Supercond. Sci. Technol. 9 (1996) 453–460. Printed in the UK
Parametrization of the niobium
thermal conductivity in the
superconducting state
F Koechlin and B Bonin
CEA, DSM/DAPNIA/Service d’Etudes d’Accelerateurs C E Saclay,
91191 Gif sur Yvette, France
Received 5 February 1996
Abstract. Thermal conductivity measurements of niobium sheets manufactured for
deep-drawing of superconducting cavities have been gathered. Due to the differing
histories of the niobium samples and a wide range of metal purities
(35 < RRR < 1750), the data demonstrate a large scatter of thermal conductivities.
An attempt was made to obtain an analytical expression with realistic parameters
for the thermal conductivity between 1.8 K and 9.25 K. The set of parameters
deduced from a least square fit of experimental data is not very different from those
yielded by the theory of superconducting metals, taken as a starting point. This
should make it possible to obtain a reasonable estimate of the thermal conductivity
of niobium in this temperature range, once the RRR and the past history of the
metal samples had been determined.
1. Introduction
In recent years, industrial suppliers of niobium have made
an increasing effort to produce very pure niobium. The
main motivation was the increasing use of superconducting
RF cavities in particle accelerators. The thermal conductivity of the metal is an important criterion, which determines
the thermal stability of the superconductor. Consequently,
several laboratories, as well as manufacturers, have studied
the improvement of niobium purity by various processes,
including vacuum melting by electron bombardment and
post-purification by solid-state gettering. We have gathered
a large experimental database of niobium thermal conductivities at cryogenic temperatures and tried to compare these
results with a theoretical model.
We then attempted to adapt a semi-empirical parametrization of the thermal conductivity profile. The aim
of this work is to demonstrate that the thermal conductivity of niobium can be estimated rather precisely from two
sets of information, namely the material residual resistivity
ratio (RRR) and the knowledge of the material history (annealings, cold working . . .). We believe that the resulting
parametrization will be useful for thermal model calculations describing the behaviour of various superconducting
niobium devices, including RF cavities for accelerators.
The theory of thermal conductivity will be summarized,
and the analytical form chosen for the parametrization
will be justified in the first part of the paper. The
experimental database will be discussed in the second and
the fit procedure in the third.
c 1996 IOP Publishing Ltd
0953-2048/96/060453+08$19.50 2. Theoretical model for niobium thermal
conductivity
The thermal conductivity of pure niobium at temperatures
lower than 9.25 K is given by the theory of metals, modified
in order to take into account the superconducting state. The
thermal conductivity of a metal is dominated by that of the
electron ‘gas’, but the lattice heat conductivity due to the
propagation of phonons, has to be added. This can be
written as:
K = Ke + Klattice .
The suffix g is used for the lattice and a suffix n or s is
added in order to specify the normal or superconducting
state.
2.1. Electron heat conductivity
2.1.1. Normal state. The dominant term in the thermal
conductivity is usually that due to electron conduction,
given for the normal state by [1, 2]:
Ken = 1/Wen = [ρ/L0 T + a · T 2 ]−1
where Wen is the electron thermal resistance term, ρ, the
residual resistivity, ρ = ρ295 K /RRR, L0 , the Lorentz
constant (L0 ≈ 2.45×10−8 W K−2 ) and a is the coefficient
of momentum exchange with the lattice vibrations. The
first term in the expression for thermal resistance is due
to the scattering of electrons by impurities and lattice
defects whereas the second term describes the scattering
of electrons by phonons.
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F Koechlin and B Bonin
Figure 1. The ratio of Kes /Ken = R (y ) as a function of reduced temperature, T /Tc = α/y .
The simple model of an electron gas in metals
(Wiedeman & Franz) gives a value of L0 ≈ 2.45 ×
10−8 W K−2 for the Lorentz constant. A more refined
theory for metals [1], verified by experiments, yields lower
values for L, depending upon temperature and metal purity.
For this reason, we shall consider L as a free parameter in
the fit calculation.
