffusion of the gas atoms as interstitial atoms. A detailed estimate of irradiation dependent diffusion of inert gas atoms depends both on the lifetime of interstitial gas atoms and on the mechanism by which the vacancies produced by irradiation are lost. Vacancies can be lost by annihilation with interstitial uranium atoms, interstitial gas atoms, or at other sinks such as dislocation lines and grain boundaries. Nelson (1968) assumes that the enhanced gas atom diffusion is of the same order of magnitude as that for enhanced self-diffusion of uranium which has been calculated by Brinkman and Johnston (1967) to be temperature independent and determined only by the defect production rate, D t o "~ 1.5 × 10-al/~. For the case of uranium dioxide, Nelson (1968) has found that at temperatures below 1000°C gas diffusion is mainly controlled by irradiation effects where a large number of irradiation-induced vacancies are present in the material. Cornell (1971) developed a relation to estimate the diffusion coefficient for fission gas from observations of bubble size and bubble concentration evolved in irradiated uranium dioxide at temperatures ranging from 800 to 1600°C. The amount of fission gas present in all the matrix bubbles during irradiation was equated to the concentration of matrix gas bubbles, n, of radius a times the number of atoms in each bubble: m =g~t/(g' +g) = 8ztya3n/3(kaT -b 2yB) (11) Ham's (1958) expression for relating the probability per second of a gas atom in the lattice being trapped in the existing bubble to the diffusion coefficient (g = 47rDan) was substituted into the preceding relation to obtain: D = 2yg'a~/[3flt(kTa + 2~,B) -- 8~ryaZn], (12)

Theories of fission gas behaviour where g' is the probability per second of a gas atom undergoing resolution. BUBBLE NUCLEATION The nucleation process may occur homogeneously as a result of chance encounters between freely migrating atoms or heterogeneously on nucleation centres such as structural defects. The estimate of the number of gas atoms needed to form a stable nucleus required for the homogeneous process has been made by Greenwood et aL (1959), based on the following concepts: The vacancies enable gas atoms to move up to one another and bring about a lowering of energy by taking up the vacancy volume. The time this small group remains together depends on the reduction in energy and the rate that further gas atoms and vacancies reach it. For example, a binding energy of 2 eV yields a lifetime of --~10-a sec, whereas with 3 eV increases this time to N100 sec. A bubble nucleus can be obtained if the gas atoms can attach enough vacancies to reduce their energy by about 3 eV. Since the energy of solution of these gas atoms may be as high as ,-~6-8 eV, a 3 eV reduction could be realized by a group containing as few atoms as from 3 to 6 and a small number of vacancies per gas atom. The atomic collision process that takes place in an irradiation environment, Nelson (1968), can cause an inert gas atom pair to dissociate. If such kinetic dissolution by irradiation occurs before a third atom is captured, then the homogeneous nucleation of bubbles is unlikely. Irradiation also enhances diffusion as well as causes dissolution; thus, the nucleation of bubbles depends on the details of the atomic collision process, the irradiation flux, the temperature of the material, and the gas concentration. Based on these arguments, the critical concentration of inert gas atoms, n e, is determined by equating the dissolution time ( , ~ F ~ ) to the capture time of a third atom. The capture time in seconds is estimated to be: tc "~ ;t0~/40(D + D')n~/3. Equating these two times and solving for the critical concentration, no, yields* n c --~ [20~/~r//20(D q- D')] 3/a.

(13)

Nelson (1969) pointed out that if the time taken for a gas atom to diffuse to a bubble nucleus is much shorter than the interval between the production * A value of r/ ~104 for collision per fission fragment in UOz has been estimated by Nelson (1968).

45

of successive gas atoms within the effective capture volume of the nucleus, the capture time will be inversely proportional to the production rate. For the heterogeneous nucleation process, the structural defects that act as nucleation centres for the gas atom clusters within the grains are precipitates, dislocations, dislocation nodes, and point defect clusters. Since the precipitate spacing is usually relatively large, precipitates cannot be reached by the majority of gas atoms. The spacing of dislocations is about the same as that at which gas atoms form into bubbles; hence, dislocations, Greenwood et aL (1959), appear to act as sites on which bubble nucleation preferentially occurs. Although Eyre and Bullough (1968) have postulated that in the high temperatures regions of the fuel both the vacancies and interstitial point defects are mobile, they combine before they can act as nucleation centres. Cornell (1968) and Turnbull (1971) have postulated that bubbles are spontaneously nucleated by fission fragments in the low temperature regions based upon the observation of bubbles lying in straight lines in Whapham and Makins (1962) electron microscope examinations of intragranular bubble distributions. A displacement spike, where the fragment makes high energy collisions with lattice atoms, causes many displacements in a small volume of crystal. This results in a high concentration of vacancies which may collapse into either a void or a dislocation loop. Calculation shows that in face-centred cubic materials such as UO2 the void is always the more stable defect. In the absence of gas atoms such a void would quickly anneal at high temperatures, but when sufficient gas exists in the lattice, the volume of the displacement spike will contain enough gas atoms to exert a pressure that will balance the surface energy. The nucleation stage thus completed, the bubble could grow by gas atoms diffusing to it. Turnbull (1971) has determined that more than four gas atoms are necessary to stabilize a void. Assuming a spike of N100/k in diameter (measured by Whapham and Makin, 1962), Cornell and TurnbuU (1971) determined that a dose of between 1.7-3.2 × 109 fissions/cm3 (0.074--0.14 at.~o burnup) is required before intragranular bubble formation in UO~becomes uniformly distributed. If 2~/;~ is the number of new bubbles nucleated per unit volume per second with ~ the number of bubbles nucleated per fission fragment and if 2~r/~2a2~ is the number of bubbles destroyed per second with 2 as the fission range (,-~6/~m, Noggle and Stiegler, 1960), then the rate of change of the bubble concentration, Turnbull (1959),

