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The effect of ionizing and displacive radiation on the thermal conductivity of alumina at low temperature
ELSEVIER
Journal of Nuclear Materials 212-215 (1994) 1069-1074
The effect of ionizing and displacive radiation on the thermal conductivity of alumina at low temperature D.P. White Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 378314376,
USA
Abstract The effects of radiation-produced vacancies and of radiation-induced electrical conductivity (RIG) on the thermal conductivity of alumina at low temperatures have been calculated. The phonon scattering relaxation times for these mechanisms have been used in the evaluation of the thermal conductivity integral in order to determine the effect each mechanism has on the lattice thermal conductivity of alumina. It is found that vacancy scattering can significantly reduce the thermal conductivity; for example, a vacancy concentration of 0.01 per atom leads to a fractional change in thermal conductivity of about 90%. It is also concluded that the scattering of phonons by electrons in the conduction band due to RIC does not lead to a large reduction in the thermal conductivity.
1. Introduction
Callaway 181,
Microwave heating of plasmas in fusion reactors requires the development of windows through which the microwaves can pass without great losses. The degradation of the thermal conductivity of alumina in a radiation environment is an important consideration in reliability studies of these microwave windows. Several recent publications have calculated the radiation induced degradation at high temperature [l-3] and at low temperature [4,5]. The current paper extends the low temperature calculations in order to determine the effect of phonon scattering by radiation produced vacancies and by conduction band electrons due to RIC on the thermal conductivity at 77 K. These low temperature calculations are of interest because the successful application of high power (> 1 MW) windows for electron cyclotron heating systems in fusion reactors will most likely require cryogenic cooling to take advantage of the low loss tangent and higher thermal conductivity of candidate window materials at these temperatures
k = (67~~)~‘~ k:, T3
[6,71. 2. Calculations
The lattice thermal conductivity of crystalline material is given by the following expression derived by
I
-_ h2 @)a[ II+
2a2
w319
(1)
where k, is Boltzmann’s constant, h is Plank’s constant divided by 2~, 0 is the Debye temperature of the material, a3 is the volume per unit cell, and I, =
1,
=
/“‘=Tc 0
-
lydxv
x4 ex
jc 0
x4 ex
(ex
7,
(ex - 1)2
dx,
x4 eX 1, = j@/Ti, dx. 0 TnTr (e’ - l)* Here x = ho/k,T, w is the circular phonon frequency, T, is the phonon relaxation time for normal processes, l/7, = l/7,, + l/7,, and l/7, = Cj l/~~, where the TV are the relaxation times of the various resistive phonon scattering processes, which include 3-phonon umklapp processes, boundary scattering, point defect scattering, and phonon-electron scattering. This formulation of the thermal conductivity takes into account the effect of phonon-phonon normal pro-
0022-3115/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0022-3115(94)00033-K
D.P. White /Journal
1070
of Nuclear Materials 212-215
cesses, which is particularly important in treating the low temperature case. In order to evaluate the expression in Eq. (l), a computer program was written to calculate the various integrals, given functional forms for the various phonon relaxation times. The forms used for the normal process relaxation time and umklapp process relaxation time are given in Eqs. (2): 7”-’ = b&P, 7;’ =b,o’T
exp( -O/cuT),
(2)
for normal and umklapp processes, respectively. The values of the parameters b, and b, for sapphire were obtained by de Goer [9] and are the values used in this case, These values were obtained by fitting the expression of Eq. (1) to experimental thermal conductivity data at high temperatures. The values obtained by de Goer are b, = 2.7 x lo-t3
K-4,
b ” =17~10-~ssK-~. .
