High Temperatures ^ High Pressures, 2002, volume 34, pages 607 ^ 616
DOI:10.1068/htjr085
Relative measurements of thermal conductivity of liquid
gallium by the transient hot-wire method
Amica Miyamura, Masahiro Susa
Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan;
fax: +81 357343141; email: [email protected]
Presented at the 16th European Conference on Thermophysical Properties, Imperial College, London,
England, 1 ^ 4 September 2002
Abstract. Thermal conductivity of liquid gallium as a function of temperature has been determined
by the transient hot-wire method with an alumina-coated probe. Theoretical analysis of the
temperature increase of the hot wire with coating indicates that the coating layer affects absolute measurements by this method and its presence is prone to yield smaller values of thermal
conductivity. Relative measurements were carried out over the temperature range 310 ^ 471 K
on liquid mercury and gallium at 310 K as standard samples to establish the correction line.
The thermal conductivities obtained for liquid gallium are as follows: 29:3 W mÿ1 Kÿ1 at
338 K, 30:5 W mÿ1 Kÿ1 at 379 K, and 31:8 W mÿ1 Kÿ1 at 471 K. These values are greater than
those derived from absolute measurements but smaller than the values predicted from the
Wiedemann ^ Franz law and smaller than several reported values.
1 Introduction
The thermal conductivity of liquid metals and alloys is one of the most important input
data for mathematical modelling of heat flow in high-temperature processes. Because of
this, many attempts have been carried out to measure thermal conductivities of liquid
metals (Mills et al 1997); however, extant data for liquid metals exhibit large scatter.
With respect to the thermal conductivity of liquid tin, for example, reported values
near the melting point range between 27 and 33 W mÿ1 Kÿ1 and the discrepancy between
the values further increases with an increase in temperature (Osipenko 1970; Zinovev
et al 1973; Hemminger 1985). This discrepancy would arise from the difference in the
measurement methods used, since physical property data often depend upon measurement methods. At the moment, it is very difficult to determine experimentally which is the
most reliable value, because there are no standard substances for thermal conductivity
measurements on liquid metals.
To obtain accurate thermal conductivities of liquid metals, it is very relevant to
correct experimental apparatuses by using standard samples the thermal conductivity
values of which are well known. In thermal conductivity measurements on organic
liquids, water, glycerol, toluene, etc are often used as calibrant liquids to correct experimental apparatuses (Powell 1991; Diguillo et al 1992). However, thermal conductivity
values of these calibrants are typically two orders of magnitude lower than those of
liquid metals and therefore these substances cannot be calibrants for measurements
on liquid metals. From the viewpoint of the magnitude of thermal conductivity, solid
metals such as nickel and iron could be used as calibrants. Actually a few workers tried
to correct their apparatuses by using these substances (Dusen 1922; Weeks and Seifert
1953), but there is the problem that published values of thermal conductivities of solid
metals also exhibit a large scatter, probably owing to the effects of crystal imperfections
such as dislocations and grain boundaries.
To avoid the effect of crystal imperfections, liquid metals could be recommended
as calibrants, and mercury and gallium, which are liquids even at room temperature,
are promising candidates for standard substances. There have been many data reported
on the thermal conductivity of mercury, and these values are in good agreement
608
A Miyamura, M Susa
with each other in the temperature range 300 ^ 573 K (Duggin 1969, Schriempf 1972;
Nakamura et al 1988; Brooks et al 1996). Also for gallium, there are several published
values of the thermal conductivity; however, the agreement among these values is not
very good, except for values near room temperature (Touloukian et al 1970; Schriempf
1973; Magomedov 1978). Accordingly, it is very important to determine accurate values
of the thermal conductivity of liquid gallium above room temperature so as to obtain
more reliable data for other liquid metals. Consequently, the aim of the present work has
been to measure the thermal conductivity of liquid gallium accurately as a function of
temperature by the transient hot-wire method as a relative technique. The transient hotwire method is one of the most reliable methods for measuring thermal conductivities
of liquid substances, because it has the advantage of avoiding the effects of convection
which is a serious problem in the measurements on liquids.
