Linear expansion coefficients have been measured for Cr, Mo and W from 230 K, 5590 K, and near room temperature. A t low temperatures, the lattice contributions for Mo and W, although small, are determined to better than 10% giving respective limiting values o f the lattice GriJneisen parameter, 3'o, o f 1.3and 1.35compared with 3,(283 K) = 1.61; their electronic components are very different, giving 3,e d In N(EF)/d In V = I. 1 (Mo) and 0.3 (W). For Cr, the "electronic" term is large and negative at low temperatures, giving 3,ern =  9; anomalies occur in ~ (7") at 124 and 311 K. =
Thermal expansion of Cr, Mo and W at low temperatures G.K. White, T.F. Smith and R.H. Carr
During various studies of solids we have gathered data on the linear expansion coefficient ot of the group 6 metals, Cr, Mo and W which are of both fundamental and technical interest. In 1961 one of us 1 determined the electronic contributions to expansion of a number of metallic elements, including three samples of chromium which gave respectively a =  ( 3 9 + 4 ) x 10t° T,  ( 3 5 + 3) x 101° T, and  (31 + 3) x 10 1° T K 1 for T < 20 K, and subsequently studied the 'anomaly' in the vicinity of the Ndel Point at 311 K. 2 The form of this 'anomaly' near 311 K varied from sample to sample although all were 'ductile' chromium of small Nz content prepared by H.L. Wain at the Aeronautics Research Laboratory in Melbourne. Later metallographic examination by Dr Wain showed that the samples were not fully recrystallized and therefore strain effects were present which could smear the transition. One was then heat treated at 1470 K and shown to be fully recrystaUized; the expansion was measured at low temperature, and also near 120 K and near 311 K as reported here. The expansion coefficients of Mo and W have been measured, by Andres, 3 who found that below 12 K 101°a  (4.3 + 0.8) T + (0.038 + 0.011) T3K 1 (Mo) and 101°a = (0.3+0.3) T + (0.045 + 0.01) T3 K 1
(W)
from which he calculated 7o = 1.1+0.3,
?e = 1.6+0.3
(Mo)
?o = 1.1 + 0.3,
3,e = 0.2 + 0.2
(W)
These lattice and electronic Griineisen parameters, 7o and ?e respectively, are determined from the T 3 and T terms in the expansion coefficient/3 and heat capacity Cp, using 3, =
Bsv/cp
= 3
sV/Cp,
where B s is the adiabatic bulk modulus and V is the atomic volume.4 Molybdenum was measured here by Carr in 1963 (unpublished) and more recently by the other authors during a study of the NbMo alloy system, s G.K. White is at the National Measurement Laboratory, CSI RO, Sydney, 2070 Australia. T.F. Smith is at Physics Department, Monash University, Clayton, Vic. 3168, Australia. R.H. Carr is at Physics Department, California State University, Los Angeles, 90032, California, USA. Received 28 November 1977.
Tungsten has also been measured here on two occasions in the past!five years and data are given below. For Mo and W, most measurements were done below 30 K, but some from 55 to 100 K are reported and compared with previous tabulated data.
Measurements All the measurements of linear thermal expansion were made using a threeterminal capacitance bridge. ~ Different cells or dilatometers were used at different times and their sensitivities in terms of A1/I varied from 10 9 to 10 1° as detailed below.
Chromium.
A rod of 100 mm length, 9 mm diameter, was prepared by Dr Wain in 1963, from electrolytic ductile material containing less than 0.001% N2 and annealed at 1470 K until fully recrystalized. 2An axial absolute dilatometer was used of rather similar geometry to that of Carr and Swenson 7 except that their detector system used inductance rather than capacitance. Data points in Figure 1 show a deep cusp in or(T) near 311 K and a less marked minimum near 124 K, the socalled spinflip temperature. The smooth curve shown is from the compilation of Corruccini and Gniewek 8 based on the work of Erring and extrapolated below 60 K. Values at rounded temperatures in Table 1 are based on present measurements. Below 20 K, or(T) is dominated by a negative linear term, ie 101°~ =  ( 3 2  + 3 ) T K  l , which obscures other contributions. From 20 to 30 K, the latter are still uncertain but may be represented within the limits of experimental error by a term 2 x 10 z2 T 3 K 1, presumably of lattice vibrational origin, leading to ?l ~ 1.0. The linear (electronic and/or magnetic) term Otem =   32 x 10 1° TK 1 together with heat capacity Cem = 1.41 TmJ mo11 K 1 and values ofBs, Vin Table 2, lead to 3,era =  9.3. Heiniger et al 9 have reviewed the heat capacity data on transition metals and we have used their values of C e m and 0 o.
Molybdenum.
In 1964 a cylinder of 19 mm diameter, Mo 1, from Murex Limited (UK) (density 10.2 g ml l and 99.9% purity) was annealed in high vacuum at 1570 K for several hours after initial machining. The expansion was measured radially in an absolute dilatometer 6 from 330 K
00112275/78/18050301 $01.00 © 1978 IPC Business Press C R Y O G E N I C S . M A Y 1978
301
5
Table 1.
3 Xf
T
. •
6
/ ¢
'4
:5
v
'..
,
2
J
I/ : /o
I
~
i
/
O • Present ~ Corrucciniand Gniewek (1961) X Kirbyond Hohn (1977) A Lisovskii (1972) I
o
IOO
Mo
W
4
 1.3
0.15
0.05
6
 1.9
0.23
0.14
8
 2.6
0.40 + 0.02
0.30 + 0.02
10
 3.2
0.60
0.58
15
 4.5
1.5 + 0.1
1.9
20
 5.4
3.5
4.8 + 0.1
25
 5.5
30
 3.8
6.5 + 0.5
10.2
20.3+ 0.5
1 2 . 5 + 1
40
7


