Table of contents

A new method is developed for deriving high-temperature series expansions of the partition function for the Heisenberg model which avoids evaluating traces of products of non-commuting operators. The method is based on a rearrangement of the series as a sum of partition functions of finite clusters of steadily increasing order. The latter are then evaluated on a computer as traces of finite matrices. The calculations have been confined to spin ½. For loose-packed lattices, two new terms have been added to the zero-field series for the partition function and three new terms to the zero-field series for magnetic susceptibility.

It is shown that the van der Waals force constant may be computed accurately and directly from oscillator strength sums S(k). In particular, for pairs of the atoms H, He, Ne, Ar, Kr and Xe the van der Waals force constant can be obtained to within an error of about 4% from accurate values of S(-2), S(-4) and S(-6), each of which may be derived from accurate refractive index measurements.

The differential cross sections for the elastic scattering of electrons by helium atoms atoms at a few selected electron impact energies, in the 10-50 eV energy range, are calculated by using the adiabatic-exchange phase shifts of LaBhan and Callaway for the l = 0, 1 and 2 partial waves, and the static-dipole phase shifts for the higher partial waves. The calculated values are in satisfactory accord with the experimental data of Bullard and Massey and of Hughes et al., and with the theoretical values of Khare and Moiseiwitsch, who included exchange only indirectly. The effect of quadrupole polarizability potential on the differential cross sections is also examined and is found to be small.

The lifetimes of the 316.5, 612.4 and 920.8 kev levels of ¹⁹²Pt, produced in the decay of ¹⁹²Ir, have been studied by the method of K electron-K electron coincidence centroid shift.

The mean life of the 612.4 kev level has been measured as (3.4 ± 1.9) × 10^-11sec and upper limits of 4.2 × 10^-11 and 3.5 × 10^-11 sec have been placed on the mean lives of the 316.5 and 920.8 kev levels respectively.

The probability distributions of intensity for thermal and ideal laser beams are derived directly from the Bose-Einstein distribution for photons in phase space, using the correspondence principle. Thus the approach of Kastler is shown to yield the same result for the intensity distribution for an ideal laser as the work of Mandel.

The possibility of producing dense high temperature plasmas by heating magnetically contained plasma with a pulsed laser beam is considered. The process requires high ion densities (> 10¹⁷ cm^-3) to be maintained during the heating period, and the magnetic field strength required to contain such a plasma has been calculated in terms of both the kinetic pressure and Bohm diffusion of the plasma. Radiation and conduction losses from the plasma are also considered, and it is concluded that temperatures of the order of 10⁶ °K should be obtainable within present-day limits of field strength and photon flux.

To attain the conditions necessary for a fusion reactor with net power output, using a hydrogen isotope, laser energies greater than 10⁴ J for a single passage of the radiation through the plasma and magnetic field strengths greater than 10⁸ G are required if the plasma is formed from a hydrogen pellet in vacuo. The magnetic field strength has been calculated on the assumption that Bohm diffusion will be the most severe plasma loss mechanism. If the plasma is formed in a gas, a total laser energy of greater than 10¹¹ J is required at a power greater than 10¹⁸ W.

Any possible future development of a high-powered ultra-violet laser will make the use of high magnetic fields unnecessary provided sufficiently short laser pulses can be obtained. In this case there is no difference between the results obtained for the vacuum- or for the gas-encased model. In each case laser energies greater than 10⁶ J at powers greater than 10¹⁵ W are required. The vacuum-encased model, with strong magnetic field, is identical with that at ruby laser frequencies.

Recent measurements of the gyromagnetic ratio of the proton have yielded results which indicate that there may be a field dependence of the magnetic moment of the proton. An attempt has been made to observe such a dependence; although none was found, a method of comparing nuclear magnetic resonance frequencies with a precision of better than 1 part in 10⁶ is described. The ratio of the proton resonance frequency to the resonance frequencies of ³H, ⁷Li, ¹¹B ¹⁹F have been measured to a precision which is in some cases as high as 1 part in 10⁷.

A detailed examination of the recent available data for dipole and non-dipole transitions in the K, L and M regions of spectra over the entire region of elements revealed the existence of a large number of new screening doublets. These, as well as those reported earlier by Sandström and by Deodhar and Padalia, have been shown to fall under one of five categories based on the nature of the initial and final levels of the constituents of a doublet.

The analysis has revealed for the first time that in x-ray emission spectra there exist many screening doublets for which both the constituents are dipole lines. Mathematical interpretation for one of the screening doublets, (L_IM_I, L_IIM_II), has been given.

