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The electronic transport properties of UO2
38
Journal of Nuclear Materials 161 (1989) 38-43 North-Holland, Amsterdam
THE ELECIRONIC
TRANSPORT
PROPERTIES
OF UO,
P.W. WINTER Safety and Reliability United Kingdom
Directorate,
UK Atomic Energv Authority,
Wigshaw Lone, Culcheth, Warrington
WA3 4NE,
Received 11 May 1988; accepted 22 September 1988
The electrical conductivity, Seebeck coefficient and electronic thermal conductivity of UO, f x are modelled using the small polarons generated by the reaction 2U4+ + U 3+ + U 5+. The reaction parameters which are found to best fit the data are a band gap of 2 eV, a vibrational entropy of 2 k and an electron-to-hole mobility ratio of unity. The electronic thermal conductivity is found to decrease sharply for large deviations from stoichiometry at high temperatures.
1. Introduction The intrinsic dc electrical conductivity o(T) of UO, has been attributed [1,2] to the thermally induced disproportionation
wherein the upper and lower Hubbard ‘bands’ reduce to two energies E, and E, separated by the Mott-Hubbard gap U equal to the internal energy required to effect the reaction (1). Thus the free energy F controlling reaction (1) may be written as
2u4+
F= U-
-+ u3+
+ u5+,
(1)
wherein the carriers (the electron on U3+ and the hole on U5+) are treated as small polarons. It has been suggested that this reaction is also largely responsible for the anomalous rise in specific heat C,(T) [3,4]. Here, a general treatment is forwarded which provides for consistent and explicit modelling of the electrical conductivity a, the Seebeck coefficient a and the electronic thermal conductivity K,,(T) of UOz+X. The model, based on reaction (l), rationalises all the available data. An earlier attempt at a unified treatment along these lines [5] achieved a measure of success but did not explicitly model the electrical conductivity and neglected the stoichiometric deviation x in the interpretation of the data for UO,. The comprehensive study of the influence of x carried out in this work affords valuable insight into the behaviour of the electronic transport processes.
It has been pointed out [S] that UO, is a Mott-Hubbard insulator rather than a classical semiconductor. The electronic energy spectrum can be described in terms of the ‘atomic limit’ of the Hubbard model
(North-Holland
(2)
where U = E,, - E, and S = S, + S,, with E, = E, - E,, E,= -(Es-E,), S,=S,-S, and S,=S,-S,. If n, p denote the molar concentrations of electrons and holes respectively we have, at equilibrium, (electroneutrality),
p = n + 2x
*P
e-BF
(1 -VP)*
(3)
(mass action),
(4)
=
where, as usual, /3= (kT)-‘. Setting S = edBF12 we obtain, p=x+
~(-262+(x2(1-4s*)+8*)1’2). l-482
(5) Thus, for x >> S, p-2x
and
n-S*(1-2~)~/2x,
(6)
for x K S,
2. The ekctronic model
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P -x+
8
-1+26
and
n-
-x+-
s 1+2S’
(7)
The following reciprocity relation is evident p(-x)
B.V.
=n(x).
(8)
39
P. W. Winter / The electronic transport properties of UO,
3. The dc electrical conductivity The electrical
conductivity
of UO,,,
may be written
as o=oi,+u,+a,,
(11)
and on, crP where uion is the ionic partial conductivity are the partial conductivities associated with electron and hole transport respectively. The electrical conductivity a,, of a carrier a of charge q is related to the diffusion coefficient D of a by the Nemst-Einstein relation a, = n,q’D/kT,
(12)
where nv is the carrier concentration The Einstein formula for random
per unit volume. diffusion gives
D = X2Pjw/6, 10*
10-3 STOICHIOMETRIC
DEVIATION
(13)
lo-'
10-Z x
Fig. 1. Defect diagram for UO,,, at T = 1500 K (solid curves) and T = 3000 K (dashed curves).
where X = =,,/fi ( a0 the lattice constant) is the jump distance, Pj the mean number of sites available for a jump and w the jump frequency. In the present context, for the electrons, Pj = 12(1 - n -p) and w = ve-ss. Thus n(1 -n-p)
In addition, if defects included we have, Oi-Vv,+X
on the oxygen
lattice
emBE/T
are with a similar relation
Co=!+
and
for up. With v - 1013 s-l,
Wm-‘K-l.