2 Experimental
2.1 Principle
In the transient hot-wire method, electric power is supplied to a thin metal wire (hot
wire) immersed at the centre of the sample, and the temperature rise of the wire, DT, is
recorded continuously. This temperature rise is related to the thermal conductivity of
the sample, l, as follows (Carslaw and Jaeger 1959):
DT ˆ
Q
…ln t ‡ A† ,
4pl
(1)
where Q is the heat generation rate per unit length of the hot wire, t is the time, and A
is a constant. In practice, the thermal conductivity of the sample is derived from the
slope of the linear portion of the relation between DT and ln t from equation (2):
Q
dDT
lˆ
.
(2)
4p d ln t
In the measurement on liquids, the linearity in the relation between DT and ln t guarantees absence of convection, and usually this applies to short-duration measurements. At
long durations, on the contrary, DT is affected by convection and therefore deviates
downwards from linearity.
When the hot-wire method is applied to electrically conducting materials such as
liquid metals, a considerable difficulty is encountered: there is electric leakage from the
hot wire to the sample. This problem can be overcome by coating the hot wire with a
thin insulating layer (Yamasue et al 1999, 2002). For the hot-wire method with an insulator-coated probe, instead of equation (1), the following equation obtains (Nagasaka
and Nagashima 1981, Yamasue et al 1999, 2002):
Q
1
DT ˆ
ln t ‡ A ‡ …B ln t ‡ C† ,
(3)
4pl
t
where B and C are constants which are determined by physical property values (thermal
conductivities and thermal diffusivities), and the dimensions of the hot wire, the coating
layer, and the sample. Comparison between equations (1) and (3) indicates that the
term (1=t)(B ln t ‡ C ) originates from the presence of the coating layer. For smaller
values of t, the term (1=t)(B ln t ‡ C ) gives a contribution to DT greater than the term
( ln t ‡ A), and this makes the effect of the coating layer more serious. On the other
hand, for larger values of t, the contribution to DT from the term (1=t)(B ln t ‡ C )
becomes progressively smaller than that from the other term, and equation (3)
approaches equation (1) even more closely as the time passes. Accordingly, calculations
based on data of long-duration measurements to which equation (2) applies could give
reasonable values of thermal conductivity. In the measurement on liquids, however, data
DT=K
Thermal conductivity of liquid metals
609
0.5 K
equation (1)
equation (3)
0.1
0.5
1
t=s
5
10
Figure 1. Changes with time in the DT calculated from equations (1) and (3).
Table 1. Values of the parameters at 310 K used for calculation for figure 1.
Sample (gallium)
Hot wire (Pt ± 13% Rh)
Coating layer (sintered alumina)
Thermal
conductivity=W mÿ1 Kÿ1
Thermal
diffusivity=m2 sÿ1
28.2
30.9
0.79
1:16 10ÿ5
1:38 10ÿ5
2:42 10ÿ7
obtained over long duration would be contaminated by convectional effects. Therefore
data obtained at short durations must be used for deriving the thermal conductivity, and
this requires the knowledge of the magnitude of contribution from the term
(1=t)(B ln t ‡ C ), namely the effect of the coating layer, at short measurement durations.
Figure 1 shows a comparison between changes with time in DT calculated from equations (1) and (3) on the following assumptions:
(i) The hot-wire probe consists of a Pt ^ 13% Rh wire of 0.15 mm diameter coated with
an Al2 O3 layer of 100 mm thickness.
(ii) The sample is liquid gallium at 310 K.
(iii) The current fed to the wire is 2.5 A.
