THERMODYNAMIC and TRANSPORT PROPERTIES of URANIUM DIOXIDE and RELATED PHASES the Following States Are Members of the International Atomic Energy Agency

Home , Seebeck coefficient, Uranium dioxide, Uranium oxide

TECHNICAL REPORTS SERIES No. 39 Thermodynamic

and Transport Properties i of Uranium Dioxide i • ¡i and Related Phases

INTERNATIONAL ATOMIC ENERGY AGENCY,VIENNA, 1965

THERMODYNAMIC AND TRANSPORT PROPERTIES OF URANIUM DIOXIDE AND RELATED PHASES The following States are Members of the International Atomic Energy Agency:

AFGHANISTAN ITALY ALBANIA IVORY COAST ALGERIA JAPAN ARGENTINA REPUBLIC OF KOREA AUSTRALIA LEBANON AUSTRIA LIBERIA BELGIUM LIBYA BOLIVIA LUXEMBOURG BRAZIL MALI BULGARIA MEXICO BURMA MONACO BYELORUSSIAN SOVIET SOCIALIST MOROCCO REPUBLIC NETHERLANDS CAMBODIA NEW ZEALAND CAMEROUN NICARAGUA CANADA NIGERIA CEYLON NORWAY CHILE PAKISTAN CHINA PARAGUAY COLOMBIA PERU CONGO (LÊOPOLDVILLE) PHILIPPINES CUBA POLAND CZECHOSLOVAK SOCIALIST REPUBLIC PORTUGAL DENMARK ROMANIA DOMINICAN REPUBLIC SAUDI ARABIA ECUADOR SENEGAL EL SALVADOR SOUTH AFRICA ETHIOPIA SPAIN FINLAND SUDAN FRANCE SWEDEN FEDERAL REPUBLIC OF GERMANY SWITZERLAND GABON SYRIA GHANA THAILAND GREECE. TUNISIA GUATEMALA TURKEY HAITI UKRAINIAN SOVIET SOCIALIST REPUBLIC HOLY SEE UNION OF SOVIET SOCIALIST REPUBLICS HONDURAS UNITED ARAB REPUBLIC HUNGARY UNITED KINGDOM OF GREAT BRITAIN ICELAND AND NORTHERN IRELAND INDIA UNITED STATES OF AMERICA INDONESIA URUGUAY IRAN VENEZUELA IRAQ VIET-NAM ISRAEL YUGOSLAVIA

The Agency's Statute was approved on 23 October 1956 by the Conference on the Statute of the IAEA held at United Nations Headquarters, New York; it entered into force on 29 July 1957. The Headquarters of the Agency are situated in Vienna. Its principal objective is "to accelerate and enlarge the contribution of atomic energy to peace, health and prosperity throughout the world".

Permission to reproduce or translate the information contained in this publication may be obtained by writing to the International Atomic Energy Agency, Kärntner Ring 11, Vienna I, Austria.

Printed by the IAEA in Austria January 1965 TECHNICAL REPORTS SERIES No. 39

THERMODYNAMIC AND TRANSPORT PROPERTIES OF URANIUM DIOXIDE AND RELATED PHASES

REPORT OF THE PANEL ON THERMODYNAMIC AND TRANSPORT PROPERTIES OF URANIUM DIOXIDE AND RELATED PHASES HELD IN VIENNA 16 -20 March 1964

INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA,. 1965 International Atomic Energy Agency. Thermodynamic and transport properties of uranium dioxide and related phases. Report of the Panel on Thermodynamic .. ., held in Vienna, 16- 20 March 1964. Vienna, the Agency, 1965. 105 p. (IAEA, Technical reports series no. 39)

541.11 546. 791. 4-31 621. 039. 543.4

THERMODYNAMIC AND TRANSPORT PROPERTIES OF URANIUM DIOXIDE AND RELATED PHASES, IAEA, VIENNA, 1965 STl/DOC/10/39 FOREWORD

Because of the growing importance of thermodynamics to nuclear technology, the International Atomic Energy Agency has initiated a project to assist in assessing and disseminating data on important nuclear materials. As a beginning, it organized a Symposium on the Thermodynamics of Nuclear Materials which was held in Vienna in May 1962. This was followed by a Panel on the Thermodynamic Properties of the Uranium-Carbon and Plutonium-Carbon Systems, held in Vienna in October 1962. The present Report is the result of a further Panel in this series, convened from 16 - 20 March 1964 to assess the thermodynamic and transport properties of the uranium dioxide phase and related uranium oxide phases. This Panel made a critical evaluation of the available data, bearing in mind the practical aspects of the use of uranium dioxide as a nuclear fuel. The findings of the Panel are presented by the Agency in this issue of the Technical Re- ports Series in the belief that they will prove to be of value for nuclear technology. The Report was compiled and edited by Dr. Charles Holley of the Division of Research and Laboratories.

CONTENTS

I. INTRODUCTION 1

II. STRUCTURAL WORK 3 1. Stable phases in the U-O system 3 2. UO2 (room temperature) .' 3 2.1. Lattice parameter. Density. 3 2.2. Atomic positions. Temperature factors 3 3. UO2 (high temperature) 5 3.1. Variation of lattice parameter with temperature 5 3.2. Temperature factors. Characteristic temperatures. Breakdown of harmonic approximation 6

4. U02 + x region 7 4.1. Variation of lattice parameter and density with composition 7 4.2. Crystal structure 9 4. 2.1. Atomic positions in statistical cell. Occupation numbers. Temperature factors .... 9 4. 2. 2. Interpretation of results for statistical cell .... 12

5. U409 5.1. Variation of lattice parameter with composition and temperature 15 5.2. Structure 15 5.2.1. X-ray studies 15 5.2.2. Neutron studies 17 6. Tetragonal phases 19 7. Conclusions 20

III. THERMODYNAMICS ... 23

1. Heat capacity measurements 23 1.1. Low temperature heat capacity data 23 1.2. High temperature heat capacity data 24 2. Lattice Dynamics of UO2 , 25 3. Free energy, enthalpy, and entropy measurements 29 3.1. Chemical thermodynamics of the UO2. oo - 2.25 region... 29

3.2. The phase diagram, U02 to U4O9 38 3.3. Hypostoichiometric U02 39 4. Vaporization processes 40 5. Theoretical treatment of U02 + x phase 42 5.1. Statistical thermodynamics of interstitials and vacancies • 42 5.2. Application of defect theory to UO2+ x 44

IV. SURFACE AND OXIDATION PROPERTIES 51

1. Adsorption properties 51 2. Oxidation processes 52 2.1. Low temperatures 52 2.2. High temperatures 53

V. PHYSICAL PROPERTIES 55 1. Thermal conductivity 55 1.1. Low temperature thermal conductivity 55 1.2. High temperature thermal conductivity 55 1.2.1. Lattice conductivity 57 1.2.2. Radiant transfer 58 1.2.3. Electronic transfer 59 2. Electrical properties 60 2.1. Normal electrical properties 60 2. 2. Effect of irradiation on electrical properties 67 3. Optical measurements 69 3.1. Intrinsic absorption edge 69 3.2. Defect absorption 70 3.3. Infra-red absorption 72 4. Magnetic measurements 73 5. Diffusion processes in UO^ 75 5.1. Oxygen diffusion 75 5.2. Uranium self-diffusion 76 5.3. Argon diffusion in calcium fluoride as a model process for fission gas transport in uranium dioxide .. 76 5.4. Fission gas release 78 6. Correlative theory of physical properties 81 6.1. Transport of energy 81 6. 2. Transport of matter 83

VI. PRACTICAL IMPLICATIONS OF THERMODYNAMIC AND TRANSPORT PROPERTIES 85

» 1. Interaction of fuel and can 85 1.1. Thermal cracking 85 1.2. Dimensional changes in UO2 under irradiation 86 2. Thermal conductivity 88 3. Phase equilibria ' 88 4. Material transport processes 89

VII. CONCLUSIONS 93

Appendix: Mathematical treatment of defect absorption 95

References 99

List of participants 103

Reports submitted to the Panel 105

I. INTRODUCTION

The high melting point of uranium dioxide and its stability under irradiation have led to its use as a fuel in a variety of types of nuclear reac- tors. A wide range of chemical and physical studies has been stimulated by this circumstance and by the complex nature of the uranium dioxide phase itself. The boundaries of this phase widen as the temperature is increased; at 2000°K a single, homogeneous phase exists from U2.27 to a hypostoichiometric (UO2-X) composition, depending on the oxygen potential of the sur- roundings. Since there is often an incentive to operate a reactor at the maximum practicable heat rating and, therefore, maximum thermal gradient in the fuel, the determination of the physical properties of the U02±xphase becomes a matter of great technological importance. In addition a complex sequence of U-O phases may be formed during the preparation of powder feed material or during the sintering process; these affect the microstructure and properties of the final product and have also received much attention. Uranium dioxide, therefore, provides an important example of a compound that exists as a single non-stoichiometric phase at high temperatures and becomes unstable as the temperature is reduced, disproportionating into phases of nearly ideal stoichiometry involving more or less complex ordered structures. Ideally, the thermodynamic stability and physical properties of U02±x should be related to the same atomic and electronic model, and its study should provide an opportunity for the correlation of a number of different properties. The International Atomic Energy Agency (IAEA) therefore called a panel meeting to discuss the thermodynamic and transport properties of the non- stoichiometric uranium dioxide phase, and this Report presents a summary of the data placed before the Panel and of the conclusions reached. A considerable amount of data on the main features of the phase diagram and on the composition limits of the various phases exists and X-ray and neutron diffraction evidence indicate some possible structural models. Chemical thermodynamic values are known with some precision for most of the region concerned. Specific discussions were held on (i) the interrelation of vibrational constants deduced from structural work and heat capacity data; (ii) the correlation of thermal conductivity with electrical conductivity and optical data; and (iii) the calculation of entropy values by the statistical treatment of simple models that are consistent with the structural, optical, and electrical properties. The outline of a generalized theory that should allow better correlation of transport and thermodynamic properties in the future was presented. The importance of making all measurements of physical properties on samples of accurately known composition, which are struc- turally well-characterized, was emphasized.

II. STRUCTURAL WORK

1. STABLE PHASES IN THE U-O SYSTEM

There are as many as 16 well-characterized uranium oxide phases, and the existence of a dozen more has been claimed. A survey of work on the uranium-oxygen phase diagram up to 1961 has been made by ROBERTS [1]. Figure 1 is a reproduction of his phase diagram. The work described in

Fig. 1

Portion of U - О phase diagram. Circles denote X-Ray results. (Reproduced by courtesy of L.E.J. Roberts [1])

this chapter is restricted to U02, U02 + x, U4O9 and the tetragonal phases with compositions in the rangfe U02 3 to U02 4 . The structures of these phases are based on the fluorite arrangement, with the additional oxygen atoms distributed either at random on the fluorite lattice or in an ordered fashion, forming a cubic or tetragonal superlattice.

2. U02 (ROOM TEMPERATURE)

2.1. Lattice parameter density

The stoichiometric oxide U02 has a cubic structure. The generally accepted value for the lattice parameter is a0 = 5.470Â (see Table I), which corresponds to a theoretical density, assuming four U02 units in the unit cell of 10.952 g/cm3. This theoretical value is close to the density of

10.950 ± 0.005 recently measured on a single crystal of vapour-grownU02 [2].

2.2. Atomic positions. Temperature factors

The crystal structure of U02 was first determined by GOLDSCHMIDT and THOMASSEN [8]. The atomic positions are those for the fluorite arrangement (Fig. 2). Interatomic distances are [9] ,

3 И - 12U = 3.868Â О - 60 = 2.735Â U - 80 = 2.369Â.

The radius ratio (cation/anion) is 0.73; according to PAULING [10] 0.73 is the critical ratio, below which the fluorite structure is less stable than the tetragonal rutile structure.

TABLE I

LATTICE PARAMETER OF U02

Parameter Reference (A)

5.4690 ± 0. 0001 [3]

5.4704 ± 0. 0008 14]

5.4703 ± 0. 0002 [5]

5.4698 i 0. 0008 [6]

5. 4720 ± 0. 0005 [7]

5.4698 ± 0. 0002 [75]

This structure has been confirmed by neutron diffraction [11] . The coherent nuclear scattering cross-section of oxygen is about one half of that for uranium; consequently the oxygen atoms can contribute appreciably to the observed neutron intensities and can be located directly with neutrons.

Moreover, the absorption cross-section of U02 for slow neutrons is at least four orders of magnitude less than for X-rays, so that more accurate measurements of the integrated intensities are possible by neutron diffraction. The neutron measurements give the following values for the isotropic

atomic temperature factors, Ви and B0 [11]: ,

2 2 Вц= 0.25 i 0.04 A , B0 = 0.43± 0.05Â .

The В's are the quantities appearing in the Debye - Waller factor exp(-2W) = exp{-2B(sin0/X)2} where 0 is the Bragg angle and X the wave length. The "R factor", giving the discrepancy between calculated and observed structure factors, is less than 2%. From Willis's study [11] it is

seen that, to a very close approximation, the structure of U02 at room temperature is described by the fluorite arrangement with isotropic thermal motion of the uranium and oxygen atoms. The temperature factor В is related to the mean-square displacement ТГ| of the atom in any direction from its mean position by the equation

В = 8тг2 XjJ .

4 Fig. 2

Unit cell of uranium dioxide. Open circles are oxygen, shaded circles are uranium, and open squares are holes in the fluorite structure. Broken lines connect oxygen atoms which form a cubic array around the central hole.

(The total mean-square displacement is 3 U^. ) Inserting the values quoted above gives rms displacements

J U^ = 0.056 Â for uranium

= 0.074 Â for oxygen.

3. U02 (HIGH TEMPERATURE)

3.1. Variation of lattice parameter with temperature

BAKER andBALDOCK[12]have observed a smooth variation of lattice parameter with temperature up to 2300°C, the highest temperature of measurement. Table II gives the linear expansion coefficient deduced by several

TABLE 11

LINEAR EXPANSION COEFFICIENT (X-RAYS) OF UO2

Temperature range Expansion coefficient Reference

6 20 - 20 0 0 9.4 X 10" /°C [12]

20 - 950 10.8 X 10-6/oC [4]

6 20 - 800 9. 9 X 10" /°C [7]

20 - 1000 10. 5 X Ю-6 /°C [13]

5 authors from measurements of the lattice parameter in the temperature range indicated. The X-ray expansion coefficient is in reasonable agreement with the expansion coefficient obtained by dilatometry [9].

3.2. Temperature factors. Characteristic temperatures. Breakdown of harmonic approximation

It is possible to relate the temperature factors В at different temperatures with the characteristic temperatures of vibration of the solid. The Debye model can be used for representing the vibrations of the uranium atoms and the Einstein model for the lighter oxygen atoms. The results, discussed in Section III.2. (Lattice dynamics) are

= 0 Debye 182°K

в = 542°K. Einstein As the temperature of UO2 increases, the intensities of the neutron reflections depart progressively from those calculated for the fluorite model assuming isotropic thermal motion of the individual atoms. The effect is shown in the differences in intensities of reflections occurring at the same Bragg angle [14]. These differences can be accounted for in a phénoméno- logie al manner by displacing the oxygen atoms along the four (111) directions towards the adjacent holes in the structure (Fig.3). This displacement can be interpreted in one of two ways: either the oxygen atoms are displaced at random along the <111) directions to give a disordered structure, or the oxygen atoms undergo anisotropic vibrations across the ideal fluorite position (Fig. 4). The general occurrence of the effect in U02 and Th02 [14] and in CaF2 [15], and its moderate dependence on temperature, favour the second interpretation. MARADUDIN and FLINN [16] have shown that the anisotropic vibration of atoms in positions with cubic point symmetry arises from anharmonic contributions to the Debye-Waller factor; it is clear that the harmonic approximation cannot be used to interpret the temperature factors of U02 at high temperature.

The neutron diffraction results for U02 can be summarized as follows: (i) The uranium atoms occupy fluorite-type positions and execute isotropic thermal motion with a mean-square amplitude increasing with temperature. At low temperatures zero-point motion leads to a departure from linearity, but above 400°C Ü| is proportional to the absolute temperature T and is given by

Üf (uranium) = 1.33 10"5 T (Â2).

At the highest temperature of observation, 1100°C, there is no evidence of anisotropic vibration. (ii) The oxygen atoms occupy fluorite-type positions but vibrate isotropically at low temperatures only. The shape of the "vibrating surface" can be described roughly in terms of four spheres (Fig. 3). At room temperature these spheres are superimposed, but at 100°C they begin to separate along the four <(111 > directions joining the oxygen atom with its

6 Fig.3

Vibrational surface of oxygen atom. Above 100°C the oxygen atom is displaced along the four [111] directions as shown. The large cross-hatched circle is the oxygen atom with the displacement along [111] shown stippled. The small shaded circles are uraniums and the open squares are holes.

Fig. 4

Two models for vibration of oxygen atoms in U02. Looking down [111]: (a) Oxygen atoms located statistically at displaced positions with each atom vibrating isotropically;and (b) Anharmonic vibration causing asymmetrical displacement of each atom across the normal position. surrounding tetrahedron of holes. At 1000°C each sphere is displaced by as much as 0.15 Â from the ideal position [14] .

4. U02 +X REGION

4.1. Variation of lattice parameter and density with composition

GRpNVOLD [4] using high temperature X-ray techniques has shown that oxygen dissolves in a homogeneous U02j.x phase above 400°C, theU02+x domain extending to U02 t7 at 950°C (Fig. 1).

This section deals with the structure of this U02+x homogeneous phase. X-ray measurements on quenched samples show that the unit cell con- tracts with increasing oxygen concentration [3, 6, 17, 18, 19, 20] . There is

7 _0 и

Fig. 5

Variation of lattice parameter with U/O ratio. relatively good agreement between different X-ray studies of compositions up to UO2.12 . but not for higher O/U ratios (Fig. 5). LYNDS et al. [20] failed an to observe the U02 + x structure between U02.13 d UO2.17 when quenching from 1100°C, whereas BELBEOCH et al. [6] succeeded in quenching theU02+x phase from a lower temperature. This discrepancy may be due to a more complex phase diagram than is generally recognized; Belbeoch suggested that for compositions higher than U02.13 the system may become diphasic again above 1000°C [6], or that there is a shift of the phase boundary between monophasic and diphasic regions towards lower O/U ratios (cf. Fig. 1)[6,20], The quenching experiments may be vitiated by the extremely rapid self- diffusion of oxygen, and it is necessary to re-investigate the U02. io~U02. 2r region above 950°C with high-temperature X-ray techniques. Even then the X-ray method, using silica capillaries, is complicated by the reactivity of Si02 with uranium oxide at high temperature. It is generally agreed that excess oxygen enters interstitiell positions in the U02 lattice, causing a contraction of the lattice spacing, and that the U4O9 structure is an ordered structure. In principle, this model could be verified exactly by comparing the increase in density actually measured with the increase calculated from the interstitial model and the cell dimensions. In practice, the exact comparison is difficult for two reasons, namely. (a) the U02 + x structure is not easily quenched-in, so that samples at room temperature tend to be mixtures of U02 and a poorly-crystalline U4O9, and (b) many U02 preparations have densities much lower than the calculated value of 10.96 g/cm3, due to microstructure faults, and these tend to alter during Oxidation. Furthermore, the densities of powders which have been exposed to air are lower than theoretical due to the chemisorption of oxygen on the surface. Some of the most complete sets of density measurements reported to date are the following: to (i) The densities of samples of composition U(\026 U°2.234 which had been prepared by oxidation at temperatures below 165°C were determined

8 at the Atomic Energy Research Establishment (AERE), Harwell. The density increased regularly with composition from 10.89 to 11.21 g/cm3; the was density of an annealed specimen of U409 (U02. 240 ) 11.18 g/cm3, giving 3 as a density increase of 0.29 g/cm from UO2.026 to UO2. 240 » compared with a calculated value of 0.28 g/стз [181]. (ii) GRJÖNVOLD [4] measured a regular density increase on samples that had been heated to high temperatures and cooled, but these were certainly mixtures of U02 and U409. The measured density difference between 3 U02 and U4O9 was 0.376 g/cm .

(iii) The densities of samples of U02 + x which had been quenched from 1000°C were measured by LYNDS et al. [20] . According to their X-ray measurements, the quench was successful. The density increased with oxidation, but quantitative comparison with theory was impossible because of the low density of the starting material, indicative of closed porosity.

(iv) Density measurements on oxidized samples of (U^Th^ )02 solid solutions, which are stable at room temperature, always indicated an increase on oxidation, but quantitative comparison is agian vitiated by microstructure faults [72] .

The evidence favours a model in which an 02 molecule dissolves in U02 to give a net gain of exactly two interstitial О atoms, but the quantitative 3 evidence is not very good except in the case of U409 where d = 11.2 g/cm [2,4,20,23].

4.2. Crystal structure

When U02 is oxidized to U02 + x no extra lines appear on the X-ray photographs. Examination of single crystals by neutron diffraction confirms that no additional reflections appear, thus the space group of U02+x is the same (Fm3m) as for UO 2. The term "space group" now refers to the symmetry properties of the "statistical cell" obtained by superimposing all the original fluorite-type cells. Provided the extra atoms are distributed randomly, the space groups of the U02 unit cell and the U02J.X statistical cell must be equivalent. Short range ordering between small groups of interstitial oxygens can occur, but any long range ordering, giving a different space group, is precluded.

4.2.1. Atomic positions in statistical cell. Occupation numbers. Temperature factors

Willis has studied single crystals of compositions U02 12 and U02 13 C at 800 C by neutron diffraction [21] . Two-dimensional Fhkk data were collected on U02.13 and three dimensional F^j data on U02 12. The analysis of both sets of data gave similar results, but because the U02 12data were more complete only these will be discussed below. The general expression for the structure factor of the hkl neutron re- flection can be written

I =n F = m b 2 2 hki ^ r r exp(2TTi) (hxr + kyr + lzr) exp(-Brsin 0/X ) (1) r = l 9 where n is the number of atoms in the statistical cell, th mr is the occupation number of the r atom,

br is the coherent scattering amplitude of the rth atom. th хгУг2г are the positional co-ordinates of the r atom, Br is the isotopic temperature factor of the rth atom, в is the Bragg angle, and X is the wave-length.

The unknowns in this expression are the number of atoms n in the (statistical) cell, the positional co-ordinates xyz of each atom, their temperature factor B, and the relative proportions m of each atom in the cell. All these parameters can be derived by a least-squares treatment in which the calculated and observed F's (38 independent hkl'sin all)are matched together. The least-squares results are summarized in Table 111. These results will be discussed in turn for each kind of atom.

TABLE III

U02_12: LEAST-SQUARES RESULTS FOR STATISTICAL CELL

Temperature Co-ordinates in Contribution factor , Atom statistical cell m to formula UOm (Á2)

X У z

Uranium 0 0 0 1.18 ± 0. 02

Oxygen О u u u m = 1. 87 ± 0. 03 1.45 ±0.04

Oxygen O' 0. 5 V V 0. 08 ± 0. 04 1. 8 ±1.4

Oxygen O" w w w 0.16 ± 0. 06 2. 0 ± 1. 6

Oxygen O'" 0.5 0.5 0. 5 -0. 02 ± 0. 02 2.0 (fixed)

u = 0.267 ± 0. 001, v = 0. 38 ± 0.01, w = 0. 41 ± 0. 01, and the "discrepancy factor" R = 3.5%.

