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The Errors of Approximation in Jeffreys's Phases

Published online by Cambridge University Press:  24 October 2008

F. L. Arnot
F. L. Arnot
Affiliation:
Trinity College; Lecturer in Natural Philosophy, The University, St Andrews
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Extract

The errors occurring in a large number of Jeffreys's phases for neutral chlorine have been determined. A discussion of the source of these errors is given, from which it appears that errors in the phases of zero order and of order unity are due to a cumulative error over the range of integration. A method of correcting for this error is obtained. This method is of easy practical application, and it reduces the error almost to zero for phases of these orders.

The error occurring in higher order phases is found to be mainly due to errors in the two π/4 terms, which take into account the exponential tails of the waves in the totally reflecting region. For phases of high order these errors in the two π/4 terms cancel each other, and consequently the total error in phases of high order is practically zero.

A correction graph for the zero-order phase is given from which the error in this phase for atoms of atomic number up to 36 can be read off. The graph is such that an extrapolation to higher atomic number would not involve much error.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 1 , January 1936 , pp. 161 - 178
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

* Jeffreys, , Proc. Lond. Math. Soc. 23 (1924), 437.Google Scholar

† Arnot, and Baines, , Proc. Roy. Soc. A, 146 (1934), 651.CrossRefGoogle Scholar

‡ Hartree, , Kronig, and Petersen, , Physica, 1 (1934), 895.CrossRefGoogle Scholar

* Hartree, , Kronig, and Petersen, , Physica, 1 (1934), 895.CrossRefGoogle Scholar

* Arnot, and Baines, , Proc. Roy. Soc. A, 146 (1934), 651.CrossRefGoogle Scholar

* In the following discussion it should be remembered that the correction term in (16) and (17) is only a first approximation correction for the neglect of F″ and f″ owing to the approximations made in determining it.

* The error has a sign opposite to that of the correction. A positive error means that Jeffreys's phase is too large.

* For l = 0 there are no π/4 terms in the expressions (14) and (15), and hence the error is entirely due to neglect of F″. See also p. 174.

* Arnot, and Baines, , Proc. Roy. Soc. A, 146 (1934), 651.CrossRefGoogle Scholar

* The error of 2·740 given by Arnot and Baines for krypton was determined using the Hartree field with Holtsmark's polarization correction. This polarization correction causes the error in δ0 for k = 0 to be much larger than when the Hartree field alone is used. Consequently δ0 for k = 0 has been recalculated using the Hartree field without any polarization correction. The error is then found to be 1·075. This value has been used in Fig. 6. The errors for the other rare gases given by Arnot and Baines were calculated using the Hartree field without any polarization correction.