We present a new type of calorimeter in which we couple an unknown heat
capacity with the aid of Peltier elements to an electrical circuit. The use of
an electrical inductance and an amplifier in the circuit allows us to achieve
autonomous oscillations, and the measurement of the corresponding resonance
frequency makes it possible to accurately measure the heat capacity with an
intrinsic statistical error that decreases as ~t^{-3/2} with measuring time t,
as opposed to a corresponding error ~t^{-1/2} in the conventional alternating
current (a.c.) method to measure heat capacities. We have built a demonstration
experiment to show the feasibility of the new technique, and we have tested it
on a gadolinium sample at its transition to the ferromagnetic state.Comment: 6 pages, 5 figure

A. Schilling

Corbino O. M.

O. Bossen

English

arXiv

Crossref

Review of Scientific Instruments

<i>LC</i>-circuit calorimetry

We present a new type of calorimeter in which we couple an unknown heat capacity with the aid of Peltier elements to an electrical circuit. The use of an electrical inductance and an amplifier in the circuit allows us to achieve autonomous oscillations, and the measurement of the corresponding resonance frequency makes it possible to accurately measure the heat capacity with an intrinsic statistical uncertainty that decreases as ∼ tm−3/2 with measuring time tm, as opposed to a corresponding uncertainty ∼ tm−1/2 in the conventional alternating current method to measure heat capacities. We have built a demonstration experiment to show the feasibility of the new technique, and we have tested it on a gadolinium sample at its transition to the ferromagnetic state

