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Low temperature thermal properties of PbI2
LOW TEMPERATURE THERMAL PROPERTIES OF PbIz W. hf. SEARSt and J. A. hiORRlSON
DepPrtwnt of Physics and Institute for Materials Research, Mchfaster University, Hamilton, Ontario, Canada (Received
I1 October 1978; accepted in revised fomt 7 December 1978)
Ab&w!-l?~~ beat capacity and tbermal conductivity of a large (56.5 g) crystal of PbI2 have been measured in the temPerahIre ~C~OO 0.5 < T < 39’K. Analysis of the heat capacity data yields a value of the limiting apparent Debye
characteristic temperature 00’ =99.4 (+0.3)“K, which corresponds to an average lattice wave velocity of 1.151 (LMHI~) x ldcm set-‘. It is consistent with a wave velocity estimated from neutron scattering experiments, but not with ooe determined from Brillouin spectra. The heat capacity data also show that dispersion of the low frequency wave3 is not unusual. a3 might have teen expected for a layer-type crystal. The apparent thermal conduction is found to be surprisingly small in the crystal.
1. INTBODIJCI’ION Many crystals display extreme anisotropy and form layer-type structures in which bonding between layers is
much weaker tban that within a layer. In some examples, bonding of the atoms in a layer may be covalent while the bonding between the layers may be largely of the van der WaaIs type. The anisotropy has consequences on the physical properties of the crystals. For example, there will be marked weakening or softening of vibrational modes in the crystallographic c-direction. This is shown in the phonon dispersion curves where the longitudinal acoustic branch is much lower in energy in the direction perpendicular to the layers [ 11. For layer-type structures, some thermodynamic properties, such as the thermal expansion, would obviously be expected to display anisotropy[2]. In other properties, such as the heat capacity, which are related to the whole vibfational spectrum, it may be more difficult to detect effects of anisotropy except under particular circumstances. For instance, t-dimensional effects of at least two kinds have been found in the heat capacity of graphite at low temperatures(3-51. In co~cction with a study of the vibrational properties of various polytypes of Pb12[6], it was found desirabk to obtain an estimate of the average wave velocity for phonons in Pblt from the limiting Debye characteristic temperature at very low temperatures. Thus, the heat capacity of a huge crystal of Pb12 was measured in the region T <4K. The experimental conditions turned out to be such that it was possible to estimate the thermal conductivity of PbI2 at the same time. The results are discussed in relation to the 2dimensiona.l character of the structure of PbL
2.1 Pmpamtion of fhe crystal specimen ‘lb crystal was grown by the Bridgman technique. Reagent gmde (98.5%) lead iodide powder was placed in Present
address:
Department
of Physics, University of
Guelph. Guelph, Ontario, Canada. 503
a glass tube which was evacuated (P < 1 x lo-. torr) and then the lead iodide was melted to drive out any trapped gas. After the tube was sealed under vacuum, it was slowly lowered through a furnace set at a temperature just above the melting point of lead iodide, i.e. at about 400°C. The rate of lowering was controlled with a low speed motor and was such that the crystal grew over a period of four days. When it was removed from the tube, the crystal was found to have an “impurity” layer on top. This part of the crystal was cut off, the remainder was resealed in a glass tube, and the melting and growth procedures were repeated. The regrown crystal did not have an obvious impurity layer on its top surface. A spectrographic analysis was made of small samples .scraped from the crystal and it showed the presence of only a few parts per million of & and Sn. The crystal was thus judged to be suitable for the calorimetric measurements. One end of the crystal was cut with a diamond saw to give the surface a smooth enough finish for mounting on the calorimeter tray (see following section). The cut was made perpendicular to the layers of the crystal. The crystal contained two boundaries and hence was composed of three crystals of the same orientation but the mosaic spread was not determined. The mass of the crystal was 56.5 g. 2.2 Calorimetric technique A tray-type calorimeter was used for the measurements. The PbL crystal was placed on a small copper tray and good thermal contact was promoted through the addition to the contact area of a measured quantity (0.01 cm3) of silicone grease. The tray was mounted in a frame suspended below the mixing chamber in a ‘He‘He dilution refrigerator. Cooling was accomplished through a heat switch, in thermal contact with the mixing chamber, which gripped a gold-plated copper spike attached to the tray. The tray carried an electrical heater and a germanium thermometer which had been intercompared with a germanium thermometer previously calibrated over the
ii’. hi. &ASSand J. A.
504
temperature range 0. I < T -c4.3K [7]. The electrical leads between the calorimeter tray and an anchoring point on the mixing chamber were of lead-covered manganin wire. The thin Nylon filaments which held the tray within the frame were spring-loaded to damp out vibration. It required several minutes for the energy supplied to the tray in each heating interval to be. distributed through the crystal. Other measurements with the same tray assembly on aluminium alloy specimens of comparable contact area181 showed that thermal transmission through the silicone grease was extremely good (equilibrium time < 1 set). Thus, the longer equilibrium time is ascribed to low thermal conductivity of the Pb12 specimen. In a separate series of measurements, the heat capacity of the tray assembly, including 0.01 cm’ of silicone grease, was determined. From the results, a smoothed heat capacity-temperature function was constructed for computing the contribution of the tray assembly to the total heat capacity of the system. Measurements were performed over the temperature range 0.5 < T < 3.9K. In this region, the temperature scale is known with an uncertainty not exceeding several millikelvin[7]. The tray assembly contributed 50% of the total heat capacity at T = 0.5K and 3% at T = 3.9K. Overall, the accuracy of the heat capacity results for Pblz is estimated to be 25% at T = O.SK, ?2% T = 1.5K and 220% at T = 3.9K.
at
2.3 Heat capacity results The measured heat capacities are listed in Table 1 and
are displayed in Fig. 1 as a plot of C,,/T3 vs T2. This is a convenient form for the determination of the coefficients of the low temperature expansion: C, =a,T3+a2T5t
........
