- Email: [email protected]
A method for the determination of the thermal resistance of metal single crystals at low temperatures
A METHOD
FOR THE DETERMINATION OF THE THERMAL RESISTANCE OF METAL SINGLE CRYSTALS AT LOW TEMPERATURES by W. J. DE HAAS and W. H. CAPEL Commumcation No. 231 d from the Kamerlingh Onnes Laboratory in Leiden
Zusarnmenfassung Es wird ein neues Messverfahren beschrieben, das besonders geeignet i s t ftir die B e s t i m m u n g d e r W ~ i r m e l e i t f i i h i g k e i t v o n M e t a l l - E i n k r i s t a l l e n bei s e h r t i e f e n T e m p e r a t u r e n . B e i s t a t i o n ~ i r e r W ~ i r m e s t r 6 m u n g w i r d die T e m p e r a t u r d i f f e r e n z z w i s c h e n b e i d e n E n d e n des zu u n t e r s u c h e n d e n StAbchens mit Hilfe zweier Dampfspannungsthermometern b e s t i m m t ; die Energie wird elektrisch zugeffihrt. Die W i i r m e l e i t f i i h i g k e i t e i n e s s e h r r e i n e n W i s m u t - E i n k r i s t a l l s (W~trmes t r o m p a r a l l e l d e r t r i g o n a l e n H a u p t a c h s e ) i s t bei 8 1 . 4 ° K . u n d b e i e i n i g e n T e m p e r a t u r e n z w i s c h e n 20 u n d 16.5°K. b e s t i m m t w o r d e n . B i s z u d e n tiefsten Telnperaturen nimmt der Wiirmewiderstand noch ab mit sinkend e r T e m p e r a t u r , i n f l 3 b e r e i n s t i m m u n g m i t d e n y o n G r ti n e i s e n u n d G o e n s u. A. g e f u n d e n e n R e s u l t a t e n ftir s e h r - r e i n e Metalle.
§ 1. Introduction. The most convenient way to measure thermal resistances at low temperatures is the method of the stationary heatcurrent: a certain amount of energy per second is supplied electrically to one end of the rod to be examined and the temperaturegradient in the rod is determined by measuring the temperature at two places on the rod. This method was first used by L e e s 1) and modified for measurements at low temperatures by S c h o t t 2) and by G r i i n e i s e n and G o e n s S ) . At very low temperatures, however, a difficulty arises in the determination of the temperaturegradient" the thermocouples with which it is measured have a much smaller sensitivity and furthermore it is desirable to use small temperature-gradients. Therefore d e H a a s and B r e m m e r 1) C.W. L e e s, Phil. Trans. Roy Soc. London (A) 208, 38i, 1908. 2) R. S c h o t t , Verh. d.D. Phys. Ges. 18,27, 1916. 3) E. G r i i n e i s e n a n d E . G o e n s , Zs.f. Phys. 44,615, 1927.
726
w.J.
D E HAAS A N D W. H. C A P E L
developed a special method for very low temperatures 1). They determined the temperatures at the ends of the rod and for this purpose one end of the rod was connected to a gas thermometer, while the other end was in direct thermal contact with the liquid in the cryostat. For the examination of bismuth single-crystals, which are broken very easily b y thermal contraction forces, this method seems to be less suitable. The vapour-tension thermometer being a very sensitive secondary thermometer, especially in the regions of liquid hydrogen and liquid helium, we have developed a method in which the temperatures at the ends of the rod are determined b y means of two vapourtension thermometers.
§ 2. Principle o/the method. The rod to be examined connects the lower ends of two tubes B1 and B2, which are filled for example with liquid hydrogen. The tube Bl contains a berating coil and is connected to a manometer; the tube B 2 is connected to a second manometer and to a pump. B y means of this pump the vapour pressure in B2 is kept at a constant value, so that one end of the rod is at a constant temperature. When now a heating current is switched on through the heating coil in B~, the temperature of the liquid in Bl will rise until the difference of temperature between the two tubes has beCome so large that the amount of energy developed per second equals the amount of energy flowing through the cross section of the rod per second. Then the equilibrium has been reached: no further evaporation of the liquid in B~ takes place and the manometer indicates a constant pressure. B y measuring the energy development in the heating coil and the temperatures of the liquid in B I and B2 the thermal resistance of the connecting rod can be determined. § 3. Description o/the apparatus. Working out this principle we have constructed an apparatus of which fig. 1 gives a general survey; fig. 2 gives a sketch of a part of the apparatus on a larger scale. A long glass tube B has been sealed in the head of the cryostat (at A) ; this tube can be connected with the high-vacuum pump H or with the fiiling-apparatus V. The tube B is closed vacuum-tight with a ground ioint S b y means of a second tube C. Three long 1) W. J. d e H a a s and H. B r e m m e r , Akad. A m s t e r d a m 34, 325, 1931. . . .
