• N S Nagendra Nath

Articles written in Proceedings – Section A

• The normal vibrations of molecules having octahedral symmetry

• The dynamical theory of the diamond lattice. I

(1) The theoretical calculations of the characteristic frequency of diamond and the experimental results of the vibration spectra of diamond are reviewed in brief; (Section 1).

(2) The vibratory equations of motion of the diamond lattice under the three forces’ system and the suitable type of intra-valence forces are formulated. A typical factor of the secular equation is then obtained and it is found that it corresponds to the vibration of the two component cubic facecentred lattices composing the diamond lattice relative to one another, each being considered as rigid. The said vibration is triply degenerate. It appears that this vibration is Raman-active but optically inactive since all the atoms are homonuclear. Hence an explanation for the origin of the principal Raman line with the shift 1332 cm.−1 and the absence of the infra-red band corresponding to 1332 cm.−1 follows.

• The dynamical theory of the diamond lattice - Part II. The elastic constants of diamond

• The dynamical theory of the diamond lattice - Part III. The diamond-graphite transformation

The transformation of the diamond structure to the graphite structure is explained in terms of three elementary operations, one of which is a definite displacement of the two cubic face-centred lattices of diamond relative to one another. It is shown here that for a certain displacement of the component lattices, diamond attains maximum energy of its configuration and becomes unstable. The temperature at which diamond becomes unstable and transforms to graphite is calculated and is shown to be in good agreement with the experimental determinations.

The ideas followed up here seem to be not very special to the diamond-graphite transformation itself but can have extensions for similar types of transformation of other substances.

• The diffraction of light by sound waves of high frequency: Part II

The theory of the diffraction of light by sound waves of high frequency developed in our earlier paper is extended to the case when the light beam is incident at an angle to the sound wave-fronts, both from a geometrical point of view and an analytical one. It is found that the maxima of intensity of the diffracted light occur in directions which make definite angles, denoted byϑ, with the direction of the incident light given by$$\sin (\theta + \phi ) - sin \phi = \pm {{n\lambda } \over {\lambda * }}, n (an integer) \ge 0$$ whereλ andλ* are the wave-lengths of the incident light and the sound waves in the medium. The relative intensity of themth order to thenth order is given by$$J_m ^2 \left( {v_0 \sec \phi {{\sin t} \over t}} \right) / J_n ^2 \left( {v_0 \sec \phi {{\sin t} \over t}} \right)$$ wherev0=2πµL /λ,t=πL tanφ /λ*,φ is the inclination of the incident beam of light to the sound waves,μ is the maximum variation of the refractive index in the medium when the sound waves are present and L secφ is the distance of the light path in the medium. These results explain the variations of the intensity among the various orders noticed by Debye and Sears for variations ofφ in a very gratifying manner.

• The diffraction of light by high frequency sound waves: Part III - Doppler effect and coherence phenomena

The theory developed in Part I of this series of papers has been developed in this paper to find the Doppler effects in the diffraction components of light produced by the passage of light through a medium containing (1) a progressive supersonic wave and (2) a standing supersonic wave.

In the case of the former the theory shows that the nth order which is inclined at an angle$$\sin ^{ - 1} \left( { - \begin{array}{*{20}c} {n\lambda } \\ {\lambda *} \\ \end{array} } \right)$$ to the direction of the propagation of the incident light has the frequencyv – nv* wherev is the frequency of light,v* is the frequency of sound andn is a positive or negative integer and that thenth order has the relative intensity$$Jn^2 \left( {\frac{{2\pi \mu L}}{\lambda }} \right)$$ where μ is the maximum variation of the refractive index, L is the distance between the faces of the cell of incidence and emergence and λ is the wave-length of light.

