2022.06.14 — bolometer icing

this post presents a quantatitive analysis of the "icing" hypothesis. the hypothesis attempts to explain the observed reduction in bolometer sensitivity with methane exposure. the argument is that the accumulation of layer of solid methane leads to an appreciable increase in the heat capacity of the bolometer, which is directly proportional to the bolometer sensivity.

bolometer lifetime

we can assign a "lifetime" $\tau$ to the bolometer by determining the exposure time required to grow a layer of solid methane whose heat capacity matches that of the diamond absorber. defining:

c_CH₄ : molar heat capacity of solid methane
c_C : molar heat capacity of diamond
f : molecular flux into bolometer
N_C : size of diamond absorber

we obtain the following expression for $\tau$: \begin{equation} \tau = \frac{c_C N_C}{c_{CH_4} f} \end{equation} from https://doi.org/10.1021/ja01355a011 I obtain a value of c_CH₄ = 0.5 cal mol^-1 K^-1 for "equilibriated" solid methane at 2 Kelvin and a value of c_CH₄ = 0.01 cal mol^-1 K^-1 for "unequilibriated" solid methane at 2 K. the authors explain that equilibriation of a pure sample of solid methane can take several hours.

using the formula found in https://doi.org/10.1080/14786435808243223 I obtain a heat capacity of diamond at 2 Kelvin of 3.2 × 10^-7 cal mol^-1 K^-1. the stark difference between the solid methane and diamond heat capacities at 2 K can be qualitatively explained in terms of the rigid nature of the diamond lattice. a much larger fraction of the phonons in diamond will be "frozen out" compared to solid methane.

the size of the diamond absorber can be determined easily from the density of diamond (3 mg mm^-3) and the absorber dimensions (4 mm diameter, 0.1 mm thickness).

the methane flux f into the bolometer is given by the flux density $\frac{df}{d\Omega}$ scattering from the surface multiplied by the solid angle $\Delta \Omega$ subtended by the bolometer. for a scattering distribution of the form $\cos^n\theta$, the peak scattered flux density $\frac{df}{d\Omega}$ is related to the incident flux $F$ by \begin{equation} \frac{df}{d\Omega} = F \frac{n+1}{2 \pi} \end{equation} the solid angle $\Delta \Omega$ subtended by the bolometer is given by \begin{equation} \Delta \Omega = \frac{\pi r^2}{R^2} \end{equation} where r=2 mm is the absorber radius and R=80 mm is the bolometer-surface distance

the incident flux $F$ can be determined from the pumping speed $S$ and the chamber pressure $P$ by the following formula \begin{equation} F = \frac{P}{P_o} \frac{S}{V_{stp}} \end{equation} where P_o = 10³ mbar and V_stp=22.4 liters. in bojung chen's thesis he calculates the pumping speed to be S = 625 l s^-1, and for neat methane at 3 bar backing pressure we observe a rise in the P₄ chamber equal of P = 6 × 10^-8 mbar.

with these numbers we obtain the following results:

F = 1.7 × 10^-9 moles s^-1
f = 6.8 × 10^-12 moles s^-1
N_C = 3.1 × 10^-4 moles
τ = 30 s (c_CH₄ = 0.5 cal mol^-1 K^-1)
τ = 24 min 40 s (c_CH₄ = 0.01 cal mol^-1 K^-1)

(see 20220614ice.py for the python script used to do these (simple) calculations).

diamond is roughly six times denser than solid methane, so the methane thickness at one bolometer lifetime is either 3 nanometers or 0.6 angstroms, depending on whether you assume the smaller or larger solid methane heat capacity, respectively.

our most recent measurements with neat methane observed at specular resulted in a lifetime closer to 70 minutes, significantly longer than even the slower estimate. off the top of my head I can list some reasons, listed in estimated order of importance, why we made observe a slower than expected sensitivity decay:

the bolometer heat capacity is the sum of the heat capacities of the diamond absorber and silicon chip. in my calculation I consider only the diamond absorber.
the molar heat capacity of the diamond absorber may in reality be larger than estimated. the diamonds used in the referenced study are "clear industrial diamond", and "none contained occlusions". perhaps our diamonds have a higher defect density, leading to an increased heat capacity.
some of the molecules striking the nickel surface may temporarily trap and desorb in a very wide angular distribution. the fraction of molecules that trap will contribute only negligibly to the growth of solid methane.
the molecules may not stick at the diamond surface with unit efficiency. the diamond surface is stiff and the molecules from the supersonic expansion are somewhat fast. if this is the case, we may expect an acceleration of the bolometer sensitivity decay once the molecules begin to impinge upon a soft layer of solid methane.

background flux

for H₂O we have the simple rule that every square inch of cold surface pumps with speed of roughly 100 l s^-1. Since methane and water have nearly the same mass, we can use this figure to estimate the flux f_bg coming into the bolometer aperture from the background pressure $P$. the result obtained is f_bg ≈ 5 × 10^-12 moles s^-1. remarkably, this is nearly as strong as the flux coming from the direct beam!

it is reasonable however to suspect that the background methane flux incident on the actual bolometer will be signficantly less than f_bg. the vast majority of the molecules entering the bolometer snout will strike the baffles. the cold baffles serve to both geometrically impede access to the bolometer and to also slow the molecules down. both of these effects will reduce the conductance between the snout aperture and the bolometer (which acts as a sort of methane pump), with the result that the true flux on the bolometer will be less than that indicated by f_bg.

in any case, it will be interesting to see what the measurements indicate. what is the bolometer lifetime subjected to only a given background methane pressure?

thermal conductivity

to wrap up, I point out that, at a fixed modulation frequency, a reduction in bolometer sensitivity can be equally well attributed to a decrease in the thermal conductivity between the absorber surface and the silicon bolometer. however, the thermal conductivity of diamond at 2 K is 80 mW cm^-1 K^-1 for natural diamond and 1 mW cm^-1 K^-1 for CVD diamond (https://doi.org/10.1007/978-1-4615-2257-7_7), while the thermal conductivity of solid methane at 2 K is 5 mW cm^-1 K^-1 (https://doi.org/10.1103/PhysRevB.55.5578). that the values for the two materials are comparable suggests that it would require an implausibly thick layer of solid methane to measurably affect the bolometer sensitivity.