Practical test procedures for setting up and calibrating Faraday Balance Systems for Testing Superconductors Follow Microbalance manuals for setting up the balance and calibrating the chart recorder and/or digital voltmeter to be used for recording the output. Connect a vacuum source and valves so that the balance can be evacuated slowly to avoid large disturbances to the balance from the outrush / inrush of gases. A gauge will be needed to read the pressure in the balance/sample area, since this pressure will determine both the thermal contact between the sample and the dewar, and the amount of noise from the convection. Set up thermometry and temperature controller Tests which require balance, but not magnet: Record balance noise with no samples, dewar at room temperature, and good vacuum in sample space. This establishes performance with no extraneous forces due to gas flow or convection. With system cooled to near lowest temperatures, establish how much gas pressure is tolerable in sample area without noticeable convection problems. (My guess is that 1mm will be okay, 100mm will be too much). repeat #2 at the high temperature end. I would expect convection to be a worse problem here since the warmest temperatures are at bottom. Magnet It would be good to measure the magnetic field versus position for the magnet, and from these data derive a map of the "force function", H * dB/dx , where x is the vertical distance. This data would establish how sensitive the instrument calibration is to the position of the sample, as well as how large a sample can be before the calibration changes because not all the sample "sees" the same H * dB/dx. However, it is not essential that this be done for the first system, and certainly not before the American Physical Society show. This is the type of information which is needed to accurately characterize the limits of performance, and to pass along to customers so that you don't get "this thing isn't working the way it's supposed to" type call from customers. For the first system we will need at least to be able to tell customers what the peak magnetic field is at the sample position, so they will know how much TC is being suppressed by the field. System Tests Alignment Experience shows that the correct vertical position for the sample should be .5 inch above the narrowest gap point of the pole pieces. Since this position is .5 inch above the center of the pole faces, the nominal sample position is 1 inch above the center of the pole faces. In the horizontal plane, the nominal position should be centered about the pole faces. In practice, we assume that there will be some variation from magnet to magnet, so the actual position of strongest force will have to be determined by adjusting the positions around the nominal one and finding the position of maximum force. The basic procedure should be to load a sample of about 10 mg of the HgCo(SCN)4 and record the change in the weight when the magnet is turned on to maximum current. The sample temperature is not important, as long as it is constant, so it might as well be room temperature., Suppose that at maximum current the product H * dB/dx was about 2 x 10**6 [Oe*gauss/cm], and at room temperature the calibrant has a mass susceptibility of about 16 x 10**-6 [cm**3/g](emu). Then if G is the acceleration due to gravity, the change in apparent mass, Dm, should be about: Dm = sm * m/G * H * dB/dx (emu) = (10-2[g] * 2 * 10**6 [Oe*Gauss/cm] * 16 * 10**-6 [cm**3/g]) / (980 [cm/sec**2]) = .33 milligram (Note that if the noise level is about .1 microgram one could measure the susceptibility of this sample with a precision of .03%!). Then turn the magnet off, adjust the position of the dewar, and sample by 1/10 of an inch or so, and again record the change in weight when the magnet is on. Repeat this procedure to map the force variation in all 3 directions, and then adjust the position to that giving the maximum weight change. Note that any change in sample temperature will also change the weight when the magnet is on, because the susceptibility is temperature dependent. For the sample we have been discussing, a .1 degree temperature change would change the weight (with field on) by about .1 microgram. Calibration The simplest way to calibrate is to use a sample of known susceptibility to establish the appropriate instrument calibration constant, c(I), for a particular magnet current, I. We do this by first measuring the weight (in dynes) of the empty sample holder (called the boat for some reason) before and after the magnet current, I, is turned on. Subtracting these two weights we get the weight change for the empty boat which we call Dwb. We then make the exact same measurement again with our sample of known mass and mass susceptibility in the boat. We calculate a weight change Dw for the sample and the boat. Dw - Db is then the weight change that the sample alone