The parameter (Y is a numerical factor which depends on the details of the zone structure; de Goer found that a = 2 gives a good fit to experimental data. The boundary scattering relaxation time has the form Tb-I
=
u/L,
(3)
where v is the phonon velocity and L is a parameter on the order of the dimensions of the crystal and is dependent on the geometry of the sample. The point defect scattering relaxation time has the form, -1
,AW4
TP
(4)
where A is a parameter which is proportional to the defect concentration and depends on the nature of the defect, which in pure unirradiated samples will include thermodynamic vacancies, and trace impurities. For a pure, single crystal, cylindrical sample 50 mm long and 5 mm in diameter (de Goer’s sample 5) de Goer varied the values of A and L to fit the calculated values of the thermal conductivity to experimental values in the temperature range 2 to 100 K. It was found that a good fit was found for L = 4.12 X 10e3 m and A = 4.08 x lo-& s3. These values will be used in the present calculations. It should be noted however that the thermal conductivity below the conductivity maximum (_ 30 K) is highly sensitive to the value of L. 2.1. Vacancy scattering In duced phire, duced
order to determine the effect of radiation-provacancies on the thermal conductivity of sapanother term, dependent on the radiation-provacancy concentration, must be added to the
(1994) 1069-1074
point defect relaxation time. For vacancies it has been shown [lO,ll] that the phonon relaxation time is given by
(5) where C, is the vacancy concentration per unit cell, AM is the change in mass of the unit cell due to a vacancy, and M is the mass of the unit cell. In this case the unit cell is chosen to be the formula unit, A&,0,, with a mass of 102 and a volume of 42.6 X 10P3” m3. The Debye temperature in this case is then 600 K, and the phonon velocity is 7 x lo3 m/s. Thus the form of the point defect relaxation time is given by
for aluminum vacancies. In this expression the first term in parentheses is that part of the relaxation time which is due to intrisic point defects, as found by de Goer, and the second term is that part which is due to radiation-produced point defects. These relaxation times may now be substituted into Eq. (1) and the thermal conductivity calculated for a range of temperatures and radiation produced vacancy concentrations. It should be noted that the relaxation time in Eq. (6) applies to isolated, randomly distributed point defects. If the vacancies cluster then this relaxation time must be appropriately modified. For example, in the case of defect clustering in a region small compared to the phonon wavelength there is a reinforcement of the scattering matrix and the scattering probability varies as the square of the number of defects in the cluster. The calculations presented here will be for isolated point defects. 2.2. Phonon-electron scattering In order to determine the effect of phonon-electron scattering on the thermal conductivity, the phonon-electron relaxation time must be included as a resistive phonon scattering process in the Callaway formulation. The phonon-electron relaxation time may be obtained though a momentum balance argument [3] and is of the form 1 -=-7
P-e
3a2
CT
p2 CT’
where u is the phonon velocity, h is the electron mobility, C is the phonon specific heat of those phonons which are allowed to interact with the conduction band electrons (due to energy and wave-vector conservation considerations), o is the electrical conductivity, and T is the temperature.
D.P. White/Journal of Nuclear Materials 212-215 (1994) 1069-1074
1071
The relaxation time given in Eq. (7) must be evaluated in order to use it in the thermal conductivity integral. Klaffky et al. [12] found evidence of a temperature-independent mobility for T < 300 K, for single crystal alumina under conditions where RIC was produced by continuous irradiation with 1.5 MeV electrons at dose rates up to 6.6 X 10’ Gy/s. Hughes [13] reported a temperature independent mobility of (3 f 1) x low4 m*/V s for an undoped aluminum sample X-ray irradiated between 100 and 350 K. In the calculations presented here it is assumed that this constant mobility holds to 77 K. The specific heat, C, of those phonons up to a cut off frequency of o, = 2v/h (2mk,T)‘/* is given, in the Debye approximation, by temperature 8,/T
x4 exp x
(exp x - lj* dx’ In these expression, m is the electron mass, n is the number of unit cells per unit volume and 0, is the phonon-electron cutoff temperature given by 0, = hoc/k, and x = ho/k,T. The expression given in Eq. (8) may be evaluated by numerical integration for any particular temperature and then this value may be used in the relaxation time of Eq. (7). It is possible to calculate the changes in the thermal conductivity due to changes in u by numerically evaluating the Callaway thermal conductivity integral of Eq. (1) for different values of D in the phonon-electron relaxation time.
3. Results and discussion The results of the thermal conductivity calculations are given in Figs. l-4. In Fig. 1 the calculated thermal conductivity versus temperature is plotted for several values of the radiation produced vacancy concentration. The largest reduction in the thermal conductivity due to these vacancies is in the region of the wnductivity maximum at approximately 30 K. The curves in Fig. 1 compare favorably with the low temperature measurements of Berman et al. [14] and of Salce et al. [15], for neutron-irradiated sapphire. For example, Berman et al. found that the thermal conductivity maximum for a sapphire sample irradiated to a dose of 50.2 x lO*l n/m* (_ 0.005 dpa) was reduced from a nonirradiated value of approximately 8000 W/m K to about 400 W/m K. From Fig. 1 it is seen that the same fractional reduction occurs at a vacancy concentration of approximately 0.002 per atom. Fig. 2 is a plot of the thermal conductivity versus vacancy concentration at 77 K. This plot shows the reduction in the thermal conductivity at liquid nitrogen temperatures, where it has been proposed to cool ECRH microwave windows to take advantage of the high thermal conductivity and low loss
(K)
Fig. 1. Plot of the thermal conductivity of single crystal alumina versus temperature. The different curves correspond to different vacancy concentrations, where c is the vacancy concentration per atom.
tangent of alumina at these temperatures [7]. In Fig. 3 the fractional change in the thermal conductivity versus RIC is plotted. The three different curves correspond to different values of the electron mobility, covering the range of uncertainty in the electron mobility. From this plot it can be seen that at the highest values of the RIC plotted (lo-* (fl m)-‘), the change in the thermal conductivity approaches a maximum of about 9%. This value corresponds to the case of all phonons below the cut off frequency being strongly scattered, and thus is the limiting value of the reduction in the thermal conductivity which can occur at this temperature due to phonon-electron scattering. In Fig. 4 the fractional change in the thermal conductivity is plotted
loo0 F
1o-7
1o-3 1o-4 1o-5 1o-G vacancy concentration (per atom)
Fig. 2. Plot of the thermal conductivity of single alumina versus vacancy concentration at 77 K.