Comm. Leiden No. 214d; Proe. Kon.
A METHOD FOR THE DETERMINATION OF THE THERMAL, ETC.
c~
20
__1_
i 0
K-..
Fig. 1. A p p a r a t u s for t h e d e t e r m i n a t i o n of t h e t h e r m a l r e s i s t a n c e of m e t a l single-crystals at low temperatures.
727
728
W . J . DE HAAS AND W. H. CAPEL
glass tubes, B1, B2 and B3, have been blown through the wall of the tube C. B3 is connected with B2. Both B 1 and B2 are joined to platinum tubes. The platinum tubes are closed off by small cylindrical copper plugs, soldered to them with tin-solder. The bismuth singlecrystals to be examined K, are soldered to the flat ends of these copper plugs with W b o d's metal. Finally the crystals are connected b y a U-shaped copper piece, soldered to the crystals with W o o d's metal. The tube B l contains the heating~coil D; this coil consists of an insulated manganin wire of about 50 ~ resistance, wound around a small copper cylinder, which is joined to the copper plug. Further the tub~ Bl contains a small stirrer F, fastened to the heating coil b y a phosphorbronze spring E. For the measurements of the energy developed four contact wires are necessary, as current and potential have to be determined. Two of the wires are soldered to the copper plug, which is in electrical contact with one end of the manganin wire, while the other two wires are soldered to a platinum wire, brought through the wall of a little side tube and connected to the other end of the manganin wire. The tube B1 is interrupted b y a spiral brass capillary L, Fig. 2. Detail of the apparatus on l a r g e r scale,
sealed to the glass b y means of D e k h ot i n s k y cement. In spite of the symmetry of the apparatus it appeared to be necessary to use this spiral spring in order to prevent breaking of the bismuth-crystals b y thermal contraction forces. A side tube connects B1 either with the filling-apparatus ,V or with the manometer M1. The tube B2 contains a stirrer of the same type as that in B1. Since during the measurement the vapour pressure in the tube B2 is
A METHOD FOR THE DETERMINATION OF THE THERMAL, ETC.
729
kept constant b y means of a pump and a regplating valve, there will be a pressure gradient along the tube and so the manometer M2 has to be connected to the space above the level of the liquid in B2 b y a separate tube B3. A side tube connects B2 either with the filling apparatus V or with the pump P. The tube B3 is connected to the mercury manometer M2 and to an oil differential-manometer for accurate regulating of the pressure. To reduce the heat supply b y radiation as much as possible the tubes B and C have been silvered and some silver screens Z, have been placed in the tube B. These screens are fixed to B2; three of them are in thermal contact with the liquid in B2, being soldered to a platinum tube N, which interrupts the tube B 2. The contact wires of the heating coil are led through the liquid in the cryostat to prevent heat supply along them. They are introduced into the cryostat through the tube R and are soldered to four platinum wires, brought through the wall of four side tubes at the bottom of B. In the tube B four very long thin copper wires T have been soldered to these platinum wires. As it must be possible to move the tube C with the whole assembly of tubes and the crystals out of B, the wires T have to be longer than the tube B. To push down these wires in putting together the apparatus a small disc W of nearly the same diameter as the tube B has been fixed to B 2 b y means of a brass rod U. In this disc have been screwed four short brass rods to which are soldered, at the bottom the wires T and at the top the contact wires of the heating coil. With this apparatus it is possible to interchange quickly the material to be investigated, it being unnecessary to open any vacuum-tight soldered joints. For the measurement of the vapour pressures, necessary for the determination of the temperatures, mercury manometers are used; the energy development in the heating coil is determined b y means. of two precise ammeters. § 4. The measurements. The tubes B1 and B2 are filled, for example with liquid hydrogen, b y condensation of hydrogen gas. In order to get a thermal contact between the cryostat and the tubes B1 and B 2 some hydrogen gas is admitted in the tube B. After the filling the Cube B is connected to the high-vacuum pomp. For complete thermal
730
W . J . DE HAAS AND W. I-I. CAPEL
insulation the pressure in this tube should be 10- s mm Hg or less. The temperature of the liquid in B2 is kept at a constant value and a heating current is switched on. Instead of waiting until equilibrium ihas been reached we bring the temperature of B~ as soon as possible to a suitable value and try then to determine the heating current for which the temperature of the liquid in B~ remains constant. This is .done b y bracketing the heating current between consecutive series of upper and lower limits. When the correct heating current has been .determined, the temperatures are measured b y determining the v a p o u r pressures. As there will always be a certain unknown heat supply to the liquid in Bl from other sources, two measurements with different heating currents have to be performed in order to eliminate this unknown amount of energy. This heat supply is caused b y con.duction along the gas column in B1 and along the glass wall of this tube and also b y radiation from above. During these two measurements the temperature of the cryostat is k e p t constant at the same mean temperature, so that no correction ihas to be applied for interchange of heat b y radiation between the •crystals and the wall of the tube B. To get the exact value of the thermal resistance of the bismuth.crystals measurements must be made both with and without the .crystals in the apparatus, the latter measurement yielding the correction-resistance, which includes not only the thermal resistance of the U-shaped copper piece, b u t also the resistance of the copper plugs and the transition-resistance between copper a n d liquid.