(1)
which the heat capacity of simple insulators is expected Table 1. Measured heat capacities at low temperatures
T (K) 0.494 0.514 0.619 0.821 0.844 1.055 1.094 1.360 1.628 1.688 1.904 I.972 2.159 2.22a 2.478 2.758 3.039 3.322 3.601 3.874
G (ml Km’mole-‘) 0.743 2 0.033 0.824 2 ohI 1.39 =o.os 3.28 +0.10 3.57 -+o.lm 7.02 20.16 7.67 ~0.16 15.2 20.2 26.0 20.5 28.6 ~0.4 41.2 +- 1.2 46.4 2 1.0 61.3 + 1.8 68.4 + 1.5 97.9 23.3 + 10 149 2 19 208 243 288 399 ~92 2 122 468
MORRISON
IO
1
ag r
‘z I
ie
n
t
i:~__--_I
1
0
I IO
5
I 15
T2(K2) Fig.
I.
CJT’ for Pb12crystal as a function of T2.
to follow in the re@on approaching the continuum limit [9]. The coefficient al is well-defined as the intercept when T2+0 and, from the graph, we obtain a, = 5.94 (ti.OS)ml Km4mole-‘. This corresponds to a limiting Debye characteristic temperature at T = OK: eo’ = (12r’NkJSa I)“‘,
(2)
where N = total number of atoms (3N, per mole for Pb12) N., = Avogadro’s number and ke = Boltzmann’s constant. In particular &= = 99.4 (ti.3) K. From a1 or 06, we can also obtain the mean wave (or sound) velocity: v:, = k&=(4nV/3N)“‘/h,
(3)
where V is the molar volume of the sample and h is Plan&s constant. The density at T = OK was estimated to be 6.23 gem-’ from measured lattice parameters at higher temperatures [6,10]. Thus, v:, = 1.151 (*0.005)x lticmsec-‘. We also obtain some information about dispersion of the lattice waves from the heat capacity data. The coefficient of the T’ term in eqn (1) should indicate how the acoustical branches of the phonon dispersion curves begin to deviate (on average) from linearity at low energies. Fiie 1 gives a value of a2. obtained from the limiting slope, of a2 = 2 (23)
x
10e2al Kd mole-‘.
We see that, within the limits assigned, a2 could be zero or negative, in which case PbI2 would be showiog a feature of 2dimensional structures such as graphite: crystauographic some dispersion in negative . . directions [ 11. More accurate measurements are needed
SO5
Low temperatore thermal properties of PbI2
where p is the density of PbL at low temperatures (6.23gcm-‘), A is the surface area of the crystal in apparent contact with the tray (3.6cm2),C, and C; are the heat capacities (as a function of temperature) of the PbIzcrystal and the copper tray respectively,and M is the massof the crystal specimen(56.5g). In the exponent, Kis c, =o;(TIeo”)‘+a;(Tleo’)J+. . . . .., (4) the thermal diffusivity and the /3,, are the roots of the equation: where a; = o,(&~)’ and n;= Use, The normalized tan(gl) = -/3/r. coefficientsa; and ai of MgOand PbIz are comparable (7) (Table 2) and the similarity of the behaviour of the two Referringto eqn (5), we note that it containsan infinite substances is further emphasized by Fig. 2 where series of exponentials. However, the contribution of (C,/T3)/(C~T’)o is plotted vs (T/&‘)*. each of them decreases rapidly for successive roots of eqn (7). In calculating the refative sizes of successive 2.4 l’7termalconductivityof PbIz The thermal lactic was determined from analy- exponent&Is, we take two extreme cases: T = 0.6K and sis of the temperature-timebehavior of the coiorimeter T = 4.2K. This leads to (from eqn 6) r = 0.4cm-’ and system after it was heated. The solution to the heat r = 12cm-‘. For the two cases, the first three roots of conduction problem for a “slab with one face in contact eqn (7) are: with a layer of perfect conductor” is given by Carslaw T = 0.6K T = 4.2K and Jaeger[lZf. The initialconditionsthat apply after the heat pulse, given that the copper tray equ~~~tes inB,=O B,=O stan~eously, are: no energy transfer other than from p2 = 0.7 82 = 1.0 the perfect conductor; initial temperature of the tray taken as TOwith respect to the temperature of the PbIz /3, = 1.6 83 = 2.0 crystal delined as T = 0 at t = 0. The equation which describes the variation of the temperature of the copper The denominator in the terms in the summation in eqn (5) becomes for p2 and &: tray with time is (eqn (3.lM) of Ref.[lZff: before more can be said about this test of Zdimensionaiity. Actually,8~is rather similarto that of the ionic crystal MgO, which can be demonstrated by comparing coelcients of a reduced expansion
(5) I@:+r’)+r= Here, I is the height of the slab taken as 3cm. The parameter I is given by r = pAC,&fC;,
(6)
1(B:+rq+r=
T = 0.6K
T = 4.2K
2.3 9.0
438 448.