1o-2 crystal
D.P. White/Journal
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of Nuclear Materials 212-215
(1994) 1069-1074
reduced by vacancies at low temperatures. From Fig. 1 it is seen that the major part of the reduction is in the area of the thermal conductivity maximum, which oc-
10.'
No
1o-3
1o-5
MC (ohm-m)-’
Fig. 3. Fractional change in the lattice thermal conductivity versus RIC at 77 K. Each curve corresponds to a different value of the electron mobility, covering the range of uncertainty in the mobility [13]. The upper curve corresponds to p = 2 x 10m4 m*/V s, the center curve to p = 3 x low4 m2/V s, and the lower cume to p = 4 x 10m4 m2/V s.
as a function of vacancy plots the fractional change
concentration. This figure with the RIC included and
with no RIC included in the calculation. The value of the RIC chosen in this plot is lo-’ (fi m)-‘. This value was chosen because it represents the maximum of the range of expected prompt RIC values in sapphire in a fusion environment [3]. It is only at vacancy concentrations less than lo-’ per atom where there is any difference between the two cases. The thermal conductivity of sapphire is significantly
loo
“““‘I
“““I
“/““1
““‘I
“I’
-I
t 1o-4 ’ 10.’
“““’ 1o-6
“‘,“‘N
“““”
1o-5
vacancy concentration
“““’
1o-4
1oe3
““’
’ 10-l
(per atom)
Fig. 4. Fractional change in the thermal conductivity versus vacancy concentration at 77 K. The upper curve corresponds to a RIC of 10P5 (fi ml-‘, and the lower curve corresponds to the case of no RIC.
curs at about 30 K. There is however significant reduction in the thermal conductivity at liquid nitrogen temperature, as can be seen in Fig. 2. From Fig. 4 it can be seen that the fractional change in the thermal conductivity for a vacancy concentration of 0.01 per atom is expected to be about 90% at 77 K. This is a very large reduction in the lattice thermal conductivity and it raises the question as to whether or not this reduction negates the increase in the thermal conductivity gained by going to low temperatures. In Refs. [1,3] it was shown that at 400 K the fractional reduction in the thermal conductivity due to a vacancy concentration of 0.01 per atom would be expected to be approximately 39% which is a much smaller fractional decrease in the lattice thermal conductivity compared to the 90% reduction expected with the same defect concentration at 77 K. The larger fractional reduction at low temperatures is due to the fact that the umklapp scattering is much weaker at low temperatures and allows the point defect scattering to be the dominant scattering mechanism at lower phonon frequencies, thus scattering a larger fraction of the phonon spectrum. However at 400 K a 39% reduction in the thermal conductivity would result in a conductivity of approximately 16 W/m K, compared to an expected conductivity of 83 W/m K after a 90% reduction in the conductivity at 77 K. Thus even at defect concentrations as high as 0.01 per atom the thermal conductivity advantage gained by going to lower temperatures would be maintained. However, as pointed out in the next section, it may be that the thermal stability of point defects at 77 K versus 400 K is the deciding factor that determines which irradiation temperature has a higher thermal conductivity for a given fluence of vacancy-producing radiation. The maximum of the range of prompt RIC values in sapphire in a fusion environment is expected to be on the order of lo-’ (a ml-‘. Using this value for the RIC the corresponding range for the fractional change in the thermal conductivity is from 0.05 to 0.2% at 77 K. It was found [1,3] that at 300 K the expected fractional change in the conductivity at this value of the RIC was about 0.01%. Thus the effect of phononelectron scattering on the thermal conductivity is larger at low temperatures but these are very small changes in the thermal conductivity and are not expected to be of importance in the design of microwave windows. Even in the extreme case of very large values of the RIC (lo-’ (a ml-‘) the maximum effect the phonon-electron interaction can have on the thermal conductivity is about 9%, as pointed out in the previous section. It should also be noted that there is essentially no change in the thermal conductivity due to the small electronic
D.P. White /Journal
component 131.
of Nuclear Materials 212-215
added by the RIC, as pointed out in Ref.
