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An earlier study 1 has demonstrated the appearance of Pendellbsung fringe structure within the Bragg reflection of neutrons from perfect crystals of silicon under suitably controlled physical conditions. This interference-fringe structure arises from the coherent addition of different traveling waves in the crystal while under Bragg reflecting conditions. It was assumed in the above study that the coherent neutron wave field was to be described as a plane wave, and accordingly the Pendellbsung fringe characteristics were interpreted quantitatively on the basis of plane wave (PW) theory. It has been shown, however, by Kato 2 that the treatment of dynamical diffraction theory with spherically coherent wave fields, spherical-wave theory (SW), leads to small but not insignificant differences relative to the PW treatment in the interference characteristics, Moreover, a series of x-ray diffraction experiments have been performed by Kato and by others which demonstrate the applicability of either SW or PW theory, depending upon experimental conditions. Accordingly, the earlier neutron study has been extended to test the applicability of the theory and to improve materially the lished. 13 The standard cofactor inversion of a matrix leads to the development of a continued fraction. 14 R. Haydock and M. J. Kelly, to be published. 15 quantitative interpretation of the fringe structure, leading to a precision determination of the silicon scattering amplitude. It has been found that, under the conditions of the experiment, indeed SW theory must be used to describe the neutron fringe characteristics. The experiments have been performed with collimated neutron radiation being delivered to a symmetrically transmitting crystal (Laue geometry) as described in the inset of Fig. 1. An exit slit opening placed on the back side of the parallel-faced crystal plate can be used to scan over the linear width of the Bragg reflection. With PW theory, the distribution of intensity across the band of Bragg reflection as a function of the parameter y = tane/tantf is given by /pw(y) =C(1y r i/2 sinlA(l-yr l/2 tan0], (1)where A = 2tNF hkl d, with t the crystal thickness, d the interplanar spacing, N the unit cell density, 9 the Bragg angle, and F hkl the crystal structure factor per unit cell. F hkl depends upon the atomic scattering amplitude b; and for the silicon (111) reflection characteristic of all of the present re-We have studied neutron Pendellosung fringe patterns within the Bragg reflection from perfect silicon crystals. The patterns as obtained with a full-spectrum incident beam show that the coherent wave front of the neutron radiation must be considered as spherical rather than planar, and also that the crystals under study possess a uniformly curved distortion. The extended fringe patterns permit a precision determination of the coherent nuclear scattering amplitude for silicon of 0.414 91(10) x 10" 12 cm per nucleus. 871

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