(a) Uranium. The uranium atoms remain fixed at the equivalent positions: 000, All attempts at refinement with the uraniums displaced from these positions were unsuccessful. The occupation number for uranium was fixed at 1.00, in accordance with the lattice parameter and density results; if this assumption is wrong the figures in the third column of Tablelll must be adjusted appropriately. The temperature factor В = 1.18Ä2 is 2 slightly higher than that (0.90 Â ) for U02 at the same temperature. The difference can be ascribed to a random rms displacement of the uranium atoms from their normal positions of about 1/10Ä.

10 (b) Normal oxygen. The oxygen atoms occupying fluorite-type sites are labelled О in Table III. As in the case of U02 at 800°C, there is an apparent displacement from to^ + ô -5+6^+6 ... where 6 = 0.017, and this displacement can be ascribed to anharmonic vibration, which causes the oxygens to vibrate asymmetrically across the normal position. The magnitude of this effect is slightly greater than in U02. The most important change in О concerns its occupation number.

Whereas all the .... positions are filled in U02 00 an appreciable proportion are empty in U02 12 . Unfortunately, the standard deviations of the occupation numbers in Table III are high, but the proportion of empty sites probably lies between five and eight per cent.

(c) Interstitial oxygen. The most natural place to accommodate the extra oxygens is at the holes Щ ... . Oxygen at these sites are labelled O"1 in Table III. However, the occupation number for O'" refines at a negative value close to zero, and this, together with the difficulty of refining the temperature factor (kept fixed in Table III), discounts the possibility of these positions being occupied. In actual fact the extra oxygens occupy two kinds of site, labelled O' and O" in Table III. These are shown in the statistical cell in Fig. 6; the O1 sites are approximately half way from the hole to the centre of the line joining two normal oxygen atoms, and the O" sites half way from the hole to the nearest normal oxygen. The displacement of the interstitial atoms from the nearest hole is roughly 1 Â.

Fig. 6

Statistical cell of U02+x showing interstitial oxygen atoms in:

(a) 0' sites; (b) 0" sites.

Open circles are oxygens in fluorite-type sites, solid circles are interstitial oxygens, and open squares are holes at • • •

In U02+x, therefore, there are three types of defect in the oxygen sublattice associated with the departure from stoichiometry, namely, interstitial oxygens át О1 sites and O" sites, and normal oxygen vacancies. These three defects cannot be distributed at random, as this inevitably brings several oxygen atoms too close together, and it is more reasonable to suppose that the defects associate together into complexes or zones.

11 4.2.2. Interpretation of results for statistical cell

It is tempting to interpret the results in Table III in terms of the local

atomic configuration in U02iX . The number of structures compatible with this Table is unlimited, but can be reduced to a manageable number if we assume (a) the formation of defect zones rather than isolated defects, and (b) that each zone contains relatively few defects. The ratios of the concentrations

normal О vacancies : O' atoms : O" atoms

are given in Table III as 13 : 8 : 16. The model to be discussed first is one in which these ratios are idealized as 2 : 1 : 2. Ideal ratios suggest that an interstitial atom enters the lattice at an O' site, and thereby causes two normal oxygens to be ejected to two O" sites. This situation seems reason- 1 able, as an O atom can be comfortably placed inside a UOi00 lattice, provided the two nearest oxygens at 1.7 A are removed (Fig. 7).

ООО

Fig. 7

0' atom at A ejects the two nearest oxygens at В, C. В, С in turn are displaced along [111] towards the adjacent 0" sites.

• - uranium at Z = J a0 ; • - hole at Z = J a0; O- normal oxygen at Z = 0, £ a0;®- 0' oxygen at Z= i a.

A possible 2 : 1: 2 structure based on this concept is shown in Fig. 8. The dotted lines in this diagram outline four cubes of side ia0, and these are labelled from the top left-hand corner as I, II, IllandlV. In the UO2.00 arrangement, oxygen atoms are located at the corners of each of the ^a0 cubes. In UO 2 + y an extra oxygen atom enters the lattice at the position marked E, where E is approximately 1Â along the [110] direction from the centre of cube II. The two oxygens at A and В are displaced along the direction [111] to positions C, D on the body diagonals of cubes I, IV respectively.

12 Using the parameters quoted in Table III the positions of the atoms in Fig. 8 are as follows. The co-ordinates are given in units of a g, and the uranium atom G is chosen as the origin of co-ordinates.

U at 0.50 0 0.50 0 -0.50 0.50 0.50 -0.50 1.00 0 0 1.00 0.50 -0.50 0 0 0 0 Oat -0.25 -0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.75 etc.

O1 at 0. 12 -0. 12 0. 50 O" at 0.09 -0.41 0.91 0.41 -0.09 0.09

The bond distances are:

(A). O1 - 2U = 2. 18 O' - 20" = 2. 75 O1 - 40 = 2. 54 O" - 3U = 2. 35 0" - O' = 2. 75 o" - ЗО = 2. 24 0" - 30 = 2. 78

A difficulty with the 2:1:2 model is that, although the relative concentrations of defects are in approximate agreement with observation, the absolute concentrations are predicted incorrectly. For U02 12 the predicted formula is UOj 76 O' 0.12 O" 0. 24. whereas the observed formula (Table III) is UOx 87 О' о 08 O" 0. i6- The standard deviations of the occupation numbers are high (these standard deviations are calculated from the inverse least-square matrix and reflect the disagreement between observed and calculated intensities), but this still fails to account for the discrepancy between 1.76 and 1.87 for the number of normal oxygens in the formula unit. The difficulty may be resolved as indicated in Fig. 8. An extra O1 atom can be inserted in cube III at the position F, where E and F are equivalent

13 Uli,

Fig. 8

Model for U02+x structure. The normal oxygens, at A and В in UO2,00, are replaced by interstitial atoms 0" at C, D and 0* atoms at E (2 : 1 : 2 structure), or at E and F (2 : 2 : 2 structure). « Uranium atoms. О Normal oxygen atoms. © Interstitial oxygen atoms 0*. ® Interstitial oxygen atoms 0". sites related by 180° rotation about the line AB. This leads to an alternative 1 model for U02 + x, in which the numbers of O and O" atoms and normal oxygen vacancies in the central defect complex are' 2, 2, 2, For this 2:2:2 structure, the predicted formula, UOi. 88 O' 0.12 O" 0.12 is in good agreement with the observed formula, UO1-87±0.03 O'0 08 ± 0 04 O"016± 0 06- The bond lengths in the 2:2:2 structure are: (A) O' - 2U = 2. 18 O' - o* = 2. 01 O1 - 20" = 2. 75 O1 - 40 = 2. 54 O" - 3U = 2. 35 O" - 20' = 2. 75 O" - 30 = 2.24

O" - ЗО = 2. 78

14 The О' - О' distance (E-F Fig. 8) is short (2.01 Â), but a change of only one standard deviation in the v parameter (Table III) of O' changes this distances to 2.16Â.

5. U409

5.1. Variation of lattice parameter with composition and temperature

The homogeneity range of U409 is not well known. PERIC [3] and GRpNVOLD [4] considered that the composition is close to U0225 from 20 to 600°C and that there is a small spread on the oxygen deficient side for higher temperatures. On the other hand, SCHANER [18] found a wide range at 20 c and of hypostoichiometry; UO2.22 ° U02 20at 900°C. In the vicinity of U02 25 two types of diffraction pattern, with or without superlattice lines, have been observed [6]. It appears that there are two structures; namely, "U409±y", with superlattice lines and extra oxygens ordered on a 4a0 cubic cell, and "U02 25-y with a unit cell of about 5.445 A A homogeneity range of 0.03 in O/U was observed in the U^Og^y structure by BELBEOCH et al. [6] . The U02. 25-y structure is disordered, or is ordered in a different way than in U409, without significant displacements of the uranium atoms. It is difficult to deduce the composition from the a0 value in the U02.25-yregion, because of the dependence of this parameter on both composition and quenching temperature.

The thermal expansion of U409 has been studied by GR0NVOLD [4] and by FERGUSON and STREET [22] . The agreement between these results is excellent. There is a small contraction of the lattice parameter between 20 and 100°C, and thereafter the lattice parameter expands uniformly with teinper ature.

5.2. Structure

5.2.1. X-ray studies

For U409±y superlattice lines have been observed on X-ray [4, 23] , electron [24, 183] and neutron diffraction patterns [25, 11] . Since the superlattice lines are prominent at high angles on X-ray photographs, the uranium atoms must be displaced from the positions they normally occupy in the fluorite structure. X-ray studies of single crystals, obtained by oxidizing UO2 with U3O8 under the equilibrium pressure of the mixture U409 + U02 6 [23], showed that the unit cell is cubic with a = 4ao = 21.77 A. The observed extinctions for the hkl reflections indicate the l'ÏSd space group. A model has been proposed by BELBEOCH, PIEKARSKI and PERIO [23]. The 64 interstitial

oxygen atoms are ordered within the large 4a0 cell which contains 832 atoms. The ideal positions are given in Table IV. The extra oxygens in this ideal arrangement are distributed very heterogeneously (Fig. 9). There are twelve tetrahedra, each with four extra oxygens as nearest neighbours, and the remaining sixteen extra oxygens' are equidistant from three of the twelve tetrahedra. The complete determination of the structure would require the knowledge of 49 positional para-

15 TABLE IV

IDEAL POSITIONS OF THE 832 ATOMS

Nature of 48 (e) positions 16(c) 12 (a) and 12 (b) 24 (d) positions the 832 atoms positions (special positions )

(x У z) X X 0 ¿

0.1875 0. 0625 0. 0625

0.4375 0. 0625 0. 0625

256 U atoms 0.3125 0. 1875 0. 0625 0. 1875

0.4375 0.1875 0.1875

0. 3125 0. 0625 0.1875

0. 125 0. 25 0 0 0 4 0. 125 0.125 0 0.125 0 4

0 0 0 î 0 0.25 0 0.125 8 i 4 0.125 0.125 0. 25 0. 125

512 О atoms 0. 125 0. 25 0. 25 i 0 4 0.375 0 0.125

0. 5 0. 125 0.125

0.357 0. 125 0.125 64 extra-oxygens 0.1875 -0.0625 0. 0625 0. 0625 m

(Ь)

y y y (С)

« 002 * 003

(d)

Fig. S

(a) U02 unit cell (the origin is at a normal oxygen atom), (b) The 4 interstitial sites are described with the repetition-law. (c) Equivalent drawing to(b). Atoms on the same Z level lie on the same diagonal.

• Interstitial site О Normal oxygen atom X Uranium atom

(d) Distribution of the 64 extra oxygen atoms at the ideal positions 48(e) and 16(c).

meters which determine the displacements of 808 atoms from their ideal positions. For иОг.гб-у no superlattice lines have been detected. When the compo-

sition U02.25 is reached short range order may take place among the interstitial oxygen atoms [31] . These ordered zones would permit one to dis-

tinguish between the UO2.25-J phase and the U02+x phase. From symmetry considerations based on 4a 0 cells with successive occupancy of the four equivalent interstitial sites seven types of zones are possible [31] . They belong to the following space groups (one of them is I "33d,

described for U409):

Cubic Tetragonal Onthorhombic

I 43d 14 Cmc

I 2г3 I 4 2d C222

C2221

5.2.2. Neutron studies

Neutron diffraction intensity measurements have been made on two U409 crystals [26]. For both crystals only fundamental (i.e. non-superlattice) reflections were measured. The restriction to fundamental reflections means that the results apply to the "composite cell" obtained by superim-

posing the 64 U02-type sub-cells in the 21.8Ä unit cell. Similar results were obtained by analysing both sets of data.

17 TADLE V

U4Og : LEAST-SQUARES RESULTS FOR COMPOSITE CELL

Temperature Co-ordinates in Contribution factor Atom composite cell m to formula UO m (A2)

X У z

Uranium 0 0 0 - 0. 56 ± 0. 04

Oxygen 0 0. 25 0. 25 0. 25 1. 77 i 0.02 1. 57 i 0. 08

Oxygen 0' 0. 5 V V 0.29 ±0.05 1. 25 ± 0. 70

Oxygen O" w w w 0. 19 è 0.04 1. 60 ± 0. 90

V = 0.372 ± 0. 005, w = 0. 378 ± 0. 005, R=3.7%.

The details of the analysis are described by ROUSE, VALENTINE and WILLIS [26] and the results are summarized below in Table V. The oxygen atoms in the composite cell occupy exactly the same kinds of position as the oxygens in the statistical cell of U02 + x (Table III). The following two points can be made concerning Table V: (a) The 2:2:2 structure assumed for the composite cell gives a formula U01 75 O'o 25 O"0 25 , in approximate agreement with U0177 О' 0 2g O"o ig given by the third column. Thus the transition from U02 + xto U4Og probably involves long-range ordering of the oxygen zones or complexes, with the configuration within each zone remaining unchanged. It appears

that zones of the U4Og structures are already present in the disordered U02 + x phase. (b) The data were recorded at room temperature and there is no evidence of anharmonic contributions to the Debye-Waller factors. This is similar to the behaviour of U02 at room temperature, but the temperature factors of both uranium and oxygen are higher than in U02. Each site in the composite cell is occupied by 64 atoms, each of which must be slightly displaced in different directions from the ideal position. Since the temperature factors are related to the mean-square displacement of the atoms from their

average positions, it is not surprising that they are higher in U409. As a first approximation WILLIS [26] assumed that the uranium atoms are not displaced from the fluorite-type positions, although the existence of superlattice lines in X-ray photographs indicates that there must be small uranium displacements. It is likely that displacements of both uranium and oxygen atoms occur and that the magnitude of these displacements can only be found by the interpretation of neutron superlattice data.

Both X-ray and neutron studies agree that the unit cell is 4a0 and the space group is I'ÎSd.

18 6. TETRAGONAL PHASES

A number of tetragonal phases have been described in the range U02 3 to U024 [1]. The a -U307 phase occurs in the early stages of oxidation of U02 at temperatures below 135°C. There is disagreement whether c/a is less than or greater than unity; published values are 0.99 [3, 27, 28] and 1.01 [29] . It is possible that the intensities are reversed, indicating c/a < 1, bv a small quantity of U02 superimposed on the tetragonal phase with c/a > 1. The O/U ratio of (Ï-U3O7 is certainly less than 2.33 and probably less than 2.30.

At 180°C, for O/U > 2.3, there is a transition to the 7a(or ß .U307) phase. The conversion is complete for O/U = 2.33. The yi phase (c/a = 1.033, с = 5.556 Â) can be preserved indefinitely below 350°C, but above this temperature it transforms to у 2 +U02 6:

Tl-T2+U02-6. (2)

The у2 phase probably exists over a range of compositions, with c/a varying from 1.017 at 350°C to 1.010 at 650°C. According to Eq. (2), y, has a higher oxygen content than 7 2. The probable compositions are:

U02 33 for уг

UO2 30 for у2 at 350°C.

UO <2.30 for y2 at 650°C.

Above 600°C reaction (3) occurs:

72-*U409+ U026. (3)

The reversibility of reactions (2) and (3) has not been confirmed, and the thermodynamic stability of the tetragonal phases has not yet been determined; it is not even known whether these phases are stable or metastable.

There is good agreement concerning the y, and y2 phases in recent studies [22, 27] on the basis of X-ray diffraction studies. WESTRUM and GRÇiNVOLD [28] have reported the existence of an oxide

U02.37 having a structure of lower than tetragonal symmetry, whereas BELBEOCH et al. [29] have interpreted the characteristic features of the diffraction lines as due to a strained tetragonal structure. A brief summary of the various observations on the tetragonal oxides is given in Table VI, using the pseudo-fluorite cell as the basis for description. The tetragonal phases are ordered structures based on the fluorite arrangement. ANDRESEN [25] has pointed out the similarity in the neutron diffraction patterns of U4O9 and a-U307. The neutron diffraction pattern of y i shows superlattice peaks [30] and the X-ray diffraction pattern of y2 shows numerous superlattice lines, but they are too close to be resolved into their components [31]. The true unit cells of the tetragonal phases are

19 TABLE VI

TETRAGONAL OXIDES

Coll edge c/a Composition Stability o/u range

(UO,) 5. 470 1 2. 00 -

У1 a = 5. 371 - 5. 384 1. 030 - 1. 033 2.33 <460"C Уг a = 5. 394 - 5.408 1. 010 - 1. 017 «2. 30 <600°C

at least as large as that of U409. No structure determination has yet been published of any of the tetragonal phases.

7. CONCLUSIONS

In UGj + x and the ordered phases based on the fluorite structure the excess oxygen occupies interstitial positions in the lattice.

In the U02 + x domain the lattice contraction with increasing O/U ratio has been determined by different workers and the agreement is fairly reasonable. At least up to 900°C there is general agreement on the positions of the phase boundaries. It is only at higher temperatures that the recent X-ray work [6] is at variance with the thermodynamic data. There is agreement between X-rays and neutron results concerning

the space group and the 4a0 cubic cell of U409 . A model of the U4Og super- cell has been proposed which gives the "ideal positions" of uranium and oxygen atoms.

Neutron diffraction work has been confined to the phases U02, U02 + x and U4O9. Two main conclusions have emerged from these studies, namely: (a) At room temperature the "vibration surface" of both uranium and oxygen atoms is spherical, but at higher temperatures this statement is true only for uranium. For oxygen the sphere becomes distorted at relatively low temperatures (100°C) to a more complex surface of the type shown in Figs. 3 and 4. This distortion arises from anharmonic contributions to the Debye-Waller factor.

(b) U02 is oxidized to U02 + x by the formation of defect zones rather than isolated point defects. Each zone contains oxygen atoms at two different kinds of interstitial site, labelled O' and O". The uranium sub-lattice is unaffected by the presence of these zones, but vacancies appear in the sublattice of normal (fluorite-type) oxygens. The most likely configuration of atoms in the zone is described by the 2:2:2 structure (Fig. 8) with two each of normal oxygen vacancies, interstitial O1 and interstitial O" atoms. This zone persists in the ordered U409 structure, and the transition to U409

20 involves simply an ordered linking together of these zones. Further speculation about the U4O9 structure must await the interpretation of neutron superlattice data. Major problems, requiring further investigation by diffraction methods, are:

(i) Determination of the homogeneity range of U409, particularly at low temperatures, and the investigation of U02 25-y structures;

(ii) Examination of U02+x above 900°C to resolve the doubts raised by BELBEOCH et al. [6] about the validity of the phase diagram shown in Fig. 1. (iii) Determination of the crystal structures of all the ordered phases with large unit cells.

III. THERMODYNAMICS

1. HEAT CAPACITY MEASUREMENTS

1. 1. Low temperature heat capacity data

In a report on isoperibol (isothermal jacket calorimeter) heat capacity

measurements on U02 by JONES, GORDON, and LONG [32] the existence of a relatively blunt lambda-type transformation at 28.7°K is indicated. Their suggestion that the thermal anomaly originated in the conversion from antiferromagnetic to paramagnetic ordering has been supported by magnetic susceptibility studies [33-36] and by neutron diffraction analysis [37] which have shown that only the lines characteristic of the fluorite structure are observed at 77°K but that at 4. 2°K additional lines consistent with antiferromagnetic ordering appear. In the course of a series of investigations on the heat capacity of both stable and metastable uranium oxidé compositions by WESTRUM et al. [28, 38-40], the presence of a very small, 0. 09 calorie per gram formular mass °K (cal/gfm °K) transition in 0-U3O7 at 30. 5°K was revealed. Since this metastable form is prepared by oxidation of UO2 at 50 to 1350C, the possible presence of a residue of UO2 in the preparation could not entirely be excluded [28, 40]; but failure of the observed transition to coincide in temperature with that reported by JONES, GORDON and LONG [32] was without explanation, since solid solution formation between the phases is

not expected. The reported analysis (99. 3% U02, 0. 7% U03) for the Jones et al. sample could probably be better interpreted in terms of a U02 + x and U4O9 - y mixture, (see Fig. 1). For these reasons and because of the rather unusual shape reported for the heat capacity curve in the transition region [33], further study of the thermal properties on well characterized UO2 was undertaken by WESTRUM and HUNTZICKER [41]. Their measurements were made by precision adiabatic calorimetry on two well-characterized samples deviating from stoichiometry by less than ± 0. 1% and contaning only a few parts per million of metallic elements. One sample consisted of U02 cylinders pressed by the Mallinckrodt Chemical Works, and the other was flame-fused material prepared for these studies by the Uranium Division of the same firm. These samples will be referred to hereafter as the "cylinder" and "flame-fused" sample, respectively. The heat capacity curves in the transition region are presented in Fig. 10 for both the above samples as well as for that of Jones et al. Values for this and other compositions are shown in Fig. 11. It is immediately evident that the transition temperature for the cylinders is 30. 4°K (in contrast to the 28. 7°C value of Jones et al. ) while that for the flame-fused material may be a few tenths of a degree lower. Moreover, the heat capacity reaches a maximum value in the vicinity of 400 cal/gfm °K or about forty times the previously reported values. These new data now accord with the suggestion of WESTRUM and

GRONV0LD [28] that the transition observed in their o?~U02e33 sample is indeed caused by the presence of the U02 phase, and it has been pointed out by BELBEOCH et al. [29] that the presence of U02 may affect the apparent

23 Fig. 10

The heat capacities of U02 samples in the region of the antiferromagnetic-paramagnetic transition.

JONES, GORDON, and LONG [32] -o-o- WESTRUM and HUNTZICKER [41], cylinders; -®-B WESTRUM and HUNTZICKER [41], flame fused.

c/a ratio, (see Chapter II). The reason for the intermediate behaviour of the flame-fused material is not evident. Thermal functions based on data taken on the cylindrical samples are summarized for selected temperatures in Table VII. The thermal properties of the transition are summarized in Table VIII. The thermal behaviour is also consistent with the magnetic data as considered in the section on magnetic properties.

1. 2. High temperature heat capacity data

New enthalpy increment determinations by CONWAY et al. [42] over the range 900 to 2000°C (with respect to 25°C) do not accord well with those of KELLE Y et al. [43] over the common range 900 to 1200°K. The heat capacity equation of RAND and KUBASCHEWSKI [44] is therefore adopted for the present purposes above 298°K and the data of WESTRUM et al. [41] for the values below that temperature. The desirability of obtaining high temperature heat capacities by the adiabatic techniques rather than by enthalpy increment determinations is obvious. Such data would permit the resolution of m any ambiguities in the present analysis.

24 Т, 'К

Fig.11

Heat capacity data on selected uranium oxides. Sources for all the data are as given in reference [40]

except that the data on U02 below 60°K are from WESTRUM and HUNTZICKER [41].