(15)
O,V, = 2 e-BFo,
where Oi is the oxygen interstitial concentration, V, the oxygen vacancy concentration and F, the frenkel energy (- 3 eV [6]). Thus,
Now from the data stoichiometric oxide, eP = 3.8 X lo6
x
of Aronson
2x(1 -2x)
e-0.3/kT/T
non-
(0 cm)-‘. (16)
We thus make the identifications, holes,
oi - +(x + (x2 + 8Fo)1’2),
[7] on highly
for the conduction
of
(10) v, - t( --x + (x’ + 8F,)“2). It transpires that the electronic reaction parameters which best fit the data are U = 2 eV and S = 2k. With these values the defect concentrations are plotted in fig 1. It is clear that throughout the range of stoichiometry the electronic defects always dominate the oxygen defects; that is, p(x) > O,(x) for x > 0 and n(x) > V,(x) for x -= 0. It will be seen that the electronic transport properties are strongly influenced by the relative values of 6, which character&es the intrinsic disorder, and x, which character&s the extrinsic disorder.
2 = 3.8 x lo6
(a
cm)-’
K
and
(17)
E=0.3eV. Now 0, = nepn
and
up = pepp ,
08)
where p signifies the carrier mobility. son data we get a hole mobility pp=(l-n-p)7
o.o97
e -0.3/k=
From
the Aron-
m2/Vs.
This is very close to that given by Dudney
et al. [lo]
which was inferred from yttria-doped some confidence in its value. We postulate that
oxide. and gives
P,, =Pp.
(19)
From analysis of the Seebeck data it will be seen in the next section that this equality is supported at high temperatures. Thus the electronic contribution to o is just a,, = C, 5k ‘(n i-p){1 C 1
--n-p)
e-s”/7
(20)
and a,/@, =
“/P .
(21)
NOW oion - q,, + ov,, since the concentrations
of the other elementary defects are too small to matter. The diffusion of the oxygen defects [11] is dominated by the diffusion of the electronic defects implicit in ref. ]14]. Since one of the electronic defects always has a concentration higher than that of the ionic defects (see fig. I), the ionic conductivity is taken to be insignificant. Thus a = uion + a,, - u,, .
(22)
As a corollary we see that a(x) - u( -x) using eq. (8). Eq. (20) is plotted in fig. 2 as a function of reciprocal temperature for various values of x with mobility parameters as in eq. (17) and electronic reaction parameters U = 2 eV and S = 2k. The data of Bates [8] and Kiileen [9] for ‘no~n~ly’ stoichiometric UO, are very similar and well represented with x - 10e3. Above 2600 K the calculated value underestimates the experimental value, the discrepancy rising to - 20% at melt. The x-dependence of the data of Dudney et al. [lo] is also weil modelled. Fig. 3 plots the ratio in eq. (21) which is always less than unity for x > 0 and greater than unity for x < 0.
Fig. 2. The electrical conductivity of UO,,,.
.2
4. The Seebeck coefficient The Seebeck coefficient example, ref. 1121)
.>_
a, = - S,*/e
i‘..
500
(Y, is given
- &(5-E,,)
by (see, for
(23)
Similarly, 1000
1500
TEMPERATURE
Fig. 3. The ratio on/ap5 assuming mobilities.
2000
2500
3000
'K
equal electron
ap = S,*/e - ;(<-
41,
(24)
and hole
where
5 is the chemical
potential
and
S,"(S,* } is the
P. W. Winter / The electronic transport properties of UO,
entropy transported mal conditions. Now
per electron
(hole) under
isother-
INTRINSIC DISORDER 10.'
