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The dependence on thickness of the thermal resistance of crystals at low temperatures
P h y s i c a V, no. 7
Juli 1938
T H E D E P E N D E N C E ON THICKNESS OF T H E T H E R M A L RESISTANCE OF CRYSTALS AT LOW T E M P E R A T U R E S by W. J. DE HAAS and TH. BIERMASZ Commun. No. 253b from the Kamerlingh Onnes Laboratory, Leiden
We have already published some measurements on the formdependence of the resistance of SiO2 and KC1 1). The explanation of the phenomenon might be that, when the free path l of the elastic waves becomes comparable with the diameter of the rod, diffuse reflections of these waves from the walls of the rod will then begin to influence the conductivity. These reflections would also account for the increase in resistance of thin rods at low temperatures. It seemed of importance to investigate more precisely the dependence of the "specific" resistance on the thickness of the rod. In the case of KC1 a rather sudden saturation with increasing thickness was found. We have now measured two other rods IIA and III, of SiO 2 _1_ the principal axis, in the same way as previously rods I and II. The dimensions of the rods were, I. length 3.20 cm, diameter IIA. ,, 4.48 . . . . . II. ,, 4.80 . . . . . III. ,, 4.40 . . . . .
0.216 0.359 0.454 0.775
cm ,, ,, ,,
Rod IIA has been ground from II. The results are given in table I and figs. 1 and 2. The points in liquid hydrogen are plotted in fig. 1. They do agree rather well, although the curve of II and IIA is somewhat steeper than that of I and III. The latter two are in very good agreement with each other. Fig. 2 shows the resistance curves in liquid helium. It is evident, that at any rate no sudden saturation for quartz exists. The curve of the thickest rod, III, lies considerably lower than -
-
619
-
-
620
w.j.
DE HAAS AND TH. BIERMASZ
the others, so that we might suspect, that the resistance w is inversely proportional to the thickness d. TABLE I T h e r m a l r e s i s t a n c e of Si02 I
T(°K) 89.4 78.0 67.6 20.12 19.07 18.89 18.80 17.17 15.82 15.38 15.11
i Wspec ( w a t t -1 em) 3.44 3.07 2.64 0.270 ~ 0.240 T 0.24000.2369 O. 1962
0;1658 0.159 0.151
IIA
T(°K) 14.90 4.62 4.33 3.81 3.45 3.10 2.73 2.37 2.20 1.80 1.67
Wspec ( w a t t -~ Cm) 0.154 0.435 0.502 '0.668
0.822 1.06o
1.374 2.098 2.56 °4,54
II
T(°K)
Wspec ( w a t t -~ em)
T(°K)
20.32 18.96 17.97 17.08 4.04 3.77 3.33 2.96 2.64
0,303 0.262 0.225 0,190 0.411 0,463 0,585 0,749,~ 0,931
19.81 18.27 17.75 16.75 15.99 15.40 4.10 3.83 3.47 3.13
Wspec ( w a t t -~ em)
watt~cm
T(°K)
Wspec ( w a t t -~ cm)
0.551 0.745
19.73 18.71 18.36 17.84 17.32 16.34 4.20 3.59 3.26 3.11 2.97
0.253 0.227 0.218 0.204 0.192 0.173 0.238 0.313 0.35'= 0.38 s
1.066
2.78
0.50 s
0.266 l 0.2363 0.213 ~ O. 1820O. 1 6 4 `= O. 1460 0.331 ~ 0.386 0.439
2.68 2.32
5,85
III
0.461
7
0.2
ol I
Y (10
14
•
16
tO
20
IX
22°K
Fig. I. Spec. thermal resistance of Sio2 _L in liquid hydrogen for rods of different cross-section In that case, the conductivity X would be proportional to d. X plotted against d will then be a straight line through the origin. For three temperatures: 2.5°K, 2.9°K and 3.3°K, we plotted X against d (fig. 3). The (;% d) curves do seem to go through the origin, b u t they are not straight lines. The resistance therefore is not inversely
THE DEPENDENCE ON THICKNESS OF THE THERMAL RESISTANCE
621
proportional to d; for d great enough a saturation will be reached. WStt-tcrn
2
4
~T
ff'K
Fig. 2. Spec. t h e r m a l resistance of SiO2 rods of different cross-section in liquid h e l i u m . 3 ~ t a t t c m "1
~ . ~
~/i-~
T= 3.3~';(
T.2~°K
Ta2.S°K
J
I
°3
0.6
O.8cm
Fig. 3. Spec. t h e r m a l c o n d u c t i v i t y of SiO2 _]_ as a f u n c t i o n of the thickness.
