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DIAMAGNETISM OF GASES AND VAPORS
Ya. Shur, Sverdlovsk
I. Introduction
Studies of the diamagnetic susceptibility of gases and vapors pursue the following two main aims.
a) As is known from the theory of magnetism (more on this in Chap. II), diamagnetism is inherent in the atoms of all substances and is proportional to the square of the effective radius of the electron shell of the atom. Owing to this latter circumstance, the measurement of the diamagnetic susceptibility of atoms and ions is one of the ways of determining their dimensions. At present the theory is at such a stage of its development that only the susceptibility of individual atoms and ions, and also of the simplest molecule—H₂—can be calculated exactly. This is because theoretical calculations of diamagnetism are possible only with the accuracy with which the distribution of charge density in the electron shell of the atom can be calculated. There exists a whole series of approximate methods that make it possible, with one or another degree of accuracy, to determine the distribution of the charge density of the atom’s electron cloud and thereby also the atomic diamagnetic susceptibility.
Comparison of the susceptibility calculated theoretically by various methods with exact experimental data can indicate how flawless a given method of theoretical calculation is. Unfortunately, up to the present time we have had at our disposal only a very small number of reliable experimental data on the diamagnetic susceptibility of atoms (Table 5), and only for those which are in the gaseous state at room temperature, i.e., the inert gases; while, for example, nothing is yet known about the diamagnetic vapors of metals.
In general, it is possible to observe the diamagnetism of a metallic atom in pure form only in the vapors of diamagnetic metals, since in the solid and liquid states of a metal its susceptibility is always further superposed with paramagnetism caused by conduction electrons. This so-called Pauli paramagnetism¹, independent of temperature just like diamagnetism, often masks the diamagnetism of the metal ions, and therefore any possibility is lost of experimentally determining, in the solid and liquid states, the magnitude of the diamagnetic susc—
susceptibility of the metal. Therefore, an accurate experimental determination of the susceptibility of atoms in the vapor state is a necessary prerequisite for the further development of the theory of magnetism.
b) Since diamagnetism depends on the geometrical dimensions of the atom, all kinds of changes undergone by the electronic shell of an atom are immediately reflected in the magnitude of diamagnetism. For example, in the case when, upon the formation of a molecule, chemical bonds somewhat distort the dimensions and shape of the electronic shells of the atoms constituting the molecule, a change in the diamagnetic susceptibility also occurs. Although we are not yet able to calculate the magnitude of this change in susceptibility, nevertheless, owing to the existence of the above-mentioned dependence, measurements of diamagnetic susceptibility (along with optical and X-ray methods) can contribute much to knowledge both of the structure of atoms and molecules and of the character of those bonds that exist between the atoms forming the molecule.
Because the density of matter in vapors is small, the forces that act on vaporous matter in a magnetic field are likewise negligible, and therefore their measurement presents great experimental difficulties. For this reason, the experimental material on the diamagnetic susceptibility of gases and especially of vapors is very limited, and even those experimental data that have already been obtained vary greatly for the same substances among different authors (Table 5).
II. Theory of Diamagnetism
1. Classical Theory of Diamagnetism
On the basis of the classical electron theory, Langevin² gave the first quantitative theory of diamagnetism. This theory was subsequently developed and refined by Pauli³ and therefore bears the name of the Langevin–Pauli theory. According to this theory, the cause of diamagnetism is the precession of the orbit around the direction of the external magnetic field.
As Larmor⁴ showed, an electron moving along a closed curve is mechanically analogous to a gyroscope. And just as a gyroscope, when rotating, precesses about the direction of the field of terrestrial gravity, so too the electron orbit, when excited by an external magnetic field, must begin to precess about it. This precession of the electron orbit, known as “Larmor precession,” will, according to Larmor’s calculations, cause a change in the angular velocity of the electron by the amount
\[ O=-\frac{eH}{2m}, \tag{1} \]
where \(e\) and \(m\) denote the absolute value of the charge and the mass of the electron, \(H\) is the intensity of the external magnetic field. But, on the other hand, an electron moving in an orbit may be regarded as an electric current flowing in a closed circuit. The magnetic moment of such a current will be equal to
\[ M=iS, \tag{2} \]
where \(i\) is the current strength, \(S\) is the area encircled by the current.
If \(\tau\) is the period of revolution of the electron, then in our case
\[ M=\frac{e}{\tau}S. \tag{3} \]
Denoting the angular velocity of the electron by \(\omega\), we obtain
\[ M=\frac{eS}{2\pi}\omega . \tag{4} \]
We are interested in the additional magnetic moment \(\Delta M\), due to the precession of the orbit, which according to equation (1) will take the following form:
\[ \Delta M=\frac{eS}{2\pi}\,O=-\frac{e^{2}SH}{4\pi m}. \tag{5} \]
Denoting by \(\overline{r_1^2}\) the time-average value of the square of the projection of the radius of the electronic orbit onto the plane perpendicular to the direction of the external magnetic field, we can write
\[ \Delta M=-\frac{e^{2}H}{4\pi m}\overline{r_1^2}. \tag{6} \]
If by \(\overline{r^2}\) we denote the square of the mean radius of the orbit, then for a spherically symmetric atom one may take
\[ \overline{r_1^2}=\frac{2}{3}\,\overline{r^2}. \]
For an atom having \(n\) electrons,
\[ \Delta M_{aт}=-\frac{e^{2}H}{6m}\sum_{n}\overline{r^2}. \tag{7} \]
And finally, for the atomic diamagnetic susceptibility we obtain the Langevin–Pauli formula
\[ \chi_A=\frac{\Delta M_A}{H}=-\frac{Ne^{2}}{6m}\sum \overline{r^2}, \tag{8} \]
where \(N\) is Avogadro’s number.
2. A Strict Classical Theory Does Not Explain the Phenomenon of Diamagnetism
If one strictly follows classical statistics, it follows that the resultant magnetic moment must be equal to zero. This proposition is proved in the following way \(^{6}\).
For a system consisting of \(n\) electrons, in the absence of an external magnetic field, the Hamiltonian function has the form
\[ E=\sum_{i=1}^{n}\frac{1}{2m}\left(p_{x_i}^{2}+p_{y_i}^{2}+p_{z_i}^{2}\right)+V(x_i,y_i,z_i), \tag{9} \]
where \(x_i,y_i,z_i\) and \(p_{x_i},p_{y_i},p_{z_i}\) are the coordinates of the electrons of our system and the corresponding momenta.
When an external magnetic field is excited along the \(z\)-axis, the Hamiltonian function, according to the laws of electrodynamics (the vector \(p_i\) in the field is replaced by the vector \(p_i-\dfrac{e}{c}A\), where \(A_x=-\dfrac{H}{2}y_i,\ A_y=\dfrac{H}{2}x_i,\ A_z=0\)), will take the form
\[ E=\sum_{i=1}^{n}\frac{1}{2m}\left[\left(p_{x_i}-\frac{He}{2c}y_i\right)^2+ \left(p_{y_i}+\frac{He}{2c}x_i\right)^2+p_{z_i}^{2}\right]+ \]
\[ +V(x_i,y_i,z_i). \tag{10} \]
From thermodynamic considerations one obtains the relation between the magnetic moment and the free energy \(F\) of the system in the form
\[ M=-\frac{\partial F}{\partial H}. \tag{11} \]
But since
\[ F=-kT\lg Z, \tag{12} \]
where \(Z\) is the statistical sum, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature, finally
\[ M=-kT\frac{\partial \lg Z}{\partial H}. \tag{13} \]
In classical statistics \(Z\) can be expressed in the following way:
\[ Z=\frac{1}{h^{3}}\int e^{-\frac{E(q,p)}{kT}}\,dq\,dp, \tag{14} \]
where \(h^{3}\) is the volume of the elementary phase cell.
In the case considered by us we have \(n\) independent atoms, and therefore
\[ Z=\left[\frac{1}{h^{3}}\int e^{-\frac{E(x_i,y_i,z_i,p_{x_i},p_{y_i},p_{z_i})}{kT}}\,dx_i\,dy_i\,dz_i\,dp_{x_i}\,dp_{y_i}\,dp_{z_i}\right]^n \tag{15} \]
It can be shown that after integration over \(p_i\), expression (15) becomes independent of \(H\), and, consequently, according to equation (13), the resulting magnetic moment of the system is equal to zero.
Thus we see that a consistent classical theory in general cannot explain the phenomenon of diamagnetism.
3. Quantum Theory of Diamagnetism\(^{5}\)
The calculation of the diamagnetic susceptibility on the basis of wave mechanics was carried out by Van Vleck.
In wave mechanics the energy operator for a system consisting of \(n\) electrons, when an external magnetic field is excited along the \(z\)-axis, may be written in the following form:
\[ E(q,p)=\frac{1}{2m}\sum_{i=1}^{n}\left(\frac{\hbar}{i}\frac{\partial}{\partial x_i}-\frac{eH}{2c}y_i\right)^2 +\left(\frac{\hbar}{i}\frac{\partial}{\partial y_i}+\frac{eH}{2c}x_i\right)^2+ \]
\[ +\left(\frac{\hbar}{i}\frac{\partial}{\partial z_i}\right)^2+V(x_i,y_i,z_i). \tag{16} \]
Let us single out from equation (16) that perturbation energy which is caused by the external magnetic field,
\[ \varepsilon(q,p)=-H\,\frac{e}{2mc}\frac{\hbar}{i} \left(x_i\frac{\partial}{\partial y_i}-y_i\frac{\partial}{\partial x_i}\right) +\frac{e^2H^2}{8mc^2}\sum_i (x_i^2+y_i^2). \tag{17} \]
Taking into account the relation between the mechanical moment \(K\) and the magnetic moment \(M\),
\[ M=-\frac{e}{2mc}K, \tag{18} \]
we may write
\[ \varepsilon(q,p)=HM+H^2\frac{e^2}{8mc^2}\sum_i(x_i^2+y_i^2); \tag{19} \]
here \(M\) is the operator of the magnetic moment of our system.