J. W. C. D i m and K. R. MERCKX

46

n is d__n = 2 ~ i : -

2 7 r F A n a 2,

(14)

dt

of individual volume f~ that together constitute a bubble enclosing gas at pressure P is NdN

= ( k T / f ~ , P ) + ( B -- f ~ ) / f ~ = ( k T / f ~ o P ) + 3.

(18)

and the equilibrium concentration of bubbles is: (15)

na ~ = ~]rry. BUBBLE G R O W T H

Gas bubbles increase in size either by accumulation of individual gas atoms or by coalescence of bubbles. Growth by the flow of gas atoms and vacancies into bubbles is the predominant mode of growth at low temperatures where the bubbles are essentially immobile. The free energy, G, of the lattice surrounding a bubble (Brown and Mazey, 1964) is =

G

4~rya ~ +

G,

__fP=P(V,T)

P4=a' da'

where it is assumed that the Van der Waals' constant, B - - 4 f ~ and f ~ N f L Thus, a bubble at constant gas pressure requires vacancies to be supplied at a rate

dt

dt (19) = (kT/f~,,P

(16)

To reach such an equilibrium condition, lattice vacancies flow to the bubble and cause a volume increase. The number of equilibrium vacancies over the two atomic volumes that are usually associated with each fission gas atom in a bubble is indicated in Table 2 (Greenwood et aL, 1959). Bubble growth is usually not restricted by the rate at which vacancies must be supplied. Examination of the conditions under which the vacancy supply is sufficient follows. The ratio, N d N , (see Table 2) of the number of vacancies of volume f ~ to the number of gas atoms

+ 3)4~raDC

where d N / d t , t h e instantaneous absorption rate of gas atoms, is given by dN -dt

JP=Pe

where the first term on the right hand side is the contribution of the surface tension; the second term is the energy associated with the strain deformation; and the third term is the free energy of the gas. When equilibrium conditions exist the compressive field is relaxed and the free energy is minimized; hence (2Yla0) -- P ( V , T ) + Pe = 0 (17)

+ 3)dN

dN~ = (kT/flvP

= 4~raDC.

Since gas pressure decreases with increasing bubble radius the vacancy-supply rate must be larger than the preceding value. In an irradiation environment this vacancy-supply comes primarily from sources due to fission and is maintained at a value c0r at a distance rl midway between the centre of the bubbles. Near the bubble surface at a 0 the concentration of vacancies is given by (Greenwood et al., 1959) Cv = c,v exp [ - ( P - 2 7 / a o ) f ~ d k T ]

(20)

where ce~ is the equilibrium concentration of vacancies appropriate to temperature T. The solution of the steady-state diffusion equation for spherical symmetry and the preceding boundary conditions is cv = [co,r1 - c,~ao exp ( - x ) ] / ( r l - ao) + [c~ exp ( - x )

- codaorx/(rx - ao)r

(21) where: x

Table 2. Number of atomic volumes per gas atom at various bubble radii (calculated for xenon at 300°C) Bubble radius (cm)

Number of atomic volume per gas atom

10-7 10-6 10-5 10-4

,-~5 10 40 400

= (P

-

2Wao)c~,,Ikr

and the flux of vacancies into the bubble is given by D,(dcv~

--

n~[co~ -- cev exp ( - - x ) ] r 1

D~(co~ -- ce~ + e~vx)rl

(22)

Xqvao(r1 -- ao) A similar solution (Greenwood e t al., 1959) for the interstitial flux is: j l = D~(co, -- cei + c, ix)r~/f~iao(rl - ao)

(23)

Theories of fission gas behaviour where ce~ is the thermal equilibrium value of the interstitial concentration and cot is the fission induced concentration. Thus, the net rate at which vacancies arrive at the bubble is given by

bubble the relaxation times for surface diffusion and for volume diffusion shape changes are ts = k Tao4124DeTvf~ 2

(26)

and t~ = C k T a o 3 l l 2 D ~ 7 ~

dN~ ----4 ~ra0(A -- A) dt = 4rraorx[D~(co~ -- ce~ + c , v x ) -- Di(co~ -- ce~ + c , i x ) ] / ~ q ( r l

