ON A METHOD FOR DETERMINING THE THERMOELECTROMOTIVE FORCE OF BULK SEMICONDUCTORS WITH AN UNKNOWN TEMPERATURE DIFFERENCE

L. G. Olert

Determining the thermoelectric e.m.f. of bulk specimens presents a difficulty because it is impossible to maintain a definite, precisely known temperature difference of the junctions. The shorter the specimen and the greater its thermal conductivity, the more difficult it is to satisfy this condition. On the other hand, not every material can be made into a wire or a strip. It is therefore desirable to have a method for determining the thermoelectric e.m.f. that makes it possible to test a specimen with an unknown temperature difference of the junctions, but that gives results for a specified temperature difference, for example, at \(t_2 - t_1 = 100^\circ\text{C}\).

Such a method exists for massive pieces of metal. Its inconvenience consists in the fact that for this it is necessary to have a set of standards, i.e., wires occupying definite positions in the thermoelectric series. For metals this is permissible, since the interval of thermoelectric e.m.f. from bismuth to antimony is not so very large and, moreover, a large part of the standards can be made of metallic wire. But if the object of study is minerals, serious difficulties arise, since it is impossible to select thermoelectric-e.m.f. standards for them.

The way out of the impasse should be sought in a differential circuit of a thermoelement with a conductor connected in parallel (Fig. 1). The thermoelectric e.m.f. of such a compound thermoelement is

\[ E=\frac{E_{nm}R_q+E_{nq}R_m}{R_m+R_q}, \tag{1} \]

where \(E_{nm}\) is the thermoelectric e.m.f. inherent in the pair \(n-m\), \(E_{nq}\) is the same for the pair \(n-q\), \(R_q\) is the resistance of branch \(q\), and \(R_m\) is the resistance of branch \(m\) [1]. Here a definite, fixed temperature difference is meant. But if the resistance of one of the branches, for example \(m\), is made variable, then two readings of the e.m.f. at different resistances \(R_m\) will quite unambiguously indicate \(E_{nq}\) for a specified temperature difference, although during the experiment it may be unknown (it is important only that in the first and second readings the temperatures be the same). If by \(E_1\) we denote the e.m.f. at resistance \(R_m\), by \(E_2\) the e.m.f. at resistance \(R_{m1}\), and by \(X\) the ratio

Fig. 1

actual temperature difference to the specified one, and by \(a=\dfrac{R_{m1}}{R_m}\), then the formula takes the following form:

\[ E_{nq}= \frac{E_1(R_m+R_q)E_{nm}R_q(a-1)} {R_m\left[aE_1(R_m+R_q)-E_2(aR_m+R_q)\right]} -\frac{E_{nm}R_q}{R_m} \tag{2} \]

We see that \(X\) does not enter the formula, that is, we do not need to know the actual temperature difference; it may be arbitrary. But if required, it can also be found:

\[ X=\frac{aE_1(R_m+R_q)-E_2(aR_m+R_q)} {E_{nm}R_q(a-1)} . \tag{3} \]

Formula (2) loses definiteness at \(a=1\), since the first term is \(0/0\), and also at \(a=\infty\). Meanwhile, the latter case is of interest, since as \(a\) increases the difference between the emf values in two readings increases, that is, the determination is refined. In practice, \(a=\infty\) can be obtained if, during the second reading, branch \(m\) is opened.

From formula (1) one can derive the following relation:

\[ E_{nq}=\frac{E_2E_{nm}R_q} {E_1(R_m+R_q)-E_2R_m}, \tag{4} \]

where \(E_1\) is the emf at resistance \(R_m\), and \(E_2\) is the emf with branch \(m\) open. The ratio of the actual temperature difference to the specified one is

\[ X_1=\frac{E_1(R_m+R_q)-E_2R_m} {E_{nm}R_q}. \tag{5} \]

Formula (4) is valid for any values of \(R_m\) and \(R_q\), but the accuracy of the determination will not be the same. The difference between the readings will be the greater, the larger the ratio \(\dfrac{R_q}{R_m}\). Therefore this method cannot be applied to good conductors, since in that case \(\dfrac{R_q}{R_m}\) will be very small; that is, the determination will be very crude, and may even fail altogether if the difference between the readings is less than the sensitivity of the galvanometer (that is, both readings give the same result). The method is intended for minerals having a high specific resistance. In this case the following construction may be recommended (Fig. 2).

The specimen under investigation \(q\) is placed between heater \(a\) and cooler \(b\), with mica spacers \(c\) and metal plates \(d\), consisting of metals \(n\) and \(m\); the latter are directly adjacent to the specimen. The halves of the plates consisting of metal \(n\) are connected by a thin wire of the same metal to the galvanometer, while the other halves are closed by wire \(m\) through switch \(k\). To achieve better contact, the system is compressed by a screw. The resistance \(R_m\) is constant for the given apparatus, while the resistance of the specimen \(R_q\) is measured in each individual case; for this purpose, with switch \(k\) open, a voltage source \(E\) is inserted into the circuit and the galvanometer reading is noted. If the specimen has a large temperature coefficient of resistance, it is necessary to determine its resistance at different temperatures; for this purpose it is placed, together with the press, in a thermostat. To determine the thermoelectromotive force, the battery \(E\) is removed, heater \(a\) is switched on, and water is let into cooler \(b\). When a con-

the constant temperature difference (which can be judged from the unchanged reading of the galvanometer), a reading is taken with the key \(k\) closed, and immediately afterward with it open. The desired thermoelectromotive force of the specimen at a temperature of \(0\)--\(100^\circ\mathrm{C}\) is calculated from formula (4), taking into account the resistance of the galvanometer and, if necessary, introducing a correction for the temperature coefficient, assuming the temperature drop from the hot end to the cold one to be uniform.

The present communication is based on an analysis of formula (1) and is preliminary. The proposed method is subject to experimental development

Fig. 2

in order to refine the details of the design and to determine the range of specific resistances for which it is applicable.

LITERATURE

1. L. G. Olert. — The thermoelement of Academician Didebulidze as a battery of thermoelements connected in series. (To be printed in the present volume of the Proceedings of TPI).