In the case of a standing supersonic wave, the diffraction orders could be classed into two groups, one containing the even orders and the other odd orders; any even order, say 2n, contains radiations with frequenciesv ± 2rv* wherer is an integer including zero, the relative intensity of thev ± 2rv* sub-component being$$J^2 n - r\left( {\frac{{\pi \mu L}}{\lambda }} \right)J^2 n + r\left( {\frac{{\pi \mu L}}{\lambda }} \right)$$; any odd order, say 2n + 1, contains radiations with frequencies$$v \pm \overline {2r + 1} v*$$, the relative intensity of the$$v \pm \overline {2r + 1} v*$$ sub-component being$$J^2 n - r\left( {\frac{{\pi \mu L}}{\lambda }} \right)J^2 n + r + 1\left( {\frac{{\pi \mu L}}{\lambda }} \right)$$. These results satisfactorily interpret the recent results of Bar that any two odd orders or even ones partly cohere while an odd one and an even one are incoherent.

• The diffraction of light by high frequency sound waves: Part IV - Generalised theory

The essential idea that the phenomenen of the diffraction of light by high frequency sound waves depends on the corrugated nature of the transmitted wave-front of light, pointed out by the authors in their first paper, has been developed on general considerations in this paper. The results in this paper can be summarised as follows:-

If progressive sound-waves travel in a rectangular medium normal to two faces and the direction of propagation of a plane beam of incident light, the incident light will be diffracted at the angles given by sin-1(-nλ/λ*) and the light belonging to the nth order will have the frequencyv–nv*.

If the sound waves are stationary, the incident light will be diffracted at the angles given by sin-1(-nλ/λ*), an even order would contain radiations with frequencies,v,v±2v*,v±4v*,....,v±2rv*,...., and an odd order would contain radiations with frequenciesv±v*,v±3v*,v±5v*,....,$$v \pm \overline {2r + 1} v*$$,.....

A differential-difference equation has been obtained for the amplitude function of the diffracted orders whose approximate solution is satisfied by the Bessel Functions already obtained by the authors in their previous papers.

• The neutrino theory of light

The standpoint assumed by Jordan is the following. Before the radioactive processes revealed the probability of the existence of the neutrinos, the only experimentally known wave fields were those which appear now as pure light fields. They turn out to be only a limiting case of a very much larger multitude of possible fields, which contain free (not compensated) neutrinos. The validity of this hypothesis could be experimentally tested by a study of a possible influence of radiation fields on the β-decay. If it is true that the β-emission is accompanied by an emission of a neutrino there should be an influence of an external neutrino field on the β-emission. But since light fields are nothing than neutrino fields (with neutrino pairs) we should also expect an influence of light radiation on the β-decay. The law of this interaction has yet to be calculated.

We wish to make another remark. Quantum mechanics was started by replacing the Fourier amplitudesqke2πivokt of co-ordinate functionq(t) by matrix elements with two indieesqkle2πivklt. In analogy, one could expect that in a quantum field theory the Fourier elementsqke2πivok(t-x/c) of a quantity representing a progressive waveq(t — x/c) should be replaced by matrix elementsqkle2πivkl(t-x/c). Here, as in quantum mechanics, one should expect the combination law v + vlm = vkm.

• The diffraction of light by high frequency sound waves: Part IV - Generalised theory

The essential idea that the phenomenon of the diffraction of light by high frequency sound waves depends on the corrugated nature of the transmitted wave-front of light, pointed out by the authors in their first paper, has been developed on general considerations in this paper. The results in this paper can be summarized as follows:—

If progressive sound-waves travel in a rectangular medium normal to two faces and the direction of propagation of a plane beam of incident light, the incident light will be diffracted at the angles given by sin−1 (−nλ/λ*) and the light belonging to thenth order will have the frequencyv–nv*.

If the sound waves are stationary, the incident light will be diffracted at the angles given by sin−1 (−nλ/λ*), an even order would contain radiations with frequencies,ν, ν ± 2ν*,ν ± 4ν*,.,ν ± 2*,., and an odd order would contain radiations with frequenciesν ±ν*,ν ± 3ν*,ν ± 5ν*,.,ν ± √2r+1ν*,..

A differential-difference equation has been obtained for the amplitude function of the diffracted orders whose approximate solution is satisfied by the Bessel Functions already obtained by the authors in their previous papers.