Our task consists in determining the magnetic moment due to the perturbation energy \(\varepsilon(pq)\). Assuming that the perturbation caused by the external field is comparatively weak, the energy of our system in the second approximation may be obtained by means of the method of perturbation theory in the following form:
\[ E=E_0+\bar{\varepsilon}_0+\sum_{s\ne 0}^{n}\frac{|\varepsilon_{s0}|^2}{E_s-E_0}, \tag{20} \]
where \(s\) denotes the index of the stationary state, \(E_0\) is the energy of the unperturbed system, \(\bar{\varepsilon}_0=\int \psi_0^* \varepsilon \psi_0\,dq\), i.e. the mean value of the perturbation energy when it is considered in the stationary state \(\psi_0\), and \(\varepsilon_{s0}\) is the matrix element of the perturbation energy.
Substituting in equation (20) the values for \(\varepsilon\) from (19),
\[ E=E_0+H\bar{M}_0+H^2\frac{e^2}{8mc^2}\sum_i(\bar{x}_i^{\,2}+\bar{y}_i^{\,2}) +H^2\sum_{s\ne 0}\frac{|M_{s0}|^2}{E_s-E_0}; \tag{21} \]
since
\[ \overline{M}_s = -\frac{\partial E_s}{\partial H}, \tag{22} \]
we finally obtain for the mean magnetic moment of the system
\[ \overline{M}=\overline{M}_0+\frac{He^2}{4mc^2}\sum_i(\overline{x_i^2}+\overline{y_i^2})+2H\sum_{s\ne0}\frac{|M_{s0}|^2}{E_s-E_0}. \tag{23} \]
Here \(\overline{M}_0\) is the mean magnetic moment in the absence of an external field. If we assume that the normal state of our system is nondegenerate, then
\[ \overline{M}_0=0. \tag{24} \]
It can be shown that, in the case of spherical symmetry of the electric field of the nucleus, the last term of equation (23) will also be equal to zero.
Considering the spherically symmetric case, we may put
\[ \sum_i(\overline{x_i^2}+\overline{y_i^2})=\frac{2}{3}\sum_i\overline{r_i^2}, \tag{25} \]
since
\[ \overline{x_i^2}=\overline{y_i^2}=\overline{z_i^2}=\frac{\overline{r_i^2}}{3}. \tag{26} \]
And finally, for the atomic diamagnetic susceptibility we obtain
\[ \chi_A=N\frac{\partial\overline{M}}{\partial H}=-\frac{Ne^2}{6mc^2}\sum_i\overline{r_i^2}. \tag{27} \]
Thus we see that formula (27) for the diamagnetic susceptibility has turned out to be of the same form as in the Langevin–Pauli theory.
4. Methods for determining \(\sum_i \overline{r_i^2}\).
As follows from formula (27), for the theoretical determination of the diamagnetic susceptibility of any atom or ion spherically symmetric with respect to the electric field of the nucleus, it is necessary to know the value of its \(\sum_i \overline{r_i^2}\). We shall now indicate a number of approximate methods by means of which the value of \(\sum_i \overline{r_i^2}\) can be calculated.
Wave-mechanical methods
a) Pauling’s method.^6 Pauling was the first, on the basis of wave mechanics, to give a method for calculating the diamagnetic susceptibility of many-electron systems. The solution of the Schrödinger equation in the nonrelativistic theory for the hydrogen atom leads to the following expression for \(\overline{r^2}\):
\[ \overline{r^2}=\frac{n^2}{Z^2}\left\{\frac{5}{2}n^2-\frac{3l(l+1)-1}{2}\right\}, \tag{28} \]
where \(n\) and \(l\) are the principal and orbital quantum numbers, and \(Z\) is the nuclear charge. For solving the many-electron problem Pauling adopted a charge distribution, similar to that of hydrogen, and obtained an expression analogous to (28),
\[ \overline{r^2}=\frac{n^2}{(Z-s)^2}\left\{\frac{5}{2}n^2-\frac{3l(l+1)-1}{2}\right\}, \tag{29} \]
where \(s\) is the screening constant.
Pauling calculated the correction \(s\) separately for various electron groups under the assumption that the influence of different subgroups is like that of charges distributed over a spherical surface. In this way Pauling was able to calculate the diamagnetic susceptibility for a large number of substances. It should be noted that in this approximate calculation method the quantity \(s\) is chosen semiempirically. Pauling’s calculation results for a number of atoms and ions are given in Table 1.
b) Stoner’s method.^7 Hartree^8 developed the most accurate (apart from Fock’s generalized method) wave-mechanical method for calculating the charge-density distribution in an atom or ion, based on ideas about the wave functions of individual electrons. For a whole series of substances Hartree calculated the charge-density distribution as a function of the distance from the nucleus.
If the radial charge density is denoted by \(\dfrac{dN}{dr}\) (charge per unit radial distance), then in this case
\[ \overline{r^2}= \frac{\displaystyle\int_{0}^{\infty} r^2\,\frac{dN}{dr}\,dr} {\displaystyle\int_{0}^{\infty}\frac{dN}{dr}\,dr}, \tag{30} \]
where
\[ \int_{0}^{\infty}\frac{dN}{dr}\,dr \]
is the number of electrons in the atom.
In Hartree’s graphs the dependence of \(\dfrac{dN}{dr}\) on \(r\) is given. If from the gra-
… \(dN/dr=f(r)\), plot the curve of the dependence of \(r^2\, dN/dr\) on \(r\), then the area of the latter gives the value of the upper integral in expression (30). For the atomic diamagnetic susceptibility we obtain
\[ \chi_A=-0.807\cdot 10^{-6}\int_0^\infty r^2\frac{dN}{dr}\,dr . \tag{31} \]
Fig. 1 shows how, from Hartree’s graphs \((1)\), giving the dependence of \(dN/dr\) on \(r\), Stoner constructed the graphs \((2)\)—the dependence of \(r^2 dN/dr\) on \(r\).
The results of the susceptibility calculations carried out by Stoner with the aid of the method indicated above are given in Table 3. As can be seen from Table 3, in this case the agreement with the experimental data is not bad. But despite the great rigor of the calculation of the charge-density distribution by Hartree’s method, its application is severely limited by the fact that obtaining the graph
\[ \frac{dN}{dr}=f(r) \]
involves an enormous amount of computational work.
Fig. 1.
c) Slater’s method⁹. The fastest wave-mechanical method of approximate calculation of \(\overline{r^2}\) was given by Slater. For the radial part of the electron wave function in a symmetric atom Slater chose an expression of the following form:
\[ \psi=r^{(n^*-1)} e^{-\frac{Z-s}{n^*}\,r}, \tag{32} \]
TABLE 1
| Electron | \(Z-s\) | |
|---|---|---|
| \(1s\) | \(17-(1\cdot 0.30)\) | 16.7 |
| \(2s,p\) | \(17-(7\cdot 0.35)-(2\cdot 0.85)\) | 12.85 |
| \(3s,p\) | \(17-(7\cdot 0.35)-(8\cdot 0.85)-(2\cdot 1.00)\) | 5.75 |
where \(n^*\) is the effective quantum number and \((Z-s)\) is the effective nuclear charge. On the basis of empirical considerations, Slater gave rules by means of which \(n^*\) and \(s\) are determined for various electron groups.
The electron density per unit radial thickness will be
\[ 4\pi r^2|\psi|^2. \]
For an electron having definite \(n^*\) and \(s\),
\[ \overline{r^2} = \frac{\displaystyle\int_0^\infty r^2(4\pi r^2)|\psi|^2\,dr} {\displaystyle\int_0^\infty 4\pi r^2|\psi|^2\,dr}. \tag{33} \]
Substituting the values of \(\psi\) from equation (32) and then integrating the resulting expression, we have
\[ \overline{r^2} = \frac{n^{*2}\left(n^*+\dfrac{1}{2}\right)(n^*+1)} {(Z-s)^2}. \tag{34} \]
From equation (34) we obtain, for the diamagnetic susceptibility of each individual electron in the atom,
\[ \chi = -0.807\, \frac{n^{*2}\left(n^*+\dfrac{1}{2}\right)(n^*+1)} {(Z-s)^2}\,10^{-6}. \tag{35} \]
Slater’s rules for finding \(n^*\) and \(s\) for various electron groups are as follows:
- For the given principal quantum numbers \(n\), the effective quantum number is determined from the following table:
| \(n=\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(n^*=\) | 1 | 2 | 3 | 3.7 | 4 | 4.2 |
- All electrons in the atom are divided into groups in the following order:
\[ 1s;\ 2s,p;\ 3s,p;\ 3d;\ 4s,p;\ 4d\ \text{and so on}. \]
The electrons in each of the indicated groups have one and the same shielding constant \(s\). The shielding constant \(s\) is calculated according to the following scheme: for \(n>n_0\), \(s=0\).
For electrons of the same group, \(s=0.35\) (except for the \(1s\) electron, for which \(s=0.30\)).
For \(s\)- and \(p\)-electrons, when \(n=n_0-1\), \(s=0.85\); when \(n<n_0-1\), \(s=1.00\).
For \(d\)- and \(f\)-electrons, \(s=1.0\) with respect to all electrons of lower groups:
Applying Slater’s second rule, we obtain \(Z-s\) for the ion \(\mathrm{Cl}^{-}\). Recall that the electronic structure of \(\mathrm{Cl}^{-}\) is as follows:
\[ 1s^{2}\,2s^{2}p^{6}\,3s^{2}p^{6}. \]
The nuclear charge of Cl is 17.
By the method thus determined (Table 1), \(n^{*}\) and \(Z-s\) are calculated, and from formula (35) the susceptibility fraction of each individual electron is obtained. The diamagnetic susceptibility of an atom or ion is obtained by summing the susceptibilities of the individual electrons.
Using this method, Brindley\(^{10}\) carried out a calculation of the susceptibilities for a whole series of atoms and ions. The results of calculating susceptibilities by Slater’s method are given in Table 3.
d) Angus’s method\(^{11}\). Angus, noting that the results of theoretical calculations of susceptibility by Slater’s method are somewhat larger than the experimental values, proposed a certain modification of the calculation of \((Z-s)\). According to Angus, \(s\)- and \(p\)-electrons having the same principal quantum number \(n\) are not equivalent in determining \((Z-s)\), as had been assumed by Slater. The distinction that Angus introduces for \(s\)- and \(p\)-electrons of the same \(n\) is best illustrated by Table 2 for the ion \(\mathrm{Cl}^{-}\), analogous to Table 1 for the same ion in the preceding Slater method.