§ 5. The calculations. A complete measurement consists of two parts. B y the index a we will denote the values relating to the first part, b y the index b those relating to the second part. Further we .denote b y Q the amount of energy developed per second in the heating coil, b y Q' the unknown heat supply per second, b y T! the temperature of the liquid in Bl, b y T 2 the temperature of the liquid in B2. If X (T) is the thermal conductivity of the connection between the two tubes at T°K., X ( T ) = 1/w(T), where w(T) is the thermal :resistance at T ° K.), we have: Q + Q' -~-- ~ ('~) ( r 1 - - r2) = .~. (T) A r
(1),
A METHOD
FOR THE DETERMINATION
OF THE THERMAL,
ETC.
7~ 1
where ~_
T1 + T 2 2 '
assuming X(T) to be a linear function of T in the temperature interval between T2 and T~. The measurement gives the values of Q~, Qb, Tla, T2~, T1 b and T2b; from equation (1) it follows that:
or, b y subtraction: ---- X (T) (A T ~ -
A Ta)
and: x (T) -
1 ~ (T)
Q~ - - Qa A ~ - - A Ta
(2).
The values of Q are deduced from the current measurements of the two ammeters. For the calculation of the temperatures from the vapour pressure of the hydrogen We use the formula of K e e s o m, ]3 ij 1 and Miss van der Horstl): 0 ---- - - 260.865 + 1.0619 1°log ib + 1.7233 1°log2 p , where 0 is the temperature in °C. and p the vapour pressure in cm Hg. To the temperatures so calculated we apply a correction for the deviations from this formula. In this way we can determine the values of A T very exactly. For the calculation of the temperatures of the liquid oxygen from the vapour pressure we use the results of measurement b y C a t h 2). Equation (2) gives the value of the total thermal resistance w' (T). B y subtraction of the value w" (T) of the correction-resistance we get the exact value w (T) of the thermal resistance of the bismuth. ) W.H. K e e s o m , A. B ij 1 a n d Miss H . v a n d e r H o r s t, c o m m . L e i d e n No. 2 1 7 a ; P r o c . K o n . A k a d . A m s t e r d a m 34, 1223, 1931. 2) P . G . C a t h, C o m m . L e i d e n No. 152d.
732
w.j.
DE HAAS AND W. H. CAPEL
§ 6. The thermal resistance o/bismuth single-crystals at low temperatures. We used very pure bismuth supplied by A d a m H i 1 g e r Ltd., London (H. S. Brand, Laboratory No. 8016). According to the report on the spectrographic analysis this bismuth has a purity of 9 9 . 9 9 5 0 , containing a trace of silver and very slight traces of a few other elements. From this b i s m u t h we prepared a large single-crystal, the long axis of the rod being parallel to the trigonal principal axis 1). Out of the large single-crystal we split two equally long pieces, length about 28 mm and cross section about 5 × 5 mm 2 (crystals Bi-III-1934 P ) ' ) . The ends of these pieces were covered electrolytically with a thin copper layer to get good solder-contacts. The two equal crystals were soldered into the apparatus in the above mentioned way. We made different measurements in the temperature region of liquid hydrogen and also one in the temperature region of liquid oxygen. The results are given in table I as a function of the mean temperature. All resistances have been expressed in watt ~1. T A B L E I. T h e r m a l resistance of single-crystals Bi-I I I- 1934 P. T °K.
