Pauli Theory Research Articles

Electrical Properties of Sodium Tungsten Bronze

With the addition of sodium atoms the ferroelectric insulator W${\mathrm{O}}_{3}$ acquires metallic properties. We have grown single crystals of ${\mathrm{Na}}_{x}{\mathrm{WO}}_{3}$ using a method developed at the Linde Air Products Company. Chemical analyses for sodium concentration showed $x=0.66$. The electrical resistivity was found to have a value of (1.9\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ohm-cm at 0\ifmmode^\circ\else\textdegree\fi{}C and to decrease approximately linearly with decreasing temperature in the range 20\ifmmode^\circ\else\textdegree\fi{}C to -160\ifmmode^\circ\else\textdegree\fi{}C. The Hall coefficient was found to have a value of (-5.1\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{3}$/coulomb at 20\ifmmode^\circ\else\textdegree\fi{}C and to decrease slightly with decreasing temperature. This value for the Hall coefficient indicates one electron carrier for each sodium atom added to the crystal. Using Pauli's theory of paramagnetism for free electrons in simple metals the calculated magnetic susceptibility is 0.53\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$. The value observed by Stubbin and Mellor for a sodium concentration corresponding to $x=0.9$ was 0.45\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$. Using the free electron theory of metals, the following properties have been calculated: electron mobility and mean free path, energy at the surface of the Fermi distribution, thermal conductivity, and electronic specific heat.

On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and $c$. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.

The Magnetic Susceptibility of Rubidium

In view of the theoretical interest involved in Pauli's theory of paramagnetism, the author has reinvestigated the magnetic susceptibility of rubidium. The metal used is subjected to double distillation in vacuum to ensure purity. The technique employed for this is fully described. The susceptibility is determined at nine different field strengths ranging from approximately 9 to 25 kilogauss. No variation in the susceptibility of more than 2 percent is observed over this range. The estimated error in the mean value of the susceptibility is between 2 and 3 percent. The uniformity of the susceptibility, accordingly, shows absence of serious ferromagnetic impurity. The mean value of the mass susceptibility is found to be ${+0.21}_{7}$ \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ at 20\ifmmode^\circ\else\textdegree\fi{}C, the Pauli theoretical value being +0.32 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$. A table is included showing the results found by all other investigators on this subject. The bearing of the results on the Pauli theory is discussed briefly.

Magnetic Moments of Atomic Nuclei

THE hyperfine structures of atomic spectra are considered as due to the interaction of the nuclear spin with the electronic orbital and spin moments. A theoretical calculation of this interaction enables one to obtain information on the magnitude of the magnetic moments of the nuclei from the separation of the hyperfine structures. This can be done for the case of the alkali atoms. Hargreaves (Proc. Roy. Soc., 124, 568; 1929) has calculated the separation due to a nuclear moment h/4π for the case of atoms with only one electron. In his calculations, however, the interaction between the electronic and nuclear spins, which is of the same order of magnitude as the other terms, has been neglected. I have therefore carried out the calculations with the following improvements. For the s-terms, which is the most important case, since they give the largest contribution to the separation, I have used Dirac's theory of the electron, since the simpler Pauli's theory gives a wrong result. For the p-terms I have used Pauli's method, taking into account the interaction of the nuclear and electronic spins. For the p-terms it is not necessary to evaluate numerically the eigenfunctions, since the constants involved in the formulae can be derived from empirical data on the separation of the electronic spin doublets. For the s-terms this is of course impossible, and I have calculated the eigenfunctions by the statistical method.