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A cryostat for measuring heat capacities from 1·2 to 300° K and measurements of the specific heat of magnesium oxide below 36° K
A C R Y O S T A T FOR MEASURING HEAT CAPACITIES FROM 1.2 TO 300° K A N D MEASUREMENTS OF THE SPECIFIC HEAT OF MAGNESIUM OXIDE BELOW 36°K E. GMELINI-
THE low temperature specific heat of the alkaline earth oxides is not well known below 50 ° K. Recent measurements by the present author on the heat capacities of beryllium oxide and calcium oxide I between 1.2 and 70 ° K showed different results from those of other workers below 50 ° K (for other work on alkaline earth heat capacities see the following references: for beryllium oxide, 2-4; magnesium oxide, 5-8; calcium oxide, 5, 9, and I0; and for strontium and barium oxides, reference 11). In this temperature range a substantial increase of the Debye temperature 0 is found, which does not fit the extrapolation of the heat capacity between 50 and 100°K to lower temperatures. But it is of interest to study the exact low temperature behaviour of 0 of these oxides. They all have a cubic structure, [except BeO, which has a hexagonal structure]. This suggests, by comparison with the relatively well established properties of the alkali halides that these oxides should have, in a first approximation, ~elatively simple interatomic forces and, as a consequence, their measured properties should be particularly suitable for a direct comparison with the predictions of lattice theory. Unfortunately, the auxiliary data required for an exact analysis (e.g. elastic constant, thermal expansion, from 0 ° K to room temperature) are not known, except for magnesium oxide where the data are reasonably well determined. Theoretical calculations of 0 near 0 ° K for magnesium oxide as obtained by Blackman ~2 by numerical integration, by Betts, Bhatia and Wyman ~a who applied Houston's method, and by means of de Launey's tables ~4 from elastic constants for magnesium oxide at 0 ° K given by Durand ~5 show an excellent agreement (Table 3) giving a value of 0_~ 946 ° K. Recent calculations by Betts ~6 by an extended Houston method disagree with the first results by 3 per cent. The first heat capacity measurements of magnesium oxide, made by Gunther 6 and Parks and Kelley s did not extend below 80 ° K. The measured values agree within t Centre National de Recherches sur les Tr~s Basses Temp6ratures, Grenoble, France, Received 28 March 1967 CRYOGENICS
C
" AUGUST
1967
l per cent with those of Barron, Berg, and Morrison s made on magnesium oxide single crystals, enclosed in a vessel sealed off with a small amount of helium gas inside. But those of Giauque and Archibald, 7 measured in the same temperature range from 20 to 300 ° K show a systematic deviation to higher specific heats (up to 100 per cent). The latter used a specimen of magnesium oxide in the form of small particles ( , ~ 1 000 A) and it was suggested by Jura and Garland ~7 that their results might be attributed to an effect of particle size on the specific heat. It might also be mentioned that heat capacity measurements by Lien and Phillips ~s on even finer magnesium oxide powder ( , ~ 100,~) have shown such a contribution, as expected from the theory of Jura and PitzerJ °
Experimental Apparatus The heat capacity measurements were made with an adiabatic calorimeter assembly which had been used for the experiments with a beryllium monocrystal, 2o with beryllium oxide and calcium oxide ~ and with the copper standard sample 42 received from Argonne National Laboratory. The cryostat will be described here in detail. In Figure 1 is shown schematically the low temperature portion of the apparatus; most of the details are given in the caption. The specimen O, the thermal mechanical contact M, and the gas thermometer bulb (R) were surrounded by a vacuum-tight thermally insulated copper can Q. This made it possible to keep the inner calorimeter (Q) at any desired temperature by means of a semiautomatic heater device (I and K). A second outer calorimeter wall (P) enclosing the inner calorimeter is immersed in the cryogenic fluid helium, hydrogen, or nitrogen) contained in a conventional nitrogen protected metallic dewar vessel. The cryoo~at holds enough liquid helium to last for more than 20 h. The inner calorimeter and the space between P and Q may be evacuated by means of a diffusion pump to a pressure of 1 latorr at room temperature.
In addition to the radiation trap of the inner calorimeter a trap (C) containing several baffles is located in the high vacuum pumping line (B). Ten~. eratures as low as about 1.2° K were obtained by ptnnpmg on the liquid helium bath. Particular attention was paid to the insulation of the cryostat from vibrations. The cylindrical shaped specimens were clamped in two rings suspended by several nylon threads L from the frame. These are illustrated in Figure 2 (H). They consisted of thin walled copper strips, 0.2 mm thick, screwed together permitting easy changing of the specimen and ensuring excellent thermal contact with the sample over the whole temperature range. A 68 f2, 0.1 W Allen Bradley carbon resistor (or a 10012 platinum resistor for the higher temperature range), used as secondary thermometer, was fitted tightly into a small cylindrical copper tube soldered on one holder J where it was kept by vacuum grease. Inside the second holder ring one or several constantan strain gauges (600-1 800 ~) (K) used as sample heaters were attached with Araldite.
The thermal relaxation time of the total sys~m for the measured specimens is less than 1-2 s at temperatures below 10° K. There was no superheating. The heat capacity of the holders (total weight 3-12 g) was always measured separately.
G
\
A B
K
C D E F G L H
I K L M N 0 P Q R S T
A B C D E
Lead wires Pumping tube Radiation trap Insulating vacuum Vapour pressure thermometer bulb F, G Stainless steel capillaries for the thermal switch and the gas thermometer bulb H Lead wires passage
I K L M N O P, R S
Heater Resistance thermometer Heat sinks Mechanical and thermal contact Frame Specimen 0 Interior and exterior calorimeter wall Gas thermometer bulb Wood's metal solder
Figure 1. The calorimeter m
A B C D E F
Lead wires passage Heat sink Lead wires Calorimeter support Helium pressure Metal bellows
G H I d K L
Copper strip Copper clamps Specimen Resistance thermometer Heater Nylon wires
Figure 2. Mechanical heat switch
In order to cool down the sample to the required temperature without any exchange gas, the bellows technique ~o-23 well established in this laboratory was used. It is schematically shown in Figure 2. A tombac bellows F (10 mm in diameter and 20 mm long) closed on one side by a copper plate was solidly attached at the other, to the calorimeter frame D and linked by a capillary to a steel cylinder containing high pressure helium 4. When it was filled with liquid helium under pressure of about 3 kg/cm 2 the copper plate pressed the thermal link against the frame. The thermal link was an annealed 0-2 mm thick copper strip G soldered on the sample holder H. To switch off this contact the helium inside the bellows was pumped off to 1 kg/cm 2 by means of the gasometer line and nearly to vacuum with the helium bath pump. There was a critical moment when the copper strip, not having broken the contact yet, touched the bellows or frame slightly. The mechanical shock liberated 5-20 erg depending on the adjustment of the thermal link in the contact region. Neither the pressure used (1 to 5 kg/cm z) for making the contact nor the releasing conditions (e.g. several seconds to 15 min cut-off time) had any detectable influence on the heat input to the sample. If the switch was clamped at room temperature, and the cryostat then cooled with liquid helium or hydrogen, it sometimes stuck so that it could not be unclamped. This inconvenience was overcome by operating the switch once at liquid nitrogen temperature. CRYOGENICS
• A U G U S T 1967
Operation of this switch involves a very small heat leak into the sample, smaller than any of the values reported by Manchester z4 and by Hill and Pickett. 25 Furthermore, the heat input by mechanical vibrations was reduced to 20-30 erg/min. The employment of a fast amplifier recorder system also contributed to ensuring excellent conditions for the measurement of very low heat capacities without exchange gas. Finally, it is an advantage that with this apparatus no force is applied directly on to the specimen. The cooling down performance is also excellent. A sample (378 g copper) was cooled from room temperature to 4 ° K within 5 h. By measurement of the cooling curve (temperature T versus time, t) of the copper sample, the temperature dependence of the thermal conductance k of the whole cooling system was calculated in the liquid nitrogen and helium regions (Figure 3). The temperature dependence from 2 to 8 ° K was similar to that found by Berman, 26 the k(T) being nearly proportional to T 2 (curve A) and gradually becoming less strongly dependent on T and almost independent at liquid nitrogen temperature. The conductance k is lower than the values measured by Berman (curve B) by a factor of two for a given applied force, but we did not yet try to get as clean surfaces as Berman. As can be seen by comparing curve A and C in Figure 3, the cooling down performance is not limited by the conductance of the copper link, which is much lighter (curve C) than that of the contact. The conductance k of the whole device was found to be proportional at 4.2 ° K to the applied load L (Figure 3, curve D). Temperatures up to 70 ° K could be measured with a 68 f2 Allen Bradley carbon resistor calibrated against the vapour pressure of liquid helium between 1.2 and 4.2 ° K and against a gas thermometer and liquid hydrogen vapour pressure in the higher temperature region. The measuring equipment (Figure 4) for the current and potential of the thermometer resistor Rm, consisted of a double microvolt potentiometer P (MECI) whose output was fed into a d.c. amplifier A (Beckmann) followed by a fast chart-recorder CR (Graphispot) used at speeds between 30"and 120 mm/min, recording off balance as a function of time. This system together with the fast response of the thermometer circuit permitted the measure of heat capacities down to 3 x 10-z mJ/deg at 1.2 ° K. For measurements above 10°K the d.c. amplifier A was replaced by a galvanometer amplifier (Sefram-Amplispot) having less than 10 laV noise and an input impedance of 250 kfl (constant gain). The constant current used in measuring the resistance of Rtn varied from 0.25 laA at 1.2°K to 10 laA at 77 ° K. With these values the current was large enough to give adequate sensitivity for dete?mining the resistance and small enough to give negligible energy dissipation in the resistor. Providing it was of this order of magnitude at the values quoted, the current had no detectable influence on the resistance measured. To eliminate most of the low frequency pick-up, the whole low voltage measuring system was screened separately. The uncertainty in resistance was equivalent to 10 mdeg at 70 ° K, to 1 mdeg at 20 ° K, and less than 0.2 mdeg for temperatures lower than 10° K. A carbon resistance thermometer does not in general behave reproducibly and it is necessary to calibrate the CRYOGENICS
