Showing papers in "Physical Review in 1928"

Thermal Agitation of Electric Charge in Conductors

Abstract: The electromotive force due to thermal agitation in conductors is calculated by means of principles in thermodynamics and statistical mechanics. The results obtained agree with results obtained experimentally.

Thermal Agitation of Electricity in Conductors

Abstract: Statistical fluctuation of electric charge exists in all conductors, producing random variation of potential between the ends of the conductor. The effect of these fluctuations has been measured by a vacuum tube amplifier and thermocouple, and can be expressed by the formula ${\overline{I}}^{2}=(\frac{2kT}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\infty}}R(\ensuremath{\omega}){|Y(\ensuremath{\omega})|}^{2}d\ensuremath{\omega}$. $I$ is the observed current in the thermocouple, $k$ is Boltzmann's gas constant, $T$ is the absolute temperature of the conductor, $R(\ensuremath{\omega})$ is the real component of impedance of the conductor, $Y(\ensuremath{\omega})$ is the transfer impedance of the amplifier, and $\frac{\ensuremath{\omega}}{2\ensuremath{\pi}}=f$ represents frequency. The value of Boltzmann's constant obtained from the measurements lie near the accepted value of this constant. The technical aspects of the disturbance are discussed. In an amplifier having a range of 5000 cycles and the input resistance $R$ the power equivalent of the effect is $\frac{{\overline{V}}^{2}}{R}=0.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$ watt, with corresponding power for other ranges of frequency. The least contribution of tube noise is equivalent to that of a resistance ${R}_{c}=1.5\ifmmode\times\else\texttimes\fi{}\frac{{10}^{5}{i}_{p}}{\ensuremath{\mu}}$, where ${i}_{p}$ is the space current in milliamperes and $\ensuremath{\mu}$ is the effective amplification of the tube.

Nuclear Motions Associated with Electron Transitions in Diatomic Molecules

Abstract: The question of nuclear motions associated with electron transitions is discussed from the standpoint of quantum mechanics. It appears that Heisenberg’s indetermination principle gives the clue to the inexactness of the earlier method based on Franck’s postulate since its strict application calls for a violation of the principle. The existence of an entirely new type of band spectrum due to the wave nature of matter is predicted and the interpretation of Rayleigh’s mercury band at 2476–2482 A.U. as of this type is suggested. Finally it is shown that while Franck’s postulate is also true for electron jumps in atoms, it is of but trivial interest because its inexactness is much greater for the electrons than for heavy nuclei.

Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative Systems

Abstract: Dirac showed that, if in the Hamiltonian $H$ momenta ${\ensuremath{\eta}}_{r}$ conjugate to the co-ordinates ${\ensuremath{\xi}}_{r}$ are replaced by $(\frac{h}{2\ensuremath{\pi}i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}{\ensuremath{\xi}}_{r}}$, the Schr\"odinger equation appropriate to the coordinate system ${\ensuremath{\xi}}_{r}$ is $(H\ensuremath{-}E){\ensuremath{\psi}}_{\ensuremath{\xi}}=0$. Applied to coordinate systems other than cartesian this usually leads to incorrect results. The difficulty is here traced partially to the way in which ${\ensuremath{\psi}}_{\ensuremath{\xi}}$ is normalized and partly to the choice of $H$. In $H$ expressions such as $\mathrm{qp}{q}^{\ensuremath{-}1}p$ and ${p}^{2}$ are not equivalent, and the simplified form is generally incorrect. A formula satisfying all the requirements of quantum mechanics for a Hamiltonian of a conservative system, in an arbitrary coordinate system, is therefore developed $H=\frac{1}{2\ensuremath{\mu}}\ensuremath{\Sigma}\stackrel{r=n}{r=1}\ensuremath{\Sigma}\stackrel{s=n}{s=1}{g}^{\ensuremath{-}\frac{1}{4}}{p}_{r}{g}^{\frac{1}{2}}{g}^{\mathrm{rs}}{p}_{s}{g}^{\ensuremath{-}\frac{1}{4}}+U$ This formula is applied to a case of plane polar coordinates and leads to correct results.