With the aid of this method of calculation, Angus computed the diamagnetic susceptibility for a large number of atoms and ions.
e) Platt’s method\(^{12}\). In Platt’s wave-mechanical method, wave functions of the following form, similar in shape to the wave functions of the hydrogen atom, are chosen for individual electrons:
\[ \left. \begin{aligned} \text{for the }K\text{-shell}\quad \psi_{1s} &= \sqrt{\frac{\alpha^{3}}{\pi}}\,e^{-\alpha r},\\[6pt] \text{for the }L\text{-shell}\quad \psi_{2s} &= \sqrt{\frac{\beta^{2}}{\pi}}\,(1+\beta r)e^{-\beta r},\\[6pt] \psi_{2p_{1}} &= \sqrt{\frac{\beta^{2}}{\pi}}\,\beta r e^{-\beta r}\cos\theta,\\[6pt] \psi_{2p_{2}} &= \sqrt{\frac{\beta^{2}}{\pi}}\,\beta r e^{-\beta r}\sin\theta\,e^{+i\varphi} \end{aligned} \right\} \tag{36} \]
and so on; the numerical values of the coefficients \(\alpha,\ \beta,\ \gamma,\ldots\) are determined by the variational method.
\(\overline{r^{2}}\) is determined in the same way as in equation (33). The results of Platt’s calculations are in good agreement with experimental data (Table 3).
TABLE 2
| Electron | \(Z - s\) | |
|---|---|---|
| \(1s\) | \(17 - (1 \cdot 0.30)\) | 16.7 |
| \(2s\) | \(17 - (1 \cdot 0.35) - (2 \cdot 0.85)\) | 14.95 |
| \(2p\) | \(17 - (7 \cdot 0.35) - (2 \cdot 0.85)\) | 12.85 |
| \(3s\) | \(17 - (1 \cdot 0.35) - (8 \cdot 0.85) - (2 \cdot 1.00)\) | 7.85 |
| \(3p\) | \(17 - (7 \cdot 0.35) - (8 \cdot 0.85) - (2 \cdot 1.00)\) | 5.75 |
Statistical methods
a) Sommerfeld’s method\(^ {13}\). Sommerfeld determined \(\sum \overline{r^{2}}\) by means of an approximate solution of the Thomas–Fermi differential equation, which gives the distribution of charge density around the nucleus, possessing spherical symmetry.
As is known, the Thomas–Fermi method is based on quantum statistics and is suitable for many-electron systems. Sommerfeld found the following formulas for the diamagnetic susceptibility:
for a neutral atom
\[ \chi_A = -3.1 Z^{\frac{1}{3}} \cdot 10^{-5}, \tag{37} \]
for a positive ion
\[ \chi_A = -3.1\left(Z^{\frac{1}{3}} - 1.846\sigma^{\frac{1}{3}}\right)\cdot 10^{-5}, \tag{38} \]
where \(Z\) is the nuclear charge, \(\sigma\) is the number of ionized electrons. The susceptibility values calculated from these formulas are excessively large in comparison with the experimental values. The reason for this lies in the fact that, in the method under consideration, the charge density, on passing to infinity, vanishes insufficiently rapidly, and therefore, in obtaining \(\sum \overline{r^{2}}\), regions far removed from the nucleus have a strong influence on the overall result. The data of Sommerfeld’s calculation are given in Table 3.
b) Gombas’s method\(^ {14}\). The advantage of the Gombas method consists in the fact that, in the statistical method of Lenz and Jensen used by him, a differential equation similar to the Thomas–Fermi equation gives a considerably more rapid decrease of the charge density in the outer regions of the atom. The statistical method of Lenz and Jensen consists in the fact that the distribution of charge density is represented by a function containing coefficients determined by the Ritz variational method. The expression for the energy is composed in accordance with Fermi statistics. For the susceptibility Gombas finds the following formulas:
for a neutral atom
\[ \chi_A=-9.41 Z^{\frac{1}{3}}\cdot 10^{-6}, \tag{39} \]
for an ion
\[ \chi_A=-2.84D(\mu,c)\frac{N}{Z^{\frac{2}{3}}}\cdot 10^{-6}. \tag{40} \]
\(D(\mu,c)\) is a function depending only on the coefficients contained in the expression for the charge density, determined by the variational method. As can be seen from Table 3, this method of calculation, like Sommerfeld’s preceding method, gives very approximate results.
TABLE 3
| Substance | Experiment | Pauling | Stoner | Slater | Angus | Platt | Sommerfeld | Tombas |
|---|---|---|---|---|---|---|---|---|
| He | \(1.9^{15}\) | 1.54 | 1.87 | 1.68 | 1.68 | 1.84 | — | — |
| Ar | \(19.7^{15}\) | 21.5 | 25.3 | 18.87 | 16.95 | 19.75 | 81 | 25 |
| Ne | \(7.6^{15}\) | 5.7 | 8.81 | 5.7 | 5.07 | 6.42 | 67 | 20 |
| Kr | \(28^{16}\) | 42 | — | 31.7 | 29.3 | — | — | — |
| Xe | \(42^{16}\) | 66 | — | 48.0 | 44.78 | — | — | — |
| Li\(^+\) | \(1.6^{17}\) | 0.6 | 0.7 | 0.665 | 0.665 | 0.731 | — | — |
| K\(^+\) | \(13.6^{17}\) | 16.7 | 17.6 | 14.40 | 13.06 | 15.37 | 26 | 20 |
| Na\(^+\) | \(7.6^{17}\) | 4.2 | 5.74 | 4.2 | 3.74 | — | — | 15 |
| Rb\(^+\) | \(27.2^{17}\) | 35 | 30.1 | 25.8 | 24.05 | — | — | 28 |
| Cs\(^+\) | \(45.75^{17}\) | 55 | — | 39.5 | 37.2 | — | — | 33 |
| Cl\(^-\) | \(20.4^{17}\) | 29 | 40.4 | 25.79 | 22.86 | 25.93 | — | 31 |
Table 3 gives a summary of theoretical calculations of the susceptibility by various methods, and the values found are compared with the most convincing experimental data. Of all the methods listed, despite all its theoretical rigor, the method of calculating the susceptibility proposed by Slater proves to be the most accessible for computation and gives good agreement with experiment.
As for molecules, a rigorous quantum-mechanical calculation exists only for the hydrogen molecule, and in this case the calculation of \(\chi\) was carried out on the basis of formula (23), taking the third term into account. The agreement with the experimental value of the susceptibility is very good (the theoretical value is \(\chi_{H_2}=3.96\cdot 10^{-6}\), and the experimental value is \(4.00\cdot 10^{-6}\)).
For calculating the susceptibility of molecules, Angus1 used the following method. He calculated separately by formula (35) the susceptibility, for example, of \(J^{-}\) and \(J^{+}\), and assumed that the sum of the susceptibilities of \(J^{-}\) and \(J^{+}\) corresponds to the susceptibility of the \(J_{2}\) molecule. In this way Angus determined the susceptibilities not only of homo-, but also of heteropolar compounds. For a whole series of molecules he obtained rather good agreement with experimental data. Nevertheless, we still must not forget that not only does Angus’s method itself for calculating the susceptibilities of ions lack a rigorous theoretical justification, but also the assumption that the diamagnetic susceptibility of a molecule is additively composed of the susceptibilities of the ions constituting it, without introducing corrections for bonds, undoubtedly introduces an error. Moreover, in the case of molecules, when calculating the susceptibility according to formula (23), we are obliged to take into account also the third term, which is neglected in Angus’s calculations.
III. Review of Experimental Studies of the Diamagnetic Susceptibility of Gases and Vapors
For the first time, Faraday, in his classical investigations of the magnetic properties of various substances, came to the conclusion that, in general, there do not exist in nature bodies neutral with respect to a magnetic field, and that in a sufficiently strong magnetic field all bodies exhibit either para- or diamagnetic properties (1845). He also showed that gases are not an exception to this law. But at that time Faraday succeeded only in observing the purely qualitative character of the magnetization of gases.
Becquerel (1850) made the first attempt at a quantitative determination of the magnetizability of gases. He employed the method of the torsion balance, which in principle remained until recent years the principal method for determining susceptibility. This method consists in the following.
If a rod of a weakly magnetic material (for example, glass) is suspended in a vacuum on a thin thread and a magnetic field is applied, then under the action of the field the rod will turn through a certain angle. The rotation of the rod is observed, and it will be proportional to its susceptibility. Then the same observation is made when the rod is surrounded by the gas under investigation. The difference between these two observations gives a quantity proportional to the susceptibility of the gas. Carrying out a series of measurements by this method, Becquerel obtained a quantitative representation of the relative magnetizability of various gases.
In 1853 Faraday, creating a scale of all substances according to the degree of their magnetizability for equal volumes, measured by the torsion-balance method a large number of different gases.
At the same time Plücker also proposed another method for the quantitative determination of the magnetizability of gases. In the method
In Plücker’s method the force was measured with which a glass sphere filled with the gas under investigation was drawn into, or repelled from, a nonuniform magnetic field. Of course, the sensitivity of such a method is very low, and its applicability was limited only to paramagnetic gases. Becquerel, who devoted much labor to the study of the magnetic properties of gases, also used Plücker’s method. The third method by which Becquerel made his observations is the determination of the magnetization of gases dissolved in water. In this case the change introduced by the dissolved gas into the magnetization of water was observed.
All these three methods have in principle survived down to our day. The results of Faraday’s and Becquerel’s investigations, however, are now of no value, since for most of the gases they measured they did not even succeed in determining correctly the sign of the susceptibility; the main reason for this was apparently the weakness of the technique for purifying the gases under investigation.
Among later investigations, attention is deserved by the measurements of the susceptibility of gases carried out by Efimov (1888). Efimov^18 made his measurements by the torsion-balance method and, for the diamagnetic gases studied, obtained correct qualitative results.
All subsequent investigations of the susceptibility of diamagnetic gases up to the careful experiments of Soné (1920) gave completely contradictory data. The best illustration of what has been said may be provided by the physicochemical tables of Landolt—Börnstein, 1923 edition, where for such “classical” diamagnetic gases as H$_2$, N$_2$, and CO$_2$ data are given from certain investigators who ascribe to them paramagnetic susceptibility. Apparently, here too the chief cause of such gross experimental errors was the poor purification of the gas from paramagnetic impurities^1).