The shape of the soft X-ray emission spectrum of lithium has been calculated in the Hartree-Fock approximation taking into account the fact that an atom about to emit a K X-ray is a point defect in the otherwise periodic potential of the lattice. The perturbation from the 1s hole was treated in the `localized approximation' of the Koster-Slater theory, its effect being expressed by a single parameter. Starting from Ham's calculated density-of-states function and gradually increasing the strength of the perturbation one observes the formation of a resonance state (virtual bound state) and then a real bound state. For reasonable values of the perturbation parameter the results show a peak about 1 ev below the Fermi energy, a feature which resembles the measurements of Crisp and Williams and others. The calculations also provide a plausible explanation of the measurements of Catterall and Trotter on the satellite K emission spectrum of lithium.

The theory of Koster and Slater for the electronic states around an impurity in a metal is formulated in terms of the Wannier functions of the host lattice. These functions are not uniquely defined, however, because they depend on how the arbitrary phase factors exp{iα(n, k)} associated with the Bloch functions are chosen. In this paper it is shown that the frequently used `localized approximation' of the Koster-Slater theory has the effect of fixing the relative phases of the Bloch functions, and consequently in this approximation the Wannier functions are uniquely defined. In the second part of the paper an approximation is described, valid in the region close to the impurity nucleus, which leads to expressions in terms of matrix elements of Bloch functions rather than Wannier functions. For two examples, the Knight shift at impurity atoms and the x-ray emission intensity, results are obtained which seem to be more acceptable physically than those previously derived.

The intensities of muon and electron neutrinos at ground level have been calculated for vertical and horizontal directions and the muon-neutrino spectrum has been found for intermediate angles. The calculations have been made for what is regarded as the most likely model of the propagation of the components of the cosmic radiation through the atmosphere, and the dispersion in the neutrino spectrum arising from uncertainties in the model has been examined.

The intensity of muon neutrinos is found to be greater in the horizontal than in the vertical direction, the ratio increasing with energy from 1.43 at 1 GeV through 2.12 at 10 GeV to 3.70 at 1000 GeV.

The uncertainty in the horizontal intensity of muon neutrinos is less than that in the vertical direction, typical values being⁺⁸_-11% at 10 GeV and ¹²_-16% at 100 GeV in the horizontal direction and ⁺¹⁷_-22% at 10 GeV and ⁺³³_-43% at 100 GeV in the vertical direction.

The phonon spectrum of solid argon has been observed directly by studying the weak impurity-induced absorption spectrum of argon containing up to 2% of neutral impurity (krypton or xenon). As the temperature of the experiment is lowered the results are in increasing agreement with the form of the calculated spectrum, modified by a frequency-dependent dipole moment.

A fairly complete rotational analysis has been made of a number of bands of the main system A-X of TeO using two separated tellurium isotopes. A new vibrational analysis is proposed and is confirmed by the isotope effect. The structure of the bands indicates that Hund's case c applies to both states, so that the transition is better described as A0⁺-X0⁺, A1-X1, rather than A³Σ^--X³Σ^-. Transitions 1-0⁺, 0⁺-1 have not been observed. Derived constants, in cm^-1, for ¹²⁸TeO are as follows:

State	T00	ωe	xeωe
A1	27641.83 +x	-	-
A0+	28036.48	ΔG1/2 = 445.25
X1	x	798.70	4.00
X0+	0	797.69	4.00
	x ≃ 2750 cm-1
	B	103α	r (Å)
1+	B0 = 0.2779±0.0002	-
A1-	B0 = 0.2775±0.0002	-	r0 = 2.066
A0+	B0 = 0.2735±0.0001	5.50 ±0.03	r0 = 2.082
1+	Be = 0.3575±0.0003	2.379±0.003
X1-	Be = 0.3572±0.0003	2.368±0.003	re = 1.821
X0+	Be = 0.3560±0.0002	2.375±0.004	re = 1.825

The upper state is perturbed, and is also predissociated in A0⁺ at ν = 9. The dissociation energy D₀" is estimated to be about 90 kcal mole^-1.