" AUGUST
lg67
resistor for each measurement. However, when no mechanical forces were applied to the resistor and it h a d been cycled several times between room temperatures and helium temperature, the successive calibration c u r v e s were different at most by 1-3 mdeg. }00
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,¢ 001
$ 1 /
/-°
Ol
o 000
A
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I I
I I I IIIJ
~o T
==
~ L-~-- (kg) I
I I I Ill
ioo (*K)
Load L x 0-75 kg • 1.75 kg © 2.00 kg A 9.50 kg Figure 3. T h e t h e r m a l c o n d u c t a n c e k of t h e mechanical heat switch as function of t e m p e r a t u r e T and Load L
The thermometer was calibrated by admitting exchange gas to the calorimeter after a specific heat measurement and taking a set of 20 to 25 calibration points over the range 1.2-4.2 ° K. The temperature of the bath was maintained constant for each point by an automatic temperature control actuated directly by the Allen Bradley resistance on the sample and dissipating energy into a 50 f2 constantan wire around the calorimeter wall. The temperature stability was better than 10-4 deg. Calibration temperatures were determined on the 1958 helium scale of NBS from the vapour pressure of helium measured in a small cavity (E in Figure 1, about 2 cm 3) on top of the calorimeter. This bulb was connected by a tube of 3 mm diameter surrounded by an insulating vacuum D, to a mercury manometer and an oil manometer (butylphthalate) for use below 2-5°K. Both manometers were read with a cathetometer with an accuracy of 2 × 10-z mm. Below the 2 point the pressure was measured from a point directly above the helium bath. The estimated error in reading the manometers corresponded to less than 0.5 mdeg except below 1.8 ° K where possible errors might amount to 4 m d e g at 1.2 ° K,
For the calibration of the temperature scale above 4 ° K we used a gas thermometer with the fixed point at 4.2 ° K. In the range of 14-20 ° K calibration was verified against the liquid hydrogen vapour pressure which was determined by the same method as the helium vapour pressure. Corrections due to ortho-para conversion were applied, if parahydrogen was not used. The
uncertainty of calibration in this temperature range was estimated to be less than 10 mdeg from the hydrogen scale of Dijk and Durieux. 27 // The calibration data between 1 and 4 ° K were fitted to a five parameter equation 1
a3
~, = at enR + a2 + e ~
a5
04
+ ~
+ (enR) '''-'-S
was about 2"66 cm 3 of which 2.50 cm 3 was aT'room temperature (copper capillary and average manometer volume) and 0.16cm 3 at a temperature varying from 4 to 300 ° K. The temperature was calculated from the expression
V(T, P) + P~ Vj PRI(I-" + B(T)/V) ~ R T I ( 1 + BI(T)/VO
constant
i
and the coefficients at calculated by a least square method by computer. The difference ( T - Te) was also determined where T is the temperature taken from the calibration tables mentioned below, and Te the calculated temperature from the equation. In a heat capacity measurement errors in the slope of the calibration curve introduce errors in the specific heat equal to the slope error. So a graph ( T - Te) versus Te shows directly this systematic error, caused by the temperature calculation. This error was always made to be less than 0-2 per cent. For the calibration points above 4 ° K an interpolation formula I/T = a + b en R was used for a first approximation. Then a polynomial T=
T a + ~ C l Tl ( i =
.
.
.
.
.
.
.
.
,
R
_-~ :
|ll
-~ -'J
I [
- -
I
i__. .........
IAI
[
,V ! " b
J
5)
was applied to fit the calibration curve. The coefficients a, b and ct were always calculated by a least-squares method. The temperature range is divided into two regions: 4--25° K and 20--80 ° K. The calculated temperatures differed from the calibration values by less than 0-2 per cent. The circuit is shown schematically in Figure 4. The asymmetrical method 2s of connecting the heater leads to the heater was adopted to eliminate errors due to the heat generated in these leads. The resistance Rh of the current leads is approximately 10f~ each (the heater resistance being greater than 1 000 f~); those for potential measurements 50 1) each. The power dissipated in the heater by a battery (2, 12, or 24 V) was measured using a double potentiometer (P) to measure both the current J through the heater and the potential drop V across the heater, in connection with an electrical clock (C), driven by a standard 50 cycle generator of frequency stability better than I0 -q, used for determining the heating period. The current was switched by a relay which provided steady loading on the current supply whether or not the current was passing through the heater. The total random error in heating energy was estimated to be less than 0.15 per cent. Heating of spurious electrical origin, at both low and high frequencies, was stopped by screening the heater system and by connecting small condensors (0.1 p.F) in parallel with the heater leads on top of the metallic dewar vessel. The gas thermometer, schematically shown in Figure 5 had a bulb (G) made of copper, wall thickness 1 mm, and a volume of 92.20 cm 3. This bulb was connected by a stainless steel capillary (D) (diameter 0.8 mm and 325 mm long) up to room temperature and then by a copper capillary (C) (diameter 1 mm, and 2 465 mm long) to an oil manometer (O). The system was used as a constant volume thermometer by always setting the lower meniscus to very nearly the same mark A. This was done by introducing or pumping gaseous helium over the butylphthalate reservoir (O) with two valves (M and N). The error was about 2 x 10-2 mm. The overall dead volume ~II
where p is the pressure, B the second virial coefficient
k-%3 Heat sink --
wire
Constonton
- - Copper wire Imm - - Copper wire O,O7mm Heater circuit (left) Rth Heater R Heater equivalence resistance C Chronometer P Potentiometer
T h e r m o m e t e r circuit (right) Rth Resistance t h e r m o meter G Constant current generator P Potentiometer A Amplifier CR Chart recorder Figure 4
for helium, and V the volume of bulb. (With
V(T, p)
=
Vo(To.Po) [I + 3 c t ( r - To) + 7(P - p 0 ) ]
where To, po, and Vo are the values at the fixed point temperature, ¢, the expansion coefficient of copper, and Y, the coefficient of compressibility). The second term represents the correction for the dead volumes. In these, the correction term in Bi is negligible. The constant depends on the amount of gas in the system and is determined by calibrating at the boiling point of helium or hydrogen. So the temperature Tcould be represented by the formula T = Ta[l + 3~t(Ta - To)] + y(p - p o ) +
(I/I/o)(B- no)
( Vo/Vo)[(Ta - To)/O] + {(Vo/Vo)[TaF(7")
-
ToF(To)]}
with Ta = (pTo/po) l'.~t
T,
T~
l fdV T(x) -- ;K, dr I K(T)dT
and F(T) = -~M
0
T,
T,
The approximate temperature Ta must be corrected by addition of 5 terms: (1) The thermal expansion of the bulb for which the values for ~t were taken from Rubin, Altmann, and Johnston ;29,30 CRYOGENICS
' AUGUST
1967
(2) The bulb deformation with pressure, which is always negligible; (3) The correction for non ideality of the helium gas, using the data of Keesom ;3t (4) The correction for the dead volume Vo at room temperature 0; (5) The correction for the dead volume VM at varying temperatures, determined by calculating the integrals F(T) with the thermal conductivity for stainless steel, a2 The correction for the thermomolecular pressure difference is negligibly small. The correction values of the items (1) to (5) considered above are shown in Table 1. TABLE 1
i Corrections for T. at
20° K
80° K
Accuracy
Calibration point at
4.2° K
20.4° K
in [%] (1)
0 mdeg K
(2) (3)
o
,,
o
,,
o
-48 + 29 + 7
,, ,, ,,
--8 + 442 + 93
,, ,, ,,
1 5 10
(4) (5)
+ 5 5 mdeg K
5 i ;
H Valve without dead
A Vacuum B Insulating vacuum C Copper capillary
volume
Glass capillary K Measuring level
d
D Stainless steel capillary E Cryostat F Calorimeter walls G Gas thermometer bulb
Accuracy
of the Gas Thermometer
The accuracy of the calibration temperatures at the boiling points of liquid helium and hydrogen, taking 10 points with an error of 10-~ torr is much better than the accuracy of the 1958 helium scale and the hydrogen scale, in which the uncertainty is about 2 mdeg K at 4-2 ° K and 10 mdeg K at 20.4 ° K. The uncertainty of the measurement of pressure on the oil manometer for the fixed points is 10-2 mm which amounts to an error of about 0-3 mdeg K for helium at 4.2 ° K and 2 mdeg K for hydrogen at 20-4 ° K; using the normal filling pressure of 1 at and 0.2 atm respectively at room temperature. The precision of .the measurement of pressure is about 4 x 10-2 mm oil, which represents 1.2 mdeg K in the range 4-40 ° K. The corrections for the gas thermometer, shown in Table 1 are believed to be accurate within the limits indicated in the Table (last column). The most serious errors arise from uncertainty in the corrections for the dead volumes. The overall accuracy is judged to be about 6 mdeg K at 2 0 ° K if we take helium as calibrating point and about 50 m ° K at 80 ° K if we take hydrogen.