Soné^19 carried out his measurements of the susceptibility of gases by the Gouy method, the principle of which is as follows.
Fig. 2.
If a rod with volume susceptibility $\chi$ and cross-section $s$, suspended vertically so that its lower end is between the poles of an electromagnet in a uniform field $H$, while its upper end is in a place where the field is almost zero (Fig. 2), is surrounded by the gas under investigation with volume susceptibility $\chi_0$, then a ponderomotive force will act on the rod
\[ F=(\chi-\chi_0)\frac{sH^2}{2}. \tag{41} \]
^1) For example, an admixture of O$_2$ of 0.5% is sufficient to cover the diamagnetism of the remaining 99.5% of the gas under investigation.
In his measurements Soné modified Gouy’s method in the following way. Instead of a rod, a glass tube was taken, divided in half by a horizontal partition into two equal parts; the tube was placed, at the point where the partition was located, between the poles of an electromagnet in a uniform magnetic field (Fig. 3). One part of the glass tube was filled with air, the other with water, and the ratio of the susceptibilities of air and water was measured. In this measurement the specific susceptibility of water served as the standard and was taken equal to \(\chi_{\mathrm{H_2O}}=-0.72\cdot 10^{-6}\). Then one compartment of the tube was filled with the gas under investigation, while the second was filled with air or evacuated. The deflections of the balance were observed; to one of its arms the glass tube was attached by a thread. For small angles of deflection one may assume that
Fig. 3.
\[ F=c\alpha, \tag{42} \]
where \(\alpha\) is the angle of rotation, and \(c\) is a constant. From equations (41) and (42) we obtain
\[ (\chi-\chi_0)=\frac{2c\alpha}{H^{2}s}. \tag{43} \]
For a constant magnetic field \(\frac{2c}{H^{2}s}\) is constant; therefore
\[ \chi-\chi_0=p\alpha. \tag{44} \]
Denoting by \(\chi_0\), \(\chi_g\), and \(\chi_w\) the susceptibilities of vacuum, gas, and water, and by \(\alpha_0\), \(\alpha_g\), and \(\alpha_w\) the corresponding deflections, we may write
\[ \chi_0-\chi'=p\alpha_0;\quad \chi_g-\chi'=p\alpha_g;\quad \chi_w-\chi'=p\alpha_w. \]
Taking \(\chi_0=0\), we finally obtain
\[ \frac{\chi_g}{\chi_w}=\frac{\alpha_g-\alpha_0}{\alpha_w-\alpha_0}. \tag{45} \]
In this way Soné determined the susceptibilities of the following diamagnetic gases: \(\mathrm{H_2}\), \(\mathrm{N_2}\), \(\mathrm{CO_2}\), and Ar.
But even into this careful investigation by Soné there crept a large experimental error in the measurement of the susceptibility of Ar, for which the measured value proved to be many times greater than the expected one1.
In 1924 Wills and Hector20, using a very ingenious apparatus based on the principle of measuring the susceptibility of gases by the Quincke method, measured several of the simplest gases.
The Quincke method consists in the following. Between the poles of an electromagnet, in a uniform magnetic field, a glass tube filled with liquid is placed, the meniscus of which is set at the center of the poles and beyond the level of which observation is made with a microscope. The force acting on the meniscus in the magnetic field will be the same as in the case of the Gouy method, for the Quincke method is a modification of the Gouy method for the case of liquids and gaseous bodies.
The apparatus used by Wills and Hector is shown schematically in Fig. 4. In the bend \(CF\) there is a paramagnetic solution. \(G\) is the narrowed part of the tube, in which a particle is suspended, beyond whose level observation is made with a microscope. The letter \(F\) denotes the boundary between the gas and the liquid, which was placed between the poles of the electromagnet.
Fig. 4.
In this apparatus the gas was magnetically balanced with respect to an aqueous solution of a paramagnetic salt of known susceptibility. Exact bringing of the gas and liquid into equilibrium was accomplished either by changing the pressure of the gas, which changed only the volume susceptibility of the gas, or by changing the temperature of the gas and of the solution and, consequently, by changing the susceptibility of the solution while the susceptibility of the diamagnetic gas remained unchanged. The susceptibility of the solution was known at any temperature, since the selected solution strictly obeyed Curie’s law. The susceptibility in these measurements was calculated in the following way.
Taking Wiedemann’s law of mixtures as valid for the solution in question,
the susceptibility is determined by the dimensions of the electron shell, then the susceptibility of an inert-gas atom should lie somewhere in the interval between the susceptibilities of the ions nearest to it. According to Pascal, the susceptibilities of the “neighbors” of Ar are as follows: \(\chi_{K^+} = -18.5\) and \(\chi_{Cl} = -20.1\). According to Sonz’s measurements, \(\chi_{Ar} = 246(!)\). Another, cruder criterion is the determination, from the value of the susceptibility, of the effective radius of the atom and comparison of it with the same quantity determined by other methods.
we have
\[ m_l\chi_l=m_w\chi_w+m_s\chi_s, \tag{46} \]
where \(m\) is mass, and \(\chi\) is the specific susceptibility, referring respectively to the solution, water, and salt.
Introducing \(R=\dfrac{m_w}{m_s}\) and taking into account that \(m_l=m_w+m_s\), we obtain
\[ \chi_l=\frac{R\chi_w+\chi_s}{1+R}. \tag{47} \]
Or, if \(R_0\) denotes the value of \(R\) at \(\chi_l=0\), we have
\[ \chi_l=\frac{R-R_0}{1+R}\chi_w. \]
Further, \(\rho_l=\dfrac{1+R}{1+R-K}\rho_w\), where \(K=\left(1-\dfrac{\rho_w}{\rho_l}\right)\dfrac{m_c}{m_s}\), where \(\rho\) is density, and the index \(c\) refers to the initial solution.
If \(\chi\) is expressed in terms of the volume susceptibility \(\kappa\), then \(\kappa_l=\rho_l\chi_l\). And finally
\[ \kappa_l=\frac{R-R_0}{1+R-K}\rho_w\chi_w. \tag{48} \]
Let us now change the temperature of the gas and the solution by a small amount \(\Delta\theta\), and compensate the resulting disturbance of the magnetic equilibrium by changing the pressure by an amount \(\Delta P\). Then
\[ \frac{\partial\kappa}{\partial p}\Delta P=-\frac{\partial\kappa_l}{\partial\theta}\Delta\theta. \tag{49} \]
For a diamagnetic gas \(\kappa=\dfrac{\rho_0^\theta}{\rho_0\theta}\kappa_0\).
After differentiation we obtain
\[ \frac{\partial\kappa}{\partial p}=\frac{\partial\kappa}{\partial P}=\frac{\theta_0}{P_0\theta}\kappa_0;\qquad \frac{\partial\kappa_l}{\partial\theta}=-\left(\rho_w\frac{dR_0}{d\theta}\frac{1}{1+R-K}\right)\chi_w \]
and finally obtain the working formula for the susceptibility of a diamagnetic gas
\[ \kappa_0=-\frac{dR_0}{d\theta}\frac{\Delta\theta}{\Delta P}\frac{P_0\theta}{\theta_0}\frac{\rho_w}{1+R-K}\chi_w. \tag{50} \]
With the aid of such a method for measuring diamagnetic gases, Wills and Hector\(^{20}\), and then Hector alone\(^{21}\), obtained quite satisfactory results for the susceptibility of a number of the simplest gases.
Measurements of the susceptibility of gases carried out by Glaser (1924–1930) were devoted to checking the dependence of the specific diamagnetic susceptibility on pressure. Glaser\(^{22}\) carried out his measurements by the torsion-balance method. In a magnetic field he placed—
DIAMAGNETISM OF GASES AND VAPORS
…was a small glass rod, having the form of an ellipsoid, suspended at its center of gravity on a thin quartz thread. The entire space around the rod was filled with the gas under investigation.
From the difference in the angles of rotation of the rod in the magnetic field, surrounded by the gas under investigation and by a gas with known susceptibility, the magnetic susceptibility of the gas under investigation was determined. If we denote by \(\chi_0\) and \(\chi_2\) the volume susceptibilities of the material of the rod and of the gas under investigation, then the torque acting on the rod will be
\[ G=(\chi_0-\chi_2)\int \frac{1}{2}(\nabla H^2\cdot r)\,dv. \tag{51} \]
On the other hand, for deflections of the rod through small angles one may take
\[ G=C\alpha_2, \tag{52} \]
where \(C\) is a constant (the torsional moment of the suspension thread) and \(\alpha_2\) is the angle of rotation of the rod.
Denoting
\[ \frac{C}{\displaystyle \int \frac{1}{2}(\nabla H^2\cdot r)\,dv}=c, \]
we have \(\chi_2=\chi_0+c\alpha_2\). If \(\alpha_0\), \(\alpha_1\), and \(\alpha_2\) refer respectively to vacuum, to a gas with known susceptibility, and to the gas under investigation, then \(0=\chi_0+c\alpha_0\), \(\chi_1=\chi_0+c\alpha_1\), and \(\chi_2=\chi_0+c\alpha_2\). Hence finally, for the volume susceptibility of the gas under investigation,
\[ \chi_2=\chi_1\frac{\alpha_2-\alpha_0}{\alpha_1-\alpha_0}. \tag{53} \]
In this way Glaser carried out relative measurements of the susceptibility of a number of gases as a function of pressure. In all the measurements indicated, the standard used was the susceptibility of \(\mathrm{H}_2\), for which the value was taken from the measurements of Soné, Wills, and Hector.
According to Glaser’s data, when the pressure is lowered, starting from atmospheric pressure, the volume susceptibility at first changes linearly with the pressure; but at a certain pressure, characteristic for each diamagnetic gas, it ceases to follow this law. In Fig. 5 one of the graphs of the dependence of the susceptibility of CO on pressure is given.