The spectra in the range 300 Å to 1400 Å, obtained from the vacuum sliding spark, indicate that, at ringing frequencies of 50 kc/s and lower when using tungsten, aluminium or steel electrodes, the insulator is the main source of ions in the discharge. High intensity vacuum ultra-violet radiation is produced if the insulator is quartz, porcelain or alumina, but if the material is easily decomposed only long wavelength radiation is produced. The effect of the electrode material can be qualitatively explained in terms of the stability and vapour pressure of the oxide, and depends to a certain extent on the sputtering rates. With aluminium, tungsten and steel the oxides are stable and non-volatile, and very low concentrations of the metallic ions are produced. With graphite carbon rapidly enters the discharge producing mainly an ionized carbon spectrum, and with brass a great deal of material enters the discharge and produces conditions which give only long wavelength radiation. At higher ringing frequencies there is some evidence for the existence of a wavelength continuum.

A microwave cavity method has been used to measure the electron loss rate in the afterglows of pulsed electrodeless discharges in oxygen and oxygen-nitrogen mixtures. The temperature of the gas was varied from 296 to 870 °K. At higher temperatures the gas pressures had to be increased (up to 114 mmHg at 753 °K) in order to reduce diffusion losses. In pure oxygen and at gas temperatures of 296 and 368 °K a departure from an apparent two-body attachment process was observed as the gas pressure was increased. The measured value of the attachment cross section Q = v_a/nv was found to be strongly dependent on gas temperature. It decreased rapidly as the gas temperature was increased from 368 to 753 °K and this is attributed to enhancement of the electron detachment process in this temperature range. Measurements in oxygen-nitrogen mixtures yielded a value of 2.3 × 10^-30 cm⁶ sec^-1 for the three-body attachment coefficient, in good agreement with the drift tube data, and value of 8.4 × 10^-15 cm³ sec^-1 for the detachment coefficient at a gas temperature of 296 °K.

The Boltzmann transport equation including an additional term for relaxation through e-e collisions has been solved for a high frequency electric field and a constant magnetic field perpendicular to it. The real and imaginary parts of the changes of resistance and Hall coefficient with magnetic field are obtained as functions of the ratio of the mean time of collisions of electrons with lattice vibrations or impurity ions to the time of relaxation through e-e collisions alone. The changes of the imaginary parts should provide a more sensitive method for investigation of the e-e collision process.

Diffuse magnetic scattering of neutrons is observed around the magnetic Bragg peaks and at the (100) position in face-centred cubic MnPd alloys. Results for the alloy containing 15 at.% Pd, which has a Néel temperature of 347 °K, are presented in detail. The diffuse scattering is discussed in terms of static short-range magnetic order, small particle size line broadening, and inelastic magnetic scattering. It is shown that the inelastic scattering model is consistent with the scattering observed. From the magnitude of the scattering an estimate is made of the `exchange energy'.

A new method is proposed for examining the dependence on the magnetization of the free energy for electrons at finite temperatures in the band model. By this method the conditions for ferromagnetism for electrons in metals with an arbitrarily complicated density of states are discussed. It is pointed out that a first-order transition in the magnetization may occur both at low temperature and at high temperatures in a case where the Fermi level is in the neighbourhood of a minimum in the density of states curve. The comparison between the theory and several experimental results which show a first-order transition, for example MnAs, is discussed.

Measurements have been made from 2 to 30 °K and 60 to 90 °K of the thermal expansion of iron, cobalt and nickel, which show that the Grüneisen parameter γ_l for the lattice decreases slightly at low temperatures and that the electronic parameter γ_e (calculated from the ratio of the electronic contributions to expansion and heat capacity) is about 2.2. Similar measurements below 30 °K on α-manganese, chromium, face-centred cubic iron-nickel (35-50% Ni) alloys, and Fe_{0 5} Mn_{0 5} indicate a negative term, proportional to T, in the expansion coefficient. This is particularly large in Invar and in α-manganese. For γ-manganese (+15% Cu) no negative term is observed in the expansion. Results are discussed in terms of electronic and magnetic contributions.

With the object of estimating dislocation densities and orientations in NaCl, a series of single crystals was prepared with up to 10% plastic deformation along the [001] axis. The ²³Na nuclear magnetic resonance was examined at room temperature using continuous wave, free induction decay and spin-echo methods. It is shown that using the spin-echo technique it is possible to separate out the quadrupole broadening from the magnetic dipole broadening and also to distinguish between the dipole broadening due to the central component of resonance and that due to the satellites. Using the continuous wave and free decay techniques this is not possible. Unfortunately the poor signal to noise ratio obtainable in the present experiments prevented a detailed analysis of the measurements, but an approximate treatment yielded useful results. Dislocation densities were obtained for the different crystal samples.

