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Thermal and electrical resistivity of some tungsten single crystals at low temperatures and in strong magnetic fields
De Nobel, J. 1957
Physica XXIII 261-269
T H E R N A L AND ELECTRICAL R E S I S T I V I T Y OF SOME TUNGSTEN SINGLE CRYSTALS AT LOW TENIPERATURES AND IN STRONG PIAGNETIC F I E L D S b y J. D E N O B E L Communication No. 306d from the Kamerlingh Onne~ Laboratorium., Leiden, Nederland
Synopsis Investigations on heat conductivity and electrical resistivity of tungsten single crystals were started in 1936 with measurements on two tl~n crystals. Results on these crystals are given in this paper. Using a single crystal with its a:ds parallel to the (1, 1, 1)direction, we failed in sepm'atingthe heat conductivitT into the lattice and electronic components by applying strong magnetic fields. Another single crystal parallel to the (I, 0, 0) direction was investigated down to 1.5°1K,and in. the liquid hydrogen region strong magnetic fields up to 36 kO were also applied. It showed large anisotropy amounting to 1.59. Results are given. 1. Intro&l, ctio,r~. In the beginning of the research on h e a t conduction in t u n g s t e n single crystals (about 1936) we studied two very thin crystals to get information about m a k i n g contacts on tungsten and about the m a g n i t u d e a n d general behaviour of h e a t c o n d u c t i v i t y of this element. Results in brief obtained with these crystals, called A and B, are given in section 2. J u s t as we investigated the he~t and electrical conductivities of a t u n g s t e n single crystal w i t h its axis parallel to the (1, 1, 1) direction 1) ~.), called crystal 1-37, ave p e r f o n n e d a similar series of measurements with a t u n g s t e n single crystal (axis parallel to the (1, 0, 0) direction), furtheron called crystal 1-38. The reason w h y this crystal was chosen, was, t h a t it showed large anisot r o p y in a transverse magnetic field and we wished to get more information about electronic and lattice h e a t conductivities (a~ and 2,g resp.) from the change of anisotropy w i t h increasing field strength. The results of this series of measurements are given in tables a n d figures in section 3. As will be shown in a theoretical n o t e a) it appeared impossible to get the information required. 2. Meas,~re~¢e,i, ts aug res¢alts with crystals A ~ , d B. T h e v e r y t h i n crystals A a n d B h a d respectively a hexagonal and a square cross-section of less t h a n 1 m m diameter. The longitudinal axes of the rods were r o u g h l y parallel to
261
1. DE NOBEL
262
the (1, 1, 1) and to the (1, 0, 0) directions. Contacts were made on these rods b y means of paper cups glued to them and filled with mercury. In these mercury gutters amalgamated copper wires were dipped. When the apparatus was cooled down, the mercury froze round the rod and very good contacts were thus established (see for a further description D e H a a s and B i e r m a s z 4)). Measurements were performed only to get a general idea and so the accuracy of the results is not as large as with the crystals 1-37 (described in earlier papers 1) 2)) and 1-38 (section 3 of this paper). In figs 1 and 2 we have 5Swatt/cmcjraa d so
f 40
3O
20 - -
I0 k
o 0
I
I
I
I
I
20
40
60
80
90°1<
T
Fig. 1. H e a t c o n d u c t i v i t y of t u n g s t e n single c r y s t a l s as a f u n c t i o n of t e m p e r a t u r e .
B []
,4
OH= oo 0 H = 540 O f
plotted the results for the heat conductivity ~ and the electrical resistivity Rspee as a function of temperature and also in a magnetic field of 540 oersted. A rotational diagram of the electrical res{stance could be made (R as a function of the angle between the transverse magnetic field and the crystal axes at constant temperature and magnetic field). The magnetic field, obtained b y means of one of the oldest electromagnets of the Kamer-
THERMAL AND ELECTRICAL RESISTIVITY OF TUNGSTEN CRYSTALS
263
lingh Onnes Lab. will certainly not have been homogeneous. The crystal A showed an anisotropy ratio of 1.13 and the crystal B of 1.23. It is evident from the figures that crystal B is the pure one. This is also evident from a plot of w T versus T3 (see ref. 3), fig. 1). The residual resistances have not been measured. Heat conduction in a magnetic field could not be investigated in the range of liquid helium since the phosphorbronze resistance thermometers were too sensitive to a magnetic field. 6.59.cm
S5
45
3.S IO~'Rsp~c
I
I 2.5 14
16
I
I
18
20
:)IOK
-T
Fig. 2. Electrical remstivity of t u n g s t e n single crystals as a function of t e m p e r a t u r e . B
[]H=
00
[ ] H = 540 13 I
A
OH=
013
(DH = 540 13 i
3. Crystal 1-38. With the crystal 1-37, parallel to the
(1, 1, 1) direction we
did not succeed in separating the total heat conductivity ~ into that of the lattice ~g and that caused b y the electrons )re. So we thought that b y investigating heat conductivity and electrical resistivity not only as a function of temperature and magnetic field but also as a function of the angle between the transverse magnetic field and the crystal axes in an anisotropic single crystal, we could draw conclusions about ~g from the change of these rotational diagrams. The assumption is that ;tg is not influenced b y a magnetic field. It is uninteresting and unnecessary to give all the data which we got in the.