This experimental result cannot be explained theoretically unless the presence of an experimental error is assumed. It is therefore not surprising that the efforts of a number of investigators were directed toward checking the “Glaser effect.” Measurements of the susceptibi-
of diamagnetic gases as a function of pressure, carried out by various authors using different methods (Lepère²³, Buchner²⁴, Hammar²⁵, Vaidyanathan²⁶, and Bitter²⁷), led to the unanimous conclusion: the “Glaser effect” does not exist in nature. Under certain special conditions, when water vapor was mixed with the gas under investigation, Bitter succeeded in observing an analogue of the “Glaser effect”—the dependence of the specific diamagnetic susceptibility on the pressure of the gas under investigation. In six subsequent papers up to 1930, Glaser defended his “effect,” but by everyone except Glaser himself this “effect” is regarded as an annoying experimental error, indicating once again what exceptional care measurements on diamagnetic gases require.
Fig. 5.
Bitter²⁷ (1930) checked the dependence of diamagnetic susceptibility on temperature. His measuring apparatus was approximately the same as Glaser’s, only, owing to the choice of a more advantageous shape of the rotating body, it had greater sensitivity. The results of his measurements of molecular susceptibility are presented in Table 4.
TABLE 4
| Gas | \(T^\circ\mathrm{K}\) | \(-\chi_M \cdot 10^6\) |
|---|---|---|
| \(\mathrm{N_2}\) | 298 | 14.8 |
| \(\mathrm{N_2}\) | 88 | 14.2 |
| \(\mathrm{H_2}\) | 298 | 5.8 |
| \(\mathrm{H_2}\) | 88 | 3.3 |
This result, like the “Glaser effect” described above, is in contradiction with theoretical ideas about the nature of diamagnetism.
A check of the “Bitter effect,” carried out by Havens¹⁵ on an apparatus analogous to Bitter’s apparatus but possessing still greater sensitivity, showed no change whatsoever in susceptibility with temperature. It may be assumed that the “Bitter effect” is the consequence of an experimental error.
Vaidyanathan²⁸ investigated the susceptibility of a number of organic compounds in vapors and found that for some of them the susceptibility changes sharply when they pass from the liquid state into the gaseous state. His method of measuring susceptibility is analogous to Glaser’s method. This result of Vaidyanathan’s, like the preceding “effects” of Glaser and Bitter, seems hardly probable. A check of Vaidyanathan’s measurements, carried out by Ya. Shur³⁵, showed that the “Vaidyanathan effect” likewise does not exist.
The most reliable of all the methods used for measuring the susceptibility of gases should be considered the method proposed by Lehrer[^23]. Lehrer’s method is based on measuring the velocity of a gas flow arising under the action of a magnetic field in a nonuniformly heated gas. In Fig. 6 Lehrer’s apparatus is shown schematically. A horizontal tube \(ABC\) passes through the interpolar space of the magnet. When the field is excited, pressure differences arise between \(B\) and \(A\), or \(C\), which are equal to one another. But if the section of the tube \(AB\) is maintained at temperature \(T_1\), and the section \(BC\) at temperature \(T_2\) \((T_1 > T_2)\), then, when the magnetic field is switched on, the pressures that arise will no longer be equal.
Fig. 6.
Consequently, a pressure difference will appear between the ends of the tube. This pressure difference is measured by means of a sensitive manometer \(M\), consisting of a light vane suspended on a thin thread (Gehse manometer[^30]). The gas jet was directed onto the vane, and the velocity of the gas flow was determined from the rotation of a mirror attached to the vane. With the aid of auxiliary apparatus the manometer could be calibrated, and this made it possible to carry out absolute measurements of susceptibility. The method for calculating the susceptibility for this method will be described in detail below.
By this method Lehrer succeeded in measuring the susceptibilities of a number of diamagnetic gases at various pressures, in showing the absence of the “Glaser effect,” and also the applicability of Wiedemann’s law of additivity of susceptibility to gas mixtures.
However, Lehrer’s apparatus, though successful in conception, at the same time, because of the low stability of the manometer, becomes an instrument very capricious in operation and, of course, cannot be used in the case of high temperatures, just as it cannot be used for chemically active gases. This apparatus, in a somewhat more improved design, although its most vulnerable part—the Gehse manometer—remained ...
unchanged, was used for further investigations of gases by Gerlach[^31] (Ar) and Mann[^16] (Ne, Kr, Xe).
TABLE 5
Molecular diamagnetic susceptibility of some gases according to data of various investigators
(measurements at room temperature) — $\chi_m \cdot 10^6$
| Gas | Soné | Wills and Hector | Hector | Glaser | Lehrer | Waihingen | Hammer | Bitter | Havens | Gerlach | Mann |
|---|---|---|---|---|---|---|---|---|---|---|---|
| He | — | 1.94 | 1.88 | — | — | — | — | — | 1.90 | — | — |
| Ne | — | — | 18.1 | 20.1 | 20.11 | 25.3 | — | — | 19.23 | 19.72 | 19.54 |
| Ar | — | — | 6.52 | 6.85 | — | — | — | — | 7.65 | — | 6.75 |
| Kr | — | — | — | — | — | — | — | — | — | — | 28.02 |
| Xe | — | — | — | — | — | — | — | — | — | — | 42.40 |
| H$_2$ | 3.99 | 3.84 | — | — | 5.08 | — | — | 5.8 | 4.00 | — | — |
| N$_2$ | 7.42 | — | 11.8 | 7.36 | — | 12.9 | 8.0 | 14.8 | 11.94 | — | — |
| CO$_2$ | 18.6 | — | — | 21.0 | 20.86 | 20.5 | 19.3 | — | 20.88 | — | — |
Table 5 gives the results of measurements of the diamagnetism of gases by various authors in recent years. From this table it is clearly evident how unreliable the experimental data on the diamagnetism even of the simplest gases are. For example, the susceptibility of H$_2$ ranges from 3.8 to 5.8 (here and below the value of the susceptibility must be multiplied by $10^{-6}$), and the susceptibility of N$_2$ from 7.3 to 14.8. The susceptibility values for the other gases vary to a lesser degree, but still quite considerably.
The reason both for the “effects” mentioned above and for the large scatter of the data in Table 5 lies in the following two defects of the experimental technique:
-
In measurements of diamagnetic gases, quite negligible admixtures of paramagnetic gases distort the result of the measurements to a very great extent.
-
In all the methods listed, inside the apparatus filled with gas there were movable parts which changed their properties under the influence of adsorption of the gas on their surfaces, temperature changes, etc.
If the first shortcoming can comparatively easily be eliminated with modern gas-purification techniques, then eliminating the second requires the creation of apparatus in which moving parts would be entirely absent.1
The apparatus proposed by R. I. Yanus and Ya. S. Shur2 contains no moving parts inside the gas under investigation, as a result of which the influence of adsorbed gas is completely excluded; this makes it possible to carry out measurements in almost saturated vapors. The applicability of the method is limited only by the chemical and thermal stability of the material of the apparatus.
The method of these authors is based on the principle of measuring the susceptibility of gases proposed by Lehrer. But in their apparatus only Lehrer’s idea remained in use, with a fundamental modification of his setup. In this method of investigating susceptibility, the gas is enclosed in a tube forming a closed rectangle (Fig. 7). This glass rectangle is placed in a vertical plane between the poles of an electromagnet in such a way that the middle of its upper horizontal part, indicated in the figure by a circle, lies in the homogeneous maximum of the magnetic field \(H\). By means of auxiliary furnaces on both sides of this region of the maximum \(H\), uniform temperatures \(T_1\) and \(T_2\) are maintained up to such distances where the field strength practically becomes close to zero. The vertical sides of the rectangle are maintained at uniform temperatures \(T_3\) and \(T_4\).
Fig. 7.
Let us create such conditions that \(T_1 > T_2\) and \(T_3 = T_4\). In this case, in the absence of a magnetic field, the gas in the tube is at rest. When the magnetic field is switched on, a pressure difference \(\Delta p_H\) arises, equal to
\[ \Delta p_H = \frac{1}{2} H^2(\varkappa_1 - \varkappa_2), \tag{54} \]
where \(\varkappa_1\) and \(\varkappa_2\) are the volume susceptibilities of the sections of the tube at temperatures \(T_1\) and \(T_2\). Substituting \(\varkappa = \frac{\chi}{\rho}\), where \(\chi\) is the specific susceptibility of the gas and \(\rho\) its density, we obtain
\[ \Delta p_H = \frac{1}{2} H^2(\chi_{T_1}\rho_{T_1} - \chi_{T_2}\rho_{T_2}); \tag{55} \]
since the specific susceptibility of a diamagnetic gas does not depend on temperature, then
\[ \Delta p_H = \frac{1}{2} H^2 \chi(\rho_T - \rho_{T_1}). \tag{56} \]
As a result of the pressure thus created, gas circulation through the tube will occur with some velocity. To measure this velocity of the gas flow, a micromanometer is used, placed in the lower horizontal part of the tube. The principle of operation of the micromanometer is as follows. At the maximum of the temperature gradient obtained between two sections of the tube having sharply different temperatures \(T_{b}\) and \(T_{e}\), a temperature indicator (a thermocouple or a bolometer) is placed. When gas circulation arises, the temperature distribution near the indicator is displaced, and this is what is recorded. Having measured the temperature for gas at rest, we can subsequently, from its change, judge both the direction and the velocity of the gas motion through the tube. Thus, in a first approximation, the change in temperature at the location of the temperature meter will be directly proportional to the change in the velocity of the circulating gas, which in turn depends on the geometrical data of the tube, the viscosity of the gas, and \(\Delta p\), i.e.
\[ \Delta t = k\Delta p, \tag{57} \]
where \(k\) is a certain constant.
Substituting \(\Delta p\) from (56) into expression (57), we obtain
\[ \Delta t_{H}=\frac{1}{2}kH^{2}\chi(\rho_{1}-\rho_{2}). \tag{58} \]
Let us switch off the magnetic field and slightly raise the temperature \(T_{3}\) relative to \(T_{4}\). In this case a gravitational pressure difference will arise between the left and right vertical parts of the tube.
The magnitude of the pressure difference \(\Delta p_{g}\) obtained in this way is as follows:
\[ \Delta p_{g}=gh(\rho_{3}-\rho_{4}), \tag{59} \]
where \(h\) is the height of the vertical part of the tube, \(g\) is the acceleration of gravity, and \(\rho_{3}, \rho_{4}\) are the densities of the gas in the corresponding vertical sections of the tube at temperatures \(T_{3}\) and \(T_{4}\). As a result of the occurrence of \(\Delta p_{g}\), gas circulation will take place in the tube, which will cause a change in the temperature of the micromanometer by the amount \(\Delta t_{g}\).