The change is a factor of 4 at the lower temperature but almost insignikant at the higher temperature. If we examine the exponential,\;e see that: exp(-f$kZt) = (exp(- Ij?n21))frrf,
(8)
where
exp(- IS$ = exP(- @I>=
T = 0.6K
T = 4.2K
0.6 0.07
0.35 0.02.
For /ltfj greater than unity, the changes in the numerator and denominatoryield minimumratios of the terms in fiz and 81 of 34 at T= 0.6K and 18 at T = 4.2K. ThiS means that, if we restrict our analysisto times such that Tabk 2. Parameters of the iOWtemperature expansions for the heat capacities of Pb12and big0 10-b;
(mJ
K-%*k-~)(ml K-’mole-‘)
i
10-6xa
(mlK-?llok-~)
99.4(&3)
5.9q-9.05)
5.83(kO.l0)
2(23)X 10-2
W6(24)
4_s8&sm59)x tO-s
3.88fr0.10)
3.6(zO.8)x IO-'
(mlK-’mole-‘) 2(+3) 2.?(%0.7)
w. hf. %UIS and J. A. MOIUUWN
5%
(KC( is greater than unity (i.e. t > 100 set), we can truncate the series in eqn (5) without sign&ant error after the term in &. Equation (5) then reduces to:
(9) and
T’=T-T<Ir) Letting (/(/?: + 3) + r), we have T’ = A,
A,, = 2rTd
eXp(-K@),
(10)
where T’ can be calculated from the measurements of T vs t by observing that, at t = m, T’ =O. The determination of the point at t = 0 is of little importance because the constant A, is not used subsequently. Therefore, by plotting lnlT’( vs 1, we can obtain K@$ from the slope (Fi. 3). The deviation from the straight line at long times is due to the small energy exchange between the calorimeter and surroundings through the thin Nylon suspension cords and electrical leads. 2.5 Numerical values of the thermal conductivity The quantity KB: is the. time constant for equilibration Bnd is listed in the second column of Table 3. Knowing the heat capacities as functions of temperature (Table 1 and Ref.[lOl), we can easily calculate fi2 for each experiment @, A and I are assumed to be constant). This allows us to calculate K and the results are given in the third column of Table 3. The thermal conductivity is defined as K = KpGIM,
Fii. 3. The relative temperature[-ItiT- T.)l as a function of time for thermalequilibrationof Pblt at T = 1.93K.
T’(K*l Fig
4. KIT for Pblz as a function of F.
(11)
and the numerical values of K are given in the fourth column of the table.
Within the framework of the Debye model, the thermal conductivity of a dielectric solid is given by K = pC,,v:.l,J3M,
Table 3. Measuredthermalconductivitiesat bw temperatures 1dx A 0.60
0.62 0.73 i:; 1.18 1.2s 1.51 1.82 1.93 E 2:34 2.44 2.65 3.07 3.32 3.79 3.98 4.21
w-
KB:
1
0.94 1.31 1.13 1.50 1.13 1.16 1.13 1.33 1.60
1.70
165 1’54 1:38 1.39 1.17 1.21 1.07 0.90 0.94 0.85
(cm’ set-‘)
K (CWcm-‘K-l)
1.68 2.28 1.77 2.0s 1.54 1.44 1.37 1.47 1.69 1.78 1.71 1.57 1.40 1.10 1.16 I.18 1.04 0.87 O.#) 0.81
0.46+0.12 0.56*0.14 1.40*0.28 1.20+0.24 1.9620.39 2.20i0.37 4.12~0.70 8.34~ 1.50 10.4 21.9 12.3 e2.2 13.5 22.4 15.0 ~2.6 17.5 zt2.6 20.2 i3.0 35.1 e5.3 4t.3 27.8 53.4 e 13.4 66.0 2 19.8 71.3 *21.4
1dxr
0.30*0.08
(P)
?I 46
$ 2 38 zl 45 41 36 36 30 31 27 z 21
(12)
where C is the phonon mean free path and v:. is the average phonon velocity. If we take v:, = 1.151(?0.005)~ l@cmsec-’ from the heat capacity measurements, we get the values of 1, given in the last column of Tabk 3. Fii 4 is a plot of the thermal conductivity in the fonn of K/T vs 7”. Thisshows that, in the limit of very low temperatures, K becomes pruportional to T’ which is consistent with the Debye model for tbe lattice viirations. A more sensitive way of displaying the results is given in Fii 5 (I,,, vs T-l). It shows that, in the higher temperature region, there is a T-’ dependence. 3. -
We shall IIrst compare estimates of tbe average wave velocities obtained from tbe Merent kinds of experiments. To obtain the average wave velocity from the neutron scatter& and Brillouin experiments, we make use of the tabks compiled by Wokott[l3], for the special case of hexagonal crystals, which require the elastic constantsasinput.TbepolytypeWofPb~istr@al amI has an extra elastic constanG cl.. Its value is small (-3 x 10” dyn cm-3[14, ls] compared to the other elas-
Low temperaturethermalpropertiesof PbI2
0 0
0 0
0.4
0.6 I/T
0.6
1.0
(K)
Pii. 5. The phone0 mean free path in WI2 as 8 function of T-l.