Heat Capacity Measurement Powder of pure magnesium oxide with a grain size of 40 to 150 p was prepared by the Pechiney method. An analysis showed the main impurities to be:
L Vacuum N Pressure O Oil (butylphthalate)
Figure 5. Gas thermometer
each plus the heating-up time of the oven of 0.5 h/100 ° C. The weight of the specimen was 52-57 + 0-02 g. All errors in the potential, the current, and the time measurements considered above are believed to contribute less than 0.1 per cent to the uncertainty in the heat capacity. The total random error, primarily due to the extrapolation of the temperature drift curves of the recorder, was believed not to exceed 0.8 per cent. The systematic errors were estimated to be less than 0.5 per cent. Owing to the relatively low specific heat C of MgO, the heat capacity of the sample holder, weighing 4-25 g, was at least 40 per cent of the total (Table 2). The heat capacity of the sample holder was measured separately. The smoothed curve of the holder is judged to have an accuracy of 0.4 per cent. This amount to a total error P on the specific heat determination, indicated in Table 2 (last column). TABLE 2
T (°K)
Contribution (%) o f sample homer
Estimated total random error (%) on C,
Error P (%) on Smoothed curve
2 4 10 20 30
70 50 40 40 40
4 2 1.6 1.6 1.6
2 1.0 0.8
of C
0.B 0.8
SiO2, 500 p.p.m. ; C203, 200 p.p.m. ; Na20, 30 p.p.m.; K20, 20 p.p.m.; and C, 7 p.p.m.
Results
The cylindrical specimen, 20 mm in diameter and 56 mm long, was formed in a press. It was then annealed three times at temperatures of 1200, 1 500, and 1 800 ° C, respectively, for about half an hour at
The heat capacity of the specimen and holder was measured f r o m 1.2 to 3 6 ° K and the smoothed heat capacity of the sample holder was deducted. The apparent Debye characteristic temperature 0 as a function of the
CRYOGENICS
• AUGUST
1967
temperature T was calculated from each measured heat capacity point c, by the formula
ci
=
#D(T/O)
23, and 38 per cent, respectively. The Debye 0 at 0 ° K is best determined directively from a plot C/T versus T 2 between 1 and 6 ° K:
•
00(MgO) = 945"0 + 1"5° K
D(T/O) is the Debye function and fl = n x 3R, where n = 2, corresponding to the fact that there are 2 atoms per molecule. The results are plotted in Figure 6. The values obtained from the elastic constants, and other measurements are likewise indicated. .%
'.
950 . . . . . . . . . .
~
I
a ~
I
L
I
I
lockmon- Betts.Bhotia,Wymon !
94O
""
:=.-~..
T l%
Discussion The measured value of 0o at 0 ° K is in excellent agreement with the theoretical calculations of 0o mentioned in the introduction and detailed below. These values are compared in Table 3 and Figure 6. At temperatures above 2 0 ° K our experimental values are in very good agreement with the measurements of Barron, Berg, and Morrison.S
930 TABLE 3
t
Befls
~ 920
Author Blackman Betts, B h a t i a , W y m a n Betts Launay Barron, Berg, M o r r i s o n
"5 "\ ". \ \
9oo
890
,
I I0
0
I
I 20 F
,
I 30
x"
"
40 (deg K)
Figure 6. Debye t e m p e r a t u r e 0 as a f u n c t i o n of absolute t e m p e r a t u r e T f r o m Cf =flD(T/8). T h e broken curve corresponds to 0o[1- 21(~) =-
ReferenceNo.
946 946 920 949 946
12 13 16 14 8
~l
• Experimental points x From Barron, Berg, and M o r r i s o n ---O by t h e o r y - - 0 calculated f r o m elastic c o n s t a n t s of MgO at 0° K
0=
Oo (°K)
The Behaviour of O(T) It would be of interest to compare not only the limiting value, Oo, of the Debye parameter O, as T - - - 0 ° K, but also the behaviour of 0 at low, non zero, temperatures with the values calculated theoretically (Figure 6). The low temperature behaviour of 0 for face centered cubic crystals was worked out in detail by Bhatia and Horton, aa using the lattice dynamical model of Bhatia a4 and an approximation to the frequency spectrum
g(to) = ~ .
104(T/0o).1
a2n/O) 2n
.
.
. (2)
I|
In the temperature range measured the data for the sample were also fitted by the least squares method to
C = ~ AtT21-~(i = 2, 3, 4, 5)
. . . (1)
that is to say by minimizing [ ( C l - C)/Ct]2. This is appropriate since the estimated total random error (Table 2) in the heat capacity in most of the measured interval is believed to be approximately constant. Using four parameters At in the above formula, the 126 measured data points could be represented by the coefficients listed below with a root mean square deviation of 0.8 per cent, as could be seen from Figure 7. There are no appreciable systematic differences between the experimental and the calculated values.
based on Houston's method. 3s They determined the connection between the elasticity data and the values of a2n. According to their theory 36 the 'equivalent Debye temperature' O(T) must be a function of temperature whose value at low T is (found to be) C,-C x I00 3 .~"
Q@
~.
"'°
" "r ~ ""
.42 = (4"618 + 0"020) x 10 -3 As = 2"77 x 10 -7 A 4 = 1"73 x 10 -l° As = 1"06 x 10 -13
°
..
~.':,~.'."
mJ/deg K 4 mole mJ/deg K s mole
".l
°
iO"..
-.
" o
•
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.: •
,
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20
. • •
°
"
"
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I
°
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40
30
.
. •
T (de9 K)
°,
mJ/deg K s mole mJ/deg K 1° mole
They are interdependent, which makes it difficult to estimate the error on A3, A4, `45. The effect of experimental error increases progressively. The error in A2 is only 0.8 per cent but in the others it is as large as 11,
-3 Figure 7 T h e difference, expressed as a percentage, between t h e measured specific heat Cl and the specific heat calculated f r o m t h e f o u r parameter f o r m u l a e CRYOGENICS
" AUGUST
1967
O(T)= OoEB2n
(7"/0o)zn (T
...