Parameters used for this calculation are summarised in table 1. The thermal conductivity
of the hot wire is a value for Pt ^ 13% Rh reported by Molgaard and Smeltzer (1968)
and the thermal diffusivity has been estimated by interpolation of thermal diffusivity
values of Pt ^ 10% Rh and Pt ^ 20% Rh alloys provided by Tanaka Precious Metals. The
thermal conductivity of the sample is the value for gallium recommended by Mills et al
(1997) and the thermal diffusivity, k, has been derived by using the density, r, and heat
capacity, Cp data for gallium (Mathiak et al 1983; Barin 1989; respectively) in the relation
k ˆ l=rCp . The thermal conductivity of the coating layer is the value for sintered alumina slurry (Toh-a Gousei Kagaku) provided by its manufacturer, leading to the thermal
diffusivity obtained from the density and heat capacity data for Al2 O3 (Winchell et al
1964; Barin 1989; respectively).
Inspection of figure 1 indicates that the difference between values calculated from
equations (1) and (3) is larger for short-duration measurements but becomes smaller for
longer-duration measurements, as expected from the preceding discussion. On the basis
of DT calculated from equation (3), thermal conductivity values of liquid gallium are
derived from the slopes obtained from measurements over time periods between 1 s and
2 s and between 9 s and 10 s from equation (2), leading to thermal conductivity values of
26.8 and 28:0 W mÿ1 Kÿ1 , respectively. The latter value is in very good agreement with
610
A Miyamura, M Susa
the input value of the thermal conductivity of gallium, but the former value is smaller
by about 5%. This indicates that the coating layer produces a systematic error in thermal
conductivity values derived from data obtained in short-duration measurements, although
these data are more relevant in avoiding convectional effects.
On the basis of the above discussion, in the present work the transient hot-wire
method has been used as a relative technique (Powell 1991), to eliminate the effect of the
coating layer. This relative measurement is based upon the fact that dDT=d ln t is proportional to 1=l according to equation (2):
dDT
ˆ …Q=4p†…1=l† .
d ln t
(4)
Prior to the measurements, a probe with coating is applied to standard samples the
thermal conductivities of which are well known, and values of dDT=d ln t are obtained to
establish the relation between dDT=d ln t and 1=l, ie the correction line. Note that the
slope of the correction line determined experimentally corresponds to the proportional
constant Q=4p in equation (4), but should have a different value from Q=4p owing to
the effect of the coating layer. Measurements with the same probe are carried out on
samples the thermal conductivity values of which are unknown in order to obtain values
of dDT=d ln t. Values of thermal conductivity are then determined by using dDT=d ln t
from the correction line. Experimental details are given in the following section.
2.2 Measurements
Figure 2 shows a schematic diagram of the experimental apparatus for the transient
hot-wire method used in the present work. It consists of a hot wire (0.15 mm diameter,
40 mm long) of Pt ^ 13% Rh, serving as a temperature sensor wire as well, to which two
platinum potential leads (0.15 mm diameter) are attached spaced apart by about 20 mm
to allow four-terminal resistance measurements of the sensor wire. These wires are
connected to platinum lead wires of 0.5 mm diameter, which are supported by alumina
tubing. To prevent electric leakage, the probe is coated with an 80 ^ 100 mm thick alumina layer deposited from alumina slurry by the electrophoretic migration method
and then dried at 473 K for 24 h.
galvanostat
crucible
digital multimeter
alumina tube
sample
Pt wire, diameter 0.15 mm
(potential leads)
Pt-13% Rh wire, diameter 0.15 mm
(hot wire)
Pt wire, diameter 0.5 mm
Figure 2. Schematic diagram of experimental apparatus.
The samples used were gallium of 99.9999% purity and mercury of 99.9% purity,
and these substances at 310 K were employed as standard samples to make the correction
line for the relative measurement, because values recorded at this temperature show
satisfactory agreement with each other for both substances, as seen in table 2. In the
present work, values recommended by NPL (Mills et al 1997) were used as standard
values for gallium and mercury at 310 K since they are consistent with other experimental values. The thermal conductivity values for gallium and mercury at 310 K are 28.2
and 7:72 W mÿ1 Kÿ1 , respectively.