tic constants. (For polytype 4H, it should be zero.) One of the elastic constants needed, c13, is not obtainable from the neutron scattering data[14] and so an estimate based on results of measurements of Brillouin spectra[lS] was taken. The calculated wave velocities are: u..(neutron, 295K) = l.tM(~O.05)x Id cm set-‘; u..(Brillouin, 2MK) = [email protected])x ld cm see-‘; and u&heat capacity, T --, OK) = 1.151(+0.005)x 1OScmsee-‘. The results from the neutron and Brillouin experiments dialer by more than their quoted uncertainties. To assess which result is to be preferred, we need to estimate the average velocity from the heat capacity at T=295K. Data for Baman spectra of polytypes of PbI2 show [6,101 that the frequencies of the transverse acoustic branch in the crystallographic c-direction decrease by about 22% when the temperature is changed from 50 to 400K. A rough caktdatiou, which takes into account the anisotropy of the crystal, indicates that the average wave velocity would decrease by about 10% for a temperature change from 0 to 29SK. Thus, it appears that the wave velocities from the neutron and heat capacity measurements a8ree well. It is not evident why the result from the BriIlouin measurements should be larger by 13%. With respect to the thermal couductivity, it has been kuown for some time [ 16) that layer-like substances such as graphhe have very low conductivities at liquid helium temperatures. For instance, the thermal conductivity of AGOT nuchar grade graplh is only 5 pW cm-’ K-’ at T= 1KI!711,This seems to be the result of weak coupling perpendimdar to the graphite layers and of effects involviug o&if-plane phonon modes(l8). Ihe temperature depetxknce of the conductivity is found to be of the form Ka7’” with n varying over the range I .8 < n <2.8[17,18].
507
As Table 3 and Fig. 4 show, the apparent thermal conductivity of Pb12 seems to be even lower than that of graphite, being about 1.2pW cm-’ K-’ at T = 1K. We should note, however, that there is a significant electronic contribution to K for graphite at low temperatures [ 191.Since the present paper was submitted for publication, a report of measurements of K of Pb12 in the direction parallel to the planes has been published[20]. The results are sampledependent but the magnitude of K is about IO’ larger than that found here. It is very improbable that the difference can be ascribed to anisotropy in K at such a low temperature. In graphite, the anisotropy in K is less than a factor of 3 in the region T I IK, although it is estimated to be as large as lo3 at T = 3OOK[21].Measurements on graphite have also shown that the magnitudes of K at low temperatures can vary enormously from specimen to specimen such as by a factor to lo’ between natural and pyrolytic graphites[21]. The results of calculations based on a simple model [6] indicate that the force constant for the interaction between the layers of PbI2 is about thirty times larger than the force constant for the same interaction in graphite. While this might lead to the expectation that K for Pb12 would be much larger than that for graphite, it should be remembered that the magnitude of K at very low temperatures is mainly determined by boundary scattering[22]. In this regard, we should note again that K for Pb12 becomes proportional to T3 in the limit of very low temperatures (Fig. 4) and that this is the appropriate temperature dependence for boundary scattering. At the higher temperatures, K increases less rapidly than as T3 (Fig. 4), indicating, perhaps the onset of specific effects of the layer type structure, as in graphite[l7,181. It might also be noted that recent measurements on another layer-type structure, boron nitride, indicate its thermal conductivity to be about 6OpWcm-’ k-’ at T = lK[23]. The small values of K have the consequence that the apparent mean free path of the phonons is also unusually small (Table 3 and Fig. 5). Even if I,,, were larger by a factor of ten, it would still be much smaller than the distance between macro boundaries in the crystal specimen or between dislocations. The density of dislocations would be unlikely to exceed 10’“cm-2. In the higher temperature region, 1, becomes approximately proportional to T-’ (Fig. 5), a result that is characteristic of phonon scattering by umklapp processes. In summary, the heat capacity results indicate that the vibrational spectrum of Pb12 is probably “normal”. They are consistent with the limited available results from other experiments except for those derived from observations of Brillouin spectra. The observed thermal conductivity seems to be unusually small but its temperature dependence is “normal”. The possibility that its magnitude is decreased because of thermal resistance between the specimen and the calorimeter tray cannot be ruled out but it is thought to be improbable. Aebrotizemaus-we sbnuld like to thank Mr. J. D. Garrett rod Dr. J. E. Greedanfor assistancewith crystal~owing ~IMJ Dr.
508
W. M. SURS and J. A. M~~IRIs~N
M. L. Klein and Dr. D. Walton for helpful discussion. The financial support of this research by tbe National Research Council of Canada is gratefully acknowledged.
I. Nicklow R., Wakabayasbi N. and Smith H. G., Phys. Reu. B!l,4951 (1972). 2. MUM R. W., J. Phys. CS, 535 (1972). 3. DeSorbo W. and Nichols G. E., 1. Phys. Chem. Solids 6,352 (1958). 4. van der Hoeven El. J. C. Jr. and Keesom P. H.. Phys. Rev. 13). 1318(1%3). 5. Fujita S. and Bugl P., Phys. Reu. MS, IOW(1969). 6. Sears W. hf., Klein M. L. and Morrison J. A., Phys. J&u. B in press (1979). 7. Lusbiin K. J. and Morrison J. A., 1. Chem. Phys. 69.4214 (1978). 8. Bansal 0. P.. Morrison 1. A. and Richards E. L., Z. Phys. B32. 379 ( 1979).