TABLE VII

THERMAL PROPERTIES OF URANIUM DIOXIDE (Units: cal, gfm, °K)

T,"K Cp S* H--HÔ -(G"-H»0)/T

10 0. 064 0.017 0.13 0. 004

25 1. 836 0.497 9. 77 0.106

35 2. 367 1. 953 53. 38 0. 428

50 3. 270 2. 938 95. 22 1. 034

100 6. 830 6. 304 348. 3 2. 821

200 12. 32 12. 90 1332. 5 6. 242

350 16. 23 20. 93 3512 10. 900

298.15 15.20 18.41 2696 9. 37

2. LATTICE DYNAMICS OF U02

This section describes the correlation of data on characteristic temperatures, derived from neutron diffraction studies, with experimental measure-

25 TABLE VIII

THERMAL PROPERTIES OF THE U02 ANTIFERROMAGNETIC-PARAMAGNETIC TRANSITION

Observer Sample Tt "K Cpmax ASt

JONES et al. [33 - 28. 7 ~ 9 0.87

cylinders 30.4 ~400 0. 86* WESTKUM and HUNTZICKER[41] 1.10**

WES TRUM and HUNTZICKER[41] flame-fused 30.1 ~ 11 0.47

* Based on resolution of the data with a somewhat arbitrary estimation of the lattice contribution.

** Based on the estimation of the lattice heat capacity of U02 as given in the lattice dynamics section of this report. ments of heat capacity. The present analysis supersedes that given by WILLIS [14]. WILLIS 111,14] and WILLIS, LAMBE and VALENTINE [45] have provided data that should be one of the best sources of information on the characteristic temperatures of stoichiometric U02. Since this is a diatomic crystal there must be a separation of the frequency distribution into at least two degenerate branches. The mass ratio between the atoms is 15:1; hence it is assumed that the vibrations of the oxygen ions can be represented by an Einstein optical branch and the vibrations of the uranium ions by a Debye accoustic branch. While such a complete separation is, at the present time, unsupported by theoretical analysis, it has led to useful values for characteristic temperatures.

The quantity BD, to be associated with the temperature weakening of the neutron diffraction lines from a monatomic cubic lattice with a Debye frequency distribution, is given by WEINSTOCK [46] in the equation:

-X wherex=0D/T, m is the mass of the scattering centre and ф(х) = (l/x)/(ß/eß-l)oß.

The corresponding quantity BE for an Einstein model with a single characteristic frequency can be derived from Weinstock1 s paper as the equation:

_hjl + exp(-eF/T) E w mkeE 1 - exp(- eE/T) '

where Eqs. (4) and (5) relate the characteristic temperatures 0D, 6e with Bd, Bg for a monatomic lattice.-

26 The experimental quantities are the temperature factors for uranium and oxygen in thé diatomic UO2 lattice, Bu and Bo respectively, which must be related to B0 and BE. There is no theoretical guidance on this point, but results in good agreement with the specific heat data are obtained by using Eq. (4), with the right-hand side multiplied by 1Д/3, for Вц and Eq. (5), with the right-hand side multiplied by 2Д/3, for B0. The equations to be used, therefore, are:

(6)

аВ 2hi_ l + expt-ft/T) K, , о /Зт0кеЕ 1 - exp(- fc /Т) '

о If the B1 s are expressed in (A)2, these become

26. 857/хп , ,, Л (8)

and 69. 081, Br ftanhif' (9)

where xd = 0D/T and xE = 0e /T. The range of observations is such that Eqs. (8) and (9) can be rewritten as

and

2 xe\ _ 69. 081 69. O8IV1 1-0. 325 (11) 2 J BqT B0T in both of which Willis's reduced temperature (T1) is to be used for (T). The results of these calculations are given in Table IX. The consistency of the values of 0D, and 0^ , derived independently at different temperatures over a range exceeding 1000°C, is striking. The average values for characteristic temperatures are

0D = 182°K,

0E = 445°K.

27 TABLE XII

CHARACTERISTIC TEMPERATURES OF U02

2 г T'(°K) BD(A ) VK) В0(А ) e^K)

293 0.31 163. 0 0.49 442.5

492 0.38 190.3 0. 75 439.3

609 0. 51 182.6 0.88 446.8

748 0. 61 185.0 1.05 450.1

915 0. 79 179. 7 1.26 452.4

1086 0. 85 188. 8 1.55 443.7

1249 1.02 184. 8 1. 78 442.6

1348 1.13 182.4 1.86 449.5

1455 1. 21 183. 1 2. 09 440.3

Average^ 182. 2 Average = 445. 2

The values of Bo in Table IX have been calculated by Willis with the assumption that the oxygen atoms are vibrating isotropically about their normal fluorite positions. He has shown [14], however, that the oxygen atoms relax in <111> directions as the temperature is raised; if this relaxation is taken into account, Bo at 1000°C is reduced from 1. 87 to 1. 26. Willis has not given values of Bo at each temperature of observation calculated on the basis of relaxation, and so the final value of 0g recommended here is the previous value multiplied by^l. 87/1. 26. This final result is

eE = 542°K. 0D remains unchanged at 182°K. These values for the characteristic temperatures are now used to calculate the lattice vibrational contribution to Cv [182], using the equation

XD x D E E Cv= 3R (12/: e -l)i-dx- 3xD/(e - 1) + 6Rxle /(e -1)2, (12)

and to compare the calculated Cv versus T curve with the observations of JONES et al. [32]. The results are shown in Fig. 12. Curve a has been calculated from Eq. (12) and curve b shows the portion arising from the acoustical branch only; the circled points represent the experimental observations.

The observations are of Cp, but the difference between Cp and Cv is not important in this temperature range. The agreement is gratifying. Only the acoustical branch is excited at very low temperatures, whereas the principal contribution comes from the optical branch at higher temperatures; hence the difficulty in obtaining agreement between Debye characteristic temperatures derived in other ways [9]. An interesting application of the heat capacities evaluated from Eq. (12) is the resolution of the several contributions to the thermal capacity in the

28 Fig. 12

Heat capacity versus tero peratme for U02. Circles denote experimental points. Full curve a is calculated from Eq.(12); curve b shows calculated contribution for acoustical branch only.

region of the magnetic transition at 30°K. As may be seen from Fig. 13, good accord obtains between the measured heat capacities of WESTRUM and HUNTZICKER [41] above 30°K and those evaluated from the lattice dynamical data. Below the transition the agreement is still adequate. Using the lattice dynamical values as an estimate of the lattice entropy below 45°K, we obtain an entropy of transition of 1.10 cal/gfm °K. This is in better accord with the theoretical value R In 2 (1. 38 cal/gfm °K) than the values

obtained by estimating the lattice contribution in U02 by the usual approximation methods. It is desirable to obtain data for estimating the (total) lattice heat capacity of the isostructural oxides thorium dioxide (diamagnetic) and plutonium dioxide (paramagnetic). Although measurements of the heat capacity of Pu02 have been reported recently by SANDENAW [47] they are complicated by the release of self-irradiation strain energy.

3. FREE ENERGY, ENTHALPY, AND ENTROPY MEASUREMENTS

3. 1. Chemical thermodynamics of the UO2. 00-2.25 region

Although a considerable number of thermochemical measurements on the uranium oxides has now been made, it will be seen below that the information available is neither sufficiently complete nor sufficiently precise to provide a really reliable and accurate set of thermochemical properties of the system. For this reason, and because of the shortness of time for

29 О 10 20 30 40 50 60 70 80 90 ТСК)

Fig. 13

Heat capacity versus temperature for U02 near transition at 30°K

the evaluations, it was decided to base the assessment of the data on that previously carried out by RAND and KUBASCHEWSKI [44], and to restrict

it to the region x= 2. 00 to 2. 25 in U02+x. For the partly disordered U02+x single phase region at high temperatures, RAND and KUBASCHEWSKI suggested the following values (Table X) for the partial entropies and heats of dissociation in the reaction

2[Oluo2+x = (02), (13)

from which values of ДН and AS at other compositions can of course be interpolated. The temperature-dependent terms imply a constant value of the

30 TABLE XII

PARTIAL MOLAR ENTROPIES AND ENTHALPIES

FOR THE REACTION 2[0]uo = 02 [44]

д Temp/range x in U02+x н0г CK)

0. 05 60. 5- 16. 1 log T 73400- 7T 1000-1750 0. 10 67. 3- 16. 1 log T 7 6 900 - 7T 1000-1750

0.15 74. 0-16.1 log T 80400- 7T 1000-1750

0. 20 80. 8 - 16. 1 log T 82 700- 7T 1290-1750

0. 22 83. 5- 16. 1 log T 83400- 7T 1355-1750

0. 24 86. 1- 16. 1 log T 83 700 - 7T 1390- 1750

increment in heat capacity of the reaction shown in Eq. (13), ДСР= - 7, independent of temperature and composition. This is an estimate because actual measurements of heat capacities have not been made. To arrive at such a tabulation it is usually desirable to start the evaluation with either the enthalpies or entropies of formation which are, however, here available only for the limiting compositions at room temperature. The evaluation was therefore based on several series of equilibrium measurements and their temperature coefficients. Fortunately the results of ARONSON and BELLE [48] (EMF: U02.0-2.2, 1150-1350°K), of BLACKBURN [49] (effusion method: UO 2.15-2.63. 12 2 0-14 2 0°K) and of ROBERTS and WALTER [50] (direct pressure measurements: UO2.1-2.3, 1325-1700°K) agree very well, and the evaluations of the partial enthalpies and entropies of dissociation were based by Rand and Kubaschewski on these results in the concentration range U02.05-2.24 . More recent equilibrium measurements by HAGEMARK [51, 52](thermogravimetric: UO 2.0-2.25, 1173-1773°K) and by MARKIN and BONES [53] (EMF: UO2.0-2.19, 800-1300°K) have produced temperature coefficients that lead to partial entropies essentially in agreement with thos estimated by Rand and Kubaschewski. It was therefore decided to accept the partial entropy curve proposed by Rand and Kubaschewski and to relate all the recent free-energy measurements in the region under review to these entropies. In addition to the two reports mentioned in the preceding paragraph measurements have been carried out recently by KIUKKOLA [54] (EMF: UO2.014-2.67, 1073-1473°K), by GERDANIAN and DODÉ [55] (thermogravimetric: UO2.00s-2.15, 1180°K) and by ANTHONY et al. [56] (C0/C02 equilibria: U02.15-2.6i, 1500-2000°K). The value found for ДН by these last authors (-97 kcal) is almost certainly too high; the technique they used was to stream gas-mixtures over a heated sample and analyse the product after equilibration. It seems possible that the equilibrium pressures differed from those assumed. To compare con- veniently the recent results with previous ones, the interpolated partial

31 entropies above, together with the temperature-dependent terms, have been used to calculate enthalpy terms correspondingto those in column 3 of TableX, according to

ДНог = AG0j + T ASÓ, + 7T - 16. lTlogT. (14)

If the agreement were perfect the enthalpies so calculated should all lie on a single line when plotted against x for UO2+X. The actual results are shown in Fig. 14 together with the curve assessed by Rand and Kubaschewski. It may be seen that in the range UO2.03-2.20 the agreement of the new results among themselves and with the assessed curve is very satisfactory, that is, within ± 1 kcal. However, some adjustment of the curve in the range UO2.20-2.25 , seems to be needed. It may also be seen from Fig. 14 that, with increasing O/U ratio, the values derived from the results obtained at the lower temperatures become

—E 3 80

2.00 2.05 2.Ю 2.15 2.20 2.25

« . 0 / U RATIO

Fig. 14

Д Hq versus X for U02+x. O- Markin and Bones, 1000°K; •- Markin and Bones, 1300°K; 2 Д- Kiukkola, 1173°K; A- Kiukkola, 1373°K; û- Aukrust, et al. , 1473°K; • - Aukrust, et al. , 1673°K; V- Gerdanian and Dodé, 1180°K¡ solid line - Rand and Kubaschewski.

32 consistently higher than those pertaining to higher temperatures of measurement. This seems to imply that the heat capacities used for the reaction shown in Eq. (13) are too negative for higher values of x, while they are of the right order of magnitude at lower values of x. ACpthus becomes less negative as x increases. It will be difficult, however, to make reasonably reliable estimates, and for this and other reasons measurements of true heat capacities in the UO2+X single phase region are very desirable (see Section III. 1. 2). A linear extrapolation of Rand' s enthalpy and entropy curves into the region UO2.00-2.03 is not justifiable, either on theoretical or on experimental grounds. The shape of the entropy curve in this region has been discussed theoretically on the basis of a simple atomic order-disorder model (see Section III. 4. 1. ). In the ideal case, that is, if the stoichiometric composition contains atoms in lattice positions only, and no lattice positions are vacant, ASoj should rise to infinity. However, the evidence suggests that this ideal case is not approached even at room temperature, and consequently the partial entropy should pass through a maximum.

O/U

Fig. 15

ASo2 versus x for U02+x in low x region.

Experimental measurements in the low O/U region have been carried out by HAGEMARK [51] and by MARKIN and BONES [53]. The partial molar entropies, Д§о2, derived from the temperature coefficients are shown in Fig. 15. It may be seen that the two sets of results disagree below x= 2.01, the maximum in Hagemark' s curve presumably occurring at a much lower x than that in Markin and Bones's curve. As the stoichiometric composition is approached, the free energy measurements become increasingly difficult,

33 as is demonstrated by the shaded area of Hagemark1 s results indicating the systematic experimental errors and by the spread of the experimental values reported by Markin and Bones. Nevertheless, the discrepancy between the two sets of measurements is probably real. It cannot be explained on the grounds of the appreciable difference in the temperature of the measurements because the maximum should shift towards UO2.000 and become lower as the temperature decreases. At this stage, it does not appear possible to resolve the differences.

HAGEMARK [51] made equilibration experiments using C0/C02 mixtures at 900 to 1500°C, changes in composition being measured by means of a quartz spring microbalance; corrections for buoyancy and for vaporization were applied. MARKIN and BONES [53] studied this composition range by galvanic cell techniques, using a coulometric titration technique to vary the composition by small amounts. The measurements were made from 700 to 1000°C. A standard state, arbitrarily defined as U02.ooo

-70-

Ud-, , , .

-4 -3 -2 log i

Fig. 16

AGq versus x for UO;+x in low x region. Markin and Bones; O- Hagemark. 34 by an absolute method involving dissolution in acid and estimation of U(VI) by polarography; in three cases out of four, values of O/U agreed to within ± 0, 001. It was therefore assumed that the stoichiometric state was known to this accuracy. Most of the measurements were made on two very different materials: (i) a flake from a sintered pellet, containing 100 to 200ppm Fe and other impurities to the 10 to 50 ppm level, and, (ii) a flake from a fused crystal of UO2 which was very pure, the highest impurity recorded being 10 ppm of Al. The results on these two specimens were in excellent agreement, and the same results were obtained on oxidation and on reduction. Actually, partial free energies derived from the two sets of measurements agree fairly well, as may be seen from Fig. 16, the discrepancy existing mostly in the temperature coefficients. They necessarily also appear in the ДНо,-х diagram which will not be reproduced here but can, if required, be constructed from Figs. 15 and 16. It is difficult to suggest a calorimetric method for this temperature range suitable for testing the equilibrium results; quantities of the order of 200 cal would have to be measured with good accuracy at relatively high temperatures. The high-temperature calorimeter developed at the National Physical Laboratory may be suitable, whereas a new Calvet calorimeter for high temperatures might be more accurate. Moreover, the differences can not be resolved by integration of partial thermodynamic quantities and comparison with the available integral quantities, because the differences involved are too small and within the errors entering this type of evaluation. Nevertheless this integration has been carried out in order to have an approximate check on the available information. Using the partial entropies of Rand and Kubaschewski at an arbitrary temperature of 1400°K for the range UO2.03-2.24 and those of Hagemark and Markin respectively for the range UO2.00-2.03 , the integral entropies between the limits x= 2. 00 and 2. 25, using the equation

AS=^jASo2dx (15) 0 have been obtained as follows: = ASi4oo 19. 35 e. u. /mole 02 using Hagemark's data, ASi400 = 19, 9 e. u. /mole O2 using Markin1 s data. The entropy change of the reaction

2=8+(02), (16) obtained from the low-temperature heat capacities of JONES, GORDON and

LONG [32] for U02 and OSBORNE, WESTRUM and LOHR [39] forU409 is:

AS298 = 32. 7 e. u. /mole 02. Using the somewhat doubtful average value, ДСР= - 7, discussed above and the entropy change at room temperature to calculate the entropy change at 1400°K, one obtains ASuoo = 26. 3 e. u. which is 6. 95 e. u. and 6. 4. e. u. respectively higher than the integrated values. The difference may be due to the entropy of disproportionation of U4O9 into иОг.244 (disordered) andU02.6. This disproportionation may be regarded essentially as an order-disorder transformation of U4O9. An entropy of transformation of 3. 5 or 3. 2 e.u./mole

35 U4O9 which follows from the above vaàues is commensurate with that of comparable metal oxides. Thus, the entropy values for the reaction shown in Eq. (16) are consistent, but further calculations should be postponed until experimental heat capacities of U4O9 and UO2+X are available. The enthalpy increment in Eq. (16) is not very well known. We may accept the enthalpy of formation of UO2 to be ДН29р = - 258. 7± 0. 6kcal/mole as obtained by HUBER and HOLLEY [57], (from the enthalpies of combustion of uranium, UO2.014 and UO2.006 extrapolated to the stoichiometric composition together with a recalculated value for U3O8), in preference to the preliminary value of - 259. 9± 2.3 kcal/mole as obtained by GERDANIAN, MARUCCO and DODÉ [58] from direct measurements in a Calvet calorimeter. No enthalpy of combustion of U4O9 is known. If one assumes that it is formed from neighbouring oxides with no evolution of heat, a value of ДНг98= + 75. 2 kcal results for equation (16). It is, of course, likely to be more positive. From heats of solution of UO2 and U4O9 in nitric acid [59], one may compute +89. 6 kcal. The value assessed by RAND and KUBA- SCHEWSKI [44], and exactly confirmed by MARKIN and ROBERTS [60] will be retained, namely, ДН298 = 84. 0 kcal. It is more rewarding to derive the data for equation (16) from free energy measurements; and in this concluding discussion the U4O9-U3O8 two- phase region may be included. Results for the two-phase regions tend to be in better agreement than those for single-phase regions, because the composition does not have to be known accurately. RAND and KUBASCHEWSKI [44] tabulated average values for the ЩОд-у-иОг+х region from 800 to 1350°K and extrapolated values at 298°K. Later results by MARKIN [53] and by KIUKKOLA [54] are in excellent agreement up to 1373°K. RAND and KUBASCHEWSKI [44] accepted the results of ROBERTS and WALTER [50] which showed the U4O9 phase decomposing peritectically at 1396 ± 5°K to UO2.6 and UO2+X. The only later work in which the region close to U4O9 has been studied is that of Kiukkola, who interprets his results as showing the U4O9 phase persisting to 1488°K. However, Kiukkola's results for compositions between UO2.2 and UO2.25 do not agree well with those of other authors, e. g. BLACKBURN [49], and results at one temperature only are reported for the compositions 0/U= 2. 245 and 2. 248. The result for O/U = 2. 245 falls on the line previously ascribed to 0/U= 2. 239, whereas that for 0/U= 2. 248 actually agrees with previous values for the U4O9-UO2.6 two-phase region. The previous results [50] can probably still be accepted. Kiukkola places the upper limit of the U4O9 phase at 0/U= 2. 250 because the results at this composition and at 0/U= 2. 300 were identical, and in excellent agreement with Roberts and Walter, who measured identical pressures for the compositions 0/U= 2. 250, 2. 257, 2. 279 and 2. 303. The rather scattered results for this phase limit determined by interpolation of effusion- cell measurements [49] give higher values. Thermodynamic values for the U4O9-UO2.6 region given by Rand and Kubaschewski are at

298-1395°K (U4O9-UO2.6)» AS(02) = 35.2 e. u. ; ДН(Ог) = 72. 8 kcal,

36 and at

1395-1750°K (UO2+X-UO2.6)» AS(02) = 36. 7 e. u. ; ДН(02) = 74. 9 kcal.

Markin's values were combined with earlier values [60] to give

ДН(С»2) = 78. 6 kcal from 873 to 1395°K

and

AS(02)= 39. 2 e.u.,

which are in excellent agreement with K1UKKOLA [54], who finds

ДН(02) = 77. 1 kcal from 1073 to 1473°K

and

Д§(02) = 38. 5 e. u..

At temperatures above 1395°K, the results of Roberts and Walter and earlier results combined to give,

ДН(02)= 81. 0 kcal

and

Д§(02) ='40. 2 e. u. .

Recalculation yields

ДН(Ог) = 79. 8 kcal

and

AS(02) = 40.1 e.u..

The numerical values given by Rand and Kubaschewski in this region therefore seem low, but all such values were chosen to give the best integrated values for ДН and ДБ, and it may be that linear extrapolation is not justified from 870 to 298°K. A re-assessment of the entire range is desirable but the task could not be accomplished by the Panel in the time available. The shape of the curve obtained when AG is plotted against T for the two-phase region indicates that the composition of the phase boundaries is not changing appreciably, but Roberts and Walter did record a small increase in the O/U ratio of the phase limit of the UO2+X phase from 1350 to 1700°K. Further results in excellent agreement have been reported by ANTHONY, KIYOURA and SATA [56] and by AUKRUST, F0RLAND and HAGEMARK [52]. These results are plotted in Fig. 17. The lower limit of the UO2.6 phase Jound by these authors was 0/U= 2. 605 to 2. 610, in agreement with many

37 RATIO О/U

Fig. 17

Phase diagram of the U-0 system for the temperature range 1000 to 1700"C. X - Anthony, Koyoura and Sato; A- Blackburn; O- Roberts and Walter; + - Hagemark. other measurements [60]; it seems theft this phase limit is not much affected by changes in temperature from 870 to 2000°K.

3. 2, The phase diagram, UO2 to XJ4O9

The present assessment of the thermodynamic data has given no reason to doubt the calcualtion of the phase boundaries made by Rand and Kubaschewski. These are shown as full lines in Fig. 18. Phase boundaries deduced by five other authors are also shown on this figure. GR0NVOLD's results [4] were deduced from high-temperature crystallography; ROBERTS and WALTER [5C§ used direct pressure measurements, and BLACKBURN [49] an effusion cell; MARKIN and BONES's [53] EMF measurements have been discussed above, and ARONSON, RULLI and SCHANER [61] deduced phase boundaries from breaks in electrical conductivity versus temperature plots. Not all the experimental points could be plotted on the figure; the dotted curve ex- presses the mean results fairly well, though there is considerable scatter. It should be noted that there must be a point of inflection in the UO2+X boundary around 0/U= 2. 15. Crystallographic data which are at variance with this phase diagram above 1000°C have been discussed in Chapter II of this Report and it may be noted again that KIUKKOLA [54] shows the U4O9 phase extending to significantly higher temperatures. The oxygen deficient boundary of the XJ^Og-y phase at high temperatures is taken from the concurrent results of Blackburn, and of Roberts andWalter.

38 Fig. IS

Phase boundary between U02+x and UOz+x + U40,.y. • - Gr^nvold; X - Roberts and Walter; о - Blackburn; Д- Aronson, Rulli, and Schauer; ® - Markin and Bones. Full line - Rand and Kubaschewski.