The theoretical value for a is given by the theory of
phonons [1]
a = (6/π )2
2
1/3
Na2/3 I5 (∞)
· [Ke (295 K) ·
22D ]−1
where Na is the effective number of conduction electrons
per atom, I5 (x) the fifth order Grüneisen integral (I5 (∞) =
124.4) and 2D is the Debye temperature of the metal. For
niobium, Na = 1, 2D = 275 K and Ke (295 K) = 54
W m−1 K−1 , giving a = 2.3 × 10−5 m W−1 K−1 . Values
of a can be deduced from experiment by measuring the
coordinates of the maximum of Ken . By measuring several
samples, we have found a = 7 × 10−6 for niobium.
Experimental values of a found by other authors [2, 3]
for various metals of roughly 3 to 10 times smaller than
the above theoretical estimate. As no confirmed value can
be assigned to a, this will be a free parameter in the fit
procedure.
2.1.2.
Superconducting state. The electron heat
conduction in the superconducting state is reduced because
the electrons which have condensed into Cooper pairs do
not contribute to any disorder or entropy transport any
more. The remaining fraction of electrons do contribute
to heat transport, but this decreases very rapidly with
454
decreasing temperature. This decrease in electron heat
conduction has been calculated by Bardeen et al [4] using
the BCS theory [5], when the thermal resistance is mainly
due to impurity scattering. It amounts to a factor
Kes /Ken = R(y)
R(y) ≤ 1
−1
R(y) = (f (0)) [f (−y) + y ln(1 + exp(−y))
+y 2 /(2(1 + exp(y)))]
y = 1(T )/kB T = (1(T )/kB Tc )(Tc /T ).
The approximation y ∼
= αTc /T is valid if T /Tc ≤ 0.6
Z ∞
zdz
f (−y) =
1 + exp(z + y)
0
with f (0) = π 2 /12 [20].
Here, Tc , is the superconductor critical temperature,
1(T ), the superconductor energy gap, kB , the Boltzmann
constant and α ≈ 1.76 in the BCS theory, but may take
values in the range 1.5 ≤ α ≤ 2 according to experiments
carried out with metals [7].
The ratio Kes /Ken = R(y) is plotted as a function of
T /Tc in figure 1. This ratio takes on very small values
when T /Tc becomes smaller than 0.3, such that electron
heat conductivity is dramatically reduced, as compared to
a metal in the normal state.
2.2. Lattice thermal conductivity
The lattice thermal conductivity is well known in the case
of insulators, since there is no electron conduction, and
particularly in the case of monocrystalline samples. In
Parametrization of the niobium thermal conductivity in the superconducting state
Figure 2. Experimental data: thermal conductivity of ‘as supplied’ niobium samples.
the case of metals, it is usually negligible compared to
the electron contribution. However, in the superconducting
state, due to the quasi vanishing electron heat conduction
at temperatures below T /Tc = 0.3 (figure 1), the lattice
heat conduction obviously plays a role and is observed
experimentally. The thermal excitation of the lattice
is usually represented by a spectrum of phonon waves.
Several mechanisms have been proposed in order to explain
the scattering of phonons in metals, but, in the case of
polycrystalline niobium, it has been difficult to find a
satisfactory model [6]. The lattice thermal conductivity is
limited by two phenomena:
(i) The scattering of phonons by the electrons where,
in a normal metal, the thermal resistance Wgen due to
interaction with electrons has a T −2 dependence [1] and
exists at all temperatures. In the superconducting state,
its contribution is strongly modified, due to electron
decoupling resulting from the condensation into Cooper
pairs. As T decreases the normal electron density falls as
exp(−1/kB T ) = exp(−y). This means that the thermal
resistance Wgen is reduced by the same factor when T
decreases from Tc to 0 [4]. The lattice thermal conductivity
can therefore be written as:
Kgen = (27/452 )Na−2 I3 (∞) · K295 K · (T /2D )2 = DT 2
for the normal metal where the symbols have already been
defined, and I3 (∞) = 7.2 and as
Kges = exp(y)H (y)DT 2
for the superconductor as calculated by Bardeen et al [4].
y has already been defined; H (y) has a flat maximum at 1
in the range 0 < T /Tc < 1 while T 2 exp(y) tends to very
large values as T decreases from Tc to zero.
