http:\trusciencetrutechnology @blogspot.com, Vol.2007,No.7, Dated 23rd July 2007.
ON THE CORIOLIS ROTATION-VIBRATION INTERACTION CONSTANTS OF
HEAVY WATER AND THEIR DISPERSION CHARACTER
Professor Kotcherlakota Lakshmi Narayana,
It is a very significant finding, that the variations of 21 , 22 are complimentary to each other. And the curves indicate similarity with familiar dispersion curves something new that was not reported earlier. This finding has a far-reaching implication. A new method of analysis is based, on an Exact Coriolis Matrix that involves three coefficients, C12, C13 and C23. The extended set of three rotation-vibration interaction constants of the heavy water is found to have the values. 21 = 0.012162, 22 =0.654505 and 23 =0.333333. These satisfy the normalization condition that the sum of squares of the values is just unity. A significant result is that the 22 is considerably now reduced. The rotation-vibration interaction constant 23 almost acquires the reduction in 22. Yet another new approach described vividly in the present article is the one that employed the mixing parameter dependent Exact Coriolis Matrix method to find the Coriolis rotation-vibration interaction constants. These values are found exhibit a distorted version of the dispersion curve nature of the constants. The asymmetric stretching mode, the bending mode and the symmetric stretch modes of the Heavy Water seem to be greatly affected by the Coriolis Effect and the mixing parameter of the underlying symmetry coordinates.
Key words: Heavy Water, D2O, thermonuclear plasma, Coriolis Matrix, Rotation-Vibration interaction, Dispersion Curves, Nuclear Moderators, Tensor Force, mixing parameter, Boson, Blue of Ocean
INTRODUCTION:
Heavy water, D2O, is water in which, both the hydrogen atoms have been substituted by the Deuterium (1). The deuterium is an isotope of hydrogen, constituted by one proton and one neutron. Heavy water has become a strategic material due its functional property as a moderator to slow down neutrons in a nuclear reactor. A typical 220 MWe pressurized heavy water reactor requires about 275 tonnes of heavy water. Nangal plant, the largest in the world, which works employing the electrolysis process, produced the first drop of heavy water in India on the August 9, 1962. Fertilizer plants have been next effectively used as the raw material to produce the deuterium. Chemical exchange between H2S and water or ammonia is the other processes adopted to produce heavy water. Manuguru plant adopted the use of fly ash in the production of heavy water in India. Deuterium has found to be effective, in substituted polymers for enhanced transmission of light.
The relative abundance (2) of deuteron in nature is only 0.0149 in atom per cent. Deuteron obeys Bose statistics, hence is a Boson, as directly confirmed by experiments on Band Spectra and, was found to be composed of just one proton and one neutron. Proton is a Fermion and hence Neutron as well is a Fermion. An outstanding peculiarity of deuteron is its small binding energy ~2.22MeV and its large average separation distance ~ 4.0E-13 cm between the constituent nucleons. These nucleons thus spend more time outside the range of attractive nuclear force.
Just as the optical doublet studies in spectroscopy of series classification and the Structure of atoms (3). Aston brought out the doublet method (4), in mass spectroscopy. And the measurements of best values, of the so-called fundamental doublets led to determine, the mass excess for Deuterium to be 14.735e-020.006amu(5). Many other investigators best values differ slightly. Non additivity of magnetic moments for the case of deuteron leads to the measured value as d=0.8576 nuclear magneton. The finite difference between d and (p+n) has been interpreted as evidence for the contribution of tensor force or the non-central force between nucleons in deuteron. When the time average, volumetric distribution of charge within deuteron deviates from spherical symmetry, then the nucleus would posses finite multipole moments. The quadrupole moment Q= 0.273e-26 cm2, for a deuteron in a quantum state (L=0,S=1) would be in harmony with magnetic moment and the mechanical moment. The existence of non-central force asserts that no longer the orbital angular momentum L is a good quantum number. The deuteron is an admixture of 4% of 3D1 level and 96% of 3S1 level that as well supports the non-additivity of magnetic moments of neutron and proton in deuteron.