Three Notes on the Quantum Theory of Aperiodic Effects

Abstract: In Section 1 it is shown that the normalization of the characteristic functions corresponding to a continuous spectrum, which has been introduced by Hellinger and Weyl, satisfies the requirements of the $\ensuremath{\delta}$-normalization of the Dirac-Jordan transformation theory It is further shown that this normalization makes the flux to and from infinity of systems for which an integral of motion $\ensuremath{\beta}$ lies in the little range $\ensuremath{\Delta}{\ensuremath{\beta}}^{\ensuremath{'}}$ equal to $(\frac{\ensuremath{\partial}E}{h\ensuremath{\partial}{\ensuremath{\beta}}^{\ensuremath{'}}})\ensuremath{\Delta}{\ensuremath{\beta}}^{\ensuremath{'}}$In Section 2 the condition for the validity of classical mechanics in the form grad $\ensuremath{\lambda}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the instantaneous wave length $\ensuremath{\lambda}=(\frac{h}{2\ensuremath{\pi}}){[2M(E\ensuremath{-}U)]}^{\ensuremath{-}\frac{1}{2}}$, is applied to establish Rutherford's formula for the scattering of $\ensuremath{\alpha}$-particlesIn Section 3 a method is developed for computing the transition probabilities between states of the same energy, and which are represented by almost orthogonal eigenfunctions The theory is applied to the ionization of hydrogen atoms in a constant electric field The period of ionization in a field of 1 volt per cm is ${10}^{{10}^{10}}$ sec The bearing of such transitions on the problem of metallic conduction is discussed

Differential Intensity Sensitivity of the Ear for Pure Tones

Abstract: The ratio of the minimum perceptible increment in sound intensity to the total intensity, $\frac{\ensuremath{\Delta}E}{E}$, which is called the differential sensitivity of the ear, was measured as a function of frequency and intensity. Measurements were made over practically the entire range of frequencies and intensities for which the ear is capable of sensation. The method used was that of beating tones, this method giving the simplest transition from one intensity to another. The source of sound was a special moving coil telephone receiver having very little distortion, actuated by alternating currents from vacuum tube oscillators. Observations were made on twelve male observers. Average curves show that at any frequency, $\frac{\ensuremath{\Delta}E}{E}$ is practically constant for intensitites greater than ${10}^{6}$ times the threshold intensity; near the auditory threshold $\frac{\ensuremath{\Delta}E}{E}$ increases. Weber's law holds above this intensity, the value of $\frac{\ensuremath{\Delta}E}{E}=\mathrm{constant}$ lying between 0.05 and 0.15 depending on the frequency. As a function of frequency $\frac{\ensuremath{\Delta}E}{E}$ is a minimum at about 2500 c.p.s., the minimum being more sharply defined at low sound intensities than it is at high. This frequency corresponds to the region of greatest absolute sensitivity of the ear. Analytical expressions are given [Eqs. (2), (3), (4) and (5)] which represent $\frac{\ensuremath{\Delta}E}{E}$, within the error of observation, as a function of frequency and intensity. Using these equations it is calculated that at about 1300 c.p.s. the ear can distinguish 370 separate tones between the threshold of audition and the threshold of feeling.

The Problem of the Normal Hydrogen Molecule in the New Quantum Mechanics

Abstract: The solution of Schroedinger's equation for the normal hydrogen molecule is approximated by the function $C[{e}^{\ensuremath{-}\frac{z({r}_{1}+{p}_{2})}{a}}+{e}^{\ensuremath{-}\frac{z({r}_{2}+{p}_{1})}{a}}]$ where $a=\frac{{h}^{2}}{4{\ensuremath{\pi}}^{2}m{e}^{2}}$, ${r}_{1}$ and ${p}_{1}$ are the distances of one of the electrons to the two nuclei, and ${r}_{2}$ and ${p}_{2}$ those for the other electron. The value of $Z$ is so determined as to give a minimum value to the variational integral which generates Schroedinger's wave equation. This minimum value of the integral gives the approximate energy $E$. For every nuclear separation $D$, there is a $Z$ which gives the best approximation and a corresponding $E$. We thus obtain an approximate energy curve as a function of the separation. The minimum of this curve gives the following data for the configuration corresponding to the normal hydrogen molecule: the heat of dissociation = 3.76 volts, the moment of inertia ${J}_{0}=4.59\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}41}$ gr. ${\mathrm{cm}}^{2}$, the nuclear vibrational frequency ${\ensuremath{ u}}_{0}=4900$ ${\mathrm{cm}}^{\ensuremath{-}1}$.

On the Quantum Theory of the Capture of Electrons

Abstract: In Section 1 the method of a previous ${\mathrm{paper}}^{1}$ is applied to find the rate at which $\ensuremath{\alpha}$ particles capture electrons from atoms. The mean free path for capture varies roughly with the sixth power of the velocity of the $\ensuremath{\alpha}$ particle, and in good agreement with Rutherford's ${\mathrm{experiments}.}^{3}$ The value of the mean free path is computed for capture in air, and agrees with the experimental value.In Section 2 the probability of radiative recombination of electrons and protons is computed. The cross section for recombination becomes infinite for small relative velocities with the inverse square of the velocity; for high velocities it is given by ${10}^{\ensuremath{-}18}{W}^{\ensuremath{-}\frac{5}{2}}$, where $W$ is the energy in volts of the incident electrons.