for Watt--1
81.44 20.01 19.83 19.53 18.33 17.63 16.54
241.6 72.4 65.3 64.5 66.7 63.1 62.3
font Watt--1 74.8 17.7 16.4 16.4 18.6 18.8 18.3
fo Watt--1 166.8 54.7 48.9 48.1 48.1 44.3 44.0
From the two measurements of the thermal resistance at 19.53°K. it follows that our accuracy was about 1.6% . In the hydrogen region we used temperature differences varying from 0.75 to 1 degree and in the measurement at 81.4°K. a difference of 3 degree; for the determination of the correction-resistance these differences were taken much smaller. l)
Cf. L. S c h u b n i k o w ,
Comm. Leiden No. 207b; Proc. Kon. Akad. A m s t e r d a m
33, 327, 1930. 2) B y P we denote the crystals grown in a direction parallel to the trigonal axis, while we d e n o t e b y S a crystal grown in a direction perpendicular to the trigonal axis.
A M E T H O D F O R T H E D E T E R M I N A T I O N OF T H E T H E R M A L , ETC .
733
In table II we give the values of the specific thermal resistance wspec. (expressed both in watt -1 cm and in ca1-1 cm gec) and of the specific heat conductivity ),spec. (expressed both in watt cm -1 and in cal cm -1 sec -1) calculated from the thermal resistance w and the dimensions of the crystals. Because of the shape of the crystals, which had a square cross section, the absolute values are not very exact; t h e y are simply mentioned for the purpose of comparison. T A B L E II. Specific thermal resistance and heat-conduction coefficient for Bi P-er zstals.
T °K.
Wsp
Wsp
;ksp
Xsp
w a t t - ~ em cal - t em see w a t t cm - ~ cal
81.44 20.01 19.53 18.33
17.63 16.54
6.01
25.2
1.97 1.75 1.73 1.60 1.59
8.3 7.3 7.3 6.7 6.6
0.166 0.507 0.572 0.577 0.626 0.631
gIll--lseg -1
0.0397 .0.121 0.137 0.138 0.150 0.151
Fig. 3 gives the graphical representation of the thermal resistance w of the crystals as a function of temperature in the liquid hydrogen region. The change of the thermal resistance of the crystals with temperature is in accordance with the general behaviour of the thermal resistance for very pure metals 1) : down to the lowest liquid hydrogen temperature the thermal resistance diminishes with decreasing 100WATT -g
50
t6
~T 47
F i g . 3. T h e r m a l
18
t9
20
21"K.
r e s i s t a n c e of s i n g l e - c r y s t a l s B i - I I I - 1 9 3 4 as a function of temperature.
P
1) C f . E . G r i i n e i s e n a n d E . G o e n s , 1 . c . W , J . d e H a a s and H. B r e m m e r , Comm. Leiden Ns. 214d, 220b and c; Proc. Kon. A k a d . A m s t e r d a m 34, 325, 1931 and 35, 131 and 323, 1932.
734
W . J . DE HAAS AND W. H. CAPEL
t e m p e r a t u r e b u t t h e r e seems to be an indication t h a t a m i n i m u m of t h e t h e r m a l resistance will be r e a c h e d at a still lower t e m p e r a t u r e . W h e n we c o m p a r e our results with the value of Xspec.o b t a i n e d b y K a y e a n d H i g g i n s 1), who find for a Bi-crystal Xsm" = 0.0129 s cal c m -1 sec -1 at 27°C., w h e n the h e a t c u r r e n t is parallel to t h e trigonal axis, it appears t h a t the q u o t i e n t ~Wec.20°K./Xspec.300°K.= = 0 . 1 2 1 / 0 . 0 1 2 9 S = 9.3 has a n o t i c e a b l y high value. This value is o n l y exceeded b y the values o b t a i n e d b y G r / i n e i s e n and G o e n s a n d others for some v e r y pure specimens of the good h e a t conductors. W e wish to c o m m e m o r a t e here respectfully the late Mr. J. H. O. van Embden Grondijs, phil. nat. cand., to w h o m we are g r e a t l y i n d e b t e d for his valuable help during the p r e p a r a t i o n s of this research.
1)
G.W.C. Kaye
andW. F. H i g g i n s ,
Phil. Mag.(7) 8,1056,1929.