264
j. DE NOBEL
series of measurements on this crystal 1-38. X-ray analysis showed that the axis of the rod made an angle of at most five degrees with the (1,0, 0) direction *). We have assembled some of the most important and striking values of ;t and Rspee in table I and II respectively. For the absolute values of Wspee, ,t and Rspee, more decimals are given than is justified, since especially in the rotational diagrams the relative values will be accurate within 1%. Results are plotted in figs 3 and 4, where for each field strength there are two curves. These two curves belong to the values in the minima and maxima. As can be seen an interchange of place between maxima and minima takes place comparing the liquid hydrogen temperatures with those 24watt/cm9mod
20
-
,///~
-
/ 16--
o
0
\
\
/
20
T
F i g . 3. J l e a r c o n d u c t i v i t y
I 40
I 60
80°K
o l c r y s t a l 1 - 3 8 in d i f f e r e n t field s t r e n g t h s a s a f u n c t i o n of temperature.
0 H = V H -
0 k() 10.30 k O
A H := 2 0 . 6 9 k ( ) ~ t f -- 32.61 k O
of liquid nitrogen. Hence the curves belonging to the extrema must cross each other between 20 and 60°K. The point of intersection shifts to higher temperature with increasing field strength. The depth and the place of the minima in the Rspee versus T-curves is uncertain, as there are no measuring points between 20 and 60°K. At this minima 2R0 < R H < 3R0. The crystal 1-38 appeared to be less pure than crystal 1-37. The value of Reo/R273 was 26 × 10.4 and 8.6 × 10.4 resp. From figure 3 can be seen that *) Mr G. De V r i e s , nat. phil. drs., was so kind as to take tile X-ray p lotographs and to analyse them.
T H E R M A L A N D E L E C T R I C A L R E S I S T I V I T Y OF T U N G S T E N CRYSTALS
165
the heat c o n d u c t i v i t y of crystal 1-38 will reach a m a x i m u m value ~f only 23 W cm -1 deg -1 at T = 16.9°K, while these values for the crystals A, B and 1-37 would be 48, 56 and 120 W cm -1 deg -1 at 11.5, 10.8 and 6.8°K. The last m e n t i o n e d values are calculated with the help of the formula w - ~T 2 + f l / T (see ref. a)). F r o m the r o t a t i o n a l d i a g r a m s (figs 5 and 6) it was not noticeable, t h a t the (1, 0, 0) direction did not coincide with the axis of the rod. The resistivities show m a x i m a and minilna e v e r y 45 degrees. In the m a x i m a , s e c o n d a r y m i n i m a in Rspee as well as in Wspee are recognisable. 13~cm
/C] /
3-7
IO~pe¢
o
o_o_~ ~ I - - - - 7 ~ J 7 tO
20
I 40
60
80 o K
--T
Fig. 4. Electrical resistivity of crystal 1-38 in different field strengths as a function of temperature. OH = 0 140 [ ] H = 32.61 kO 7, H :- 10.301¢0 ..", H = 36.27 k¢) ~\ H = 20.69 kO I n view of the 90 degree periodicity we have fitted Wspee and Rspee to the following forms: Ze;spee = ao + al cos 40~ + a2 cos 8~ + aa cos 12~ + a4 cos 160~ + a5 cos 20~ + a6 cos 24e. Rspee = b0 + bl cos 4~ + be cos 8e + ba cos 12~ + b4 cos 16~ + b5 cos 20e + b6 cos 24~. The values of a0 to a6 and of b0 to b6 for different t e m p e r a t u r e s and field strengths are given in table I I I . In the last column the value of the anisot r o p y ratio is mentioned.