The Assignment of Quantum Numbers for Electrons in Molecules. I

Abstract: Quantum numbers, notation, closed shells, molecular states.---The problem of making a complete assignment of quantum numbers for the electrons in a (non-rotating) diatomic molecule is considered. A tentative assignment of such quantum numbers is made in this paper (cf. Table III) for most of the known electronic states of diatomic molecules composed of atoms of the first short period of the periodic system. The assignments are based mainly on band spectrum, and to a lesser extent on ionization potential and positive ray, data. The methods used involve the application and extension of Hund's theoretical work on the electronic states of molecules. Although the actual state of the electrons in a molecule, as contrasted with an atom, cannot ordinarily be expected to be described accurately by quantum numbers corresponding to simple mechanical quantities, such quantum numbers can nevertheless be assigned formally, with the understanding that their mechanical interpretation in the real molecule (obtainable by an adiabatic correlation) may differ markedly from that corresponding to a literal interpretation. With this understanding, a suitable choice of quantum numbers for a diatomic molecule appears to be one corresponding to an atom in a strong electric field, namely, quantum numbers ${n}_{\ensuremath{\tau}}$, ${l}_{\ensuremath{\tau}}$, ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$, and ${s}_{\ensuremath{\tau}}({s}_{\ensuremath{\tau}}=\frac{1}{2} \mathrm{always})$ for the $\ensuremath{\tau}'\mathrm{th}$ electron, and quantum numbers $s$, ${\ensuremath{\sigma}}_{l}$, and ${\ensuremath{\sigma}}_{s}$ for the molecule as a whole (${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$ and ${\ensuremath{\sigma}}_{s}$ represent quantized components of ${l}_{\ensuremath{\tau}}$ and $s$, respectively, with reference to the line joining the nuclei). These quantum numbers may be thought of as those associated with the imagined "united atom" formed by bringing the nuclei of the molecule together. A notation is then proposed whereby the state of each electron and of the molecule as a whole can be designated, e.g. ${(1{s}^{s})}^{2}{(2{s}^{p})}^{2}{(2{s}^{s})}^{2}(2{p}^{p})$, $^{2}P$ for a seven-electron molecule with ${\ensuremath{\sigma}}_{l}=1$, $s=\frac{1}{2}$; in a symbol such as $2{s}^{p}$ the superscript denotes ${l}_{\ensuremath{\tau}}$, the main letter, ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$, thus $2{s}^{p}$ means that the electron in question has ${n}_{\ensuremath{\tau}}=2$, ${l}_{\ensuremath{\tau}}=1$, ${\ensuremath{\sigma}}_{l\ensuremath{\tau}}=0$. Electrons with ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}=0, 1, 2, \ensuremath{\cdots}$, are referred to as $s, p, d, \ensuremath{\cdots}$, electrons. It is shown that in a molecule it is usually natural to define a group of equivalent electrons giving a resultant ${\ensuremath{\sigma}}_{l}=0$, $s=0$ as a closed shell; in this sense, two $s$ electrons, or four $p, or d, f, \ensuremath{\cdots}$, electrons form a closed shell. The possible molecular states corresponding to various electron configurations are deduced by means of the Pauli principle (cf. Table I, and Appendix).Promoted electrons, binding energy, bonding power, and relation of molecular to atomic electron states.---As Hund has shown, some of the electrons must undergo an increase in their $n$ values (principal quantum numbers) when atoms unite to form a molecule. Such electrons are here called promoted electrons. The electrons in a molecule may be classified according to their bonding power, positive, zero, or negative. Electrons whose presence tends to hold a molecule together, as judged by the fact that their removal from a stable molecule causes a decrease in the energy of dissociation $D$ or an increase in the equilibrium internuclear separation ${r}_{0}$ may be said to have positive bonding power, and are identified with, or defined as, bonding electrons. The definitions of bonding power in terms of changes of $D$, and of changes of ${r}_{0}$, are unfortunately not in general equivalent, and we must accordingly distinguish "energy-bonding-power" and "distance-binding-power". On the whole, promoted electrons should tend to show negative energy-bonding-power, unpromoted