10
where SC is the configurational
6 10.'
l(r'
10.'
entropy
(26)
SC = k In Therefore,
Similarly,
We also have, 09) The relation
total [12]
Seebeck
coefficient
a is given
by
the 10
5 DIMENSIONLESS
(30)
I:=0.5
I
15
ENERGY
A.
tn('/,z)
Fig. 5. Electronic
conductivity normal&d to x = 0 as a function of /SF for different values x. The shaded region represents the high temperature regime, 2000 K < T < 3000 K, for U =
-0.3
2eVandS=2k. - 0.2
With our assumption of equal mobilities we have, using eq. (4),
-0.1
electronic
fa=ln(l_~_p)-~+--$$
and hole
(31)
=-ln(l_~_p)+~-$-,$.
(32)
Or again,
s,- s u+p-“v k
p+nkT
1 .
(33)
We note that, for x B 6, eq. (32) gives e -LYk
0.5 ---
KILLEEN
DATA'
-ln-
e --a-1500 TEMPERATURE
Fig.
4. The
2000
2500
T OK
Seebeck coefficient
of UO,,,
k
(34)
S, - S”
k 1000
+%
For x -=K6, eq. (33) gives
0.6
0.1_. 500
2x
1-2x
3000
2k
This relation temperatures Pn = Pp.
’
(35)
was observed in ref. [S] for x = 0 at high since the appearance of saturation forces
P. W. Winter / The electronic transport properties of U02
42
The formulae for u can now be used to analyse the data of K&en [9]. The experimental Seebeck coefficient appears to be independent of temperature at low temperatures, taking the value a - 0.6 mV K-‘. This is consistent with eq. (34) and gives S, as a function of x. The data is best fitted with x = low4 which in practice is appropriate for the dry hydrogen atmosphere used. This yields S,= -1.56/c. The data also appears to saturate at high temperatures to a value of (Y- -0.22 mV K-‘. This is consistent with eq. (35) and yields S, = 3.56k giving S = 2k. Again, taking U = 2 eV completes the best parameterisation. Fig. 4 depicts the graph of a as a function of T for various values of X. In particular we note that, from eq. (35), the Seebeck coefficient of stoichiometric UO, is independent of temperature assuming a constant value of - 0.22 mV K-l. The mobility ratio Q = f~,/p’, at x = 0 may be derived from s, - S” -+-2k
ee= k
1-Q l+Q
U 2kT’
Unfortunately, we do not know the Seebeck o(O) of stoichiometric UO, experimentally, assumption that Q = 1 is equivalent to a(O) stant. Finally, from eqs. (33) and (8) we deduce a(x)
+ a( -x)
yielding
= 2o(O),
the Seebeck coefficient
coefficient hence our being conthat (36)
Using eq. (13) we may write
where C, is defined in eq. (17) having a value of 2.83 W m-i K-‘. The effect of x is determined by the ratio P=K,(X),K
t
to)=
el
v+2v2 “P(l-n-P) n+P
8
Thus p is a function of the disorder only. From eqs. (6) and (7) we have (1 - 2X)3 P-
S(1 + 2s)*
_l-;&?x~
parameters
for
x%S,
for
x
(4) 6 and x
(42)
In fig. 5 p is plotted as a function of the dimensionless energy PF for various values of x. Clearly p is a decreasing function of x and an increasing function of S with p --) 0 as x * + or S --* 0. The shaded area in the figure locates the high temperature regime corresponding to our reaction parameters lJ = 2 eV and S = 2k. Clearly, oxidation has a very dramatic effect on the electronic thermal conductivity as eq. (41) shows. The same holds for reduction, since %(-X)
=‘%I(~).
for UO,_,. 4
5. The thermd
conductivity
As a consequence of the hopping nature of their transport the small polarons associated with the U3+ electrons and U 5+ holes separately contribute zero [13]. Thus there is no counterpart to the Weidemann-Franz term is classical semiconductors. The electronic contribution to the thermal conductivity is just the ambipolar term given by (141
;; K,I =
(37)
2
E 1
But, from eqs. (23) and (24)
1500
(38)
2500
2000 TEMPERATURE
3000
1 OK
Fig. 6. The thermal conductivity of U02; “Round-Robin” data 1151 (upper curve); and the calculated electronic component (lower curve).