Extrapolating one finds this thickness to be about three or four cm. It is possible to think, that since the hydrogen curve of n and IIA
622
W. J. DE HAAS AND TH. BIERMASZ
shows a slight deviation, the points & and W in fig. 3 are somewhat too high. It seems to us, t h a t nevertheless in that case too the (;% d) curves will be no straight lines. Moreover it is clear, that the points of I and I I I do not lie on a straight line through the origin. It is to be remarked t h a t the curvatures of the three (),, d) curves relative to the conductivities, are constant. This means t h a t at the lowest temperaturesa saturation will still exist. For a rod with saturation, the free path l will be much smaller than d. We may think t h a t then the value of l is determined by the rate of scattering of the elastic waves by partial reflections from the walls of the mosaicstructure crystals. The reflections from the mosaic-crystals might cause the increase in resistance for the infinite lattice. If the mosaiccrystals are small, the scattering is great, which means that saturation will be soon reached. The mosaic crystals of quartz being much larger than those of KC!, the saturation of the latter substance will perhaps therefore be reached at a smaller thickness than in the case of quartz. Quartz shows, in fact, saturation' at three, KC1 at about one cm. However, the mosaic crystal of quartz is generally a thousand times greater than that of KC1. It is possible, though, that the reflection coefficients for the two substances are different. The existence of a saturation perhaps can be considered as a proof of the reality of the reflections of the elastic waves from the mosaic crystals. It is possible to calculate the value of l and its temperature dependence, by applying D e b ij e's formula:
~qcl X--
4
(1)
In this formula c represents the specific heat per gram, p the density and q the velocity of the elastic waves; p and q are almost independent to the temperature, hence l will show the same temperature dependence as Z/c. For] temperatures higher than 0.5 ® (69 is the D e b ij e tem-perature) c is constant. At these temperatures ;~ ~-~ 1/T; and therefore l ~ 1/T. If the temperature is taken lower, c begins to decrease while ;~ remains' about ~-~ 1/T, Iso that the increase of l with decreasing temperature will be still stronger. By substituting for KC1 the values of c=0.00235/37.3 cal/degr, at 4°K2), p = 1.98; q = 3.2.105 cm/sec and k in (1), one finds 1 to be 1 ram. For k has been taken 4 × 0.24 cal/cm3), which is about the conductivity there
T H E D E P E N D E N C E ON T H I C K N E S S OF T H E THERMAL R E S I S T A N C E
623
would be, if no reflexions from the wall were present to. cause an increase in resistance. For temperatures lower than the temperature at which the minimum of the (wT) curve is found, this calculated value of l is quite comparable with the thickness of the rod (2.5 mm), in agreement with the fact, that for these temperatures the dependence of the "specific" resistance upon the thickness begins. We have to expect, that in this temperature region, the dependence of l on temperature would be less strong. In the case of saturation the dissipation of energy of the waves namely would be determined only b y the scattering on the mosaic crystals. This can be supported b y (1). In effect, for quartz X ~-~ T 2"s, at the lowest hehumtemperatures even ~-~ T 3, as well as c. TtXerefore l is nearly constant. In the case of KC1, l will show, however, a temperature dependence. As to diamond 1), previous considerations show a connection between the specific heat and the relatively high value (24°K) of the temperature at which the increase begins. The specific heat of diamond diminishes rapidly, owing to its high ® value. As the experiment proves ),, and therefore 1 c to be constant, l must increase with decreasing temperature at the