• G N Ramachandran

Articles written in Proceedings – Section A

• Reflection of light by a periodically stratified medium

• On the transmission of light through a cloud of randomly distributed particles

The phenomenon of the transmission of light through a cloud of particles distributed at random in a transparent medium is theoretically investigated on the basis of wave-optics. The cases of both transparent and of opaque particles are considered, and it is found that the transmitted beam is progressively attenuated, the intensity diminishing exponentially with the increase in the thickness of the medium. The actual values of the attenuation coefficient are calculated in both cases. It is found that for an opaque particle, the diminution of intensity in the forward direction is actually double what would be expected from simple geometric considerations. For transparent particles, the transmitted intensity shows spectral variations, and this explains some of the phenomena hitherto not well understood, such as the colours shown by the transmitted light, and the complementary nature of this colour and the colour of the light scattered by the cloud in the forward direction.

• The theory of coronae and of iridescent clouds

The commonly accepted theory of coronae, based on the idea that water droplets act as opaque disks, is not only theoretically unsound, but is not also in accord with experimental facts. The theory due to Mecke is also not satisfactory as it based partly on geometrical optics and partly on the theory of diffraction. In this paper, a new theory of the phenomenon is developed, using only the principles of wave-optics, and taking into consideration the portions of the wave-front transmitted through the droplets. The integrals so obtained are integrated by a suitable method, and expressions are obtained for the intensity distribution in the corona. A discussion of these expressions shows that the theory satisfactorily explains most of the phenomena exhibited by coronae, such as the extreme susceptibility of the ring system to even small changes in the radius of the drops, and the oscillation of the ring system as the radius steadily increases.

• Optical theory of chromatic emulsions and of the Christiansen experiment

A theory of the optical phenomena exhibited by chromatic emulsions as also those observed in the Christiansen experiment, has been worked out,de novo, on the basis of the diffraction of light by a sphere immersed in a medium of nearly the same refractive index. Expressions are derived both for the intensity of the transmitted light, and of the light diffracted in other directions. These are discussed in relation to the intensity and the spectral nature of the light and it is shown that the theory can satisfactorily account for the various phenomena observed by Sethi and Sogani.

• Fluctuations of light intensity in coronæ formed by diffraction

• Modes of atomic vibration in the fourteen Bravais lattices

In this paper, the degeneracies and the directions of the normal modes of vibration in the fourteen Bravais lattices are worked out from symmetry considerations, making use of Sir C. V. Raman’s theory. Denoting the ratio of the displacements of adjacent atoms along the primitive translations by α, β, γ, these can have only the values + 1 or −1, so that the vibrations fall into eight types in the general case of no symmetry. In lattices possessing symmetry, however, some of the types could be brought over into others by the application of symmetry operations, and thus would be equivalent with the latter. Also, the directions of motion in any particular type of vibration may be equivalent. This is determined by selecting a group of operations which brings the atoms of the same phase to coincide for that type of vibration, and finding the directions of motion which satisfy the symmetry requirements of this group. In this way, both the directions of motion and their degeneracies are known. The number of distinct modes of vibration computed for the various lattices, are forΓtr 21;Γm 21,Γm′ 15;Γ0 21,Γ0′ 15,Γ0″ andΓ0′″ 12;Γt 12,Γt′ 8;Γrh 8,Γh 8;Γc 5,Γc′ 4,Γc″ 4. The modes are also described in as complete a manner as could be done by using considerations of symmetry alone.

• Modes of vibration of the hexagonal close-packed lattice

The modes of vibration of the hexagonal close-packed structure is worked out on the basis of the Raman theory of crystal vibrations. The character tables are drawn up, and the vibrations are derived by a method due to E. V. Chelam without using the character table of the space-group. It is found that there are 13 vibrations with degeneracies 1, 2, 2, 4, 3, 3, 3, 3, 3, 3, 6, 6, 6. The planes which take part in each mode and the directions of vibration for these are also described.