P. W. Winter / The electronic transport properties of UO,
remains the calculation of the value of K,, at the stoichiometric composition. This value is plotted in fig. 6 and compared with the measured total thermal conductivity [15]. The electronic component is seen to be significant at high temperatures contributing over 30% of the total conductivity at melt. The shape of the curves in fig. 6 suggests that the whole of the increase in conductivity could well be attributable to the electronic component. There
6. Conclusions It has been shown that the behaviour of the electrical conductivity (I and Seebeck coefficient a of UO,,, is consistent with the small polaron hopping process (1). The data is consistent with equal electron and hole mobilities, an activation energy E = 0.3 eV and reaction parameters CJ= 2 eV and S = 2 k. The behaviour of the electronic transport properties is determined by the relative values of the disorder parameters S and x. There is a p-type extrinsic regime characterised by x B 6, an n-type intrinsic regime characterised by x < S and a p-n transition when S - x. A clear prediction is made of the electronic thermal conductivity IQ. It appears that K,, suffers a dramatic decrease with departure from stoichiometry. The onset of this reduction occurs at the p-n transition 8 - x. This effect was first pointed out in ref. [16] in the context of failed AGR fuel. It should be noted that complications arise with grossly non-stoichiometric oxide. For T < 2600 K mixed phases are present whilst for T > 2600 K a liquid phase may develop. Nevertheless it is likely that the process (1) will still be operative in these circumstances. Furthermore, defect-defect interaction will become increasingly significant with increasing x.
43
Finally, at very high temperatures, the thermal generation of multiply charged small polarons [4] may enhance the calculated values of u and K,, and account for the underestimate of u.
References
PI F.A. Kroger, Z. Phys. Chem. 49 (1966) 178. PI C.R.A. Catlow, Point Defects and Electronic Properties of
[31
[41 151 161
[71 PI [91 WI Pll P21 1131 [I41
[I51 [I61
Uranium Dioxide, Proc. R. Sot. London, Ser. A353 (1977) 533. D.A. Macinnes, in: Proc. Int. Symp. on the Thermodynamics of Nuclear Materials, Jiilich, FRG, 1979 (IAEA, Vienna, 1979) paper IAEA-SM 236/37. P.W. Winter and D.A. Macinnes, J. Nucl. Mater. 137 (1986) 161. G.J. Hyland and J. Ralph, High Temp.-High Press. 15 (1983) 179. Hj. Matzke, Diffusion in nonstoichiometric oxides, in: Nonstoichiometric Oxides, Ed. 0. Toft Sorenson (Academic Press, New York, London, 1981). S. Aronson, J.E. Rulli and B.E. Shaner, J. Chem. Phys. 35 (4) (Oct. 1961). J.L. Bates, C.A. Hinman and T. Kawada, J. Am. Ceram. Sot. 50 (1967) 652. J.C. Killeen, J. Nucl. Mater. 88 (1980) 185. N.J. Dudney, R.C. Coble and H.L. Tuller, J. Am. Ceram. Sot. 64 (1981) 627. W. Breitung, J. Nucl. Mater. 74 (1978) 19. A. Haug, Theoretical Solid State Physics, Vol 2 (Pergamon, Oxford, 1972). K.D. Schotte, Z. Phys. 196 (1966) 393. T.C. Harman and J.M. Honig, Thermoelectric and Thermomagnetic Effects and Applications (McGraw-Hill, New York, 1967). J.B. Conway and A.D. Feith, General Electric Report GEMP-715 (1969). P.W. Winter and D.A. Macinnes, UKAEA internal report, UO, BPC/P(87)285 (1987).