If \(T_{3}\) is increased only slightly relative to its initial value \(T_{4}\), then the influence of small changes in the friction of the gas against the walls of the instrument may be neglected and equation (57) may be applied with the same constant \(k\):
\[ \Delta t_{g}=kgh(\rho_{3}-\rho_{4}). \tag{60} \]
Solving equations (58) and (60) together and assuming that in practice the accuracy of the Clapeyron formula is quite sufficient, we obtain the final expression for the specific diamagnetic susceptibility
\[ \chi= \frac{ 2gh\left(\dfrac{1}{T_{3}}-\dfrac{1}{T_{4}}\right) }{ H^{2}\left(\dfrac{1}{T_{2}}-\dfrac{1}{T_{1}}\right) } \frac{\Delta t_{H}}{\Delta t_{g}}. \tag{61} \]
This same method is, of course, also applicable to paramagnetic gases. It is easy to see that, for the case of a paramagnetic gas obeying Curie’s law, i.e. in which \(\chi T=C\) (\(C\) is Curie’s constant), we obtain
\[ C=\frac{2gh}{H^2}\, \frac{\left(\dfrac{1}{T_4}-\dfrac{1}{T_3}\right)} {\left(\dfrac{1}{T_2^2}-\dfrac{1}{T_1^2}\right)} \frac{\Delta t_H}{\Delta t_g}. \tag{62} \]
In what follows, in describing investigations of various gases and vapors by this method, it will be indicated in detail how the present principle of measurement was implemented in practice in solving one or another experimental problem. Common to all designs of instruments realized on this principle is the absence of any moving parts whatsoever, which made it possible to reduce to a minimum the influence of external disturbances and to eliminate completely the influence of adsorption phenomena. The operations for measuring magnetic susceptibility in this method are reduced mainly to the precise measurement of the temperature of individual sections of the instrument (\(T_1, T_2, T_3\), and \(T_4\)) and of the temperature change of the micromanometer. In comparison with all previous methods, in which the test weakly magnetic body, surrounded by the gas under investigation, was suspended on the finest threads and whose rotation had to be observed, this method is considerably simpler in operation and therefore can readily be used for large-scale physicochemical investigations.
To test the method, the susceptibility of CO\(_2\) was measured. Table 5 gives values for the molecular susceptibility of CO\(_2\), ranging from \(-18.6\cdot10^{-6}\) to \(-21.0\cdot10^{-6}\). In this case an average value equal to \(-20.24\cdot10^{-6}\) was obtained. This value is in good agreement with the most reliable experimental results. The sensitivity of the method at atmospheric gas pressure in the instrument makes it comparatively easy to determine \(\chi\) with an accuracy of up to \(5\cdot10^{-9}\). The method described has considerably expanded the possibilities for measuring the most diverse gases and vapors, since thanks to it measurements at high temperatures, as well as measurements of vapors of chemically active substances, have become possible. We shall dwell in somewhat more detail on a number of measurements carried out by means of this method.
A. Determination of the character of the bond in the carbon monoxide molecule from the value of the magnetic susceptibility\(^{33}\)
The character of the chemical bond can be judged from the magnitude of the diamagnetic susceptibility of the given chemical compound. As is known from Chapter II, diamagnetic susceptibility depends on the effective dimensions of the atom. It was indicated above that effective
the dimensions of atoms change noticeably with the presence of a bond; consequently, this must to some extent be reflected also in the magnitude of the diamagnetic susceptibility. This dependence was first noted by Pascal\(^{34}\) and, over a number of years (1908–1913), was studied in greater detail on the basis of measurements of a vast number of different organic compounds. By generalizing an enormous body of experimental material, Pascal found that the diamagnetic susceptibility of most organic compounds and salts of monovalent metals is obtained by adding the constituent elements of the given chemical compound, with the addition of a certain constant \(\lambda\), depending on the character of the chemical bonds present in the given compound,
Fig. 8.
\[ \chi_M = \Sigma \chi_A + \lambda, \tag{63} \]
where \(\chi_M\) is the molecular susceptibility, and \(\chi_A\) the atomic susceptibilities of the elements entering into this compound.
The aim of this investigation was to measure the susceptibility of CO and to compare the value found with the values found by Pascal; this will make it possible to determine \(\lambda\), and thereby also the character of the chemical bond in the given molecule.
The apparatus for measuring magnetic susceptibility consisted of glass tubes forming a closed rectangle (Fig. 8). The width of the vertical tubes of the apparatus was 20 mm, that of the horizontal tubes, with the exception of the section of tube located inside the magnetic field, was 12 mm. The section of tube located in the interpolar space had an elliptical cross-section with axes of 15 and 5 mm. This was done in order to reduce the distance between the poles of the electromagnet and thereby obtain a higher magnetic-field strength, without at the same time increasing the resistance to the gas flow inside the apparatus. The sections of the tubes at the temperatures \(T_1\), \(T_2\), \(T_3\), and \(T_4\) were surrounded by copper jackets (2 mm thick) fitting tightly against the glass. Copper–constantan thermocouples were soldered to these copper jackets.
In the elliptical part of the apparatus there were two molybdenum tubes, which ensured the creation of a uniform temperature along the individual temperature sections inside the magnetic field. The temperatures \(T_1\), \(T_2\), \(T_3\), and \(T_4\) were produced by external furnaces, insulated from the metal jackets by a layer of sheet asbestos 3–5 mm thick. A special check established that, with a sufficient holding time (from 1 to 2 hours), the temperature of the gas inside the apparatus fully corresponded to the temperature
of the susceptibilities of hydrogen and nitrogen, determined by Wills and Hector^20: \(\chi_{\mathrm{H}_2}=-3.94\) and \(\chi_{\mathrm{N}_2}=-11.8\) (both here and below all susceptibility values must be multiplied by \(10^{-6}\)). In a subsequent work, Bitter^28 measured the susceptibility of the standards he had previously used and found for them the following, very different values: \(\chi_{\mathrm{N}_2}=-14.8\) and \(\chi_{\mathrm{H}_2}=-5.8\) (at room temperature). At the same time he found (as was indicated earlier) a sharp change in the diamagnetic susceptibility upon lowering the temperature: \(\chi_{\mathrm{H}_2}=-3.3\) and \(\chi_{\mathrm{N}_2}=-14.2\) at \(88^\circ\mathrm{K}\). Both the absolute values of the magnetic susceptibility of normal gases obtained by Bitter and the temperature anomaly he found for these diamagnetic gases are refuted by later investigators, who made their observations by the most irreproachable methods^15.
According to Bitter’s measurements^37, the molecular susceptibility of methane and ethylene is as follows: \(\chi_{\mathrm{CH}_4}=-12.2\) and \(\chi_{\mathrm{C}_2\mathrm{H}_4}=-12.0\). The molecular susceptibility of these same substances in the liquid state^38 is considerably greater: \(\chi_{\mathrm{CH}_4}=-40.1\) and \(\chi_{\mathrm{C}_2\mathrm{H}_4}=-44.8\). Since the technique of measuring magnetic susceptibility in the liquid state is very reliable, there is hardly any reason to doubt the validity of these experimental data. On the other hand, there are no grounds to suppose that the magnetic susceptibility itself changes so strongly when a substance passes from the liquid into the vapor state. The fact that, according to Bitter’s measurements, the molecular susceptibility of organic compounds as strongly different from one another as methane, ethylene, and acetylene is approximately one and the same also arouses doubt: \(\chi_{\mathrm{CH}_4}=-12.2\); \(\chi_{\mathrm{C}_2\mathrm{H}_2}=-12.5\) and \(\chi_{\mathrm{C}_2\mathrm{H}_4}=-12.0\). Apparently, in carrying out all his measurements, Bitter did not succeed in eliminating the influence of adsorbed gas located inside the apparatus, which distorted both the absolute value of the magnetic susceptibility and, with particular sharpness, manifested itself under changes of temperature and produced the phenomenon that Bitter took to be a change in the diamagnetic susceptibility of a gas with temperature. All this, taken together, casts doubt on the quantitative results obtained by Bitter’s measurements of the magnetic susceptibility of vaporous organic compounds.
Of considerably greater importance for knowledge of the magnetic properties of vapors of organic compounds is the investigation of Vaidhyanathan^29. Vaidhyanathan set himself the task of tracing whether the magnetic susceptibility of organic compounds changes when they pass from the liquid state into vapor. The results of his measurements showed that almost all the substances he measured (11 out of 12) had, in the vapor state, a susceptibility somewhat greater (\(\sim 10\%\)) than in the liquid, with the exception of carbon disulfide and benzene. For these last two substances this difference is especially large: the molecular susceptibility for the liquid state is \(\chi_{\mathrm{C}_6\mathrm{H}_6}=-57\) and \(\chi_{\mathrm{CS}_2}=-45\), while for the gaseous state it is \(\chi_{\mathrm{C}_6\mathrm{H}_6}=-83\) and \(\chi_{\mathrm{CS}_2}=-75\).
of the outer linings in all sections far from the ends of the linings. Noticeable deviations of the internal temperature from the external temperature are found near the ends of the linings at distances comparable with the transverse dimensions of the apparatus at that point. In view of this, the linings of the vertical sections of the apparatus with temperatures \(T_3\) and \(T_4\) were lengthened so much that they also covered the nearest horizontal sections of the apparatus over a distance of \(40\) mm.
The micromanometer, placed at the center of the lower tube (Fig. 8), had the following construction. Between the end of a massive metal sleeve and a wire tungsten spiral heated by current from an auxiliary battery, a large temperature gradient was created (about \(400^\circ\) per \(1\) cm). At the point of maximum of this gradient there was placed a copper—constantan thermocouple (\(\varnothing = 0.05\) mm), which measured the temperature at this point. The current passed through the heating spiral of the micromanometer was required to be very constant; with a current strength from \(0.2\) to \(0.5\) A its fluctuations did not exceed \(0.001\) A.
These measurements gave, for the molecular susceptibility of gaseous CO, the value \(-118 \cdot 10^{-7}\). Let us compare it with the value of the susceptibility calculated according to Pascal.
