The fuel density, p is an important property of the fuel and is a function of the following factors: fuel composition, temperature, amount of porosity, O/M ratio, and burnup. The theoretical densities of the materials (pT) can be calculated from the knowledge of lattice type and values of lattice parameter. Assuming that the elements form a solid solution, the theoretical density of the material can be calculated from the following relation [12]:

Pt = MsystemN/ (VNa), (1)

where, Msystem is the atomic weight of the system, N the number of atoms per unit cell, V the volume of the lattice, and Na the Avogadro constant. Thus, in the case of
a (Th1-xPux)O2 solid solution, which is a fcc fluorite-type structure, with a lattice parameter a, the theoretical density can be estimated as :

pT — 4 [(1 — x) Mxh + xMpu + 2Ma/a3Na. (2)

The density of a material as a function of temperature can be calculated using linear thermal expansion data obtained from a pushrod dilatometer, which mea­sures thermal elongation of a material with respect to temperature (T). The relation between linear thermal expansion and density is expressed [40] as p0/p = (L/L0)3, L/L0 = (1 ? AL/L0) (see Eq. (A.4) in Appendix 1 for details), so that one writes the fractional change in density as

where Ap = p-p0 is the difference between densities at temperatures T and T0.

In fluorite-type solid solutions, when the lattice parameter, a, and the molecular weight, M, are known, the theoretical density (Mg/m3) can be calculated using the relation [12]:

pT — 4M/(NAa3) . (4)

For pure ThO2, the volume of the unit cell is (0.55975 nm)3 which is equal to 1.75381 x 10_22 cm3. Density of pure ThO2 at 298 K may, therefore, be calcu­lated as 10.00 g/cm3 [12]. The theoretical density of the ThO2-UO2 solid solution can be calculated from the following equation that makes use of Eq. (4) consid­ering the additive rule for the molecular weights and cell volumes [39]:

p(Th1-yUyO2) — 4[M2 + y(M1 — M2)/[NA(a3 + y(a3 — a|)], (5)

where, M1 and M2 are the molecular weights of UO2 and ThO2, respectively, y is the molar fraction of UO2 and a1 and a2 are the lattice parameters of UO2 and ThO2, respectively. The theoretical density of the ThO2-UO2 solid solution as a function of UO2 content at 298 K is shown in Fig. 4.

The recommended equations for the density of solid uranium dioxide are based on the lattice parameter value of 0.54704 nm reported by Gronvold [46] at 293 K and thermal expansion data by Martin [47]. The above lattice parameter values are in good agreement with measurements by Hutchings [49] and are in full agreement with the recommendations of Harding et al. [48]. Assuming the molecular weight of UO2 is 270.0277, this lattice parameter gives a density at 293 K as 10.956 g/cm3. Applying the thermal expansion values of Martin [47], the density at 273 K is 10.963 ± 0.070 g/cm3. The values reported by Benedict et al. [51] and MATPRO for solid UO2 are 10.970 ± 0.070 and 10.980 ± 0.020 g/cm3, respectively. Densities of UO2 and PuO2 given by various authors are shown in Table 4.

Fig. 4 Theoretical density of the ThO2-UO2 solid solution as a function of UO2 content at 298 K [52]. (permission from Elsevier)

L(273) (9.9734 • 10-1 + 9.802 • 10-6 • T — 2.705 • 10-10 • T2 + 4.391 • 10-13 . t3)

(7)

For 923 K < T < 3,120 K,

L(T) = L(273)(9.9672 • 10-1 + 1.179 • 10-5 • T — 2.429 • 10-9 x T2 + 1.219

• 10-12 • T3). (8)

The density of solid stoichiometric UO2 or mixed oxide (MOX) with a com­position of UO2-4 % PuO2 as a function of temperature for the temperature range of 273-923 K is given by ORNL as [6]:

q(T) = q(273) (9.9734 • 10-1 + 9.802 • 10-6 • T — 2.705 • 10-10 • T2 + 4.391 • 10-13 • T3)-3

(9)

and the density of UO2 or MOX for the temperature range of 923 K to the melting temperature,

q(T) = p(273) (9.9672 x 10-1 + 1.179 x 10-5 • T — 2.429 x 10-9 • T2 + 1.219 x 10-12 • T3)-3

(10)

The recommended uncertainty in the density value is 1 % in the entire tem­perature range.

