calibration would be in error by about 2 percent if the effect of mass loading is not taken into account for a test accelerometer of about 19-gram mass.

2.2 Calibration of the BTB Accelerometer
by the Comparison Method

The BTB accelerometer is calibrated by a comparison method in which a reference accelerometer (about 19 gram) is mounted on top of the BTB accelerometer (fig. 2). The reference accelerometer has been calibrated by comparison to NBS reference shakers (which have been reciprocity calibrated, 10-3500 Hz) and by interferometric displacement measurement (4000-10,000 Hz) [2]. The comparison calibration is performed by energizing the shaker at a given test frequency, and measuring the voltage ratio of the output of the BTB accelerometer to the output of the reference accelerometer. This ratio is then multiplied by the sensitivity of the reference accelerometer, thereby thereby yielding the sensitivity of the BTB accelerometer at this test frequency.

Similarly, calibrations are performed at the other test frequencies to obtain the complete calibration of the BTB accelerometer. The results of this comparison calibration of the BTB accelerometer are shown, together with the previously described absolute

Figure 4-A back-to-back accelerometer with a dummy mass mounted for optical calibration.

calibration data, in figure 7. The data in figure 7 indicate agreement between the absolute interferometric method and the comparison method to

لسلـ

21

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20.6

20.2

19.8

19.8

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19.4

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1 2 3

12 4 5 6 7 8 9 10 FREQUENCY kHz

15

Sensitivity

of Accelerometer

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21.40

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Figure 7-Optical and comparison calibration of a back-to-back accelerometer.

be within about 1 percent. Thus, the simple comparison method provides an accurate calibration of the BTB accelerometer, but does not yield the mass loading characteristics of the BTB accelerometer. One study [3] attempts to measure the mass loading by the comparison method by inserting masses between the BTB accelerometer and the reference accelerometer. This does give some qualitative results but does not take into account the relative motion between the accelerometers introduced by the inserted mass.

3. The Use of the BTB Accelerometer as a Laboratory Standard

Once calibrated, the BTB accelerometer, mounted on a suitable shaker, can be used as a reference standard, suitable for calibrating accelerometers up to a frequency of 15 kHz. The calibration is valid only for accelerometers of about 19-gram mass (since this is the calibration mass). However, this mass is typical of

4.

Conclusion

The technique presented here eliminates one step in the calibration of a BTB accelerometer. Instead of using a two-step method of first calibrating a singleended accelerometer and then calibrating the BTB accelerometer by comparison to it, a one-step absolute method is used. It is desirable to have a standard calibrated by an absolute method in terms of fundamental units (e.g., the wavelength of light). Further data need to be collected on BTB accelerometers which are typically in use as laboratory standards. With these additional data, this technique will result in improved accuracy calibrations in the 3- to 15-kHz frequency range. In addition to improved accuracy, the absolute method described above yields additional information about the accelerometer (mass loading characteristics) which the comparison method does not yield.

The BTB accelerometer, when properly calibrated under a loaded condition, can be an accurate and repeatable calibration standard. This standard, when mounted on a suitable shaker, will provide a convenient and accurate setup for performing comparison accelerometer calibrations.

5. References

[1] Payne, B. F.

Absolute

calibration

of back-to-back accelerometers, Proceedings of the 27th international instrumentation symposium; 1981 May; Indianapolis, IN; 1981. 483-488.

[2] Payne, B. F.; Koyanagi, R. S.; Federman, C.; Jones, E. Accelerometer calibration at the National Bureau of 21st Standards, Proceedings of the international instrumentation symposium ASD/TMD; 1975 May 19-21; Philadelphia, PA; 1975. 1–17.

[3] Koyanagi, R. S.; Pollard, J. D.; Ramboz, J. D. A systematic study of vibration transfer standards—mounting effects, Natl. Bur. Stand. (U.S.) NBSIR 73-291; 1973 September. 42p.