electrons positive energy-bonding-power, but much should depend on "orbit dimensions."Certain rules governing the relations of the electronic states of a molecule to those of its dissociation products are discussed; in addition to theoretical rules established by Hund in regard to ${\ensuremath{\sigma}}_{l}$ and $s$ values, another, presumably less strict, rule is here proposed, namely that the ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$ values of all the atomic electrons before union should be preserved in the molecule (${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$ conservation rule). Selection rules for electronic transitions are also discussed; in addition to rules given by Hund, the following are proposed: $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}=\ifmmode\pm\else\textpm\fi{}1$ for intense transitions; $\ensuremath{\Delta}{\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}=0, \ifmmode\pm\else\textpm\fi{}1$.Results.---The key to the assignment of quantum numbers made here is found in the fact that the molecules BO, C${\mathrm{O}}^{+}$, and CN show an inverted $^{2}P$ state instead of the normal $^{2}P$ which should occur if this state were analogous to the ordinary $^{2}P$ states of the Na atom. The existence of such a low-lying inverted $^{2}P$ indicates that in these molecules there exists a closed shell of $p$ electrons from which one is easily excited. It is concluded that this is a ${(2{p}^{p})}^{4}$ shell. The identification of two other closed shells, of $s$ electrons, very likely ${(3{s}^{p})}^{2}$ and ${(3{s}^{s})}^{2}$, follows; the electrons in these and the ${(2{p}^{p})}^{4}$ shell are roughly equal in energy of binding. According to this interpretation, the electron jumps involved in the band spectra of BO, CN, C${\mathrm{O}}^{+}$, and ${\mathrm{N}}_{2}^{+}$ are more analogous to X-ray than to optical electron transitions. From this beginning, proceeding to CO, ${\mathrm{N}}_{2}$, NO, ${\mathrm{O}}_{2}$, ${\mathrm{O}}_{2}^{+}$, ${\mathrm{F}}_{2}$, ${\mathrm{C}}_{2}$, etc., a self-consistent assignment of quantum numbers is built up for most of the known states of the various molecules treated in this paper. The spectroscopic analogies of CN, ${\mathrm{N}}_{2}$, NO, etc., to Na, Mg, Al are justified and the partial failures of these analogies, such as the chemical resemblance of CN to a halogen, are explained. Nearly all the hitherto observed ionization potentials of the molecules discussed can be accounted for by the removal of a single electron from one or another of the various closed shells supposed to be present. The ${\mathrm{N}}_{2}^{+}$ band fluorescence produced by short wave length ultraviolet light (Oldenberg) is accounted for as the expected result of photo-ionization of a $3{s}^{p}$ electron. The steadily decreasing heat of dissociation in the series ${\mathrm{N}}_{2}$-NO-${\mathrm{O}}_{2}$-${\mathrm{F}}_{2}$ is accounted for by the successive addition of promoted $3{p}^{p}$ electrons with strong negative bonding power. Starting from ${\mathrm{N}}_{2}$, whose normal state corresponds to a $^{1}S$ configuration of closed shells, we add one $3{p}^{p}$ electron to give the $^{2}P$ normal state of NO and ${\mathrm{O}}_{2}^{+}$, two to give the $^{3}S$ normal state of ${\mathrm{O}}_{2}$, four to give a closed shell, ${(3{p}^{p})}^{4}$, which accounts for the $^{1}S$ normal state of ${\mathrm{F}}_{2}$.In ${\mathrm{N}}_{2}$ (probably also in ${\mathrm{O}}_{2}$ and the other homopolar molecules, but data are too few), band systems for which $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}\ensuremath{ e}1$ are notably lacking, thus giving support to Hund's predicted selection rule for homopolar molecules; in the analogous heteropolar molecule CO, many systems occur with $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}=0$, although they are probably weaker, as expected, than those for which $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}=\ifmmode\pm\else\textpm\fi{}1$. On account of this strict selection rule in ${\mathrm{N}}_{2}$, certain levels should be metastable, in particular the final level of the afterglow ($\ensuremath{\alpha}$) bands of active nitrogen. There is evidence for the existence of a strict selection rule $\ensuremath{\Delta}s=1$ in homopolar molecules.