FOR THE DETERMINATION OF THE THERMAL RESISTANCE OF METAL SINGLE CRYSTALS AT LOW TEMPERATURES by W. J. DE HAAS and W. H. CAPEL Commumcation No. 231 d from the Kamerlingh Onnes Laboratory in Leiden
Zusarnmenfassung Es wird ein neues Messverfahren beschrieben, das besonders geeignet i s t ftir die B e s t i m m u n g d e r W ~ i r m e l e i t f i i h i g k e i t v o n M e t a l l - E i n k r i s t a l l e n bei s e h r t i e f e n T e m p e r a t u r e n . B e i s t a t i o n ~ i r e r W ~ i r m e s t r 6 m u n g w i r d die T e m p e r a t u r d i f f e r e n z z w i s c h e n b e i d e n E n d e n des zu u n t e r s u c h e n d e n StAbchens mit Hilfe zweier Dampfspannungsthermometern b e s t i m m t ; die Energie wird elektrisch zugeffihrt. Die W i i r m e l e i t f i i h i g k e i t e i n e s s e h r r e i n e n W i s m u t - E i n k r i s t a l l s (W~trmes t r o m p a r a l l e l d e r t r i g o n a l e n H a u p t a c h s e ) i s t bei 8 1 . 4 ° K . u n d b e i e i n i g e n T e m p e r a t u r e n z w i s c h e n 20 u n d 16.5°K. b e s t i m m t w o r d e n . B i s z u d e n tiefsten Telnperaturen nimmt der Wiirmewiderstand noch ab mit sinkend e r T e m p e r a t u r , i n f l 3 b e r e i n s t i m m u n g m i t d e n y o n G r ti n e i s e n u n d G o e n s u. A. g e f u n d e n e n R e s u l t a t e n ftir s e h r - r e i n e Metalle.
§ 1. Introduction. The most convenient way to measure thermal resistances at low temperatures is the method of the stationary heatcurrent: a certain amount of energy per second is supplied electrically to one end of the rod to be examined and the temperaturegradient in the rod is determined by measuring the temperature at two places on the rod. This method was first used by L e e s 1) and modified for measurements at low temperatures by S c h o t t 2) and by G r i i n e i s e n and G o e n s S ) . At very low temperatures, however, a difficulty arises in the determination of the temperaturegradient" the thermocouples with which it is measured have a much smaller sensitivity and furthermore it is desirable to use small temperature-gradients. Therefore d e H a a s and B r e m m e r 1) C.W. L e e s, Phil. Trans. Roy Soc. London (A) 208, 38i, 1908. 2) R. S c h o t t , Verh. d.D. Phys. Ges. 18,27, 1916. 3) E. G r i i n e i s e n a n d E . G o e n s , Zs.f. Phys. 44,615, 1927.
726
w.J.
D E HAAS A N D W. H. C A P E L
developed a special method for very low temperatures 1). They determined the temperatures at the ends of the rod and for this purpose one end of the rod was connected to a gas thermometer, while the other end was in direct thermal contact with the liquid in the cryostat. For the examination of bismuth single-crystals, which are broken very easily b y thermal contraction forces, this method seems to be less suitable. The vapour-tension thermometer being a very sensitive secondary thermometer, especially in the regions of liquid hydrogen and liquid helium, we have developed a method in which the temperatures at the ends of the rod are determined b y means of two vapourtension thermometers.
§ 2. Principle o/the method. The rod to be examined connects the lower ends of two tubes B1 and B2, which are filled for example with liquid hydrogen. The tube Bl contains a berating coil and is connected to a manometer; the tube B 2 is connected to a second manometer and to a pump. B y means of this pump the vapour pressure in B2 is kept at a constant value, so that one end of the rod is at a constant temperature. When now a heating current is switched on through the heating coil in B~, the temperature of the liquid in Bl will rise until the difference of temperature between the two tubes has beCome so large that the amount of energy developed per second equals the amount of energy flowing through the cross section of the rod per second. Then the equilibrium has been reached: no further evaporation of the liquid in B~ takes place and the manometer indicates a constant pressure. B y measuring the energy development in the heating coil and the temperatures of the liquid in B I and B2 the thermal resistance of the connecting rod can be determined. § 3. Description o/the apparatus. Working out this principle we have constructed an apparatus of which fig. 1 gives a general survey; fig. 2 gives a sketch of a part of the apparatus on a larger scale. A long glass tube B has been sealed in the head of the cryostat (at A) ; this tube can be connected with the high-vacuum pump H or with the fiiling-apparatus V. The tube B is closed vacuum-tight with a ground ioint S b y means of a second tube C. Three long 1) W. J. d e H a a s and H. B r e m m e r , Akad. A m s t e r d a m 34, 325, 1931. . . .