1. Introduction

One of the fundamental problems recurring in low temperature physics is the accurate determination of the temperature of the sample. Fortunately the nuclear orientation (NO) technique offers us one of the most direct and accurate means of measuring temperatures below 1 K. In the NO method, which is essentially a measurement of the degree of ordering of a nuclear spin system in thermodynamic equilibrium, the temperature is derived from the Boltzmann factor and, in principle, is absolute (thermodynamic). Thus a measurement of the anisotropic emission of y-rays or B- or a-particles from an assembly of oriented radioactive nuclei has the same potential of yielding the absolute temperature as does the scattering or absorption of photons or particles by an oriented nuclear target. An obvious restriction is, of course, that the nuclei must have a nonvanishing spin. In addition, it should be kept in mind that the temperature obtained with a NO thermometer is that of the nuclear spin system and not that of the lattice. If the lattice temperature is to be measured, then the spin-lattice relaxation time must be relatively short.

About the Author: H. Marshak is with NBS' Center for Absolute Physical Quantities.

Most of the work in NO thermometry has been done using y-ray anisotropy (y-RA) thermometers. In addition to being primary thermometers, these types of NO thermometers have many advantages over conventional thermometers: they are usually physically small and metallic, and thus can be easily attached (soldered) to the experimental package with good thermal contact; for some the self-heating due to the radioactivity is quite small (~0.02 erg/min); no wires are attached to it; the readout is digital, and the (counting) equipment needed can be relatively inexpensive. Moreover, some can operate in zero magnetic field as well as in a magnetic field. Although y-RA thermometers have been used in many different types of low temperature experiments, e.g., studies of nuclear properties (spins, moments, multipolarities, etc.), calibration of secondary thermometers (paramagnetic salts, resistance thermometers, etc.) and development of a low temperature scale, and, have been the subject of several reviews [1-3]', their use is still not as extensive as it could be. This perhaps stems from the nonuser's having to learn a new technique foreign to his own field of research. One of the objectives here is to provide sufficient information to

Figures in brackets indicate literature references at the end of this paper.

those unfamiliar with this technique so that they too can use it. Thus, details are given which would normally not be given.

A general discussion of NO thermometry is given in the next section, thereafter we will deal almost entirely with y-RA thermometry. In section 2 the theory of y-RA thermometry is given along with some general remarks. This is followed by a discussion of practical y-RA thermometry which is divided into two sections, section 3 dealing with the general problem, and section 4, 4, with specific y-RA thermometers. Section 5 contains recent experimental results. The conclusions are included in the last section.

2. Theory of Nuclear Orientation Thermometry

2.1 Basic Concepts

The degree of orientation of an ensemble of nuclei of spin I can be specified in various ways. The most general description is given in terms of a spin-density matrix Ρ with (21+1)2 matrix elements. For an ensemble of nuclear spins with cylindrical symmetry the description is considerably simplified. In this case the symmetry axis is the axis of quantization of the nuclear spin system and p is a diagonal matrix with (21+1) matrix elements. These diagonal matrix elements, Pmm are just the relative populations, am of the nuclear spin substates m (m=I, I−1, ---, -I). Since it is convenient to normalize the populations am such that Σa=1, we are left with only 27 independent values of am

m

In the theoretical interpretation of the nuclear process studied, it is usually more convenient to work with the [(27+1)2-1] independent quantities, B(I), called statistical tensors, which are defined in terms of the density matrix. For a spin ensemble with cylindrical symmetry the number of statistical tensors, or orientation parameters, is reduced to 21. The explicit expression for these orientation parameters is given by