9. Barron T. H. K. and Morrison J. A., Con. 1. Phys. 35, 799 (1957). IO. Sears W. M., Ph.D. Thesis, McMaster University (1978). Il. Barron T. H. K., Berg W. T. and Morrison J. A., Pm. R. sm. A250.70 (1959). 12. Carslaw H. S. and Jaeger J. C.. Conduction of Hear in Solids 2nd Edn, p. 128. Oxford University Press (1959). 13. Wolcott N. M.. J. C&m. Phys. 31,536 (1959). 14. Domer B.. Gbosh R. E. and Ha&eke G., Phvs. Status Solidis (b) 73,65j (1976). IS. Sandercock J.. Fesfh&perproblunc 15, 183(1975). 16. Berman R., Ado. Phys. 2, 103(1953). 17. Edwards D. 0.. Sarwinski R. E., S&man P. and Tough 1. T., Cryogenics II, 392 (1968). 18. Kellv B. T.. Phil. Ma. 15. 1005(1967). 19. Klein C. A.‘and Holl&d M. G., ihys:Reu. 136, A575 (1964). 20. Anders E. E., Volcbok I. V. and Sukharevski B. Ya., FIrike Niz&ihhTemperatur 4, 1202(1978). 21. Slack G. A., Phys. Reu. 127,694 (1962). 22. Rosenberg H. M., Low Tcmperaturr Solid State Physics, Cbap.3, Oxford University Press (1963). 23. Sichel E. K., Miner R. E., Abrabams M. S. and Buioccbi C. J., Phys. Reu. 813,4&T! (1976).
DepPrtwnt of Physics and Institute for Materials Research, Mchfaster University, Hamilton, Ontario, Canada (Received
I1 October 1978; accepted in revised fomt 7 December 1978)
Ab&w!-l?~~ beat capacity and tbermal conductivity of a large (56.5 g) crystal of PbI2 have been measured in the temPerahIre ~C~OO 0.5 < T < 39’K. Analysis of the heat capacity data yields a value of the limiting apparent Debye
characteristic temperature 00’ =99.4 (+0.3)“K, which corresponds to an average lattice wave velocity of 1.151 (LMHI~) x ldcm set-‘. It is consistent with a wave velocity estimated from neutron scattering experiments, but not with ooe determined from Brillouin spectra. The heat capacity data also show that dispersion of the low frequency wave3 is not unusual. a3 might have teen expected for a layer-type crystal. The apparent thermal conduction is found to be surprisingly small in the crystal.
1. INTBODIJCI’ION Many crystals display extreme anisotropy and form layer-type structures in which bonding between layers is
much weaker tban that within a layer. In some examples, bonding of the atoms in a layer may be covalent while the bonding between the layers may be largely of the van der WaaIs type. The anisotropy has consequences on the physical properties of the crystals. For example, there will be marked weakening or softening of vibrational modes in the crystallographic c-direction. This is shown in the phonon dispersion curves where the longitudinal acoustic branch is much lower in energy in the direction perpendicular to the layers [ 11. For layer-type structures, some thermodynamic properties, such as the thermal expansion, would obviously be expected to display anisotropy[2]. In other properties, such as the heat capacity, which are related to the whole vibfational spectrum, it may be more difficult to detect effects of anisotropy except under particular circumstances. For instance, t-dimensional effects of at least two kinds have been found in the heat capacity of graphite at low temperatures(3-51. In co~cction with a study of the vibrational properties of various polytypes of Pb12[6], it was found desirabk to obtain an estimate of the average wave velocity for phonons in Pblt from the limiting Debye characteristic temperature at very low temperatures. Thus, the heat capacity of a huge crystal of Pb12 was measured in the region T <4K. The experimental conditions turned out to be such that it was possible to estimate the thermal conductivity of PbI2 at the same time. The results are discussed in relation to the 2dimensiona.l character of the structure of PbL
2.1 Pmpamtion of fhe crystal specimen ‘lb crystal was grown by the Bridgman technique. Reagent gmde (98.5%) lead iodide powder was placed in Present
address:
Department
of Physics, University of
Guelph. Guelph, Ontario, Canada. 503
a glass tube which was evacuated (P < 1 x lo-. torr) and then the lead iodide was melted to drive out any trapped gas. After the tube was sealed under vacuum, it was slowly lowered through a furnace set at a temperature just above the melting point of lead iodide, i.e. at about 400°C. The rate of lowering was controlled with a low speed motor and was such that the crystal grew over a period of four days. When it was removed from the tube, the crystal was found to have an “impurity” layer on top. This part of the crystal was cut off, the remainder was resealed in a glass tube, and the melting and growth procedures were repeated. The regrown crystal did not have an obvious impurity layer on its top surface. A spectrographic analysis was made of small samples .scraped from the crystal and it showed the presence of only a few parts per million of & and Sn. The crystal was thus judged to be suitable for the calorimetric measurements. One end of the crystal was cut with a diamond saw to give the surface a smooth enough finish for mounting on the calorimeter tray (see following section). The cut was made perpendicular to the layers of the crystal. The crystal contained two boundaries and hence was composed of three crystals of the same orientation but the mosaic spread was not determined. The mass of the crystal was 56.5 g. 2.2 Calorimetric technique A tray-type calorimeter was used for the measurements. The PbL crystal was placed on a small copper tray and good thermal contact was promoted through the addition to the contact area of a measured quantity (0.01 cm3) of silicone grease. The tray was mounted in a frame suspended below the mixing chamber in a ‘He‘He dilution refrigerator. Cooling was accomplished through a heat switch, in thermal contact with the mixing chamber, which gripped a gold-plated copper spike attached to the tray. The tray carried an electrical heater and a germanium thermometer which had been intercompared with a germanium thermometer previously calibrated over the
ii’. hi. &ASSand J. A.