(ii) The scattering by grain boundaries, lattice
dislocations, and sample boundaries in the case of a
monocrystal. In a general way, this implies a mean free
path for phonons and a T −3 dependence with temperature
[7]. This term was calculated by Casimir [8] in the case of
scattering by crystal boundaries as:
Kgb = BlT 3
where l is the phonon mean free path, and B a constant
depending on the mechanism of scattering and the material.
For the monocrystal case, l, is the sample’s shortest
dimension, Casimir gives (in W m−2 K−4 )
B = 3 · 6 × 109 /(a0 2D )2
where a0 is the metal lattice parameter (in Å). For the
general purpose, this gives only an order of magnitude of B.
Because the electrons are not implied in these mechanisms,
this term is not dependent on the solid being normal or
superconducting.
Because the thermal resistance mechanisms have not
been clearly identified in the case of the lattice heat
conductivity, it is assumed that the functional form given
by the theory is still valid (i.e. the power dependence with
T ), but that the values of D and B may depart from their
theoretical estimates. In the fit calculation, these were
chosen as free parameters.
The resulting lattice thermal conductivity is obtained by
adding the thermal resistances due to both phonon scattering
455
F Koechlin and B Bonin
Figure 3. Experimental data: thermal conductivity of ‘annealed’ niobium samples.
mechanisms
Kgs ∼
=
1
1
+
exp(y)DT 2
BlT 3
−1
.
The addition of two resistances of opposing temperature
dependence results in a maximum in the lattice thermal
conductivity curve. This has been observed in many
experiments with superconductors [2, 3, 6, 9–13], but, as
will be discussed later, is not systematically visible with
niobium. However, the existence of this maximum in the
K(T ) curve gives strong support to the first limitation of
lattice thermal conductivity, namely the scattering of the
phonons by the normal electrons in the metal, reduced by
the condensation into Cooper pairs (term exp(y)).
use the theoretical estimates or try to obtain more realistic
values by fitting these five parameters to experimental
results.
Taking into account the basic data for niobium
(ρ295 K = 14.5 × 10−8  m, 2D ∼
= 275 K, a0 = 2.6 Å,
K295 K = 54 W m−1 K−1 ) these parameters are easily
calculated as:
L = 2.45 × 10−8 W K−2
a = 2.30 × 10−5 m W−1 K−1
α = 1.76
B = 7.0 × 103 W m−2 K−4
1/D = 300 m K−3 W−1
2.3. Total thermal conductivity
The total heat conductivity of the superconducting metal
is obtained by adding the electron term Kes (T ) and
the lattice term Kgs (T ). This of course is valid for
temperatures T lower than the critical temperature, because
y = 1(T )/kB T is defined only in this domain.
h ρ
i−1
295 K
Ks (T ) = R(y)
+ aT 2
L RRR T
−1
1
1
+
+
.
D exp(y)T 2
BlT 3
In order to obtain Ks (T ) using this model, it is necessary
to give experimental values to three variables: temperature
T , residual resistivity ratio RRR and phonon mean free
path l. On the other hand the following five theoretical
parameters should be defined: a, L, α, B, D. One can
456
3. Experimental database
3.1. Summary
Measurements of the thermal conductivity of the niobium
sheets prepared in the fabrication of superconducting high
frequency cavities have been carried out at Desy laboratory
[15] and at Saclay [13, 14].
16 experimental curves, K = f (T ), were obtained for
a large range of residual resistivity ratios, varying from 35
to 1750. Amongst these curves, 3 were measured at Desy
(RRR = 244, 379, 1071), 13 were measured at Saclay,
with 13 different samples, including 2 samples of high
purity Russian niobium [16] (RRR = 1750, 830) and 1
sample of American (Wah-Chang) niobium post-purified
(1350 ◦ C, 20 h) at the Wupperthal University laboratory
Parametrization of the niobium thermal conductivity in the superconducting state
Figure 4. A comparison of the experimental and theoretical plots of thermal conductivity, for niobium with RRR = 350.