4. Conclusions The reduction in the thermal conductivity due to the scattering of phonons by vacancies is more pronounced at low temperatures than at high temperatures. It has been shown however that the thermal conductivity advantage gained by going to low temperature is maintained. However the vacancy concentration depends on the thermal stability of the vacancies. If many of the vacancies recombine at 400 K but are stable at 77 K, then the low temperature thermal conductivity could be less than the 400 K thermal conductivity for a given fluence of vacancy producing irradiation. The work of Rohde and Schulz [16] has shown that irradiation temperature has an important effect on the number of remaining stable defects in irradiated alumina. This effect has also been seen in BeO, Pryor et al. [17] compared the thermal resistance per fission neutron dose of samples irradiated at 348 K with samples irradiated at 90 K by McDonald 1181.The thermal resistance of the samples irradiated at low temperature was about 16 times larger than the resistance of the high temperature samples for a given neutron dose. This problem points to the need of performing in-situ cryogenic thermal conductivity tests or postirradiation thermal conductivity measurements on samples irradiated at cryogenic temperatures. An important consideration if postirradiation measurements are to be performed is that there is evidence that significant annealing of defects may occur at room temperature. This was seen by McDonald [18] in Be0 which was irradiated at 95 K, after a final anneal at 310 K approximately 25% of the radiation induced thermal resistance had recovered. Further evidence that significant room temperature annealing effects are important in ceramics is given by Coghlan et al. [19] and Buckley et al. 1201.Coghlan et al. found that neutron-irradiated spine1 continued to swell after irradiation at room temperature, perhaps indicating continuing diffusion and clustering of defects. Buckley et al. found that the dielectric loss tangent of proton-irradiated ceramics recovered significantly on the timescale of hours at room temperature. Thus, in order to rule out postirradiation thermal effects on ceramics irradiated at low temperature it is important to maintain the samples at low temperature until the measurements are made. It should also be noted that the flexibility of ECRH transmission systems will allow the windows to be located in low fluence positions, with the aim of maintaining the low temperature thermal conductivity advantage. However the magnitude of tolerable fluence levels must still be identified experimentally.
(1994) 1069-1074
1073
The changes in thermal conductivity due to phonon-electron scattering have been shown to be insignificant at the RIC levels expected in the fusion environment. These changes are particularly insignificant when the temperature dependence of the thermal conductivity at low temperatures is considered. In this low temperature region the thermal conductivity is strongly temperature-dependent; for example, in the region around 77 K calculations show that a 1% change in the temperature leads to a 4% change in the thermal conductivity. Thus the effects of temperature fluctuations on the thermal conductivity are expected to be at least as important as any effects due to phononelectron scattering. The small changes expected in the thermal conductivity due to phonon-electron scattering suggest that there is no need for in-situ cryogenic thermal conductivity measurements or postirradiation measurements of samples held at low temperature in order to study this effect. However this does not rule out the need for performing these tests in order to study the effects of thermally unstable radiation-produced defects.
Acknowledgment This research was performed while the author held a fellowship in the Oak Ridge National Laboratory Postdoctoral Research Program, administered through the Oak Ridge Institute for Science and Education.
References [l] D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/H, ORNL (1991) p. 277. [2] D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/12, ORNL (1992) p. 298. [3] D.P. White, J. Appl. Phys. 73 (1993) 2254. [4] D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/13, ORNL (1992) p. 326. [51 D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/14, ORNL (19931, in press. [6] G.R. Haste, H.D. Kimrey and J.D. Proisie, Report ORNL/TM-9906 (1986). [7] R. Heidinger, J. Nucl. Mater. 179-181 (1991) 64. [8] J. Callaway, Phys. Rev. 113 (1959) 1046. [9] A.M. de Goer, J. Phys. (Paris) 30 (1969) 389. [lo] P.G. Klemens, in Solid State Physics, vol. 7, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1958) p. 1. [ill C.A. Ratsifaritana, in Phonon Scattering in Condensed Matter, ed. H.J. Maris (Plenum, New York, 1980) p. 259. [12] R.W. Klaftky, B.H. Rose, A.N. Goland and G.J. Dienes, Phys. Rev B 21 (1980) 3610.
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of Nuclear Materials 212-215
[13] R.C. Hughes, Phys. Rev. B 19 (1979) 5318. [14] R. Berman, E.L. Foster and H.M. Rosenberg, in Report of the Conf. on Defects in Crystalline Solids, University of Bristol, 1954 (The Physical Society, London, 1955) p. 321. [15] B. Salce and A.M. de Goer, Proc. Conf. Digest of the Int. Conf. on Defects in Insulating Materials, Parma, 1988, p. 499.
(1994) 1069-1074
[16] M. Rohde and B. Schulz, J. Nucl. Mater. 173 (1990) 289. [17] A.W. Pryor, R.J. Tainsh and G.K. White, J. Nucl. Mater. 14 (1964) 208. [18] D.L. McDonald, Appl. Phys. Lett. 2 (1963) 175. [19] W.A. Coghlan, F.W. Clinard, Jr., N. Itoh and L.R. Greenwood, J. Nucl. Mater. 141-143 (1986) 382. [20] S.N. Buckley and P. Agnew, J. Nucl. Mater. 155-157 (1988) 361.