• Errata

• Diffraction coronæ due to non-spherical particles

It is pointed out that the Fraunhofer diffraction due to an obstacle of arbitrary shape can be replaced by that given by a linear distribution of sources along its boundary. Most of the intensity at any point in the pattern can again be supposed to originate from a finite number (usually two) of point sources, called “opposed points” or “poles”, situated on the boundary. In this way, if there are a large number of non-spherical particles distributed at random, then a ring system will be formed whose size will correspond to the distance between the opposed points. If this distance is a constant over an appreciable region of the boundary, then the rings will stand out from the background intensity. This explanation of the formation of the rings, has been verified using spores ofPinus longifolia and ofLycopodium. It is also shown that variations in the size of the particles affect the clarity of the rings detrimentally, the rings becoming less and less clear as the range of particle sizes increases. This is also verified by using two types ofLycopodium.

• X-Ray topographs of diamond

• X-ray reflection and the structure of diamond

Laue photographs with the X-ray beam normal to the surface (111) planes have been taken for two typical bluc-fluorescent diamonds exhibiting widely different intensities of fluorescence, but similar in other respects. Microphotometry of the peak intensity of the various spots shows that although the intensity of all the spots is greater with the more fluorescent diamond, the ratio (r) of the intensities of the corresponding spots varies. Empirically, it is found that (r−1) is proportional to the product of the structure factor, the wave-length reflected and a function of the angle of incidence.

• A new derivation of the Darwin-Prins formula of X-ray reflection

In this paper is described a new derivation of the formulæ for the reflection of X-rays by perfect crystals, which forms an alternative approach to the problem to that adopted by Darwin and Prins. It consists in obtaining a solution of the difference equations which occur in the problem for a crystal containing a finite number (n) of laminations. The case of a crystal of infinite depth is obtained from this by proceeding to the limit whenn→∞. The formulæ thus obtained are identical with those of Prins for an absorbing crystal, while for a non-absorbing crystal, the width of the region of perfect reflection is in agreement with Darwin’s value. However, there is a difference in the formulæ for the variation of intensity with angle outside this region. This has been shown to be due to an assumption made by Darwin which is not altogether justifiable.

• The angular divergence of the x-ray reflections by diamond

• On the radiation from the boundary of diffracting apertures and obstacles

Neglecting the obliquity factor, which is justified when one is considering only small angle diffraction, it is shown that the surface integral which is usually employed for the determination of the disturbance at any point can be easily converted into a line integral along the boundary of the diffracting screen. The formulæ thus obtained show that with either an aperture or an obstacle the illumination in the region of shadow can be completely represented as the effect of radiations arising from the boundary, while in the region of light the disturbance due to the direct light is superposed on this. The phase of the boundary radiation is determined by the region (of light or shadow) to which the ray towards the point of observation proceeds from the boundary, being opposite to that of the incident light in the former case, and being the same in the latter case. It is however shown that this leads to no discontinuity in the illumination as the point of observation passes from the region of light into the region of shadow. The boundary radiation can again be effectively replaced by the radiations arising from a finite number of point-sources situated on the boundary called ‘poles’, for which the path to the observation pointvia the boundary is a maximum or a minimum. The phase of the resultant disturbance due to regions of the boundary including and lying on either side of a pole is shown to lead over or lag behind that of the wave from the pole by the quantity π/4, according as the pole is one of maximum or minimum path. Applying these ideas to the diffraction pattern of a circular disc, it is shown that the calculated radii of the rings in the region of shadow agree well with those deduced from Lommel’s theory.

• On the crystal symmetry of diamond and its X-ray reflections

The problem of the symmetry of the diamond structure in relation to its X-ray behaviour is considered in a formal manner. It is shown that the presence or absence of the 200 or the 222 reflection cannot uniquely decide whether the symmetry is tetrahedral or octahedral. The 200 reflection is shown to be absent if the structure is either completely symmetric or antisymmetric with respect to the centre of inversion at 1/8, 1/8, 1/8 or if the two distributions are superposed in any arbitrary ratio. The 222 reflection is, however, absent only in the fully antisymmetric case. Making use of these results, the nature of the structures that are possible for the tetrahedral and the octahedral modifications of diamond are discussed.

• On the nature and origin of the laminations observed in diamond

• The luminescence of diamond excited by X-radiation

• X-ray topographs of diamond—Part II

• Luminescence as “forbidden” electronic transitions in diamond

Arguments are adduced to show that the doublet centred at 4152 Å occurring in the spectrum of all fluorescing diamonds arises from ‘forbidden’ transitions analogous to the forbidden3P-1S transitions in the spectrum of C I. On the basis of this analogy, fluorescence lines are also expected to occur at about 8816 Å and 7849 Å, analogous to the3P-1D and1D-1S transitions in C I. Of these, the former should also occur in absorption, while the latter should not occur in absorption and should be emitted only if the exciting radiation has a wavelength shorter than 4152 Å. A line has actually been found at 7930 Å, satisfying the latter conditions. The former line could not be recorded being outside the limit of sensitivity of the photographic plate.