Journal of Nuclear Materials 161 (1989) 38-43 North-Holland, Amsterdam
THE ELECIRONIC
TRANSPORT
PROPERTIES
OF UO,
P.W. WINTER Safety and Reliability United Kingdom
Directorate,
UK Atomic Energv Authority,
Wigshaw Lone, Culcheth, Warrington
WA3 4NE,
Received 11 May 1988; accepted 22 September 1988
The electrical conductivity, Seebeck coefficient and electronic thermal conductivity of UO, f x are modelled using the small polarons generated by the reaction 2U4+ + U 3+ + U 5+. The reaction parameters which are found to best fit the data are a band gap of 2 eV, a vibrational entropy of 2 k and an electron-to-hole mobility ratio of unity. The electronic thermal conductivity is found to decrease sharply for large deviations from stoichiometry at high temperatures.
1. Introduction The intrinsic dc electrical conductivity o(T) of UO, has been attributed [1,2] to the thermally induced disproportionation
wherein the upper and lower Hubbard ‘bands’ reduce to two energies E, and E, separated by the Mott-Hubbard gap U equal to the internal energy required to effect the reaction (1). Thus the free energy F controlling reaction (1) may be written as
2u4+
F= U-
-+ u3+
+ u5+,
(1)
wherein the carriers (the electron on U3+ and the hole on U5+) are treated as small polarons. It has been suggested that this reaction is also largely responsible for the anomalous rise in specific heat C,(T) [3,4]. Here, a general treatment is forwarded which provides for consistent and explicit modelling of the electrical conductivity a, the Seebeck coefficient a and the electronic thermal conductivity K,,(T) of UOz+X. The model, based on reaction (l), rationalises all the available data. An earlier attempt at a unified treatment along these lines [5] achieved a measure of success but did not explicitly model the electrical conductivity and neglected the stoichiometric deviation x in the interpretation of the data for UO,. The comprehensive study of the influence of x carried out in this work affords valuable insight into the behaviour of the electronic transport processes.
It has been pointed out [S] that UO, is a Mott-Hubbard insulator rather than a classical semiconductor. The electronic energy spectrum can be described in terms of the ‘atomic limit’ of the Hubbard model
(North-Holland
(2)
where U = E,, - E, and S = S, + S,, with E, = E, - E,, E,= -(Es-E,), S,=S,-S, and S,=S,-S,. If n, p denote the molar concentrations of electrons and holes respectively we have, at equilibrium, (electroneutrality),
p = n + 2x
*P
e-BF
(1 -VP)*
(3)
(mass action),
(4)
=
where, as usual, /3= (kT)-‘. Setting S = edBF12 we obtain, p=x+
~(-262+(x2(1-4s*)+8*)1’2). l-482
(5) Thus, for x >> S, p-2x
and
n-S*(1-2~)~/2x,
(6)
for x K S,
2. The ekctronic model
OO22-3115/89/$03.50
TS,
0 Elsevier
Physics
Publishing
Science
Publishers
Division)
P -x+
8
-1+26
and
n-
-x+-
s 1+2S’
(7)
The following reciprocity relation is evident p(-x)
B.V.
=n(x).
(8)
39
P. W. Winter / The electronic transport properties of UO,
3. The dc electrical conductivity The electrical
conductivity
of UO,,,
may be written
as o=oi,+u,+a,,
(11)
and on, crP where uion is the ionic partial conductivity are the partial conductivities associated with electron and hole transport respectively. The electrical conductivity a,, of a carrier a of charge q is related to the diffusion coefficient D of a by the Nemst-Einstein relation a, = n,q’D/kT,
(12)
where nv is the carrier concentration The Einstein formula for random
per unit volume. diffusion gives
D = X2Pjw/6, 10*
10-3 STOICHIOMETRIC
DEVIATION
(13)
lo-'
10-Z x
Fig. 1. Defect diagram for UO,,, at T = 1500 K (solid curves) and T = 3000 K (dashed curves).
where X = =,,/fi ( a0 the lattice constant) is the jump distance, Pj the mean number of sites available for a jump and w the jump frequency. In the present context, for the electrons, Pj = 12(1 - n -p) and w = ve-ss. Thus n(1 -n-p)
In addition, if defects included we have, Oi-Vv,+X
on the oxygen
lattice
emBE/T
are with a similar relation
Co=!+
and
for up. With v - 1013 s-l,
Wm-‘K-l.