[17] (RRR = 560)—these three samples were sent from
Wupperthal to Saclay in order to be characterized. The
experimental procedure used at Saclay has been described
elsewhere [13, 14]. The total number of experimental points
included in these curves was 322.
3.2. Experimental results
A part of these results are shown in figure 2 for samples of
niobium sheets kept ‘as received’ from the manufacturer,
and figure 3 for which may have been modified by a heat
treatment, called ‘annealed’ samples. The criterion for use
of the label ‘annealed’ is associated with the possibility
of recrystallization during the heat treatment, by applying
a temperature higher than 850 ◦ C for at least one hour.
Indeed, it has been observed that below this temperature,
negligible growth of the crystals occurred. So far, we
have not used the label ‘annealed’ when an outgassing
was performed at 700–800 ◦ C in order to remove the
interstitial hydrogen and to relax mechanical constraints.
The continuous lines between the experimental points are
sketched here only for a better understanding of the picture.
As can be seen in figure 2 (‘as received’ niobium), the
curves for RRR = 40 and 265 show a pronounced bump
in thermal conductivity between 1.5 and 4 K. This unusual
feature, as compared to the other curves, could be due to
an enhanced lattice thermal conductivity. Both samples
come from a supplier, Teledyne Wah-Chang, from which
we have no other ‘as received’ sample. This feature may be
due to a process of fabrication different from the others, for
instance a stronger annealing or outgassing at one of the last
stages of fabrication which could have increased the size
of the grains. However, under observation, no significant
difference with samples from other suppliers was found.
In figure 3 (‘annealed’ niobium), similar bumps, which
may be due to high temperature annealing and associated
recrystallization, are hardly visible between 1.5 and 3 K,
and probably lay within the experimental error. Indeed, for
large values of RRR (> 300) the electron conductivity may
be large enough, even with R(y) = 8×10−3 (at T = 1.8 K),
to overtake the phonon peak. Nevertheless, the correlation
between high temperature annealing producing larger grain
sizes and the existence of a bump on the curves due
to enhanced lattice thermal conductivity, is not perfectly
established. This may be due to our poor knowledge of the
past history of the ‘as received’ samples.
On the other hand, the value of the total heat
conductivity at 4.2 K is roughly equal to RRR/4, as was
stated several years ago by Padamsee [18]. This can easily
be explained using the theoretically derived equation given
in this paper for the electron heat conductivity, Kes , and
figure 1 to deduce the ratio R(y) for a temperature of 4.2 K.
With α = 1.76, R(y) = 0.309.
A computer program was written in order to calculate
the thermal conductivity of niobium, using the equations
given in this paper. The coefficients had those values
estimated from theory (except α = 1.98), the RRR was
measured independently and the phonon mean free path
was assumed to equal the average grain size, i.e. 50 µm.
No adjustment was performed and figure 4 shows the
comparison between the curve calculated for a RRR of 350
and the experimental points for a sample of RRR equal
to 356. Quantitative agreement was good at 4.2 K, but
the slope of the theoretical curve was steeper than the
experimental profile. This result was quite encouraging
457
F Koechlin and B Bonin
Figure 5. A comparison of the calculated curves K (T ), with the best fitted parameters (full line) and with experimental plots
(line joining symbols), for typical RRR values.
in our attempts to fit the parameters of the model with the
experimental data.
already given at the end of section 2. The criterion was the
minimization of the quantity
Q =
2
3.3. Data associated with an experimental point
322 X
Kexp − Kcal 2
l
322 experimental points, each being a separate measurement, were recorded. The following information had to be
included in the database for each point: (i) the temperature T , (ii) the thermal conductivity K, (iii) the RRR of the
sample (measured separately) and (iv) the information ‘HT
annealed’ or ‘as received’.