Bossen, O

Schilling, A

ZORA

LC-Circuit CalorimetryO. Bossen1 and A. Schilling1Physik-Institut der Universita¨t Zu¨rich, Winterthurerstrasse 190, 8057 Zu¨rich,SwitzerlandWe present a new type of calorimeter in which we couple an unknown heat capacitywith the aid of Peltier elements to an electrical circuit. The use of an electrical in-ductance and an amplifier in the circuit allows us to achieve autonomous oscillations,and the measurement of the corresponding resonance frequency makes it possibleto accurately measure the heat capacity with an intrinsic statistical uncertaintythat decreases as ∼ t−3/2m with measuring time tm, as opposed to a correspondinguncertainty ∼ t−1/2m in the conventional alternating current (a.c.) method to measureheat capacities. We have built a demonstration experiment to show the feasibility ofthe new technique, and we have tested it on a gadolinium sample at its transition tothe ferromagnetic state.To be published in Review of Scientific InstrumentsOnline Preprint: http://arxiv.org/abs/1107.2801I. INTRODUCTIONCalorimetry is an old field in experi-mental physics, and numerous experimen-tal techniques have been developed for theaccurate measurement of heat capacities1.The frequently used a.c. technique inventedby Corbino2 and modified by Sullivan andSeidel3, for example, measures a heat capac-ity by monitoring the temperature amplitudeof a sample that is subject to an oscillatingheating power , or, although less common,by measuring an oscillating heat current pro-duced by a Peltier element4. Such a.c.techniques have numerous advantages overother methods, e.g., heat-pulse, differential-thermal-analysis5 or relaxation techniques1.A certain robustness to noise and the possi-bility to use very small temperature ampli-tudes stems from the fact that the signal ofinterest can be separated from the instrumen-tal noise by using a frequency selective filter,such as a lock-in amplifier. To increase thetotal accuracy even further, the duration of ameasurement tm can be increased at will, andthe resulting statistical uncertainty decreasesas t−1/2m .If a calorimeter could be built where the heat1capacity is related to the frequency insteadof the amplitude of an oscillation one wouldretain most of the advantages of the conven-tional a.c. techniques, but the statistical un-certainty would decrease much more rapidly6,i.e., as t−3/2m . Therefore, the accuracy of sucha method would be mainly limited by thestability of the oscillator components, ratherthan by statistical constraints.In the following we describe the realizationof an oscillating thermo-electrical circuit us-ing Peltier elements which makes an electri-cal inductor to effectively act as a “thermalinductor“. The heat capacity can be calcu-lated from the measured resonance frequencyof the circuit, and its statistical uncertaintydrops according to t−3/2m as expected.II. OPERATING PRINCIPLE OFTHE THERMO-ELECTRICALOSCILLATORIt is known that the flow of heat Qt (witha corresponding heat current It = Q˙t) be-tween entities with a certain heat capacity Ctthrough links with a finite thermal conductiv-ity can be mapped onto the flow of charge Qe(with the electrical current Ie = Q˙e) in elec-trical circuits that is composed of capacitorsCe and resistors, respectively, where the tem-perature T in the thermal case correspondsto the electrostatic potential V in the equiv-FIG. 1. An electrical inductor Le is connectedto a Peltier element that separates a heat capac-ity Ct from a thermal bath T . Changes in theelectrical current Ie induce a temperature differ-ence ∆T = LeI˙e/α (where α denotes the Seebeckcoefficient) across the Peltier element that coun-teracts the variation in the flow of heat I˙t.alent electrical circuit. The differential equa-tions describing the time evolution of Qt (t)or Qe (t), respectively, are of first order intime t. However, a simple thermal analogueof an electrical inductor involving a higherderivative Q¨t = I˙t does not exist, althoughthe inclusion of terms including I˙t is justifiedin certain rare cases7. Therefore it is impos-sible to realize autonomously oscillating ther-mal circuits by using only standard thermalelements.In our experiment we connect an electricalinductor Le to a Peltier element as shown in2Fig. 1. If the heat current It through thePeltier element to the thermal bath varieswith time, an associated electrical current Iein the circuit changes according to ΠI˙e = I˙t, where Π is the Peltier coefficient of thePeltier element. It induces a voltage dropLeI˙e across the inductor that opposes thechanges in the heat current It through thePeltier element via the Seebeck effect, i.e.,a resulting additional temperature difference∆T = LeI˙e/α (where α denotes the