(15)
O,V, = 2 e-BFo,
where Oi is the oxygen interstitial concentration, V, the oxygen vacancy concentration and F, the frenkel energy (- 3 eV [6]). Thus,
Now from the data stoichiometric oxide, eP = 3.8 X lo6
x
of Aronson
2x(1 -2x)
e-0.3/kT/T
non-
(0 cm)-‘. (16)
We thus make the identifications, holes,
oi - +(x + (x2 + 8Fo)1’2),
[7] on highly
for the conduction
of
(10) v, - t( --x + (x’ + 8F,)“2). It transpires that the electronic reaction parameters which best fit the data are U = 2 eV and S = 2k. With these values the defect concentrations are plotted in fig 1. It is clear that throughout the range of stoichiometry the electronic defects always dominate the oxygen defects; that is, p(x) > O,(x) for x > 0 and n(x) > V,(x) for x -= 0. It will be seen that the electronic transport properties are strongly influenced by the relative values of 6, which character&es the intrinsic disorder, and x, which character&s the extrinsic disorder.
2 = 3.8 x lo6
(a
cm)-’
K
and
(17)
E=0.3eV. Now 0, = nepn
and
up = pepp ,
08)
where p signifies the carrier mobility. son data we get a hole mobility pp=(l-n-p)7
o.o97
e -0.3/k=
From
the Aron-
m2/Vs.
This is very close to that given by Dudney
et al. [lo]
which was inferred from yttria-doped some confidence in its value. We postulate that
oxide. and gives
P,, =Pp.
(19)
From analysis of the Seebeck data it will be seen in the next section that this equality is supported at high temperatures. Thus the electronic contribution to o is just a,, = C, 5k ‘(n i-p){1 C 1
--n-p)
e-s”/7
(20)
and a,/@, =
“/P .
(21)
NOW oion - q,, + ov,, since the concentrations
of the other elementary defects are too small to matter. The diffusion of the oxygen defects [11] is dominated by the diffusion of the electronic defects implicit in ref. ]14]. Since one of the electronic defects always has a concentration higher than that of the ionic defects (see fig. I), the ionic conductivity is taken to be insignificant. Thus a = uion + a,, - u,, .
(22)
As a corollary we see that a(x) - u( -x) using eq. (8). Eq. (20) is plotted in fig. 2 as a function of reciprocal temperature for various values of x with mobility parameters as in eq. (17) and electronic reaction parameters U = 2 eV and S = 2k. The data of Bates [8] and Kiileen [9] for ‘no~n~ly’ stoichiometric UO, are very similar and well represented with x - 10e3. Above 2600 K the calculated value underestimates the experimental value, the discrepancy rising to - 20% at melt. The x-dependence of the data of Dudney et al. [lo] is also weil modelled. Fig. 3 plots the ratio in eq. (21) which is always less than unity for x > 0 and greater than unity for x < 0.
Fig. 2. The electrical conductivity of UO,,,.
.2
4. The Seebeck coefficient The Seebeck coefficient example, ref. 1121)
.>_
a, = - S,*/e
i‘..
500
(Y, is given
- &(5-E,,)
by (see, for
(23)
Similarly, 1000
1500
TEMPERATURE
Fig. 3. The ratio on/ap5 assuming mobilities.
2000
2500
3000
'K
equal electron
ap = S,*/e - ;(<-
41,
(24)
and hole
where
5 is the chemical
potential
and
S,"(S,* } is the
P. W. Winter / The electronic transport properties of UO,
entropy transported mal conditions. Now
per electron
(hole) under
isother-
INTRINSIC DISORDER 10.'
