• S Parthasarathy

Articles written in Pramana – Journal of Physics

• Probability distributions of the fractional differences of the structure amplitudes and intensities of two related crystals

The probability distributions of the fractional intensities and amplitudes of x-ray reflections from a pair of imperfectly related structures are derived when both the structures satisfy the requirements of a given type of basic Wilson distribution. These two distributions are used to obtain theoretical expressions for 2 new fractional type ofR-indices which are expected to be useful in the final stages of refinement. The theoretical distributions are also used to deduce some theoretical distributions which are useful as tests for centrosymmetry via the random permutation method. The theoretical values of the relevant semi-cumulative functions are also tabulated.

• Overall values of conventional discrepancy indices for crystals with heavy atoms. Part I—Results for a triclinic crystal when the model consists of the heavy atoms and a part of light atoms

The joint probability density functions of the normalized structure amplitudes of the structure and the model (i.e.,yN andypc) are derived for triclinic crystals containing heavy atoms (1, 2 and many) by taking the model to consist of the heavy atoms and a part of the light atoms in the unit cell. These functions are derived for the two cases where the model is completely correct (i.e., the related case) and where the model is completely wrong (i.e., the unrelated case) in terms of the fractional contributions to the local mean intensity from the heavy atoms and all known atoms (i.e., σ1h/2 and σ12) as parameters. These functions are then used to obtain the theoretical local values of 〈yN〉 and 〈|yNn − σ1n(yPc)n|〉,n=1, 2. A method of using these results to compute the theoretical overall values ofR(F) andR(I) for the related and unrelated cases is briefly described. A comparison of the observed values of these indices with their theoretical values for the related and unrelated cases would help in determining the correctness of the proposed trial structure.

• Overall values of conventional discrepancy indices for crystals with heavy atoms. Part II. Results for monoclinic and orthorhombic crystals when the heavy-atom part alone constitutes the model

Theoretical expressions of 〈yN〉, 〈|yN − σ1yPc |〉 and 〈|yN2σ12 (yPc)2|〉 (whereyN andyPc are the normalized structure amplitudes of the structure and the model respectively) are derived in terms of the heavy atom contributionσ12 for monoclinic and orthorhombic crystals containing a few (i.e., 1 or 2) heavy atoms of the same kind per asymmetric unit by taking the heavy atom part alone as the model. Results are obtained for both the related and unrelated cases. The local values of 〈yN〉 and 〈|yNn − σ1n (yPc)n|〉, (n=1, 2) calculated from these expressions can be used to calculate the overall values of the conventionalR-indicesR(F) andR(I) for the related and unrelated cases. These overall values could be used to check the correctness of heavy atoms located in the structure.

• Overall values of conventional discrepancy indices for related and unrelated models of a non-centrosymmetric crystal with a centrosymmetric group

Theoretical expressions for the overall values of the conventional discrepancy indicesR(F) andR(I) are derived for a non-centrosymmetric crystal with a centrosymmetric group by taking the centrosymmetric group and a part of the other atoms in the unit cell as the trial structure. These results are used to obtain tables of values of these indices in terms of the parameter σ1c2 and σ12 which define the fractional contribution to the local mean intensity from the centrosymmetric group and all the known atoms respectively.

• Theoretical evaluation of the overall values of unnormalized discrepancy indices for truncated data in crystals obeying Wilson distributions

Theoretical expressions for the local values of six types of unnormalized R-indices are derived for an imperfectly related incomplete model of a crystal (centrosymmetric and non-centrosymmetric) with truncated data which is characterized by a truncation limit yt. These indices depend on the parameters σ1, D and yt. In the situations of practical interest (i.e., σ12&gt;0·3 and yt&lt;0·2) R-indices for the centro-symmetric case decrease as ytincreases while these for the non-centro-symmetric case remain more or less constant.

• Theoretical evaluation of the probability of success of the quasi-anomalous method

Theoretical expressions for the probability of success of the quasi-anomalous method have been derived for triclinic, monoclinic and orthorhombic crystals containing a few (1, 2 or 3) heavy atoms per asymmetric unit besides a large number of light atoms. The results derived take into account data-truncation due to unobserved reflections. Using the theoretical expressions, tables of probability values for the success of the quasi-anomalous method are obtained as a function of the relevant parametersk andδ12. Corresponding results for triclinic crystals containing many heavy atoms (i.e.P = MN and MC cases) have also been obtained. It is seen that, using suitable heavy atoms to prepare the heavy-atom derivative, probability of success as high as 0.7 could be obtained in the case of proteins containing 1000 to 1500 atoms.

• Theoretical evaluation of the overall values of Booth type R-indices based on intensity variables

Theoretical expressions for the overall values of three Booth type R-indices based on intensity variables are derived. The results are applicable to crystals of any space group containing any number and type of atoms at general positions in the asymmetric unit. The theoretical results were tested in the case of models of a few crystal structures.

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