266
J. D E NOBEL TABLE
H kO
T Oh:
Wspec [ ~]position [ w a t t -x c m w a t t c m - * head deg deg-* cryostat
74.42 70.40 65.77 19.95 19.14 18.18 4.09 3.84 3.61 3.58 3.38
0.423 0.403 0.374 0.0456 0.0452 0.0450 0.125 0.138 0.151 0.152 0.165
75.9 76.1
0.445 0.456
2.25 2.19
71.2 71.4
0.430 0.446
2.33 2.24
67.5 67.6 67.5 67.5
0.382 0.404 0.401 0.404
2.62 2.48 2.49 2.48
58.1 58.5 58.5
0.307 0.383 0.381
20.126 19.172 18.052 17.018 16.092 15.162
0.1792 0.1936 0.214 x 0.2285 0.2384 0.2460
5.59 8.17 4.67 4.38 4.20 3.91
20.112 19.164 19.164 18.015 17.002 16.075 15.092
0.402t 0.4436 0.4454 0.491' 0.5460 0.593 s 0.627 r
2.49 2.25 2.25 2.04 1.83 1.69 1.59
20.093 19,047 18.073 16.980 16.100
0.5624 0.6282 0.7250 0.8030 0.871 s
1.78 1.59 1.38 1.25 1.15
20.192 19.118 18.056 17.043 16.143 15.240
0.7399 0.8194 0.9360 1.057 1.167 1.256
1.38 1.22 1.07 0.949 0.940 0.797
20.109 17.770 15.245
0.8550 1.142 1.50'
1.17 0.876 0.666
I
3.26 2.61 2.63
H kO 20.42 20.39
2.364 2.481 2.674 21.93 22.12 i 22.22 8.00 7.25 6.62 6.58 , 6.06
i
i
20.37 20.35 20.34 20.32
i
1575
20.31 20.28
1575
20.27 20.28
l I
1575 121 114
32.64
1570 114 153
32.63
32.63
7" °K
Wspec I ~' [position w a t t - I c m [ w a t t era-* head deg I deg-* cryostat 0.4104 0.4600 0.477 v 0.4710 0.4790 0.4710 0.4496 0.405 o 0.386 s 0.3913 0.3967 0.4522 0.4745 0.4694 0.4722 0.4514 0.3990 0.3930 0.4041
2.44 2.17 2.09 2.12 2.09 2.12 2.23 2.47 2.59 2.56 2.52 2.21 2.11 2.13 2.12 2,22 2.80 2.54 2.48
150 1355 121 114 107 100 93 785 71 675 64 495 35 27 19 3 348 333 150
0.7602 0.873* 0.9230 0.911 s 0.9237 0.9078 0.8622 0.8225 0.7624 0.7197 0.7465 0.8022 0.855 o 0.896"0.9178 0.907 s 0.9141
1.32 1.15 1.08 1.10 1.08 I.I0 I. 16 1.22 1.31 1.39 1.34 1.25 1.17 1.12 1.09 I.I0 I.II
150 135 s 121 114 107 I00 93 85~ 785 71 64 56~ 49 s 42 35 27 19
1.24 I 1.331 1.54 s 1.64 s 1.690 1.676 1.699 1.65' 1.53 R 1.313 1.23 o 1.332
0.806 0.750 0.647 0.607 0.591 0.596 0.589 0.606 0.652 0.763 0.810 0.750
157 s 150 1355 128 121 I14 107 1O0 93 785 71 64
i
THERMAL
AND ELECTRICAL
RESISTIVITY TABLE
H kO 0
position head
T °K
10 y .R s pec Q cm
77.35
592.1 620.6 655.7 655.2
114
508.7 580.2
165
429.7 509.1
165
355.1 445.8 440.3
165 114
20.88
32.61
0 32.62
72.97
0 32.61
68.51
0 32.61
63.95
0 10.65 20.88
55.35
32.61
215.5 233.6 273.0 268.4 333.5 326.2
cryostat
20.025
10.30 10.29
19.044 17.053 14.209
88.7 112.3 92.1 94.7 97.8 124.5
157 s 114 157 s
237.9 244.3 249.4 356.0 254.1 368.7 266.9 390.5 297.6 407.4
165
376.5 381.6 565.1 392.7 414.0 617.6
1575
531.7 546.2 563.4
165 1575
20.69
20.484 20,043
19.047 17.047
20.67
15,080
26.23
20.410 20.053
26.17 26.16
19.053 17.027 17.025
32.61
20.357
20.064
T cK
10 °.A'spec Q cm
32.59
572.4 630.6 638.0 641.2
187 s
32.58 32.57
19.023 17.053 15.138 14.160
32.61
20.064
555.1 563.4 596.9 . 589.3 728.6 797.8 836.1 845.7 845.3 842.4 847.7 845.6 834.1 805.1 729.0 611.4 545.3 566.5 698.7 764.0 801.2 835.4 849.2 848,5 845.9 850.3 844.8 817.6 748.3
165 1575 150 150 1315 128 1246 121 117s 114 1105 107 1035 100 93 786 71 64 495 42 386 35 31 27 23 19 15 11 3
36.27 36.23
20.020
161¼ 150 121 114
36.27
19.016 16.998 15.092
657.6 743.5 1051.9 1046.5 682.4 730.9 767.9 854.0 1207.7 1212.9 782.7 787.8 887.1 1231.3 1234.1
114 165 114
10.35
114
1575 114 1575 114 1576 114 1575 114
114 1575 114
position head
H kO
165
14.1 12.3
267
CRYSTALS
II
165
20.42 14.50
20.95 20.78
OF TUNGSTEN
14.195
cryostat
161¼
150 121 114 161¼ 157 s 150 121 114
268
J. DE NOBEL
To calculate the coefficients a it is flecessary to know the values of Wspee for one temperature and one field strength. The difficulty is that during the investigation of a rotational diagram of heat conduction the mean temperature of the two resistance thermometers changes considerably (at 15.4°K 1.8 c m
1.5
graad/twatt
--
IOQcm (.2 - -
Q6
--
2
IO?Rspec.
o3
I ao
I 60
I 90
1 ,20
,so°
F i g . 5. Thermal resistivity as a function of the angle between crystal axes and magnetic field at different temperatures and field strengths. H
=
Io
I 0
I 30
I
I
90
t2o
I t5o °
F i g . 6. Electrical resistivity as a function o f the angle between crystals axes and magnetic field at different field strenghts at approx. 2 0 a n d 15°K (the highest curve belonging t o the lower temperature).