(3)
n
temperature range B4 = (1 + 0.2) x l04, giving for the O(T) of MgO
where B0 = l; 6'2 = (20n2/21) (a4/a2) (KOo/h) 2
O = Oo [l -- 21(T/O0) 2 - lO4 (T/O0)4]
B4 = 16 7r4(a6/a2) (KOo/h) 4 and
which differs less than 1 per cent from the experimental curve up to T = 0 . 0 5 0 o _ 5 0 ° K . This is shown in Figure 6 by the dashed curve.
00 = (9No/a2)ll3(h/K)
Retaining only the first two terms of the expansion O(T) = Oo[1 +/5'2(T/0o) 2]
...
(4)
the connection with the elasticity data can be made at least to a first approximation through a parameter a, on which B2 depends, cr, whose value varies between 0 and 2, is defined by
Comparison with Other Alkaline Earth Oxides C u r v e 2 in F i g u r e 8 gives o u r r e c e n t results f o r C a O (1)
in the low temperature region. There difference between these data and those which emphasizes that the ratios of the (or-the interatomic forces) are rather oxides.
is only a small for MgO, a fact elastic constants similar in both
= [(C,,/C44) - (C,2/C44) - 2] in which Cu are the standard symbols for the elastic constants of a cubic crystal. For instance at a critical value a = - 1.39, B2 is zero and the curve of O(T) is initially fiat. Using Houston's method 3s Betts, Bhatia, and Wyman ~3 have evaluated numerically the factor a2 to calculate the Debye 0o at 0 ° K for several cubic crystals and found, for instance, 0o(MgO) = 946 ° K in excellent agreement with our measured value (Table 3). The detailed calculation of the coefficient B2 (a2, a4) based on the theory was done by Horton and Schiff 37 for several face centered cubic crystals (A1, Ag, Cu, Pb) using their room temperature elastic constants. These results are shown in Figure 8, where 0/00 versus T/Oo is plotted. The values of B2 and the corresponding a are likewise indicated at each curve. We also added in Figure 8 curves for copper and silver (dotted lines) corresponding to values of B2, as calculated recently also by Horton and Schiff as up to a higher approximation; the discrepancy between the corresponding dotted curves and the full lines shows the limit of validity of the theory we are exploiting. Now the parameter a for MgO can be calculated taking data for Cu.given by Durand~5; which yields a(MgO) = - 0.653. Then B2 for MgO may be determined from Figure 8 by a simple graphical interpolation between the known values of Bz and a, which is much easier than the detailed application of the exact formula and sufficiently accurate B2(MgO) = 21.2 + 1 Another theoretical determination of 6'2 can be made. Recently Markus 39 has carried out numerical calculations of B2 (r~, r2) where r~ = ( C , , - G 2 / 2 C , ) and r2 = (C44/C~,) for a face centered cubic lattice with nearest neighbour interaction. Using for the calculations of r~ and r2 the relevant elastic constants of MgO ~5, B2 (r,, r2) is found from the curves of Markus to be: B2 = 2 1 + 1, in very good agreement with the value quoted above. Figure 8 (curve 1) shows on the same scale the experimental curve for MgO. At least up to T = 0.03 0o the agreement with formula (4) is within 1 per cent. At higher temperatures the deviation becomes considerable, formula (4) having been restricted to the first two terms only. An experimental value of B4 would be in the lowest C R Y O G E N I C S - A U G U S T 1967
r
~/
pb~o- :--I.50
I02
/ O'OZ
Pb ....... .~T'_.-""004
006
T ~-o
,00
%, :-::.Z>X..", O98
096
094
- ...... ------
\
\ \\ X..~4colculoted,
\';CL
~
L+a4z
Theoretical value from Bhatia Theoretical values from Born and Begbie Experimental values: 1, MgO; 2, CaO; 3, Cu; and 4, Ag. 3 and 4 are taken from references 43 and 44
Figure 8.0/0o as a several values of ponding B=. 0 and ture
function of T/8oat low temperatures for a = (Cn(C44--Cn/C44-2) and corres0o denote the effective Debye temperaat T and 0° K respectively
This is confirmed by the fact that another parameter has the same value for CaO and MgO: Blackman 42 has discussed the variation of 0 with temperature for ionic crystals with special reference to the ratio 0o/0~o, where 0 is the Debye 0 at high temperatures, T > 00/2, which is nearly constant. If the interatomic forces of the crystal are similar the results for face centered cubic crystals show that the relation 0o/0~ is proportional to the square root of a mass factor 0o/0~ "~ [(4m, mz)/(ml + m2)2]1/2 = [1 - r/2]' t2 where
q = (ml - m2) (ml + m2)
and m~, m2 are the atomic masses in the molecule. 231
Indeed the product C = (Oo/&o) (1 - ~2)1/2 is found to be a characteristic constant for many series of compounds 4~ (e.g. potassium halides, sgdium halides, lithium halides) where the elastic constants are nearly unchanged and the greater part of the change in the relation (0/00) can be described by the effect of the change in the mass ratio. The values of 0oo for MgO and CaO are best estimated by a plot of 0 against 1/T 2 from the high temperature heat capacity data of both oxides 5,s. The results for C are in very good agreement (0~(MgO)= 779 + 5° K; 0~(CaO) = 530:1:25 ° K) C (MgO) = 0-810 + 0-005 C (CaO)
= 0.795 + 0.025
It seems that the alkaline earth oxides except BeO show similar properties as regards the elastic constants, interatomic forces, and the frequency spectra; it would be of interest to determine their elastic constants, especially near 0 ° K, and to study in detail the behaviour of the Debye temperature. Comparing the known heat capacity data for these oxides on the basis of the present lattice dynamical theories, SrO and BaO should show a deep minimum near 0/10 which has not yet been verified. The author is indebted to Professor L. Weil for many helpful suggestions and for his stimulation, interest, and encouragement. He also wishes to thank Professor A. Lacaze and Drs. K. H. Gobrecht, S. Marcucci, J. Souletie, and J. J. Veyssie, for many helpful discussions in the development of the apparatus, to M. Vitter for the sample preparation, to Miss E. Nedelka for her kind help in preparing the calculation programmes for the computer, and to Mr. J. G. Gilchrist for revising the manuscript. REFERENCES
1. GMELIN,E. C.R. Acad. Sci. Paris 262, 1452 (1966) 2. ASLANIAN, J., CAILLA'r, R., SALESSE, M., and WElL L. C.R. Acad. Sci. Paris 253, 1032 (1961) 3. KELLEY,K. K. J. Amer. chem. Soc. 1, 1217 (1939) 4. DE COMnAPJEUA. private Communication (C. E. N. GRENOnLE)
212