The reasons for placing the oxygen-rich boundary of the U4O9 phase at UO2.25, and for rejecting the higher values obtained by Blackburn, have been given in the previous section. The region of the phase diagram above 1123°C has been plotted in Fig. 17 and has already been discussed.

3. 3. Hypostoichiometric UO2

The behaviour of UO2 at temperatures above 1800°C has been the subject of some confusion in the literature. The formation of a hypostoichiometric uranium oxide, as detected by the presence of free uranium in UO2 after cooling of the sample, has been questioned since contamination (e. g. with carbon) may have influenced the results. Recent investigations on uncontaminated samples by ROTHWELL [62] indicate that U02_x can be formed above 1800°C at sufficiently low partial pressures of oxygen provided that the vaporization of UO2 is not the pre- dominant process:

= +|x(02).

After cooling in an inert atmosphere, inclusions of metallic uranium have been found in the UO2 matrix. Indirect evidence that this uranium may have precipitated from a homogeneous high temperature, UO2-X- phase on cooling.

39 arises from the fact that a rhenium wire in contact with UO 2_x at this high temperature showed no tendency to melt and break, which should have occurred in the presence of free uranium [63]. AITKEN et al. [63] obtained a limiting value for the solubility of U in UO2-X by vaporization studies in hydrogen and found 0/U= 1. 89. (An earlier ratio of 1. 96 by ANDERSON et al. [64] seems doubtful due to the difficulty of determining the composition from the lattice parameter). The melting point of the congruently vaporizing UO2-X was found to be 100 to 150° higher than that of the stoichiometric oxide [65]. Measurements of the electrical conductivity and of the Seebeck coefficient of UO2-X at 1800°C [65] and 2050°C are consistent with an n-type metal excess semi-conductor. The large values of the Seebeck coefficient (- 2500 to 5000 ßV/°С) cannot be attributed solely to an electronic contribution from the free uranium at room temperature. An increase of thermal conductivity on irradiation has been ascribed to the formation of UO2-X [66]. Because the formation of the иОг-х phase may have appreciable effects upon the transport phenomena and perhaps on fission gas release, furtheir studies of its stability and properties are desirable.

4. VAPORIZATION PROCESSES

In the vapour phase in equilibrium with the dioxide phase above 2000°C three gaseous oxides, UO, UO2 and UO3 have been observed. The standard free energies of formation of all these oxides have been determined with varying degrees of accuracy. The thermodynamic properties of the gaseous dioxide have been calculated from vapour pressure measurements [75] of the congruently subliming dioxide, the composition of which is very near the stoichiometric composition. The result is that

AG? (U02, g) = - 125 400 + 6. 22 T cal/mole. (17)

The gaseous trioxide is observed under oxidizing conditions and its thermodynamic properties have been obtained [76] from a study of the equilibrium.

Í+Í(02)= (UO3). (18)

The stoichiometry of the gaseous molecule was demonstrated to be that corresponding to the trioxide. Reasonable estimates of the absolute entropy

of gaseous U03 compared with known values of monomeric and polymeric tungsten trioxide indicate that uranium trioxide is monomeric. The standard free energy of formation is

AG° (UO3, g) = - 198 500+ 19.0 T cal/mole. (19)

De MARIE et al. [77] have reported a mass spectrometric study of vapori-

zing compositions yielding gaseous UO, U02 and U03. The gaseous monoxide was first observed by CHUPKA [78]. Because of uncertainties in

40 ionizational cross-sections the properties of the monoxide are not as well known as those for the dioxide and the trioxide.

AGf (UO) = (- 16 800 - 10. 0 T) cal/mole. (20)

1900 °K

Fig. 19

Suggested description of the variation of total pressure with composition in the uranium-oxygen system between uranium and uranium dioxide: (a) Vapour pressure of pure uranium by Rauh and Thorn; (b) Effect of oxygen observed by Rauh and Thorn; (c) System studied by de Maria et al. ; and (d) Vapour pressure of uranium dioxide measured by Ackermann et al., (Reproduced by courtesy of R.J. Ackermann, E.G. Rauh and R.J. Thorn [79])

All of the uranium oxides except UO2.00 vaporize incongruently, i. e. the composition of the vapour is considerably richer in oxygen than is the condensed phase. The vapour is predominantly oxygen with a small amount of gaseous trioxide. Hence, any of the oxides with O/U > 2 will ultimately yield UO2.Û0 in vacuo. The schematic phase diagram (Fig. 19) of pressure versus composition illustrates the general behaviour [79]. At compositions with a ratio of oxygen to uranium less than two, the vapour is predominantly gaseous UO. It has been established by absolute effusion measurements [80] and by mass-spectrometric observations [79] that small amounts of dissolved oxygen at the lower temperatures suppress the vapour pressure of uranium. Hence, there appears to be a congruently vaporizing composition near the uranium rich end as indicated in Fig. 19. The general character of the diagram at higher temperatures is not yet known. But above about 2100°C the vapour pressure of the uranium contaminated with oxygen exceeds that of

41 pure uranium so that the oxygen contaminant can be removed. At all temperatures the minimum at UO2.00, however, appears to remain. If this is true, then the production of UO2-X is accomplished under conditions where there is a preferential loss of oxygen out of the system so that the composition of the solid can "ride" up on the indicated boundary.

5. THEORETICAL TREATMENT OF U02+x PHASE

5. 1. Statistical thermodynamics of interstitials and vacancies

The model used in this section is essentially that developed by ANDERSON [67]* with the following exceptions: (a) a distinction is made between the contributions made to the vibrational partition function of the crystal by an ion in a regular lattice position and one in an interstitial position; and (b) it is assumed that the metal sublattice is fixed and perfect and that the defects are in the oxygen sublattice. Further, the calculations have been extended here to the actual evaluation of partial thermodynamic quantities using recognized sources for whatever auxiliary quantities and constants are required. The relationship between the activity, Xo, of an oxygen atom in the gas phase and the (partial) pressure of oxygen gas is

Po2 = XjjkTQ exp(D/RT). (21)

The activity is to be equated separately to that associated with interstitial oxygen ions and to that associated with vacancies in the fluorite oxygen positions in the condensed phase. Thus the numbers of those defects in equilibrium with the same pressure of oxygen can be found and the corresponding value of x in UO2+X found from their differences. Then the partial quantities are found directly from the following equations:

AGot - RT In P0j

= 2 RT In X0 + RT ln(kTQ) + D, (22)

/ -ASÖ!= 2R^lnX0 + T^^/x^+R( lnQ + T^^)+R + RlnkT, (23)

ДНо2= Дбог + ТДН0!. (24) The semigrand partition function for the defects, from which the two expressions for the activity in the condensed phase are developed, is -E r(T,Nu, XoHp^Nu.Nt.NJqN'q^ eRT . (25)

NfNv

* Anderson's original equations arenot logically correct, since differentiation was performed under the incorrect assumption that the number of lattice sites was fixed.

42 In this equation Nu is the number of uranium atoms present, N¡ is the number of interstitial ions, Nv is the number of regular oxygen lattice vacancies, q¡ and qv are the contributions of an interstitial ion and a regular lattice ion, respectively, to the crystal partition function and E is the excess energy introduced into the crystal by the presence of the defects. This energy is given by

2 2 E = NvEv-N1Ei - EvvNv /(evNu) - E„ N /(ÚÍNu), (26) where Ev is the energy required to remove a regular lattice ion, as an atom, to a position of rest far from the crystal, E¡ is the same for an interstitial ion, Evv is related to the energy decrease brought about by the formation of a pair of vacancies and correspondingly for Ej¡. Following Anderson, no account has been taken of electronic disorder. The distribution function, П, of Eq. (25) is given by

(gj Nu)! (o-yNu): ~ Ni !(a¡Nu - Nj)'. Nvi{evN0-Nv)l * ' in which ai is the number of possible interstitial sites per uranium ion and a y (s 2) is the possible number of oxygen vacancies per uranium ion. Equation (25) is developed by the usual procedure [67] to give

i \ i 1 - 0y Ev , 2 6 Ew . /nn\ lnXo=ln-g RT RVT 4v* * ' and

in lnT E¡ 2 0i Eii lnqi (29mn)\ lnX0= JTg~~iyr~ RT " ' in which the convenient quantities

ev = Nv/(o„Nu) and

6i = N¡/(eiNu) have been introduced.

The evaluation of parameters such as Ev, Ei, etc., that is necessary before computations can be carried out, was made in the following manner. For the present work the principal interest was at small values of x. Even though interstitial oxygen is the principal defect over most of the range of x, this cannot be the only defect as x-> 0 because the entropy would then increase without bound. Further, the occurrence of vacancies can be used to explain the formation of a lower phase boundary near x = 0 via the preci- pitation of metallic uranium. Hence Ev and Evv have been determined by setting x = 0 exactly at the lower phase boundary at about 1250°K and, in addition, by requiring that the optical analysis (Section V. 3. ) be satisfied at

43 960°C. The energy Eu has been set by requiring that a two phase region via the formation of a higher oxide be no longer possible above 1125°C. This is the temperature observed by ROBERTS and WALTER [68] above which U4O9 can no longer exist. Because of the interest in small values of x, a¡ has been set equal to unity since this would be expected to be (nearly) the filling density when the actual density of interstitials is small. Values of the q1 s have been calculated from

(30) with an appropriate characteristic temperature 0. Since Eq. (30) is independent of any defect density, its effect is as though в were an Einstein temperature regardless of what might be thought to be the true nature of the frequency distribution. Since, in fact, it is the light atom defects which are being discussed, this is not serious. It was found however that, given a choice of two characteristic temperatures, making a supposedly proper choice of the higher for both q¡ and qv yielded entropy results nowhere near the observations at larger values of x(~0. 05). By introducing a separate q¡ and qv a "proper" 0 can be used at least for the regular oxygen lattice and still allow a closer agreement with observation than is otherwise possible. Finally, E¡ was chosen by requiring that the value of - Д0о, at 0 = 0. 03 and T= 800°C be that observed by MARKIN and BONES [69] at this temperature and x= 0. 03. The illustrative calculations of - Д§0г and - ДЙо2 shown in Figs. 20 and 21 were made with the parameters given in Table XI. The entropy maximum shown in Fig. 19 bears a qualitative resemblance to that observed by MARKIN and BONES [69] and is reminiscent of similar behaviour in alloys [70]. However, the size and location of the calculated maximum is sharply dependent on temperature, contrary to the observations of Markin and Bones. In alloys, a minimum can be observed [70] at what would here correspond to hypostoichiometry and, in fact, it can be shown by differentiating Eq. (23) that extrema should be expected at values of x given to a good approximation by

(31)

Since v/x2=± x, an entropy extremum (a minimum) is also to be expected in the hypostoichiometric region. Its actual observation could be prevented by the intervention of a lower phase boundary, determined by the value of Evv. The appearance of such a critical temperature for the existence of multiple values of x with the same partial entropy would be an interesting repetition of the phenomenon of a critical temperature for two phase formation associated with Eqs. (28) and (29).

5. 2. Application of defect theory to UOo+x

The formal theory of a nonstoichiometric oxide containing both oxygen vacancies and interstitials has been outlined above in Section 5. 1. We point

44 o/u о/и Fig. 20 Fig. 21

Calculated partial molar entropy of oxygen in U02+x Calculated partial molar enthalpy of oxygen in Ю2+х TABLE XII

PARAMETERS USED IN CALCULATION OF PARTIAL MOLAR ENTROPIES AND ENTHALPIES

j V i

Oij 2 1

9j(°K) 870* 200

E;(kcal/mole) 132.887 94.723

Ejj(kcal/mole) 15. 036 5. 56

* Although all the present calculations were done prior to development of the analysis for ©E shown in Section III 2., iis use would not alter the results appreciably out here three complications that affect the application of the theory to U02 + X( dealing first with the region UO2.01 to UO2.25» where the concentration of vacancies can be neglected. Presumably some account must be taken of the electronic disorder arising because of the oxidation of a portion of the U atoms. Although the theory of the electrical properties of U02+x is not yet sufficiently developed to give positive guidance, it seems that a reasonable approximation at the present time is that the extra charges are localized on U ions. (Section V). ARONSON and CLAYTON [71] have applied a model involving localized electronic disorder to both U02+x and (U, Th)02+x solid solutions. The reaction assumed was

4+ 2 5+ 02(gas) + 4 U + 2 interstitial sites = 2 O " + 4 U . (32)

The choice of U0+ rather than U°+ was made on the basis of the crystallographic data on (U, Th)02+x [72], which shows a regular but non-linear lattice contracting with increasing x up to a mean U valency of 5. 0, and thereafter a lattice expansion. There is also fragmentary data on paramagnetic resonance 5+ spectra which indicate that U is present in UO:+x and (U, Th)02+x [73],though it does not prove that U5+ is formed exclusively; magnetic evidence also lends some support (see below), though it is, in part, conflicting. The model is also in accord with later measurements of the electrical conductivity and thermoelectric power on sintered plates [61]. Aronson and Clayton deduced the following expression for the configurational entropy contribution:

Nil Nç к In W = к In (33) [_(N¡ - N0¡)'. NQ¡ '. NUS:NU4:

where N¡, N0i, Nc, Nus, and Nu< are the numbers of interstitial sites, interstitial oxygen ions, cation sites, U5+ and U4+ ions respectively. They find

46 __ v 2v AS(02)=-2R lnyf-^-4R Inj^ + Q. (34) where Q includes a term for the decrease in entropy on converting one mole of gas to the solid and a term for the contribution to the vibrational entropy of the solid of the added interstitial oxygen ions. Eq. (34) is plotted on Fig.22 with Q = - 49 e. u., and the term expressing electronic disorder on this model of localized U5+ ions is also plotted; it can be seen that nearly the entire entropy change can be accounted for in this way. It is worth noting that at low values of x, the term for the entropy Д§(Ог) becomes - 6R In x, which predicts a too rapid change of AS with x.

/ / /„ //S •

EO. <34—yf tri«5

EO. (35a)

s i ^ x * EO. (36)

/ * / / ^ ^ EXPE OMENTAL CURVE W,' III/ III/

1 . I 1 • i i i I.LI 2.0 2.1 2.2 o/u

Fig. 22

Curves from theoretical equation for ASn

A different approach was outlined by HAGEMARK [74]. Taking the view that interstitial oxygen occupied (i, sites in the fluorite cell, he noted that, in the limit, only one per unit cell was occupied, but that they must all be equally available when x-> 0. This is compatible with the assumption that, when x is small, each 0¡ will block the occupancy of the 12 nearest interstitial sites, but, as x increases, the average number of interstitial sites blocked per Oí ion must decrease owing to neighbouring Oi ions blocking some of the same sites. He writes for the configurational entropy term

te № H (a Nu - N^ ) ' (Not)I where a is the number of available interstitial sites /U ion, and a = -—^-r— ' 1 + 12 x 47 Then,

x AS(0 ) = - 2R In (35a) 2 a - x and this equation also is plotted in Fig. 22; the value assumed for Q= 21.4e.u.

Recently WINSLOW [184] has shown that the expression evaluated for Д§ог in Eq. (35a) is not consistent with the defect structure model proposed by Hagemark. When ex - a(No¡ ) the partial molar entropy of oxygen is

AS , = - 2R In — + Q. (35b) 0 1 /VIVI - Y

The extra term in Eq. (35a) is due to the incomplete expression for the configurational term W. Using the a(x) given above, Winslow has calculated values for ASo2 in Eq. (35b) (Q set equal to - 24. 7 e. u. ). The result is shown in Fig. 22. All previous treatments have assumed that there is only one type of interstitial position. We must now consider the implications for the thermodynamic treatment of the actual structure of U02+x as determined by neutron diffraction. The structure assumed is the one involving "complexes" of O', O" and lattice vacancies discussed in an earlier section of this Report. The unit of the structure, on this interpretation, involves two excess О ions and

six U ions and could be written Uß014 (see Section II. 4. 2. 2. ). If all U atoms took part in units of U6014, the formula weight would correspond to U02.33, the composition of the most stable tetragonal oxide, and the limiting compo-

sition of (U, Th)02+x solid solutions. The important point is that it is not necessary to allow for the independent introduction of О', O" and lattice vacancies, but rather for the occurrence of "zones" or "complexes" of some type, whose number is governed by the number of O' ions, equal to x. A configurational term Zx must be added to allow for the Z equivalent configurations of a "complex" around any given O' site. The problem then becomes formally similar to that already treated: the number of "zones" is equal to half the number of O1 ions if we consider that the O1 ions enter in pairs. The basic problems remain the same,namely, (i) the calculation of the number of available sites as a function of number occupied and (ii) the contribution of the electronic disorder to the entropy. As one example of a very simple treatment, consider the case of O' ions entering in pairs, with the occupancy of each pair blocking 36 similar sites. Further, take the localized charge model with two U5+ ions trapped on the two U ions nearest to the two O' ions, and the other two U5* ions free to move in the lattice. We then have, for an approximate treatment,

and

x (1-х) Д3(0 ) = Q - г log - R In - 2R In, X„ + In (36) 2 z 12 - 36x 1 - 2x 48 Equation (36) is also plotted in Fig. 22; the experimental results are fitted fairly well up to x= 0. 15, with Q - \ logz= - 32. 4 e. u. The conclusion of this section is that the variation of enthalpy and entropy in this range can be understood on the basis of reasonable models, but that the experimental data are not precise enough to allow a definitive choice between several models. Confirmation is needed of the exact values of the partial quantities at the extremities of the composition range. The most important consideration, however, is the proper treatment of the disorder due to electronic effects, which must be related to the mechanism of conduction and charge localization in UO2+X at high temperatures. The range UO2.00 to UO2.01. The discrepancy between the two sets of experimental results available for this región [53, 74] has been noted in Section III. 3.1. The results obtained by Hagemark are in reasonable agreement with a model in which the process is still the incorporation of interstitial oxygen with large positive values of AS due to the behaviour of terms such as lnx as x-»0. By contrast, Markin's results indicate a rapid change in ДЙ to more negative values as x-»0; a new process is occurring, and the obvious hypothesis is that oxygen vacancies are being filled. The formal treatment of a model involving vacancies and interstitials given in Section III. 4. 1. above shows the same qualitative features as found by Markin, (see Fig. 15). There is one important difference. Any model based on a vacáncy concentration which changes with temperature will predict that the position of the maximum in the AS versus x curve will change with temperature, and this was not found by Markin and Bones. Their results are better fitted by an assumption of a static concentration of oxygen vacancies, independent of temperature. Since they found the same behaviour on studying a number of samples of different impurity concentration, an explanation based on impurities of lower valent cations present in UO2 is unlikely. It is possible that the oxygen vacancies are associated with uranium vacancies quenched-in from high temperature treatment, though the slender experimental evidence available does not support this. It may be noteworthy that the two sets of results [53, 74] were obtained in very different temperature ranges, though the theoretical model in Section III. 5. 1. would actually predict a maximum in the AS curve at higher values of x at higher temperatures. No further advance can be made in the understanding of the results in this region at present. What is certain is that AG(02) varies very rapidly with composition below 0/U= 2. 01, (see Fig. 16) and this emphasizes the need for very careful control of ambient conditions if meaningful values of any physical properties are to be obtained.

IV. SURFACE AND OXIDATION PROPERTIES

The actual state of UO2 powder feed material used for preparing pellets can affect sintering rates and, through the microstructure of the final compact, even important properties during use. One of the most important parameters is the state of aggregation of the powder itself - the degree of crystallinity, crystallite size, type of crystallite agglomerate, and porosity present. These matters are referred to briefly in a later section of the report but were not specifically discussed by the Panel. Further important variations can be due to chemical changes caused by the presence of adsorbed species on the surface and a concentration gradient of oxygen through the particles, with the possible formation of shells of different stable or metastable structures, occurring whenever UO2 has been exposed to air. These chemical considerations are summarized here. Although the influence of the surface is of greatest effect when the particle size is small, it is worth noting that many of these effects have been shown to be quite general and the composition of surface layers of even a large single crystal will differ from that of the bulk.

1. ADSORPTION PROPERTIES

The adsorption properties of U02 are briefly summarized in Table XII below [81] ; the amount of gas reacting is recorded in units of Vm, the volume of 02 necessary to form one physically-absorbed monolayer on the UO^ surface. The low temperature chemisorption of CO and of 02 mutually interfere, while the prior adsorption of H2 has little effect on subsequent 02 adsorption. A simple hypothesis is that O2 and CO adsorption occurs on U atoms, but H2 reacts with О ions, possibly forming OH". The decomposition of CO [82], which occurs on a fully reduced surface of U02 above 500°C, proceeds to a limit which depends on the surface area but not on the temperature. There is no deposition of bulk carbon, but rather the formation of a type of surface carbide, which is accompanied by some loss of oxygen from the U02 surface; the chemisorption of oxygen at -183°C is inhibited by this "carbide" layer. The deposition reaction stops if a few per cent of CO2 are added to the CO.

A calorimetric study of the adsorption of O2 on U02 at -183°C showed that the enthalpy of adsorption fell from ~55 kcal/mole to ~6 kcal/mole as the surface was progressively covered [81] . These observations agree with others that indicate that some, but not all, of the oxygen chemisorbed on the surface is removed from the surface during evacuation at room temperature [83, 84] .

Since U02 powders are commonly prepared by reduction of higher oxides at 500 to 1000°C, it is worth noting that such preparations may contain absorbed hydrogen or a stable surface 'carbide1. The formation of surface carbide during reduction by CO can be prevented by the addition of a few per cent of CO2. The bulk of the hydrogen can be removed by pumping above 600°C, but no measurements have been reported of the residual hydrogen content. Simi- larly, CO^ and H¿0 adsorbed at low temperatures can be removed by pumping above 200°C. Adsorbed hydrogen can react with oxygen subsequently ab-

51 TABLE XII

ADSORPTION PROPERTIES OF U02

Temperature Gas reacting Gas Reaction CC) (V/Vm)

0, - 195 to - 183 Chemisorption 0.3-0.6

- 138 to 50 Surface oxidation s 2.2

> 80 Bulk oxidation

H2 - 183 Chemisorption ~0. 3

20 - 0

> 400 Chemisorption 1. 0-1. 6

CO - 183 to 20 Chemisorption 0. 7-0. 8

275 Chemisorption ? 0. 08.

> 500 Carbon deposition ~2. 0

C02 20 to 200 Chemisorption >0.4

sorbed and be desorbed as water. A considerable portion of the oxygen chemisorbed on the surface is mobile at room temperature and above; since the maximum enthalpy of adsorption is 55 kcal/mole, it might be predicted that all the chemisorbed oxygen should be mobile at 700°C. However, con- tinual oxidation of the surface may proceed at all temperatures, since the oxygen partial pressures of a normal "vacuum system" or of "pure argon" are far higher than those reported in Section III. for the equilibrium pressures of O2 over UO2+X close to stoichiometry; the extent of oxidation will, of course, be limited by the maximum quantity of oxygen available.