-- ao)

(24)

where fl ,-, flv N f~. If all sinks are equally good for both vacancies and interstitials, D~(co~ - ce~) = Di(coi -- cei), equation (24) reduces to

where C is a correlation factor and • is the number o f diffusing atoms per unit surface area. Nichols (1966) found t s ~ tv for metal fuel with bubbles as large as 2/~m in dia, which supported his conclusion that the first stage of "coalescence" is a surface diffusion shaping process. The bubble growth by the accural of vacancies is given by 2 D ~ c , ~ f ~ 7 ( P a o 1 2 7 - 1) (27)

da.__~o

d N ~ = 4 ~ r a o r l x ( D~ce~ + Dice~)/ f~(rl _ a° ) dt -- 4 r r D v c ~ a o r l ( P

47

dt

-- 2 ~ l a o ) / k T ( r 1 -- ao).

Comparing the required supply rate with the arrival rate, the amount by which P must exceed 2~,la o at a particular stage during growth is P - 2~/a o = k T ( r x -- a o ) D c [ ( k T / f ~ P ) q- 3]/Ce~r~D ~

(25) A significant departure from equilibrium conditions is usually defined to occur when the gas pressure exceeds twice the surface-tension restraining pressure (Speight, 1968). If at no time throughout bubble growth this situation arises, it is reasonable to equate P with 2~,/a o during the entire growth sequence. BUBBLE GROWTH BY BUBBLE

COALESCENCE

Coalescence consists of the merger of separate bubbles into a smaller number of larger bubbles. Barnes and Mazey (1963) observed helium bubbles in copper films migrating, colliding, and coalescing. The total bubble surface area was unchanged in these experiments hence the total bubble volume increased. Nichols (1969) has postulated that the apparent conservation of total bubble surface is the net result of two stages of the collision process, namely "coalescence" in which the two bubbles rapidly sinter together by a constant volume process into one bubble by surface diffusion and "volume adjustment" in which vacancies migrate to the bubble to re-establish equilibrium. When equilibrium is established in bubbles containing an ideal gas within a matrix exerting no uncompensated pressure then surface area is conserved. Nichols and Mullins (1965) and Nichols (1966) calculated that for small-amplitude spherical harmonic perturbations on the surface of a nearly spherical

= D(jv -ji)

=

kTao 2

(1 - aolrx)

[see derivations of equations (22-25) and note a 0 is equilibrium bubble radius]. Rate equation (27) is integrated between 21/3 a and a/~a assuming two bubbles originally of radius a collided, surface area was maintained, 8 ~ras = 4 rra0S, and pressure P = 2~,]a was constant. The resultant time, t~, for the volume change to occur is t v -----0.1(1 - ao/rl)kTaoa/D~cevf~T

(28)

which is equivalent to that given by equation (26). Externally applied stresses affect the volume occupied by bubbles in those regions where bubble collision and growth mechanisms arc active. The compressive effect of externally applied hydrostatic pressures from 0 to 1000 bar at 900°C on the fission gas induced swelling in uranium has been studied by Kucinski e t al. (1969). Their observations indicated that the average diameter of the bubbles reduced from 3300 to 1800 A and the total number increased from 0.9 to 4.2 × 101S/cma as the pressure was increased from 0 to 110 bar, respectively. Furthcr increases in pressure from 110 to 1000 bar showed no effect on the average bubble size and density. One way of looking at process of applied compressive stresses on bubble compression is to consider the decrease in ratio of vacancies captured by each gas atom, N d N , as given by equation (18). That is NdN

= k T l f 2 ( P ~ + 2~la) + 3

At Pe = 0, this ratio is proportional to a and the number of vacancies captured per gas atom increases as the bubble grows but when P~ is much greater than 2~,la, the ratio is bounded by an upper limit of k T / ( P ~ f ~ ) + 3 and is independent of the bubble size. In this case, even if coalescence occurs, the resulting larger bubble will not require more

48

J. W. C. DIAS and K. R. MERCKX

vacancies to reach equilibrium. Also when an external restraint exists, the total surface area no longer remains constant but decreases as coalescence takes place. The coalescence of two bubbles of the same radius under pressure at constant temperature is described by

(P~ + 27/ao)47rao3/3 = 2(P e + 27/a)47rai3/3 (29) where a 1 is the radius of the initial bubbles and a 0 is the radius of the final bubble. The application o f pressure not only reduces the size of bubbles but increases their mobility. F o r example, migration by a surface diffusion mechanism has been shown to be proportional to a -4, so that the size reduction by pressure also increases mobility and tends to reduce the total number of bubbles.

u = -(F/F)In (NI/N2)