Seebeckcoefficient) across the Peltier element is im-posed that counteracts the variation in theflow of heat, I˙t . We can therefore define aneffective thermal inductanceLt :=∆TI˙t=LeαΠ. (1)However, as a consequence of the signifi-cant losses produced by the non-zero thermalconductivity and electrical resistance of thePeltier element, the idealized element shownin Fig. 1 allows in general only for an over-damped variation of the heat current. Thisobstacle can be avoided by including an am-plifier with a suitable gain in the electri-cal circuit, and autonomous oscillations caneventually be achieved in this way. In theideal case of a stationary oscillation with con-stant amplitude where all losses are exactlycompensated by the amplifying element, theelectrical circuit behaves like a simple LC cir-cuit with thermal inductance Lt and a heatcapacity Ct, or with an electrical inductanceLe and an equivalent electrical capacitanceCe =QeV=∫Ie dtα∆T=Ct∫Ie dtα∫ΠIe dt=CtαΠ(2)and with resonance frequency ω2 = 1/LtCt =1/LeCe. Therefore, the heat capacity can bederived from ω according toCt =αΠLeω2, (3)notably without any need to quantify thethermal and electrical losses in the Peltier ele-ment. To the best of our knowledge, the cor-respondences between an Le and a thermalanalogue Lt, or Ct and a Ce as expressed inEqs. (1) and (2), respectively, are novel andhave also never been exploited in any calori-metric technique.III. EXPERIMENTAL SETUPWe have chosen the circuit shown in Fig. 2containing two Peltier elements (MPC-D701,Micropelt Inc.) based on Bi2Te3, with lat-eral dimensions of ≈ 3.5mm and a thicknessof ≈ 1mm. While the left element acts, to-gether with the sample to be measured, as theequivalent electrical capacitance as describedabove, the right element is used to compen-sate for thermal and electrical losses and tomaintain a constant amplitude of the tem-perature oscillation. For this purpose, the3FIG. 2. Circuit diagram showing the Peltier element system, the Gyrator to mimic a large electricalinductance, and the amplifier to compensate for thermal and electrical losses (see text). The voltageVG controls the gain of the amplifier. The two Peltier elements have the same polarity, and thesample is attached to their “top side“. The measured signal is tapped at Vout.0 10 20 30 40 50 60time [s] 0.10 0.050.000.050.100.150.200.250.300.35T [K]FIG. 3. Thermal oscillations of the LC-circuit. The solid line is a least-squares fit to a sinusoidalvariation of ∆T in time around an offset value that is caused by lost heat dissipated in the Peltierelements.voltage across the right element is amplifiedand then fed onto the electrical contacts ofthe left element. To achieve a stable oscilla-tion of the desired amplitude, the gain of theamplifier can be controlled using a voltage-controlled gain amplifier (AD630, Analog De-vices). An additional clamp with Schot-tky diodes (BAT43) and two potentiome-ters forms a nonlinear resistor that reducesthe gain at large oscillation amplitudes andtherefore forces the circuit to self-stabilize.For a design heat capacity Ct ≈ 20mJ/K tobe measured near room temperature (whichcorresponds to 2.87 · 10−4mol of a Dulong-Petit solid), and using α ≈ 29mV/K andΠ ≈ αT ≈ 8.4W/A for the used Peltier el-4ements, the equivalent electrical capacitancebecomes Ce ≈ 82mF. Depending on the heatcapacities and the thermal conductivities in-volved in the thermal part of the system,the frequency of the temperature oscillationmust be small enough to allow for a quasi-stationary thermal equilibrium. If we choosef = ω/2pi ≈ 0.18Hz we need an inductanceLe ≈ 10H which would require impracticallylarge passive elements. Therefore we replacedthe inductance by a gyrator, the circuit dia-gram of which is also shown in Fig. 2. Theeffective induction of the gyrator is given by8Le = (RP +RG1)RG2Cgyr, where the internalresistance RP ≈ 6.5Ω of the Peltier elementhas to be included here. With RG1 = 1Ω ,RG2 = 200 kΩ and Cgyr = 6.8µF we indeedobtain Le ≈ 10H.IV. TEST MEASUREMENTSA test sample (45.2 mg or 2.87 · 10−4molof gadolinium with 99.99% purity that hadbeen previously annealed at T = 1123Kin vacuum to reduce lattice strains9) wassandwiched between the two Peltier elementsand placed into a commercial PPMS sys-tem (Physical Property Measurement Sys-tem, Quantum Design) for control of thebase temperature. The gain of the ampli-fier was chosen in a way to achieve temper-ature oscillations with an amplitude of the0.1800.1850.1900.1950.2000.2050.2100.215f=/2 [Hz]a)280 285 290 295 300Temperature [K]0.0090.0100.0110.0120.0130.0140.0150.0160.017Ct [J/K]b)FIG. 4. a) Measured frequency of the oscillatorand b) heat capacity of a Gd sample, as functionsof temperature. Crosses represent the data thatwe