I
50
31


57.5
60
101
136
•
2 0
65
90
135
158
t T. "
I
j v T,
J[ /
Cr
5
2
=, o_
Values of (z, 10 8 K 1 at rounded temperatures
O'
I
I
200
300
75
143
189
208
85
200
228
242
100
270
295

283
520
510
434
T, K
Table 2.
Physical data for Cr, Mo and W
Fig. 1
Linear e x p a n s i o n c o e f f i c i e n t s f o r tungsten, molybdenum a n d c h r o m i u m . Present d a t a are s h o w n as O, • and previous d a t a are __8 X12 A13
Cr
V, ml (at OK)
and 65100 K. The avalues are small compared to copper at low temperatures and various experimental runs were not very reproducible, the two principal sets of data (for T < 15 K) giving 101°or = 2.4 T + 0.037 T a K I and 101°a = 3 . 4 T + 0 . 0 3 3 T 3 K  t . Later three runs (in 1975 and 1977) were made in the differential copper cell ~° on Mo 2, this cylinder being 51 mm long x 19 mm diameter and of similar origin and heat treatment to Mo 1. Data below 15 K gave values of the coefficients for T and T 3 varying from 2.7 to 3.1 x 10 1° K ~ and 31 to 37 x 10 ~2 K ~ respectively. Our weighted mean is 101oa
= (2.8_+0.2) T + ( 0 . 0 3 5 T  0 . 0 0 2 ) T 3 K  t ,
cf Andres a who found 10~°~ = (4.3 + 0.8) T + (0.038 T0.011) T a K a . Taking values Ce = 1.83 T m J tool l K 1, ~ (515 K) 455460 K, 9 and for B s and V from Table 1, we calculate 7e =
1.1+0.1
and 3,0 = 1.30+ 0.05. Pressure derivatives of elastic constants l~ give 3,o(elastic) = 1.3. Values of a between 15 and 30 K from the various runs on Mo 2 differed by up to 10% as shown by the error limits in Table 1, and the earlier values for Mo 1 generally lie between the extremes. The errors, which seem to be relatively large for these low expansion coefficent metals
302
7.2
Bs, GPa (at 0 K) 0o, K
Mo
W
9.36
9.53
190
265
313
~600
470
390
101° ~ e / T
32+3
3,e
9.3+0.9
2.8+0.2
0.3+0.2
1.1 +0.1
0.3
1012al/T 3
~+2
3.5+0.2
5.3_+0.2
3,0
~ + 1
1.3
1.35
3'0 (elastic)

1.3
3'283

1.61
1.61
(eg 1 x 10s at 30 K, 0.1 x 10SK ~ at 15 K),are within the usual error range of the dilatometer. In Fig. 1 our values (see also Table 1) from 55100 K and at 283 K agree reasonably with the curve from Corruccini and Gniewek s based on the work of Erfling and of Nix and MacNair. The value a(300) = 509 x 10 a given by Lisovskii ~3 also fits well. His other data between 60 and 3 0 0 K are only reported on a small graph and values copied from this graph and appearing in Thermophysical Properties of Matter ~aare significantly smaller and are inconsistent with the single tabulated value.
Tungsten. Two series of measurements (1973, 1977) were made on a 51 mm long x 19 mm diameter cylinder (99.9% pure), also from Murex Limited, of density 18.9 g m11. After initial machining it was annealed at 1570 K in vacuo. Each series was made in the differential copper dilatometer. Below 12 K, data could be represented by 10~°a = (0.3 + 0.2) T + ( 0 . 0 5 3 + 0.003) T 3 K t (1973) 10~°a = (0.4 + 0.2) T + (0.054 T0.005) T 3 K t (1977);
C R Y O G E N I C S . M A Y 1976
N(EF)
A 1973 Tungsten 0 1977
^ A/,
'~
= Nb(1 + X)
whence dlnN(Er)_
oi 0
dlnNb
dlnV
~ 0 ~ ~ . ~ ~.~ ~ "
+
dlnV
X
dlnX
I+X
dlnV
Griessen 17 has estimated that
C
~"'~'"~NAndres
e~
dlnNb
(1964)
0.7 (Mo), = 0.2 (W).
din V I
I
50
I00 T 2 , Kz
I
150
20(
alT versus 72 for tungsten
Fig. 2
as illustrated in Fig.2. The mean leads to %
= 0.3+0.2,
3'0 = 1  3 5 + 0 . 1 ,
using Ce = 0.90 T m J mol t K t, 0 (412 K) = 380 K, and values for Bs and V given in Table 1. Our data in Fig. 1 from 5590 K and at 283 K agree reasonably with the curve of Corruccini and Gniewek a (based on Nix and MacNair) and with the NBS values) 2 There is a value from Lisovskii la of a(300) = 456 x 10 s which agrees but his other data have again only been reported graphically; values listed 14 copied from his small graph are significantly smaller near 300 K than his tabulated value.
Discussion These three elements each have a large bulk modulus and Debye characteristic temperature so that their expansion coefficients are small compared to copper for which a(283) = 1653 x 10 a K t and a(30 K) = 100 x 10 s K 1. It follows that errors of measurement are relatively more important for these elements. At liquid oxygen temperatures and 283 K our values generally agree with those in the NBS compilation, s There are apparent discrepancies in the range 100 to 300 K between this compilation, Lisovskii ~a and NBS reference data on tungsten ~2 which need to be resolved. At low temperatures, despite the fact that relative errors approach 10%, it is clear that for Mo and W the lattice Grtineisen parameter decreases from its roomtemperature value of 1.6 to the zero temperature limit 7o ~" 1.3. This difference is common for many metallic elements. Our values are of somewhat higher accuracy but not significantly different from those reported by Andres. The electronic contribution for Mo is rather smaller than given by Andres a but confirms that Te
We estimate from the pressure dependences of the superconducting transition Is' t9 Tc that