This was sufficient to give a crude estimation of the
phonon mean free path l, which was itself assumed to be
comparable to the average grain size (AGS). The purchased
niobium sheets all had an AGS corresponding to the ASTM
index 5.5 to 7.5, i.e. approximately 50 µm. On the other
hand, niobium annealed during several hours at 1100 to
1300 ◦ C (with or without titanium getter) showed AGS
of 0.3 to 1.5 mm. In order to simplify the situation we
assumed that l was equal to 50 µm in the case of ‘as
received’ and 500 µm in the case of ‘annealed’ niobium.
Kexp + Kcal
where
Kexp = measured K(T , RRR, l)
Kcal = computed K(T , RRR, l, {L, a, α, B, D}).
4.2. Results
The results of the calculation are shown in table 1 including
the initail and the best fit values obtained for the five
parameters which were allowed to vary. Moreover, the
comparison of the curves K = f (T ) calculated with these
best values (solid lines) and the experimental curves (lines
joining the experimental points), is shown in figure 5 for
typical values of the RRR.
4.3. Influence of cold work
4. Fit calculations and results
4.1. Method
A Monte-Carlo least-square fit procedure in a five
dimension space was used. The five components of the
randomly varied vector were the five parameters already
mentioned in section 2 of this paper, namely L, a, α, B,
D. The starting point was the set of theoretical values
458
Figure 6 shows the special case of a niobium sample with
an initial RRR of 400, whose thickness was reduced by 23%
by rolling. As a result of this process, the material’s RRR
was reduced to 300. As can be seen, the parametrization
using the new value of RRR, is still in agreement with the
measured thermal conductivity for this sample. This seems
to indicate that the present parametrization is valid also for
cold worked samples.
Parametrization of the niobium thermal conductivity in the superconducting state
Figure 6. Comparison of a curve K (T ) ‘predicted’ by the fit (using the measured RRR ), with the plot of experimental
measurements. New sample, not included in the database.
Table 1. Comparison between the best fit and the initial
values for the parameters.
4.4. Discussion
(i) L: the best fit value is slightly lower than the
theoretical value L0 . As explained in section 2, this was
expected.
(ii) a: the value given by the fit is 10 times smaller
than the experimental one and 30 times smaller than the
theoretical one.
(iii) α: a value lower than that given by theory is
unusual [19], but it can be explained by the need for a
reduced slope in the plot of K versus T as can be seen in
figure 4.
(iv) B: the fitted value is not far from the estimated
one (60%) if one takes into account the uncertainties
surrounding the definition of the phonon mean free path and
the assumption that the functional form found by Casimir
for a monocrystal could be kept.
(v) D: the discrepancy between the two values (20%),
is unexpectedly small.
4.5. Quality of the fit
From the 322 measured points, the minimum value of Q2
is 9.8. The resulting uncertainty on Kcal is
h(1K/K)2 i1/2 ∼
= 35%.
This relatively large error bar can be appreciated in
figures 5 and 6, where it should also be noted that the
discrepancy between experimental and best fit data is
large for temperatures lower than 2 K. Meanwhile, for
the temperatures higher than 2 K and up to 9 K, the
thermal conductivity approximation by the best fit is quite
satisfactory.
Parameter
L
a
α
B
1/D
Initial value
(theory)
−8
2.45 × 10
2.30 × 10−5
1.76
7.00 × 103
3.00 × 102
Best fit value
−8
2.05 × 10
7.52 × 10−7
1.53
4.34 × 103
2.34 × 102
Unit
W K−2
m W−1 K−1
W m−2 K−4
m K3 W−1
5. Conclusion
The resulting parametrization reproduces the experimental
data reasonably well in a wide temperature range and for
a variety of niobium samples. The parametrization is
less valid for temperatures below 2 K. In this range, the
thermal conductivity is influenced by the lattice contribution
which depends strongly on the past metallurgical history
of the material. Nevertheless, the overall precision of the
parametrization (35%) is sufficient for most thermal model
calculations. We hope it will be a valuable tool for future
applications of superconductivity.
Acknowledgments
The authors would like to thank Claire Antoine for her help
in measuring the grain sizes for a large number of samples,
and Yves Boudigou who performed the largest part of the
thermal conductivity measurements.
459
F Koechlin and B Bonin
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