Irradiation-induced re-solution The existence of an efficient re-solution process o f fission gases from intergranular bubbles in oxide fuel under neutron irradiation conditions has been demonstrated experimentally by several workers Whapham, 1965; Ross, 1969; Turnbull and Cornell, 1970, 1971; Pati, 1971. The removal of gas atoms from bubbles can be effected by various energetic particles resulting directly and indirectly from fission events. However, the fission fragments, which are emitted with energies of as much as 100 MeV, are the most predominant in interacting with gas bubbles. Alternate mechanisms for resolution have been proposed by Turnbull (1968) and Nelson (1969). Turnbull favours a sputtering process in which fission fragments that pass close to a bubble lose energy by electron excitation, cause a block of material to traverse the bubble, and thereby entrap gas atoms into the matrix on the far side. On the other hand, Nelson suggests that the major re-solution mechanisms are either from atomic collisions between gas atoms and fission fragments or indirectly as a consequence of interactions of gas atoms with matrix material that is ionized by the displacement cascades. Based on his proposed sputtering re-solution mechanism, Turnbull (1968) derived the following expression for u (the ratio of the number of gas atoms being knocked out of a bubble per second, --dN]dt, to the number of gas atoms per bubble, N,

u=--(dN~/N=4~r2a'Ffr/3 \ dt//

of the cross-sectional area of a block of material sputtered across a bubble and the fraction of gas atoms colliding with the block that are actually trapped and inversely proportional to the bubble radius. Electron-microscope examinations of irradiated uranium dioxide have enable Tumbull and Cornell (1970, 1971) to estimate values for u based upon measurements of the sizes of bubbles in a post-irriadation annealed specimen and subsequent measurements of the sizes of the bubbles after the same specimen was reirradiated and then calculating the number of gas atoms required to be in equilibrium with these bubble sizes. Hence, if F = P At is the dose required to reduce the number of gas atoms in a bubble from N1 to Nz, then integration of equation (30) yields

(30)

where 2 is the length of a fission fragment track, and f r is the fraction of gas atoms trapped per sputtering event, f r is proportional to the product

(31)

A t 200°C and P = 10lz fissions/cmZ-sec, Turnbull and Cornell (1970) observed bubble sizes ranging from 50 to 75 A. u was calculated to be 10- s < u < 10-4sec -1, where the lower and upper limits corresponded to doses F = 3.3 x 10i8 fission/cmz and F = 7 × 1017 fissions/cmz, respectively. A t 1200°C and P = 10ia fissions/cmZ-sec, the bubble sizes ranged from (Turnbull, 1971) 50 to 100/~ and u was calculated to be 1.8 × 10-a < u < 3.6 × 10-4sec - i , where the lower and upper limits corresponded to doses F = 1.63 × 1017 fissions/cmz and F = 8 × 1016 fissions/cm~. To explain these results, the block would have to have an area of ,--1600 A 2 (Whapham and Makin, 1962) and this block would entrap four monolayers of gas (Turnbull and Cornell, 1970). The re-solution process itself should be substantially temperature-independent, hence the numerical evaluation of u from low-temperature experiments is believed to be in error (TurnbuU and Cornell, 1970). Nelson's (1969) collision or interaction model uses a quantity ~ (the number of gas atoms that receive energy greater than a certain minimum level Emin per atom per second with energy greater than Emm which can be represented by ~ ~(Emin, E) = (9.g~t,

(32)

where E is the energy with which the fission fragment is emitted, cg.g is either the cross section for collisions with fission fragments or an effective cross section for indirect interaction of fission

49

Theories of fission gas behaviour fragments with gas atoms via collisions with lattice atoms and 4I is the effective fission fragment flux. A struck gas atom within a bubble must receive at least Emin N200-300 eV and not suffer a large single collision with another gas atom before reaching the surface if it is to be trapped in the lattice. Thus, only a fraction of gas atoms within a critical distance from the bubble surface, 8, have a good chance of escaping from atoms with radii larger than the critical distance. ~ is dependent on the minimum escape energy of the struck gas atoms and on the gas atom density. The fraction f ~ of gas atoms escaping from a bubble of radius a per collision is a 4~rrl 2 dr, --6

f.~(a)

~.a3/3

a > ~

3~5/a,

(33)

and r E ( a ) = 1,

a _<

For small bubbles containing a Van der Waal's gas, the number of gas atoms per bubble N is 4~raZ[3B. The total bubble re-solution rate, - d N l d t , is this number of gas atoms times the probability of being in the critical distance times the ~ or dN dt

4~ra s ~ / B ,

a > d

N~,

a

(34)

Bubble mobility

Collisional coalescense of bubbles occurs in oxide fuel by the bubbles moving bodily inside the fuel. Bodily motion of inert-gas-filled bubbles was first observed by Barnes and Mazey (1963) in pulse-heated copper foils previously injected with helium. The bubbles were observed to migrate under the influence of the temperature gradient and coalesnce with other bubbles which lay in their path. The variation of the velocity of bubbles moving under the influence of a driving force has been shown by Shewmon (1964) to depend on the dominant mechanisms of matter transport around and through the bubble. The potential gradients such as a thermal gradient tend to move the atoms (a) around the bubble via surface-diffusion, (b) through the bubble via the formation of vacancies on the trailing side of the bubble, volume diffusion of vacancies through the matrix to the leading side, and their annihilation at the leading side, and (c) through the bubble via fuel vaporization on the hot side of the bubble and condensation on the cold side. The functional relationship between the bubble velocity, v, and the potential gradient or driving force, F, is given by v = D bF/kT