5. PARKS,G. S., and KELLEY, K. K. J. Phys. Chem. 30, 47 (1926) 6. GUNTHER,P. Ann. Phys. 51, 838 (1916) 7. GIAUQUE,VV. R., and ARCHIBALD,R. C. J. Amer. chem. Soc. 59, 561 (1937) 8. BARRON,T. H. K., BERG, W. T., and MORRiSON,J. A. Proc. roy. Soc. A250, 70 (1959) 9. NERNST, W., and SCHWERS, F. Sitzungsber. d. Deutschen Akademie d. Wissenschaften, Berlin, 355 (1914) 10. KOREEF,K. Attn. Phys. Berlin 36, 36 (1911) I 1. ANDERSON,A. J. Amer. Chem. Soc. 57, 429 (1935) 12. BLACKMAN,M. Proc. roy. Soc. 164, 62 (1938) 13. BETrs, D. D., BHA'rtA, A. B., and WYMAN, M. Phys. Rev. 104, 37 (1956) 14. DE LAUNEY,J. J. chem. Phys. 24, 1071 (1956) 15. DURAND, M. A. Phys. Rev. 50, 499 (1936) 16. BETTS,D. D. Can. J. Phys. 39, 233 (1961) 17. JURA, G., and GARLAND, C. W. J. Amer. chem. Soc. 74, 6033 (1952) 18. LIEN, H., and PmLLIPS, N. E. J. chem. Phys. 29, 1415 (1958) 19. JURA, G., and PrrZER, K. S. J. Amer. chem. Soc. 74, 6030 (1952) 20. GMEUN, E. C.R. Acad. Sci. Paris, 259, 3459 (1964) 21. ASLANIAN,J., and WEIL, L. C.R. Acad. Sci. Paris, 251, 1468 (1960) 22. SOULETIE,J. Thesis (Grenoble, 1967) 23. GOaRECHr, K. H., VEYSSIE,J. J., WEIr, L. Ann. Acad. sci.fenn. 210, 63 (1966) 24. MANCHESTER,F. P. Can. J. Phys. 37, 989 (1959) 25. HmL, R. W., and PICKETT, G. R. Ann. Acad. sci. fenn. 210, 40 (1966) 26. BERMAN,R. J. appl. Phys. 27, 318 (1956) 27. VAN DDK, H., and DtJRIEUX, M. Physica 24, 1 (1958) 28. LOGAN, J. K., CLEMEr,rr, J. R., and JEFFERS, H. R. Phys. Reo. 105, 1435 (1957) 29. RtJBIN, T., AL'rMANN, H. W., and JOHNSTON, H. L. J. Amer. chem. Soc. 76, 5289 (1954) 30. SERMONS,R. O., and BALUFFI,R. W. Phys. Reo. 108, 278 (1957); BENNAKKER,J. J. M., and SWENSON,C. A. Rev. sci. lnstrum. 26, i 204 (1955) 31. KE~OM, W. H. Helium (Elsevier, Amsterdam, 1959) 32. POWELL, P. L., and BLANPIED,W. A. Nat. Bur. Stand Circular 556 33. BHAXIA,A. B., and HORrON, G. K. Phys. Reo. 98, 1715 (1955) 34. BHATIA,A. B. Phys. Reo. 97, 363 (1955) 35. HOUSTON,W. V. Reo. Mod. Phys. 20, 161 (1948) 36. SErrz, F., and TURNBULL, D. Solid Stat. Phys. Suppl. 3, p. 121 (Academic, New York, 1963) 37. HORTON,G. K., and StraFE, N. Phys. Rev. 104, 32 (1956) 38. HORTON,G. K., and SCruFF, N. Can. J. Phys. 36, 1127 (1958) 39. See ref. 36, p. 125 40. BARRON, T. H. K., BERG, W. T., and MORRISON, J. A. Proc. roy. Soc. 242, 478 (1957) 41. BLACKMAN,M. Proc. roy. Soc. 181, 58 (1942) 42. To be published 43. AHLERS,G. Rev. sci. Instrlrm. 37, 477 (1966) 44. AHLERS,G. to be published in J.phys. Chem. Soc. (1967)
CRYOGENICS
• AUGUST
1967
THE low temperature specific heat of the alkaline earth oxides is not well known below 50 ° K. Recent measurements by the present author on the heat capacities of beryllium oxide and calcium oxide I between 1.2 and 70 ° K showed different results from those of other workers below 50 ° K (for other work on alkaline earth heat capacities see the following references: for beryllium oxide, 2-4; magnesium oxide, 5-8; calcium oxide, 5, 9, and I0; and for strontium and barium oxides, reference 11). In this temperature range a substantial increase of the Debye temperature 0 is found, which does not fit the extrapolation of the heat capacity between 50 and 100°K to lower temperatures. But it is of interest to study the exact low temperature behaviour of 0 of these oxides. They all have a cubic structure, [except BeO, which has a hexagonal structure]. This suggests, by comparison with the relatively well established properties of the alkali halides that these oxides should have, in a first approximation, ~elatively simple interatomic forces and, as a consequence, their measured properties should be particularly suitable for a direct comparison with the predictions of lattice theory. Unfortunately, the auxiliary data required for an exact analysis (e.g. elastic constant, thermal expansion, from 0 ° K to room temperature) are not known, except for magnesium oxide where the data are reasonably well determined. Theoretical calculations of 0 near 0 ° K for magnesium oxide as obtained by Blackman ~2 by numerical integration, by Betts, Bhatia and Wyman ~a who applied Houston's method, and by means of de Launey's tables ~4 from elastic constants for magnesium oxide at 0 ° K given by Durand ~5 show an excellent agreement (Table 3) giving a value of 0_~ 946 ° K. Recent calculations by Betts ~6 by an extended Houston method disagree with the first results by 3 per cent. The first heat capacity measurements of magnesium oxide, made by Gunther 6 and Parks and Kelley s did not extend below 80 ° K. The measured values agree within t Centre National de Recherches sur les Tr~s Basses Temp6ratures, Grenoble, France, Received 28 March 1967 CRYOGENICS
C
" AUGUST
1967
l per cent with those of Barron, Berg, and Morrison s made on magnesium oxide single crystals, enclosed in a vessel sealed off with a small amount of helium gas inside. But those of Giauque and Archibald, 7 measured in the same temperature range from 20 to 300 ° K show a systematic deviation to higher specific heats (up to 100 per cent). The latter used a specimen of magnesium oxide in the form of small particles ( , ~ 1 000 A) and it was suggested by Jura and Garland ~7 that their results might be attributed to an effect of particle size on the specific heat. It might also be mentioned that heat capacity measurements by Lien and Phillips ~s on even finer magnesium oxide powder ( , ~ 100,~) have shown such a contribution, as expected from the theory of Jura and PitzerJ °
Experimental Apparatus The heat capacity measurements were made with an adiabatic calorimeter assembly which had been used for the experiments with a beryllium monocrystal, 2o with beryllium oxide and calcium oxide ~ and with the copper standard sample 42 received from Argonne National Laboratory. The cryostat will be described here in detail. In Figure 1 is shown schematically the low temperature portion of the apparatus; most of the details are given in the caption. The specimen O, the thermal mechanical contact M, and the gas thermometer bulb (R) were surrounded by a vacuum-tight thermally insulated copper can Q. This made it possible to keep the inner calorimeter (Q) at any desired temperature by means of a semiautomatic heater device (I and K). A second outer calorimeter wall (P) enclosing the inner calorimeter is immersed in the cryogenic fluid helium, hydrogen, or nitrogen) contained in a conventional nitrogen protected metallic dewar vessel. The cryoo~at holds enough liquid helium to last for more than 20 h. The inner calorimeter and the space between P and Q may be evacuated by means of a diffusion pump to a pressure of 1 latorr at room temperature.
In addition to the radiation trap of the inner calorimeter a trap (C) containing several baffles is located in the high vacuum pumping line (B). Ten~. eratures as low as about 1.2° K were obtained by ptnnpmg on the liquid helium bath. Particular attention was paid to the insulation of the cryostat from vibrations. The cylindrical shaped specimens were clamped in two rings suspended by several nylon threads L from the frame. These are illustrated in Figure 2 (H). They consisted of thin walled copper strips, 0.2 mm thick, screwed together permitting easy changing of the specimen and ensuring excellent thermal contact with the sample over the whole temperature range. A 68 f2, 0.1 W Allen Bradley carbon resistor (or a 10012 platinum resistor for the higher temperature range), used as secondary thermometer, was fitted tightly into a small cylindrical copper tube soldered on one holder J where it was kept by vacuum grease. Inside the second holder ring one or several constantan strain gauges (600-1 800 ~) (K) used as sample heaters were attached with Araldite.
The thermal relaxation time of the total sys~m for the measured specimens is less than 1-2 s at temperatures below 10° K. There was no superheating. The heat capacity of the holders (total weight 3-12 g) was always measured separately.
