Ramesh Narayan

A new look at pulsar statistics - birthrate and evidence for injection

We make a statistical analysis of the periodsP and period-derivativesP of pulsars using a model independent theory of pulsar flow in theP-P diagram. Using the available sample ofP andP values, we estimate the current of pulsars flowing unidirectionally along theP-axis, which is related to the pulsar birthrate. Because of radio luminosity selection effects, the observed pulsar sample is biased towards lowP and highP. We allow for this by weighting each pulsar by a suitable scale factor. We obtain the number of pulsars in our galaxy to be 6.05−2.80+3.32 × 105 and the birthrate to be 0.048−0.011+0.014 pulsars yr−1 galaxy−1. The quoted errors refer to 95 per cent confidence limits corresponding to fluctuations arising from sampling, but make no allowance for other systematic and random errors which could be substantial. The birthrate estimated here is consistent with the supernova rate. We further conclude that a large majority of pulsars make their first appearance at periods greater than 0.5 s. This ‘injection’, which runs counter to present thinking, is probably connected with the physics of pulsar radio emission. Using a variant of our theory, where we compute the current as a function of pulsar ‘age’ (1/2P/P), we find support for the dipole braking model of pulsar evolution upto 6 × 106 yr of age. We estimate the mean pulsar braking index to be 3.7−0.8+0.8.

A new approach to repulsion in ionic crystals

Abstract An empirical approach is attempted to make the repulsion potential of ions in an ionic crystal structure independent and crystal independent so that once the repulsion parameters for an ion are evaluated in one compound, in one structure, they could be used for that ion in any crystal. The repulsion between ions is postulated to be due to the increase in the internal energy of the ions arising from the distortion and the compression at the points of contact with their neighbours. Using an exponential form for the compression energy involving two parameters per ion, a repulsion potential for ionic crystals is proposed which includes the repulsion between nearest and next nearest neighbours. The repulsion parameters for the alkali and halogen ions have been determined to fit the behaviour of 20 alkali halides over the pressure range 0–45 kbars. The r.m.s. percentage deviations from experiment, of the calculated values of lattice spacing and compressibility are respectively 0.997% and 6.76%. The calculated radii of the ions in the various compounds compare well with the experimental values deduced from electron density maps. The advantages of the present form of the repulsion potential over earlier forms are discussed.

Estimation of van der waals dipole-quadrupole interactions

Abstract A formula has been derived based on a variational approach for the van der Waals dipole-quadrupole interaction coefficient between two atoms or ions. The coefficients for the various ion pairs in the alkali halides have been estimated on the basis of this formula. The results agree quite well with those estimated by Mayer ( J. Chem. Phys. 1 , 270 (1933)) using a perturbation approach. The present formula is shown to have a practical advantage over the perturbation formula.

Repulsion parameters of ions and radicals-application to perovskite structures

Abstract The compressible ion approach to repulsion which has been shown to work well for the alkali halides ( J. Phys. Chem. Solids 37 , 395 (1976) ; Curr. Sci. 46 , 359 (1977)) has been extended to other cubic ionic crystals. Repulsion parameters have been refined for a number of ions and radicals viz., Cu + , Ag + , Tl + , Mg 2+ , Ca 2+ , Sr 2+ , Ba 2+ , Zn 2+ , Cd 2+ , Hg 2+ , Mn 2+ , Fe 2+ , Co 2+ , Ni 2+ , Sm 2+ , Eu 2+ , Yb 2+ , Pb 2+ , H − , O 2− , S 2− , Se 2− , Te 2− , NH 4 + , SH − , SeH − , BrO 3 − , ClO 3 − , ClO 4 − , CN − , NH 2 − , NO 3 − , BH 4 − , BF 4 − , SO 4 2− , NH 2− . Using these parameters, calculations have been made on the lattice spacings and compressibilities of a number of perovskite-like crystals of the form A + B 2+ C 3 − . The predicted values agree well with experiment. In the case of four crystals viz., LiBaF 3 , LiBaH 3 , LiEuH 3 and LiSrH 3 , there were large discrepancies between the calculated and observed lattice spacings. When these crystals were assumed to be of the inverse perovskite structure, calculations showed good agreement with the experimental data.

Maximum Entropy Image Reconstruction-A Practical Non- Information-Theoretic Approach

