# Thermal Conduction

There are three ways to conduct heat:  radiation, conduction, and convection.

In our cryostats, we remove the air from the vessel so convection is (near) zero;  we usually only deal with conduction and radiation.

P = net power flow (from hot to cold)
A = area of surface
T1 = temperature of  hot surface (in Kelvin)
T2 = temperature of cold surface (in Kelvin)
sigma = 6e-8  (Watts/m^2/K^4;  Stefan-Boltzmann constant)
e = emissivity of surface (0=shiny, 1=black)

The power radiated by a surface at temperature T is

P_emit = e*sigma*A*T^4

Usually the environment around that surface is emitting as well, and the so the surface will be absorbing power (in the case of a black environment at T=Tenv)

P_abs = e*sigma*A*Tenv^4.

The net power flow is the important thing, and is

P_net = P_emit – P_abs = e*sigma*A*(Ts^4 – Tenv^4)

Things get a little more complicated if the “environment” isn’t a blackbody;  this is often the case in our cryostats, where we have shiny walls (to reduce the heat transfer) on both the hot and cold surfaces viewing eachother.  In the case of two parallel surfaces, both with emissivity e, the net power flow is (see for example G.K. White, “Experimental Techniques in Low-Temperature Physics“, Ch 5)

P_net = sigma*A*(T1^4 – T2^4)*(e1*e2)/(e1+e2-e1*e2),

which for e1=e2=e<<1 (often applicable in our cryostats) reduces to

P_net = sigma*A*(T1^4 – T2^4)*e/2.

Conduction

Heat is transferred through solids by conduction.  Imagine a rod of length L, cross sectional area A, thermal conductivity kappa, and a temperature difference from one end of the other of delta_T.  The heat flowing through the rod (due to that temperature difference) is

P = kappa * (A/L)*delta_T

For a given delta_T, we see that the power is related to the material (kappa), and the geometry (A/L).   If we want high power (eg if we are trying to conduct heat well) then we want to choose a good heat conductor, and a large (A/L).   If we want low power (to try and reduce the heat conduction, usually trying to isolate something, temperature wise) then we choose a poor heat conductor and a low (A/L).

The thermal conductivity kappa is, for solids, related to the heat transfer by phonons and by electrons.  Phonons travel well (eg without scattering) in good crystal lattices without impurities or defects.  The same is true for electrons.  For that reason, the best thermal conductors tend to be pure metals – such as copper, aluminum, gold, and silver.  In some cases, pure crystal insulators, such as silicon, can be good thermal conductors by virtue of their good phonon transport.

Alloys, such as brass and steel (including stainless steel) and copper-nickel, are full of impurities that impede phonon and electron transport.  While they are electrical conductors, they are poor ones.  This is true for heat as well.

In general, the thermal conductivity is very temperature-dependent due to the importance of temperature on scattering of phonons and electrons. The factor kappa above is therefore really given by an integral of kappa(T) over the temperature range between the two ends.  You can often use tabulated values of these integrals over convenient cryogenic temperatures… but often we’re interested in rough numbers and can ballpark them, rather than actually doing the integral.

A great source for cryogenic thermal properties is the NIST database, at http://cryogenics.nist.gov/MPropsMAY/materialproperties.htm