20.74 kO
T = 20.0°K
'.~ H
= 32.63 kO
7" = 2 0 . 0 ° K
0 H
:
© H
= 32.62kO
7" =
A H
= 20.69 kO
15.5°K
I 60
10.30 kO
~
H =
[] H
26.23 kO
= 32.61 kO
in a field of 32.63 kQ for instance it changed by as much as 0.37°K). Even if we neglect the temperature variation of the liquid hydrogen bath, the temperature of the colder thermometer is different ~t each position of the crystal with respect to the magnetic field, because the heat resistance of the part of the crystal between the bath and that thermometer changes considerably, when the crystal is turned on its axis (anisotropy latio of Wspee ~ 1.59).
T H E R M A L A N D E L E C T R I C A L R E S I S T I V I T Y OF T U N G S T E N CRYSTALS
9-69
TABLE III H kO
T OK
32.62 32.62 32.63 26.39 26.36 20.76 20.32 10.30 10.30
15.50 15.50 20.00 15.30 20.10 15.15 20.00 15.13 19.97
l
[ a0
aniso-
a,
a2
aa
a4
as
an
tropy factor
1.521 1.518 0.848 1.083 0.614 0.752 0.442 0.267 0.192
0.250 0,253 0.112 0.162 0.090 0.106 0,047 0.022 0,007
--0.027 --0.029 --0.026 --0.023 --0.018 --0.007 --0.013 0 0.002
--0.004 0.002 --0.001 0 0.001 --0.004 --0.002 0 --0.001
--0.021 --0.017 --0.006 --0.013 --0.008 --0.003 --0.004 --0.001 0
0.007 0 --0.002 0.012 --0.001 --0.002 0.001 --0.001 0
--0.010 --0.007 --0.001 --0.001 --0.004 0.001 0 0 . 0
1.45 1.45 1.36 1.38 1.31 1.35 1.28 1.25 1.10
H kO
oK
T bo
bx
b2
b3
b4
b5
be
anisotropy factor
36.27 36.27 36.27 32.61 32.61 26.23 26.23 20.69 20.69 10.30 10.30 10.30
14.195 15.092 20.020 15.138 20.064 15.075 20.053 15.080 20.043 14.209 19.000 20.025
10.47 10.18 8.87 8.25 7.16 5.55 4.86 3.48 3.09 1.124 1.042 1.01 s
2.12 2.12 1,85 1.52 1.49 1.09 0.97 s 0.65 s 0.552 0.14 s 0.121 O. 124
--0.21 --0.12 --0.16 0.05 --0.12 --0.07 --0.10 --0.034 --0.044 --0.009 --0.01 o --0.01'
0. I I 0. I I 0.08 --0.08 0.02 0.05 0.02 0.031 0.02 t 0.00* 0.00~ 0
--0.12 --0.09 --0.12 0.I0 --0.085 --0.07 --0.05 --0.532 --0.032 --0.00 e --0.006 --0.006
0.02 --0.01 0 --0.17 --0.01 --0.00 s --0.005 --0.007 --0.002 --0.002 --0.00' --0.002
--0.05 --0.04 --0.05 0.05 --0.02 --0.03 --0.13 --0.015 --0.012 --0.00 a --0.00 t --0.00 i
1.59 1.59 1.61 1.59 1.57 1.57 1.57 1.51 1.50 1.35 1.33 1.32
A second complication is, that on a measuring day the current through the magnet slowly falls off. To reduce a rotational diagram for the thermal resistivity to one temperature and one field strength, it was necessary to construct plots of the thermal resistivity as a function of temperature for different field strengths for one position of the crystal. From these plots one could deduce the thermal resistivity as a function of the magnetic field at different temperatures. Repeating this treatment for other positions of the crystal the corrections on the measured thermal resistances for variations of mean temperature and field strength are known. A part of this research was done under the auspices of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)" and was made possible by a financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk onderzoek (Z.W.O.)" and from the "Nederlandse Centrale Organisatie voor Toegepast-Natuurwetenschappelijk Onderzoek (T.N.O.)". Received 11-1-57. REFERENCES 1) De H a a s , W. J. and De N o b e l , J., Commun. Kamerlingh Onnes Lab., Leiden No. 251d; Physica 5 (1938) 449. 2) De N o b e l , J., Commun. No. 278b; Physica 15 (1949) 532. 3) De N o b e l , J., Commun. Suppl. No. l13a; Physica ,°3 (1957) 273. 4) De H a a s , W. J. and B i e r m a s z , Th., Commun. No. 249a; Physica 4 (1937) 752.