According to Pascal, the atom C enters into the composition of a molecule with a susceptibility of \(-60.0 \cdot 10^{-7}\); the atom O correspondingly has \(-46.0 \cdot 10^{-7}\). A double bond always considerably lowers diamagnetism. A triple bond lowers diamagnetism to a much smaller degree than a double bond. A quadruple bond leaves the diamagnetism of the molecule almost unchanged or even slightly increases it. Thus, for example, the bond \((\mathrm{C}=\mathrm{N}-)\) lowers the diamagnetism of a molecule by the amount
\[ \lambda = +81.5 \cdot 10^{-7}, \]
whereas the bond \((\mathrm{C}\equiv\mathrm{N}-)\) only by
\[ \lambda = +8.0 \cdot 10^{-7}. \]
The double bond \(\mathrm{C}=\mathrm{O}\), according to Pascal, lowers the diamagnetism by
\[ \lambda = 63.5 \cdot 10^{-7}. \]
Hence, if a double bond were present, we should have had for the CO molecule the value
\[ \chi_M = -(60.0 + 46.0 - 63.5)\cdot 10^{-7} = -42.5 \cdot 10^{-7} \]
instead of the measured
\[ -118 \cdot 10^{-7}. \]
Agreement with experiment requires the introduction into the calculation of
\[ \lambda = -12 \cdot 10^{-7}. \]
Thus the magnetic properties of CO undoubtedly reject the presence of a double bond and, just as the results of measurements of the effective radii of the atoms in the CO molecule and the dissociation energy of the CO molecule, speak in favor of the presence of a triple bond.
B. Magnetic susceptibility of vapors of organic compounds \(^{35}\)
Measurements of the magnetic susceptibility of organic substances in vapors have been carried out only twice by two investigators: Vaidyanathan \(^{29}\) and Bitter \(^{36}\).
Bitter measured the magnetic susceptibility of a whole series of organic substances, using as the standard the value of the molecular
The latter result is completely incomprehensible, for in order to produce a considerable change in the magnetic susceptibility, it would be necessary for some very substantial changes to occur in the dimensions of the molecule itself upon transition from one aggregate state to another. But precisely in the case of these two molecules (\(\mathrm{CS}_2\) and \(\mathrm{C}_6\mathrm{H}_6\)) such changes are least to be expected. For example, a change in magnetic susceptibility may occur if the molecules of the liquid are in an associated state (in the form of molecular complexes), and upon transition to the vapor these molecular complexes break up. The magnetic susceptibility in this process should increase.
A convincing example of such a change in magnetic susceptibility as a result of the breakup of molecular complexes may be furnished by water. Study of the Raman spectrum of water\(^{37}\) has shown that, as the temperature is raised, it passes ever more from the state \((\mathrm{H}_2\mathrm{O})_3\), \((\mathrm{H}_2\mathrm{O})_2\), and \(\mathrm{H}_2\mathrm{O}\) into the state \(\mathrm{H}_2\mathrm{O}\) owing to the destruction of the complexes \((\mathrm{H}_2\mathrm{O})_3\) and \((\mathrm{H}_2\mathrm{O})_2\). A number of studies have shown that only \(\mathrm{H}_2\mathrm{O}\) exists in vapors. Along with this, the specific susceptibility of water, equal for ice to \(\chi = -0.699^{38}\), reaches at room temperature \(\chi = -0.72^{39}\). If one compares the change in the state of water with increasing temperature, obtained with the aid of the Raman spectrum, with the change in magnetic susceptibility corresponding to these same temperatures, then for the individual states of water the following values of the susceptibility are obtained: \(\chi_{\mathrm{H}_2\mathrm{O}} = -0.775\); \(\chi_{(\mathrm{H}_2\mathrm{O})_2} = -0.722\), and \(\chi_{(\mathrm{H}_2\mathrm{O})_3} = -0.704^{40}\). But, as is known, water possesses a considerable dipole moment \((\mu_{\mathrm{H}_2\mathrm{O}} = 1.8 \cdot 10^{-18,41})\), and therefore in the liquid state it forms molecular complexes. As for molecules of benzene and carbon disulfide, these molecules have no dipole moment\(^{42}\), and therefore it is also difficult to expect any appreciable association of them in the liquid state.
Vaidyanathan\(^{29}\), in his work, indicates that a change in magnetic susceptibility depending on the aggregate state of a substance also occurs for a number of pure elements upon their transition from the solid to the liquid state, as was shown by Honda\(^{43}\) and Owen\(^{44}\). But the measurements to which Vaidyanathan refers cannot claim great precision, since they were made 25 years ago on preparations of insufficient purity, and therefore simple purification of the substance under study at the melting point from absorbed gases may quite well cause a noticeable change in susceptibility. In addition to this fact, there is also possible an actual change of susceptibility at the melting point, for example when the valence electrons of an atom lose the possibility of encompassing several atoms simultaneously (destruction of the anomalous diamagnetism of bismuth).
All the possibilities enumerated for a change of susceptibility upon transition from one aggregate state to another cannot occur in the case of the transition of organic compounds from liquid to vapor. Using the new method for measuring the magnetic susceptibility of gases and vapors, Ya. Shur determined anew
the susceptibility of benzene vapor and carbon disulfide³⁵.
For this study an apparatus was made of molybdenum glass (Fig. 9). The temperature indicator was a bolometer consisting of a thin tungsten wire (thickness 0.023 mm). The temperature gradient was produced by a tungsten furnace located inside the apparatus at a distance of 10 mm from the bolometer. All temperatures \((T_1, T_2, T_3\) and \(T_4)\) were measured by means of Cu—const thermocouples soldered to the outer copper plates (thickness 0.5 mm), which covered the corresponding sections of the apparatus. The part of the apparatus placed inside the magnetic field had an elliptical cross-section (major semiaxis 6 mm, minor 3 mm).
Fig. 9.
Inside this part of the tube there were two elliptically shaped tubes \((K_1\) and \(K_2)\) made of tantalum sheet, which ensured the creation of a uniform temperature along the individual sections of the apparatus \(B_1\) and \(B_2\).
Benzene. Chemically pure benzene was taken for the investigation. Before the apparatus was filled, \(C_6H_6\) was repeatedly distilled in vacuum with the aid of the purifying unit \(L\) (Fig. 10). The purification was carried out in the following order: apparatus \(A\), as well as the purifying unit \(L\) soldered to it, was first degassed at a temperature of \(500^\circ\text{C}\) for several hours until a vacuum of \(\sim 10^{-6}\) mm was reached. Then benzene was poured through opening \(a\) into branch \(b\), and the opening was quickly sealed. With the aid of liquid air the benzene was frozen, and the entire purifying apparatus \(L\), together with apparatus \(A\), was pumped down to a vacuum of \(\sim 10^{-6}\) mm. After this, without stopping the pumping, trap \(c\) is immersed in liqu-
air, while allowing the benzene to evaporate from branch \(b\). In this process the gases absorbed by the benzene, such as oxygen, nitrogen, and hydrogen, will be completely pumped out. After the stopcock is closed, the benzene is again distilled into branch \(a\), and the purification process is resumed. At the end of the fivefold distillation carried out in the order described above, the benzene was collected by freezing in branch \(e\) of the apparatus, and then, after prolonged pumping, apparatus \(A\) was thawed at the constricted place of tube \(f\).
The purification carried out in this way guarantees the complete absence of oxygen, which is the most dangerous impurity in measuring the susceptibility of diamagnetic gases, and reduces the content of other impurities, such as \(\mathrm{H_2O}\) and \(\mathrm{CO_2}\), to negligible amounts.
Fig. 10.
Carbon disulfide. Carbon disulfide was also taken chemically pure and was then purified with pure mercury from traces of sulfur. Further purification of \(\mathrm{CS_2}\) was carried out similarly to benzene. At high temperatures of the order of \(800^\circ\mathrm{C}\), appreciable dissociation of \(\mathrm{CS_2}\) occurs \((\sim 8\%)^{45}\). In order to guarantee complete chemical stability of the carbon disulfide molecules while carrying out the measurements, the highest temperature in the apparatus did not exceed \(100^\circ\mathrm{C}\).
The measurements gave the following value for the specific susceptibility of carbon disulfide:
\[ \chi = -0.53 \cdot 10^{-6} \]
and for the specific susceptibility of benzene
\[ \chi = -0.75 \cdot 10^{-6}. \]
Measurement of the magnetic susceptibility of benzene and carbon disulfide in the liquid state was carried out many times, and the results
the studies of various authors differ very little from one another. Therefore we believe that the data are quite reliable, and there is no need to verify them. The specific susceptibility of liquid benzene and carbon disulfide is, respectively,
\[ \chi_{\mathrm{C_6H_6}}=-0.71-0.73^{46} \quad\text{and}\quad \chi_{\mathrm{CS_2}}=-0.56-0.59^{47}. \]
If these values are compared with the data that we found in the vapor state, we see that, within the limits of experimental error, the magnitude of the magnetic susceptibility remains unchanged.
As was indicated at the beginning, for these two substances it was difficult to expect any other result.
Thus we can now fully justifiably draw the conclusion that, in the case of all organic substances investigated up to the present time, the magnetic susceptibility in the vapor state remains approximately of the same magnitude as in the liquid state.
C. Magnetic susceptibility of bromine vapor[^48]
This investigation pursued two aims: 1) to construct an apparatus suitable for measuring the susceptibility of chemically active vapors and, thereby, to extend the range of applicability of the new method for determining susceptibility; 2) to compare the value of the susceptibility of vapor-phase bromine both with approximate theoretical calculations and with the experimental value of the susceptibility of liquid bromine.
Fig. 11.
The apparatus consists of a closed rectangle made of refractory glass (Fig. 11). The thermostats, which ensure with the necessary accuracy the constancy of temperature in separate sections of the tube, are indicated by dashed lines. Through the center of the upper part of the tube passes a capillary \(A\), sealed into the walls of the tube, in which a thermocouple is placed (copper—constantan, diameter \(0.04\ \mathrm{mm}\)). The readings of this thermocouple can serve to compare the velocities of gas motion inside the apparatus caused by the action on the gas of either a magnetic or a gravitational field. To concentrate the gas stream and to strengthen the temperature gradient near the capillary \(A\), nozzles \(B_1\) and \(B_2\) are sealed inside the tube,
ends of which come very close to capillary \(A\). The section of the tube located in the magnetic field has an elliptical cross-section with axes of 15 and 6 mm. The entire instrument was placed in a thermostat in order to produce the necessary vapor pressure inside the tube.
