scholarly journals On the form and potential energy of the isomorphous crystals, ruby (Al 2 O 3 ) and hæmatite (Fe 2 O 3 )

1. 1. Recently, much work has been done in attempting to explain the observed size and shape of crystals in terms of their interionic forces. To do this, a highly idealised form of the crystal has been postulated in which the ions of the crystal (in accordance with Kossel’s theory) are regarded as pointcharges, of magnitude equal to the resultant ionic charge and situate at the points of the crystal lattice determined by X-ray analysis. For ions which are surrounded symmetrically by other ions, these points doubtless correspond to the nuclei of the respective atoms, but for unsymmetrically situated ions there is some uncertainty as to the exact position to which to assign the pointcharge. Moreover, in order that the crystal may be in equilibrium, other repulsive forces, in addition to the electrostatic forces of attraction and repulsion between the ionic point-charges, are assumed to be operative between the various ions. These are termed “intrinsic repulsive” forces, and have been represented by an inverse power law varying with the distance according to the n th power, and expressible in the form μ n r - n . Consequently, in order to determine the equilibrium configuration of the crystal, as given by the minimum value of the potential energy of the crystal, it is necessary to determine (i) the electrostatic potential energy of an infinite array of point-charges, arranged according to the crystal pattern, and (ii) the potential energy due to the intrinsic repulsive forces between the ions.

§1. This paper is one of a series, planned by Prof. S. Chapman and one of the present authors in collaboration with the object of calculating theoretically the properties of the carbonate and nitrate crystals. A necessary preliminary to such an investigation is the calculation of the electrostatic potential energy of an infinite array of electrical charges distributed according to the scheme determined by X-ray analysis. This calculation has been made by Prof. Chapman and his collaborators, and the results have just been published. The results of this investigation were, however, communicated to us before publication, for which we offer grateful acknowledgment. The results of this paper have similarly been conveyed to Prof. Chapman, who with Mr. Topping is using them in a theoretical determination of some of the physical properties of the carbonates and nitrates.


This paper is a continuation of a former paper, in which the electrostatic potential energy of the carbonate and nitrate crystals of the calcite type was calculated, for a series of values of two parameters which determine the size and shape of the unit crystal cell. In addition to the electrostatic forces of attraction and repulsion between the atomic ions of which these crystals are supposed to consist, there are certain other repulsive forces which will be distinguished by the term “intrinsic.” Since the former paper was written, data concerning these repulsive forces have been published by Lennard-Jones and his collaborators, for many of the ions included in the crystals referred to. The part of the crystal energy which corresponds to these forces can therefore now be calculated in many cases, and, in particular, for the crystals CaCO 3 , MgCO 3 , and NaNO 3 . By mutual arrangement, our values of the electrostatic potential energy for carbonate crystals of the calcite type were communicated to Dr. Lennard-Jones before publication, and he, with Miss Dent, calculated the energy corresponding to the repulsive forces for the series of values of the parameters which we had used. It was thus possible to determine the whole energy of the crystals of CaCO 3 and MgCO 3 , measured relatively to a state of infinite dispersion of the metallic ions and the ionic groups CO 3 – – (as wholes); such values of the energy were found for various configurations, and the stable configuration of minimum energy could be shown to lie on a certain linear series of configurations. The stable configuration cannot be found completely at present owing to lack of knowledge of the forces determining the size of the CO 3 ionic group itself. The parameter in the series of configurations just mentioned is the size of the CO 3 group, as measured by the distance ( b ) from the C to the O ion. The series as determined by Lennard-Jones and Dent did not include the actual measured configuration, but the difference between the measured and nearest calculated configurations was not great, and afforded an estimate of the distance b , viz.. 1·08 Å. In our previous paper we attempted to deduce the value of b without a knowledge of the repulsive forces, by finding the configuration of minimum electrostatic energy in a virtual displacement of the crystal chosen so as to eliminate, as far as possible, any change in the energy corresponding to the repulsive forces. The CO 3 ion was maintained of constant size, and the distance between each oxygen ion and the nearest metallic ion was also kept constant. For such a displacement there will be no change in the energy due to the repulsive force between nearest neighbours among the ions, and since the repulsive forces vary according to a high inverse power of the distance, it was assumed that there would be no change in the total energy of the repulsive forces. The value of b deduced on this assumption was 0·92 Å. The slight difference between this and the value found by Lennard-Jones and Dent is investigated, for the parallel case of NaNO 3 , in 6 of this paper, and found to be due to a slight variation, during the above virtual displacement, in the energy of the repulsive forces between oxygen ions not of the same CO 3 group.


2016 ◽  
Vol 4 (1) ◽  
pp. 27
Author(s):  
Hasmukh K. Tank

<p>We first find here that the ratio of: (loss in energy of cosmologically red-shifting photon) and (loss in electrostatic potential-energy of an electron at the same distance <em>D</em>) remains equal to the famous ratio (G m<sub>e </sub>m<sub>p</sub>) / e<sup>2</sup> leading us towards a possibility that ‘cosmological red-shift’ may be due to gravitational effect. Also the ratio <em>h H<sub>0</sub> / m<sub>e</sub> c<sup>2 </sup>= (G m<sub>e </sub>m<sub>p</sub>) / e<sup>2</sup></em>. Starting with Mach’s principle, that ‘mass’ of an object is because of its ‘cosmic gravitational potential energy’, we arrive at a possibility that every moving chunk of matter and energy should experience a fixed value of acceleration <em>H<sub>0 </sub>c</em>. For the purpose of comparison, we express the ‘cosmological red shift’ as deceleration of the photon, and find that the deceleration experienced by the photon matches perfectly with the expected value. Then it is argued that if such a deceleration is true for a chunk of energy called photon, then it must be true for every particle of matter too. Strikingly, the decelerations experienced by the space-probes Pioneer-10, Pioneer-11, Galileo and Ulysses, as carefully measured by Anderson J.D. ET. Al. Match perfectly with the deceleration of the ‘cosmologically red-shifting photons’; thus providing supportive evidence for the new explanation proposed here.</p>


2006 ◽  
Vol 19 (3) ◽  
pp. 277-280 ◽  
Author(s):  
Zhe-ming Ni ◽  
Guo-xiang Pan ◽  
Li-geng Wang ◽  
Wei-hua Yu ◽  
Cai-ping Fang ◽  
...  

The differential equation d 2 y / dx 2 + 1 dy / x dx — y = 0, which differs from Bessel’s Equation of zero order only in the sign of the third term ( — y ), has two solutions denoted by I 0 ( x ) and K 0 ( x ): these solutions tend exponentially to infinity and zero respectively as x → ∞ by positive values. The function K 0 ( x ) is of physical importance, particularly in connection with the electrostatic potential of a periodic linear series of charges. It has been much used recently in calculations of the electrostatic potential energy of certain crystals, for which it was found necessary to construct the following tables. It appears that the earliest tables of K 0 ( x ) are due to Aldis (‘Roy. Soc. Proc.,’ vol. 64, pp. 219-221 (1899)), who gave the values of K 0 ( x ) to 21 decimal places for values of x from x = 0 to x = 6·0 at intervals of 0·1, and also to between 7 and 13 significant figures from x = 5·0 to x = 12·0 at intervals of 0·1. These tables are reprinted in the ‘Treatise on Bessel Functions’ by Gray, Mathews and MacRobert (Macmillan, 1922). Jahnke and Emde, ‘Functionentafeln,’ pp. 135-6 (Leipzig, 1923), also tabulate the function K 0 ( x )—there denoted by ½ i πH 0 ( 1 ) ( ix )—over the same range of x , but only to four significant figures.


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