• Photoelastic constants of diamond

All the photoelastic constants of diamond have been determined for the first time. The three stress-optic coefficients areq11=4·2×10−11,q12=−3·6×10−11,q44=2·6×10−11, from which the elasto-optic coefficients have been deduced to bep11=0·125,p12=−0·325,p44=0·11. It is found thatq11q12 andq44 are both positive and also that the refractive index of the crystal should decrease when subjected to a hydrostatic pressure, both of which are unique for diamond among the cubic crystals studied so far. In the course of the investigation, a new technique has been developed for determining the absolute path retardation, based on the production of localised interference fringes by the light coming from the two surfaces of the crystal.

• Thermo-optic behaviour of solids - I. Diamond

A phenomenological theory has been put forward for the variation of the refractive indices of solids with temperature, based on the idea that the refractive index can be expressed in terms of the number of dispersion centres and the polarizabilities of these centres, which are again dependent on the dispersion frequencies. The refractive index variation can then be represented as the sum of two terms, one arising from the change in the number of dispersion centres and the other from the variation of the frequencies. Accurate data of the thermal variation of the refractive index of diamond have been obtained for the first time from — 180° C. to 450° C. for three wavelengths, 4358, 5461 and 5893 Å.U. A new method was employed for this purpose which measuresdn/dt directly and is capable of general application. When applied to diamond, the theoretical ideas bring out the interesting fact that the rate of change of the dispersion frequencies with temperature is correlated to the rate of change of the fundamental lattice frequency of the crystal.

• Thermo-optic behaviour of solids - II. Fused quartz

The theory outlined in Part I of these series has been applied to calculate the thermal variation of refractive index of fused silica. It is found that the theory can account quantitatively for the experimental values ofdn/dt over the range of wavelengths from 1850 Å.U. to 6000 Å.U. Using the fact that the frequencies of vitreous silica are practically the same as those of crystalline quartz, the variation of the refractive index with temperature from −130°C. to 500°C. has been calculated and found to fit well with the experimental data.

• Thermo-optic behaviour of solids - III. Fluorspar

Making use of the dispersion formula given in the International Critical Tables, with absorption bands at λλ 0·0942 and 35·48 μ, and using the theory developed in Part I, the course of the variation ofdn/dt with wave-length has been satisfactorily explained from 0·185 to 6·5 μ. The dispersion formula however fails below 0·185 μ, and consequently two new formulæ have been developed holding down to 1300 Å. U., one of which uses dispersion frequencies at λλ 0·082, 0·1115 and 35·48 μ together with a non-unity constant in the expression forn2, while the other is of the Ketteler-Helmholtz type with four frequencies at λλ 0·045, 0·088, 0·1115 and 35·48μ. All the formulæ explain the variations ofdn/dt, in particular, as to why it increases algebraically as one proceeds both into the ultraviolet and the infra-red. The interesting fact emerges that the extreme ultraviolet frequency at λ 0·045 μ does not vary with temperature, while the proportionate variation of the one at 0·0888 μ is much less than that of the one at 0·1115 μ. This result becomes intelligible when one remembers that the deeper levels in the crystal would be less affected by temperature than the low-lying ones. Even the last one, although it is of the same order, is only one half of the proportionate variation of the infra-red lattice frequency. It is also found that the rate of variation of the ultraviolet frequencies is constant at different temperatures. In these respects, fluorspar differs from diamond and vitreous silica. The difference has been attributed to the fact that the binding in fluorspar is not purely covalent, but is also partly electrovalent.

• Thermo-optic behaviour of solids - IV. Zinc-blende

The refractive index of zinc-blende increases appreciably on heating for all wavelengths in the visible region the increase being 0·0212 and 0.0107 respectively for wavelengths 4358 Å.U. and 7320 Å.U. on heating from 0° to 205° C. It is shown that by applying the general theory of Part I one can satisfactorily explain both the variation ofdn/dt with wavelength and the courses of the temperature—refractive index curves from −80° to 700° C. for different wavelengths. The calculations show that the ultra-violet dispersion frequency at 2532 Å.U. alters appreciably with temperature, the proportional rate of change χ=−d (log ν)/dt being 70×10−6. This is nearly ten times the corresponding rate for diamond. The rate of change becomes less and less on lowering the temperature. It is pointed out that, for diamond, χ actually vanishes at very low temperatures and this is also probably the case for zinc-blende.