Our previous investigation (6) on the molecular force fields of heavy water revealed that its force field ellipse is three times more in area than that for W2O molecule. The mixing parameter value describes an orthogonal linear combination of two mass weighted vibration cartesian coordinates S1 and S2 of the A1-species, of heavy water. We have already found that the constraint technique yields a mixing parameter value +1.52 or –0.9136 for gaseous heavy water.
Smith & Linnett (7) gave the frequency assignments for the heavy water as w1=2758.06cm-1, w2=1210.25 cm-1, w3=2813.79 cm-1 with DO bond length as 0.957Angstroms. The restricted rotations or the rocking motions (librations) of the molecule of heavy water play an important role in ice and liquid water than in the Vapor State (8). The librations arise due to restrictions imposed by hydrogen bonding on the heavy water. Since the heavy water has larger moments of Inertia, its libration frequencies are reduced by a factor of two, relative to those of ordinary water H2O. At 250C the liquid D2O has L2 major libration band at 345.6 cm-1 and with a minor band L1 around 187.75 cm-1. The infrared absorption and the Stokes and anti-Stokes shifts of excitation spectral lines become more complex due to overtones and combinations of vibration frequencies with libration bands.
Liquid water D2O does not absorb in the Red region but absorbs more in the infra red region. It is blue in color solely, due to scattering of light, unlike the ordinary water or the hydrothermal colloidal silica. Its interesting to note that liquid ordinary water has w1=3280 cm-1, w2=1644cm-1 and w3= 3490cm-1 according to Martin (8). Ordinary water capability of hydrogen bonding with anions gives rise to possible ionic kosmotrophes or chotrophes.
Broadening and movement of frequencies to lower wavenumbers characterize the ionic kosmotrophes (8) whereas the chotrophes exhibit narrowing and movement towards the higher wavenumbers. When symmetry force constant (mdyn/A) F120.0 was used, then the values of F11=7.711 F22=0.722 F33=7.97 have been obtained, by us for D2O by employing the Green’s function formalism and partitioning (GFP) technique. Our values of the wave number assignments are w1=2671 cm-1, w2= 1178.0 cm-1, w3=2788.0cm-1. A finite value of F12=0.2273 was found by us to yield F11=7.783,F22=0.722 and F33=7.97. While the work of Venketeswarulu and Thanalakshmi reported (9), the values F11=7.601 F22=0.918 F12=-0.0035 F33=7.5725. The values of wavenumbers for the D2O molecule adopted in these investigations were almost the same as w1=2669.4 cm-1, w2=1178.38 cm-1 and w3= 2787.92 cm-1 those recently considered by Martin Chaplin (8).
MOLECULAR VIBRATION OF HEAVY WATER:
The molecular vibration and the rotation-vibration interaction would be of great interest in view of its possible presence, in the oceans and seas of other cosmic entities or in the liquid and gaseous matter of extra terrestrial planetary atmospheres. Also the D replacement of H in ordinary water, that gives the heavy water structure, lowered the values of wavenumbers of heavy water relative to the ordinary water. Also the ratios of w1: w2 in heavy water and the ordinary water, viz., 2.265:2.293 differ considerably.
The action of heavy water as a moderator, to slow down the fast neutrons in fission chain nuclear reactor has been successfully implemented world over. The deuterium available in high concentration in sea water nearly 30mg in one liter, which in the form of heavy water, is a constitutive material for sustained thermonuclear plasma experiments.
To the author’s knowledge, only cursory accounts of the study of molecular vibrations of heavy water are available in literature. In this respect that the present work report that the molecular force fields ellipse of heavy water molecular vibrations, is three times more in area than that for W2O molecule seems to be a physically significant finding (6). The object of the present work is to highlight the applicability of the new method of analysis of rotation-vibration interaction features of the heavy water, using the Exact Coriolis Matrix and the enlarged set of Coriolis rotation-vibration interaction constants (10). As has been emphasized, in a previous document it is desirable to assert that unlike, the just two Coriolis components, three constants have become necessary to describe the heavy water rotation-vibration interaction. The three distinct values for the product, biproduct and the sum rules characterize the molecular force field of the heavy water.