Journal of Nuclear Materials 212-215 (1994) 1069-1074
The effect of ionizing and displacive radiation on the thermal conductivity of alumina at low temperature D.P. White Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 378314376,
USA
Abstract The effects of radiation-produced vacancies and of radiation-induced electrical conductivity (RIG) on the thermal conductivity of alumina at low temperatures have been calculated. The phonon scattering relaxation times for these mechanisms have been used in the evaluation of the thermal conductivity integral in order to determine the effect each mechanism has on the lattice thermal conductivity of alumina. It is found that vacancy scattering can significantly reduce the thermal conductivity; for example, a vacancy concentration of 0.01 per atom leads to a fractional change in thermal conductivity of about 90%. It is also concluded that the scattering of phonons by electrons in the conduction band due to RIC does not lead to a large reduction in the thermal conductivity.
1. Introduction
Callaway 181,
Microwave heating of plasmas in fusion reactors requires the development of windows through which the microwaves can pass without great losses. The degradation of the thermal conductivity of alumina in a radiation environment is an important consideration in reliability studies of these microwave windows. Several recent publications have calculated the radiation induced degradation at high temperature [l-3] and at low temperature [4,5]. The current paper extends the low temperature calculations in order to determine the effect of phonon scattering by radiation produced vacancies and by conduction band electrons due to RIC on the thermal conductivity at 77 K. These low temperature calculations are of interest because the successful application of high power (> 1 MW) windows for electron cyclotron heating systems in fusion reactors will most likely require cryogenic cooling to take advantage of the low loss tangent and higher thermal conductivity of candidate window materials at these temperatures
k = (67~~)~‘~ k:, T3
[6,71. 2. Calculations
The lattice thermal conductivity of crystalline material is given by the following expression derived by
I
-_ h2 @)a[ II+
2a2
w319
(1)
where k, is Boltzmann’s constant, h is Plank’s constant divided by 2~, 0 is the Debye temperature of the material, a3 is the volume per unit cell, and I, =
1,
=
/“‘=Tc 0
-
lydxv
x4 ex
jc 0
x4 ex
(ex
7,
(ex - 1)2
dx,
x4 eX 1, = j@/Ti, dx. 0 TnTr (e’ - l)* Here x = ho/k,T, w is the circular phonon frequency, T, is the phonon relaxation time for normal processes, l/7, = l/7,, + l/7,, and l/7, = Cj l/~~, where the TV are the relaxation times of the various resistive phonon scattering processes, which include 3-phonon umklapp processes, boundary scattering, point defect scattering, and phonon-electron scattering. This formulation of the thermal conductivity takes into account the effect of phonon-phonon normal pro-
0022-3115/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0022-3115(94)00033-K
D.P. White /Journal
1070
of Nuclear Materials 212-215
cesses, which is particularly important in treating the low temperature case. In order to evaluate the expression in Eq. (l), a computer program was written to calculate the various integrals, given functional forms for the various phonon relaxation times. The forms used for the normal process relaxation time and umklapp process relaxation time are given in Eqs. (2): 7”-’ = b&P, 7;’ =b,o’T
exp( -O/cuT),
(2)
for normal and umklapp processes, respectively. The values of the parameters b, and b, for sapphire were obtained by de Goer [9] and are the values used in this case, These values were obtained by fitting the expression of Eq. (1) to experimental thermal conductivity data at high temperatures. The values obtained by de Goer are b, = 2.7 x lo-t3
K-4,
b ” =17~10-~ssK-~. .
