= Low-temperature heat capacity measurement
Low-temperature <calorimetry> measures a small sample's <heat capacity> by resolving a small <temperature> change after a known electrical <energy> input. The experimental challenge is that the support, heater and <thermometer> also warm up. Their <calorimeter addenda> must be removed before interpreting the sample's <phonons> or <electronic heat capacity>. The <Debye T-cubed test> turns the corrected response into a test of the low-<temperature> vibrational spectrum.
<Manuel Brando>'s 2009 experiment used a sapphire platform suspended by four alloyed platinum wires from a <copper> ring fixed to a cryostat stage. Grease attached the sample; a chip heater and ruthenium-oxide <resistance thermometer> served platform PLS-2. A controller stabilized the bath; a lock-in amplifier supplied heating pulses and read the calibrated sensor. The steady rise calibrated <thermal conductance> as $G=P/\Delta T$. A separate $59.1\,\mathrm{mg}$, $99.99\%$ pure <gold> sample, compared with reference <heat capacities>, established the <calorimeter addenda>. Small pulses limited property variation; imperfect sample contact required a second relaxation time. The platform contribution rose below $1.5\,\mathrm K$, making simple extrapolation unsafe.<Development of a Relaxation Calorimeter for Temperatures between 0.05 and 4 K by Manuel Brando>[Brando, Development of a relaxation calorimeter for temperatures between 0.05 and 4 K, §§II–IV.B, Table I, Figs. 4, 5 and 8: construction, calibration, contact correction and low-temperature platform upturn]{ref}
For an original idealization, assume sample and platform share one <temperature> $T$, the bath remains at $T_b$, and both the total <heat capacity> and the weak-link <thermal conductance> $G$ are approximately constant across one small pulse. Ignore other <heat transfer> and any <phase transition>. The <first law of thermodynamics> becomes
$$
(C_s+C_a)\frac{dT}{dt}=P(t)-G(T-T_b),\qquad
\tau=\frac{C_s+C_a}{G}.
$$
Here $C_s$ is sample <heat capacity>, $C_a$ is <calorimeter addenda>, $P$ is deposited electrical <power> from <Joule heating>, and $\tau$ is a <time constant>. This is the same balance as <calorimeter heat-loss correction>, used here with a deliberately weak cryogenic link and a separately subtracted platform. For a constant pulse $P_0$ beginning at bath <temperature>, then switched off at $t_h$,
$$
T(t)-T_b=\begin{cases}
\dfrac{P_0}{G}(1-e^{-t/\tau}),&0\le t\le t_h,\\
\dfrac{P_0}{G}(1-e^{-t_h/\tau})e^{-(t-t_h)/\tau},&t>t_h.
\end{cases}
$$
The cooling <exponential decay> determines $\tau$, while the heating plateau determines $G=P_0/(T_\infty-T_b)$. Together with <calorimeter addenda>, they give $C_s=G\tau-C_a$. Cooling alone determines a ratio, not an absolute <heat capacity>. With finite $t_h$, the observed end rise is smaller than the plateau; fit the full response or apply the displayed correction rather than identifying the two.
\Image[/low-temperature-calorimeter-pulse-response.png]
{title=A cryogenic pulse probes sample plus <calorimeter addenda> through a weak thermal link}
{description=Original constructed one-<temperature> model, not a drawing of Brando's apparatus or measured data. The bath is fixed at $2.000\,\mathrm K$, the heater supplies $2\times10^{-9}\,\mathrm W$ for $100\,\mathrm s$, and the total <heat capacity> is $10^{-6}\,\mathrm{J\,K^{-1}}$. Internal sample-platform <thermal equilibrium> is assumed instantaneous; the actual experiment can require two relaxation times.}
For the stipulated figure parameters, $G=5.00\times10^{-8}\,\mathrm{W\,K^{-1}}$, $\tau=20.0\,\mathrm s$, and $C_a=2.00\times10^{-7}\,\mathrm{J\,K^{-1}}$ give $C_s=8.00\times10^{-7}\,\mathrm{J\,K^{-1}}$. The asymptotic rise is $0.0400\,\mathrm K$, only $2\%$ of bath <temperature>. At $100\,\mathrm s$, the rise is $0.0397305\,\mathrm K$. Of $2.00\times10^{-7}\,\mathrm J$ input, only $3.97305\times10^{-8}\,\mathrm J$ remains stored: the other $1.60270\times10^{-7}\,\mathrm J$ has flowed to the bath. Treating this long pulse as an isolated $Q/C$ experiment would overestimate total <heat capacity> by a factor of $5.034$.
Suppose independent <standard uncertainties> are $u(G)=0.05\times10^{-8}\,\mathrm{W\,K^{-1}}$, $u(\tau)=0.2\,\mathrm s$ and $u(C_a)=0.10\times10^{-7}\,\mathrm{J\,K^{-1}}$. First-order <propagation of uncertainty> gives
$$
u(C_s)^2=\tau^2u(G)^2+G^2u(\tau)^2+u(C_a)^2,
\qquad C_s=(8.00\pm0.17)\times10^{-7}\,\mathrm{J\,K^{-1}}.
$$
The uncertainty is one <standard deviation> under those assumptions, about $2.2\%$ of sample <heat capacity>. Shared <thermometer calibration> errors can correlate fitted $G$ and $\tau$; then their <covariance> contributes too. Smaller samples amplify the relative influence of <calorimeter addenda>. Proposed checks include varying pulse amplitude, repeating with a reference sample, checking sensor self-heating, and comparing single- and two-time-constant fits. A drifting bath, slow internal equilibration or a transition invalidates this simple reduction even if a cooling curve looks smooth.
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