same rate as c decreases. The increase of l therefore being very rapid too, we have to expect, that already for a rather high temperature, 24°K, l becomes comparable with the thickness of the rod. This really is the case, for the calculated value of l, if the values of X, c, g and q at 24°K is substituted. For still lower temperatures the increase in resistance would again be caused by reflections of elastic waves b y the walls of the diamond rod. As a rule, the minimum in the (wT) curve will shift to higher temperatures, if the specific heat per cm at low temperatures of the substance is smaller. For thin rods, the 0,, d) curves of quartz are straight lines. For d -+ 0, X -+ 0, so that in that case the "specific" resistance becomes infinite. The behaviour of SiO 2// in liquid helium is in agreement with the form dependence of Si02 _k. The specific resistance of the previously measured thin rod IA 3) in liquid hydrogen is two times smaller than that of SiO 2 _i_. By multiplying the resistances of IA b y two, the hydrogen curves of both orientations are brought into agreement. If one corrects in this way for T -~ 2.9°K and 3.3°K the corrected values of the resistance of IA, and then calculates the conductivities ~, it is found, that the latter lie on the (X, d) curves of quartz J_, if for d is taken the diameter of I,. In fig. 3 these points
624
T H E D E P E N D E N C E ON T H I C K N E S S OF T H E THERMAL R E S I S T A N C E
are denot.ed by + . The diameter of IA being small (1.3 mm), it is to be expected that for very small diameters the "specific" resistance will really increase enormously. Summarizing, we may say that the hypothesis of reflections of elastic waves, from the walls of the rod, or from the walls of the mosaic-crystals, has made it possible to describe the different phenomena from one point of view. Received May 28th 1938.
REFERENCES 1) W. J. d e H a a s and T h . B i e r m a s z . Physica, 's-Grav. 5, 47, 1938. Commun. Kamerlingh Onnes Lab., Leiden No. 251a. W.J. de Haas andTh. Biermasz. Physica,'s-Grav. 5, 320,1938 ; Commun. No. 251b. 2) C. W. C l a r k , Diss. Leiden. 3) W . J . d e H a a s and T h . B i e r m a s z . Physica, 's-Grav. ~, 673, 1935.
Juli 1938
T H E D E P E N D E N C E ON THICKNESS OF T H E T H E R M A L RESISTANCE OF CRYSTALS AT LOW T E M P E R A T U R E S by W. J. DE HAAS and TH. BIERMASZ Commun. No. 253b from the Kamerlingh Onnes Laboratory, Leiden
We have already published some measurements on the formdependence of the resistance of SiO2 and KC1 1). The explanation of the phenomenon might be that, when the free path l of the elastic waves becomes comparable with the diameter of the rod, diffuse reflections of these waves from the walls of the rod will then begin to influence the conductivity. These reflections would also account for the increase in resistance of thin rods at low temperatures. It seemed of importance to investigate more precisely the dependence of the "specific" resistance on the thickness of the rod. In the case of KC1 a rather sudden saturation with increasing thickness was found. We have now measured two other rods IIA and III, of SiO 2 _1_ the principal axis, in the same way as previously rods I and II. The dimensions of the rods were, I. length 3.20 cm, diameter IIA. ,, 4.48 . . . . . II. ,, 4.80 . . . . . III. ,, 4.40 . . . . .
0.216 0.359 0.454 0.775
cm ,, ,, ,,
Rod IIA has been ground from II. The results are given in table I and figs. 1 and 2. The points in liquid hydrogen are plotted in fig. 1. They do agree rather well, although the curve of II and IIA is somewhat steeper than that of I and III. The latter two are in very good agreement with each other. Fig. 2 shows the resistance curves in liquid helium. It is evident, that at any rate no sudden saturation for quartz exists. The curve of the thickest rod, III, lies considerably lower than -
-
619
-
-
620
w.j.