Thermal conductivity of liquid metals
611
Table 2. Thermal conductivities of mercury and gallium at 310 K.
Authors
Mercury,
l=W mÿ1 Kÿ1
Authors
Gallium,
l=W mÿ1 Kÿ1
Schriempf (1972)
Nakamura et al (1998)
Brooks et al (298 K) (1996)
NPL recommended value
(1997)
7.79
7.97
7.8
7.72
Touloukian et al (1970)
Magomedov (1978)
Schriempf (1973)
NPL recommended value
(1997)
28.1
28.4
28.4
28.2
Measurements were carried out in the temperature range 310 ^ 471 K in the following
manner. The sample was placed in a cylindrical vessel (26 ^ 30 mm diameter, 150 mm
deep) in a water or oil bath, depending on temperature. The temperatures of the baths
were controlled within 0:1 K and 1 K, respectively, by PID control. The use of the
baths led to uniform temperature distribution in the sample, which reduced convectional
effects. For further reduction of the convectional effects, the vessel was positioned in the
bath so that the temperature at the sample surface was higher by about 2 K than that
at its bottom. The temperature was measured at the sample surface with a Chromel ^
Alumel thermocouple with an accuracy of 1 K.
The hot-wire probe was placed in the centre of the sample, and the current (2.5 ^ 3 A)
was fed to the hot wire via a galvanostat. The temperature rise, DT, of the hot wire was
continuously monitored as the voltage change, DV, between the potential wires of the
hot wire, on the basis of the principle of four-terminal resistance measurements.
Accordingly, instead of dDT=d ln t, the value of dDV=d ln t was determined from the
slope of the linear portion of the relation between DV and ln t, and equation (4) was
also modified to read:
dDV I 3 aT R273 XT 1
. ,
ˆ
d ln t
l
4p
(5)
where I is the supplied current, aT is the temperature coefficient of electric resistivity of
the hot wire at the absolute temperature T, R273 is the resistance between the potential
leads of the hot wire at 273 K, and XT is the resistance per unit length of the hot wire at
T. In accordance with equation (5), values of dDV=d ln t obtained in the measurements
on liquid mercury and liquid gallium at 310 K were plotted against 1=l to establish the
correction line. Furthermore, when this was applied to temperatures different from
310 K, the slope of the correction line was modified by taking into account changes in
the temperature-dependent terms in equation (5), such as aT and XT. In this way, values
of the thermal conductivity of gallium at higher temperatures were determined on the
basis of the modified correction line by using measured values of dDV=d ln t. In addition,
an absolute measurement was also carried out for comparison, in which the thermal
conductivities were derived directly from equation (5) with the use of values of dDV=d ln t.
3 Results and discussion
3.1 Correction line
Figure 3 shows a typical voltage change, DV, as a function of the natural logarithm of
time, ln t, obtained in measurements on mercury at 310 K with a current of 2.5 A. It can
be seen that there is a linear relation between DV and ln t in the time period 0.7 s ^ 1.5 s,
and that DV deviates from linearity in the time periods below 0.7 s and above 1.5 s. The
deviation in the former period is due to complex effects of the IR drop of the hot wire
and the heat capacities of the hot wire and the coating layer, whereas in the latter period
it is due to the effect of convection. The value of dDV=d ln t is obtained from the slope
of the linear portion and thereby convectional effects are avoided. This value is used to
612
A Miyamura, M Susa
0.654
DV=V
0.653
0.652
0.651
0.650
0.1
0.5
1
5
10
t=s
Figure 3. Typical voltage change as a function of time for mercury at 310 K.
DV=V
0.645
0.644
0.643
0.642
0.1
0.5
1
t=s
5
10
Figure 4. Typical voltage change as a function time for gallium at 310 K.
make a correction line for relative measurements. On the other hand, on using the
absolute technique, the value of dDV=d ln t leads to a thermal conductivity value of
7:46 W mÿ1 Kÿ1 calculated from equation (5). However, this value is lower than those
in table 2 and contains a systematic error arising from the coating layer, as mentioned in
section 2.1.