10
where SC is the configurational
6 10.'
l(r'
10.'
entropy
(26)
SC = k In Therefore,
Similarly,
We also have, 09) The relation
total [12]
Seebeck
coefficient
a is given
by
the 10
5 DIMENSIONLESS
(30)
I:=0.5
I
15
ENERGY
A.
tn('/,z)
Fig. 5. Electronic
conductivity normal&d to x = 0 as a function of /SF for different values x. The shaded region represents the high temperature regime, 2000 K < T < 3000 K, for U =
-0.3
2eVandS=2k. - 0.2
With our assumption of equal mobilities we have, using eq. (4),
-0.1
electronic
fa=ln(l_~_p)-~+--$$
and hole
(31)
=-ln(l_~_p)+~-$-,$.
(32)
Or again,
s,- s u+p-“v k
p+nkT
1 .
(33)
We note that, for x B 6, eq. (32) gives e -LYk
0.5 ---
KILLEEN
DATA'
-ln-
e --a-1500 TEMPERATURE
Fig.
4. The
2000
2500
T OK
Seebeck coefficient
of UO,,,
k
(34)
S, - S”
k 1000
+%
For x -=K6, eq. (33) gives
0.6
0.1_. 500
2x
1-2x
3000
2k
This relation temperatures Pn = Pp.
’
(35)
was observed in ref. [S] for x = 0 at high since the appearance of saturation forces
P. W. Winter / The electronic transport properties of U02
42
The formulae for u can now be used to analyse the data of K&en [9]. The experimental Seebeck coefficient appears to be independent of temperature at low temperatures, taking the value a - 0.6 mV K-‘. This is consistent with eq. (34) and gives S, as a function of x. The data is best fitted with x = low4 which in practice is appropriate for the dry hydrogen atmosphere used. This yields S,= -1.56/c. The data also appears to saturate at high temperatures to a value of (Y- -0.22 mV K-‘. This is consistent with eq. (35) and yields S, = 3.56k giving S = 2k. Again, taking U = 2 eV completes the best parameterisation. Fig. 4 depicts the graph of a as a function of T for various values of X. In particular we note that, from eq. (35), the Seebeck coefficient of stoichiometric UO, is independent of temperature assuming a constant value of - 0.22 mV K-l. The mobility ratio Q = f~,/p’, at x = 0 may be derived from s, - S” -+-2k
ee= k
1-Q l+Q
U 2kT’
Unfortunately, we do not know the Seebeck o(O) of stoichiometric UO, experimentally, assumption that Q = 1 is equivalent to a(O) stant. Finally, from eqs. (33) and (8) we deduce a(x)
+ a( -x)
yielding
= 2o(O),
the Seebeck coefficient
coefficient hence our being conthat (36)
Using eq. (13) we may write
where C, is defined in eq. (17) having a value of 2.83 W m-i K-‘. The effect of x is determined by the ratio P=K,(X),K
t
to)=
el
v+2v2 “P(l-n-P) n+P
8
Thus p is a function of the disorder only. From eqs. (6) and (7) we have (1 - 2X)3 P-
S(1 + 2s)*
_l-;&?x~
parameters
for
x%S,
for
x
(4) 6 and x
(42)
In fig. 5 p is plotted as a function of the dimensionless energy PF for various values of x. Clearly p is a decreasing function of x and an increasing function of S with p --) 0 as x * + or S --* 0. The shaded area in the figure locates the high temperature regime corresponding to our reaction parameters lJ = 2 eV and S = 2k. Clearly, oxidation has a very dramatic effect on the electronic thermal conductivity as eq. (41) shows. The same holds for reduction, since %(-X)
=‘%I(~).
for UO,_,. 4
5. The thermd
conductivity
As a consequence of the hopping nature of their transport the small polarons associated with the U3+ electrons and U 5+ holes separately contribute zero [13]. Thus there is no counterpart to the Weidemann-Franz term is classical semiconductors. The electronic contribution to the thermal conductivity is just the ambipolar term given by (141
;; K,I =
(37)
2
E 1
But, from eqs. (23) and (24)
1500
(38)
2500
2000 TEMPERATURE
3000
1 OK
Fig. 6. The thermal conductivity of U02; “Round-Robin” data 1151 (upper curve); and the calculated electronic component (lower curve).