2. OXIDATION PROCESSES

2. 1. bow temperatures

Oxygen is not ordinarily mobile in the UO¿ lattice at temperatures below about 70°C, but more than one chemisorbed layer is absorbed at temperatures above about -100°C. Between these temperature limits, if isothermal conditions are maintained, the amount of O2 absorbed at any temperature and pressure increases as the logarithm of the time of exposure to O2 [85] to a limit of about 2.2 Vm. Density and X-ray evidence on the low-temperature oxidation of larger particles of U02 suggest the penetration of the lattice by oxygen and the formation of a surface skin of one of the tetragonal oxides 185, 29) . Infra- red lines characteristic of amorphous UO3 have been observed when very small (IOOÂ) particles of U02 are oxidized at room temperature [86, 87] . There is also evidence that higher oxides are formed on larger particles (surface area 2 m2/g) from enthalpy of wetting experiments, which indicate

52 a different dependence on oxygen content for heat of wetting in water and in organic liquids [88] and from the behaviour of some of the tetragonal oxides in the atmosphere [29] . Slow transformation to higher oxides on the surface may then be a general phenomenon.

2. 2. High temperatures

Small particles oxidize exothermically on exposure to air or O2 giving partial or complete conversion to U3O8. Complete oxidation to UO2 can be effected at 1 atm O2 for small particle preparations and at higher pressures for large particle preparations. Preparations with the usual range of particle sizes.oxidize at low temperatures in two stages, the first to about UO2.33, the next step, to U3Oe, commencing at about 220°C or higher. Oxidation occurs above 80°C by a process which follows a diffusion law with an activation energy of 20 to 25 kcal/mole [1] . The various structures that are found at moderate temperatures have been summarized in Section II. The first product is normally "aUA" formed as a layer which thickens as oxidation proceeds, with eventual conversion to Y1U3O7 . The final composition and rate of oxidation depends on the O2 pressure, so the amount of oxygen diffusing inwards is controlled by surface processes. The structures formed must be in a state of strain, since the contraction in the volume of the unit cell which takes place is not accompanied by any particle break-up or change in surface area. The rate of diffusion is determined by diffusion through the product layer, and strain at the interphase boundaries is indicated by the profiles of the X-ray lines [29] . The structures initially formed may be thermodynamically metastable as well as being in a state of strain. The oxidation process is thus even more complex than has been previously reported and the structures actually present in a given specimen are not a simple function of O/U ratio; they will depend also on the particle size, temperature and O2 pressure, rate of oxidation and duration of the experiment. Further complex changes take place on annealing, some of which have been summarized in Section II. For the present purpose it is only necessary to note that samples of UO2+X which have been quenched or cooled from high temperatures usually consist of a mixture of phases. Approximate calculations of the rate of migration of oxygen through the UO2+X structure can be made using the diffusion results collected elsewhere (Section V) and agree with the experimental observation that an extremely rapid quench is necessary for the high- temperature structure to be preserved. It has frequently been found that the final product of oxidation of UO2 below200°Chas a O/U ratio above 2.33. The explanation may be that a new structure of composition UgO^ is formed with lower symmetry than tetragonal [28], but the X-ray evidence has also been discussed in terms of lattice strain [29] and the possible formation of thin layers of higher oxides noted above.

V. PHYSICAL PROPERTIES

1. THERMAL CONDUCTIVITY

The thermal conductivity of uranium dioxide at elevated temperatures has been studied more intensively than that of any other oxide [9/ 89] because of its practical importance. Only recently have single crystals sufficiently large to permit accurate conductivity measurements become available. These single crystal data have proved very interesting and form the backbone of the work to be discussed which, for convenience, is divided into two temperature ranges.

1.1. Low temperature thermal conductivity

Thermal conductivity measurements in the low temperature range from 4 to 300°K have been made by BETHOUX et al. [90] on a sintered polycrystalline material of apparent density 9.97 and by PENNINCKX [91] on a single crystal (see Fig. 23). The general shape of both curves is the same; in particular, a minimum in the thermal conductivity corresponding to a paramagnêtic-antiferromagnetic transition has been observed in both determinations at the same temperature, 30°K (see Section III. 1. 1.). Between 4 and 200°K the single crystal values coincide to a good approximation with those given by Bethoux et al. for the sintered polycrystal. From 200°K the two values start to diverge and at 300°K the thermal conductivity of the single crystal is about 20% less than that of the polycrystal. In both measurements the thermal conductivity decreases again below 12°K. It is somewhat surprising that in the very low temperature range (4 to 30°K) the curves for a single crystal and for a polycrystal correspond so well. This suggests that the grain size is not the factor limiting the mean free path of the phonons at these very low temperatures. The single crystal had an oxygen content certainly larger than 2.001, as could be deduced from its electrical conductivity {<7= 1.7 X 10"3 ohm"1 cm'1 at 300°K). Transmission electron micro-

scopic observations have shown that in such a crystal small U4O9 precipitates are homogeneously distributed in the UO2 matrix, their mean distance being about 1000 Â . One can expect that such precipitates were also present in the sintered material. It is possible that these precipitates are responsible for the scattering of the phonons at very low temperatures. Some data on the thermal conductivity of a single crystal are shown in Table XIII.

1.2. High temperature thermal conductivity

Earlier experimental work on measurements of thermal conductivity of UO2 and UO2+X have been adequately summarized in several publications [9,89]. These studies showed that sintered stoichiometric UO2 obeys a relationship of the type К = A/(B+T), where К is the thermal conductivity over the temperature range 200 to ~1300°C. Factors found to affect the conductivity were excess oxygen, which had a large and deleterious effect, and the shape and distribution of the porosity, which had a minor effect.

55 "К

Fig. 23

Thermal conductivity of U02. © Single crystal. ® Polycrystal.

TABLE XIII

THERMAL CONDUCTIVITY OF SINGLE CRYSTAL U02

T(°K) К ( mW/'K cm)

300 83

77 40

30 9

12 22

4 11

More recent experiments have been concerned with measurements of the thermal conductivity of U02 single crystals and of hypostoichiometric polycrystalline material. The work on single crystals was largely stimulated by Hanford workers who advanced the hypothesis that energy transfer by photons was likely to make an appreciable contribution to overall conductivity under certain conditions [93] . Determination of the thermal conductivity of single crystals showed that there was a minimum in the plot of conductivity against temperature at about 600°C [94], and a maximum in the curve around 1200°C followed by a continuously falling conductivity with further increase of temperature [95] . Attempts to verify this hypothesis in the practical case of a fuel element by workers at the Atomic Energy of Canada, Limited (AECL) failed to reveal any enhanced thermal conductivity occurring as a result of in-pile columnar

56 grain growth [66] . These investigators re-confirmed the Battelle Memorial Institute (BMI) results [94] on enhancement in thermal conductivity of single crystals of fused U02; however they also showed that the improvement in thermal conductivity was lost by oxidation, which also eliminated the free uranium originally present. They then demonstrated that polycrystalline hypostoichiometric sinters also displayed the increased conductivity shown by single crystal hypostoichiometric material. The AECL workers [66] suggested that the increase in thermal conductivity is due to the inclusion of a large electronic conduction component at elevated temperatures when the excess uranium dissolved to form hypostoichiometric uranium dioxide. The thermal conductivity of hypostoichiometric material at ~100°C is the same as that of stoichiometric material [96] . Examination of the rods that Scott used (ROTHWELL [62]) in his thermal conductivity determinations showed some free uranium to be present, so his results, which agree with other workers for stoichiometric U02, were obtained on slightly hypostoichiometric material. Further experimental work on hypostoichiometric material is obviously necessary. Stoichiometric and near-stoichiometric U02 are semi- conductors. Contributions to thermal conductivity will arise from energy transfer by phonon/phonon interactions, by photons and by current carriers. These modes of transfer can interact with one another and, if strong, such interactions can be of importance. However, since experimental data on such interactions are limited, the approach adopted will be to consider each of the modes of energy transfer separately and deal with interactions as a perturbation. This approach is essentially that adopted in the most recent and comprehensive analysis of the existing data [97] .

1.2.1. Lattice conductivity

Over the major portion of the temperature range of practical interest and hence of experimental measurements, the thermal resistance should arise from phonon/phonon interactions. Existing theory shows the lattice contribution to the thermal conductivity (K) to be inversely proportional to temperature (T) whereas in fact К a l/(T + constant). At 1300°C the discrepancy may be as much as 15%. It must be remembered that the existing theory is based on small deviations from harmonic vibrations in a perfect lattice. KLEMENS [98] has pointed out that "there exists, as yet, no full treatment of highly imperfect lattices". In Section II. it is shown that at room temperature both uranium and oxygen can be treated as harmonic oscillators but that at temperatures above 100°C the harmonic approximation breaks down for oxygen. Under these circumstances one may query whether existing lattice conductivity theory is applicable to UO2 at high temperatures and if not, whether a great deal of emphasis should be placed on the other possible contributions to thermal conductivity. Theory developed exclusively for phonons will not indicate at present qualitatively in which direction the enhanced anharmonic behaviour of the oxygen atoms will affect thermal conductivity; however, one of the effects of anharmonic terms is to bring about equipartition of energy and thereby to increase the contributions of other modes to the transfer of heat (Section V. 6. ). In this connection the structure of UO2-X obviously should be investigated with respect to the 'vibration surface' of its

57 oxygen atoms. It will be of great interest to examine whether the anharmonic contributions to the vibration surface are greater in UO2-X than in UO2.00.

1.2.2. Radiant transfer

The radiant transfer contribution (Kr) to conduction is given by [99]

16 к n2 T3 Kr = — , 3 a where к is the Stefan-Boltzmann constant, n the index of refraction, T the temperature in °K, and a the absorption coefficient for incident radiation. Although this equation predicts the increase in transmission of energy with decreasing a, it does not say anything about the equipartition of energy. Con- sequently, even though a large value of a resulting from equipartition would predict a small radiative transfer, the net effect which accompanies the

large a would be to increase the total transfer of energy. For U02 the value of n(~2.2) is virtually independent of temperature, but as normal stoichiometric UO2 is an extrinsic semi-conductor up to 900 to 1100°C [100], free carrier absorption is to be expected so that a will be a function of temperature as well as wave-length. Bates and several other investigators have demonstrated that single crystal UCb is transparent over a large portion of the infra-red (3-13 цт) with minimum values of the absorption coefficient at circa 6 cm"1 [101] and this transparency has been invoked to explain the large difference in thermal conductivity observed with UO2 single crystals [93] compared to normal sintered material. It has been observed that a single crystal thin section which appeared bright red at room temperature became opaque at 600°C [102] but this effect could be due to broadening of an absorption peak. CHRISTENSEN [971 has derived values for absorption coefficients for material deviating slightly from stoichiometry in the direction of excess metal and in addition extrapolated to the behaviour of trulv stoichiometric material (defined as being an intrinsic conductor at room temperature). For UC>2.ooothe radiation contribution would peak sharply at 600°C, 0.016 watts/cm°C, for UO2.003 at 900°C,0.008 watts/cm°C, and for UO2.03 at ~1500°C, 0.001 watts/cm°C. In this temperature range the lattice contribution would vary from 0.04 watts/cm°C at 600°C to 0.02 watts/cm°C at ~1500°C. According to these calculations there will be little effect of radiant transfer on UO2.003 but truly stoichiometric material would show an appreciable improvement of about 35% at 600°C. This is to be compared with the observed improvement of thermal conductivity for single crystals of UO2 compared with sintered polycrystalline material of ~90% at 1200°C. The actual existence of a "UO2" which is intrinsic down to room temperature has

yet to be demonstrated. Slightly hyperstoichiometric U02 is reported to be p-type extrinsic at room-temperature whereas hypostoichiometric material shows n-type metal excess conductivity [65] . Data on the behaviour of hypostoichiometric material at higher temperatures are not yet available. In the light of the interpretation of electrical conduction in terms of a small polaron transport process (Section V.2. ) the validity of such calculations

58 may be queried, but no quantitative alternative treatment can at present be offered. A further factor which increases the absorption coefficient in the transparency region is grain boundary scattering: the coefficient for polycrystalline UO2 at wave-lengths less than 8 цт is reported to be an order of magnitude greater than that for single crystal material [101] . The general situation with regard to possible radiation contribution is seen to be confused. Direct measurements of absorption coefficients as functions of temperature, wave-length, and composition in the range UO2-X to UO2.005 both for single and polycrystalline material are needed before further advance is possible. In a sense, such measurements are direct experimental determinations of the interaction Hamiltonian function (Section V. 6. ).

1.2.3. Electronic transfer

In a semi-conductor exhibiting mixed conduction the conductivity Kei due to mixed hole and electron conduction is given in [103], by the equation

(37)

where k= Boltzmann's constant, e = carrier charge, T = temperature °K,

On = electronic conductivity, crp = hole conductivity, and total conductivity a = <7n + ap , Eg = activation energy for conduction.

For extrinsic semi-conduction either crn >>0P or сгр» crn and in either case the second term in brackets in Eq. (37) will be very small compared with the first. Nearly stoichiometric UO2 has a conductivity of about 10"1 iîcrn1 at tí 1200°K so that Kei is ~2 X 10" watts/cm°C. To raise this contribution to significance a 103 to 104 increase in extrinsic conduction is needed. It has been stated that the high temperature electrical conductivity of hypostoichiometric material is greater than that of stoichiometric material, but no suggestion has been made that the increase is several orders of magnitude. An explanation of the enhanced conductivity of UOo-x must be sought elsewhere. If electrical conduction in this range is due to small polaron transport this approach may not be valid but no alternative approach can be offered at present.

The term стп стр/ст has a maximum value of cr/4 when стп = crp, i. e. when intrinsic semi-conduction occurs. If Eg is of the order of 1 eV then the second term in brackets in Eq. (37) will dominate the first by about two orders of magnitude. There has been no thorough investigation of the intrinsic semi-conduction behaviour of nearly stoichiometric polycrystalline UO?. WOLFE [100] has tentatively suggested that intrinsic conduction starts in such material at about 900°C with an activation energy of 0.95 eV and that the band gap is double this, i.e. 1.9 eV. BRIGGS [104] has determined the thermal activation energy (Eg) for intrinsic conduction as 1.26 eV starting at 1100°C. Thus in the case of intrinsic conduction, to a first approximation 2 Kei =2T (k/e) IOOCJ. At 1200°K where

59 significant. Data for the electrical conductivity of single crystal UO2.000 (Section V. 2. ) show that it becomes intrinsic at ~800°K with Eg~2.9 eV. Specific conductivities are 0.5- 1 X 10"2 ohm"1 cm"1 at 800°K and 1-2 ohm"1 cnr1 4 at 1000°K. These values give KeJ~0.4 X 10" watts/cm°C at 800°K and ~10"2 watts/°C at 1000°K. This latter value represents a contribution which is 20% of the experimental value for polycrystalline 'stoichiometric' UO2 at this temperature (1000°K). If therefore hypostoichiometric material showed a lower temperature of transition to intrinsic conduction and greater than an order of magnitude increase in intrinsic conduction, the electronic contribution to thermal conduction could be significant in polycrystalline material and even more so in single crystals. It should be remembered that these latter calculations assume equal mobilities for the two carriers but this has not been shown to be true. To summarize, it would seem at present that insufficient data are available to assess accurately the radiant and electronic contributions to the thermal conductivity of either stoichiometric or hypostoichiometric UO2 whilst the application of standard phonon scattering theory to UO2 could bear closer study. The enhancement of conductivity of single crystal fused UO2 (actually UO2-1O is of considerable practical interest and similar studies need to be made on polycrystalline U02-xto establish the thermal conductivity behaviour of this material more firmly.

2. ELECTRICAL PROPERTIES

2. 1. Normal electrical properties

The electrical properties of UO¿ have been studied by several investigators. Adequate reviews have been written by MEYER [105] for the work up to 1940, and by WILLARDSON and MOODY [106] for the work up to 1961. From these reviews, it is clear that UO2 is a p-type extrinsic semi-conductor below ~800°C, its conductivity arising from the positive holes due to deviations from the stoichiometric composition. Above ~800°C intrinsic semi-conduction has been observed. The results of WILLARDSON et al. [107], which were the most significant before 1960, were interpreted by them on the assumption that UOs isa classical semi-conductor and that a band picture is valid. It has been shown, however, by HEIKES and JOHNSTON [108] that band theory is inadequate in describing the properties of a large number of ionic semi-conductors, such as the transition metal oxides. These authors explain the conduction mechanism by a jumping of electrons (or holes) from one cation to a neighbouring cation, the activation energy being associated with the mobility rather than with a carrier production. The electrons are thus not free to move through the lattice, but are localized at the metal ions. The conduction by this hopping mechanism should be considered [108, 109] as a thermally activated diffusion process, with a mobility

ß=ed~VQ exp (-E/kT) (38)

in which d is the jump distance, v0 the jump frequency and E the activation energy. The mobility of the charge carriers is very low («1 cm2/Vs) in

60 all these compounds. The conductivity can be expressed as:

const X exp (-E/kT) (39) a = WС T in which W is the probability that an electron, upon attempting a jump will find a site unoccupied.

In a study of the electrical properties of UO2+X, ARONSON et al. [61] used the ideas of Heikes and Johnston in order to explain the electronic conduction in UO2. They assumed an activated jumping of localized holes between U5+ and U4+ in UO2+X and thus introduced the concept of the hopping mechanism of conduction in UO2 . Recent measurements, both on sintered pellets [100] and on single crystals [110] have not revealed any Hall-effect, and thus indicate very low mobilities. The reason that the usual concept of energy bands cannot be used is that the electron-phonon interaction in these ionic crystals is so strong that polarons of small radius are created. The treatment of the small polaron motion that is used to describe the current transfer in these compounds has led to a hopping model at sufficiently high temperatures (the carriers move by random jump), characterized by an activated mobility:

It* e"E/kT •

It therefore provides a theoretical basis for the current transfer mechanism of Heikes and Johnston. Measurements of the electrical properties of UOjmade before 1960 are incomplete and made on poorly characterized samples. HASIGUTI and KIYOURA [llll forinstance, observed irreversible hysteresis when measuring the conductivity and stated that this was accounted for by the absorption of excess oxygen during the measurements because of the poor vacuum. Such effects have also been observed by WILLARDSON et al. [107], who noted that during measurements on oxidized samples changes in these samples occur, due to the appearance of U409.y and U307.z: the conductivity decreases and the material becomes n-type. These authors thus related the magnitude and the type of conductivity found to the phase relationships in the UO2 to U4O9 system. They further pointed out that caution must be exercised in any quantitative interpretation of the conductivity data because of the effects of grain boundaries in sintered UOc pellets. In fact, the electrical conductivity data of poly crystalline samples have only a limited significance as the conductivity is at least partly determined by the resistance of the intergranular boundaries. The measurements of the electrical conductivity and of the thermoelectric power by ARONSON et al.[61] at temperatures of 500 to 1150°C have resulted in a much better understanding of the electrical properties of иОг+х. They showed that the electrical conductivity in the single phase region can be represented by

a = (3.8 X 106/T) X 2x (1 - 2x) exp (-E/kT). (40)

61 A plot of log aT against l/T was shown to be a straight line for samples of a constant composition, the activation energy E being 0.30 eV. Breaks in these plots appeared to correspond to the transition of the single-phase region to the two-phase UO?+x - U4O9.J. region. ARONSON et al. showed that their thermoelectric power measurements as a function of oxygen excess can be represented by the equation:

(41) in which a is the Seebeck coefficient. Measurements of the conductivity and the Seebeck coefficient in the temperature range from 25 to 1100°C on samples with large grain sizes (60 цт and 150 цт have recently been reported by WOLFE [100] . The crystals he used had compositions deviating only slightly from stoichiometry (O/U = 2.003). In Fig. 24, the electrical conductivity is plotted against 1 /Т. From this figure it is evident that the grain size has a large influence on the magnitude of the electrical conductivity. From the slope of the log oT versus l/T lines an activation energy of 0.20 eV is calculated, being somewhat lower than the value obtained by ARONSON et al. (0.30 eV). The low mobilities calculated from the conductivity data by Wolfe are in agreement with the proposed mechanism of jumping of localized holes and the observation that no Hall coefficient could be measured. Between 400 and 800°C the Seebeck coefficient has a constant value (Fig. 25), which means, to the first approximation, that there is a constant number of carriers in this region. Further, from the fact that the Seebeck coefficient is positive in the whole temperature range investigated by Wolfe, it is concluded that the conductivity is p-type. Above about 800°C a marked decrease in the thermoelectric power was observed which may be interpreted as a transition to intrinsic conductivity. This is also confirmed by the change in slope of the conductivity curves. From the slope of the plot of log ст versus l/T an activation energy of 0.95 eV is calculated, corresponding to an energy gap of 1.9 eV. Evidence for the existence of intrinsic conduction, found by Wolfe at high temperatures, has also been found previously by WILLARDSON et al. [107] . Measurements of the electrical conductivity of UO2 single crystals of different composition as a function of temperature have been reported by NAGELS et al. [110] in Table XIV. Interpretation of their results was done on the basis that the charge carriers in UO2 are small polarons. Hall-effect measurements performed between 200 and 950°K on their single crystals yielded no measureable response, indicating a Hall mobility smaller than 0.06 cm2/V s in the whole temperature range (ß< 0.015cm2/V s at room temperature). The low mobility in UO2 can only be explained if the effective mass of the polaron is very large. Nagel1 s electrical conductivity measurements between 90 and 800°K for crystals with compositions varying

between U02.oooand иОг.мбаге plotted in Fig. 26 as InaT versus l/T. Ac- cording to the theory of small polarons, which in its most simplified form leads to

62 —- ю3/т

Fig. 24

Electrical conductivity of U02 003 as a function of reciprocal temperature

the curves should be straight lines with slopes determined by the activation energy. The activation energy from 0.34 - 0.19 eV with increasing conductivity is shown in Table XV. This effect could be explained on the basis of Nagaev's theory of small polarons [112] which predicts a decrease of the mobility activation energy with decreasing degree of defectiveness. As can be seen from Fig. 26 the conductivity curves of the samples with different oxygen content display a remarkable difference at low temperatures. For the specimens with a very low oxygen content a linear relationship between InaT and l/T exists over about eight decades. However, when the oxygen excess increases the InaT curves bend off at low temperatures. The complete

63 temperature (°c)

Fig. 25

Seebeck coefficient of U02 ш

data for these slightly oxidized U02 single crystals can be separated into two components: E E (тТ = А1 X exp(-pjr) + A2Xexp(-j^). (43)

The values of E2 (high temperature) vary from 0.19 - 0.22 eV and Ej^ (low temperature) from 0.065 to 0.080 eV.

TABLE XIV

SPECIFIC ELECTRICAL CONDUCTIVITY OF U02.ooo*

Temperature 1 о(П cm)" CK)

7 298 5. 6xl0"

350 5. 0X10"6

400 2. 0X10"5

450 6. 3xl0"5

500 1. 6X10"4

550 3.1X10"4

600 5. 4X10"4

* Lowest O/U - ratio.

64 T

Fig. 26

Electrical conductivity of single crystals of U02. Results of Nagels et al.