Figure 4 shows a typical DV as a function of ln t obtained in the measurement on
gallium at 310 K, with the current of 2.5 A. The value of dDV=d ln t for gallium is determined from the slope of the linear portion in the time period 0.7 s ^ 3 s, and is also used
to make a correction line for the relative measurement. It seems that the onset of convection is delayed, as compared with figure 3 for mercury. This would be so because the
temperature gradient in gallium was smaller than that in mercury because of the higher
thermal conductivity of gallium. In the absolute measurement, the value of dDV=d ln t
leads to a thermal conductivity value of 25:8 W mÿ1 Kÿ1 obtained from equation (5).
The values of dDV=d ln t thus obtained are plotted against the reciprocal of the
corresponding thermal conductivities (1=l), resulting in the correction line shown in
figure 5, where values of 7.72 and 28:2 W mÿ1 Kÿ1 have been used as the thermal conductivities of mercury and gallium at 310 K, respectively. Figure 5 also includes the
theoretical relation between dDV=d ln t and 1=l predicted from equation (5). The theoretical line passes through the origin and has a slope of 5:41 10ÿ3 W V mÿ1 Kÿ1 under the
following conditions: I ˆ 2:5 A, aT ˆ 1:54 10ÿ3 Kÿ1, R273 ˆ 0:24 O, XT ˆ 11:7 O mÿ1,
as calculated from equation (5). On the other hand, the correction line has an intercept
on the ordinate, and its slope can be obtained as 5:35 10ÿ3 W V mÿ1 Kÿ1 from figure 5, which is smaller than the slope of the theoretical line. Because of these differences,
thermal conductivities derived from the theoretical lineöthis corresponds to the absolute measurementöare smaller than those derived from the correction line. The same is
Thermal conductivity of liquid metals
613
dDV=d ln t=104 V
8
6
4
2
0
correction line
theoretical line
0
0.05
0.1
(1=l)=m K Wÿ1
0.15
Figure 5. Correction line.
true for other hot-wire probes. This confirms that the absolute measurement is affected
by the coating layer, because the presence of the coating layer leads to reduced values
of thermal conductivity, as mentioned in section 2.1.
As a consequence, we can represent the correction line in the form:
dDV
I 3 aT R0 XT 1
. ‡D ,
ˆC
4p
d ln t
l
(6)
where C and D are probe constants, depending on the coating layer. The correction line
in figure 5 applies only to measurements at 310 K but can be extended to measurements
at other temperatures by introducing changes in the temperature-dependent terms in
equation (6). Substitution of aT and XT for a310 and X310 produces a new correction line
for the absolute temperature T, with the probe constants assumed to be independent of
temperature.
0.722
0.807
0.721
0.806
DV=V
DV=V
3.2 Thermal conductivity of liquid gallium
Figures 6a and 6b show typical voltage changes as functions of ln t obtained in the
measurements on liquid gallium at 379 K and 471 K, respectively, where the current fed
to the wire was 2.5 A. Even in measurements at higher temperatures, linear portions are
obtained in the time period 0.9 ^ 2.0 s. At longer durations, DV deviates upwards from
linearity. This is due to the effect of heat reflection at the vessel wall, since vessels having
smaller diameter were used in these measurements and, furthermore, the thermal conductivity of gallium increases with increasing temperature. In figure 6a, for example, the
slope of the linear portion gives a value of 28:9 W mÿ1 Kÿ1 on the basis of the correction
line for 379 K when using the relative technique, and another value of 26:6 W mÿ1 Kÿ1
from equation (5) when using the absolute technique. The former value is larger by about
8% than the latter.
0.720
0.719
0.718
0.1
(a)
0.805
0.804
0.5
1
t=s
5
0.803
0.1
10
(b)
0.5
1
t=s
5
10
Figure 6. Typical voltage changes as functions of time for gallium at (a) 379 K and (b) 471 K.
