Beta (velocity)

Abstract: We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.

Abstract: Analysis of 24 years of lunar laser ranging data is used to test the principle of equivalence, geodetic precession, the PPN parameters beta and gamma, and G/G. Recent data can be fitted with a rms scatter of 3 cm. (a) Using the Nordtvedt effect to test the principle of equivalence, it is found that the Moon and Earth accelerate alike in the Sun's field. The relative accelerations match to within 5 x 10(exp -13) . This limit, combined with an independent determination of y from planetary time delay, gives beta. Including the uncertainty due to compositional differences, the parameter beta differs from unity by no more than 0.0014; and, if the weak equivalence principle is satisfied, the difference is no more than 0.0006. (b) Geodetic precession matches its expected 19.2 marc sec/yr rate within 0.7%. This corresponds to a 1% test of gamma. (c) Apart from the Nordtvedt effect, beta and gamma can be tested from their influence on the lunar orbit. It is argued theoretically that the linear combination 0.8(beta) + 1.4(gamma) can be tested at the 1% level of accuracy. For solutions using numerically derived partial derivatives, higher sensitivity is found. Both 6 and y match the values of general relativity to within 0.005, and the linear combination beta+ gamma matches to within 0,003, but caution is advised due to the lack of theoretical understanding of these sensitivities. (d) No evidence for a changing gravitational constant is found, with absolute value of G/G less than or equal to 8 x lO(exp -12)/yr. There is significant sensitivity to G/G through solar perturbations on the lunar orbit.

Abstract: Bragg , Got t f r i ed and Wes t 3 ) have attempted the X-ray analysis of this crystal on the assumption of a formula \ N a j J · 11 \ A I Z 0 3 , agreeing with the best chemical analysis then available. They were not able to devise a structure which was completely satisfactory, but were led to suggest an ideal structure with a composition Ν α 2 0 · i l A l 2 0 3 to which "beta alumina" might tend. Since such a structure is now satisfactory from chemical analysis and from the density, a test can be made of its fit with X-ray intensities. This paper deals with the carrying out of such a test on N a 2 0 · 11 A l 2 0 3 itself, and on the isomorphous K 2 0 · H A 1 2 0 3 which has been prepared by the Norton Company. The unit cells of N a 2 0 · 11 A l 2 0 3 and K 2 0 · 11 A l 2 0 3 are hexagonal with a0 = 5.584, c0 = 22.45Ä, and a0 = 5.584, c0 = 22.67A, respectively. The space group is C6/mmc (Z)gA). The only point which is in doubt in the ideal structure suggested by Bragg , G o t t f r i e d and W e s t is the position of the Ν a (or K) atoms in the mirror planes. If the origin is taken at one of the centres of symmetry the Ν a may be at either (00 J ) or (§ 3 έ)· The present work suggests that the Ν a is actually upon the second of these two positions, so that the parameters of the structure become:

Abstract: This paper contains several graphs and nomographs which make it possible to obtain, very quickly, $log(\mathrm{ft})$ values for most $\ensuremath{\beta}$-decays. The use of these figures is discussed.