Comm. Leiden No. 214d; Proe. Kon.
A METHOD FOR THE DETERMINATION OF THE THERMAL, ETC.
c~
20
__1_
i 0
K-..
Fig. 1. A p p a r a t u s for t h e d e t e r m i n a t i o n of t h e t h e r m a l r e s i s t a n c e of m e t a l single-crystals at low temperatures.
727
728
W . J . DE HAAS AND W. H. CAPEL
glass tubes, B1, B2 and B3, have been blown through the wall of the tube C. B3 is connected with B2. Both B 1 and B2 are joined to platinum tubes. The platinum tubes are closed off by small cylindrical copper plugs, soldered to them with tin-solder. The bismuth singlecrystals to be examined K, are soldered to the flat ends of these copper plugs with W b o d's metal. Finally the crystals are connected b y a U-shaped copper piece, soldered to the crystals with W o o d's metal. The tube B l contains the heating~coil D; this coil consists of an insulated manganin wire of about 50 ~ resistance, wound around a small copper cylinder, which is joined to the copper plug. Further the tub~ Bl contains a small stirrer F, fastened to the heating coil b y a phosphorbronze spring E. For the measurements of the energy developed four contact wires are necessary, as current and potential have to be determined. Two of the wires are soldered to the copper plug, which is in electrical contact with one end of the manganin wire, while the other two wires are soldered to a platinum wire, brought through the wall of a little side tube and connected to the other end of the manganin wire. The tube B1 is interrupted b y a spiral brass capillary L, Fig. 2. Detail of the apparatus on l a r g e r scale,
sealed to the glass b y means of D e k h ot i n s k y cement. In spite of the symmetry of the apparatus it appeared to be necessary to use this spiral spring in order to prevent breaking of the bismuth-crystals b y thermal contraction forces. A side tube connects B1 either with the filling-apparatus ,V or with the manometer M1. The tube B2 contains a stirrer of the same type as that in B1. Since during the measurement the vapour pressure in the tube B2 is
A METHOD FOR THE DETERMINATION OF THE THERMAL, ETC.
729
kept constant b y means of a pump and a regplating valve, there will be a pressure gradient along the tube and so the manometer M2 has to be connected to the space above the level of the liquid in B2 b y a separate tube B3. A side tube connects B2 either with the filling apparatus V or with the pump P. The tube B3 is connected to the mercury manometer M2 and to an oil differential-manometer for accurate regulating of the pressure. To reduce the heat supply b y radiation as much as possible the tubes B and C have been silvered and some silver screens Z, have been placed in the tube B. These screens are fixed to B2; three of them are in thermal contact with the liquid in B2, being soldered to a platinum tube N, which interrupts the tube B 2. The contact wires of the heating coil are led through the liquid in the cryostat to prevent heat supply along them. They are introduced into the cryostat through the tube R and are soldered to four platinum wires, brought through the wall of four side tubes at the bottom of B. In the tube B four very long thin copper wires T have been soldered to these platinum wires. As it must be possible to move the tube C with the whole assembly of tubes and the crystals out of B, the wires T have to be longer than the tube B. To push down these wires in putting together the apparatus a small disc W of nearly the same diameter as the tube B has been fixed to B 2 b y means of a brass rod U. In this disc have been screwed four short brass rods to which are soldered, at the bottom the wires T and at the top the contact wires of the heating coil. With this apparatus it is possible to interchange quickly the material to be investigated, it being unnecessary to open any vacuum-tight soldered joints. For the measurement of the vapour pressures, necessary for the determination of the temperatures, mercury manometers are used; the energy development in the heating coil is determined b y means. of two precise ammeters. § 4. The measurements. The tubes B1 and B2 are filled, for example with liquid hydrogen, b y condensation of hydrogen gas. In order to get a thermal contact between the cryostat and the tubes B1 and B 2 some hydrogen gas is admitted in the tube B. After the filling the Cube B is connected to the high-vacuum pomp. For complete thermal
730
W . J . DE HAAS AND W. I-I. CAPEL
insulation the pressure in this tube should be 10- s mm Hg or less. The temperature of the liquid in B2 is kept at a constant value and a heating current is switched on. Instead of waiting until equilibrium ihas been reached we bring the temperature of B~ as soon as possible to a suitable value and try then to determine the heating current for which the temperature of the liquid in B~ remains constant. This is .done b y bracketing the heating current between consecutive series of upper and lower limits. When the correct heating current has been .determined, the temperatures are measured b y determining the v a p o u r pressures. As there will always be a certain unknown heat supply to the liquid in Bl from other sources, two measurements with different heating currents have to be performed in order to eliminate this unknown amount of energy. This heat supply is caused b y con.duction along the gas column in B1 and along the glass wall of this tube and also b y radiation from above. During these two measurements the temperature of the cryostat is k e p t constant at the same mean temperature, so that no correction ihas to be applied for interchange of heat b y radiation between the •crystals and the wall of the tube B. To get the exact value of the thermal resistance of the bismuth.crystals measurements must be made both with and without the .crystals in the apparatus, the latter measurement yielding the correction-resistance, which includes not only the thermal resistance of the U-shaped copper piece, b u t also the resistance of the copper plugs and the transition-resistance between copper a n d liquid.