THE EXACT CORIOLIS MATRIX FOR HEAVY WATER ( D2O)
The exact Coriolis Matrix for description of rotation-vibration interaction of XY2 type molecule, which involves three Coriolis Coefficients, C12, C13 and C23 instead of, the just two as extensively mentioned and used variedly in the literature.
The characteristic equations involved are as given below.
| F C - E | | - E |=0.
The F signifies the force field symmetry matrix of C2v symmetry of the Heavy Water. Here C stands for the Exact Coriolis Matrix of the molecule.
The usual definition of the Coriolis rotation-vibration interaction constants, namely the -values, is by the expression,
= L-1 C (L-1 )T. For details of notation see reference 10.
An aim of the present work is to report the new findings on the Coriolis rotation-vibration constants, adopting the above secular equations for the heavy water. These also turn out to be three in numbers and on normalization sum of their squares would be unity. The CF matrix of the secular equation, similar to the have been explicitly worked out and the results are presented below in detail. The use of the force constant values reported by both Smith & Linnett and the values given by the present author, has led to two different sets of the Coriolis rotation-vibration constants, . Accordingly the distinct product, biproduct and the sum rule values turn out to be discrete. For the heavy water D2O molecule present work values are F1=7.783 F2=0.722, F12=0.2273, F3=7.93, F21=F12 with mass of Oxygen 16.0 and mass of Deuteron =2.014735 and bond distance of DO is r=0.957000 Angstroms.
The bond angle used is 104.523 degrees and the best value of mixing parameter from our previous reported data is -0.913617. The wavenumbers are w1=2671 cm-1, w2=1178 cm-1, w3=2788 cm-1. The total mass of the molecule is M=20.029470 and d1=0.223442 equals the reciprocal of square root of the total mass M of the molecule. We need to use the value of d2= (2.0*mx*Sin2(/2)+my) -0.5 =0.232361 for obtaining the B=TM-1/2S matrix of the GFP technique. The notation adopted is the same as in the references given below (10,11).
NUMERICAL DATA AND THE ANALYSIS
The inverse L- Matrix for the heavy water molecular vibration has the following non-vanishing values. Li00=-1.352425 Li01=0.025314, Li10=0.184355, Li11=0.938038, and Li22=1.319265. These have been used to get the Coriolis rotation-vibration interaction constants 21=0.018243 22=0.981757 and with the normalization condition that
21 + 22 = 1.0.
Its worth investigating the graphs of the 21 , 22 values variation with the mixing parameter. The results are presented in a graphical form below (Fig.1). From the Fig.1 we note that the variations of 21 , 22 are complimentary to each other. And the curves indicate similarity with familiar dispersion curves of ferrite material (13). This finding has a far-reaching implication. At the mixing parameter value 0.19, the squares of Coriolis rotation-vibration interaction constants seem to be almost the same, since their values are found to be 21=0.507171, 22=0.492829.
THE NEW METHODS OF ANALYSIS:
A new method of analysis is based, on an Exact Coriolis Matrix that involves three coefficients, C12, C13 and C23. The extended set of three rotation-vibration interaction constants of the heavy water are found to have the values 21 = 0.012162 22 =0.654505 and 23 =0.333333. These satisfy the normalization condition that the sum of squares of the values is just unity.
A significant result is that the 22 is now considerably reduced. The new rotation-vibration interaction constant 23 almost acquires the reduction in 22. The variation of 21 , 22 continue to be complimentary to each other, in spite that their magnitudes have changed considerably. At the value 0.2 of the mixing parameter it has been found that the values of the 21 =0.344535, , 22 =0.322132 are approximately equal as one expects from a dispersion curve nature of variation of the constants and the 23 =0.333333.