Physica XXIII 261-269
T H E R N A L AND ELECTRICAL R E S I S T I V I T Y OF SOME TUNGSTEN SINGLE CRYSTALS AT LOW TENIPERATURES AND IN STRONG PIAGNETIC F I E L D S b y J. D E N O B E L Communication No. 306d from the Kamerlingh Onne~ Laboratorium., Leiden, Nederland
Synopsis Investigations on heat conductivity and electrical resistivity of tungsten single crystals were started in 1936 with measurements on two tl~n crystals. Results on these crystals are given in this paper. Using a single crystal with its a:ds parallel to the (1, 1, 1)direction, we failed in sepm'atingthe heat conductivitT into the lattice and electronic components by applying strong magnetic fields. Another single crystal parallel to the (I, 0, 0) direction was investigated down to 1.5°1K,and in. the liquid hydrogen region strong magnetic fields up to 36 kO were also applied. It showed large anisotropy amounting to 1.59. Results are given. 1. Intro&l, ctio,r~. In the beginning of the research on h e a t conduction in t u n g s t e n single crystals (about 1936) we studied two very thin crystals to get information about m a k i n g contacts on tungsten and about the m a g n i t u d e a n d general behaviour of h e a t c o n d u c t i v i t y of this element. Results in brief obtained with these crystals, called A and B, are given in section 2. J u s t as we investigated the he~t and electrical conductivities of a t u n g s t e n single crystal w i t h its axis parallel to the (1, 1, 1) direction 1) ~.), called crystal 1-37, ave p e r f o n n e d a similar series of measurements with a t u n g s t e n single crystal (axis parallel to the (1, 0, 0) direction), furtheron called crystal 1-38. The reason w h y this crystal was chosen, was, t h a t it showed large anisot r o p y in a transverse magnetic field and we wished to get more information about electronic and lattice h e a t conductivities (a~ and 2,g resp.) from the change of anisotropy w i t h increasing field strength. The results of this series of measurements are given in tables a n d figures in section 3. As will be shown in a theoretical n o t e a) it appeared impossible to get the information required. 2. Meas,~re~¢e,i, ts aug res¢alts with crystals A ~ , d B. T h e v e r y t h i n crystals A a n d B h a d respectively a hexagonal and a square cross-section of less t h a n 1 m m diameter. The longitudinal axes of the rods were r o u g h l y parallel to
261
1. DE NOBEL
262
the (1, 1, 1) and to the (1, 0, 0) directions. Contacts were made on these rods b y means of paper cups glued to them and filled with mercury. In these mercury gutters amalgamated copper wires were dipped. When the apparatus was cooled down, the mercury froze round the rod and very good contacts were thus established (see for a further description D e H a a s and B i e r m a s z 4)). Measurements were performed only to get a general idea and so the accuracy of the results is not as large as with the crystals 1-37 (described in earlier papers 1) 2)) and 1-38 (section 3 of this paper). In figs 1 and 2 we have 5Swatt/cmcjraa d so
f 40
3O
20 - -
I0 k
o 0
I
I
I
I
I
20
40
60
80
90°1<
T
Fig. 1. H e a t c o n d u c t i v i t y of t u n g s t e n single c r y s t a l s as a f u n c t i o n of t e m p e r a t u r e .
B []
,4
OH= oo 0 H = 540 O f
plotted the results for the heat conductivity ~ and the electrical resistivity Rspee as a function of temperature and also in a magnetic field of 540 oersted. A rotational diagram of the electrical res{stance could be made (R as a function of the angle between the transverse magnetic field and the crystal axes at constant temperature and magnetic field). The magnetic field, obtained b y means of one of the oldest electromagnets of the Kamer-
THERMAL AND ELECTRICAL RESISTIVITY OF TUNGSTEN CRYSTALS
263
lingh Onnes Lab. will certainly not have been homogeneous. The crystal A showed an anisotropy ratio of 1.13 and the crystal B of 1.23. It is evident from the figures that crystal B is the pure one. This is also evident from a plot of w T versus T3 (see ref. 3), fig. 1). The residual resistances have not been measured. Heat conduction in a magnetic field could not be investigated in the range of liquid helium since the phosphorbronze resistance thermometers were too sensitive to a magnetic field. 6.59.cm
S5
45
3.S IO~'Rsp~c
I
I 2.5 14
16
I
I
18
20
:)IOK
-T
Fig. 2. Electrical remstivity of t u n g s t e n single crystals as a function of t e m p e r a t u r e . B
[]H=
00
[ ] H = 540 13 I
A
OH=
013
(DH = 540 13 i
3. Crystal 1-38. With the crystal 1-37, parallel to the
(1, 1, 1) direction we
did not succeed in separating the total heat conductivity ~ into that of the lattice ~g and that caused b y the electrons )re. So we thought that b y investigating heat conductivity and electrical resistivity not only as a function of temperature and magnetic field but also as a function of the angle between the transverse magnetic field and the crystal axes in an anisotropic single crystal, we could draw conclusions about ~g from the change of these rotational diagrams. The assumption is that ;tg is not influenced b y a magnetic field. It is uninteresting and unnecessary to give all the data which we got in the.