§ 5. The calculations. A complete measurement consists of two parts. B y the index a we will denote the values relating to the first part, b y the index b those relating to the second part. Further we .denote b y Q the amount of energy developed per second in the heating coil, b y Q' the unknown heat supply per second, b y T! the temperature of the liquid in Bl, b y T 2 the temperature of the liquid in B2. If X (T) is the thermal conductivity of the connection between the two tubes at T°K., X ( T ) = 1/w(T), where w(T) is the thermal :resistance at T ° K.), we have: Q + Q' -~-- ~ ('~) ( r 1 - - r2) = .~. (T) A r
(1),
A METHOD
FOR THE DETERMINATION
OF THE THERMAL,
ETC.
7~ 1
where ~_
T1 + T 2 2 '
assuming X(T) to be a linear function of T in the temperature interval between T2 and T~. The measurement gives the values of Q~, Qb, Tla, T2~, T1 b and T2b; from equation (1) it follows that:
or, b y subtraction: ---- X (T) (A T ~ -
A Ta)
and: x (T) -
1 ~ (T)
Q~ - - Qa A ~ - - A Ta
(2).
The values of Q are deduced from the current measurements of the two ammeters. For the calculation of the temperatures from the vapour pressure of the hydrogen We use the formula of K e e s o m, ]3 ij 1 and Miss van der Horstl): 0 ---- - - 260.865 + 1.0619 1°log ib + 1.7233 1°log2 p , where 0 is the temperature in °C. and p the vapour pressure in cm Hg. To the temperatures so calculated we apply a correction for the deviations from this formula. In this way we can determine the values of A T very exactly. For the calculation of the temperatures of the liquid oxygen from the vapour pressure we use the results of measurement b y C a t h 2). Equation (2) gives the value of the total thermal resistance w' (T). B y subtraction of the value w" (T) of the correction-resistance we get the exact value w (T) of the thermal resistance of the bismuth. ) W.H. K e e s o m , A. B ij 1 a n d Miss H . v a n d e r H o r s t, c o m m . L e i d e n No. 2 1 7 a ; P r o c . K o n . A k a d . A m s t e r d a m 34, 1223, 1931. 2) P . G . C a t h, C o m m . L e i d e n No. 152d.
732
w.j.
DE HAAS AND W. H. CAPEL
§ 6. The thermal resistance o/bismuth single-crystals at low temperatures. We used very pure bismuth supplied by A d a m H i 1 g e r Ltd., London (H. S. Brand, Laboratory No. 8016). According to the report on the spectrographic analysis this bismuth has a purity of 9 9 . 9 9 5 0 , containing a trace of silver and very slight traces of a few other elements. From this b i s m u t h we prepared a large single-crystal, the long axis of the rod being parallel to the trigonal principal axis 1). Out of the large single-crystal we split two equally long pieces, length about 28 mm and cross section about 5 × 5 mm 2 (crystals Bi-III-1934 P ) ' ) . The ends of these pieces were covered electrolytically with a thin copper layer to get good solder-contacts. The two equal crystals were soldered into the apparatus in the above mentioned way. We made different measurements in the temperature region of liquid hydrogen and also one in the temperature region of liquid oxygen. The results are given in table I as a function of the mean temperature. All resistances have been expressed in watt ~1. T A B L E I. T h e r m a l resistance of single-crystals Bi-I I I- 1934 P. T °K.