The parameter (Y is a numerical factor which depends on the details of the zone structure; de Goer found that a = 2 gives a good fit to experimental data. The boundary scattering relaxation time has the form Tb-I
=
u/L,
(3)
where v is the phonon velocity and L is a parameter on the order of the dimensions of the crystal and is dependent on the geometry of the sample. The point defect scattering relaxation time has the form, -1
,AW4
TP
(4)
where A is a parameter which is proportional to the defect concentration and depends on the nature of the defect, which in pure unirradiated samples will include thermodynamic vacancies, and trace impurities. For a pure, single crystal, cylindrical sample 50 mm long and 5 mm in diameter (de Goer’s sample 5) de Goer varied the values of A and L to fit the calculated values of the thermal conductivity to experimental values in the temperature range 2 to 100 K. It was found that a good fit was found for L = 4.12 X 10e3 m and A = 4.08 x lo-& s3. These values will be used in the present calculations. It should be noted however that the thermal conductivity below the conductivity maximum (_ 30 K) is highly sensitive to the value of L. 2.1. Vacancy scattering In duced phire, duced
order to determine the effect of radiation-provacancies on the thermal conductivity of sapanother term, dependent on the radiation-provacancy concentration, must be added to the
(1994) 1069-1074
point defect relaxation time. For vacancies it has been shown [lO,ll] that the phonon relaxation time is given by
(5) where C, is the vacancy concentration per unit cell, AM is the change in mass of the unit cell due to a vacancy, and M is the mass of the unit cell. In this case the unit cell is chosen to be the formula unit, A&,0,, with a mass of 102 and a volume of 42.6 X 10P3” m3. The Debye temperature in this case is then 600 K, and the phonon velocity is 7 x lo3 m/s. Thus the form of the point defect relaxation time is given by
for aluminum vacancies. In this expression the first term in parentheses is that part of the relaxation time which is due to intrisic point defects, as found by de Goer, and the second term is that part which is due to radiation-produced point defects. These relaxation times may now be substituted into Eq. (1) and the thermal conductivity calculated for a range of temperatures and radiation produced vacancy concentrations. It should be noted that the relaxation time in Eq. (6) applies to isolated, randomly distributed point defects. If the vacancies cluster then this relaxation time must be appropriately modified. For example, in the case of defect clustering in a region small compared to the phonon wavelength there is a reinforcement of the scattering matrix and the scattering probability varies as the square of the number of defects in the cluster. The calculations presented here will be for isolated point defects. 2.2. Phonon-electron scattering In order to determine the effect of phonon-electron scattering on the thermal conductivity, the phonon-electron relaxation time must be included as a resistive phonon scattering process in the Callaway formulation. The phonon-electron relaxation time may be obtained though a momentum balance argument [3] and is of the form 1 -=-7
P-e
3a2
CT
p2 CT’
where u is the phonon velocity, h is the electron mobility, C is the phonon specific heat of those phonons which are allowed to interact with the conduction band electrons (due to energy and wave-vector conservation considerations), o is the electrical conductivity, and T is the temperature.
D.P. White/Journal of Nuclear Materials 212-215 (1994) 1069-1074
1071
The relaxation time given in Eq. (7) must be evaluated in order to use it in the thermal conductivity integral. Klaffky et al. [12] found evidence of a temperature-independent mobility for T < 300 K, for single crystal alumina under conditions where RIC was produced by continuous irradiation with 1.5 MeV electrons at dose rates up to 6.6 X 10’ Gy/s. Hughes [13] reported a temperature independent mobility of (3 f 1) x low4 m*/V s for an undoped aluminum sample X-ray irradiated between 100 and 350 K. In the calculations presented here it is assumed that this constant mobility holds to 77 K. The specific heat, C, of those phonons up to a cut off frequency of o, = 2v/h (2mk,T)‘/* is given, in the Debye approximation, by temperature 8,/T
x4 exp x
(exp x - lj* dx’ In these expression, m is the electron mass, n is the number of unit cells per unit volume and 0, is the phonon-electron cutoff temperature given by 0, = hoc/k, and x = ho/k,T. The expression given in Eq. (8) may be evaluated by numerical integration for any particular temperature and then this value may be used in the relaxation time of Eq. (7). It is possible to calculate the changes in the thermal conductivity due to changes in u by numerically evaluating the Callaway thermal conductivity integral of Eq. (1) for different values of D in the phonon-electron relaxation time.
3. Results and discussion The results of the thermal conductivity calculations are given in Figs. l-4. In Fig. 1 the calculated thermal conductivity versus temperature is plotted for several values of the radiation produced vacancy concentration. The largest reduction in the thermal conductivity due to these vacancies is in the region of the wnductivity maximum at approximately 30 K. The curves in Fig. 1 compare favorably with the low temperature measurements of Berman et al. [14] and of Salce et al. [15], for neutron-irradiated sapphire. For example, Berman et al. found that the thermal conductivity maximum for a sapphire sample irradiated to a dose of 50.2 x lO*l n/m* (_ 0.005 dpa) was reduced from a nonirradiated value of approximately 8000 W/m K to about 400 W/m K. From Fig. 1 it is seen that the same fractional reduction occurs at a vacancy concentration of approximately 0.002 per atom. Fig. 2 is a plot of the thermal conductivity versus vacancy concentration at 77 K. This plot shows the reduction in the thermal conductivity at liquid nitrogen temperatures, where it has been proposed to cool ECRH microwave windows to take advantage of the high thermal conductivity and low loss
(K)
Fig. 1. Plot of the thermal conductivity of single crystal alumina versus temperature. The different curves correspond to different vacancy concentrations, where c is the vacancy concentration per atom.
tangent of alumina at these temperatures [7]. In Fig. 3 the fractional change in the thermal conductivity versus RIC is plotted. The three different curves correspond to different values of the electron mobility, covering the range of uncertainty in the electron mobility. From this plot it can be seen that at the highest values of the RIC plotted (lo-* (fl m)-‘), the change in the thermal conductivity approaches a maximum of about 9%. This value corresponds to the case of all phonons below the cut off frequency being strongly scattered, and thus is the limiting value of the reduction in the thermal conductivity which can occur at this temperature due to phonon-electron scattering. In Fig. 4 the fractional change in the thermal conductivity is plotted
loo0 F
1o-7
1o-3 1o-4 1o-5 1o-G vacancy concentration (per atom)
Fig. 2. Plot of the thermal conductivity of single alumina versus vacancy concentration at 77 K.