504
temperature range 0. I < T -c4.3K [7]. The electrical leads between the calorimeter tray and an anchoring point on the mixing chamber were of lead-covered manganin wire. The thin Nylon filaments which held the tray within the frame were spring-loaded to damp out vibration. It required several minutes for the energy supplied to the tray in each heating interval to be. distributed through the crystal. Other measurements with the same tray assembly on aluminium alloy specimens of comparable contact area181 showed that thermal transmission through the silicone grease was extremely good (equilibrium time < 1 set). Thus, the longer equilibrium time is ascribed to low thermal conductivity of the Pb12 specimen. In a separate series of measurements, the heat capacity of the tray assembly, including 0.01 cm’ of silicone grease, was determined. From the results, a smoothed heat capacity-temperature function was constructed for computing the contribution of the tray assembly to the total heat capacity of the system. Measurements were performed over the temperature range 0.5 < T < 3.9K. In this region, the temperature scale is known with an uncertainty not exceeding several millikelvin[7]. The tray assembly contributed 50% of the total heat capacity at T = 0.5K and 3% at T = 3.9K. Overall, the accuracy of the heat capacity results for Pblz is estimated to be 25% at T = O.SK, ?2% T = 1.5K and 220% at T = 3.9K.
at
2.3 Heat capacity results The measured heat capacities are listed in Table 1 and
are displayed in Fig. 1 as a plot of C,,/T3 vs T2. This is a convenient form for the determination of the coefficients of the low temperature expansion: C, =a,T3+a2T5t
........
(1)
which the heat capacity of simple insulators is expected Table 1. Measured heat capacities at low temperatures
T (K) 0.494 0.514 0.619 0.821 0.844 1.055 1.094 1.360 1.628 1.688 1.904 I.972 2.159 2.22a 2.478 2.758 3.039 3.322 3.601 3.874
G (ml Km’mole-‘) 0.743 2 0.033 0.824 2 ohI 1.39 =o.os 3.28 +0.10 3.57 -+o.lm 7.02 20.16 7.67 ~0.16 15.2 20.2 26.0 20.5 28.6 ~0.4 41.2 +- 1.2 46.4 2 1.0 61.3 + 1.8 68.4 + 1.5 97.9 23.3 + 10 149 2 19 208 243 288 399 ~92 2 122 468
MORRISON
IO
1
ag r
‘z I
ie
n
t
i:~__--_I
1
0
I IO
5
I 15
T2(K2) Fig.
I.
CJT’ for Pb12crystal as a function of T2.
to follow in the re@on approaching the continuum limit [9]. The coefficient al is well-defined as the intercept when T2+0 and, from the graph, we obtain a, = 5.94 (ti.OS)ml Km4mole-‘. This corresponds to a limiting Debye characteristic temperature at T = OK: eo’ = (12r’NkJSa I)“‘,
(2)
where N = total number of atoms (3N, per mole for Pb12) N., = Avogadro’s number and ke = Boltzmann’s constant. In particular &= = 99.4 (ti.3) K. From a1 or 06, we can also obtain the mean wave (or sound) velocity: v:, = k&=(4nV/3N)“‘/h,
(3)
where V is the molar volume of the sample and h is Plan&s constant. The density at T = OK was estimated to be 6.23 gem-’ from measured lattice parameters at higher temperatures [6,10]. Thus, v:, = 1.151 (*0.005)x lticmsec-‘. We also obtain some information about dispersion of the lattice waves from the heat capacity data. The coefficient of the T’ term in eqn (1) should indicate how the acoustical branches of the phonon dispersion curves begin to deviate (on average) from linearity at low energies. Fiie 1 gives a value of a2. obtained from the limiting slope, of a2 = 2 (23)
x
10e2al Kd mole-‘.
We see that, within the limits assigned, a2 could be zero or negative, in which case PbI2 would be showiog a feature of 2dimensional structures such as graphite: crystauographic some dispersion in negative . . directions [ 11. More accurate measurements are needed
SO5
Low temperatore thermal properties of PbI2
where p is the density of PbL at low temperatures (6.23gcm-‘), A is the surface area of the crystal in apparent contact with the tray (3.6cm2),C, and C; are the heat capacities (as a function of temperature) of the PbIzcrystal and the copper tray respectively,and M is the massof the crystal specimen(56.5g). In the exponent, Kis c, =o;(TIeo”)‘+a;(Tleo’)J+. . . . .., (4) the thermal diffusivity and the /3,, are the roots of the equation: where a; = o,(&~)’ and n;= Use, The normalized tan(gl) = -/3/r. coefficientsa; and ai of MgOand PbIz are comparable (7) (Table 2) and the similarity of the behaviour of the two Referringto eqn (5), we note that it containsan infinite substances is further emphasized by Fig. 2 where series of exponentials. However, the contribution of (C,/T3)/(C~T’)o is plotted vs (T/&‘)*. each of them decreases rapidly for successive roots of eqn (7). In calculating the refative sizes of successive 2.4 l’7termalconductivityof PbIz The thermal lactic was determined from analy- exponent&Is, we take two extreme cases: T = 0.6K and sis of the temperature-timebehavior of the coiorimeter T = 4.2K. This leads to (from eqn 6) r = 0.4cm-’ and system after it was heated. The solution to the heat r = 12cm-‘. For the two cases, the first three roots of conduction problem for a “slab with one face in contact eqn (7) are: with a layer of perfect conductor” is given by Carslaw T = 0.6K T = 4.2K and Jaeger[lZf. The initialconditionsthat apply after the heat pulse, given that the copper tray equ~~~tes inB,=O B,=O stan~eously, are: no energy transfer other than from p2 = 0.7 82 = 1.0 the perfect conductor; initial temperature of the tray taken as TOwith respect to the temperature of the PbIz /3, = 1.6 83 = 2.0 crystal delined as T = 0 at t = 0. The equation which describes the variation of the temperature of the copper The denominator in the terms in the summation in eqn (5) becomes for p2 and &: tray with time is (eqn (3.lM) of Ref.[lZff: before more can be said about this test of Zdimensionaiity. Actually,8~is rather similarto that of the ionic crystal MgO, which can be demonstrated by comparing coelcients of a reduced expansion
(5) I@:+r’)+r= Here, I is the height of the slab taken as 3cm. The parameter I is given by r = pAC,&fC;,
(6)
1(B:+rq+r=
T = 0.6K
T = 4.2K
2.3 9.0
438 448.