obtained by the present thermal LC-method,while circles denote the values from a measure-ments on the same sample using a commercialheat capacity system (see text).order of 0.2K. For each data point, the os-cillating voltage was read out during 1200 swith a 10 Hz-triggered Keithley 2001 Multi-meter in equal time intervals of 0.1 s (see Fig.3), and the resulting data were then least-squares fitted to obtain the oscillation fre-quency (see Fig. 4a). To compensate for the5heat capacity of the empty system (≈ 15% ofthe total heat capacity), corresponding mea-surements have been done without the sam-ple installed. The resulting values of theheat capacity are shown in Fig. 4b, togetherwith heat-capacity data obtained on the samesample using the commercial heat-capacityoption of the PPMS system and with tem-perature steps of ≈ 0.5K. While the heat-capacity data using our thermal LC- methodare neither smoothed nor further averaged,the PPMS data were averaged over 6 individ-ual measurements. The data-acquisition timefor each of the LC- method data points was1200 s, while collecting an averaged PPMSdata point took about one hour of time, andthe temperature amplitudes for both tech-niques were comparable (0.3 K - 0.5 K). Thetransition of the Gd sample to the ferromag-netic state takes place at the Curie tempera-ture TC ≈ 290K, and the absolute values forthe heat capacity agree well with the PPMSdata (see Fig. 4b).V. ACCURACYCONSIDERATIONSThe accuracy of any heat-capacity mea-surement is limited by systematic and sta-tistical uncertainties. Sources of systematicerrors (such as finite instrumental resolution,limited accuracy by a large background sig-0.17400.17450.17500.17550.1760A [V]a)1.44501.44551.44601.4465 [Hz]b)10-710-610-510-410-3relative errorc)t1/2mt3/2mA /A/	20 50 100 200 500 1000 2000 5000tm ,  [s]10-610-510-4yd)1/2FIG. 5. a) Amplitudes and b) frequencies fora selected data point together with the respec-tive standard deviations from the Levenberg-Marquardt fit, as functions of measurement timetm. c) The respective relative standard devia-tions on a log-log scale to illustrate the power-law behavior discussed in the text Solid lines arefits according to t−1/2 for σA/A and to t−3/2 forthe σω/ω data, respectively. d) The overlappingAllan deviation σy, plotted against the observa-tion time τ . The solid line illustrates the ap-proximate τ−1/2 dependence of σy (see text).6nal, errors in the temperature calibrationetc.) are common to all techniques for mea-suring a heat capacity, and we shall not fur-ther discuss these shortcomings here. Themain advantage of the present thermal LC-circuit technique is that it significantly re-duces the statistical uncertainty with increas-ing measuring time.The statistics behind the problem of esti-mating the amplitude A, the frequency ω andthe phase ϕ of an oscillation A sin(ωt+ϕ) outof a set of N samples of finite length T hasbeen analyzed in detail in Ref. 6. Assuminga white noise with variance σ2, the result-ing relative standard deviation σA/A in theamplitude A from a measurement of N oscil-lation cycles with length T = 2pi/ω isσAA≥ σA√N. (4)In a conventional a.c. technique to mea-sure heat capacities, the relative standard de-viation in the heat capacity Ct from the meanvalue is proportional to the respective stan-dard deviation in the measured amplitudeA of a temperature oscillation2, and there-fore the statistical uncertainty in Ct decreaseswith t−1/2m where tm = NT is the total timeto take such a measurement (in all other con-ventional methods, the statistical uncertaintyalso decreases as N−1/2, where N is the num-ber of measurements to calculate the averageof a data point).The corresponding relative standard de-viation σω/ω in the frequency ω , however,varies with N according toσωω≥√3σpiA√N(N2 − 1), (5)i.e., it decreases with N−3/2 (or t−3/2m ) forlarge N . As the heat capacity in our tech-nique is proportional to ω−2 (see Eq. 3), therelative standard deviation σCt/Ct ≈ 2 ·σω/ωalso decreases with N in the same fashion. InFig. 5 we show the amplitudes and the fre-quencies for a selected data point, togetherwith the respective relative standard devia-tions, as functions of the measurement timetm which is proportional to the number of os-cillation cycles N . The standard deviationsσA and σω have been taken from a standardLevenberg-Marquardt routine used for least-squares fitting the experimental data. TheσA/A data decrease relatively slowly withN−1/2 , while the σω/ω data (and along withthem σCt/Ct ) drop much faster with N−3/2as expected. Therefore, very high accuraciesfor Ct can, in principle, be achieved in a com-parably very short measuring time. For ex-ample, to reduce the statistical uncertaintyin Ct by a factor of 10, conventional