P. W. Winter / The electronic transport properties of UO,
remains the calculation of the value of K,, at the stoichiometric composition. This value is plotted in fig. 6 and compared with the measured total thermal conductivity [15]. The electronic component is seen to be significant at high temperatures contributing over 30% of the total conductivity at melt. The shape of the curves in fig. 6 suggests that the whole of the increase in conductivity could well be attributable to the electronic component. There
6. Conclusions It has been shown that the behaviour of the electrical conductivity (I and Seebeck coefficient a of UO,,, is consistent with the small polaron hopping process (1). The data is consistent with equal electron and hole mobilities, an activation energy E = 0.3 eV and reaction parameters CJ= 2 eV and S = 2 k. The behaviour of the electronic transport properties is determined by the relative values of the disorder parameters S and x. There is a p-type extrinsic regime characterised by x B 6, an n-type intrinsic regime characterised by x < S and a p-n transition when S - x. A clear prediction is made of the electronic thermal conductivity IQ. It appears that K,, suffers a dramatic decrease with departure from stoichiometry. The onset of this reduction occurs at the p-n transition 8 - x. This effect was first pointed out in ref. [16] in the context of failed AGR fuel. It should be noted that complications arise with grossly non-stoichiometric oxide. For T < 2600 K mixed phases are present whilst for T > 2600 K a liquid phase may develop. Nevertheless it is likely that the process (1) will still be operative in these circumstances. Furthermore, defect-defect interaction will become increasingly significant with increasing x.
43
Finally, at very high temperatures, the thermal generation of multiply charged small polarons [4] may enhance the calculated values of u and K,, and account for the underestimate of u.
References
PI F.A. Kroger, Z. Phys. Chem. 49 (1966) 178. PI C.R.A. Catlow, Point Defects and Electronic Properties of
[31
[41 151 161
[71 PI [91 WI Pll P21 1131 [I41
[I51 [I61
Uranium Dioxide, Proc. R. Sot. London, Ser. A353 (1977) 533. D.A. Macinnes, in: Proc. Int. Symp. on the Thermodynamics of Nuclear Materials, Jiilich, FRG, 1979 (IAEA, Vienna, 1979) paper IAEA-SM 236/37. P.W. Winter and D.A. Macinnes, J. Nucl. Mater. 137 (1986) 161. G.J. Hyland and J. Ralph, High Temp.-High Press. 15 (1983) 179. Hj. Matzke, Diffusion in nonstoichiometric oxides, in: Nonstoichiometric Oxides, Ed. 0. Toft Sorenson (Academic Press, New York, London, 1981). S. Aronson, J.E. Rulli and B.E. Shaner, J. Chem. Phys. 35 (4) (Oct. 1961). J.L. Bates, C.A. Hinman and T. Kawada, J. Am. Ceram. Sot. 50 (1967) 652. J.C. Killeen, J. Nucl. Mater. 88 (1980) 185. N.J. Dudney, R.C. Coble and H.L. Tuller, J. Am. Ceram. Sot. 64 (1981) 627. W. Breitung, J. Nucl. Mater. 74 (1978) 19. A. Haug, Theoretical Solid State Physics, Vol 2 (Pergamon, Oxford, 1972). K.D. Schotte, Z. Phys. 196 (1966) 393. T.C. Harman and J.M. Honig, Thermoelectric and Thermomagnetic Effects and Applications (McGraw-Hill, New York, 1967). J.B. Conway and A.D. Feith, General Electric Report GEMP-715 (1969). P.W. Winter and D.A. Macinnes, UKAEA internal report, UO, BPC/P(87)285 (1987).