(35)

where the effective bubble diffusion coefficients for the preceding three transport mechanisms, Nichols (1967), (1969), are

and dN

- - -

=

Db = 3Ds~nf~/2~ra 4 (surface),

(36)

Db = 3 D~f2[4~ra 3 (volume)

(37)

Db = 3Dgf2~p'/4,~kTa 3 (vapor)

(38)

_<

dt For Emi n 300 ,..~eV, Nelson (1969) estimates a value of 10--15 A for the critical distance 8. Nelson (1969) also determined that for a 100 MeV Xe fragment and 4~ = 101° fission fragments/cm2-sec, dynamic re-solution occurs primarily as a result of collision cascades intersecting the bubbles if Emi, is below 10z eV. But if Emin is in excess of this value, dynamic re-solution occurs predominantly as a consequence of direct encounters between fission fragments and gas atoms. Nelson's model can be used to explain the results of Turnbull and Cornell (1970) measured at an irradiation temperature of ,~200°C. To agree with Turnbull and Cornell's (1971) results measured at 1200°C, the gas atoms would have to undergo resolution with an energy of N50 eV. Since a gas atom with this energy is not believed, Kornelson (1964), to be able to pass through the lattice surface, Nelson's model underestimates the re-solution rate.

and where p ' is the effective vapor pressure of the rate controlling species. A thermal gradient causes a primary heat flux and also creates a material flux, Jm, of a mobile species. This material flux can be related to the heat of transport, Q*, Nichols (1969), Jm = D E ( - - Q * V T / T ) I k T .

(39)

To facilitate treating bubble motions, the relation (39) is converted to bubble velocity by multiplying the material flux times the number of atoms in a bubble (4~ra3]3f~c). The effective local thermal gradient, VT, is not the average thermal gradient in the matrix, (dT/dx); but is that due to the perturbation in heat transfer around the bubble. This perturbation effect is handled by a numerical

50

3. W . C . DIAS a n d -6

I

t

l

i

I

i

I

'~

-9

g

-12

-15

_

Evaporation

/

condensation

/

Volume

_

-18I -21

-6

-5

r

I

1

-4

p

diffusion

I

-5

I

I

-2

I

-I

Fig. 2. Migration velocity as a function of bubble radius for spherical bubbles in UO2 at T = 2000°C and dT/dx = 1000°C/cm. factor, 0, given by Nichols, (1969) tit = [3K/(2K + K')] - \dx ] dx /tiT\

V T --- 01--:-" /

(40)

where K and K' are the thermal conductivities of the matrix and bubble. The combination of converting flux to bubble velocity and local gradient to matrix gradient converts equation (39) into the form of equation (35) with the resultant force on the bubble of Ft~ = (4~raa/3~)(a./T) 0 d__T dx

(41)

Similar relations can be developed for any potential gradient where the free energy is reduced by the bubble moving through the potential gradient. Leiden and Nichols (1971) formulated such a bubble driving force for an isothermal tensile stress gradient field. The dependence of the migration velocity in a thermal gradient on the bubble size is shown in Fig. 2 for UO2, assuming T = 2000°C and dT/dx = 1000°C/cm (Sha and Hughes, 1970). As indicated in Fig. 2, the surface diffusion and evaporation-condensation mechanisms play the dominant roles for small and large bubble migration in UO 2, respectively. The results indicate that the change in the dominant mechanism occurs at a bubble radius of about 16/~m. In-reactor measurements on actual bubble velocities are limited to the works of Michels (Michels and Poeppel, 1973; Michels, Poeppel and Niemark, 1970). Michels measured the distance micronsized bubbles moved in a temperature gradient in a known time interval. By assuming the bubbles were moving by surface diffusion and guessing a