1o-2 crystal
D.P. White/Journal
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of Nuclear Materials 212-215
(1994) 1069-1074
reduced by vacancies at low temperatures. From Fig. 1 it is seen that the major part of the reduction is in the area of the thermal conductivity maximum, which oc-
10.'
No
1o-3
1o-5
MC (ohm-m)-’
Fig. 3. Fractional change in the lattice thermal conductivity versus RIC at 77 K. Each curve corresponds to a different value of the electron mobility, covering the range of uncertainty in the mobility [13]. The upper curve corresponds to p = 2 x 10m4 m*/V s, the center curve to p = 3 x low4 m2/V s, and the lower cume to p = 4 x 10m4 m2/V s.
as a function of vacancy plots the fractional change
concentration. This figure with the RIC included and
with no RIC included in the calculation. The value of the RIC chosen in this plot is lo-’ (fi m)-‘. This value was chosen because it represents the maximum of the range of expected prompt RIC values in sapphire in a fusion environment [3]. It is only at vacancy concentrations less than lo-’ per atom where there is any difference between the two cases. The thermal conductivity of sapphire is significantly
loo
“““‘I
“““I
“/““1
““‘I
“I’
-I
t 1o-4 ’ 10.’
“““’ 1o-6
“‘,“‘N
“““”
1o-5
vacancy concentration
“““’
1o-4
1oe3
““’
’ 10-l
(per atom)
Fig. 4. Fractional change in the thermal conductivity versus vacancy concentration at 77 K. The upper curve corresponds to a RIC of 10P5 (fi ml-‘, and the lower curve corresponds to the case of no RIC.
curs at about 30 K. There is however significant reduction in the thermal conductivity at liquid nitrogen temperature, as can be seen in Fig. 2. From Fig. 4 it can be seen that the fractional change in the thermal conductivity for a vacancy concentration of 0.01 per atom is expected to be about 90% at 77 K. This is a very large reduction in the lattice thermal conductivity and it raises the question as to whether or not this reduction negates the increase in the thermal conductivity gained by going to low temperatures. In Refs. [1,3] it was shown that at 400 K the fractional reduction in the thermal conductivity due to a vacancy concentration of 0.01 per atom would be expected to be approximately 39% which is a much smaller fractional decrease in the lattice thermal conductivity compared to the 90% reduction expected with the same defect concentration at 77 K. The larger fractional reduction at low temperatures is due to the fact that the umklapp scattering is much weaker at low temperatures and allows the point defect scattering to be the dominant scattering mechanism at lower phonon frequencies, thus scattering a larger fraction of the phonon spectrum. However at 400 K a 39% reduction in the thermal conductivity would result in a conductivity of approximately 16 W/m K, compared to an expected conductivity of 83 W/m K after a 90% reduction in the conductivity at 77 K. Thus even at defect concentrations as high as 0.01 per atom the thermal conductivity advantage gained by going to lower temperatures would be maintained. However, as pointed out in the next section, it may be that the thermal stability of point defects at 77 K versus 400 K is the deciding factor that determines which irradiation temperature has a higher thermal conductivity for a given fluence of vacancy-producing radiation. The maximum of the range of prompt RIC values in sapphire in a fusion environment is expected to be on the order of lo-’ (a ml-‘. Using this value for the RIC the corresponding range for the fractional change in the thermal conductivity is from 0.05 to 0.2% at 77 K. It was found [1,3] that at 300 K the expected fractional change in the conductivity at this value of the RIC was about 0.01%. Thus the effect of phononelectron scattering on the thermal conductivity is larger at low temperatures but these are very small changes in the thermal conductivity and are not expected to be of importance in the design of microwave windows. Even in the extreme case of very large values of the RIC (lo-’ (a ml-‘) the maximum effect the phonon-electron interaction can have on the thermal conductivity is about 9%, as pointed out in the previous section. It should also be noted that there is essentially no change in the thermal conductivity due to the small electronic
D.P. White /Journal
component 131.
of Nuclear Materials 212-215
added by the RIC, as pointed out in Ref.