The conductivity of single crystals with O/U ratio smaller than 2.001 have values ranging from 6 X lO"? to about 2X10-3 (Пет)-1 at room temperature. The latter value is obtained for samples having an O/U ratio equal to 2.001. No further change in the conductivity occurs as the excess oxygen content increases still further to UO2. об- This feature indicates that the upper limit of the UO2+X phase is UO2.001 at room temperature, close to the limit indicated by the recent thermodynamic data. It should be pointed out that the measurements by Wolfe and by Nagels differ markedly, especially for the electrical conductivity. Measurements of the thermoelectric power of a series of single crystals [110] are shown in Fig. 27. From the sign of the Seebeck coefficient it could be deduced that the conduction is p-type for these crystals. From 120 to 300°K the variation of the thermoelectric power with temperature is small indicating a rather constant number of charge carriers. In the same temperature range the conductivity increases exponentially over about four decades.

65 TABLE XV

SPECIFIC ELECTRICAL CONDUCTIVITY OF URANIUM DIOXIDE AT ROOM TEMPERATURE (290°K)

Sample о Activation energy Observer Grain size (Пет)"1 (eV) Preparation (jim) ОД] - ratio

ARONSON et al. [61] sintered pellet 60 2. 003 2. 4 X lo"4 0. 30 (98-99%)

WOLFE [100] sintered pellet 60 2. 003 7. 8 XIО"4 0. 20. (98.5%) 150 2.003 9. 3 X Ю"4 0. 20

3 NAGELS et al. [110] single crystal - < 2. 001 6. 10-7- 2.10' * 0. 34 - 0.19 2. 003 2.10"3 0. 22

* Due to variations in composition between UO2,00!l and U02.ooi • _ 700 gl о 600- о CT © >a . CD . —— Д x. x x"x" —-о 1 » >-"X- û: 500' ф—ъ sï. ш /X оS © © о- 400- а -X Л СЕ CURVE о/и о 300- UJ © - ш © - g 200- 2.002 о: © U1 © 2.00 2 Í 100 © 2.11 0 350 250 150 Ó -50 ' "150 © TEMPERATURE (°C) -100

Fig.27

Seebeck coefficient of single crystal U02 as a function of temperature

This behaviour can be explained if one assumes that the mobility is thermally activated. However, here the question arises whether the formula of the thermoelectric power deduced for band semi-conductors may be used for semi-conductors where the current transfer occurs by small polarons. Above 300° the thermoelectric power decreases; this probably corresponds to an increase in the number of charge carriers, which can be explained by a change in the ratio UO2/U4O9, the formation of UC>2+X being favoured at increasing temperature.

2.2. Effect of irradiation on electrical properties

The influence of fission fragment damage on the electrical conductivity of UO2 single crystals of different composition after successive periods of reactor exposure has been studied by NAGELS, [113] . The main effect observed on the electrical behaviour of UO2 is an appreciable decrease in

conductivity for non-stoichiometric U02 samples (Fig. 28). A constant value is reached at an integrated thermal neutron flux of about 4 X 1010 n/cmS (or 4 X 1015 fissions/cm3), suggesting a saturation of damage in UO2. The electrical conductivity of a stoichiometric crystal remained practically unchanged after exposures up to 1.6 X 1016 fissions/cm3.

The fact that the conductivity of the non-stoichiometric crystals drops markedly shows that the defect centres produced by irradiation are mainly hole-traps, which reduce the initial free hole concentration.

As stated above, the conductivity of the non-stoichiometric crystals reaches a constant value at about 4 X 1Ö15 fissions/cm3. This effect can easily be understood if one considers that there is some overlapping of tracks,

67 Fig. 28

Decrease in electrical conductivity for non-stoichiometric U02 during irradiation especially at rather large doses. The fraction of volume affected by the fission tracks, taking overlapping into account, is given by the formula:

v N fvol=l-e" o (44)

3 where N is the number of tracks per cm and v0 the volume of a track. In- 16 troducing v0 =4.5 X 1СГ (d= 100 Â, 1 =6 цт), one finds that 97% of the volume is affected by fission tracks for N = 8 X 10l5/cm3(or 4 X 1015 fissions/cm3), and 100% for N = 1016/cm3 . There is a good agreement between the experimentally observed and the calculated value of the number of fission tracks at which saturation of damage occurs. Recovery experiments, performed in the temperature-range from -150 to 850°C using the method of BALARIN and ZETZSCHE [92], showed the existence of two distinct annealing stages. A reverse annealing process occurs between 350 and 500°C. It has been observed by transmission electron microscopy that the superlattice structure of the U4O9 precipitates is destroyed by the passage of fission fragments leading to a homogenization of the precipitated U4O9 phase in the UCb matrix [113] . The further decrease in conductivity, occurring upon subsequent heating to 350°C, probably indicates that the excess oxygen is rearranged in the ordered U4O9 superstructure. As a consequence of this reprecipitation, electrically active oxygen ions are removed. Recovery of the conductivity starts only at about 650°C, with an annealing peak at 725°C. This second annealing process obeys second order kinetics, the associated activation energy of defect migration being 2.5 eV. By comparing the energy of motion determined in this way with the activation energy for uranium and oxygen self-diffusion (3.8 to 4.7 eV for U and 1.3 eV

68 for О), one finds that the most plausible explanation for the 2.5 eV process and thus for the recovery stage at 725°C is uranium point defect migration. This suggests that the defect centres, which act as hole traps, are uranium vacancies.

3. OPTICAL MEASUREMENTS

Optical properties, correctly interpreted, can throw much light on the electronic structure of a material. There have been few determinations of such properties for uranium dioxide, possibly because its high optical density necessitates the use of thin specimens with attendant handling problems. When the optical properties of UO5 were last reviewed, far from light being thrown on its electronic structure, it was pointed out that a large difference existed between the band gap derived from optical absorption data and that derived from electrical conductivity data [9] . In the following sections, an attempt is made to resolve this difference; existing data are interpreted in terms of defect absorption and new data on the optical properties of single crystals are reviewed.

3. 1. Intrinsic absorption edge

Uranium dioxide is generally recognized as a semi-conductor which can become intrinsic at sufficiently high temperatures. Despite the discussion of the recent electrical data above in terms of polaron theory, the conventional language of semi-conductor theory will be used here. Values for the energy gap of 2.09 eV [101], 2.18 eV[114], 3.0 eV[107], 1.9 eV[100], 1.26 eV[104] and 5.25 eV [ 104, 115] , have been derived from different types of experimentation, in particular from measurements of optical obsorption and electrical conductivity. Of the optically observed energies, the highest is most likely tobe correct since absorption due to other sources can be so large in this material as to appear to be the intrinsic absorption unless films are used or the crystal is extremely thin. It has been observed by RALPH [102] and by WILLIAMS [116] that it is possible to prepare single crystals sufficiently thin to appear yellow by transmitted light. Hence, even in single crystals, the intrinsic gap, when observed optically, cannot be as low as ~2 eV and the film results [104, 115] are accepted. Thus it would appear that this edge, when observed optically, is not less than 5.25 eV. It is not to be expected that there should be direct numerical agreement between electrical and optical observations of the intrinsic band gap. For a polar semi-conductor it has been pointed out that the optical and thermal activation energies should not coincide since the optical transition occurs without relaxation whereas the thermal transition corresponds to the difference in equilibrium configurations [117], (see Section V. 6.1.), Under such circumstances the ratio of optical to thermal activation energy is approximately that of the static to high frequency dielectric constants. For

U02 this latter ratio is 24:5.8 [104] so that the thermal activation energy for intrinsic conduction corresponding to the 2400 Â edge is 1.25 eV. It has been pointed out in Section V. 2. that band theory is inappropiate for U02

69 and that small polaron theory should be applied. Correlation of the intrinsic absorption edge with the electrical properties as interpreted in terms of small polarons does not appear possible at present.

3. 2. Defect absorption

In addition to providing an estimate of the energy gap between the valence band and the conduction band, the optical observations made by ACKERMANN et al. [115] on thin films of uranium dioxide show two partially resolved peaks with centres near 300 and 400 nm. Both were present in films annealed at ~960°C for two weeks at an ion gauge reading of 10~6 mm Hg or less. When these films were heated in successively higher pressures of oxygen the optical density at the shorter wave-length peak grew and that at the longer wave-length peak diminished. Since the absorption coefficients in the region of these peaks are cnr1, verification of the situation for bulk material is very difficult and detailed numerical verification almost impossible. The reflectance measurements on uranium dioxide powders made by COMPANION and WINSLOW, [118] however, do verify the character of these two peaks to the extent possible by this method. This verification, plus consistency with measured thermodynamic properties, indicate that the results of the thin film measurements are applicable to bulk material, that the short wave-length peak is to be associated with interstitial oxygen ions and that the long wave-length peak is to be associated with the presence of vacancies in the regular fluorite oxygen lattice. Actual numerical associations were used in the application of the defect theory of nonstoichiometry. The details of the determination of these numbers will be given here and in the Appendix to this Report. A relationship, given by Eq. (8) of the Appendix has been derived between the sum of distributions of oscillators of two types, each of which gives rise to several absorption bands, and the resulting index of refraction and extinction coefficient. The resolution actually observed is a partial, but definite, one into two peaks. If only two (Gaussian) distributions are used with the values of the optical constants determined for their well annealed films by ACKERMANN et al. [115] the following constants for the distributions give the fit shown in Fig. 29.

.i Njfj/Nu Ej(eV) Wj(eV) V 0.00866 3.25 0.603 i 0.0439 4.25 1.098

Here, V means "vacancy," i means "interstitial," Nj/Ny is the ratio of the number of defects of the jth type to the number of uranium atoms, fj is the oscillator strength, Ej is the centre, and Wj a measure of the width of the jth type of oscillator. An object of this analysis was a direct verification of the defect theory of nonstoichiometry by a direct determination of a coincident pair of interstitial and vacancy densities. If this is to be done it is necessary to know the oscillator strengths which, in turn, requires knowledge of the actual nature

70 E ( ELECTRON VOLTS)

Flg. 29

Resolution of observed optical absorption by thin films of U02 into two broad bands. The circles are the observations cast into a form required by Eqs. (52) and (53), and the line shows the fit obtained. The ordinate is (volunie)/(polarizability) per unit energy calculated in (eV)"1.

36m En(E)K(E) 0 e2h2[n2(E)+2]

rtilî • Г LL_ e j = lNu/irWj

of the absorbing centres. As far as actual knowledge deduced from independent determinations is concerned, there is none, of course. In order to determine if these results are within reason, then, certain further sup- positions, which must be reasonable, must be made. Since the assumption of localized centres was made, a vacancy must contain at least one electron.

Because of the rather strong ionic character of U02 it has been supposed here, indeed, that it contains two electrons so that the electrical neutrality in the region of the vacancy is maintained. It should be noted that a weak

absorption at about 3 eV has been observed in ThOs [120] where the same 2-electron centre at an oxygen vacancy, but no interstitial centres, could be

expected. It is further argued in the Appendix that Wv is excessive for it to be descriptive of a single transition.

71 In the case of the interstitial peak the width is evengreater. The simplest argument seems to be that an interstitial entity is also doubly charged (nega- tively) and the wide band contains many unresolved transitions derived from the six 2p electrons of the oxygen ion. It is clear, of course, that all these transitions which are suggested as contributing to the width of two peaks must be of energies in the neighbour- hood of 2 - 5 eV. They must be from the ground states of th'ese centres to high excitation levels, or to the conduction band, or from the full band to (empty) high lying levels of the "impurity" centres, It is quite likely that some of the infra-red levels [101] could be transitions between closely neighbouring excitation levels of these centres, for each of which there is an oscillator strength. Nevertheless, present considerations have been made on the basis that the total oscillator strength to be assigned to these two supposedly complex peaks is equal to the number of electrons assumed to be in the centre. Thus fv = 2 and f¡ = 6, which would mean a composition of these films of U02.oo3- The arguments given here and in the Appendix seem to lend reasonable support to this analysis and interpretation of these two peaks. Substanti- ation would require, however, much work in the way of measurements of the optical absorption at low temperatures, measurements of photoconductivity, electron spin resonance analysis, and so on. A comparatively low absorption peak has been observed [115, 121, 122] at about 665 nm, corresponding to an energy of 1.86 eV. KIKUCHI andNASU [122] have suggested its association with interstitial oxygen. COMPANION and WINSLOW [118] however, found it to be most pronounced at that oxygen concentration where one would expect the greatest humber of vacancies. The magnitude of the energy involved would also make it easier to associate with the oxygen vacancy absorption centre as proposed earlier. It is difficult to draw conclusions on this aspect of the problem from GRUEN'S [121] work since his plates of solid solutions of Th02-U02 were, apparently, of variable thickness and he only gave optical density in arbitrary units, and since he used pure Th02 plates as blanks. In either case, such an association would be another factor which would reduce the total oscillator strength of one or the other of the defect peaks by some unknown amount below those assigned in that analysis.

3.3. Infra-red absorption

BATES [101] has observed many infra-red peaks in U02 in an initial study. He has associated them in a general way with uranium metal inclusions and impurity centres in addition to those to be expected in pure well annealed uranium dioxide of varying stoichiometry. As he recognizes, further sorting and refinement of these observations is required. An important development has been the discovery in this region of the absorption spectrum of U02 of a window in the range 3-13 jum [101], (see Section V. 1.2.2. ).

72 4. MAGNETIC MEASUREMENTS

There is reasonable qualitative agreement between three sets of measurements of magnetic susceptibilities of (U, Th)C>2 solid solutions 1123, 124, 180] . The Curie-Weiss law is obeyed with the magnetic moment p of the U(IV)ion falling from 3.1 to 3.2 Bohr magnetons (BM) in U02 to 2.85 to 3.0 BM at infinite dilution, and the Weiss constant Д decreases from 210 to 230 for U02 to 26 for 2% UO2 in Th02; some residual value would be anticipated for Д at infinite dilution. The magnetic moment for the gaseous U4+ ion in the 5f 2 configuration is 3.58 BM (L-S coupling) or 3.84 BM (j-j coupling). The experimental results were therefore interpreted as showing that the U(IV) ion had the 6d2 configuration, the susceptibility being close to the 'spin-only' value for two unpaired electrons (=2.83 BM) with the orbital contribution nearly, but not quite, quenched by the crystal field.

The susceptibility results for U02 follow the Curie-Weiss law well from ~100°K upwards. A maximum in the susceptibility occurs about 30°K with a corresponding maximum value of [d(XT)/dt] between 30 and 31°K. This temperature is in excellent accord with the observed anomaly in the heat capacity reported to this Panel [41], (cf. Section III.1.1. ) as indeed it should be according to the theory of FISHER [125] . The two sets of data [36, 126] presented in Fig. 30 as a plot of XT against T are in good accord. Below 24°K, the susceptibility is reported [35, 36] to be constant and equal to 1.59 X 10"5/g or 3900 X 10"6/mole whereas preliminary data on another sample [126] reveal a more complex dependence on temperature.

A different interpretation of the results for U02 and (Ü, Th)C^ was proposed by HUTCHINSON and CANDELA [127]. They calculated the crystalline field energies of an ion surrounded by eight oxygen ions in a cubic array, as in UOs, and showed that the observed values above 100°K were consistent with a 5f2 configuration and a triplet state being energetically favoured. The magnetic susceptibility was calculated as X= N{(g3)-/KT} x(25/6), with g = 4/5, equal to the spin only value g = N(ß2/3KT) 4S(S + 1), with S = 1. DAWSON and LISTER [180] studied a series of oxides made by low temperature oxidation. The susceptibility decreased as oxygen was added but a Curie-Weiss law was obeyed fairly well with large positive values of Д. The results were compared with predictions made by assuming that the U(IV) was oxidized to U(VI) which was assumed tobe diamagnetic. After correcting for the diamagnetism of U(.IV) and 0=, and neglecting any contribution due to U(VT), the susceptibility of the remaining U(IV) in each oxide was calculated.

Effective moments calculated from цец = 2.83\/YU(IV) (Т + Д) were as follows:

Composition 2.00 2.06 2.11 2.18 2.25 2.30 Meff 3.20 3.33 3.09 3.01 2.97 2.94

This decrease in jueff was attributed to increased quenching of the orbital angular momentum. It is probable, however, that the compositions of the samples used were somewhat in error (i. e. low in O/U by ~0.05) and the concentration gradient of oxygen through the samples would probably alter at the higher temperatures of measurement (563°K).

The susceptibilities of UO2.1, U4O9 and UO2.33 have been measured from 1.5 to 44°K[36] . All these specimens showed a maximum in the X - T plot

73 т CK)

Fig. 30

Magnetic measurements on U02. x - Leask, et al. ; O- Westrum, et al. The inflection at 30°K occurs at the magnetic transition point. at 6.4°K, the magnitude of which was larger the higher the oxygen content; the behaviour of the sample of UO;,i between 0 and 20°K was not much affected by the thermal history and state of annealing. Extrapolated values at 0°K were as follows:

X/g 105 UO2.00 1.59 UO2.1 2.7 U4O9 3.5 UO2.33 ' 4.5

The results of ARROTT and GOLDMAN [35] are marred (as they recognized) by uncertainties regarding the composition and phases present in their samples, but they too report a magnetic anomaly at about 4°K for many of the oxidized samples. They could not decide bétween a model based onU(VI) or U(V) on the basis of their results. Their conclusion that a two-phase

74 region exists between UO2.31 and UO0.43 is certainly incorrect and probably reflects the unsatisfactory nature of their samples.

Some results on oxidized (U, Th)C^+x solid solutions were reported by DAWSON and ROBERTS [129]. Solid solutions containing 14.8, 27.9and43.1% of UO2 were oxidized to mean U valencies between 4.3 and 5.46. The susceptibility at 300°K decreased approximately linearly with increasing oxidation until the U oxidation number was 5.0, and more slowly thereafter. The results were interpreted as favouring the introduction of U(V) in the early stages of the oxidation, but it was noted that the earlier results for

UO2+X did not fall into the same pattern. Since oxidation of (U, Th)02 solid solutions offers the possibility of preparing homogeneous ЛЮп+х phases which are quite stable at all temperatures, these measurements could well be extended, and susceptibilities measured over a wider temperature range. A rather surprising aspect of the comparison of magnetic behaviour and thermal properties is the absence of any appreciable magnetic heat capa-

city contribution in the data reported on a-U3O7, ß-U307, U4O9, and U308 below 10°K. Further work is necessary to confirm this or to resolve the apparent discrepancy. It appears that the magnetic transition at 6.4°Kmight be associated with local magnetic ordering within the complexes or "zones"

of the ЩОд structure, which occur in the U02+x and other structures, (see Section II).

5. DIFFUSION PROCESSES IN U02

In this section the data available on the diffusion of oxygen, uranium and

rare gas fission products through U02 are discussed. It is evident that the mechanism of release of fission product gases depends on the state of the

U02 lattice following irradiation, which is briefly discussed in Section VI; it may therefore be misleading to make too close comparisons between fission product migration and self-diffusion processes. Of the latter, recent evidence suggests that the self-diffusion of uranium through the bulk lattice is slower than has previously been reported and, indeed, that reliable values of diffusion coefficients have not been established yet. We include also brief mention of a detailed study of the diffusion of argon in calcium fluoride crystals which has shown that in this case the lattice diffusion of a rare gas atom is not affected by the concentration of vacancies and interstitials present, but much affected by radiation effects; i.e. by defects of higher order. The strong similarities between Ca F2 and UO> with regard to rare gas diffusion make it seem likely that the same holds true also in the latter case.

5.1. Oxygen diffusion

AUSKERN and BELLE [128] studied the exchange of O18 between CO^ and samples of иОг and U02+x at temperatures from 400 to 800°C. At the lower temperatures studied, some of the UOo+x samples should have been in the two-phase region, but no account was taken of this. Analysis of the

samples before and after exposure to СОг indicated no appreciable oxidation. This result may indicate that the oxidation equilibrium between CO? and U02

75 is not readily established at temperatures below 800°C although the exchange of Ö18 between gas and solid surface is rapid; however, the CO content of the gas was not measured and it is impossible to calculate how much oxidation should have occurred. The results indicated that diffusion is assisted by the presence of inter-

stitial oxygen. A sample analyzing as U02 wi (average particle radius = 2.65 дт) showed D = 1.2 X 10"3 exp (-65 300/RT). D values for the non-stoichiometric oxides were higher, rising with x, and the activation energy given is 29 700 cal/mole, though a lower value might be preferable. Values for two samples were:

0/U = 2.004, a = 0.5 цт; D = 7. 0 X 10"6 exp (-29 700/RT) 0/U = 2.063; a = 1.3 цш; D = 2.06 X 10"3 exp (-29 700/RT).

The increase of D with x (at low values of x) is compatible with an "interstitialcy" (indirect interstitial) mechanism of diffusion, as had been suggested on the basis of earlier exchange experiments [129] . Values for

the diffusion of oxygen down a concentration gradient in UO^+x cannot readily be obtained from oxidation experiments at low temperatures because of the formation of tetragonal structures under these conditions, (see Section II). 12 2 It is worth noting that Q.e]f reached 10" cm' /s at 750°C for O/U = 2.002 and at 420°C for O/U = 2.063. The results are shown in Fig. 31.

5. 2. Uranium self-diffusion

AUSKERN and BELLE [131] measured the diffusion of U:33 in sintered

pellets of U02 of > 98% theoretical density held in a H2 stream, estimating the penetration by a-emission from the surface exposed after successive grinding operations. They found D = 4.3 X 10"* exp (-88 000/RT) from 1450 to 1850°C, but they noted some decrease in the apparent diffusion coefficient with time of anneal, and the activity profile differed somewhat from the calculated profile for volume diffusion. Similar, but rather higher, values were

reported by Lindner for diffusion anneals carried out in H2 [132]; he found D = 0.23 exp [-104 600/RT] . In later work ALCOCK and Me ÑAMARA [133] found D = 1.18 exp [ -108 000/RT], but they also examined two single crystal specimens and found much less U penetration at 1600°C, the diffusion coefficient being ~1X10"15 cm2/s; the corresponding value for polycrystalline UO2 is ~2X10"14 cm2/s. This is direct evidence that grain boundary diffusion accounts for the major part of the tracer penetration on polycrystalline ag- gregates and the variability of the results obtained by different investigators may be understood on this basis.

5. 3. Argon diffusion in calcium fluoride as a model process for fission gas transport in uranium dioxide

After introducing argon atoms into calcium fluoride single crystals by reactor irradiation, by which the argon is created from calcium in an (n, a) reaction, one can study a process analogous to that of xenon diffusion in UO2I134].

76 TEMPERATURE (°C)

Fig. 31

Self-diffusion of oxygen in uranium dioxide (Reproduced by courtesy of A. B. Auskem and J. Belle [128])

The release of Ar from CaF2 has been extensively studied with respect to the following tasks: (i) To test the assumption that a diffusion process is responsible for the gas release; to establish quantitatively the conditions under, which the release is due to diffusion alone, or to diffusion combined with evaporation of the specimen surface, and to account for the effect of evaporation [135] . (cf. Section V. 5. 4. ); (ii) To determine the atomic mechanism of the diffusion; (iii) To establish the connection between radiation damage in the crystals and the rate of the gas release. The diffusion mechanism was investigated on crystals which were doped with sodium and yttrium respectively in order to vary the concentration of Frenkel-defects in the fluorite lattice [136] (Fig. 32). It turned out that lattice diffusion predominates between 600 and 1400°C with argon atoms jumping from interstitial to interstitial (enthalpy of migration 67.4 kcal/mole), whereas the transport below about 600°C occurs along dislocations (enthalpy of migration 8 kcal/mole).