The change is a factor of 4 at the lower temperature but almost insignikant at the higher temperature. If we examine the exponential,\;e see that: exp(-f$kZt) = (exp(- Ij?n21))frrf,
(8)
where
exp(- IS$ = exP(- @I>=
T = 0.6K
T = 4.2K
0.6 0.07
0.35 0.02.
For /ltfj greater than unity, the changes in the numerator and denominatoryield minimumratios of the terms in fiz and 81 of 34 at T= 0.6K and 18 at T = 4.2K. ThiS means that, if we restrict our analysisto times such that Tabk 2. Parameters of the iOWtemperature expansions for the heat capacities of Pb12and big0 10-b;
(mJ
K-%*k-~)(ml K-’mole-‘)
i
10-6xa
(mlK-?llok-~)
99.4(&3)
5.9q-9.05)
5.83(kO.l0)
2(23)X 10-2
W6(24)
4_s8&sm59)x tO-s
3.88fr0.10)
3.6(zO.8)x IO-'
(mlK-’mole-‘) 2(+3) 2.?(%0.7)
w. hf. %UIS and J. A. MOIUUWN
5%
(KC( is greater than unity (i.e. t > 100 set), we can truncate the series in eqn (5) without sign&ant error after the term in &. Equation (5) then reduces to:
(9) and
T’=T-T<Ir) Letting (/(/?: + 3) + r), we have T’ = A,
A,, = 2rTd
eXp(-K@),
(10)
where T’ can be calculated from the measurements of T vs t by observing that, at t = m, T’ =O. The determination of the point at t = 0 is of little importance because the constant A, is not used subsequently. Therefore, by plotting lnlT’( vs 1, we can obtain K@$ from the slope (Fi. 3). The deviation from the straight line at long times is due to the small energy exchange between the calorimeter and surroundings through the thin Nylon suspension cords and electrical leads. 2.5 Numerical values of the thermal conductivity The quantity KB: is the. time constant for equilibration Bnd is listed in the second column of Table 3. Knowing the heat capacities as functions of temperature (Table 1 and Ref.[lOl), we can easily calculate fi2 for each experiment @, A and I are assumed to be constant). This allows us to calculate K and the results are given in the third column of Table 3. The thermal conductivity is defined as K = KpGIM,
Fii. 3. The relative temperature[-ItiT- T.)l as a function of time for thermalequilibrationof Pblt at T = 1.93K.
T’(K*l Fig
4. KIT for Pblz as a function of F.
(11)
and the numerical values of K are given in the fourth column of the table.
Within the framework of the Debye model, the thermal conductivity of a dielectric solid is given by K = pC,,v:.l,J3M,
Table 3. Measuredthermalconductivitiesat bw temperatures 1dx A 0.60
0.62 0.73 i:; 1.18 1.2s 1.51 1.82 1.93 E 2:34 2.44 2.65 3.07 3.32 3.79 3.98 4.21
w-
KB:
1
0.94 1.31 1.13 1.50 1.13 1.16 1.13 1.33 1.60
1.70
165 1’54 1:38 1.39 1.17 1.21 1.07 0.90 0.94 0.85
(cm’ set-‘)
K (CWcm-‘K-l)
1.68 2.28 1.77 2.0s 1.54 1.44 1.37 1.47 1.69 1.78 1.71 1.57 1.40 1.10 1.16 I.18 1.04 0.87 O.#) 0.81
0.46+0.12 0.56*0.14 1.40*0.28 1.20+0.24 1.9620.39 2.20i0.37 4.12~0.70 8.34~ 1.50 10.4 21.9 12.3 e2.2 13.5 22.4 15.0 ~2.6 17.5 zt2.6 20.2 i3.0 35.1 e5.3 4t.3 27.8 53.4 e 13.4 66.0 2 19.8 71.3 *21.4
1dxr
0.30*0.08
(P)
?I 46
$ 2 38 zl 45 41 36 36 30 31 27 z 21
(12)
where C is the phonon mean free path and v:. is the average phonon velocity. If we take v:, = 1.151(?0.005)~ l@cmsec-’ from the heat capacity measurements, we get the values of 1, given in the last column of Tabk 3. Fii 4 is a plot of the thermal conductivity in the fonn of K/T vs 7”. Thisshows that, in the limit of very low temperatures, K becomes pruportional to T’ which is consistent with the Debye model for tbe lattice viirations. A more sensitive way of displaying the results is given in Fii 5 (I,,, vs T-l). It shows that, in the higher temperature region, there is a T-’ dependence. 3. -
We shall IIrst compare estimates of tbe average wave velocities obtained from tbe Merent kinds of experiments. To obtain the average wave velocity from the neutron scatter& and Brillouin experiments, we make use of the tabks compiled by Wokott[l3], for the special case of hexagonal crystals, which require the elastic constantsasinput.TbepolytypeWofPb~istr@al amI has an extra elastic constanG cl.. Its value is small (-3 x 10” dyn cm-3[14, ls] compared to the other elas-
Low temperaturethermalpropertiesof PbI2
0 0
0 0
0.4
0.6 I/T
0.6
1.0
(K)
Pii. 5. The phone0 mean free path in WI2 as 8 function of T-l.
