The problem that brought about the need for this project was that we did not know how efficient the atom detector is. The detector consists of a rhenium wire, heated by a current. When atoms hit the wire, they become ionized, and the resulting charged particles accelerate by way of an electric field towards a channel electron multiplier, which counts them. In order to know how many are counted out of how many hit the wire, I had to obtain an independent measurement of atom flux. The method by which I accomplished this was through the use of a quartz crystal oscillator. A thin slab of quartz crystal, which is a piezoelectric material, has a specific frequency at which it oscillates in a driving circuit. If a small amount of mass is deposited onto the crystal, its frequency will decrease, just as a mass-spring system’s frequency decreases with added mass. Thus, by putting a quartz crystal oscillator in the path of the atom beam, the atoms get deposited onto the crystal and its frequency will decrease at a measurable amount. From this, calculations reveal the atom flux, resulting in the detector efficiency.

Inside the chamber, the circuit platform with the crystal on it was put on a simple mechanical rotating stage, which could be turned so that the crystal went in and out of the beam. The geometry of the system is illustrated in figure 2, and a picture of the region between the skimmer and collimating slit is shown in figure 3.

Throughout all my calculations, I assumed that the atom beam behaves as if there exists an extended source that radiates in all directions. To get the dimensions and position of this effective source, I scanned the detector wire laterally in order to get the shape of the beam. For a second measurement, the detector was kept stationary, and the collimating slit was scanned back and forth. Taking the positions of half-maximum intensity in both cases, a simple geometric calculation with similar triangles gives the position and size (half-max width) of the effective source.

To calculate the mass deposited onto the crystal, I assumed that the crystal acted like a massive spring. I also assumed that it extended and compressed in the direction of its thinnest dimension as its main mode of vibration, which is supported by literature on quartz crystals. The Lagrangian of such a spring with rigid mass (M) added on to one side becomes

Applying Euler’s formula gives an equation of motion of the form

with a frequency of

where M is the deposited mass, m_o is the original mass, and ν_o is the original frequency. Thus, the deposited mass onto the crystal is described by

However, this derivation neglects the electrodes on the crystal, and assumes that their mass is part of the mass of the crystal itself.

The last necessary calculation is the derivation of the flux ratios, or finding out what kind of flux onto the detector wire the flux onto the crystal corresponds to. To do this, I considered the two transverse dimensions of the beam separately. The height of the beam is unhindered by the first collimating slit, and gets cropped off by a 20 micron horizontal slit just in front of the detector. Because of this distance, I assumed the beam in that dimension be from a point source, meaning that the vertical intensity distribution would be a constant value. The horizontal distribution comes from the fact that the first slit is relatively close to the extended source. The scanning of the detector wire revealed this distribution, and a Gaussian curve fit gave the numerical formula. A graphical depiction of the 2-D distribution is shown in figure 4.

The shaded region depicts the "shadow" cast by the detector wire, and the 20 micron vertical slit has been left out for simplicity. The height on the slit that corresponds to the 20 micron vertical distance at the wire, times the width of the slit, I assumed to be the effective area of the slit. Thus, all the atoms that pass through this effective area are part of the beam profile. This profile, acquired from scanning the detector laterally, could be fit by a Gaussian, resulting in an empirical formula. The percentage of atoms that the wire eclipses can then be calculated by integrating the curve over the width of the wire. This, therefore, gives the percentage of atoms passing through the effective slit that hit the wire. On the other side of the collimating slit, I assumed that the beam diverges as 1/r², and thus calculated the ratio of the fluxes from the exposed area of the crystal to the effective area of the slit. By combining this ratio with the ratio of wire to beam profile, the final flux ratio turned out to be

The physical setup of the system was a bit tricky. It turned out to be easier to superimpose the crystal oscillations onto a signal from a function generator and then measure the beat frequency. However, the two frequencies could not be too close to each other, because the generator would start driving the crystal. After the two were combined, a diode to ground cropped the signal, and a low-pass filter cleaned it up. After an amplifier, the signal was sent to a frequency meter and a speaker. Another amplifier then sent the signal off to a computer. The setup can be seen in figure 5. The crystal driving circuit inside the chamber has been simplified so as not to clutter the diagram too much. As the crystal was rotated into the beam, one could actually hear the pitch of the beat frequency changing.

The actual data taken by the computer showed clearly the changing frequency when the crystal was in the beam. Figure 6 shows such a sample, where the crystal was taken out of the beam at 114 seconds and put back in at 217 seconds.

Frequency vs. Time

Frequency Change Over a Longer Time Period.

The regions of increasing frequency do not fit on perfectly straight lines, although it seems that they always start out that way. Since crystal frequencies are highly dependent on temperature, this could be due to a heating of the crystal by the atoms that condense on it. When the crystal is taken out, it radiates this heat away, which would explain the slight downslope of the frequency seen in figures 6 and 7.

Using these data and the calculations from before, the final result of the efficiency of the detector becomes

However, from the uncertainties involved in the assumptions, and from the fact that the detector might just be working more efficiently on some days than on others, it is probably more accurate to state that the efficiency lies somewhere between 3 and 4%.