77 т[°с] 1200 1000 800 600 400 300 200

—• fV']

Fig. 32

Argon diffusion in CaFz as a function of temperature. The branch with the highest slope was obtained on crystals with a surface-to-volume ratio s/v <10 cm"1. The crystals used in the low temperature region are characterized by s/v к2х102 cm"1. 3 o = CaF2/Na+; O? CaF2/Y + ;- = CaFz.

Even a moderate irradiation has the effect that measured diffusion coefficients are not representative of the real diffusion process. The apparent D-value always lies below the real one, the difference exceeding a factor of 1000 at high neutron doses, (Fig. 33). Since this difference is temperature dependent, all important transport data obtained with highly irradiated crystals

are distorted, e.g. activation energies and D0-values. The results are in reasonable agreement with analogous observations on UO2 which have recently been published by MacEWAN and STEVENS [137] and may indicate a trapping of rare gas atoms in defect clusters due to radiation damage.

5.4. Fission gas release

A great deal has been written about fission gas release from U02 and is summarized in two recent reviews [138, 139] . Gas release may occur by recoil, lattice diffusion, grain growth and evaporation. Gas released by recoil constitutes only a very minor contribution to gas release in the practical case. Some results attributed to recoil losses for in-pile measurements could also be interpreted by diffusion mechanisms with very low activation energy (diffusion along dislocations) [136] . Gas release by diffusion after light irradiation has been extensively studied. The values for the activation energy fall into three groups centred at 49, 71 and 129 kcal/mol, the value of 71 being dominant in most cases [140, 141] . However it has been shown that the release kinetics due to continuous evaporation of the surface is very similar to that caused by diffusion [135] . With the aid of a simple model for the competition between diffusion and evaporation the corres-

78 F»ST NEUTRONS/cm2(E »O.SMeV)

2 FAST NEUTRONS/cm (En >0.5 MeV)

Fig.33

Argon diffusion in CaF2 as a function of fast neutron dose

(a) "Macroscopic" diffusion coefficient for argon diffusion

in CaF2 at 750°C as a function of the fast neutron dose, (b) Fraction of all argon atoms which are free to diffuse in the lattice at 750°C as a function of the fast neutron dose. ponding contributions to the release have been accounted for quantitatively for different meterials [135] . The results show that in the case of U02 the release is probably merely due to evaporation at temperatures above 1500°C. But one has to bear in mind that at about 1600°C equiaxial grain-growth begins to occur in sintered UO2 and this change in gas release behaviour may be also associated with such structural changes. However, at slightly higher temperature (~1800°C), in a fuel rod much more marked structural changes occur, viz. columnar grain growth, accompanied by virtually complete gas release [150] . The only likely hypothesis so far advanced to explain these phenomena is based on the continued movement of macroscopic lenti- cular voids up the temperature gradient to the axis of the rod by a vapour transport process [151] . It may be significant that at temperatures approxi- mating that which marks the lower limit of columnar grain formation UO2

79 becomes appreciably plastic and ÚO2-X is formed by heating at low oxygen partial pressures. In irradiated U02+x the columnar grains extend much farther towards the circumference of the fuel rod [152] but it is not clear whether columnar grain growth occurs at a lower temperature in such material which has a lower thermal conductivity than stoichiometric U02 [89] . Also found in this region are second phases which have been reported as uranium for irradiated UO2 clad in Zircalloy [163] and as (molybdenum+ ruthenium) or (barium + cerium) containing phases of U02 clad in stainless steel [159] . The presence of liquid phases can lead to columnar grain growth of the type described. Phase relationships in the UOo.x/U02 ranges as well as the UO2/fission oxide are obviously relevant to an understanding of this situation. For temperatures below 1600°C the agreement for actual values of diffusion coefficients from polycrystalline sintered material is not quite as good as for activation energies, and D values have generally been derived from experimental diffusion coefficients (D1) by the equivalent sphere concept, the sphere'radius having been derived from surface area measurements. A diffusion coefficient (D) of about 10"14 cm2/s at 1400°C is fairly typical and if one assumes uniform release from the whole surface for a year, this would correspond to release from a layer only about 10"6 cm thick. Slight departures from stoichiometry due to surface oxidation are known to give rise to a considerably enhanced heating burst [143, 148], and gas is known to be released more rapidly in UO=ix [144], which may be attributable to an evaporation process [135] or to the effect of oxidation on microstructure. The importance of adequate characterization and control of the physico chemical nature of the surface during such experiments must be emphasized. A rapid physical non-destructive testing method for determining minor departures from stoichiometry might therefore prove a very useful practical tool. The effect of departure from stoichiometry on the metal-rich side upon diffusion behaviour has as yet received little attention apart from observation of cooling bursts which nevertheless could constitute an appreciable amount of fission gas [146] . This behaviour is not repeated in hypostoichiometric single crystals [147] . Heating bursts may be due to a number of causes and evidence has been advanced that fission xenon may be im- mobilized by closed porosity in the initial material [137, 136] . Diffusion coefficients for single crystal and fused material appear to be lower than those for sintered materials but recent work has shown the rate of gas release to be a function of dose [137, 136] as well as type of UO?. The difference in behaviour between the single crystal and sintered material in the latter investigation has been explained in terms of trapping by various types of line defects, e.g. grain boundaries which would vary from single crystal to polycrystalline material. However, fused polycrystalline material is similar in behaviour to single crystals [149] so a more sophisti- cated explanation must be sought; it is suggested that vacancy clusters may be responsible [137] . To summarize, in the temperature range below 1600°C release is pre- dictable although the mechanism of release must still be regarded as im- perfectly understood. From a practical viewpoint it seems desirable to have

UÓ2 very close to stoichiometric composition, of low surface area and possibly having closed porosity or vacancy clusters as traps for fission product

80 gases. The only statement that can be made about gas release for the temperature range above 1600°C is that the mechanisms of structural changes occurring are insufficiently understood to know which physical properties are involved and hence to enable any prediction of how fission gas release may be minimized. The obvious practical answer to this problem is to'limit centre temperatures as far as practicable. This in turn means limiting fuel ratings. The best method of removing this rating limitation would at present seem to be by increasing the thermal conductivity of the oxide.

6. CORRELATIVE THEORY OF PHYSICAL PROPERTIES

It is helpful to summarize here some of the correlative theories which should serve to coordinate many of the physical properties of the uranium dioxide phase in the future. In particular, we outline here general theories concerned with the transport of energy and of matter.

6.1. Transport of energy

The transport of energy within a solid like the uranium dioxide phase (not at equilibrium) is accomplished via several mechanisms such as phonons, excitons, polarons, free charges, etc. These coextensive subsystems can be coupled to each other to various extents and consequently can be isolated from each other, partially isolated, or equally partitioned energy wise. To obtain an insight into the problem one must adopt a perspective or frame- work which recognizes this possible partial equipartition. The following treatment sketches the theoretical aspects. Although the problem of the electron-phonon interaction in this particular case of ionic or covalent bonds has not been treated, to our knowledge, the equivalent one in a metal where the continuum of the free electrons tends to establish equipartition has been investigated. An excellent summary is presented in ZIMAN's [154] book on transport phenomena in solids. In the case of metals, one initiates the study-by constructing the Hamiltonian function for a system composed of ions and electrons. The form, however, is essentially the same as that for a molecule, so that the question is the same as that involved in the separation of the vibrational part of the wave function from the electronic part. The Schrödinger equation for the molecular system is

(59)

= e^XiXj), in which i refers to the co-ordinates of the nuclei, j refers to the electronic co-ordinates, and V(X4Xj) is the potential energy which depends upon the positions of both the nuclei and the electrons. In the Born-Oppenheimer procedure of handling this equation, one assumes that the motions of the electrons are adiabatically isolated from the motions of the nuclei. To ac-

81 complish this formally, one writes the wave function as the product of two functions. Thus

(MX¡Xj) =u (XjXj)v(Xj). (60)

Substituting the right side of this equation into the Schrödinger equation, one can then separate out the equation, v +E v(Xi)=e,v(Xi) (61)

{"Iâ;i f } ' in which

E + +v +u (62) - ñpñrtf ^ ¿xt ЛЕ?

If the first two terms in the sum are identified as

i 1 and if it is set equal to zero, then substitution of e from Eq. (62) into Eq. (59) with ф = uv yields the results:

J+vtX^jufX^HE^MX^); (64) j and e1 = e.

Hence the original Eq. (59) has been divided via He= 0 into two equations, one involving only the nuclear co-ordinates, Eq. (61), and another involving both nuclear and electronic co-ordinates, Eq. (64). However, the latter contains the kinetic energy of only the electronic co-ordinates and therefore permits Х; to be treated as a parameter. Consequently its solutions yield eigenvalues, E(X¡), which contain Xt as a parameter. The value of E(X;) so obtained can be employed in the equation containing the kinetic energy of the nuclei, Eq. (61). It is apparent that this procedure has separated the two kinetic energies so that it contains no mechanism for effecting the transfer of kinetic energy between phonons and electrons. The means for the inter- change is contained in the interactional Hamiltonian function, Eq.(63). This expression is an extremely important one with respect to the question of transfer of energy via radiation because it is the one describing the partitioning of energy. It is the equation which should be satisfied in expressing the extent to which the electrons "follow" the nuclear motion and to which the nuclei "follow" the electronic motion. Under reversible conditions these two would be equal. However, there is no assurance that the process is reversible because of the large difference in the masses. As the nuclei move, the electrons can probably also move, but the electronic motion

82 can be so much faster than the nuclear motion that the nuclei cannot keep up with the electrons. Consequently, although the transfer of energy from phonons to electrons is probably accomplished during one period of oscil- lation, that in the reverse direction is not. The process may therefore be an irreversible one, unless continua such as free electrons are available to establish equipartition.

6. 2. Transport of matter

Because of the extreme difficulties of the experimentation, the data obtained by Auskern and Belle, (see Section V.5. ) are not precise, but they do nevertheless characterize reliably the general behaviour and thereby are in accord with the observed thermodynamic properties. A qualitative description can be effected by reference to the statistical model for diffusion developed by RICE [155] who has evaluated the frequency Г, with which an atom jumps a distance Ax in the expression for the diffusion coefficient:

D = |T(AX)2. ' (65)

To evaluate Г Rice describes a situation in which a central atom vibrates about its equilibrium position while the surrounding shell of atoms vibrates as a group. When the two frequencies are in phase the central atom can pass over a saddle point into the available vacant sites (lattice vacancy or interstialcy). Thus

r=Sp..(2j P({6}), (66) in which p^ty is a distribution function describing the probability of the co- existence of an atom at site i and a vacancy at site j,and P({6}) is the frequency with which there occurs a vibrational amplitude sufficiently great for the atom to surmount the saddle point and a simultaneous distortion of the surrounding lattice so that the atom can pass over the col. The product of these functions should be summed over all nearest pairs. The function рЛ2) , however, is approximated by the site fraction of vacancies so that

(2) . Рц = в. (67)

If U0 is the energy which an atom centrally located within a shell of neighbours must acquire to oscillate with a critical amplitude (q0 ), Uj is the energy needed to expand the shell sufficiently, and gkl(2) is a function cor- relating pairs к and 1, P ({6}) can be evaluated in terms of these quantities. The result is

J D = ra 0i/ exp ^ RT J exp (-^j ^ ) (68) k>l in which r is a geometrical parameter such that r = z (Дх)2/2а2 where z is the number of nearest neighbours, a is the lattice parameter, and v is a

83 "weighted mean frequency" given by a normal mode analysis. The function Wki is the free energy associated with the pair correlation function. Thus

Ww = - RT In gwl2> = ДНк1 - TASkl. (69)

Since it has been assumed that the addition of oxygen to the uranium dioxide phase does not change the lattice dynamics (see Section III.2.), one is justified, to the same order of approximation, in assuming that U0, Uj, and W|d are independent of composition. Consequently, the general course of the diffusion coefficient is dictated by the variation of the function в or the probability of finding an atom at one site and a vacancy at an adjacent site, summed over all possible pairs, which is a function of the energies Ev, Evv, E¡ and EH (see Section III.4.1. ). Although numerical values are not available for Nv and N;, at least values at stoichiometry are indicated by the optical properties (see Section V.3.), and the general trend is suggested by the variation of the partial molar enthalpy and entropy, (see Fig. 13). Near stoichiometry the diffusion is controlled by the small number of oxygen ions available for migration. To migrate the ion must come from a lattice site and move into either a vacancy or interstitial position, most probably the latter because there is an extremely large number of them available. Hence the activation energy for diffusion at compositions near stoichiometry is determined essentially by the energy needed to create a vacancy. With increasing excess oxygen the number of ions available for diffusion increases so that the diffusion rate increases. As the number of occupied interstitial sites approaches the total number available, the diffusion coefficient un- doubtedly attains a maximum value and thereafter decreases. The experimental results clearly show an increased diffusion rate as x in U02+x increases, but results at large values of x have not yet been reported, (see Section V. 5. ).

84 VI. PRACTICAL IMPLICATIONS OF THERMODYNAMIC AND TRANSPORT PROPERTIES

The limitations of uranium dioxide as a reactor fuel are now fairly well known as also are the factors reponsible for these limitations. A basic understanding of the properties detërmining these latter factors should enable an assessment to be made of whether the practical limitations might be ameliorated. In the following sections such an assessment is attempted based upon the thermodynamic and transport properties which have been discussed in previous chapters. The limitations of UO2 manifest themselves primarily as difficulties with the cladding material and arise because of various modes of interaction of the fuel with the clad. After discussions of these interactions the relevant thermodynamic and transport properties are examined.

1. INTERACTIONS OF FUEL AND CAN

1.1. Thermal cracking

The can of a UO2 -fuelled element has to act as a structural member as well as a physical barrier between coolant and fuel because of the in- ability of UO2 to accomodate the thermal strain imposed in service which in turn results in cracking. This strain is equal to the product of the linear coefficient of thermal expansion X and the temperature difference across

the fuel (0i~02) and reaches a maximum value at the cool surface in the case of a fuel rod. The strain can be accomodated either elastically or plastically. The maximum linear strain that can be accomodated will be the ratio of ultimate strength (a) to Young's modulus (E); thus the maximum temperature difference which can be accomodated will be this ratio divided by the

thermal expansion coefficient. Using typical values of X, E and a, for U02 at room temperature (0i~02) has a value of ~70°C, which has been confirmed experimentally. These values for a, E and X vary little with temperature up to 800°C. Above 800°C there are few data available on the elastic properties of UO2, but the strength varies little up to 1300°C as does also the coefficient of thermal expansion, so that up to 1300°C when plastic flow manifests itself in stoichimetric UO2, no major increase in the value of (01-02) can be accomodated. At typical gas-cooled reactor fuel ratings, a surface/centre temperature difference would be about 1000°C and this would rise to ~ 1600°C for a more highly rated fuel in a liquid cooled reactor. Thus at least an order of magnitude increase in ст/ЕХ is needed to

approach the possibility of eliminating thermal cracking of U02. Appreciable changes in X and E are not likely to be effected by minor compositional

changes and no improvements in strength of U02 sufficiently great to influence significantly the thermal cracking of UO2 in the elastic range of behaviour can be expected. Accomodation of strain by plastic deformation becomes of practical interest at temperatures above ~ 1300°C in UO2 [153] and above ~850°C in UO(2+x) [156, 157]. The factor of practical interest is how much strain can be accomodated plastically before cracking occurs. A strain of ~ lO"3 has

85 to be accommodated for each 100°C temperature difference. Failure in UO2 seems to have occurred by grain boundary parting in both types of oxide but at a much lower strain in UO2 (~0. 2%) for comparable flow rates than in UO2+X • The mechanism of flow in these two materials is not clearly established. To summarize, under a steady thermal gradient it is unlikely that anything can be done to eliminate cracking in U02 at temperatures below which it behaves elastically. It might be possible to extend the range of plastic behaviour and the extent to which plastic strain could be accommodated by minor compositional changes. Unfortunately, even if une racked material exists under steady state conditions, on reactor shut-down this core will be subjected to quenching stresses, the magnitudes of which cannot be assessed in a general manner. To a certain extent, the core would be insulated by the cracked peripheral steady state case. However, as the temperature is continually decreasing, the material will certainly be unable to support this elastic strain without cracking. The last way in which the problem of thermal cracking could be allevi- ated is by reduction of the strain, i. e. by reducing (0i~02) by increasing thermal conductivity. An order of magnitude increase would be necessary to eliminate thermal stress cracking completely under a steady temperature gradient and is obviously out of the question. Minor increases might however be significant in preventing cracking of a plastic core during stress reversal on cooling. All in all there would seem to be no possibility of making improvements in the resistance to thermal cracking of UO2 other than minor ones which might relieve the can of a small part of its function as a structural member.

1.2. Dimensional changes in UO2 under irradiation

Dimensionally, U02 is a relatively stable material under irradiation, but it does show á definite, increasing macroscopic change of dimensions with increasing burn-up and may thus exert a pressure on the can. Most of the evidence relating to this topic has come from a series of irradiations on plate type specimens carried out by the Westinghouse Atomic Power De- velopment Corp. (WAPD) [158]. Surface temperatures were about 300°C and centre temperatures were calculated to be normally lower than 1000°C. Up to about 12X1020 fissions/cm3 a rate of volume increase of ~0. 2% per 1020 fissions/cm3 is obtained for UO2 of 95°o theoretical density; for greater burn-ups the rate of expansion rapidly increases. The effect of irradiation temperature, specimen shape, and rate of burn-up are as yet ill-defined. Thus cylindrical specimens irradiated at the Atomic Energy Research Es- tablishment, Harwell (AERE), to ~5X1020 fissions/cm3 at centre temperatures of up to 1600°C showed no major dimensional changes [159], however thicker plates irradiated by WAPD to about 6X1020 fissions/cm3 showed quite marked volume increases [158]. Restraint, which is closely associated with specimen shape, might therefore be very important. Such large swellings were accompanied by the presence of large bubbles in theUC>2 [158].

86 This macroscopic instability is not reflected on the sub-microscopic scale by changes in lattice parameter. At a burn-up of — 3. 5 X 10'2i fissions/ cm3 and a centre temperature of ~ 950°C, UO2 is reported to retain its crystallinity with little change in lattice parameter [179]. Experiments using UO2 in both polycrystalline and single crystal .form [160] have shown that at 65°C the increase of lattice parameter reaches a maximum of ~0.1% at~6X10l6 and 5X1017 fissions/cm3 respectively. With increasing dose there is a decrease in lattice parameter to that representing 0. 05% expansion, which value remains unchanged up to the maximum exposure examined (3X1020 fissions/cm3). At 400°C there is no difference in behaviour between single and polycrystalline material, the maximum value of 0. 05% occurring at 4X1017 fissions/cm3, the saturation value being ~0.01%. Post irradiation annealing studies on irradiated UO2 single crystals show that the structural changes have been eliminated after 24 h at ~ 900°C whilst there is still some residual damage ( — 0. 01%) in the polycrystalline material even after 24 h at 1000°C. The inference to be drawn from the disparity in sub-microscopic and macroscopic behaviour is presumably that the solid fission products are rejected completely from the UO2 lattice, as are the gaseous fission products. This being so, there must be an unavoidable expansion, which for the solid fission products will be determined by their state of chemical com- bination, possibly modified by minor compositional changes in the UO2. No practical lead on this latter point is available since comparisons of the macroscopic and sub-microscopic dimensional changes during isothermal irradiation of UO2 and UO2+X have yet to be made. The position with respect to fission gas agglomeration is somewhat different since there is now a body of information associated with swelling of uranium metal [161, 162]. In simple terms it seems desirable to retain the fission gases in the form of bubbles sufficiently small for the internal gas pressure balanced by the (2>/r) force to be high. (Where y is the surface tension and r is the radius of the bubble. ) This problem reduçes to one of controlling the nucleation and initial growth of the bubble, which in the metal case has been shown technologically to be brought about by metallic additions, initially present in the uranium microstructure as compound precipitates. The precise role of these precipitates with respect to nucleation and growth is not yet clearly defined although their practical efficacy is not in doubt [161]. However in drawing analogies it must be borne in mind that fission gas release from uranium metal does not become appreciable until quite large amounts (~10%) of swelling have occurred in marked contrast to the UO2 case. It therefore needs to be substantiated that the growth of inert gas bubbles in UO2 provides an appreciable contribution to the macroscopic growth of UO2 on irradiation. Within the range of linear growth this appears doubtful. Recent work on xenon diffusion in UO2 has been interpreted on the basis of the trapping of this gas at two possible sites, one of which appeared to be closed porosity in the original material and the second "small vacancy clusters" [137] . Evidence has been acquired for the trapping of argon in such clusters in irradiated CaF2 [136]; other evidence on gas release from UO2 has been considered in a previous chapter. Electron microscope studies of irradiated UO2 revealed a very limited number of bubbles at regions which had been at temperatures above 1400°C and below 1800°C

87 where columnar growth commences [162]. Transmission electron microscope studies of irradiated and annealed material are still at too early a stage to be definitive, but nothing comparable to the uranium case has yet appeared. Whether this absence of macroscopic growth is due to a naturally more homogenous bubble nucleation in UO2 (because of either the inherent properties of the UO2 in this temperature range or because of the lack of plasticity of the UO2), is a matter for speculation. If the latter is true then further speculation must revolve around possible differences in the behaviour of gas agglomerates at grain boundaries and in the body of the crystal. At higher temperatures, within the grain growth ( 1600-1800°C) and columnar growth (> 1800°C) ranges, gas release is so great that it seems unlikely that bubble growth as previously conceived can contribute appreciably to macroscopic swelling. At burn-ups where gross swelling occurs this is associated with the presence of large bubbles in the UO2 and also high gas release [158]. Be- haviour in this region is as yet insufficiently well-defined to permit discussion although "irradiation induced plasticity" or some such equivalent phrase is often encountered in reports dealing with this subject. To summarize it seems possible to define some areas in which a basic understanding of the properties or processes involved might enable some of the present limitations of UO2 as a fuel to be eased. These areas are thermal conductivity, phase equilibria and material transport processes.

2. THERMAL CONDUCTIVITY

An increase of thermal conductivity, should, for a given fuel rating and geometry reduce the centre temperature and hence ease practical problems associated with fission gas release and possibly thermal cracking. The results quoted for single crystals [95] translated into practical terms mean that the rod-power (Jkd0) for the temperature interval 500 to 1600°C is 50% greater than for polycrystalline material. Compared with sintered material this is a very desirable improvement, of considerable practical significance particularly if polycrystalline material having similar properties can be developed. The basic understanding of the process of thermal conduction in иОг at elevated temperatures is not sufficiently advanced to enable predictions of practically possible improvements tobe made. Having stated this, the probable effects of radiation induced damage on the processes which contribute to thermal conduction need to be assessed. The easiest way of settling speculation on this point involves direct in-pile measurements of thermal conductivity of single and polycrystalline U02-x- The results of such experiments may also throw further light on the conduction processes in unirradiated material.