B ̧(D)=(21+1)12pò(D= έ (−1)1"[(2λ+1) (27+1)]12

m=-1

where Em are the energies of the nuclear m-states, k the Boltzmann constant and T the absolute temperature. At temperatures T>Em/k the populations are essentially equal and are given by (27+1)1 (only at T= ∞ are the populations exactly equal). For

temperatures TSE/k the populations are unequal, resulting in nuclear orientation. The lower the temperature, the greater is the degree of nuclear orientation. In figure 1 we show some population distributions for a nuclear spin system with I=3 and Em/k=(0.013 K)m, which are the values for the 54MnNi y-RA thermometer. The first three, (a), (b), and (c) are Boltzmann distributions at temperatures of 1.0, 0.1, and 0.02 K respectively. As one can see, there are hardly any differences between the populations at 1.0 K, the ratio of the populations for the lowest and highest m-states being 1.07. At 0.1 and 0.02 K the distributions are skewed to favor the lower states with the ratios now being approximately 2 and 50 respectively. In the distribution shown in (d), the Boltzmann distribution (c) was perturbed so that the lowest two m-states were equally populated, thus yielding a non-Boltzmann distribution. For all four distributions, values of B(I) can be calculated; however, only the first three can be associated with meaningful values of the absolute temperature. We nuclear will designate these, i.e., orientation parameters for a system of nuclear spins in (b) (c)

m= 3

2

(a)

(d)

thermodynamic equilibrium by B(I,T) rather than by BA(I). If the perturbation is turned off in (d) and the spins can relax (by interacting among themselves or with their environment, e.g., the lattice) so that thermal equilibrium is achieved, then the populations will again by governed by a Boltzmann distribution reflecting this new equilibrium temperature.

m

Thus, we see that the entire theoretical basis for NO thermometry is contained in eqs (1) and (2): one determines the value of any nonzero B(I,T) for a nuclear spin system in thermodynamic equilibrium where Em is known, and from this a unique value of the absolute temperature is obtained. How well one can determine the value of the temperature depends upon the accuracy with which one knows E, and how accurately B (I,T) can be measured. The latter will contain all the statistical and almost all the systematic errors of the measurement. As we shall see later on, a complete understanding of all the systematic errors associated with deducing a temperature from an ensemble of oriented nuclear spins is nontrivial and will cause us the greatest concern about how well one can measure the absolute temperature.

In figure 2 we show the six B (I,T) plotted as a function of temperature for the nuclear spin system that was used in calculating the populations for (a), (b), and (c) of figure 1, namely, I=3 and Em/k= (0.013 K)m. As one can see each B1(I,T) is a singlevalued function of the temperature.

It is often the case that the measurement made on an ensemble of nuclear spins will depend upon more than one B (I,T) value; however, as we mentioned previously, once Em is known, any one of the nonzero B(I,T) will yield the temperature. Once the temperature is known, all the other B(I,T) are

m

uniquely determined. This follows directly from eq (2), i.e., once E, and T are known, all of the a, can be calculated and hence each B(I,T). Thus, we see that the condition that the populations be governed by the Boltzmann distribution is very restrictive. Whereas in the general case of an ensemble of oriented nuclear spins with cylindrical symmetry the am were independent (except for normalization), now they are dependent such that if any one a, is known along with Em then the remaining a can be obtained. For the general case above where we do not have a Boltzmann distribution, each of the B(I) would have to be measured in order to obtain all the am

m

It is important to realize that having the populations follow a Boltzmann distribution does not necessarily imply that the spin system is in thermal equilibrium. A spin system can be prepared with the a, being the same as those given by a Boltzmann distribution; however, if there are no spin-spin or spin-lattice interactions (the latter being important for very dilute systems, e.g., y-RA thermometers) to achieve thermal equilibrium a spin temperature cannot be defined. The most obvious example of this is a nuclear spin system with I=1/2. In this case every distribution (of the two states) corresponds to a Boltzmann distribution; however, the spin system need not be in thermal equilibrium.

Although the concept of spin temperature is fundamental to NO thermometry, and leads to some interesting properties (e.g., negative temperatures) not found in other thermodynamic systems, we will not be concerned with those here since many excellent articles have been written on this subject [4–6].

B1(I,T)