Martin [47] recommends from assessment of the available data on hyperstoi­chiometric uranium dioxide (UO2+x), using the same equations for the linear thermal expansion of UO2 and of UO2+x for x in the ranges 0-0.13 and 0.23-0.25. Therefore, Eqs. (9) and (10) are recommended for the density of UO2+x for x in the ranges 0-0.13 and 0.23-0.25. No data on the effect of burnup on density or thermal expansion of UO2 are currently available. In the absence of data, Eqs. (9) and (10) are recommended for UO2 during irradiation, in accord with the recommendation of Harding et al. [48]. The density of UO2 as a function of temperature is shown in Fig. 5.

As mentioned earlier, the phase diagram shows a continuous series of solid solutions between UO2 and ThO2. This is supported by the fact that deviations from Vegard’s law are within the uncertainties of the lattice parameter measure­ments. Also, densities of ThO2-UO2 solid solution at room temperature can be calculated using additive rule:

q(273) = 9.99003 + 0.00953 • y (in g/cm3), (11)

where y is mol% UO2.

The calculated theoretical densities, obtained from the measured lattice con­stants, for some ThO2-UO2 solid solution are given below in Table 5.

Fig. 5 Density of UO2 [6, 31] and ThO2 [40] as a function of temperature

Table 5 Room temperature lattice constants and theoretical densities of ThO2-UO2 solid solutions Composition (mol% UO2) Lattice parameter (nm ± 0.00003 nm) Theoretical density (g/cm3) 0.0 0.55975 10.00 10.1 0.55846 10.09 20.2 0.55726 10.18 30.1 0.55590 10.28 40.3 0.55475 10.37 50.1 0.55355 10.46 60.1 0.55225 10.55 69.9 0.55098 10.65 80.1 0.54969 10.75 90.0 0.54841 10.85 100 0.54705 10.96

Belle and Berman [12] calculated theoretical density of ThO2-UO2 solid solution for different UO2 content (x) from the lattice parameter data of Cohen and Berman [52]. The following equation shows the relationship between the theo­retical density and UO2 contents.

p(T) (g/cm3) = 9.9981 + 0.0094 • (x) — 8.7463 • 10-6(x)2+1.1192 • 10-7 • (x)3,

(12)

where, x is the UO2 content.

Density of (Th, U)O2 system has also been calculated as a function of tem­perature by many authors [12, 40, 53]. Momin and Venketeswarulu [54], Momin and Karkhanwala [55], Kempter and Elliott [56] and Springer et al. [57] reported the densities from the lattice and bulk expansion data. The density of (Th, U)O2 system as a function of temperature and UO2 content has been estimated from a
linear relationship of lattice parameters of (Th, U)O2 as a function of UO2 [12, 38, 57] at 298 K.

a298(nm) = 0.55972 — 1.27819 x 10-4[%UO2] (13)

A relationship for the average coefficient of linear thermal expansion in the temperature range 298-1,600 K as a function of UO2 content was obtained from the literature using the high temperature lattice parameter measurements [12, 58-61]. Theoretical density was calculated as a function of UO2 content using Eq. (12). Subsequently, the theoretical density was derived as a function of tem­perature and UO2 content from the basic mass balance equation, i. e.,

Pt. vt = Po • Vo, (14)

where, pT, p0, VT, and V0 are the densities and volumes at temperatures T and T0, respectively.

With the coefficient of thermal expansion, the following equation was derived for the theoretical density [40]:

p(T) (g/cm3) = 10.087 — 2.891.10-4 x T — 6.354.10-7(x) x T + 9.279.10-3(x) + 5.111.10-6 x (x)2,

where x is UO2 content. It is observed that the variation in density obtained from Eq. (15) and that from the literature is within ±0.28 %. The theoretical densities of UO2 and ThO2 at different temperatures are given in Table 6.