3. PHASE EQUILIBRIA

The importance of minor deviations from stoichiometry, in the direction of oxygen excess or deficiency, upon the pratical performance of UO2 as a fuel has been made clear in the preceeding sections. Interest at present

88 centres on hypostoichiometric material largely because of the enhanced thermal conductivity which it is reported to have [66, 95], but also because of its possible presence in high temperature regions of irradiated material [163]. The general deficiencies in knowledge of phase equilibria in the region UO2-UO2-X are such as to render the interpretation of experimental observations very doubtful. Minor deviations from stoichiometry in the direction of oxygen excess may give rise to quite marked differences in fission gas release. Thus non-destructive methods for the close characterization of the 0:U ratio in the range 2-х to 2.01, prior to irradiation, are needed to render irradiation experiments more meaningful. A detailed knowledge of phase equilibria is also essential for adequate control of the preparative and fabrication processes leading to the production of a closely characterized UO2 body. These aspects are discussed in the next section.

4. MATERIAL TRANSPORT PROCESSES

The importance, for a variety of reasons, of obtaining a high density low surface area sintered UO2 was realized at an early stage of its development. Technological answers were arrived at well in advance of a basic understanding of the sintering process. An understanding of the transport processes likely to be involved is, at this point in time, only of interest in- asmuch as they may throw light, in a general manner, on the sintering process. An understanding of material transport under the influence of stress at elevated temperature does still have some immediate interest in the con- text of strain relaxation and bubble growth. Sintering theory for oxides is not on a particularly firm footing at present. The case for material transport during sintering by a bulk diffusion mechanism has been strongly urged and widely accepted over the past decade. Models for sintering based on diffusion processes have been subject to considerable study, and theoretical development has reached a stage where quantitative comparison of calculated and observed sintering behaviour has become possible. As the volume diffusion coefficients for anion and cation generally differ it has been normal to assume that the movement of the ion having the lower diffusion coefficient would be rate controlling. This assumption has led to serious discrepancies between theory and experiment, with sintering being more rapid than would be predicted from measured diffusion coefficients [164]. Thus the apparent diffusion coefficients calculated from data on the later stages of alumina sintering are five orders of magnitude greater than the measured oxygen diffusion coefficient in single crystal alumina [165]. This discrepancy can be reduced by two orders of magnitude if the diffusion coefficient typical of polycrystalline material is used. A rationalization of the situation has been attempted based on the suggestion that during sintering movement of the slower volume diffusiong ion may not be rate-controlling as this ion may choose to move more rapidly by a grain boundary process [164]. As BURKE and COBLE [164] state, "the final explanation of this puzzling discrepancy in apparent diffusion coefficients is one of the most important problems in sintering theory today". This statement refers to the behaviour of relatively uncomplicated oxides. The pre-

89 vious sections have demonstrated that "uranium dioxide" is a very complex material. Oxygen in excess of stoichiometry will invariably be present in powders before sintering: it will be non-uniformly distributed at room - temperature and this distribution will vary in a complex manner during the sintering cycle as a function of temperature and atmosphere. Thus the surface area of a "UO2" powder has been found to vary closely with change and direction of change of its oxygen content [167]. By varying temperature and oxygen pressure independently, it was found that there was a pronounced decrease of surface area during reduction but a much smaller change during oxidation. The structures and properties of the crystallite agglomerates will vary according to the method of chemical preparation and are difficult to characterize [166]; variations in these properties can result in the development of optimum compacting pressures which will differ from preparation to preparation [167, 168, 169]. Because of inadequacy of characterization of starting material with respect to all these factors, comparison of the results of different investigators is virtually impossible. However, in studies of the sintering of "UO2" an effect has béen observed which may ultimately throw further light on the sintering process. De- parture from stoichiometry has a beneficial effect on sintering behaviour over the composition range UO2-UO2.02; increase of oxygen content beyond UO2.03 has no further beneficial effect [170, 171]. Both enhancement and saturation effects have to be explained. It must however be remebered that the effect of microagglomerate state upon this optimum 0:U ratio has not been examined. Much more needs to be known about self-diffusion processes in иОг-ЩОд before discussions of these phenomena can be raised above the level of pure speculation. With respect to the creep of polycrvstalline oxides at elevated temperatures the situation again is far from satisfactory. Polycrystalline alumina [172, 173] and beryllia [174] are found to deform in a way which is in accord with the Herring-Nabarro model for diffusional creep. However, the diffusion coefficients calculated from experimental creep data are comparable to the measured values for the cation which diffuses in these cases more rapidly than the anion by several orders of magnitude [175, 176]. The situation is virtually identical with that for sintering theory and the same solution of the difficulty has been proposed. No serious attempt has been made to interpret the sintering behaviour of U02 and UO2+X in terms of the Coble models due largely to the known complexity of the processes. Attempts have been made to interpret the creep behaviour of UO2 and UO2+X on the basis of the diffusion model [156]. The activation energy for the deformation of UO2 is ~91 kcal/mole [153, 156] which is to be compared with ~88 kcal/mole [9] for uranium self-diffusion and ~ 65 kcal/mole for oxygen [177]. This suggests that the rate-controlling ion is uranium but the diffusion coefficients calculated from experimental data using the Herring-Nabarro model are greater than the measured values by about three orders of magnitude [156]. This situation is analogous to that already described for alumina and it is tempting to take refuge in the hypothesis that uranium might in fact be diffusing via grain boundaries. How- ever the recent experiments of ALCOCK and McNAMARA [178] suggest that the uranium diffusion coefficient used in these comparisons was a grain

90 boundary value, and that the volume diffusion coefficient could be an order of magnitude lower. An alternative explanation to diffusional creep is obviously necessary and a little qualitative information in favour of grain boundary sliding has been obtained. Failure occurs in UO2 under flexural creep by grain boundary separation after about 0. 2% maximum fibre strain. Non-stoichiometric oxide deforms much more readily and at lower temperatures than the stoichiometric material with an activation energy of ~ 65 kcal/mole [156, 157] which is to be compared with ~ 30 kcal/mole for oxygen diffusion [131]. No well substantiated value for uranium self diffusion in UO2+X exists at present although an activation energy of ~52 kcal has been reported for U02 heated in argon and hence probably non-stoichiometric [132]. Grain boundary sliding definitely occurred in UO2 + X and maximum fibre strains of up to 2% in bend have been observed prior to grain boundary failure. If swelling due to gas pressure occurs by a Herring-Nabarro mechanism, on the evidence presented, the behaviour of a bubble within a grain could be orders of magnitude different from that at a grain boundary and markedly influenced by specific impurities. The importance to an understanding of these creep processes of establishing a comprehensive qualitative picture of cation and anion diffusion in both UO2 and U02+x in single and polycrystalline form, is apparent. Material transport via a vapour phase appears to be of practical importance only in the columnar growth process. Adequate agreement exists between experiment and calculation using U02 vaporization data [150, 151]. However the overwhelming technological importance of this process with respect to fission gas release has been commented upon already. Suppres- sion of columnar growth would lead to substantial reduction in fission gas release at centre temperatures above 1800°C, so that a very clear and detailed understanding of the basic mechanisms involved appears desirable.

VII. CONCLUSIONS

It will appear from the detailed discussion of the evidence in the body of the Report that a great deal remains to be done before the various properties can be correlated into a single, coherent model of the nature of the non- stoichiometric uranium dioxide phase. However, an encouraging start has been made. It is not desirable to attempt to summarize the conclusions of the preceeding sections here, since they would be misleading unless quali- fied by a detailed discussion of much of the experimental evidence, and this has been given already. Instead, we shall list again the immediate questions which seem to demand further attention and which were emphasized during our discussions. These topics were the following: (i) There is a need for X-ray or neutron diffraction work at high temperatures (> 1000°C) particularly in the region 2. 15< 0/U< 2. 25, in order to resolve the questions raised by X-ray work on quenched samples in this region. (ii) Some further work on structural topics is desirable, in particular: (a) neutron diffraction data on compositions for which 2.15<0/U<2.25, to refine the values for site occupancy and hence the model for

U02+x; (b) the complete structure for U4O9; (c) neutron diffraction studies of the tetragonal phases; and (d) attempts to decide the status of the UO2.25 phase which does not show the superstructure lines characteristic of well-crystallized

U4O9. (iii) The hypostoichiometric UO2-X region is not well characterized. It is necessary to define: (a) preparative conditions, (b) the variation of x with temperature and oxygen pressure, (c) the phase limits, and (d) structural and physical properties of well-characterized samples. The frfee energy of formation of gaseous UO is not well established. (iv) More accurate values for the partial molar enthalpy and entropy of oxygen in UO2« are required for a rigorous test of theoretical models. It is unlikely that equilibrium measurements at high temperatures can be made with sufficient accuracy. Enthalphy measurements in an adiabatic calorimeter at high temperatures might be preferable. It is particularly necessary to measure AS values close to the stoichiometric composition by a new technique. Measurements of heat capacities in the UO2+X single-phase region should be made.

(v) Magnetic susceptibility measurements on U4O9 and U3O7 show anomalies at 6°K, which are not apparent in specific heat measurements. More magnetic or thermal studies are required to resolve this. (vi) More experimental data on thermal conductivity at high temperatures is required; single crystal and polycrystalline material should be studied in the stoichiometric and hypostoichiometric states. The experiments should be conducted in-pile as well as out-of-pile and the effect of pro- longed irradiation better established. (vii) Measurements of the electrical conductivity of both single crystal and polycrystalline specimens should be made at temperatures up to 1200°C

93 at least; the compositions should be carefully controlled and extend to the hypostoichiometric range if possible, in view of possible important contributions to the thermal conductivity. (viii) There is much scope for work on optical and infra-red absorption as a function of temperature and composition. These measurements are also relevant to the theory of thermal conductivity. (ix) Self-diffusion coefficients of U and О in UO2+X need to be better established, especially at higher values of x. More work is required on the mechanism of foreign gas diffusion in UO2 and иОг+х- (x) The effects of thermal gradients both on mass and heat transfer were discussed only briefly from a theoretical stand-point. This subject does not seem to have received the attention it deserves in view of its great technological importance. Finally, we stress that this Panel was not constituted to discuss irradiation effects in any detail, though the major variations in properties due to irradiation effects have been noted in the foregoing sections. Our discussions do serve to draw attention to the potentially variable nature of the unirradiated material and to the need for the very complete characterization of samples before irradiation.

94 APPENDIX

MATHEMATICAL TREATMENT OF DEFECT ABSORPTION

The complex polarizability, at energy E (or frequency v, where E=hi/), of an oscillator of mass m whose resonance energy is Ej and which, because of damping, has a natural line width, y¡, is

e2h2 m E?-E2 + i7jE where fj is the transition probability or oscillator strength. The following development from Eq. (1) of an expression which will relate the numbers of absorbing centres to observed optical density is an extension of arguments made by DEXTER [119]. In a solid, because of the thermal motion of the atoms of the crystal, the absorption by a given type of centre will not be sharp; there will be a distribution of values of Ej. If, for example this normalized distribution is given by some function of energy, Gj(E), then an oscillator which gives rise to a single absorption band in the solid will have a polarizability

G E )dE ttj(E)= —f ] J f 2 ^2 ' ' . (2) m J E'2 -E2 + Í7j E

If, on the other hand, this oscillator has several levels which give rise to more than one absorption band, its polarizability will be

a (E) 2 2 (3) i -^TZAe fi V . /Г E'' 2G* (E')dE-E« ' +. íy'E ' J о J i.e. there will be a distribution of energy, due to the thermal motion, for each band. If, in addition, oscillators of more than one type are present, there being N¡ of type j, the polarizability of the system becomes

<*(E)=^Nj0j(E). (4) i For the present application there are two types of absorbing centres to be considered, those due to randomly distributed vacancies in the regular cubic oxygen lattice and those due to randomly distributed interstitial oxygen ions. For this case, then, Eq. (4) becomes

a(E) = Nvav(E) + Niai(E). (5)

Many of the arguments and assumptions to be made in the following development are those of DEXTER [119]. Moreover, many of these plus the ones added for the present application are not justified a priori; the principal justification is the consistency of the result with other known or sur- mised facts. Hence they will merely be stated.

95 It is assumed that the absorbing centres are well localized and that the Lorentz local field correction is applied. Then the relation between the complex index of refraction, (n-iK), and the complex polarizability, as given by Dexter, is

j where n is the index of refraction and К is the extinction coefficient at energy E. Equation (6) is as given by Dexter, who associates no with the index of refraction of a pure, ideal, transparent host crystal. His e is inserted to correct the polarizability of the host for the possible necessary removal of its atoms to allow (substitional) impurity atoms to be present. For his application, where the values of Nj can be treated as small compared to the, numbers of atoms in the host, he can also assume n to be close to no. In the present case no e is to be used since no direct comparison with an unadulterated crystal is to be made and, indeed, cannot be made. Instead of the n0, the equation as used here would contain ne, the very high frequency index of refraction of the actual crystal. If one chooses to work with the imaginary parts of Eq. (6),. the result is

2 2 6nK _4T e -h V V . Г TfEGf(EMdE' (n2 + К2)2 + 4(n2 - K2) + 4 ~ 3 m L ¡L j J -2 - E2 )2 + ( .{Е)2 ' ico (E ТJ where n and К on the left are, of course, functions of the energy, E. Here it is argued, with Dexter, that the natural widths, 7J, are many orders of magnitude less than the actual band widths so that each integration can be performed by setting

TjE , (E|2 — E2)2 +(T¿Ef 2F6(E'"E)'

б being the Dirac deltá function. Further, the present analysis has been made by neglecting K2 compared to n2. When this simplification is made, the integrations are performed and the concentrations of the absorbing centres are written as fractions of the uranium atoms present, Nu , so that Eq. (7) becomes

2 2 6n(E)K(E) e h NTIV /N¡ NVflpt . .„.

If only two distributions are assigned (see text), it is implied that one can write

.^fjGjtE)- G,IE). (9) 1

96 An analysis has been made on this basis by using the same Gaussian distribution as proposed by Dexter:

Gj(E) = (•JwWj)'1 exp [-(E - Ej)2/Wj2], (10) which is normalized to the degree that the width measure, Wj , is small compared to Ej. That is

. /E-Ej Г « i M—= . dt 17 J e W. J Г 0 Ej /2 —1L Wi and the integral on the right is only 10'4 for Ej/Wj as small as 2. 57. Some further consideration to details of substitution (9) must now be made. If G®(e) is Gaussian, of the form of Eq.(10), it can be shown that the integration corresponding to normalization gives the result Ef? , and i ' that the mean energy is Si-'Í^'W. i where 'l-I'i- t Thus Ej would be associated with Ej of the text, and the latter cannot be directly associated with any level spacing. The width measure of one of these composite peaks can be found from the average of 2(E-Ë)2 to be

W2 ^f/tW» )2 + 2^Tf|(Ej - E)2

If it is assumed that every W?, the width measure for a single transition, has the same value, than

W2= fjtWf)2 + 2^f'(E® -E)2. t If is estimated by using results for an F centre in an alkalihalide, it is found to be ~0. 3 eV at room temperature. If, in this case, it is assumed that all values of E?-E are small compared to the widths and fy= 2, f¡ = 6 are used, one finds the common W® to be 0.43 eV and the common W¡ to be 0.45 eV.

REFERENCES

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102 LIST OF PARTICIPANTS

The members attending the Panel Meeting, held on 16 - 20 March 1964 at the Headquarters of the International Atomic Energy Agency in Vienna to assess the thermodynamic and transport properties of the uranium dioxide and related phases, were:

ROBERTS, Dr, L.E.J. Chemistry Division (Chairman) Atomic Energy Research Establishment Harwell, Didcot, Berks. England

BELBEOCH, Dr. Bella, Commissariat à l'energie atomique C.E.N, de Saclay B.P. No. 2, Gif-sur-Yvette Seine et Oise, France CORDFUNKE, Dr. E.H.P. Reactor Centrum Nederland Petten (N.H.) Netherlands FELIX, Dr. F. Hahn-Meitner-Institut für Kernforschung Glienickerstrasse 100 Berlin 39 Federal Republic of Germany HAGEMARK, Dr. K. Institute of Silicate Science Technical University of Norway Trondheim, Norway JAKES, Dr. and Mrs. D. Nuclear Research Institute of the Czechoslovak Academy of Sciences Reí u Prahy Czechoslovak Socialist Republic KUBASCHEWSKI, Dr. O. National Physical Laboratory Teddington, Middlesex United Kingdom KOLAR, Dr. D. Nuclear Institute "J. Stefan" Jam ova 39 Ljubljana, Yugoslavia LAGERWALL, Dr. T. Hahn-Meitner-Institut für Kernforschung Glienickerstrasse 100 Berlin 39 Federal Republic of Germany

103 NAGELS, Dr. P. Studiencentrum voor Kernenergie Laboratorie van het S. C.K. te Mol-Donk Belgium

THORN, Dr. R.J. Argonne National Laboratory 9700 South Cass Avenue Argonne, 111. United States of America WAGENER, Dr. K. Hahn-Meitner-Institut für Kernforschung Glienickerstrasse 100 Berlin 39 Federal Republic of Germany WESTRUM, Dr. E.F. Department of Chemistry University of Michigan Ann Arbor, Mich. , United States of America WILLIAMS, Dr. J. Metallurgy Division Atomic Energy Research Establishment Harwell, Didcot, Berks. United Kingdom WILLIS, Dr. B.T.M. Metallurgy Division Atomic Energy Research Establishment Harwell, Didcot, Berks. United Kingdom WINSLOW, Dr. G.H. Argonne National Laboratory 9700 South Cass Avenue Argonne, 111. United States of America

SECRETARIAT

Scientific Secretaries: HARA, Dr. R. \ Division of Research HOLLEY, Dr. C.E.J and Laboratories, IAEA

104 REPORTS SUBMITTED TO THE PANEL

The following reports were submitted to the Panel and the assessment contained in this publication is based on these reports and discussions held among the Panel members:

[1] BELBEOCH, B. U02-like structures [2] CORDFUNKE, E.H.P. The electrical properties of uranium dioxide [3] HAGEMARK, К. The partial free enthalpy, enthalpy

and entropy of oxygen in the U02+x phase at higher temperatures [4] JAKES, D. About the possibility of appreciat- ing the influence of non-stoichiometric oxygen on structure, sinter- ability and some other properties

of U02 [5] KUBASCHEWSKI, О. Physicochemical properties of uranium dioxide [6] LAGERWALL, T. Argon diffusion in calcium fluoride as a model process for fission gas transport in uranium dioxide [7] NAGELS, P. Preparation and physical properties

of U02 single crystals [8] ROBERTS, L.E.J. Thermodynamic, chemical and magnetic properties of the non- stoichiometric uranium dioxide phase [9] THRON, R.J. and Some aspects of the thermo- WINSLOW, G.N, dynamic and transport properties

of the U02 phase [10] WILLIAMS, J. Notes on the practical significance of certain thermodynamic and transport properties of uranium dioxide [11] WILLIS, B.T.M. Neutron and X-ray diffraction

studies of U02, U02+x and U4O9 [12] MÖLLER, P. Thermogravimetric tests of urani- WAGENER, К, and um dioxide and interpretation of ZIMEN, K.E. the non- ideal rare gas release on annealing of irradiated samples

105 IAEA SALES AGENTS

Orders for Agency publications can be placed with your bookseller or any of our sales agents listed below:

AUSTRALIA ITALY Hunter Publications, Agenzia Editoriale Internazionale 23 McKillop Street Organizzazioni Universali (A.E.I.O.U.) Melbourne, C.l Via Meravigli 16 Milan AUSTRIA Georg Fromme & Co. JAPAN Spengergasse 39 Maruzen Company Ltd. Vienna V 6, Tori Nichome BELGIUM Nihonbashi (P.O. Box 605) Office international de Übmirie Tokyo Central 30, avenue Mnrnix Brussels 5 MEXICO BRAZIL Libraría Internacional Livraria Kosmos Editora Av. Sonora 206 Rua do Rosario, 135-137 Mexico 11, D.F. Rio de Janeiro NETHERLANDS Agencia Expoente Oscar M. Silva N.V. Martinus Nijhoff Rua Xavier de Toledo, 140- Io Andar (Caixa Postal No. 5.614) Lange Voorhout 9 Sfio Paulo The Hague BYELORUSSIAN SOVIET SOCIALIST REPUBLIC NEW ZEALAND See under USSR Whitcombe & Tombs, Ltd. CANADA G.P.O. Box 1894 The Queen's Printer Wellington, C.l Ottawa, Ontario NORWAY CHINA (Taiwan) Johan Grundt Tanum Books and Scientific Supplies Service, Ltd., Karl Johans gate 43 P.O. Box 83 Oslo Taipei PAKISTAN DENMARK Karachi Education Society Ejnar Munksgaard Ltd. Haroon Chambers 6 Norregade South Napier Road Copenhagen К (P.O. Box No. 4866) Karachi, 2 FINLAND 4» Akateeminen Kirjakauppa POLAND Keskuskatu 2 Osrodek Rozpowszechniana Helsinki Wydawnictw Naukowych FRANCE Polska Akademia Nauk Office international de Pafac Kultury i Nauki documentation et librairie Warsaw 48, rue Gay-Lussac e SOUTH AFRICA Paris 5 Van Schaik's Bookstore (Pty) Ltd. GERMANY, Federal Republic of Libri Building R. Oldenbourg Church Street Rosenheimer Strasse 145 (P.O. Box 724) 8 Munich 8 Pretoria

ISRAEL SPAIN Heiliger and Co. Librería Bosch 3 Nathan Strauss Street Ronda de la Universidad 11 Jerusalem Barcelona SWEDEN UNITED KINGDOM OF GREAT BRITAIN C.E. Fritzes Kungl. Hovbokhandel AND NORTHERN IRELAND Fredsgatan 2 Her Majesty's Stationery Office Stockholm 16 P.O. Box 569 London, S.E.I SWITZERLAND Librairie Payot UNITED STATES OF AMERICA Rue Grenus 6 National Agency for 1211 Geneva 11 International Publications, Inc. 317 East 34th Street TURKEY New York, N.Y. 10016 Librairie Hachette 469, Istiklâl Caddesi VENEZUELA Sr. Braulio Gabriel Chacaree Beyoglu, Istanbul Gobernador a Candilito 37 Santa Rosalia UKRAINIAN SOVIET SOCIALIST REPUBLIC (Apartado Postal 8092) See under USSR Caracas D.F. UNION OF SOVIET SOCIALIST REPUBLICS YUGOSLAVIA Mezhdunarodnaya Kniga Jugoslovenska Knjiga Smolenskaya-Sennaya 32-34 Terazije 27 Moscow G-200 Belgrade

IAEA publications can also be purchased retail at the United Nations Bookshop at United Nations Headquarters, New York, at the news-stand at the Agency's Headquarters, Vienna, and at most conferences, symposia and seminars organized by the Agency. In order to facilitate the distribution of its publications, the Agency is prepared to accept payment in UNESCO coupons or in local currencies. Orders and inquiries from countries where sales agents have not yet been appointed may be sent to :

Distribution and Sales Unit, International Atomic Energy Agency, Kärntner Ring 11, Vienna I, Austria INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1965 PftlCE: USA and Canada: US $2.50 Austria and elsewhere: S 52,50 (15/-stg; F.Fr. 10,-; DM 8,-)