K . R . MERCKX

reasonable value for Q*, a relationship between the bubble diffusion coefficient and Michels' results could be found with equations (35)-(41). Michels' experiments, however, did not provide enough evidence to support the assumption of surface diffusion for bubble migration. Out-of-reactor measurement have been made, either in a temperature gradient produced in a thin UO2 foil by beam heating within the electron microscope, or by the measurements of random bubble motions which occur during an isothermal anneal of similar samples. Connell and Williamson (1965) have studied directed bubble motion due to a temperature gradient in foils injected with krypton ions of energy 100 keV. Analysis of the radius dependence of the unidirectional drift showed the bubble velocity to vary as a -z, as predicted by surface diffusion. Similar experiments by Manley (1968) produced a wide variation in bubble behaviour, but measurements on one area confirmed the radius dependence of bubble velocity as a -z. In this experiment, some of the bubbles were associated with small precipitates thought to contain solid fission products, and there was a strong suggestion that these bubbles were less mobile than those without associated precipitates. In contrast to the preceding results, work reported by Barnes and Mazey (1964) on UO2 foils containing helium bubbles showed no discernible dependence of mobility on bubble size. The authors suggested that the specimens were far from stoichiometric since they were prepared by the oxidation of thin uranium foils in the electron microscope. Two studies of random migration of bubbles during isothermal anneals of UO2 samples at 1500 and 1600°C by Cornell and Bannister (1967) used samples containing krypton injected as 100keV ions and precipitated into bubbles of radius 30-109 A. Their results suggested that bubble migration ranges vary directly with bubble radius, in contradiction to the predictions of all the predicting theories. An additional effect, perhaps due to the presence of precipitates in their samples, was suggested to have modified the relationship between bubble velocity and radius. Further experiments by Gulden (1967) used neutron irradiated material, followed by isothermally annealed at 1400, 1450 and 1500°C. After these treatments, the bubble diameters ranged from 25 to 140/~. Bubbles with radii greater than 37/~ had a dependence of bubble migration distance on bubble radius which was characteristic of a migration mechanism controlled by volume diffusion. Smaller bubbles migrated at a slower rate than predicted by a volume-diffusion-controlled mechanism. All the

Theories of fission gas behaviour bubbles studies by Gulden were associated with spherical precipitates of diameter ,-,20-50 A which may have inhibited surface diffusion by forming a layer of impurities. Gulden speculated that vacancy diffusion through a layer of contaminants (presumably solid fission products) is the rate-controlling process. Nichols (1969) suggests an alternate interpretation that the bubbles migrated by surface diffusion but faceting effects and interactions with dislocations and solid fission products, cause the low apparent mobilities and anomalous sizedependencies. The theories of diffusion controlled bubble motion in oxide fuel during neutron irradiation by either Brownian motion or by a driving mechanism has been considerably jeopardized by recent microscope observations suggesting that gas bubbles are essentially immobile. The first significant measurements supporting the bubble immobilization theory were undertaken by Cornell et aL (1969) from examination of intergranular bubbles in irradiated UO~ material (which acquired a burnup of 3.2 x 10t9 fissions/cm3 in 40 days) where the estimated irradiation temperature lay in the range 8601580°C. Their results showed no detectable variation in either bubble size or density across a grain diameter in the direction of the temperature gradient, suggesting that directed motion of these bubbles was not occurring. Many of the bubbles were aligned, presumably along fission fragment tracks. The theory that the displacement spikes produced along the track act either as nucleation or pinning sites for gas bubbles was proposed. This theory has been substantiated by Cornell's (1971) examination of samples from UO~ pellets irradiated to doses between 3.2 x 1019-4.6 x 1030 fissions] cm3 at temperatures between 700-1600°C. No evidence for the directed motion of gas bubbles towards the grain boundaries was found. Moreover, the uniformity in size of the gas bubbles throughout the grain led Cornell to confirm Speight's (1969) postulate that the random motion of bubbles was an unimportant process in the redistribution of fission gas within a grain. Only the investigations carried out on ion-bombarded material suggested that bubbles migrate by surface diffusion, whereas those on materials containing solid fission products indicated substantial anomalies in bubble motion behaviour with possible immobilization of small bubbles by structural defects (e.g. dislocations, dislocation loops) and by faceting (Warner and Nichols, 1971). Furthermore, due to the re-solution effect, small bubbles can be completely destroyed before they move far enough to collide with each

51

other, The preceding examinations were performed at temperatures no higher than 1600°C. Fortemperatures higher than 1800°C, the increasing agitation with temperature of matrix ions around the bubble may offset the immobilization and]or retarding effects of gas re-solution, structural defects, and solid fission product precipitates. Retarding forces on bubbles due to structural defects

Non-homogeneous spatial distributions of bubbles and bubble concentration near to structural defects (dislocations, grain boundaries, particulates, etc.) can be treated by developing retarding forces due to interaction with the structural defects. If a bubble becomes attached to a dislocation line the energy of the system can be lowered (Barnes and Nelson, 1965). When the bubbles move relative to the dislocation line, the dislocation lines become disturbed and a Fa force is exerted on the bubble due to the line tension of the bowed dislocation (Nichols, 1969) Fa = 2, cos $ ~ #b 2 cos $,

(42)

where 25 is the angle between the two dislocation segments at their point of intersection with the bubble surface,/~ is the shear modulus and b is the magnitude of the Burger's vector. This force reaches its maximum value for ~ = 0 at which time the dislocation segments come together. A moving dislocation can drag a bubble along with it while a stationary dislocation line (e.g. due to pinning at some points distant from the bubble) will retard the motion of a bubble. Thus, dislocations can provide a restraining force which will balance a driving force on the bubble (e.g. due to a thermal gradien0 up to the maximum value /~b~. If this force exceeds/~bz then the bubble will be able to move. In the case of a thermal gradient the driving force is proportional to the volume of bubble and when the bubble reaches critical size the bubble will pull off the dislocation (equation 41 is equated to equation 42). For example, if T ,-~ 1000°K and VT ,-, 1000°C/cm, Barnes and Nelson (1965) have estimated a value of ,-~600/~ for an. Since the bubble-dislocation interaction is of sufticient strength to bow a dislocation, dislocations which are not pinned can remain attached to migrating bubbles (Li et al., 1970) or a dislocation dipole can be formed behind a migrating bubble. Such dislocation interactions may significantly reduce bubble velocity (Weeks, Scattergood, and Pati, 1970). The influence of an attached dipole on the velocity of a migrating bubble can be