G
\
A B
K
C D E F G L H
I K L M N 0 P Q R S T
A B C D E
Lead wires Pumping tube Radiation trap Insulating vacuum Vapour pressure thermometer bulb F, G Stainless steel capillaries for the thermal switch and the gas thermometer bulb H Lead wires passage
I K L M N O P, R S
Heater Resistance thermometer Heat sinks Mechanical and thermal contact Frame Specimen 0 Interior and exterior calorimeter wall Gas thermometer bulb Wood's metal solder
Figure 1. The calorimeter m
A B C D E F
Lead wires passage Heat sink Lead wires Calorimeter support Helium pressure Metal bellows
G H I d K L
Copper strip Copper clamps Specimen Resistance thermometer Heater Nylon wires
Figure 2. Mechanical heat switch
In order to cool down the sample to the required temperature without any exchange gas, the bellows technique ~o-23 well established in this laboratory was used. It is schematically shown in Figure 2. A tombac bellows F (10 mm in diameter and 20 mm long) closed on one side by a copper plate was solidly attached at the other, to the calorimeter frame D and linked by a capillary to a steel cylinder containing high pressure helium 4. When it was filled with liquid helium under pressure of about 3 kg/cm 2 the copper plate pressed the thermal link against the frame. The thermal link was an annealed 0-2 mm thick copper strip G soldered on the sample holder H. To switch off this contact the helium inside the bellows was pumped off to 1 kg/cm 2 by means of the gasometer line and nearly to vacuum with the helium bath pump. There was a critical moment when the copper strip, not having broken the contact yet, touched the bellows or frame slightly. The mechanical shock liberated 5-20 erg depending on the adjustment of the thermal link in the contact region. Neither the pressure used (1 to 5 kg/cm z) for making the contact nor the releasing conditions (e.g. several seconds to 15 min cut-off time) had any detectable influence on the heat input to the sample. If the switch was clamped at room temperature, and the cryostat then cooled with liquid helium or hydrogen, it sometimes stuck so that it could not be unclamped. This inconvenience was overcome by operating the switch once at liquid nitrogen temperature. CRYOGENICS
• A U G U S T 1967
Operation of this switch involves a very small heat leak into the sample, smaller than any of the values reported by Manchester z4 and by Hill and Pickett. 25 Furthermore, the heat input by mechanical vibrations was reduced to 20-30 erg/min. The employment of a fast amplifier recorder system also contributed to ensuring excellent conditions for the measurement of very low heat capacities without exchange gas. Finally, it is an advantage that with this apparatus no force is applied directly on to the specimen. The cooling down performance is also excellent. A sample (378 g copper) was cooled from room temperature to 4 ° K within 5 h. By measurement of the cooling curve (temperature T versus time, t) of the copper sample, the temperature dependence of the thermal conductance k of the whole cooling system was calculated in the liquid nitrogen and helium regions (Figure 3). The temperature dependence from 2 to 8 ° K was similar to that found by Berman, 26 the k(T) being nearly proportional to T 2 (curve A) and gradually becoming less strongly dependent on T and almost independent at liquid nitrogen temperature. The conductance k is lower than the values measured by Berman (curve B) by a factor of two for a given applied force, but we did not yet try to get as clean surfaces as Berman. As can be seen by comparing curve A and C in Figure 3, the cooling down performance is not limited by the conductance of the copper link, which is much lighter (curve C) than that of the contact. The conductance k of the whole device was found to be proportional at 4.2 ° K to the applied load L (Figure 3, curve D). Temperatures up to 70 ° K could be measured with a 68 f2 Allen Bradley carbon resistor calibrated against the vapour pressure of liquid helium between 1.2 and 4.2 ° K and against a gas thermometer and liquid hydrogen vapour pressure in the higher temperature region. The measuring equipment (Figure 4) for the current and potential of the thermometer resistor Rm, consisted of a double microvolt potentiometer P (MECI) whose output was fed into a d.c. amplifier A (Beckmann) followed by a fast chart-recorder CR (Graphispot) used at speeds between 30"and 120 mm/min, recording off balance as a function of time. This system together with the fast response of the thermometer circuit permitted the measure of heat capacities down to 3 x 10-z mJ/deg at 1.2 ° K. For measurements above 10°K the d.c. amplifier A was replaced by a galvanometer amplifier (Sefram-Amplispot) having less than 10 laV noise and an input impedance of 250 kfl (constant gain). The constant current used in measuring the resistance of Rtn varied from 0.25 laA at 1.2°K to 10 laA at 77 ° K. With these values the current was large enough to give adequate sensitivity for dete?mining the resistance and small enough to give negligible energy dissipation in the resistor. Providing it was of this order of magnitude at the values quoted, the current had no detectable influence on the resistance measured. To eliminate most of the low frequency pick-up, the whole low voltage measuring system was screened separately. The uncertainty in resistance was equivalent to 10 mdeg at 70 ° K, to 1 mdeg at 20 ° K, and less than 0.2 mdeg for temperatures lower than 10° K. A carbon resistance thermometer does not in general behave reproducibly and it is necessary to calibrate the CRYOGENICS
" AUGUST
lg67
resistor for each measurement. However, when no mechanical forces were applied to the resistor and it h a d been cycled several times between room temperatures and helium temperature, the successive calibration c u r v e s were different at most by 1-3 mdeg. }00
!
i
i
i
rig
T~It/
I0
sS s/
O~
03 • klmW/degK)
/"
o2
,¢ 001
$ 1 /
/-°
Ol
o 000
A
I
I
I I
I I I IIIJ
~o T
==
~ L-~-- (kg) I
I I I Ill
ioo (*K)
Load L x 0-75 kg • 1.75 kg © 2.00 kg A 9.50 kg Figure 3. T h e t h e r m a l c o n d u c t a n c e k of t h e mechanical heat switch as function of t e m p e r a t u r e T and Load L
The thermometer was calibrated by admitting exchange gas to the calorimeter after a specific heat measurement and taking a set of 20 to 25 calibration points over the range 1.2-4.2 ° K. The temperature of the bath was maintained constant for each point by an automatic temperature control actuated directly by the Allen Bradley resistance on the sample and dissipating energy into a 50 f2 constantan wire around the calorimeter wall. The temperature stability was better than 10-4 deg. Calibration temperatures were determined on the 1958 helium scale of NBS from the vapour pressure of helium measured in a small cavity (E in Figure 1, about 2 cm 3) on top of the calorimeter. This bulb was connected by a tube of 3 mm diameter surrounded by an insulating vacuum D, to a mercury manometer and an oil manometer (butylphthalate) for use below 2-5°K. Both manometers were read with a cathetometer with an accuracy of 2 × 10-z mm. Below the 2 point the pressure was measured from a point directly above the helium bath. The estimated error in reading the manometers corresponded to less than 0.5 mdeg except below 1.8 ° K where possible errors might amount to 4 m d e g at 1.2 ° K,
For the calibration of the temperature scale above 4 ° K we used a gas thermometer with the fixed point at 4.2 ° K. In the range of 14-20 ° K calibration was verified against the liquid hydrogen vapour pressure which was determined by the same method as the helium vapour pressure. Corrections due to ortho-para conversion were applied, if parahydrogen was not used. The
uncertainty of calibration in this temperature range was estimated to be less than 10 mdeg from the hydrogen scale of Dijk and Durieux. 27 // The calibration data between 1 and 4 ° K were fitted to a five parameter equation 1
a3
~, = at enR + a2 + e ~
a5
04
+ ~
+ (enR) '''-'-S
was about 2"66 cm 3 of which 2.50 cm 3 was aT'room temperature (copper capillary and average manometer volume) and 0.16cm 3 at a temperature varying from 4 to 300 ° K. The temperature was calculated from the expression
V(T, P) + P~ Vj PRI(I-" + B(T)/V) ~ R T I ( 1 + BI(T)/VO
constant
i
and the coefficients at calculated by a least square method by computer. The difference ( T - Te) was also determined where T is the temperature taken from the calibration tables mentioned below, and Te the calculated temperature from the equation. In a heat capacity measurement errors in the slope of the calibration curve introduce errors in the specific heat equal to the slope error. So a graph ( T - Te) versus Te shows directly this systematic error, caused by the temperature calculation. This error was always made to be less than 0-2 per cent. For the calibration points above 4 ° K an interpolation formula I/T = a + b en R was used for a first approximation. Then a polynomial T=
T a + ~ C l Tl ( i =
.
.
.
.
.
.
.
.
,
R
_-~ :
|ll
-~ -'J
I [
- -
I
i__. .........