meth-ods require a 100 times longer duration of ameasurement, while the thermal LC methodneeds only a ≈ 5 times longer data acquisi-tion time. As both σA/A and σω/ω are pro-portional to the same quantity σ/A, there is7no hidden drawback in the equation 5 thatwould put this tremendous statistical advan-tage into perspective.However, in order to tap the full potentialof this technique it is essential that the oscil-lation frequency ω is sufficiently stable duringthe time of the measurement so that the po-tentially minuscule statistical uncertainty isnot impaired by extrinsic fluctuations, e.g.,by an insufficient stabilization of the tem-perature of the sample or by an unwantedvariability in the electronic circuit. While aLevenberg-Marquardt fit always finds someleast-squares minimum, with a σω that de-creases for increasing tm, the actual oscilla-tor frequency can be unstable in such a waythat repeated measurements under identicalexperimental conditions lead to different re-sults for the fitted parameters. The quan-tity σω can therefore not be readily inter-preted as the true measurement error of ω.A standard measure for the stability of anoscillator is provided by the Allan deviationσy10. Assuming white noise in the oscilla-tor frequency ω, for example, σy is expectedto drop as τ−1/2, where τ denotes the timeover which the frequency is observed. Atlarge observation times σy can saturate oreven increase again as soon as other sourcesof noise (Flicker noise, random-walk noise)become dominant11. In Fig. 5d we have plot-ted the overlapping Allan deviation as a func-tion of observation time τ on a log-log scale.While σy initially drops approximately withτ−1/2, is saturates in our experiment aroundσy ≈ 4 · 10−6 for τ ≈ 1000 s and increases forlarger vales of τ . In this sense we interpretthe minimum in σy as an estimate for theachievable uncertainty in ω for a given setof experimental data, while the σω/ω valuesfrom Eq. 5 and shown in Fig. 5c representthe statistical uncertainty assuming a per-fectly stable frequency with white noise onlyin the temperature amplitude. In our demon-stration experiment, no special care has beentaken to minimize extrinsic sources for vari-ations in ω. Such sources of error are the de-cisive factors that are limiting the maximumachievable accuracy in the present thermalLC technique but they are to a large extentcontrollable, while the intrinsic statistical un-certainty should play a significantly less im-portant role than in conventional techniquesto measure heat capacities.VI. CONCLUSIONSWe have described a method where a heatcapacity is coupled to an electrical circuitwith the aid of Peltier elements, which makesan electrical inductor to effectively act as a“thermal inductor“. The use of an amplifierin the circuit allows us to achieve autonomousoscillations, and the measurement of the cor-8responding resonance frequency makes it pos-sible to calculate the heat capacity. While thestatistical uncertainty in conventional tech-niques to measure heat capacities scales ac-cording to t−1/2m with measuring time tm, therespective uncertainty in the present thermalLC technique decreases significantly faster,i.e., with t−3/2m , and is ultimately limited bythe stability of the thermal and electroniccomponents used in the experiment .ACKNOWLEDGMENTSThis work was supported by the Schweiz-erische Nationalfonds zur Fo¨rderung derWissenschaftlichen Forschung, Grant. No.20-131899REFERENCES1E. Gmelin, Thermochim. Acta 29, 1 (1979);G. R. Stewart, Rev. Sci. Instrum. 54, 1(1983); Y. Kraftmakher, Physics Reports356, 1 (2002).2O. M. Corbino, Physik. Zeitschr. XI, 413(1910); XIII, 375 (1912).3P. F. Sullivan and G. Seidel, Phys. Rev.173, 679 (1968).4D. Jung, I. Moon, and Y. Jeong, Ther-mochim. Acta 391, 7 (2002); R. Lortz,S. Abe, Y. Wang, F. Bouquet, U. Tutsch,and A. Junod, Rev. Sci. Instrum. 76,103902 (2005).5A. Schilling and M. Reibelt, Rev. Sci. In-strum. 78, 033904 (2007).6D. C. Rife and R. R. Boorstyn, IEEE Trans.Inform. Theory IT-20, 591 (1974).7R. C. L. Bosworth, Nature 158, 309 (1946);T. F. McNelly, S. J. Rogers, D. J. C. R. J.Rollefson, W. M. Goubau, G. E. Schmidt,J. A. Krumhansl, and R. O. Pohl, Phys.Rev. Lett. 24, 100 (1970); B. M. Weedy,IEEE Electric. Pwr. Syst. Res. 15, 197(1988); D. D. Joseph and L. Preziosi, Rev.Mod. Phys. 61, 41 (1989).8D. F. Berndt and S. C. Dutta Roy, IEEE J.Solid-State Circ. SC-4, 161 (1969).9D. S. Simons and M. B. Salamon, PhysicalReview B 10, 4680 (1974).10P. Allan, Proc. of IEEE 54 54, 221 (1966).11W. Riley, Handbook of Frequency Stabil-ity Analysis, NIST Special Publication No.1065 (NIST, 2008).9

LC-circuit calorimetry

LC-Circuit Calorimetry

LC-Circuit Calorimetry

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