DE HAAS AND TH. BIERMASZ
the others, so that we might suspect, that the resistance w is inversely proportional to the thickness d. TABLE I T h e r m a l r e s i s t a n c e of Si02 I
T(°K) 89.4 78.0 67.6 20.12 19.07 18.89 18.80 17.17 15.82 15.38 15.11
i Wspec ( w a t t -1 em) 3.44 3.07 2.64 0.270 ~ 0.240 T 0.24000.2369 O. 1962
0;1658 0.159 0.151
IIA
T(°K) 14.90 4.62 4.33 3.81 3.45 3.10 2.73 2.37 2.20 1.80 1.67
Wspec ( w a t t -~ Cm) 0.154 0.435 0.502 '0.668
0.822 1.06o
1.374 2.098 2.56 °4,54
II
T(°K)
Wspec ( w a t t -~ em)
T(°K)
20.32 18.96 17.97 17.08 4.04 3.77 3.33 2.96 2.64
0,303 0.262 0.225 0,190 0.411 0,463 0,585 0,749,~ 0,931
19.81 18.27 17.75 16.75 15.99 15.40 4.10 3.83 3.47 3.13
Wspec ( w a t t -~ em)
watt~cm
T(°K)
Wspec ( w a t t -~ cm)
0.551 0.745
19.73 18.71 18.36 17.84 17.32 16.34 4.20 3.59 3.26 3.11 2.97
0.253 0.227 0.218 0.204 0.192 0.173 0.238 0.313 0.35'= 0.38 s
1.066
2.78
0.50 s
0.266 l 0.2363 0.213 ~ O. 1820O. 1 6 4 `= O. 1460 0.331 ~ 0.386 0.439
2.68 2.32
5,85
III
0.461
7
0.2
ol I
Y (10
14
•
16
tO
20
IX
22°K
Fig. I. Spec. thermal resistance of Sio2 _L in liquid hydrogen for rods of different cross-section In that case, the conductivity X would be proportional to d. X plotted against d will then be a straight line through the origin. For three temperatures: 2.5°K, 2.9°K and 3.3°K, we plotted X against d (fig. 3). The (;% d) curves do seem to go through the origin, b u t they are not straight lines. The resistance therefore is not inversely
THE DEPENDENCE ON THICKNESS OF THE THERMAL RESISTANCE
621
proportional to d; for d great enough a saturation will be reached. WStt-tcrn
2
4
~T
ff'K
Fig. 2. Spec. t h e r m a l resistance of SiO2 rods of different cross-section in liquid h e l i u m . 3 ~ t a t t c m "1
~ . ~
~/i-~
T= 3.3~';(
T.2~°K
Ta2.S°K
J
I
°3
0.6
O.8cm
Fig. 3. Spec. t h e r m a l c o n d u c t i v i t y of SiO2 _]_ as a f u n c t i o n of the thickness.
Extrapolating one finds this thickness to be about three or four cm. It is possible to think, that since the hydrogen curve of n and IIA
622
W. J. DE HAAS AND TH. BIERMASZ
shows a slight deviation, the points & and W in fig. 3 are somewhat too high. It seems to us, t h a t nevertheless in that case too the (;% d) curves will be no straight lines. Moreover it is clear, that the points of I and I I I do not lie on a straight line through the origin. It is to be remarked t h a t the curvatures of the three (),, d) curves relative to the conductivities, are constant. This means t h a t at the lowest temperaturesa saturation will still exist. For a rod with saturation, the free path l will be much smaller than d. We may think t h a t then the value of l is determined by the rate of scattering of the elastic waves by partial reflections from the walls of the mosaicstructure crystals. The reflections from the mosaic-crystals might cause the increase in resistance for the infinite lattice. If the mosaiccrystals are small, the scattering is great, which means that saturation will be soon reached. The mosaic crystals of quartz being much larger than those of KC!, the saturation of the latter substance will perhaps therefore be reached at a smaller thickness than in the case of quartz. Quartz shows, in fact, saturation' at three, KC1 at about one cm. However, the mosaic crystal of quartz is generally a thousand times greater than that of KC1. It is possible, though, that the reflection coefficients for the two substances are different. The existence of a saturation perhaps can be considered as a proof of the reality of the reflections of the elastic waves from the mosaic crystals. It is possible to calculate the value of l and its temperature dependence, by applying D e b ij e's formula:
~qcl X--
4
(1)
