By Clyde A. Morrison

From December 1985 via March 1986 the textual content of this booklet shaped the foundation of an in-hours path taught by means of the writer at Harry Diamond Laborato ries. massive assistance in revising and organizing the 1st draft used to be given by way of John Bruno. the unique draft of those notes used to be according to a suite of lectures introduced on the Universidade Federal de Pernambuco, Recife, Brazil, among 2 November 1981 and a pair of December 1981. The stopover at to Recife was once a reaction to an invi tation of Professor Gilberto F. de Sa of the Physics division. within the guidance of those notes I made many requests of my coworkers for prior resul ts and memories of our early paintings. between these consul ted have been Donald Wortman, Nick Karayianis, and Richard Leavitt. extra, a few .suggestions from my Brazilian colleagues helped make the lectures extra transparent. specific between those have been Professor Oscar Malta and Professor Alfredo A. da Gama either one of whom I desire to thank for his or her aid. Encouragement and advice with investment for far of this paintings got here from Leon Esterowitz of the Naval learn Laboratory and Rudolph Buser and Albert Pinto of the guts for evening imaginative and prescient and Electro-Optics.

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Using the above results calculate and obtain the following table for the matrix elements of the crystal field. m' j' j 1 3 3 m B20 1 B40 2" 2" 2" 2" 1/5 0 2" 2" 2" 2" 8/35 2/21 2" 2" 2" -/6/35 -2/6/21 1 5 5 1 3 2" 3 5 3 2" 3 -1/5 0 2" 2" 2" 3 2/35 -117 2" 2" 2" 5 -217 1/21 2" 2" 3 -6/35 2/21 2" 2" 2" 5 5 3 5 5 5 5 3 3 2" 1 2" 2" 3 1 2" To obtain the energy levels we let and Then, for the m H 11 + H s-o (j = 1) 2 = 1 2" levels, we have the secular equation E = 57 0 with a similar result for m = ~.

For K = 0, 1, 2, 3. 2 1/2 Bibliography and References Brink, D. , and G. R. , pp 40-45 and 116-118. Edmonds, A. R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, NJ, pp 92-100. Rose, M. E. (1957), Elementary Theory of Angular Momentum, Wiley, New York, NY, chapter VI, also pp 225-227. , R. Bevins, N. Metropolis, and J. K. Wooten, Jr. (1969), The 3-j and 6-j Symbols, MIT Press, Cambridge, MA. Sobelman, 1. 1. (1979), Atomic Spectra and Radiative Transitions, SpringerVerlag, New York, NY, pp 66-72.

28). 8). enient to build spherical tensors of higher rank by coupling products of angular momentum operators. I. I. a q-a (a) where + Tkq = in which, by the properties of the C-G coefficient, k is restricted to 0 ~ k ~ 2. The Tk~ thus constructed is patently a spherical tensor of rank k, projection q. 1» gives (b) with m' = m + q. 1. I. IIR. and the C-G coefficient requires mil = m' - a. Substituting these results into (d) gives where we have used the same technique used in (e) on the second matrix element in (d).