264
j. DE NOBEL
series of measurements on this crystal 1-38. X-ray analysis showed that the axis of the rod made an angle of at most five degrees with the (1,0, 0) direction *). We have assembled some of the most important and striking values of ;t and Rspee in table I and II respectively. For the absolute values of Wspee, ,t and Rspee, more decimals are given than is justified, since especially in the rotational diagrams the relative values will be accurate within 1%. Results are plotted in figs 3 and 4, where for each field strength there are two curves. These two curves belong to the values in the minima and maxima. As can be seen an interchange of place between maxima and minima takes place comparing the liquid hydrogen temperatures with those 24watt/cm9mod
20
-
,///~
-
/ 16--
o
0
\
\
/
20
T
F i g . 3. J l e a r c o n d u c t i v i t y
I 40
I 60
80°K
o l c r y s t a l 1 - 3 8 in d i f f e r e n t field s t r e n g t h s a s a f u n c t i o n of temperature.
0 H = V H -
0 k() 10.30 k O
A H := 2 0 . 6 9 k ( ) ~ t f -- 32.61 k O
of liquid nitrogen. Hence the curves belonging to the extrema must cross each other between 20 and 60°K. The point of intersection shifts to higher temperature with increasing field strength. The depth and the place of the minima in the Rspee versus T-curves is uncertain, as there are no measuring points between 20 and 60°K. At this minima 2R0 < R H < 3R0. The crystal 1-38 appeared to be less pure than crystal 1-37. The value of Reo/R273 was 26 × 10.4 and 8.6 × 10.4 resp. From figure 3 can be seen that *) Mr G. De V r i e s , nat. phil. drs., was so kind as to take tile X-ray p lotographs and to analyse them.
T H E R M A L A N D E L E C T R I C A L R E S I S T I V I T Y OF T U N G S T E N CRYSTALS
165
the heat c o n d u c t i v i t y of crystal 1-38 will reach a m a x i m u m value ~f only 23 W cm -1 deg -1 at T = 16.9°K, while these values for the crystals A, B and 1-37 would be 48, 56 and 120 W cm -1 deg -1 at 11.5, 10.8 and 6.8°K. The last m e n t i o n e d values are calculated with the help of the formula w - ~T 2 + f l / T (see ref. a)). F r o m the r o t a t i o n a l d i a g r a m s (figs 5 and 6) it was not noticeable, t h a t the (1, 0, 0) direction did not coincide with the axis of the rod. The resistivities show m a x i m a and minilna e v e r y 45 degrees. In the m a x i m a , s e c o n d a r y m i n i m a in Rspee as well as in Wspee are recognisable. 13~cm
/C] /
3-7
IO~pe¢
o
o_o_~ ~ I - - - - 7 ~ J 7 tO
20
I 40
60
80 o K
--T
Fig. 4. Electrical resistivity of crystal 1-38 in different field strengths as a function of temperature. OH = 0 140 [ ] H = 32.61 kO 7, H :- 10.301¢0 ..", H = 36.27 k¢) ~\ H = 20.69 kO I n view of the 90 degree periodicity we have fitted Wspee and Rspee to the following forms: Ze;spee = ao + al cos 40~ + a2 cos 8~ + aa cos 12~ + a4 cos 160~ + a5 cos 20~ + a6 cos 24e. Rspee = b0 + bl cos 4~ + be cos 8e + ba cos 12~ + b4 cos 16~ + b5 cos 20e + b6 cos 24~. The values of a0 to a6 and of b0 to b6 for different t e m p e r a t u r e s and field strengths are given in table I I I . In the last column the value of the anisot r o p y ratio is mentioned.
