To investigate the spectacular physical phenomena induced by the magnetic field in (TMTSF)₂ClO₄, it is necessary to work with single crystal samples, so as to be able to orient the magnetic field along a precise crystallographic direction. The available single crystals have a mass of a few milligrams and their heat capacity hardly reach about 10 nanojoules per Kelvin at a temperature of 0.3K.

Classical calorimetry methods [1] are not well suited to measuring such heat capacities. They consist in delivering a known heat amount to a sample that has been thermally isolated from its environment, and in measuring its resulting temperature rise. As far as small samples are concerned, with masses smaller than 100 mg, satisfactory adiabatic conditions are difficult to achieve.

On the contrary, the so-called non adiabatic techniques accommodate the existence of a thermal coupling between the heat sink and the sample. In addition, a heat sink enables to submit the sample to a periodical thermal regime, to average the received signals, and thus to improve the resolution power of the method. Alternative calorimetry belongs to the non adiabatic techniques [2,3]. Compared with other methods of the same family, it is characterised by a great sensitivity to low temperature heat capacity variations. Thanks to this property, the alternative method is perfectly well suited to heat capacity measurements performed as a function of an external parameter, in a continuous mode. So it is particularly convenient to study the phase transitions induced by a magnetic field in Bechgaard salts (see the following sections).

The thermal model of figure 1 describes the sample thermal behaviour and the heat exchanges with its environment.

FIG. 1. Thermal model of the alternative nanocalorimetry.

The sample is attached to the nanocalorimeter thanks to a constraint gauge glue. The nanocalorimeter is equipped with a thermometer and a heating element (Figure 2).

FIG. 2. Nanocalorimeter enabling the measurement of tiny single crystals, with a heat capacity Cp smaller than 500pJ/K at a 0.3 K temperature.

The total heat capacity is the sum of the sample contribution and the nanocalorimeter contribution: C_tot = C_s+C_nano. A sinusoidal power P(t)=P₀(1+coswt) is dissipated in the nanocalorimeter. In the steady state regime, the zero frequency component P₀ is evacuated to the thermal sink, of temperature T_b, through the thermal link, of thermal conductance K_b. This component acts so as to maintain the mean temperature of the sample-calorimeter set at the value: T₀=T_b + P₀/K_b. The alternative component dP_w(t) produces a temperature oscillation dT_w(t) at the same frequency, in direct relation to the heat capacity C_tot. The sample temperature T(t) is written as the solution to the thermal balance:

It means that the heat amount per time unit used to rise the sample temperature is the difference between the total supplied power and the evacuated power toward the heat sink. A steady state solution to this equation is as follows:

where the response time t_b of the nanocalorimeter-sample set toward the heat sink is written as:

and where the phase shift is expressed as:

Thus the temperature T(t)=T₀ + dT_w(t) is the sum of a constant component:

and an alternative component at the same frequency f=w/2p as the supplied power:

where the temperature oscillation amplitude has the following form:

So the thermal oscillation amplitude dT is a complicated function of C_tot. If the frequency and the sink thermal conductance are chosen so that wt_b>>1, then the second term inside the root can be neglected. The total heat capacity, sum of the sample contribution and the one from addenda, simply writes:

However the thermal conductivity inside the sample may not necessarily be considered as infinite. It can be shown [2] that the diffusion influence can be neglected if the temperature oscillation period, 2p/w, is large as compared with the sample internal relaxation time, t_s=C_s/K_s (where K_s is the sample effective internal thermal conductance), and at the same time small as compared with the sample relaxation time toward the heat sink, t_b. The latter condition can be checked during the experiment because it is equivalent to: dT/(T-T_b)<<1.

Besides the temperature oscillation phase shift may be used to indicate a possible departure from the theoretical model: in the ideal case of a linear response, the oscillation has a quadrature phase shift retardation with respect to the supplied power, but the finite thermal conductivity effects are revealed by an additional phase shift. The latter is for instance related to the internal relaxation time t_s as far as the sample thermal conductivity itself is concerned. Simultaneously measuring this phase shift and the oscillation amplitude enables to determine the thermal conductivity and the heat capacity in the same experiment, if a suitable thermal model is available for that (see the following, and section 4-b) [publi 15].

[1] G.R. Stewart, Rev. Sci. Instrum. 54, 1 (1983).
[2] P.F. Sullivan and G. Seidel, Phys. Rev. 173, 679 (1968).
[3] P. Manuel and J.J. Veyssié, Rev. Gén. Therm. Fr. 171, 231 (1976).