• Thermo-optic behaviour of solids - V. Alkali halides

A direct verification of the general theory of thermo-optic behaviour has been obtained by applying it to the alkali halides. Rocksalt, sylvine and potassium iodide have been considered in detail. For this purpose, new dispersion formulæ embodying observed absorption frequencies have been developed for NaCl and KI. With KI, it is found that the proportionate variation χ [=−d (logv)/dt] of the first ultra-violet frequency at 2190 Å.U. measured by Fesefeldt agrees with what is calculated from thedn/dt data, thus verifying the fundamental basis of the author’s theory. The theory successfully explains the whole course of the variation ofdn/dt with wavelength in the case of rocksalt, and in particular the positive values ofdn/dt near about 2000 Å.U. and its reversal in sign with increase of wavelength. The first three ultra-violet frequencies have χ’s of the order of +100×10−6, +25×10−6 and − 30×10−6 respectively, the last one near 500 Å.U. not varying with temperature. With sylvine,dn/dt measurements in the visible and near infra-red are satisfactorily explained with similar values of χ as in rocksalt. The paper also contains a general review of the dispersion data for the alkali halides. It is shown that a dispersion formula of the Drude form is more appropriate for these salts than one of the Lorentz-Lorenz form.

• Thermo-optic behaviour of solids - VI. Optical glasses

In this paper, the author’s general theory of thermo-optic behaviour is applied to the case of optical glasses. Making use of the dispersion formulæ of Huggins and co-workers, which employs one dispersion frequency for each oxide component, formulæ are developed, which give the rate of change of the refractive index with temperature,dn/dt. Computations made on the basis of these formulæ give fairly correct values ofdn/dt for a large variety of multicomponent optical glasses studied by Pulfrich and by Peters. Although the formulæ givedn/dt fairly correctly for any particular wavelength in the visible, they do not give accurate values for the dispersion ofdn/dt. The discrepancy is shown to be due to the inadequacy of using one dispersion frequency for each component in the dispersion formula. Using two frequencies, one of which is in the remote ultra-violet, and gives only a constant contribution to refractive index, new dispersion formulæ are derived and on their basis bothdn/dt for any wavelength as well as its dispersion are correctly obtained. In addition, the theory also explains qualitatively other observations made on the thermooptic behaviour of optical glasses, such as the anomalous variations ofdn/dt near the annealing and softening temperatures and broadly the correlation between composition and the value ofdn/dt of the glasses. It is inferred that the proportionate variation of the dispersion frequencies of all the components diminishes with fall of temperature, and probably vanishes at very low temperatures, as is the case with crystals like diamond.

• Birefringence of crystals and its temperature-variation - Part I. Calcite and aragonite

It is suggested that the birefringence of crystalline bodies can be explained as arising from the existence of polarised electronic transitions, so that the probability of transition is different for different directions of the incident electric vector. As a result, in the dispersion formulæ for the three principal refractive indices of a biaxial crystal, the oscillator-strengths will be different, although the dispersion frequencies are the same. The application of the idea to the cases of calcite and aragonite (two strongly birefringent crystals) enables one to construct dispersion formulæ involving three ultra-violet frequencies at 1535, 1000 and 500 Å.U. The formulæ, which fit the dispersion data for both rays of calcite from 0·2 to 3 μ, show that a large part of the birefringence arises from the large anisotropy in the strength of the nearest ultra-violet frequency at 1535 Å.U. A similar result is also found for aragonite.

The dispersion formulæ are also successful in explaining the thermooptic behaviour of both calcite and aragonite, when they are utilised in the author’s theory of thermo-optic behaviour, which has so far been applied only for isotropic solids. For anisotropic solids, an additional principle has to be used,viz., that, while the total oscillator strength does not alter with temperature, the individual strengths along the three principal directions can change, resulting in a mutual transfer. It is suggested that this transfer is related to the differences in thermal expansion along the three directions, which finds some support in the case of aragonite. A physical explanation of the transfer can be found, wholly or partly, in the tilting oscillations of the CO3 ion.