614
A Miyamura, M Susa
50
Magomedov (1978)
Schriempf (1973)
l=W mÿ1 Kÿ1
absolute technique
relative technique
Peralta et al (2000)
Gamazov (1979)
Touloukian et al (1970)
Mills et al (1997)
Wiedemann ^ Franz
40
30
20
300
400
500
600
T=K
Figure 7. Thermal conductivities of liquid gallium.
Table 3.Values of thermal conductivity, l, and thermal diffusivity, k, at 471 K used for the calculations.
Sample (gallium)
Hot wire (Pt-13% Rh)
l=W mÿ1 Kÿ1
k=m2 sÿ1
38.4
41.2
1:67 10ÿ5
1:65 10ÿ5
Figure 7 shows thermal conductivity values of liquid gallium determined both by
relative and by absolute techniques as functions of temperature, along with several
published values. Values predicted from the Wiedemann ^ Franz law are also included, for
which electric conductivity values reported by Monaghan (1999) have been used. The
values derived from relative measurements in the present work are greater by 5% ^ 10%
than those from absolute measurements. This is because relative measurements can
eliminate the effect of the coating layer. The scatter in the thermal conductivity values
from relative measurements is 4% at most. It has also been found that the thermal
conductivity values are not dependent on the supplied current.
The thermal conductivity values from relative measurements are in good agreement
with reported values near room temperature, and both increase as temperature increases,
but the discrepancy between them becomes larger at higher temperatures. At higher
temperatures, the values from relative measurements are very close to those reported by
Gamazov (1979) but smaller than those reported by Magomedov (1978) who used the
axial heat flow method, Schriempf (1973) who used the laser pulse method, and Peralta
et al (2000) who used the transient hot-wire method; and also smaller than those recommended by Touloukian et al (1970) and Mills et al (1997), and the ones predicted from
the Wiedemann ^ Franz law. To explain the discrepancy between the present and the
reported values, two reasons can be considered: (i) the values derived from relative
measurements are still affected by the coating layer, and (ii) the values reported in the
literature are affected by convection. In the present work, the probe constants in equation (6) have been assumed to be independent of temperature, when the correction line
for 310 K is converted to correction lines for higher temperatures. However, if this
assumption is not reasonable and the thermal conductivity and diffusivity of the coating
layer decrease drastically with increasing temperature, there is a possibility that the
relative technique is also affected by the coating layer at higher temperatures.
To investigate this possibility, the calculation based on equation (3) has been carried
out again. Let us assume that the true thermal conductivity of gallium at 471 K is
Thermal conductivity of liquid metals
615
38 W mÿ1 Kÿ1 , but the apparent value has been measured as 31 W mÿ1 Kÿ1 by the
present method, owing to the effect of the coating layer. In this calculation, various
values are substituted for thermal conductivity and diffusivity of the coating layer given
in table 3, whilst other parameters are the same as those given in table 1. This calculation
gives DT as a function of ln t, which is used to derive the apparent thermal conductivity
of gallium from the slope obtained in the time period 1 s ^ 2 s from equation (2). As a
result, values of thermal conductivity and diffusivity of the coating layer should be
smaller by a factor of about 0.3 than those given in table 1 in order to obtain an apparent
thermal conductivity value of 31 W mÿ1 Kÿ1 for gallium. However, it is unlikely that
the thermal conductivity of ceramics would show such a steep decrease with a temperature increase of only 160 K. It is therefore difficult to regard the values derived from
relative measurements as being still affected by the coating layer. Therefore the values
reported in the present work are likely to be more accurate, and the reported values are
probably affected by convection.
4 Conclusions
The thermal conductivity of liquid gallium has been measured by the transient hot-wire
method with an alumina-coated probe as a relative technique over the temperature range
310 ^ 471 K, with liquid mercury and gallium at 310 K used as standard samples. Theoretical analysis has also been attempted to investigate the effect of the coating layer on
absolute measurements.