J. W. C. D~AS and K. R. MERCKX

52

accounted for by modifying the driving force to include the loss of energy per unit length of dipole created (a reduction of ,-,(/~b2/2~r) In (a]b - 1) for a dipole with screw dislocation segments). Grain boundary tension Tab (Nichols, 1969) will exert a force of 2~rTgb a sin 4 on a bubble along the surface of the cone defined by the intersection of the boundary with the bubble. The resolved force Fab is exerted along the cone axis Fgb = ~raygbsin 2~,

(43)

where ff is the half angle of this cone. For the case of a thermal gradient driving force on the bubble, the critical radius, a~g, for bubble trapping by the boundary occurs when Fgb reaches its maxim u m OraTgb) and is equal to the thermal gradient force. This critical radius is agb "~ (3 f~TubT/4Q* AT) 1/2

For T ,-, 1000°K and VT N 1000°C/cm, one finds a critical bubble radius, agb to be a few thousand A compared to aa = 600 ~ previously determined for pull off from a dislocation. Observations of grain-boundary formation behind large migrating lenticular bubbles (Nichols, 1968; Burke, 1968) leads to the formation of dislocation dipoles and reduced migrating velocity. The effective driving force may be modified (Weeks, Scattergood, Pati, 1970) to be the thermal force, equation (41), minus the grain boundary retarding potential (4' = *r/2 in equation 43) where the grain boundary cylinder trailing the bubble is assumed to have a circumference 2~ra which agrees with Burke's (1968) observations of migrating lenticular pores having trailing grain boundaries attached to the major circumference of the bubble. When large bubbles become pinned on grain boundaries they tend to become lenticular in shape because of the high energy of the grain boundary. The lenticular bubble may be approximated as an oblate spheroid with minor axis ~ and with major axis perpendicular to the thermal gradient. The driving force due to a thermal gradient as affected by the new geometry has the numerical factor, 0 in equation (41) given by Weeks, Scattergood and Pati (1970).

cipitate can only move by atoms diffusing around the surface or by volume diffusion, it is likely to move more slowly than the bubble. This behaviour has been substantiated with post-irradiation examinations of UO2 fuel pellets (Manley, 1968; Michels and Poeppel, 1973) where bubbles containing precipitates have been observed to move more slowly than those without precipitates. The attachment of solid fission product precipitates to bubbles versus bubbles with included precipitates could explain the discrepancy between the observations by Cornell and Bannister (1967) and Gulden (1967). Cornell and Bannister, for example, proposed that the attachment of bubbles to very small precipitates (diameter > 2 0 A ) may impede the migration of small bubbles more than the large ones thus explaining their observation that bubble migration ranges varied directly with bubble radius. Gulden (1967) noticed that the small precipitates generally lay within the bubbles and could move with the bubbles by a non-diffusional process if solid fission product precipitates are small enough. Reduction in bubble mobility in this case is by the suppression of surface diffusion of matrix atoms due to the presence of a layer of solid fission product contaminants. REFERENCES Barnes, R. S. and S. J. Mazey (1963) Proc. Roy. Soc. A-275, 47-57. Barnes, R. S. and D. J. Mazey (1964) Proceedings of the 3rd European Regional Conference on Electron Microscopy, pp. 127--139. Barnes, R. S. and R. S. Nelson (1965) Atomic Energy Research Establishment, AERE-R 4952.

Booth, A. H. (1957) AECL-496 (CRDC-721). Brinkman, J. A. and W. V. Johnston (1967) Metallurgical Society Conferences 37, 275-301. Brophy, ~I. H. et al. (1964) Thermodynamics of Structure. Wiley, New York. Brown, L. M. and D. J. Mazey (1964) Phil. Mag .10, 1081-1085. Burke, J. E. (1968) Science 161, (3847), 1205-1212. Chandrasekhar, S. (1943) Rev. Mod. Phys. 15, 1-89. Childs, B. G. (1963) J. nucl. Mat. 9, 217-244. Cornell, R. M. (1969) Phil. Maff. 19, 539-554. Cornell, R. M. (1971) J. nucl. Mat. 38, 319-328. Cornell, R. M. and G. H. Bannister (1967) Proc. Br. ceram. Soc. 7, 355-362. Cornell, R. M., M. V. Speight and B. C. Masters (1969) J. nucl. Mat. 30, 170--178. Cornell, R. M. and J. A. Turnbull (1971) J. nucl. Mat. 0 = { 1 - [~2/(~2 _ 7/2)] [ 1 - - [ r ~ / ( ~ 2 -- ~2]1/~ 41, 87-90. x cot -1 [~/(~2 _ ~211/~]}-1 Cornell, R. M. and G. K. Williamson (1965) J. nucl. Mat. 17, 200. Davies, D. and G. Long (1963) AERE-R 4347. where a 3 was replaced by ~2v. Bubbles will also adhere to any internal boundary Duncombe, E., C. M. Friedrick, and J. K. Fischer, (1970) Bettis Atomic Laboratory, WAPD-TM-961. including that at the interface of a precipitate Eyre, B. L. and R. Bullough (1968) J. nucl. Mat. 26, particle (Barnes and Nelson, 1965). Since a pre249-266.