IAI
[
,V ! " b
J
5)
was applied to fit the calibration curve. The coefficients a, b and ct were always calculated by a least-squares method. The temperature range is divided into two regions: 4--25° K and 20--80 ° K. The calculated temperatures differed from the calibration values by less than 0-2 per cent. The circuit is shown schematically in Figure 4. The asymmetrical method 2s of connecting the heater leads to the heater was adopted to eliminate errors due to the heat generated in these leads. The resistance Rh of the current leads is approximately 10f~ each (the heater resistance being greater than 1 000 f~); those for potential measurements 50 1) each. The power dissipated in the heater by a battery (2, 12, or 24 V) was measured using a double potentiometer (P) to measure both the current J through the heater and the potential drop V across the heater, in connection with an electrical clock (C), driven by a standard 50 cycle generator of frequency stability better than I0 -q, used for determining the heating period. The current was switched by a relay which provided steady loading on the current supply whether or not the current was passing through the heater. The total random error in heating energy was estimated to be less than 0.15 per cent. Heating of spurious electrical origin, at both low and high frequencies, was stopped by screening the heater system and by connecting small condensors (0.1 p.F) in parallel with the heater leads on top of the metallic dewar vessel. The gas thermometer, schematically shown in Figure 5 had a bulb (G) made of copper, wall thickness 1 mm, and a volume of 92.20 cm 3. This bulb was connected by a stainless steel capillary (D) (diameter 0.8 mm and 325 mm long) up to room temperature and then by a copper capillary (C) (diameter 1 mm, and 2 465 mm long) to an oil manometer (O). The system was used as a constant volume thermometer by always setting the lower meniscus to very nearly the same mark A. This was done by introducing or pumping gaseous helium over the butylphthalate reservoir (O) with two valves (M and N). The error was about 2 x 10-2 mm. The overall dead volume ~II
where p is the pressure, B the second virial coefficient
k-%3 Heat sink --
wire
Constonton
- - Copper wire Imm - - Copper wire O,O7mm Heater circuit (left) Rth Heater R Heater equivalence resistance C Chronometer P Potentiometer
T h e r m o m e t e r circuit (right) Rth Resistance t h e r m o meter G Constant current generator P Potentiometer A Amplifier CR Chart recorder Figure 4
for helium, and V the volume of bulb. (With
V(T, p)
=
Vo(To.Po) [I + 3 c t ( r - To) + 7(P - p 0 ) ]
where To, po, and Vo are the values at the fixed point temperature, ¢, the expansion coefficient of copper, and Y, the coefficient of compressibility). The second term represents the correction for the dead volumes. In these, the correction term in Bi is negligible. The constant depends on the amount of gas in the system and is determined by calibrating at the boiling point of helium or hydrogen. So the temperature Tcould be represented by the formula T = Ta[l + 3~t(Ta - To)] + y(p - p o ) +
(I/I/o)(B- no)
( Vo/Vo)[(Ta - To)/O] + {(Vo/Vo)[TaF(7")
-
ToF(To)]}
with Ta = (pTo/po) l'.~t
T,
T~
l fdV T(x) -- ;K, dr I K(T)dT
and F(T) = -~M
0
T,
T,
The approximate temperature Ta must be corrected by addition of 5 terms: (1) The thermal expansion of the bulb for which the values for ~t were taken from Rubin, Altmann, and Johnston ;29,30 CRYOGENICS
' AUGUST
1967
(2) The bulb deformation with pressure, which is always negligible; (3) The correction for non ideality of the helium gas, using the data of Keesom ;3t (4) The correction for the dead volume Vo at room temperature 0; (5) The correction for the dead volume VM at varying temperatures, determined by calculating the integrals F(T) with the thermal conductivity for stainless steel, a2 The correction for the thermomolecular pressure difference is negligibly small. The correction values of the items (1) to (5) considered above are shown in Table 1. TABLE 1
i Corrections for T. at
20° K
80° K
Accuracy
Calibration point at
4.2° K
20.4° K
in [%] (1)
0 mdeg K
(2) (3)
o
,,
o
,,
o
-48 + 29 + 7
,, ,, ,,
--8 + 442 + 93
,, ,, ,,
1 5 10
(4) (5)
+ 5 5 mdeg K
5 i ;
H Valve without dead
A Vacuum B Insulating vacuum C Copper capillary
volume
Glass capillary K Measuring level
d
D Stainless steel capillary E Cryostat F Calorimeter walls G Gas thermometer bulb
Accuracy
of the Gas Thermometer
The accuracy of the calibration temperatures at the boiling points of liquid helium and hydrogen, taking 10 points with an error of 10-~ torr is much better than the accuracy of the 1958 helium scale and the hydrogen scale, in which the uncertainty is about 2 mdeg K at 4-2 ° K and 10 mdeg K at 20.4 ° K. The uncertainty of the measurement of pressure on the oil manometer for the fixed points is 10-2 mm which amounts to an error of about 0-3 mdeg K for helium at 4.2 ° K and 2 mdeg K for hydrogen at 20-4 ° K; using the normal filling pressure of 1 at and 0.2 atm respectively at room temperature. The precision of .the measurement of pressure is about 4 x 10-2 mm oil, which represents 1.2 mdeg K in the range 4-40 ° K. The corrections for the gas thermometer, shown in Table 1 are believed to be accurate within the limits indicated in the Table (last column). The most serious errors arise from uncertainty in the corrections for the dead volumes. The overall accuracy is judged to be about 6 mdeg K at 2 0 ° K if we take helium as calibrating point and about 50 m ° K at 80 ° K if we take hydrogen.
Heat Capacity Measurement Powder of pure magnesium oxide with a grain size of 40 to 150 p was prepared by the Pechiney method. An analysis showed the main impurities to be:
L Vacuum N Pressure O Oil (butylphthalate)
Figure 5. Gas thermometer
each plus the heating-up time of the oven of 0.5 h/100 ° C. The weight of the specimen was 52-57 + 0-02 g. All errors in the potential, the current, and the time measurements considered above are believed to contribute less than 0.1 per cent to the uncertainty in the heat capacity. The total random error, primarily due to the extrapolation of the temperature drift curves of the recorder, was believed not to exceed 0.8 per cent. The systematic errors were estimated to be less than 0.5 per cent. Owing to the relatively low specific heat C of MgO, the heat capacity of the sample holder, weighing 4-25 g, was at least 40 per cent of the total (Table 2). The heat capacity of the sample holder was measured separately. The smoothed curve of the holder is judged to have an accuracy of 0.4 per cent. This amount to a total error P on the specific heat determination, indicated in Table 2 (last column). TABLE 2
T (°K)
Contribution (%) o f sample homer
Estimated total random error (%) on C,
Error P (%) on Smoothed curve
2 4 10 20 30
70 50 40 40 40
4 2 1.6 1.6 1.6
2 1.0 0.8
of C
0.B 0.8
SiO2, 500 p.p.m. ; C203, 200 p.p.m. ; Na20, 30 p.p.m.; K20, 20 p.p.m.; and C, 7 p.p.m.
Results
The cylindrical specimen, 20 mm in diameter and 56 mm long, was formed in a press. It was then annealed three times at temperatures of 1200, 1 500, and 1 800 ° C, respectively, for about half an hour at
The heat capacity of the specimen and holder was measured f r o m 1.2 to 3 6 ° K and the smoothed heat capacity of the sample holder was deducted. The apparent Debye characteristic temperature 0 as a function of the
CRYOGENICS
• AUGUST
1967
temperature T was calculated from each measured heat capacity point c, by the formula
ci
=
#D(T/O)
23, and 38 per cent, respectively. The Debye 0 at 0 ° K is best determined directively from a plot C/T versus T 2 between 1 and 6 ° K:
•
00(MgO) = 945"0 + 1"5° K
D(T/O) is the Debye function and fl = n x 3R, where n = 2, corresponding to the fact that there are 2 atoms per molecule. The results are plotted in Figure 6. The values obtained from the elastic constants, and other measurements are likewise indicated. .%
'.
950 . . . . . . . . . .
~
I
a ~
I
L
I
I
lockmon- Betts.Bhotia,Wymon !
94O
""
:=.-~..
T l%
Discussion The measured value of 0o at 0 ° K is in excellent agreement with the theoretical calculations of 0o mentioned in the introduction and detailed below. These values are compared in Table 3 and Figure 6. At temperatures above 2 0 ° K our experimental values are in very good agreement with the measurements of Barron, Berg, and Morrison.S
930 TABLE 3
t
Befls
~ 920
Author Blackman Betts, B h a t i a , W y m a n Betts Launay Barron, Berg, M o r r i s o n
"5 "\ ". \ \
9oo
890
,
I I0
0
I
I 20 F
,
I 30
x"
"
40 (deg K)
Figure 6. Debye t e m p e r a t u r e 0 as a f u n c t i o n of absolute t e m p e r a t u r e T f r o m Cf =flD(T/8). T h e broken curve corresponds to 0o[1- 21(~) =-
ReferenceNo.
946 946 920 949 946
12 13 16 14 8
~l
• Experimental points x From Barron, Berg, and M o r r i s o n ---O by t h e o r y - - 0 calculated f r o m elastic c o n s t a n t s of MgO at 0° K
0=
Oo (°K)
The Behaviour of O(T) It would be of interest to compare not only the limiting value, Oo, of the Debye parameter O, as T - - - 0 ° K, but also the behaviour of 0 at low, non zero, temperatures with the values calculated theoretically (Figure 6). The low temperature behaviour of 0 for face centered cubic crystals was worked out in detail by Bhatia and Horton, aa using the lattice dynamical model of Bhatia a4 and an approximation to the frequency spectrum
g(to) = ~ .