The theory is successful in explaining the remarkable fact that, whiledn/dt for both indices of calcite is positive, that for all the indices of aragonite is negative, the difference being attributable to the much larger coefficient of thermal expansion of the latter.

• Birefringence of crystals and its temperature-variation - II. Sodium and potassium nitrates

• Photoelastic constants of diamond corrections

• Photo-elasticity of diamond

The photo-elastic constants of diamonds have been redetermined and are found to beq11=−5·05,q12=+2·15,q44=−2·8×10−14 cm.2 dyne−1 andp11=−0·31,p12=+0·09,p44=−0·12. These lead to a decrease in refractive index when diamond is subjected to a hydrostatic pressure.

• Photo-elastic constants of sodium chlorate

Sodium chlorate is the first crystal belonging to the tetrahedrite class (T) of the cubic system for which photoelastic constants have been measured. Since the crystal exhibits optical activity and no birefringence in the absence of stress, special techniques have to be adopted for measuring the birefringence introduced by stress. This has been done by the use of a petrological microscope in conjunction with the stressing apparatus and measuring the ellipticity and other characteristics of the light transmitted by the crystal. A particularly simple method is to use the “elliptic analyser”. Since the crystal does not possess four-fold axes, the relative orientation of the X- Y- and Z-axes was determined by means of X-rays. From observations on crystals compressed along [100], [110] and [111] directions,q11q12,q11q13 andq44 were evaluated. All the four constants were independently obtained by combining these with polarisation measurements of light diffracted by ultrasonic waves in the crystal. The values are:q11=1·48,q12=3·88,q13=2·89,q44=−1·58×10−13 cm.2 dyne−1;p11=0·173,p12=0·258,p13=0·223,p44=−0·0187.q12 andq13 are different as is to be expected from Bhagavantam’s theory for crystal classes T and Th. For stress along X-axis, the values of birefringence for observation along the Y- and Z- axes differ by as much as 70%, which is the largest observed so far for a cubic crystal.

• Theory of optical activity of crystals - I. General ideas

The paper contains a classical description of the first order terms in the polarisability theory of optical activity. Optical activity essentially arises because the dipoles induced by the light wave in the medium are not all in the same phase. As a result of their mutual influences, the resultant induced moment will not be in phase with the electric field of the light wave and would lead to a rotation (ρ). By comparing the results calculated from the structure with the phenomenological theory of light propagation in the crystal, the magnitude ofρ is obtained. This method has been applied to a hypothetical tetragonal crystal having a spiral structure. It leads to the interesting result that the rotation at right angles to the optic axis is opposite in sign to that along the axis, as in quartz. The theory also predicts that, in general, the rotation will vary faster than 1/λ2, as is found to be the case with many substances.

• The theory of optical activity of crystals - II. Calculation of the rotatory power of NaClO3 and NaBrO3

The theory of optical activity outlined in Part I has been applied to the cases of NaClO3 and NaBrO3. The calculated values for the rotatory power agree reasonably well with measurements previously reported.

• The theory of optical activity of crystals - III. Calculation of the rotatory power ofβ-quartz

Using the general theory of Part I, the rotatory power ofβ-quartz along and perpendicular to the axis are calculated to be 236° and 115°. These agree well with the value 252° observed for the former and the value 0·50 observed for the ratio of the two inα-quartz.

• Photoelastic constants of sodium chlorate from ultrasonic diffraction

Details are given of the theory and technique of measuring photoelastic constants in optically active cubic crystals from ultrasonic diffraction of light. The results thus obtained, in conjunction with the measurements of relative path retardation produced by stressing the crystal, enable one to obtain all the four constants independently.

• X-Ray anti-reflections in crystals

The paper deals with the theory of an interesting phenomenon (which has been designated as “anti-reflection”) that the intensity of the transmitted beam may exhibit a peak larger than the background when a Bragg reflection occurs in an absorbing crystal. The theory is based on the dynamical theory of Ewald and Laue. It comes out that the effect is due to a decrease in the effective absorption coefficient of the crystal near the Bragg reflection, and to the consequent increase in the transmitted intensity predominating over the loss of energy by reflection. The anti-reflection peak becomes more prominent, the greater the thickness of the crystal. The results of the theory are found to be in accord with the previous observations of Borrmann and Campbell. The theory further predicts that the peaks in the reflected and transmitted beams would not be coincident and this requires further verification.

