Results are presented for the lattice parameter and magnetic susceptibility as a function of temperature for a series of solid solutions of SmS with SmP. At room temperature the SmS~"P"alloys with x «0.06 exhibit a metallic appearance and a much smaller lattice constant and magnetic susceptibility as compared with SmS. For x «0.15 the susceptibilities do not show a Curie-law divergence or magnetic ordering. These properties are consistent with the Sm ions being in a homogeneous intermediate-valence state similar to that of SmS under pressure. The intermediate valence estimated from the lattice parameter is 2.8 at x =0.06 and increases toward the pure trivalent value with increasing phosphorous concentration. Also, the color of the alloys changes from gold to red to silver grey. The alloys with 0.03 & x (0.06 exhibit a varying mixture of two distinct phases, the gold intermediate-valence phase and the black phase characteristic of the x (0.03 alloys. The properties of the black phase are consistent with a mixture of Sm + and Sm + ions where Sm ions with one or more P ions as nearest neighbors are trivalent. The 0.04~x «0.08 alloys exhibit a transition from the gold phase to the black phase with decreasing temperature. This transition which is continuous at higher concentrations changes to a discontinuous one near x =0.06 as seen in the temperature variation of the lattice parameter.
The intensity distribution in a Fraunhofer single-slit diffraction pattern is described by transcendental equations of the form x=C sinx and x=C tanx. The first equation appears in the solution for the width of the central maximum of the pattern. The second results from an attempt to locate the positions of the secondary maxima. Students who are used to explicit (closed) solutions of equations are often distressed by the apparent need for trial-and-error methods. Graphical solutions give a good ’’feel’’ for the situation, but are not satisfying to those who would like greater precision. Numerical methods, which rely only on introductory calculus, can be easily applied. Algorithms employing pocket calculators are presented.
Experiments designed to illustrate the principles embodied in Ampere's Law and Faraday's Law often depend on the knowledge of the number of turns of wire on various types of laboratory coils. The lack of direct measurement of this parameter can be overcome with the use of inexpensive ͑Ͻ$85͒ digital clamp-on ammeters. The meters themselves illustrate the connection between the current enclosed by a closed path and the line integral of the magnetic intensity H around that contour. In this paper we present laboratory exercises that make essential use of clamp-on ammeters.
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