266
J. D E NOBEL TABLE
H kO
T Oh:
Wspec [ ~]position [ w a t t -x c m w a t t c m - * head deg deg-* cryostat
74.42 70.40 65.77 19.95 19.14 18.18 4.09 3.84 3.61 3.58 3.38
0.423 0.403 0.374 0.0456 0.0452 0.0450 0.125 0.138 0.151 0.152 0.165
75.9 76.1
0.445 0.456
2.25 2.19
71.2 71.4
0.430 0.446
2.33 2.24
67.5 67.6 67.5 67.5
0.382 0.404 0.401 0.404
2.62 2.48 2.49 2.48
58.1 58.5 58.5
0.307 0.383 0.381
20.126 19.172 18.052 17.018 16.092 15.162
0.1792 0.1936 0.214 x 0.2285 0.2384 0.2460
5.59 8.17 4.67 4.38 4.20 3.91
20.112 19.164 19.164 18.015 17.002 16.075 15.092
0.402t 0.4436 0.4454 0.491' 0.5460 0.593 s 0.627 r
2.49 2.25 2.25 2.04 1.83 1.69 1.59
20.093 19,047 18.073 16.980 16.100
0.5624 0.6282 0.7250 0.8030 0.871 s
1.78 1.59 1.38 1.25 1.15
20.192 19.118 18.056 17.043 16.143 15.240
0.7399 0.8194 0.9360 1.057 1.167 1.256
1.38 1.22 1.07 0.949 0.940 0.797
20.109 17.770 15.245
0.8550 1.142 1.50'
1.17 0.876 0.666
I
3.26 2.61 2.63
H kO 20.42 20.39
2.364 2.481 2.674 21.93 22.12 i 22.22 8.00 7.25 6.62 6.58 , 6.06
i
i
20.37 20.35 20.34 20.32
i
1575
20.31 20.28
1575
20.27 20.28
l I
1575 121 114
32.64
1570 114 153
32.63
32.63
7" °K
Wspec I ~' [position w a t t - I c m [ w a t t era-* head deg I deg-* cryostat 0.4104 0.4600 0.477 v 0.4710 0.4790 0.4710 0.4496 0.405 o 0.386 s 0.3913 0.3967 0.4522 0.4745 0.4694 0.4722 0.4514 0.3990 0.3930 0.4041
2.44 2.17 2.09 2.12 2.09 2.12 2.23 2.47 2.59 2.56 2.52 2.21 2.11 2.13 2.12 2,22 2.80 2.54 2.48
150 1355 121 114 107 100 93 785 71 675 64 495 35 27 19 3 348 333 150
0.7602 0.873* 0.9230 0.911 s 0.9237 0.9078 0.8622 0.8225 0.7624 0.7197 0.7465 0.8022 0.855 o 0.896"0.9178 0.907 s 0.9141
1.32 1.15 1.08 1.10 1.08 I.I0 I. 16 1.22 1.31 1.39 1.34 1.25 1.17 1.12 1.09 I.I0 I.II
150 135 s 121 114 107 I00 93 85~ 785 71 64 56~ 49 s 42 35 27 19
1.24 I 1.331 1.54 s 1.64 s 1.690 1.676 1.699 1.65' 1.53 R 1.313 1.23 o 1.332
0.806 0.750 0.647 0.607 0.591 0.596 0.589 0.606 0.652 0.763 0.810 0.750
157 s 150 1355 128 121 I14 107 1O0 93 785 71 64
i
THERMAL
AND ELECTRICAL
RESISTIVITY TABLE
H kO 0
position head
T °K
10 y .R s pec Q cm
77.35
592.1 620.6 655.7 655.2
114
508.7 580.2
165
429.7 509.1
165
355.1 445.8 440.3
165 114
20.88
32.61
0 32.62
72.97
0 32.61
68.51
0 32.61
63.95
0 10.65 20.88
55.35
32.61
215.5 233.6 273.0 268.4 333.5 326.2
cryostat
20.025
10.30 10.29
19.044 17.053 14.209
88.7 112.3 92.1 94.7 97.8 124.5
157 s 114 157 s
237.9 244.3 249.4 356.0 254.1 368.7 266.9 390.5 297.6 407.4
165
376.5 381.6 565.1 392.7 414.0 617.6
1575
531.7 546.2 563.4
165 1575
20.69
20.484 20,043
19.047 17.047
20.67
15,080
26.23
20.410 20.053
26.17 26.16
19.053 17.027 17.025
32.61
20.357
20.064
T cK
10 °.A'spec Q cm
32.59
572.4 630.6 638.0 641.2
187 s
32.58 32.57
19.023 17.053 15.138 14.160
32.61
20.064
555.1 563.4 596.9 . 589.3 728.6 797.8 836.1 845.7 845.3 842.4 847.7 845.6 834.1 805.1 729.0 611.4 545.3 566.5 698.7 764.0 801.2 835.4 849.2 848,5 845.9 850.3 844.8 817.6 748.3
165 1575 150 150 1315 128 1246 121 117s 114 1105 107 1035 100 93 786 71 64 495 42 386 35 31 27 23 19 15 11 3
36.27 36.23
20.020
161¼ 150 121 114
36.27
19.016 16.998 15.092
657.6 743.5 1051.9 1046.5 682.4 730.9 767.9 854.0 1207.7 1212.9 782.7 787.8 887.1 1231.3 1234.1
114 165 114
10.35
114
1575 114 1575 114 1576 114 1575 114
114 1575 114
position head
H kO
165
14.1 12.3
267
CRYSTALS
II
165
20.42 14.50
20.95 20.78
OF TUNGSTEN
14.195
cryostat
161¼
150 121 114 161¼ 157 s 150 121 114
268
J. DE NOBEL
To calculate the coefficients a it is flecessary to know the values of Wspee for one temperature and one field strength. The difficulty is that during the investigation of a rotational diagram of heat conduction the mean temperature of the two resistance thermometers changes considerably (at 15.4°K 1.8 c m
1.5
graad/twatt
--
IOQcm (.2 - -
Q6
--
2
IO?Rspec.
o3
I ao
I 60
I 90
1 ,20
,so°
F i g . 5. Thermal resistivity as a function of the angle between crystal axes and magnetic field at different temperatures and field strengths. H
=
Io
I 0
I 30
I
I
90
t2o
I t5o °
F i g . 6. Electrical resistivity as a function o f the angle between crystals axes and magnetic field at different field strenghts at approx. 2 0 a n d 15°K (the highest curve belonging t o the lower temperature).