Fields currents from points

Abstract: The laws governing the extraction of electrons from metals in high vacua by fields, first developed through experiments with crossed wires, then with fine wire cathodes discharging to cylindrical anodes, have been now found to hold throughout for field currents between points and planes. The theory needed for the quantitative determination of the potential gradients at points is here given, and critical gradients then determined experimentally. The generality of the linear relation between log i and the reciprocal of field-strength is experimentally established.

The Normal State of Helium

Abstract: An approximate wave function for normal helium is calculated, by using theoretically determined functions for the limiting cases of large and small $r'\mathrm{s}$ and interpolating between them. The charge density computed from this wave function is in good agreement with that found independently by Hartree. The diamagnetism of normal helium is calculated, and agrees with observation within the experimental error. The repulsive forces between two helium atoms are calculated by the method of Heitler and London, and the attractive Van der Waals forces are roughly estimated from Wang's results with hydrogen. The potential curve so found gives a "molecular diameter" in agreement with experiment, and the minimum of the curve leads to approximately correct density and boiling point for the liquid.

On the Quantum Theory of Electronic Impacts

Abstract: It is shown that the previous treatment of electronic collisions has been incomplete; the error consists in the neglect of terms in the solution which correspond to an interchange of the colliding electron with one of those in the atom. The corrected first order cross section for elastic collisions is evaluated by Dirac's method for atomic hydrogen and helium. The complete solution for hydrogen is set up by Born's method for hydrogen; it is shown that the elastic cross section becomes infinite, for low velocities, with the reciprocal of the velocity; it is further shown that the first order cross section reduces to that already obtained. For hydrogen this is a monotonically increasing function; for atoms with completely paired electrons the monotonic increase is broken by a minimum at velocities corresponding to about a volt; the higher the azimuthal quantum number of the paired valence electrons, the more marked the minimum, and the lower the voltage at which it occurs.

The Self Consistent Field and the Structure of Atoms

Abstract: The method proposed by Hartree for the solution of problems in atomic structure is examined as to its accuracy as a method of solving Schr\"odinger's equation. A wave function is set up at once from his method, and the matrix of the energy computed with respect to it. The non-diagonal terms are shown to be small, indicating that the function is a good approximation to a real solution. The energy levels are found by perturbation theory from this matrix, and are compared with the term values as found by Hartree. His values should be corrected for three reasons: he has neglected the fact that electron distributions are not really spherical; he has not considered the resonant interactions between electrons; and he has made an approximation which amounts to neglecting the polarization energy. The sizes of these corrections are estimated, and they are found to be of the order of the errors actually present in the numerical cases he has worked out.

The Recombination of Argon Ions and Electrons

Abstract: Afterglow spectrum in argon due to recombination.---The arc spectrum of argon is found to persist approximately 0.001 sec. after an arc of 0.4 amp. in pure argon at 0.5 mm pressure is cut off. Lines involving jumps from high energy levels are relatively much stronger in the afterglow than in the arc. When sodium vapor is present the $D$ lines are strong in the arc but absent in the afterglow showing that the electron speeds in the afterglow are too low to excite the spectrum of argon by direct electron impact. The highly excited argon atoms must therefore be produced by the recombination of ions and electrons. The presence of 0.001 mm of hydrogen does not affect the intensity of the afterglow so that the persistence of metastable atoms is not involved in the production of the afterglow.Effect of applied potentials on the afterglow.---Accelerating voltages of 3 to 10 volts quench the afterglow during the period of application of 0.001 sec. Retarding after-voltages up to 90 volts have practically no effect. Measurements with an intermittently connected exploring electrode show that the velocities of the electrons are increased by the applied accelerating voltage but unaffected by the retarding voltage. The quenching is apparently the result of the decreased probability of recombination because of the higher velocities of the electrons. When low accelerating voltages have been applied for 0.001 sec. the intensity of the afterglow and the conductivity of the arc space in the period immediately following this application are both greater than they would be had the voltage not been applied. The intensity of the afterglow is thus directly related to the concentration of positive ions and the quenching of the afterglow shown to be connected with a saving up of ions as it should be if the effect of the after-voltages is to prevent recombination.Ordinary arc spectra are obtained under conditions which are here most unfavorable to the occurrence of recombination. They are presumaly primarily excitation spectra. The use of intermittent discharges for obtaining recombination spectra and strong high series members is suggested.The measurements show positive ion concentrations in the afterglow of the order of ${10}^{12}$ per cc. The mean energy of the electrons is 0.4 volt. From the measured rate of change of the concentration of ions, there results a value of 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}10}$ for the coefficient of recombination. This value may be in error by a factor of 5 because of several unavoidable errors.

On Dielectric Constants and Magnetic Susceptibilities in the New Quantum Mechanics Part III-Application to Dia- and Paramagnetism