dinN(EF) din V
differs markedly between Mo and W. For Mo, 7e ~ 1.1 is larger than the free electron value of 2]3 is and lies in the same range (1 to 2.5) as do most other transition (and polyvalent) metals, 4 whereas 3' ~ 0.3 for tungsten is significantly smaller. McMillan t6 has linked the observed densityofstates N ( E F ) to the bandstructure densityofstates Nb via an electronphonon interaction parameter
C R Y O G E N I C S . M A Y 1978
X
dlnX
I+X
dlnV

0.4(Mo), 0.05 (W).
Together these give values for 7e = dlnN(EF)/dln V which appear remarkably consistent with our results, considering the experimental uncertainties. Nothing very new can be said about chromium for which 7era ~  9. The antiferromagnetic interactions obviously are important and, at present, cannot be unscrambled from electronic effects discussed above. The large negative value is consistent with the large negative pressure dependence of the Ndel temperature (eg review a of Collins and White, also McWhan and Rice 2° and magneto strictive data 21); that is, it arises largely from the volume dependence of the antiferromagnetic exchange interactions. The magnetic interactions play such a major role in chromium that it is difficult to identify the magnitude o f the lattice contribution with reliability: 71 approximates to 1.0 at low and normal temperatures. We thank Dr J.G. Collins for helpful discussions and Dr R. Griessen for sending his unpublished calculations.
References 1 2 3 4
White,G.K. Phil Mag 6 (1961) 815 White,G.K.Australian JPhys 14 (1961) 359 Andres, K. Phys Kondens Materie 2 (1964) 294 Collins,3.G., White, G.K. Progr Low Temp Phys 4 (1964) 450
5
White, G.K., Collins, J.G., Birch, J.A., Smith, T.F., Finlayson, T.R. Proc Int Conf Physics of Transition Metals (1978)
E 7 8 9
Toronto Cart, R.H., McCammon, R.D., White, G.K. Proc Roy Soc A 280 (1964) 72 Carr, R.H., Swenson, C.A. Cryogenics 4 (1964) 76 Corruecini, R.J., Gniewek, JJ. Thermal Expansion of Solids (1961) NBS Monograph 29, US Govt Printing Office, Washington DC Heiniget, F., Buchet, E., Mullet, J. Phys. Kondens. Materie 5 (1966) 243
10 11
White, G.K., Coilins, J,G.J Low Temp Phys 7 (1972) 43 Katahara, K.W., Manghnani, M.H., Fisher, E.S. Proc Int Conf Physics of Transition Metals (1978) Toronto
12
Kirby, R.K., Hahn, T.A. private communication (1977) Interim Certificate for SRM 737, Nat Bur Stands, Washington DC Lisovskii,Yu. A. Soy Phys  Sol State 14 (1973) 2015 (transl.) Touloukian, Y.S. (Ed.) Thermophysical Properties of Matter 12 Thermal Expansion (1975) Plenum Press New York Vatley, J.H.O. Proc Roy Soc A 237 (1956) 413 McMillan,W.L. PhysRev 167 (1968) 331 Grieuen, R. private communication (1976); Smith, T.F., Shelton, R.N. J Phys F: Metal Phys 5 (1975) 911 Smith, T.F., Finlaymn, T.R. J Phys F: Metal Phys 6 (1976) 709 McWhan,D.B., Rice, T.M. Phys Rev Lett 19 (1967) 846 Fawcett, E. Phys Lett 32A (1970) 117
13 14 15 16 17 18 19 20 21
303