for Watt--1
81.44 20.01 19.83 19.53 18.33 17.63 16.54
241.6 72.4 65.3 64.5 66.7 63.1 62.3
font Watt--1 74.8 17.7 16.4 16.4 18.6 18.8 18.3
fo Watt--1 166.8 54.7 48.9 48.1 48.1 44.3 44.0
From the two measurements of the thermal resistance at 19.53°K. it follows that our accuracy was about 1.6% . In the hydrogen region we used temperature differences varying from 0.75 to 1 degree and in the measurement at 81.4°K. a difference of 3 degree; for the determination of the correction-resistance these differences were taken much smaller. l)
Cf. L. S c h u b n i k o w ,
Comm. Leiden No. 207b; Proc. Kon. Akad. A m s t e r d a m
33, 327, 1930. 2) B y P we denote the crystals grown in a direction parallel to the trigonal axis, while we d e n o t e b y S a crystal grown in a direction perpendicular to the trigonal axis.
A M E T H O D F O R T H E D E T E R M I N A T I O N OF T H E T H E R M A L , ETC .
733
In table II we give the values of the specific thermal resistance wspec. (expressed both in watt -1 cm and in ca1-1 cm gec) and of the specific heat conductivity ),spec. (expressed both in watt cm -1 and in cal cm -1 sec -1) calculated from the thermal resistance w and the dimensions of the crystals. Because of the shape of the crystals, which had a square cross section, the absolute values are not very exact; t h e y are simply mentioned for the purpose of comparison. T A B L E II. Specific thermal resistance and heat-conduction coefficient for Bi P-er zstals.
T °K.
Wsp
Wsp
;ksp
Xsp
w a t t - ~ em cal - t em see w a t t cm - ~ cal
81.44 20.01 19.53 18.33
17.63 16.54
6.01
25.2
1.97 1.75 1.73 1.60 1.59
8.3 7.3 7.3 6.7 6.6
0.166 0.507 0.572 0.577 0.626 0.631
gIll--lseg -1
0.0397 .0.121 0.137 0.138 0.150 0.151
Fig. 3 gives the graphical representation of the thermal resistance w of the crystals as a function of temperature in the liquid hydrogen region. The change of the thermal resistance of the crystals with temperature is in accordance with the general behaviour of the thermal resistance for very pure metals 1) : down to the lowest liquid hydrogen temperature the thermal resistance diminishes with decreasing 100WATT -g
50
t6
~T 47
F i g . 3. T h e r m a l
18
t9
20
21"K.
r e s i s t a n c e of s i n g l e - c r y s t a l s B i - I I I - 1 9 3 4 as a function of temperature.
P
1) C f . E . G r i i n e i s e n a n d E . G o e n s , 1 . c . W , J . d e H a a s and H. B r e m m e r , Comm. Leiden Ns. 214d, 220b and c; Proc. Kon. A k a d . A m s t e r d a m 34, 325, 1931 and 35, 131 and 323, 1932.
734
W . J . DE HAAS AND W. H. CAPEL
t e m p e r a t u r e b u t t h e r e seems to be an indication t h a t a m i n i m u m of t h e t h e r m a l resistance will be r e a c h e d at a still lower t e m p e r a t u r e . W h e n we c o m p a r e our results with the value of Xspec.o b t a i n e d b y K a y e a n d H i g g i n s 1), who find for a Bi-crystal Xsm" = 0.0129 s cal c m -1 sec -1 at 27°C., w h e n the h e a t c u r r e n t is parallel to t h e trigonal axis, it appears t h a t the q u o t i e n t ~Wec.20°K./Xspec.300°K.= = 0 . 1 2 1 / 0 . 0 1 2 9 S = 9.3 has a n o t i c e a b l y high value. This value is o n l y exceeded b y the values o b t a i n e d b y G r / i n e i s e n and G o e n s a n d others for some v e r y pure specimens of the good h e a t conductors. W e wish to c o m m e m o r a t e here respectfully the late Mr. J. H. O. van Embden Grondijs, phil. nat. cand., to w h o m we are g r e a t l y i n d e b t e d for his valuable help during the p r e p a r a t i o n s of this research.
1)
G.W.C. Kaye
andW. F. H i g g i n s ,
Phil. Mag.(7) 8,1056,1929.