Abstract: 1.General mathematical theory.---Modifications are given in the general derivation of the Langevin-Debye formula by means of quantum mechanics published in part I which are required by the moment now being magnetic rather than electric. The main additions are the appearance of a diamagnetic term, and allowance for the fact that the magnetic moment consists of two parts arising respectively from orbital electronic motions and from the electrons' internal spins. These parts cannot always be treated as rigidly coupled to form a "permanent" resultant moment. The Hamiltonian function used for the internal spin is that of a spherical top.2.Diamagnetism.---The writer's previous claim that Pauli's formula for the diamagnetic susceptibility can be applied to molecules as well as atoms is shown to be invalid because of the anomalous fact that the square of the angular momentum, unlike the average angular momentum, does not vanish even in $S$ states when there is more than one nucleus. Pauli's formula is instead an upper limit to the diamagnetism in non-monatomic molecules and is a good approximation when the Schr\"odinger wave function has nearly as much symmetry as in an atomic $S$ state.3.Paramagnetism of atoms.---Limiting values for the paramagnetic susceptibilities of atoms are $\ensuremath{\chi}=N\frac{[4s(s+1)+k(k+1)]{\ensuremath{\beta}}^{2}}{3kT}$ and $\ensuremath{\chi}=[\frac{N{g}^{2}j(j+1){\ensuremath{\beta}}^{2}}{3kT}]+N\ensuremath{\alpha}$ where $\ensuremath{\beta}$ is the Bohr magneton and $\ensuremath{\alpha}$ is a constant. These two formulas are rigorously applicable when the multiplet intervals are respectively very small or very large compared to $\mathrm{kT}$, and are valid regardless of whether the magnetic field is strong enough to change the quantization by producing a Paschen-Back effect. The formula for small multiplets yields susceptibilities slightly different from those given by Laporte and Sommerfeld's expression for this case, and is much simpler, as they overlooked the contribution of the portion of the magnetic moment which is perpendicular to the invariable axis.4.Paramagnetism of molecules.---The susceptibility is calculated on the basis of the Hund theory of molecular quantization. Formulas are given applicable to his couplings of type (a) and type (b) provided in the former the multiplet intervals are either very large or very small compared to $\mathrm{kT}$. The experimental susceptibilities for ${\mathrm{O}}_{2}$ and Cl${\mathrm{O}}_{2}$ are in accord with the assumption that the normal states are respectively $^{3}S$ and $^{2}S$ terms. In the particular case of $S$ terms the numerical results are the same in the atomic and molecular formulas, but, unlike previous theories, it is not necessary to suppose the orbits are as freely oriented in molecules as in atoms. Polyatomic molecules may have lower paramagnetic susceptibilities than diatomic ones because the dissymmetry causes large fluctuations in electronic angular momentum.5.Paramagnetism of nitic oxide.---Spectroscopists have recently found that the normal states of the NO molecule are $^{2}P$ terms separated by 120.9 ${\mathrm{cm}}^{\ensuremath{-}1}$. This permits an unambigous calculation of the susceptibility of NO which agrees with the experimental value within 1.5 per cent. Deviations from Curie's law are calculated which result from the doublet interval being comparable with $\mathrm{kT}$. These deviations should be detectable experimentally if the susceptibility of NO could be measured over a wide temperature range.

Central Fields and Rydberg Formulas in Wave Mechanics

Abstract: The analogy in wave mechanics is set up for the non-penetrating part of the electron orbits on the older theory, in which the motion of the outer electron is like that in the field of a point charge. The Rydberg formula follows from this as in the older theory. The separate Rydberg series for the various multiplets also are explained. The method previously applied by the writer to the helium problem appears as a next approximation, in which the outer electron moves, not in a hydrogen-like field, but in a somewhat more accurate central field. It is shown that this method should give a good approximation to the energy levels, and also to the wave function, except at very small distances. The attempt to use this function as a starting point for perturbation calculations is shown, however, not to be justified, for the matrix of the Hamiltonian function $H$, which is used in the perturbation method, greatly magnifies the errors in the wave function. Suggestions are made as to the next step in improving the method, by increasing the accuracy of the wave function for small $r'\mathrm{s}$.

Interpretation of the Atmospheric Absorption Bands of Oxygen

Abstract: It is shown that the atmospheric oxygen absorption bands can be attributed to a $^{3}S\ensuremath{\rightarrow}^{1}S$ transition from the normal ($^{3}S$) to a metastable $^{1}S$ excited state of ${\mathrm{O}}_{2}$. This accounts for all the strong lines, and explains missing lines, without conflict with existing theory. Certain very weak series such as the ${A}^{\ensuremath{'}}$ band are, however, not yet explained. Of the three rotational levels for each value of ${j}_{k}$ in the $^{3}S$ normal state, the two for which $j={j}_{k}\ifmmode\pm\else\textpm\fi{}1$ show only a very small separation, which increases slowly with ${j}_{k}$, while the third is separated from the other two by an interval of about 2 wavenumbers which does not change with ${j}_{k}$ (cf. Fig. 2 and Table I). The $^{3}S$ and $^{1}S$ states involved in the atmospheric bands may perhaps be attributed both to the same electron configuration, in agreement with a suggestion made in a previous paper. If this is the case, it is likely that a metastable $^{1}D$ state derived from the same configuration also exists, and that infra-red atmospheric bands corresponding to the transition $^{3}S\ensuremath{\rightarrow}^{1}D$ should be found.