(i) The coating layer affects absolute measurements by this method and its presence is
prone to yield smaller values of thermal conductivity.
(ii) The thermal conductivities derived from relative measurements are as follows:
29:3 W mÿ1 Kÿ1 at 338 K, 30:5 W mÿ1 Kÿ1 at 379 K, and 31:8 W mÿ1 Kÿ1 at 471 K.
(iii) The above values are greater than those derived from absolute measurements but
smaller than the values predicted from the Wiedemann ^ Franz law, and smaller than
several values reported in the literature.
References
Barin I, 1989 Thermochemical Data of Pure Substances (New York, Weinheim: VCH)
Brooks R F, Monaghan B J, Barnicoat A J, Mccabe A, Mills K C, Quested P N, 1996 Int. J.
Thermophys. 19 1151 ^ 1161
Carslaw H S, Jaeger J C, 1959 Conduction of Heat in Solids 2nd edition (Oxford: Clarendon
Press)
Diguillo R M, McGregor W L, Teja S, 1992 J. Chem. Eng. Data 37 242 ^ 245
Duggin M J, 1969, in Proceedings of the 8th International Thermal Conductivity Conference
Eds C Y Ho, R D Taylor (New York: Plenum) pp 727 ^ 735
Dusen M S, 1922 J. Opt. Soc. Am. 6 739 ^ 743
Gamazov A A, 1979 Sov. Phys. J. 22 113
Hemminger W, 1985 High Temp. ^ High Press. 17 465 ^ 468
Magomedov A M, 1978 Tezisy Nauchn. Soobshch. Vses. Konf. Str. Svoistvam Met. Shlakovykh
Rasplavov, 3rd volume 2, pp 21 ^ 24
Mathiak E, Nistler W, Waschkowski W, Koester L, 1983 Z. Metallkde. 74 793 ^ 796
Mills K C, Monaghan B J, Keene B J, 1997 Thermal Conductivities of Molten Metals part 1 Pure
Metals (Teddington, UK: National Physical Laboratory)
Molgaard J, Smeltzer W W, 1968 J. Less-Common Met. 16 275 ^ 278
Monaghan B J, 1999 Int. J. Thermophys. 20 677 ^ 690
Nagasaka Y, Nagashima A, 1981 Trans. Jpn. Soc. Mech. Eng. B 47 1323 ^ 1331
Nakamura S, Hibiya T, Yamamoto F, 1988 Rev. Sci. Instrum. 59 2600 ^ 2603
Osipenko V P, 1970 Sov. Phys. J. 12 1570 ^ 1573
Peralta V, Dix M, Wakeham W A, 2000, in Thermal Conductivity 25 (Thermal Expansion 13)
pp 333 ^ 339
Powell J S, 1991 Meas. Sci. Technol. 2 111 ^ 117
Schriempf J T, 1972 High Temp. ^ High Press. 4 411 ^ 416
Schriempf J T, 1973 Solid State Commun. 13 651 ^ 653
616
A Miyamura, M Susa
Touloukian Y S, Powell R W, Ho C Y, Klemens P G (Eds), 1970 Thermal Conductivity of Metallic
Elements and Alloys volume 1 of Thermophysical Properties of Matter Eds Y S Touloukian,
C Y Ho (New York: IFI/Plenum)
Weeks J L, Seifert R L, 1953 Rev. Sci. Instrum. 24 1054 ^ 1057
Winchell A N, Winchell H, 1964 The Microscopical Characters of Artificial Inorganic Solid Substances (New York: Academic Press)
Yamasue E, Susa M, Fukuyama H, Nagata K, 1999 Metall. Mater. Trans. A 30 1971 ^ 1979
Yamasue E, Susa M, Fukuyama H, Nagata K, 2002 J. Cryst. Growth 234 121 ^ 131
Zinovev V E, Baskakova A A, Korshunova N G, Baronikhina N A, Zagrevin L D, 1973 Inzh.
Fiz. Zh. 25 490 ^ 494
ß 2002 a Pion publication