• Investigation of the degree of perfection of a crystal by means of polarized X-rays

The paper describes an investigation of the intensity of Bragg reflection when the incident X-rays are polarized and the azimuth of the electric vector is varied with respect to the plane of reflection. It is observed, using natural and ground (211) faces of NaNO3, that the variation of intensity with azimuth of polarization is different for a mosaic and a perfect crystal. Such a difference is in fact to be expected from theoretical considerations. The actual behaviour of both the ground and the natural faces was found to be intermediate between what is predicted by theory for an ideally perfect and an ideally mosaic crystal. By comparing the observed azimuthal variation of the integrated reflection with the theoretical expectation for the two limiting cases, it is possible to assess the degree of perfection of the crystal.

• X-ray anti-reflections in crystals - Part II. Calculation of the integrated reflection and integrated anti-reflection for an internal reflection

Making use of the theory developed in Part I, the integrated values of the reflected and the anti-reflected intensity have been obtained analytically for an internal reflection of a perfect crystal. Three special cases are considered, namely a symmetrical reflection, an asymmetrical reflection and also when absorption is very heavy. It is found that when absorption is large, the formula for integrated reflection reduces to that for a mosaic crystal, which may be physically explained by the fact that multiple reflections are not allowed to play a prominent part owing to the beam being quickly attenuated by absorption.

• Studies on collagen - I. Structure of the collagen group of proteins

The paper reports the details of the revised structure of collagen. It is composed of three helical polypeptide chains, each of which has ten residues in three turns of a left-handed helix. The three chains are further wound into a superhelix in the form of interwined coiled coils. The major helix is right-handed and makes one turn in thirty residues. The structure has reasonable hydrogen bonds, two for every three residues. The shree-chain cylindrical rods are stacked together in hexagonal array and stabilised by cross-linkages through hydroxyproline side-groups. In trying to fit these in the lattice, a slight uncoiling of the major helix is required, resulting in a repeat of 618 A along the fibre axis. The proposed structure is in good agreement with the infra-red and X-ray data and also fits in broadly with the amino-acid composition and other properties of collagen.

• Structure of elastin

• Studies on collagen - II. Cylindrical lattice structure of collagen

It is suggested that the structure of collagen is based on a cylindrical lattice. The protofibrils, consisting of triple chains, are packed together in a hexagonal array close to the centre, and are extended outwards in the form of cylindrical sheets. Each sheet has a pseudohexagonal symmetry about the common axis. This cylindrical lattice explains (a) the large number of equatorial reflections of long spacing observed in the diffraction pattern of native fibres and (b) the absence of the 110 reflection and its analogue in all layer lines. The number of sheets in a single cylindrical rod is about 7 and its diameter about 200 Å.

The above results were verified by model optical diffraction experiments. Also, from these experiments it is shown that the structure can also belong to the roll type with the added advantage that the formation of a roll in the form of a continuous sheet is more probable from the point of view of fibrogenesis.

• “Līlāvatī”—A new analogue computer for solving linear simultaneous equations and related problems - Part I. General principles and design of model I

• A problem in probability related to the passage of light through a cloud of particles

• Analysis of the x-ray diffraction pattern of helical structures

The paper deals with the theory of the diffraction pattern of helical structures having the number of units per turn (n) neither integral nor rational. The conventional treatment suffers from the defect that the repeat spacing along the axis of the helix is taken as the standard of reference, and this does not exist, being infinite, whenn is irrational. The difficulty is got over in this paper by focussing attention on the ‘unit height’ (h = resolved component of a unit along the axis) and ‘unit twist’ (t = fraction of a complete rotation for one unit, = l/n), which vary continuously irrespective ofn being rational or irrational. Explicit formulæ are obtained in terms of their Bessel indices for the observed layer line-spacings which turn out to be very simply related to the reciprocals of the unit height and the pitch of the helix, A technique of analysing, the observed diffraction pattern for the elements of the helical structure is also given, with examples. The case of a coiled-coil is seen to have the same general features as the simple coil, the layer line-spacing being now related to two pitches, namely, those of the major and the minor helices, and the unit height. The relationship of the diffraction pattern of a helix in its uncoiled and its coiled-coil form is also found to be rather simple, being similar to the multiplet splitting produced by a magnetic field in spectral lines.