The Shape and Intensities of Infra-Red Absorption Lines

Abstract: An expression for the shape of an infra-red absorption line is developed on the basis that the principal factor in the broadening of a line is the limitation of the length of wave train a molecule may absorb due to its perturbation by thermal collisions. The shape of the line is accordingly found by expanding the finite wave train with a Fourier integral and then integrating over the distribution of lengths of wave train given by the kinetic theory of gases. The absorption coefficient as found in this way may be expressed to a high approximation by means of two damping curves involving the number of molecules per unit volume, the temperature, and $\ensuremath{\sigma}$ the effective diameter.To apply this result to the analysis of observed infra-red spectra allowance must be made for the low spectrometer resolution due largely to the wide slits employed. Two expressions are developed, holding for all but very weak lines, which relate the area under the absorption line $\mathrm{Abs}$, the minimum value of the transmission ${T}_{min}$ and the true intensity $\ensuremath{\alpha}$ with the slit width $a$, the cell length $l$, and the molecular constants. $\mathrm{Abs}=\frac{{[5.412\ensuremath{\alpha}n{\ensuremath{\sigma}}^{2}\mathrm{l}]}^{\frac{1}{2}}}{{[\ensuremath{\pi}hm]}^{\frac{1}{4}}}$ $\mathrm{Abs}=\ensuremath{-}2.42a {log}_{10} {T}_{min}$It is shown that these formulae are capable of interpreting the absorption lines of the infra-red spectrum of HCl observed by R. F. Paton and yield a value of 10.8\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ cm for $\ensuremath{\sigma}$. The meaning of $\ensuremath{\sigma}$, the distance to which two molecules may approach without altering each others phases, as well as the range of validity of the assumptions is discussed and a correction to the absorption area formula for faint lines is deduced. In connection with a consideration of the absorption measurements of HCl by Kemble and Bourgin, a computation is made of the effective moving charge of the molecule giving the value $\ensuremath{\epsilon}=(.199) 4.77\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ E.S.U.

The Physical Pendulum in Quantum Mechanics

Abstract: It is pointed out that the Mathieu functions of even order are the characteristic functions of the physical pendulum in the sense of Schrodinger’s wave mechanics. The relation of various properties of the functions, as known from purely analytical investigations of them, to the pendulum problem is discussed.

The Element of Time in the Photoelectric Effect

Abstract: Time of appearance and cessation of the photoelectric effect from a potassium hydride surface.---A method has been devised which has made possible the study of the time variation of the photoelectric emission from a metal surface illuminated by light flashes of ${10}^{\ensuremath{-}8}$ sec. duration. The experimental arrangement has also yielded information on the speed of operation of the Kerr cell electro-optical shutter described in earlier work and has made possible for the first time the observation of the steepness of wave fronts traveling along wires resulting from spark discharges. Photoelectric emission begins in less than 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}9}$ sec. after the beginning of the illumination of a potassium hydride surface. The light shutter closes less abruptly than it opens and the experimental observations indicate that the sum of the time required for the shutter to close plus the time during which the photoelectric emission persists after cessation of irradiation is less than ${10}^{\ensuremath{-}8}$ sec. A wave traveling along a wire resulting from the sudden change of potential of one end by a spark discharge is so steep that the time necessary for about half the wave-front to pass a point 6 meters along the wire is 4.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}9}$ sec. Theoretical considerations bearing on these results are discussed.

X-Ray Diffraction in Liquid Normal Paraffins

Abstract: X-ray diffraction as a function of angle in liquid pentane, hexane, heptane, octane, nonane, decane, dodecane, tetradecane and pentadecane.---In the ionization diffraction-angle curve, there was one diffraction peak only, instead of two as found by Stewart, Morrow and Skinner with the primary alcohols and normal fatty acids. The absence in the $n$-paraffins of the second peak, which, in the alcohols and fatty acids, indicated a longitudinal spacing, is explained qualitatively by consideration of the longitudinal molecular forces. These could not produce as uniform a longitudinal arrangement as occurs in the other compounds. The single peak was located at the same angle with each $n$-paraffin, excepting, at first, pentane and decane. Subsequent purification of the pentane produced no alteration in the peak position but when the sample was finally made synthetically, the peak came into coincidence with that of the other seven. The decane is believed to contain isomers. The lateral separation of molecules is approximately 4.6A and is essentially the same as in the liquids previously studied and referred to in this paper. In six of nine cases, the diffraction intensity does not decrease as one approaches 0\ifmmode^\circ\else\textdegree\fi{}. The significance is not clear though one is reminded of total reflection and also of refraction. If one assumes the C atom to occupy a length in the chain of 1.3A, as found by other observers of x-ray diffraction in solids, and the length of H, 1.0A, and if one assumes the volume occupied by a molecule to be the square of the lateral spacing times the molecular length, then the computed values of densities of the seven normal paraffins are correct to within less than 4 percent or a mean of 2 percent. Two additional but faint peaks in the diffraction intensity curves are found with pentadecane and tetradecane. The corresponding spacings are 2.1A and 1.23A. The nearness of these to the repetitive values of 2.0A and 1.26A which are found in a diamond, is suggestive.