20.74 kO
T = 20.0°K
'.~ H
= 32.63 kO
7" = 2 0 . 0 ° K
0 H
:
© H
= 32.62kO
7" =
A H
= 20.69 kO
15.5°K
I 60
10.30 kO
~
H =
[] H
26.23 kO
= 32.61 kO
in a field of 32.63 kQ for instance it changed by as much as 0.37°K). Even if we neglect the temperature variation of the liquid hydrogen bath, the temperature of the colder thermometer is different ~t each position of the crystal with respect to the magnetic field, because the heat resistance of the part of the crystal between the bath and that thermometer changes considerably, when the crystal is turned on its axis (anisotropy latio of Wspee ~ 1.59).
T H E R M A L A N D E L E C T R I C A L R E S I S T I V I T Y OF T U N G S T E N CRYSTALS
9-69
TABLE III H kO
T OK
32.62 32.62 32.63 26.39 26.36 20.76 20.32 10.30 10.30
15.50 15.50 20.00 15.30 20.10 15.15 20.00 15.13 19.97
l
[ a0
aniso-
a,
a2
aa
a4
as
an
tropy factor
1.521 1.518 0.848 1.083 0.614 0.752 0.442 0.267 0.192
0.250 0,253 0.112 0.162 0.090 0.106 0,047 0.022 0,007
--0.027 --0.029 --0.026 --0.023 --0.018 --0.007 --0.013 0 0.002
--0.004 0.002 --0.001 0 0.001 --0.004 --0.002 0 --0.001
--0.021 --0.017 --0.006 --0.013 --0.008 --0.003 --0.004 --0.001 0
0.007 0 --0.002 0.012 --0.001 --0.002 0.001 --0.001 0
--0.010 --0.007 --0.001 --0.001 --0.004 0.001 0 0 . 0
1.45 1.45 1.36 1.38 1.31 1.35 1.28 1.25 1.10
H kO
oK
T bo
bx
b2
b3
b4
b5
be
anisotropy factor
36.27 36.27 36.27 32.61 32.61 26.23 26.23 20.69 20.69 10.30 10.30 10.30
14.195 15.092 20.020 15.138 20.064 15.075 20.053 15.080 20.043 14.209 19.000 20.025
10.47 10.18 8.87 8.25 7.16 5.55 4.86 3.48 3.09 1.124 1.042 1.01 s
2.12 2.12 1,85 1.52 1.49 1.09 0.97 s 0.65 s 0.552 0.14 s 0.121 O. 124
--0.21 --0.12 --0.16 0.05 --0.12 --0.07 --0.10 --0.034 --0.044 --0.009 --0.01 o --0.01'
0. I I 0. I I 0.08 --0.08 0.02 0.05 0.02 0.031 0.02 t 0.00* 0.00~ 0
--0.12 --0.09 --0.12 0.I0 --0.085 --0.07 --0.05 --0.532 --0.032 --0.00 e --0.006 --0.006
0.02 --0.01 0 --0.17 --0.01 --0.00 s --0.005 --0.007 --0.002 --0.002 --0.00' --0.002
--0.05 --0.04 --0.05 0.05 --0.02 --0.03 --0.13 --0.015 --0.012 --0.00 a --0.00 t --0.00 i
1.59 1.59 1.61 1.59 1.57 1.57 1.57 1.51 1.50 1.35 1.33 1.32
A second complication is, that on a measuring day the current through the magnet slowly falls off. To reduce a rotational diagram for the thermal resistivity to one temperature and one field strength, it was necessary to construct plots of the thermal resistivity as a function of temperature for different field strengths for one position of the crystal. From these plots one could deduce the thermal resistivity as a function of the magnetic field at different temperatures. Repeating this treatment for other positions of the crystal the corrections on the measured thermal resistances for variations of mean temperature and field strength are known. A part of this research was done under the auspices of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)" and was made possible by a financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk onderzoek (Z.W.O.)" and from the "Nederlandse Centrale Organisatie voor Toegepast-Natuurwetenschappelijk Onderzoek (T.N.O.)". Received 11-1-57. REFERENCES 1) De H a a s , W. J. and De N o b e l , J., Commun. Kamerlingh Onnes Lab., Leiden No. 251d; Physica 5 (1938) 449. 2) De N o b e l , J., Commun. No. 278b; Physica 15 (1949) 532. 3) De N o b e l , J., Commun. Suppl. No. l13a; Physica ,°3 (1957) 273. 4) De H a a s , W. J. and B i e r m a s z , Th., Commun. No. 249a; Physica 4 (1937) 752.