Another entirely different type of method of analysis explicitly developed in the present work, is the one that employs the symmetry mixed matrix U to determine the Exact Coriolis Matrix. Unlike the previously reported methods of constructing the Coriolis Matrix for example, by Meal & Polo (14) and Cyvin (15), in the present work an entirely new approach has been used to get the Coriolis Matrix. The concept of using the U-matrix to find the Coriolis Matrix involves the fact that the Coriolis Matrix, itself might be dependent on the mixing parameter of the symmetry vibrations. The parity operation that was reported in a previous paper (10) by the present author has been retained to get the Exact Coriolis Matrix. In the new approach, the proper matrix that transforms the set of internal symmetry coordinates into the cartesian representation has been constructed intuitively. The aim of the present construct is to get the Exact Coriolis Matrix. Wolfram and DeWames (12) aimed at construction of U matrix with the idea of obtaining the Green’s functions, for the perturbed and the unperturbed molecules.
At mixing parameter values -0.1 and 0.2, the typical rotation-vibration interaction constants values are found to be as given below. At the mixing parameter value -0.1: zeta1s=21 =0.125195, zeta2s=22 =0.494506, zeta3s=23 =0.380299 and at mixing parameter value 0.2 : zeta1s=21 =0.421127, zeta2s= 22=0.310027, zeta3s= 23 =0.268846 respectively.
The graphical representation is reproduced in the Fig.2.
From Fig.2 it may be noted that the dispersion nature of the constants relative to the mixing parameter is distorted, though the general trend of S-type curves may be found.
THE NEW PRODUCT AND SUM RULES
What is of interest is the nature of the new product and sum rules, for the Coriolis interaction constants 1, 2 and 3. Nature of these values in different environments is of paramount significance.
The characteristic given below becomes a third order for the FC product matrix.
| F C - E | | - E |=0.
For details of notation reference maybe made to 10,14,15 listed below. The determinant of the secular equations would readily yield the product, diproduct and sum rules for the case of adoption of a perfectly skew-symmetric and an Exact Coriolis matrix that involves three non-vanishing coefficients. Here diproduct rule is the sum of products of two individual roots chosen in cyclic order, out of the three possible roots of a third secular determinant equation. It has been already demonstrated, for the case of NO2 in a KBr crystal environment that the said three rules, involve explicitly three magnitude values. In the present work the mentioned magnitude values dependence on the adopted or the chosen mixing parameter, is explicitly brought out for the example, of heavy water.
The characteristic determinant of the matrix secular equation below has ambiguous signs for is elements, unless one restricts himself to the skew-symmetric character of the matrix.
| - E |=0.
Redressal of this secular matrix in to different forms, with different signs for the elements, though has no physical support, leads to possible different rotation-vibration constants. The correct procedure mathematically is, to strictly adhere to the skew-symmetric property of the matrix. An arbitrary assignment of signs of the secular matrix elements needs to be avoided, unless one wants adopt some simulation studies. Given below are the correct rules:
First the product rule for all the three roots of the secular equation for the case of
XY2 molecule is given below. In the case of heavy water D2O molecule, product vanishes. Hence the expression:
2*21 * 22* 23 * 1*2 * 3 =
-22* 22*23 *(22* 1 - 22* 3)- 22* 23*21*(21*1 + 22* 2) -
23* 21*21*(23* 2 + 21* 3)+(23* 1 + 22* 3)*(23* 2 + 21* 3)* (21* 1 + 22* 3)
Secondly, the sum of products of two individual roots out of the three roots of the secular equation, which the later is a polynomial in third degree, is given for the said example as follows. I term this as the diproduct sum and values for it, by different three methods have been tabulated below:
(21* 1 + 22* 2)*(23* 2 +21* 3 +22* 1 - 22* 3) +
(22* 2 + 21* 3)*(23* 1 - 22* 3)-
22* 23*21-23* 21*21 - 22* 22*23
Finally, the sum of all the three possible roots of the third order secular equation of the Coriolis interaction product matrix FC is obtained as below:
-(21* 1 + 22* 2 +23* 2 +21* 3 +23* 1 + 22* 3)
It may be noted some of the signs of the expressions given here differ from those given in an earlier article by me, reference 10 listed below.