In this formula c represents the specific heat per gram, p the density and q the velocity of the elastic waves; p and q are almost independent to the temperature, hence l will show the same temperature dependence as Z/c. For] temperatures higher than 0.5 ® (69 is the D e b ij e tem-perature) c is constant. At these temperatures ;~ ~-~ 1/T; and therefore l ~ 1/T. If the temperature is taken lower, c begins to decrease while ;~ remains' about ~-~ 1/T, Iso that the increase of l with decreasing temperature will be still stronger. By substituting for KC1 the values of c=0.00235/37.3 cal/degr, at 4°K2), p = 1.98; q = 3.2.105 cm/sec and k in (1), one finds 1 to be 1 ram. For k has been taken 4 × 0.24 cal/cm3), which is about the conductivity there
T H E D E P E N D E N C E ON T H I C K N E S S OF T H E THERMAL R E S I S T A N C E
623
would be, if no reflexions from the wall were present to. cause an increase in resistance. For temperatures lower than the temperature at which the minimum of the (wT) curve is found, this calculated value of l is quite comparable with the thickness of the rod (2.5 mm), in agreement with the fact, that for these temperatures the dependence of the "specific" resistance upon the thickness begins. We have to expect, that in this temperature region, the dependence of l on temperature would be less strong. In the case of saturation the dissipation of energy of the waves namely would be determined only b y the scattering on the mosaic crystals. This can be supported b y (1). In effect, for quartz X ~-~ T 2"s, at the lowest hehumtemperatures even ~-~ T 3, as well as c. TtXerefore l is nearly constant. In the case of KC1, l will show, however, a temperature dependence. As to diamond 1), previous considerations show a connection between the specific heat and the relatively high value (24°K) of the temperature at which the increase begins. The specific heat of diamond diminishes rapidly, owing to its high ® value. As the experiment proves ),, and therefore 1 c to be constant, l must increase with decreasing temperature at the same rate as c decreases. The increase of l therefore being very rapid too, we have to expect, that already for a rather high temperature, 24°K, l becomes comparable with the thickness of the rod. This really is the case, for the calculated value of l, if the values of X, c, g and q at 24°K is substituted. For still lower temperatures the increase in resistance would again be caused by reflections of elastic waves b y the walls of the diamond rod. As a rule, the minimum in the (wT) curve will shift to higher temperatures, if the specific heat per cm at low temperatures of the substance is smaller. For thin rods, the 0,, d) curves of quartz are straight lines. For d -+ 0, X -+ 0, so that in that case the "specific" resistance becomes infinite. The behaviour of SiO 2// in liquid helium is in agreement with the form dependence of Si02 _k. The specific resistance of the previously measured thin rod IA 3) in liquid hydrogen is two times smaller than that of SiO 2 _i_. By multiplying the resistances of IA b y two, the hydrogen curves of both orientations are brought into agreement. If one corrects in this way for T -~ 2.9°K and 3.3°K the corrected values of the resistance of IA, and then calculates the conductivities ~, it is found, that the latter lie on the (X, d) curves of quartz J_, if for d is taken the diameter of I,. In fig. 3 these points
624
T H E D E P E N D E N C E ON T H I C K N E S S OF T H E THERMAL R E S I S T A N C E
are denot.ed by + . The diameter of IA being small (1.3 mm), it is to be expected that for very small diameters the "specific" resistance will really increase enormously. Summarizing, we may say that the hypothesis of reflections of elastic waves, from the walls of the rod, or from the walls of the mosaic-crystals, has made it possible to describe the different phenomena from one point of view. Received May 28th 1938.
REFERENCES 1) W. J. d e H a a s and T h . B i e r m a s z . Physica, 's-Grav. 5, 47, 1938. Commun. Kamerlingh Onnes Lab., Leiden No. 251a. W.J. de Haas andTh. Biermasz. Physica,'s-Grav. 5, 320,1938 ; Commun. No. 251b. 2) C. W. C l a r k , Diss. Leiden. 3) W . J . d e H a a s and T h . B i e r m a s z . Physica, 's-Grav. ~, 673, 1935.