tic constants. (For polytype 4H, it should be zero.) One of the elastic constants needed, c13, is not obtainable from the neutron scattering data[14] and so an estimate based on results of measurements of Brillouin spectra[lS] was taken. The calculated wave velocities are: u..(neutron, 295K) = l.tM(~O.05)x Id cm set-‘; u..(Brillouin, 2MK) = [email protected])x ld cm see-‘; and u&heat capacity, T --, OK) = 1.151(+0.005)x 1OScmsee-‘. The results from the neutron and Brillouin experiments dialer by more than their quoted uncertainties. To assess which result is to be preferred, we need to estimate the average velocity from the heat capacity at T=295K. Data for Baman spectra of polytypes of PbI2 show [6,101 that the frequencies of the transverse acoustic branch in the crystallographic c-direction decrease by about 22% when the temperature is changed from 50 to 400K. A rough caktdatiou, which takes into account the anisotropy of the crystal, indicates that the average wave velocity would decrease by about 10% for a temperature change from 0 to 29SK. Thus, it appears that the wave velocities from the neutron and heat capacity measurements a8ree well. It is not evident why the result from the BriIlouin measurements should be larger by 13%. With respect to the thermal couductivity, it has been kuown for some time [ 16) that layer-like substances such as graphhe have very low conductivities at liquid helium temperatures. For instance, the thermal conductivity of AGOT nuchar grade graplh is only 5 pW cm-’ K-’ at T= 1KI!711,This seems to be the result of weak coupling perpendimdar to the graphite layers and of effects involviug o&if-plane phonon modes(l8). Ihe temperature depetxknce of the conductivity is found to be of the form Ka7’” with n varying over the range I .8 < n <2.8[17,18].
507
As Table 3 and Fig. 4 show, the apparent thermal conductivity of Pb12 seems to be even lower than that of graphite, being about 1.2pW cm-’ K-’ at T = 1K. We should note, however, that there is a significant electronic contribution to K for graphite at low temperatures [ 191.Since the present paper was submitted for publication, a report of measurements of K of Pb12 in the direction parallel to the planes has been published[20]. The results are sampledependent but the magnitude of K is about IO’ larger than that found here. It is very improbable that the difference can be ascribed to anisotropy in K at such a low temperature. In graphite, the anisotropy in K is less than a factor of 3 in the region T I IK, although it is estimated to be as large as lo3 at T = 3OOK[21].Measurements on graphite have also shown that the magnitudes of K at low temperatures can vary enormously from specimen to specimen such as by a factor to lo’ between natural and pyrolytic graphites[21]. The results of calculations based on a simple model [6] indicate that the force constant for the interaction between the layers of PbI2 is about thirty times larger than the force constant for the same interaction in graphite. While this might lead to the expectation that K for Pb12 would be much larger than that for graphite, it should be remembered that the magnitude of K at very low temperatures is mainly determined by boundary scattering[22]. In this regard, we should note again that K for Pb12 becomes proportional to T3 in the limit of very low temperatures (Fig. 4) and that this is the appropriate temperature dependence for boundary scattering. At the higher temperatures, K increases less rapidly than as T3 (Fig. 4), indicating, perhaps the onset of specific effects of the layer type structure, as in graphite[l7,181. It might also be noted that recent measurements on another layer-type structure, boron nitride, indicate its thermal conductivity to be about 6OpWcm-’ k-’ at T = lK[23]. The small values of K have the consequence that the apparent mean free path of the phonons is also unusually small (Table 3 and Fig. 5). Even if I,,, were larger by a factor of ten, it would still be much smaller than the distance between macro boundaries in the crystal specimen or between dislocations. The density of dislocations would be unlikely to exceed 10’“cm-2. In the higher temperature region, 1, becomes approximately proportional to T-’ (Fig. 5), a result that is characteristic of phonon scattering by umklapp processes. In summary, the heat capacity results indicate that the vibrational spectrum of Pb12 is probably “normal”. They are consistent with the limited available results from other experiments except for those derived from observations of Brillouin spectra. The observed thermal conductivity seems to be unusually small but its temperature dependence is “normal”. The possibility that its magnitude is decreased because of thermal resistance between the specimen and the calorimeter tray cannot be ruled out but it is thought to be improbable. Aebrotizemaus-we sbnuld like to thank Mr. J. D. Garrett rod Dr. J. E. Greedanfor assistancewith crystal~owing ~IMJ Dr.
508
W. M. SURS and J. A. M~~IRIs~N
M. L. Klein and Dr. D. Walton for helpful discussion. The financial support of this research by tbe National Research Council of Canada is gratefully acknowledged.
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