The Internal Pressure of Pure and Mixed Liquids

Abstract: Measurement of internal pressures. Defining internal pressure as $T{(\frac{\ensuremath{\partial}p}{\ensuremath{\partial}T})}_{v}$, designated $T\ensuremath{\gamma}$, it can be measured on the principle of a constant volume thermometer. This was done for 8 pure liquids and 12 mixtures of 50 mole percent composition. Values for the former are given at 15\ifmmode^\circ\else\textdegree\fi{}, 20\ifmmode^\circ\else\textdegree\fi{}, 25\ifmmode^\circ\else\textdegree\fi{} and 35\ifmmode^\circ\else\textdegree\fi{}, and for the latter at 20\ifmmode^\circ\else\textdegree\fi{}, 25\ifmmode^\circ\else\textdegree\fi{} and 35\ifmmode^\circ\else\textdegree\fi{}. The values of $\ensuremath{\gamma}$ in atmospheres per degree at 20\ifmmode^\circ\else\textdegree\fi{} are as follows: 1 heptane, 8.66; 2 acetone, 11.22; 3 carbon tetrachloride, 11.47; 4 benzene, 12.58; 5 carbon disulfide, 12.67; 6 ethylene chloride, 14.17; 7 ethylene bromide, 15.20; 8 bromoform, 15.32. The values of $\ensuremath{\gamma}$ for the mixtures at 20\ifmmode^\circ\else\textdegree\fi{} are, designating the components by the foregoing numbers, 1-2, 9.27; 1-3, 9.68; 1-4, 9.86; 1-5, 9.84; 1-7, 10.66; 1-8, 10.86; 3-4, 12.19; 4-5, 12.32; 4-6, 12.92; 4-8, 14.05; 2-5, 11.77; 5-7, 13.98.Relations satisfied by ${(\frac{\ensuremath{\partial}p}{\ensuremath{\partial}T})}_{v}$.---It was found, further, (1) that $\ensuremath{\gamma}$ is a function of the specific or molal volume only; (2) that for each pure liquid ${\mathrm{v}}^{2}{T}_{1}$ $\ensuremath{\gamma}$ is a constant, $a$, v being the molal volume, and ${T}_{1}$ the temperature at which the pressure is 1 atmosphere (a function of molal volume); (3) that the values for the mixtures are less than additive, less than those calculated from the equation of Biron, $\ensuremath{\gamma}=\frac{{\ensuremath{\gamma}}_{1}{\ensuremath{\gamma}}_{2}}{({\ensuremath{\gamma}}_{1}{\mathrm{N}}_{1}+{\ensuremath{\gamma}}_{2}{\mathrm{N}}_{2})}$ where ${\mathrm{N}}_{1}$ and ${\mathrm{N}}_{2}$ are the mole fractions of the components, here 0.5; (4) that they are given within 1 or 2 percent by the relation $a={({a}_{1}{a}_{2})}^{\frac{1}{2}}$ except in some of the mixtures of carbon disulfide, acetone and ethylene chloride---the first of these is in other respects irregular, and the last two are polar; and (5) that a still better agreement is given by considering that $\ensuremath{\gamma}\mathrm{v}$ is additive.Indirect determination of compressibility.---The compressibility, $\ensuremath{\beta}$, can be calculated by combining $\ensuremath{\gamma}$ with the coefficient of expansion, $\ensuremath{\alpha}=\ensuremath{\beta}\ensuremath{\gamma}$. In most instances, the agreement with the directly determined values is satisfactory.

Application of the Fermi Statistics to the Distribution of Electrons Under Fields in Metals and the Theory of Electrocapillarity

Abstract: It is assumed that each atom in mercury is ionized into a positive ion and an electron. Because of the crowded state of the positive ions it is supposed that they cannot move in an electric field, while, following Sommerfeld and Pauli, the electrons are assumed to act like a completely degenerate gas following the Fermi statistics. The distribution of electrons under an electric field due to a charge on the surface of the metal is discussed, and a relation derived which gives the charge on the surface in terms of the potential difference between surface and interior. To a first approximation the charge and potential difference are proportional to each other, as if there were a condenser of constant capacity at the surface. In order to find the capacity an estimate must be made of the dielectric constant of the mercurous ions of the mercury. This is done with the aid of measurements of the refractive index of mercurous ions. The magnitude of the equivalent capacity is such that, when considered in conjunction with the diffuse layer of ions in the solution, electrocapillary curves can be explained.