Theories of fission gas behaviour Greenwood, G. W., A. J. E. Foreman and D. E. Rimmer (1959) J. nucl. Mat. 4, 305-324. Greenwood, G. W., and M. V. Speight (1963) d. nucl. Mat. 10, 140-144. Gulden, M. E., (1967) d. nucl. Mat. 23, 30-36. Ham, F. S. (1958) J. Phys. Chem. Solids, 6, 335-351. Hurst, D. G. (1962) AECL-1550 (CRRP-1124). Jankus, V. A. and R. W. Weeks (1972) Nucl. Engr. & Design 18, 83-96. Kornelson, E. V. (1964) Can. d. Phys. 42, 364-381. Kulcinski, G. L., R. D. Leggett, C. R. Harm and B. Mastel (1969) d. nucl. Mat. 30, 303-313. Kulcinski, G. A. and R. D. Leggett (1969) d. nucl. Mat. 31, 279-287. Leiden, S. H. and F. A. Nichols (1971) J. nucl. Mat. 38, 309-313. Li, C., S. R. Pati, R. B. Poeppel, and R. D. Scattergood (1970) Nucl. Appl. & Tech. 9, 188-194. MacEwan, J. R. and P. A. Morel (1966) Nucl. AppI. 2, 158-170. MacEwan, J. R. and W. H. Stevens (1964) J. nucl. Mat. 11, 77-93. Manley, A. J., (1968) d. nucl. Mat. 27, 216-224. Marlowe, M. O., (1970) GEAP-12148. Nelson, R. S. (1968) J. nucl. Mat. 25, 227-232. Nelson, R. S. (1969) J. nucl. Mat. 31, 153-161. Michels, L. C. and R. B. Poeppel (1973) J. appl. Phys. 44, 1003-1008. Michels, L. C., R, B. Poeppel and L. A. Neimark (1970) Trans. Am. nucl. Soc. 13, 601-602. Nichols, F. A. (1966) J. appl. Phys. 37 (7), 2805-2808. Nichols, F. A. (1967) J. nucl. Mat. 22, 214-222. Nichols, F. A, (1968) J. nucl. Mat. 27, 137-146. Nichols, F. A. (1969) J. nucl. Mat. 30, 143-165.

53

Nichols, F. A. and W. W. Mullins (1965) Trans. met Soc. A1ME 233, 1840-1848. Noggle, T. S. and J. O. Steigler (1960) J. apFl. Phys. 31, 2199-2208. Pati, S. R. (1971) Trans. Am. nucl. Soe., 14 (2), 580-581. Ronchi, R. and H. Matzke (1971) Europaisches Institut fur Transurane, EURATOM, Karlsruhe, Germany, Interner Berichter Nr. 179. Ross, A. M. (1969), J. nucl. Mat. 30, 134-142. Sha, W. T. and T. H. Hughes, (1970) Argonne National Laboratory, ANL-7701. Shewmon, P. G. (1964) Trans. met. Soc. AIME 230, 1134-1137. Speight, M. V. (1968) Met. Sci. 3". 2, 73-76. Speight, M. V. (1969) Nucl. Sci. Engng 37, 180-185. Turnbull, J. A. (1968) Berkeley Nuclear Laboratories, RD/B/N 1112. Turnbull, J. A. (1971) J. nucL Mat. 38, 203-212. Turnbull, J. A. and R. M. Cornell (1970) 2. nucl. Mat. 36, 161-168. Turnbull, J. A. and R. M. Cornell (1970) d. nucl, Mat. 37, 355-357. Turnbull, J. A. and R. M. Cornell (1971) d. nucl, Mat. 41, 156-160. Warner, H. R. and F. A. Nichols (1970) NucL AFFlic. Technol. 9, 148-166. Warner, H. R. and F. A. Nichols (1971) Proceedings of the Conference on Fast Reactor Fuel Technology, pp. 267-289. Weeks, R. W., R. D. Scattergood and S. R. Pati (1970) J. nucL Mat. 36, 223-229. Whapham, A. D. (1965) Trans. Am. nucl. Soc. 8, 21-22. Whapham, A. S. and M. J. Makin (1962) Phil. Maff. 7, 1441-1455.