• The neutrino theory of light

• Neutrinos and light quanta

It is found necessary in this paper to introduce the spin of the neutrino in the neutrino theory of light in order to obtain two photon operators for each energy state of photons and derive the Planck formula. The derivation by Jordan without the idea of spin for the neutrino is not free from objection for he obtains half the Planck formula for the radiation density for his photons have no polarisation states. This difficulty disappears in our derivation of the Planck formula for we obtain two photon operators for each energy state of photons. These two operators are connected with the polarisation states of light.

The author is highly grateful to Prof. Dr. Max Born for the very valuable discussions during his stay here and for his very kind correspondence. He is also grateful to Professor C. V. Raman for his interest in this work.

• The diffraction of light by high frequency sound waves: Part V - General considerations—oblique incidence and amplitude changes

The essential idea that the phenomenon of the diffraction of light by high frequency sound waves depends on the corrugated nature of the transmitted wave-front of light has been developed on general considerations in this paper to apply for the case of the oblique incidence of the incident light to the sound waves. It is found that the intensity distribution will hot be symmetrical in general thus explaining the results of Debye and Sears, Lucas and Biquard, Bär and Parthasarathy. The consideration of the amplitude changes of the traversing beam of light explains the results of Hiedemann, Bär and Lucas.

We are highly thankful to Prof. Dr. R. Bär of Zürich for having kindly sent us a copy of the proof of a paper by him describing experimental tests of our theory which is now in course of publication in theHelvetica Physica Acta.

• The diffraction of light by high frequency sound waves: Generalised theory - The asymmetry of the diffraction phenomena at oblique incidence

• The visibility of ultrasonic waves and its periodic variations

• The neutrino theory of light—II

• The diffraction of light by supersonic waves

• Quantum theory of X-ray reflection and scattering - Part I. Geometric relations

When X-rays fall upon a crystal, the characteristic vibrations of the crystal lattice may be excited thereby, in much the same way as in the phenomenon of the scattering of light in crystals with diminished frequencey, the excitation being a quantum mechanical effect. From the equations for the conservation of energy and momentum, the geometrical relations entering in this effect are deduced theoretically for the two cases in which the lattice vibrations fall within (1) the acoustic range of frequency and (2) the optical range. In the first case, the incident X-rays are scattered in directions falling within, a cone having the incident ray as axis and with a semi-vertical angle 2 sin−1 λ 2γ* whereγ* is the minimum acoustical wave-length. In the second case, we have a quantum-mechanical reflection of the X-rays with diminished frequency in a direction which generally follows the geometric formula 2d sin 1/2(θ+ϕ)=nλ where θ and ϕ are the glancing angles of incidence and reflection on the crystal spacings. For crystals with specially rigid bindings, the alternative fomulad sin (θ+ϕ)=nλ cosϕ is indicated as being more appropriate. In either case, the intensity of the reflection should fall off rapidly as ϕ and ϕ diverge.

• The two types of X-ray reflection in crystals

• Errata

• The intensities of the Raman lines in carbon dioxide

The polarisability of a molecule is assumed to be made up of the bond polarisabilities as functions of the inter-nuclear distances. It is pointed out that certain normal co-oridnates could be quadratic functions of some of the variations in the inter-nuclear distances which fact accounts for the appearance of overtone Raman lines assuming only first order variations in the bond polarisabilities. These ideas have been applied to the case of CO2 and the intensity ratio of the Fermi split lines has been calculated which is in good accord wih experimental observations.

• Errata

• The vibrations of an infinite linear lattice consisting of two types of particles

The solutions for the displacements of an infinite linear lattice consisting of two types of particles under some initial disturbance have been obtained and their asymptotic nature has been investigated by the method of “steepest descents” which shows that the displacements are asymptotically time dependent superposition of the normal vibrations according to Raman.

• Vibrations of an infinite linear lattice consisting of two types of particles—II

• # Proceedings – Section A

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