4. Conclusions The reduction in the thermal conductivity due to the scattering of phonons by vacancies is more pronounced at low temperatures than at high temperatures. It has been shown however that the thermal conductivity advantage gained by going to low temperature is maintained. However the vacancy concentration depends on the thermal stability of the vacancies. If many of the vacancies recombine at 400 K but are stable at 77 K, then the low temperature thermal conductivity could be less than the 400 K thermal conductivity for a given fluence of vacancy producing irradiation. The work of Rohde and Schulz [16] has shown that irradiation temperature has an important effect on the number of remaining stable defects in irradiated alumina. This effect has also been seen in BeO, Pryor et al. [17] compared the thermal resistance per fission neutron dose of samples irradiated at 348 K with samples irradiated at 90 K by McDonald 1181.The thermal resistance of the samples irradiated at low temperature was about 16 times larger than the resistance of the high temperature samples for a given neutron dose. This problem points to the need of performing in-situ cryogenic thermal conductivity tests or postirradiation thermal conductivity measurements on samples irradiated at cryogenic temperatures. An important consideration if postirradiation measurements are to be performed is that there is evidence that significant annealing of defects may occur at room temperature. This was seen by McDonald [18] in Be0 which was irradiated at 95 K, after a final anneal at 310 K approximately 25% of the radiation induced thermal resistance had recovered. Further evidence that significant room temperature annealing effects are important in ceramics is given by Coghlan et al. [19] and Buckley et al. 1201.Coghlan et al. found that neutron-irradiated spine1 continued to swell after irradiation at room temperature, perhaps indicating continuing diffusion and clustering of defects. Buckley et al. found that the dielectric loss tangent of proton-irradiated ceramics recovered significantly on the timescale of hours at room temperature. Thus, in order to rule out postirradiation thermal effects on ceramics irradiated at low temperature it is important to maintain the samples at low temperature until the measurements are made. It should also be noted that the flexibility of ECRH transmission systems will allow the windows to be located in low fluence positions, with the aim of maintaining the low temperature thermal conductivity advantage. However the magnitude of tolerable fluence levels must still be identified experimentally.
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The changes in thermal conductivity due to phonon-electron scattering have been shown to be insignificant at the RIC levels expected in the fusion environment. These changes are particularly insignificant when the temperature dependence of the thermal conductivity at low temperatures is considered. In this low temperature region the thermal conductivity is strongly temperature-dependent; for example, in the region around 77 K calculations show that a 1% change in the temperature leads to a 4% change in the thermal conductivity. Thus the effects of temperature fluctuations on the thermal conductivity are expected to be at least as important as any effects due to phononelectron scattering. The small changes expected in the thermal conductivity due to phonon-electron scattering suggest that there is no need for in-situ cryogenic thermal conductivity measurements or postirradiation measurements of samples held at low temperature in order to study this effect. However this does not rule out the need for performing these tests in order to study the effects of thermally unstable radiation-produced defects.
Acknowledgment This research was performed while the author held a fellowship in the Oak Ridge National Laboratory Postdoctoral Research Program, administered through the Oak Ridge Institute for Science and Education.
References [l] D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/H, ORNL (1991) p. 277. [2] D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/12, ORNL (1992) p. 298. [3] D.P. White, J. Appl. Phys. 73 (1993) 2254. [4] D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/13, ORNL (1992) p. 326. [51 D.P. White, in Fusion Reactor Materials Semiannual Progress Report DOE/ER-0313/14, ORNL (19931, in press. [6] G.R. Haste, H.D. Kimrey and J.D. Proisie, Report ORNL/TM-9906 (1986). [7] R. Heidinger, J. Nucl. Mater. 179-181 (1991) 64. [8] J. Callaway, Phys. Rev. 113 (1959) 1046. [9] A.M. de Goer, J. Phys. (Paris) 30 (1969) 389. [lo] P.G. Klemens, in Solid State Physics, vol. 7, eds. F. Seitz and D. Turnbull (Academic Press, New York, 1958) p. 1. [ill C.A. Ratsifaritana, in Phonon Scattering in Condensed Matter, ed. H.J. Maris (Plenum, New York, 1980) p. 259. [12] R.W. Klaftky, B.H. Rose, A.N. Goland and G.J. Dienes, Phys. Rev B 21 (1980) 3610.
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[13] R.C. Hughes, Phys. Rev. B 19 (1979) 5318. [14] R. Berman, E.L. Foster and H.M. Rosenberg, in Report of the Conf. on Defects in Crystalline Solids, University of Bristol, 1954 (The Physical Society, London, 1955) p. 321. [15] B. Salce and A.M. de Goer, Proc. Conf. Digest of the Int. Conf. on Defects in Insulating Materials, Parma, 1988, p. 499.
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[16] M. Rohde and B. Schulz, J. Nucl. Mater. 173 (1990) 289. [17] A.W. Pryor, R.J. Tainsh and G.K. White, J. Nucl. Mater. 14 (1964) 208. [18] D.L. McDonald, Appl. Phys. Lett. 2 (1963) 175. [19] W.A. Coghlan, F.W. Clinard, Jr., N. Itoh and L.R. Greenwood, J. Nucl. Mater. 141-143 (1986) 382. [20] S.N. Buckley and P. Agnew, J. Nucl. Mater. 155-157 (1988) 361.