J. Phys. F: Metal Phys., Vol. 4, April 1974. Printed in Great Britain. @ 1974.
LETTER TO THE EDITOR
The low-temperature thermal conductivity of
niobium-zirconium alloys
N Morton, B W James, G H Wostenholm and R J Nicholls
Department of Pure and Applied Physics, University of Salford, Salford, M5 4WT,
Lancs
Received 11 February 1974
Abstract. The thermal conductivities of two samples of nominal 75 % Nb-25 % Zr alloy
have been measured below 20 K. The measured conductivities are substantially higher
than found previously for segregated niobium-zirconium alloys, and the phonon
carrier component is found to be completely dominant at all temperatures. The effective
number of electrons scattering the phonons is unusually small.
The measurement and analysis of the low-temperature thermal conductivities of a series
of superconducting niobium-zirconium alloys have been described recently by Morton
er a1 (1973). These alloys were quenched from 1500 "C and annealed at 800 "C to induce
segregation into two distinct phases. The matrix and inclusion phases probably contained
about 17 % and 66 % of zirconium atoms respectively, providing a magnetic flux pinning
structure able to support high-density supercurrents. We report here further measurements, between 4 and 20 K, made by the technique employed previously, for two strip
samples of nominally single-phase 75 %Nb-25 %Zr alloy, one in the as-rolled condition,
and the second following an anneal for 1 h at 1500 "C and an oil quench. The measurements are estimated to have a typical uncertainty of & 7 %.
The measured thermal conductivities are shown in figure 1, and comparison shows that
the present values are approximately 20 times higher than for the two-phase samples
measured previously. This large increase was unexpected, although some variation was
proposed by Morton (1968) to account for the widely varying resistance to flux jumping
instabilities of these alloys. Additional measurements of the low-temperature electrical
resistivity and superconducting critical temperature Tc were made by a four-terminal
method. The estimated residual resistivity po and Tc values are shown in table 1. A
calculation employing the resistivity data and the Wiedemann-Franz law shows that the
electron carrier contribution amounts to no more than approximately 3 % of the total
thermal conductivity, and so it may reasonably be neglected in any further analysis.
Accordingly, the low-temperature thermal conductivity K g is due to phonon carriers,
and may be represented as a function of temperature by the expression
Kg = (AT-2h-l
CmP)-I
(1)
+2
m
where A and C, are constants which characterize the electron and defect scattering
mechanisms respectively, A is related to the effective electron per atom number N , the
Debye temperature OD, and the electronic component of the thermal conductivity at
high temperature Kern by the expression
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Letter to the Editor
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60
-
7
6 50V
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- 0
'
IO
5
0
15
T(K)
Figure 1. Variation of thermal conductivity Kg as a function of temperature for quenched
(open circles) and as-rolled (full circles) 75 %Nb-25 %Zralloy samples. The full curves
represent the behaviour of equation (3) above the critical temperature.
Table 1
Sample
treatment
Te (K)
PO
Quenched
10.71
As-rolled
1046
(@2 m)
A (W-I m dega)
CI x lo4 (W-1 m)
0.19
2.22 & 0.07
5.30 f 0.16
0.26
2.21 & 0.18
4.10 f 0.38
A = 0*224N28~2~,co-1.
(2)
Within the range of temperatures used here, the likely scattering defects are point defects
(mainly the zirconium atoms) and dislocations, for which m takes the values 1 and -2
respectively. The factor h is a unique function of the reduced temperature TTc-l,which is
unity for temperatures above Tc. Klemens and Tewordt (1964) have calculated values for
h, but these appear to be much too high for the extreme type 11 superconductive niobiumzirconium alloys, although satisfactory for niobium-molybdenum alloys according to
the work of Sousa (1969).
For the normal-state region, rearranging equation (1) yields the expression
T 2 ~ - 1=
(A
+ 12-2) + CIT~.
(3)
Hence graphs of T2tcg-1 against T3 should be linear, as shown plausibly in figure 2. Since
dislocations will not be present in the quenched sample, the intercept of the graph yields
a value for A directly, and inspection shows no noticeable increase in the intercept
value for the as-rolled sample. Accordingly C-2 = 0 for both samples. The graph for
the quenched sample, on the other hand, exhibits a greater slope, corresponding to an
increase in C1. This effect may possibly be associated with a more completely random
distribution of the zirconium atoms in this sample, with a correspondingly shorter mean
free path and enchanced scattering of phonons. Values for A and C1, and the corresponding standard errors, determined from a least-mean-squares program, are shown in
table 1. The curves shown in figure 1 at temperatures above T, are drawn with these
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Letter to the Editor
1
7t
I
1
2
I
I
3 4 5
IO-' T' (K')
1
6
7
Figure 2. Graphs of T 2 ~ g against
-1
T8above the critical temperature. The straight lines
represent least-mean-square fits to equation (3), omitting one of the full circle data
points.
values, and agree tolerably well with the data. The main difference between the present
results and those obtained previously lies in the much smaller value of the constant A as
compared with values in the region of 100 W-1 m deg3 for two-phase niobium-zirconium
alloys and approximately 330 W-l m deg3 for niobium obtained by Sousa (1969). Since
OD changes relatively little in the alloy system according to Morin and Maita (1963), and
Keto is approximately constant in most alloy series (due essentially to a close adherence
to Matthiessen's rule), this suggests, following equation (3), that Nchanges by an order of
magnitude. In principle N may be derived from band structure calculations in a similar
way to the density of electron states, and it is suggested that detailed calculations would
be of value. Thus, in the present case, the very low value of N for single-phase 75 %Nb25 %Zr alloy compared with pure niobium appears to be related to the well documented
peak in the density of states occurring for niobium-zirconium alloys
The factor h has been calculated from equation (1) assuming the values for the
constants given in table I , and the results are shown in figure 3. The low values obtained
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
09
T 7E-'
Figure 3. Variation of the factor h as a function of reduced temperature.
Letter to the Editor
L97
are close to those found previously for two-phase niobium-zirconium alloy, and are
orders of magnitude below the theoretically predicted figures.
References
Klemens P G and Tewordt L 1964 Rev. mod. Phys. 36 118-20
Morin F J and Maita J P 1963 Phys. Reo. 129 1115-20
Morton N 1968 Cryogenics 8 79-81
Morton N, James B W, Wostenholm G H, Sanderson R J and Black M A 1973 Cryogenics 13 665-70
Sousa J B 1969 J. Phys. C: Solid St. Phys. 2 629-39