Thus, in this instrument there is not a single moving or mechanical part, which makes it possible to preserve the substance being measured in an entirely pure state. Therefore, despite the somewhat reduced sensitivity of this instrument in comparison with the one described earlier, in which the thermocouple was in direct contact with the gas enclosed in the instrument, it is fully possible to measure, with sufficient accuracy, the magnetic susceptibilities of such substances as until now could not be studied.
Bromine, obtained from the firm Kahlbaum, was repeatedly distilled at a vacuum of \(10^{-5}\) mm Hg (the evaporation temperature of bromine being room temperature, the condensation temperature that of liquid air), and then the bromine was collected in the instrument, which was subsequently sealed off. Before being filled, the instrument was degassed for several hours at a temperature of about \(400^\circ\), likewise under high vacuum. All these precautions, together with the absence in the instrument of any metallic parts, valves, and ground joints, give grounds for believing that in the measured bromine vapors there were no appreciable impurities of other substances (H\(_2\)O and other readily condensing substances); gases that at the temperature of liquid air still have appreciable elasticity (O\(_2\) and others) were entirely absent. The lowest temperature in the entire instrument was \(60^\circ\)C. This was done in order to obtain a bromine vapor pressure not lower than atmospheric (the boiling point of bromine is \(59^\circ\)).
The accuracy of the measurements depended, above all, on the accuracy of determining the gas temperature \(T_1, T_2, T_3, T_4\) in the corresponding sections of the instrument; therefore special attention was paid to temperature measurements. The temperature of the metallic jacket, which closely fitted the glass of the instrument and was insulated from the heating furnaces of the thermostats by a layer of asbestos 3–5 mm thick, was measured directly with an accuracy up to \(0.02^\circ\)C. The section of uniform magnetic field in the junction of temperatures \(T_1\) and \(T_2\) had a length of 50 mm, with the distance between the ends of the jackets being 10 mm; the error arising from assigning to the gas at the boundaries of the uniform magnetic field the temperatures \(T_1\) and \(T_2\) remains here below 5%.
The insulation of the micromanometer thermocouple from the gas flow by a glass capillary also creates in it a certain thermal inertia: it does not immediately show the temperature that arises in the gas surrounding the capillary as a result of mixing of the temperature gradient when the magnetic field is switched on or when the temperatures \(T_3\) and \(T_4\) are changed. A corresponding check showed that, in the case of the instrument described, 7 min are necessary for the final stabilization of the micromanometer thermocouple readings after each change in the operating regime of the instrument (with \(T_5\) and \(T_6\) unchanged).
As was indicated in Chapter II, rigorous theoretical calculations of the diamagnetic susceptibility of molecules, except for H₂, have not been carried out up to the present time; an approximate method for calculating molecular susceptibility was proposed by Angus¹¹. According to Angus, the susceptibility of bromine is equal to the sum of the susceptibilities of Br⁺ and Br⁻. Table 6 gives the values of the molecular susceptibility calculated in this way from the values of the ionic susceptibility, computed by the approximate methods of Pauling, Slater, and Angus, and compares them with the experimental value of the susceptibility of bromine vapor.
TABLE 6
| Pauling | Slater | Angus | Measurements | |
|---|---|---|---|---|
| $-\chi_m \cdot 10^6$ | 89.81 | 67.56 | 62.24 | 74 |
As is evident from Table 6, the approximate calculations agree quite well with the experimental value of the susceptibility of bromine vapor.
Measurements for bromine in the liquid and solid states were carried out by Owen⁴⁴, who obtained for the specific susceptibility
$$ \chi = -0.40 \cdot 10^{-6}. $$
This shows that the specific susceptibility of bromine remains unchanged within the limits of experimental error upon transition from the liquid to the gaseous state.
D. Magnetic susceptibility of mercury vapor
As was indicated in Chapter I, the diamagnetism of a metallic atom can be determined only in the vapor of the metal. The magnitude of the susceptibility of a metallic atom can be calculated by several approximate methods for calculating susceptibility (Chapter II). Comparison of the experimental value with the theoretical one can indicate how accurate the theoretical methods for calculating magnetic susceptibility are. By comparing the susceptibility of a metallic vapor with the susceptibility of its ion, one can estimate the share of the susceptibility caused by the valence electrons of the atom. Mercury was chosen as the object of investigation as a diamagnetic metal with the lowest boiling point (360° C at atmospheric pressure).
The apparatus for measuring the susceptibility of mercury was made of refractory glass (softening temperature ~850° C) and had approximately the same form and dimensions as the apparatus described for measuring the susceptibility of vapors of organic compounds. Its difference from the preceding apparatus consisted in the special construction of the micromanometer and thermostat. Figure 12 schematically shows the arrangement of the micromanometer located in the center.
DIAMAGNETISM OF GASES AND VAPORS
of the upper horizontal tube of the apparatus (Fig. 9). The letter \(A\) (Fig. 12) denotes a glass tube, to the edges of which glass holders shaped like small mushrooms were welded. A thin tungsten wire (\(\varnothing = 0.023\ \mathrm{mm}\)), having a zigzag form, as can be seen from the upper drawing of Fig. 12, served as the bolometer. Inside the glass tube \(A\) there was a thick-walled tantalum tube \(C\), serving to create the fixed “cold part” of the micromanometer. The heating furnace of the micromanometer was made of a thin tungsten spiral fastened on hooks like those of the bolometer (tube \(B\), Fig. 12). The edges of tubes \(A\) and \(B\) were at a distance of \(8\ \mathrm{mm}\) from one another.
Fig. 12.
After the apparatus had been filled with mercury and sealed off from the purification installation, the whole apparatus was wound with a uniform heating winding (nichrome \(\varnothing = 0.3\ \mathrm{mm}\)). This furnace made it possible to maintain the lowest temperature in the apparatus at about \(360^\circ\mathrm{C}\). The whole apparatus was thermally insulated with a \(5\ \mathrm{cm}\) layer of asbestos. Heating of the external furnace was monitored by eight thermocouples. All thermocouples in this apparatus were made of Pt—(Pt—Rh) \(\varnothing = 0.1\ \mathrm{mm}\) and were calibrated with the aid of a standard thermocouple. In all other details the apparatus did not differ in any way from those described earlier.
For the investigation, mercury was taken that had been purified of metallic impurities by nitric acid and of all other contaminants by distillation in vacuum. The apparatus, together with the purification installation analogous to Fig. 10 (instead of trap \(C\) there was simply a sealed-off side arm of the tube), was degassed for 12 hours at a temperature of \(700^\circ\mathrm{C}\). Then mercury was poured into side arm \(b\), and tube \(a\) was quickly sealed. At high vacuum (\(\sim 10^{-6}\ \mathrm{mm}\ \mathrm{Hg}\)) the mercury was repeatedly distilled (the evaporation temperature of mercury was about \(100^\circ\mathrm{C}\)). Finally, the mercury was distilled into the apparatus, which was then sealed off.
Chemical analysis showed that the metals tungsten and tantalum do not amalgamate while being for many days in the vapor of boiling mercury. Therefore all metallic parts located inside the apparatus were made only of tungsten and tantalum. The measurements gave, for the specific susceptibility of mercury,
\[ \chi = -(0.39 \pm 0.04)\cdot 10^{-6}. \]
In the solid and liquid states the susceptibility of mercury was measured by Owen^44, Oxley^50, and Foote^51. As can be seen from Table 7, the measurement results are fairly close to one another and therefore may be accepted as sufficiently reliable.
TABLE 7
Specific susceptibility of mercury \((\chi \cdot 10^6)\)
| Owen | Oxley | Foote | ||
|---|---|---|---|---|
| −0.15 | −0.155 | — | Solid (polycrystal) | |
| — | — | −0.122 | ∥ to the principal axis | Solid (single crystal, hexagonal lattice) |
| — | — | −0.115 | ⟂ to the principal axis | Solid (single crystal, hexagonal lattice) |
| −0.18 | −0.185 | — | Liquid |
Comparing the data of Table 7 with the results of measurements of mercury vapor, we see how sharply the paramagnetism of the conduction electrons lowers its resultant diamagnetic susceptibility in the solid or liquid state of mercury.
TABLE 8
Atomic susceptibility of mercury \((\chi_{\mathrm{Hg}}\cdot 10^6)\)
| Slater | Angus | Sommerfeld | Tombas | Experiment |
|---|---|---|---|---|
| −84.6 | −84.2 | −133.3 | −40.4 | −78.2 |
Table 8 presents theoretical calculations of the susceptibility carried out by several methods. In this case as well, the approximate calculation of the susceptibility by Slater’s method gives quite good agreement with experiment.
TABLE 9
| Pauling | Slater | Angus | Experiment^17 |
|---|---|---|---|
| −55 | −47.8 | −47.5 | −40.4 |
Table 9 compares the theoretical and experimental susceptibility of the mercury ion.
The value of the susceptibility of the mercury atom obtained by Ya. S. Shur^49 is almost twice as large as the experimental value of the susceptibility of its ion. This shows that the two valence electrons of mercury account for almost one half of the entire susceptibility of the mercury atom.
This result is very significant, since for the first time it has been possible to determine experimentally the magnitude of the diamagnetic susceptibility of the valence electrons of an atom.
IV. Conclusion
The present review shows that our knowledge in the field of the diamagnetic susceptibility of gases and vapors is still far from sufficient. Nevertheless, even the few results that have been obtained are of considerable scientific interest. In connection with the development of a new method for measuring the susceptibility of gases and vapors^32, thanks to which the investigation of a considerable number of substances that previously could not be studied (metal vapors, chemically active substances) has become possible, one may hope that in the coming years our knowledge in this field will be significantly increased.
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The method of Wills and Hector, described in detail above, cannot lay claim to this, since the presence in the apparatus of direct contact between the gas and the paramagnetic solution makes possible both absorption of the gas by the solution and evaporation of the solution itself, which cannot but distort the results of the measurements. Moreover, this method is completely inapplicable at high temperatures. ↩↩↩
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Reference marker as printed on the page. ↩