undergoes when we turn on our particular magnet current, I. Let H be the magnetic field generated by I and let x be in the vertical direction. Then Dw-Db is the force exerted by the imposed field H on our sample which has known mass m, and known susceptibility, sm. From equation (3) we have Dw-Db = sm * m * H * dB/dx, (4) where dB/dx is the gradient of the B field corresponding to H. (We use dB/dx rather than dH/dx so the units will formally balance.) Since sm and m are known and Dw-Db has been measured, we can solve for H * dB/dx = (Dw-Db)/(sm * m). H * dB/dx in the volume occupied by our sample is constant for a fixed magnet current I and so becomes our calibration constant. Accordingly we let c(I) = 1/(H * dB/dx) = sm*m/(Dw-Db). (5) For any other sample occupying roughly the same volume as the calibration sample and having a known mass m' we can calculate its susceptibility sm' from its measured weight change Dw' (obtained by turning I on and off while its in the boat) as follows: sm' = c(I)*(Dw' - Db)/ m'. To sum up, we calibrate our Faraday Balance Susceptometer at a given magnet current I by first measuring the weight change of the boat when I is turned on and then measuring the weight change of the boat loaded with a sample S, of known mass and susceptibility. These two measurements are subtracted obtaining a measurement of the force magnetic field generated by I exerts on S. Using S's susceptibility and mass we then calculate a calibration constant c(I) which allows us to determine the susceptibility of any other sample of known mass which occupies roughly the same volume as S when it is put in the boat. A typical calibration sample material is HgCo(SCN)4. This has a mass susceptibility of +16.44 x 10-6 [cm**3/g] at 20° C. This susceptibility is strongly temperature dependent, being approximately proportional to 1/T (T in Kelvin). Therefore a 1 degree error in the absolute temperature will cause a susceptibility variation in the calibration sample of about .3%. In principle, one would expect that both the magnetic field and the gradient in the field would vary linearly with the magnet current. If this were true, the force would vary as the square of the current, and the calibration factor would therefore vary as 1/I**2. However, the data sent to me by Walker suggest that the force will not increase exactly this way, presumably due to saturation effects in the iron core. For the purposes of a demo at the APS meeting, it would be sufficient to run a calibration at one current, and then plot out a "weight change versus current" plot for that sample, so the approximate calibration constant at any current will be known. Ultimately, customers will want to calibrate to their own satisfaction, anyway, so for production systems only a single current calibration will need to be done, mostly for the purpose of recording the basic properties of each system, and in order to spot whether something is way out of line before a system is shipped. Linearity Here one wants to demonstrate that, for a single material at constant temperature and magnet current, the weight change is linear with the sample mass. What is needed is a plot of "net weight change (i.e., corrected for the weight change of the empty boat) versus sample weight" for 5-10 samples covering a range of, say, 5-50 mg of the calibrant. These data should fall on a straight line through the origin, and the scatter is a good indication of overall instrument precision. This data would not be needed as a test of each instrument; it would however make one more confident that the system works well. Curie-Weiss Law Curve This phenomenon gives a useful and readily performed test of the precision of the instrument and how well it performs through variations of sample temperature. If one graphs the susceptibility (sm) of a sample against absolute temperature (T) one gets a curve which according to the Curie Weiss law should have an equation of the form: sm = C / (T - Tc ) where C and Tc are constants and is the temperature in Kelvin. (If there is no interaction between neighboring paramagnetic ions in a given molecule of the sample, the "Weiss constant," Tc, will be zero, and the sample will said to have simple "Curie law 1/ behavior"). This data gives a feel for what the instrument will do as a whole system, in a situation where temperature fluctuation occurs. A technique for checking that the Curie Weiss law is being "seen by the instrument is to measure a sample of the HgCo(SCN)4 at 6 to 10 temperatures spanning the limits of the system. This data is converted to mass susceptibilities using the calibration constant derived using this same sample and 1/ each value is plotted versus temperature. The result should be a straight line. The intersection of this line with the temperature axis gives the Weiss constant, which is about 10K for this compound.

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