The maximum entropy method (MEM) of image reconstructtion is discussed in the context of incomplete Fourier information (as in aperture synthesis). Several current viewpoints on the conceptual foundation of the method are analysed and found to be unsatisfactory. It is concluded that the MEM is a form of model-fitting, the model being a non-linear transform of a band-limited function. A whole family of ’entropies’ can be constructed to give reconstructions which (a) are individually unique, (b) have sharpened peaks and (c) have flattened baselines. The widely discussed 1nB and - B1nB forms of the entropy are particular cases and lead to Lorentzian and Gaussian shaped peaks respectively. However, they hardly exhaust the possibilities-for example, B1/2 is equally good.The two essential features of peak sharpening and baseline flattening are shown to depend on a parameter which can be controlled by adding a suitable constant to the zero spacing correlation ρ00. This process, called FLOATing, effectively tames much of the unphysical behaviour noted in earlier studies of the MEM. A numerical scheme for obtaining the MEM reconstruction is described. This incorporates the FLOAT feature and uses the fast Fourier transform (FFT), requiring about a hundred FFTs for convergence. Using a model brightness distribution, the MEM reconstructions obtained for different entropies and different values of the resolution parameter are compared. The results substantiate the theoretically deduced properties of the MEM.To allow for noise in the data, the least-squares approach has been widely used. It is shown that this method is biased since it leads to deterministic residuals which do not have a Gaussian distribution. It is suggested that fitting the noisy data exactly has the advantage of being unbiased even though the noise appears in the final map. A comparison of the strengths and weaknesses of the MEM and CLEAN suggests that the MEM already has a useful role to play in image reconstruction.

The structures of the ammonium halides

A comparison with the alkali halides suggests that all the ammonium halides should occur in the NaCl centre-of-mass structure. Experimentally, at room temperature and atmospheric pressure, only NH,1 crystallizes in this structure, while NH,F is found in the ZnO structure, and NH&I and NH,Br occur in the CsCl structure. We show that a distributed charge on the NH,+ ion can explain these nstructures. Taking charges of + 0.2e on each of the five atoms in NH,+, as suggested by other studies, we have recomputed the Madelung energy in the cases of interest. A full ionic theory including electrostatic, van der Waals and repulsive interactions then explains the centre-of-mass structures of all the four ammonium halides. The thermal and pressure transitions are also explained reasonably well. The calculated phase diagram of NH,F compares well with experiment. Barring the poorly understood NH,F(II) phase, which is beyond the scope of this work, the other features are in qualitative agreement. In particular, the theory correctly predicts a pressure transition at room temperature from the ZnO structure directly to the CsCl structure without an intermediate NaCl phase. A feature of our approach is that we do not need to invoke hydrogen bonding in NH,F.

Optimum choice of wavelengths in the anomalous scattering technique with synchrotron radiation

A formula has been derived for the mean-square error in the phases of crystal reflections determined through the multiwavelength anomalous scattering method. The error is written in terms of a simple function of the positions in the complex plane of the centres corresponding to the different wavelengths. For the case of three centres, the mean-square error is inversely proportional to the area of the triangle formed by them. The theoretical values are in good agreement with those obtained by earlier workers from computer simulations. The present method makes it easier to optimize the number and the actual wavelengths to be employed in the multiwavelength method. The maximum benefits of this method are expected in experiments employing synchrotron radiation or neutrons.

Ionic compressibilities and ionic radii - systematic trends

Ionic radii and compressibilities have been calculated for a number of monovalent and divalent ions and radicals on the basis of the compressible ion theory. In this theory, the compression energy of an ion is given as a two-parameter function of its radius,A exp (−r/p), the radius and compressibility of the ion being monotonically decreasing functions of the compressing force acting on it. Choosing a standard force reflecting the average environment in the alkali halides, univalent radii and compressibilities have been calculated. This is the first theory to estimate ionic compressibilities. The values show systematic trends among groups of related ions. Anions are found to be significantly more compressible than cations (e.g., the compressibilities of Ca++, K+, Cl− and S− − are respectively 0.8530, 1.342, 2.952 and 5.150 × 10−12 cm2/ dyne). Multivalent or ‘crystal’ radii and compressibilities have also been calculated by scaling the standard force by the square of the ionic charge. The calculated ionic radii are closer to experimental values than the classical empirical radii.

A Method of Reducing Termination Errors in Radial Distribution Functions

The radial distribution function [g(r)] of a liquid can be obtained from an integral transform of its X-ray scattering intensity I(/~). Experimentally, I(#) can be measured over only a limited range, leading to termination errors in g(r). Using the strict positivity of g(r), an iterative method is proposed to reduce these errors. The negative portions of each successive distorted approximation of g(r) are replaced by the value zero and this function is used to generate l(p) in the non-measured range. Computer calculations on a model function yield encouraging results. The method appears to have an advantage over current approaches as it does not require a knowledge of the hard-core diameter of the atoms. It is also expected to work in the case of molecular liquids where there are some difficulties in applying the current methods.

Compressive energy of ions in ionic crystals

The possibility of writing the repulsive energy in the Born model of binary ionic crystals as a sum of two separate contributions from the two ions has been investigated. Such an approach leads to two identities, one connecting the lattice spacings of a family of ionic crystals and the other connecting their compressibilities. These identities have been tested on the alkali halide crystals over a range of pressures. The agreement is found to be quite satisfactory. Some further predictions with respect to crystals which exist as two polymorphs have also been tested. In all cases, the deviations of the experimental values from the exact identities can be traced to the fact that second neighbour repulsions in the crystals have been neglected. It is hence concluded that individual compressive energies for ions in ionic crystals is a very attractive possibility.