From the Fig.3 given below it is very evident the dispersion nature of the
product appearing in the secular determinant. The typical values are listed in a tabular form. Table 1. Presents the sum and the diproduct sum of the secular equation characteristic values.
Table 1. Heavy Water Data on Product & Sum Rules
Mixing Sum Diproduct Sum
-1.0 -8.005674 8.924532
-0.5 -7.963586 8.635751
0.0 -8.988364 15.667048
0.5 -10.490393 25.972907
1.0 -11.164184 30.595980
Table 2. Squres of Rotation-Vibration Constants
Mixing 21 22
-1.0 0.032260 0.967740
-0.5 0.019794 0.980206
0.0 0.323311 0.676689
0.5 0.768178 0.231821
1.0 0.967740 0.032260
An important achievement of the present research is the determination of the rotation-vibration constants of heavy water adopting an entirely different approach. The idea is to construct a newer Exact Coriolis Matrix using the so-called U-matrix, which in turn gives the dependence of the Exact Coriolis Matrix directly on the mixing parameter. Earlier workers thought of studying these constants, as derived from the L-Matrix that of course, yields the G=LLT. The merit of the present method is that skew-symmetric exact Coriolis Matrix, itself involves the mixing commensurate with the mixing of symmetry coordinates. Secondly, in the method we use the B={TM-1/2S} matrix truncated to arrive at the . This procedure avoids the construction of L-matrix unless one wants to construct it specifically. The Table.3 describes the data obtained by using the U-matrix construct of the Exact Coriolis Matrix.
Table. 3 . Squares of Rotation-Vibration Constants by a newer method.
Mixing 21 22 23
-1.0 0.001662 0.380341 0.617997
-0.5 0.000823 0.287924 0.711253
0.0 0.008768 0.181003 0.810229
0.5 0.048040 0.117417 0.834543
1.0 0.110409 0.095230 0.794361
RESULTS:
The main results may be summarized as below. A comparative study of three different methods of utilizing the Coriolis Matrix to determine the rotation-vibration interaction constants has been presented. The results of the mixing and the values of entities 21 , 22, 23 have been tabulated. A significant finding is that the dispersion nature of the rotation-vibration interaction constants arises, as well from the inherent character of the Coriolis matrix.
REFERENCES:
1. K. S. Parthasarathy, Former Secr. AERB, The Hindu, Thursday, April 19, 2007.
2. Robley D. Evans The Atomic Nucleus, McGraw-Hill Book Company, Inc. p.252, 1955
3. K. Rangadhama Rao, D.Sc. Thesis, Madras Univ., 1925
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& Paper No. 51 at 64th Sess. Ind. Sci. Cong. Bhubaneswar, 1977
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10. K.L.Narayana, trusciencetrutechnology@blogspot.com Vol.2007, No.7 dt.
Friday, July 13, 2007 (Note:The product and sum rules expressions need to be examined.)
11 K. L. Narayana et al, J. Inorg. Nucl. Chem. 39, pp. 19-24, 1977
12.T. Wolfram, R.E. DeWames, Bull. Chem. Soc. of Japan, 39, p. 207-214, 1966
13. K. L. Narayana,” Lead Ferrite aluminate doped Dielectric Waveguides”
6th Intern. Conf. On Ferrites,(ICF6),Tokyo, Japan, Paper No. 29, PaI-7. 29-9-92.
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15. S. J. Cyvin, L.Kristiansen, Acta. Chemica. Scand. 16, p.2453-2454, 1962.
7/23/2007 11:24:54 AM
ACKNOWELEDGEMENT:
The author is greatly indebted to (late) Professor K. Rangadhama Rao D.Sc. (Madras). D.Sc. (London) for inspiring guidance and support.
Res Address:
Prof. Dr. K. Lakshmi Narayana,
Retd. Prof. of Physics (SU),
17-11-10, Narasimha Ashram,
Official Colony, Maharanipeta. P. O.
Visakhapatnam-530002, India
Kotcherlakota_l_n@hotmail.com
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