• On faltung and correlation of functions and their application in physical problems

This paper deals with a number of applications of the correlation and faltung functions, their Fourier transforms and their integrals. It is possible to show that various types of distortions produced by a recording instrument do not affect the value of the integral of the quantity recorded. This should be of great interest to designers of recording instruments. The advantage of using the F.T. in compounding probability distribution functions is pointed out with an illustration giving a short derivation of Kluyver’s famous distribution for the problem of random walk in two dimensions by using this method. Finally, the relation of the correlation function to the Patterson function of a crystal structure is also pointed out.

• Reconstruction of substance from shadow - 1. Mathematical Theory with Application to Three-Dimensional Radiography and Electron Microscopy

• Potential functions for hydrogen bond interactions - III. Empirical Potential Function for the Peptide N-H…O=C Hydrogen Bond

A careful comparison of the distribution in the (R, θ)-plane of all NH … O hydrogen bonds with that for bonds between neutral NH and neutral C=O groups indicated that the latter has a larger mean R and a wider range of θ and that the distribution was also broader than for the average case. Therefore, the potential function developed earlier for an average NH … O hydrogen bond was modified to suit the peptide case. A three-parameter expression of the form$$V_{hb} = V_{min} + p_1 \Delta ^2 + q_1 e^{p_3 \Delta } \theta ^2$$, with △ = R - Rmin, was found to be satisfactory. By comparing the theoretically expected distribution in R and θ with observed data (although limited), the best values were found to bep1 = 25,p3 = − 2 andq1 = 1 × 10−3, with Rmin = 2·95 Å and Vmin = − 4·5 kcal/mole. The procedure for obtaining a smooth transition from Vhb to the non-bonded potential Vnb for large R and θ is described, along with a flow chart useful for programming the formulae. Calculated values of ΔH, the enthalpy of formation of the hydrogen bond, using this function are in reasonable agreement with observation. When the atoms involved in the hydrogen bond occur in a five-membered ring as in the sequence a different formula for the potential function is needed, which is of the form Vhb = Vmin +p12 +q1x2 wherex = θ − 50° for θ ≥ 50°, withp1 = 15,q1 = 0·002, Rmin = 2· Å and Vmin = − 2·5 kcal/mole.

• Potential functions for hydrogen bond interactions - IV. Minimum Energy Conformation of the α-Helical Structure of PoIy-L-Alanine

Making use of the empirical potential functions forpeptide NH .. O bonds, developed in this laboratory, the relative stabilities of the rightand left-handed α-helical structures of poly-L-alanine have been investigated, by calculating their conformational energies (V). The value of Vmin of the right-handed helix (αP) is about — 10.4 kcal/mole, and that of the left-handed helix (αM) is about — 9.6 kcal/mole, showing that the former is lower in energy by 0.8 kcal/mole. The helical parameters of the stable conformation of αP aren ∼ 3.6 andh ∼ 1.5 Å. The hydrogen bond of length 2.85 Å and nonlinearity of about 10° adds about 4.0 kcal/ mole to the stabilising energy of the helix in the minimum enregy region. The energy minimum is not sharply defined, but occurs over a long valley, suggesting that a distribution of conformations (ϕ, ψ) of nearly the same energy may occur for the individual residues in a helix. The experimental data of a-helical fibres of poly-L-alanine are in good agreement with the theoretical results for αP. In the case of proteins, the mean values of (ϕ, ψ) for different helices are distributed, but they invariably occur within the contour for V = Vmin + 2 kcal/mole for αP.

• Stabilization of the collagen structure by hydroxyproline residues

The molecular structure of collagen is now accepted to be based on a triple-stranded coiled-coil, in which the three strands are held together predominantly by hydrogen bonds. Recent experimental evidence has shown that the presence of hydroxyproline residues in the third position of the repeating tripeptide unit lends additional stability to the collagen structure. In this paper, we report a model structure, which is supported by these observations. In a model structure proposed earlier, there are two hydrogen bonds per tripeptide unit, one of which is a direct interchain hydrogen bond, while the second hydrogen bond can be formedvia a water molecule. It has now been shown that the same water molecule can also form a hydrogen bond with the oxygen of theγ-hydroxyl group of hydroxyproline in the third position in the sequence (Gly-R2-R3).

This hydroxyl group can also take part in an inter-triple-helix hydrogen bond. Our studies thus show the role played by hydroxyproline residues in the structure and stability of collagen.

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