104(T/0o).1
a2n/O) 2n
.
.
. (2)
I|
In the temperature range measured the data for the sample were also fitted by the least squares method to
C = ~ AtT21-~(i = 2, 3, 4, 5)
. . . (1)
that is to say by minimizing [ ( C l - C)/Ct]2. This is appropriate since the estimated total random error (Table 2) in the heat capacity in most of the measured interval is believed to be approximately constant. Using four parameters At in the above formula, the 126 measured data points could be represented by the coefficients listed below with a root mean square deviation of 0.8 per cent, as could be seen from Figure 7. There are no appreciable systematic differences between the experimental and the calculated values.
based on Houston's method. 3s They determined the connection between the elasticity data and the values of a2n. According to their theory 36 the 'equivalent Debye temperature' O(T) must be a function of temperature whose value at low T is (found to be) C,-C x I00 3 .~"
Q@
~.
"'°
" "r ~ ""
.42 = (4"618 + 0"020) x 10 -3 As = 2"77 x 10 -7 A 4 = 1"73 x 10 -l° As = 1"06 x 10 -13
°
..
~.':,~.'."
mJ/deg K 4 mole mJ/deg K s mole
".l
°
iO"..
-.
" o
•
• i" "
.: •
,
""
20
. • •
°
"
"
"
I
°
I
40
30
.
. •
T (de9 K)
°,
mJ/deg K s mole mJ/deg K 1° mole
They are interdependent, which makes it difficult to estimate the error on A3, A4, `45. The effect of experimental error increases progressively. The error in A2 is only 0.8 per cent but in the others it is as large as 11,
-3 Figure 7 T h e difference, expressed as a percentage, between t h e measured specific heat Cl and the specific heat calculated f r o m t h e f o u r parameter f o r m u l a e CRYOGENICS
" AUGUST
1967
O(T)= OoEB2n
(7"/0o)zn (T
...
(3)
n
temperature range B4 = (1 + 0.2) x l04, giving for the O(T) of MgO
where B0 = l; 6'2 = (20n2/21) (a4/a2) (KOo/h) 2
O = Oo [l -- 21(T/O0) 2 - lO4 (T/O0)4]
B4 = 16 7r4(a6/a2) (KOo/h) 4 and
which differs less than 1 per cent from the experimental curve up to T = 0 . 0 5 0 o _ 5 0 ° K . This is shown in Figure 6 by the dashed curve.
00 = (9No/a2)ll3(h/K)
Retaining only the first two terms of the expansion O(T) = Oo[1 +/5'2(T/0o) 2]
...
(4)
the connection with the elasticity data can be made at least to a first approximation through a parameter a, on which B2 depends, cr, whose value varies between 0 and 2, is defined by
Comparison with Other Alkaline Earth Oxides C u r v e 2 in F i g u r e 8 gives o u r r e c e n t results f o r C a O (1)
in the low temperature region. There difference between these data and those which emphasizes that the ratios of the (or-the interatomic forces) are rather oxides.
is only a small for MgO, a fact elastic constants similar in both
= [(C,,/C44) - (C,2/C44) - 2] in which Cu are the standard symbols for the elastic constants of a cubic crystal. For instance at a critical value a = - 1.39, B2 is zero and the curve of O(T) is initially fiat. Using Houston's method 3s Betts, Bhatia, and Wyman ~3 have evaluated numerically the factor a2 to calculate the Debye 0o at 0 ° K for several cubic crystals and found, for instance, 0o(MgO) = 946 ° K in excellent agreement with our measured value (Table 3). The detailed calculation of the coefficient B2 (a2, a4) based on the theory was done by Horton and Schiff 37 for several face centered cubic crystals (A1, Ag, Cu, Pb) using their room temperature elastic constants. These results are shown in Figure 8, where 0/00 versus T/Oo is plotted. The values of B2 and the corresponding a are likewise indicated at each curve. We also added in Figure 8 curves for copper and silver (dotted lines) corresponding to values of B2, as calculated recently also by Horton and Schiff as up to a higher approximation; the discrepancy between the corresponding dotted curves and the full lines shows the limit of validity of the theory we are exploiting. Now the parameter a for MgO can be calculated taking data for Cu.given by Durand~5; which yields a(MgO) = - 0.653. Then B2 for MgO may be determined from Figure 8 by a simple graphical interpolation between the known values of Bz and a, which is much easier than the detailed application of the exact formula and sufficiently accurate B2(MgO) = 21.2 + 1 Another theoretical determination of 6'2 can be made. Recently Markus 39 has carried out numerical calculations of B2 (r~, r2) where r~ = ( C , , - G 2 / 2 C , ) and r2 = (C44/C~,) for a face centered cubic lattice with nearest neighbour interaction. Using for the calculations of r~ and r2 the relevant elastic constants of MgO ~5, B2 (r,, r2) is found from the curves of Markus to be: B2 = 2 1 + 1, in very good agreement with the value quoted above. Figure 8 (curve 1) shows on the same scale the experimental curve for MgO. At least up to T = 0.03 0o the agreement with formula (4) is within 1 per cent. At higher temperatures the deviation becomes considerable, formula (4) having been restricted to the first two terms only. An experimental value of B4 would be in the lowest C R Y O G E N I C S - A U G U S T 1967
r
~/
pb~o- :--I.50
I02
/ O'OZ
Pb ....... .~T'_.-""004
006
T ~-o
,00
%, :-::.Z>X..", O98
096
094
- ...... ------
\
\ \\ X..~4colculoted,
\';CL
~
L+a4z
Theoretical value from Bhatia Theoretical values from Born and Begbie Experimental values: 1, MgO; 2, CaO; 3, Cu; and 4, Ag. 3 and 4 are taken from references 43 and 44
Figure 8.0/0o as a several values of ponding B=. 0 and ture
function of T/8oat low temperatures for a = (Cn(C44--Cn/C44-2) and corres0o denote the effective Debye temperaat T and 0° K respectively
This is confirmed by the fact that another parameter has the same value for CaO and MgO: Blackman 42 has discussed the variation of 0 with temperature for ionic crystals with special reference to the ratio 0o/0~o, where 0 is the Debye 0 at high temperatures, T > 00/2, which is nearly constant. If the interatomic forces of the crystal are similar the results for face centered cubic crystals show that the relation 0o/0~ is proportional to the square root of a mass factor 0o/0~ "~ [(4m, mz)/(ml + m2)2]1/2 = [1 - r/2]' t2 where
q = (ml - m2) (ml + m2)
and m~, m2 are the atomic masses in the molecule. 231
Indeed the product C = (Oo/&o) (1 - ~2)1/2 is found to be a characteristic constant for many series of compounds 4~ (e.g. potassium halides, sgdium halides, lithium halides) where the elastic constants are nearly unchanged and the greater part of the change in the relation (0/00) can be described by the effect of the change in the mass ratio. The values of 0oo for MgO and CaO are best estimated by a plot of 0 against 1/T 2 from the high temperature heat capacity data of both oxides 5,s. The results for C are in very good agreement (0~(MgO)= 779 + 5° K; 0~(CaO) = 530:1:25 ° K) C (MgO) = 0-810 + 0-005 C (CaO)
= 0.795 + 0.025
It seems that the alkaline earth oxides except BeO show similar properties as regards the elastic constants, interatomic forces, and the frequency spectra; it would be of interest to determine their elastic constants, especially near 0 ° K, and to study in detail the behaviour of the Debye temperature. Comparing the known heat capacity data for these oxides on the basis of the present lattice dynamical theories, SrO and BaO should show a deep minimum near 0/10 which has not yet been verified. The author is indebted to Professor L. Weil for many helpful suggestions and for his stimulation, interest, and encouragement. He also wishes to thank Professor A. Lacaze and Drs. K. H. Gobrecht, S. Marcucci, J. Souletie, and J. J. Veyssie, for many helpful discussions in the development of the apparatus, to M. Vitter for the sample preparation, to Miss E. Nedelka for her kind help in preparing the calculation programmes for the computer, and to Mr. J. G. Gilchrist for revising the manuscript. REFERENCES
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212
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CRYOGENICS
• AUGUST
1967