The frequencies of the latrice vibrations of metallic sodium have been calculated by Toya's method at 1661 points within the irreducible sector of the first Brillouin zone. From these the vibrational frequency distribution has been obtained and this shows good agreement with the distribution obtained by Dixon et al. by fitting force constants to the results of neutron scattering experiments. The distribution has been used to obtain values of the specific heat.

A simple quasi-lattice model, in which no pair of neighbouring sites may be occupied, can be solved exactly. The `lattice' consists of triangles of sites connected at corners, and can be split into three equal sublattices, one of which is preferentially occupied when the overall density exceeds a certain value. This model reproduces the behaviour of the continuum rigid-sphere model much more closely than do lattice models possessing only two sublattices. There is more than one configuration that makes the free energy a local minimum, and we have to select the most stable one. When this is done, the transition is first order, and the isotherm consists of two distinct portions, closely resembling what is obtained for the rigid-sphere model.

A new approximation to the theory of a classical fluid of number density ρ in equilibrium at absolute temperature T is investigated. Although the model takes some account of the elementary diagrams, it is nevertheless such that all clusters retained in the number-density expansions of the total correlation function h(r) and the direct correlation function c(r) are of simple hyper-netted chain type. The approximation, which is defined by the integral equation h(r₁₂) - c(r₁₂) = y^m(r₁₂) ln y (r₁₂) = ρ∫c(r₁₃)h(r₂₃)dr₃ where y(r) = {1 + h(r)} exp{u(r)/kT} and u(r) is the interaction potential between two atoms at distance r apart, reduces to the hyper-netted chain approximation when m = 0.

Cluster-integral expressions have been obtained for all the `compressibility' and `pressure' virial coefficients up to and including F_c and F_p, and these have been evaluated in the particular case of a hard-sphere gas. When the adjustable parameter m is so chosen that D_c = D_p, the numerical estimates of the virial coefficients E_c and F_c are found to be identical with the true values of the fifth and sixth virial coefficients, respectively, of the hard-sphere gas. In addition, for parameter values in the range 0.4 ⩽ m ⩽ 0.45 the present model approaches overall internal consistency much more closely than is the case with any previously studied approximation of comparable simplicity.

Using some of the basic ideas of information theory, an approximate expression is derived for the contribution to the entropy of a real monatomic liquid arising from the bond-length variation between nearest neighbouring atoms. Some further simplifying approximations enable an equation of state to be derived. This is compared with experimental results for liquid argon. When the equation of state is adjusted to fit one experimental isotherm, the calculated values of C_P are found to lie within 20% of the experimental values.

An interferometric method for measuring the modulus and phase of the optical transfer function is described. A simple lens having seven wavelengths of primary coma, and negligible values of other aberrations, has been designed for use in monochromatic light. Using this lens, the computed modulus and phase of the transfer function have been measured for amounts of primary coma equal to 0.63, 1.26, 2.0 and 3.5 wavelengths. Good agreement with theory is obtained.

It is predicted that superconductive semimetals or semiconductors should have a further phase transition in very strong magnetic fields, characterized by vanishing electric resistivity in the direction of the magnetic field.

The effect of biquadratic exchange on the transition temperature of a ferromagnet or antiferromagnet is calculated using a classical Bethe-Peierls-Weiss method. The effect on the Curie-Weiss constant θ is also given.

The long-range interaction energy between two normal hydrogen atoms has been calculated directly, without recourse to laborious variational methods. The result E₂(R) = -12 998R^-6-248.80R^-8-6571R^-10-... confirms that of Chan and Dalgarno as opposed to that of Hirschfelder and Löwdin.

It is shown from first principles that the phase difference between two waves is independent of the motion of the observer. The different errors in two earlier treatments of the problem by Fürth are pointed out.

Recently Fürth proposed an experiment to test the theorem of the relativity of simultaneity of events. Using this theorem he shows that the proposed experiment will result in a measurable phase shift. He neglects, however, the Lorentz transformation for spatial coordinates, and it is shown here that when this transformation is included the phase shift will actually be zero.

A discrepancy in the analysis of Corinaldesi and Trainor is pointed out, and it is shown that the correct procedure leads to a non-vanishing expression for the exchange scattering amplitude given by the Born-Oppenheimer approximation for excitation of the m = ± 1 levels of the 2p state of atomic hydrogen by electron impact. This expression is employed to correct earlier calculations of 1s-2p excitation total cross section.