By Clyde A. Morrison

From December 1985 via March 1986 the textual content of this booklet shaped the root of an in-hours path taught by means of the writer at Harry Diamond Laborato ries. substantial information in revising and organizing the 1st draft was once given by means of John Bruno. the unique draft of those notes was once in keeping with a set of lectures added on the Universidade Federal de Pernambuco, Recife, Brazil, among 2 November 1981 and a pair of December 1981. The stopover at to Recife used to be 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 past resul ts and memories of our early paintings. between these consul ted have been Donald Wortman, Nick Karayianis, and Richard Leavitt. additional, 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 support. Encouragement and tips with investment for far of this paintings got here from Leon Esterowitz of the Naval study Laboratory and Rudolph Buser and Albert Pinto of the heart for evening imaginative and prescient and Electro-Optics.

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**Example text**

1 we xonsidered the crystal-field Hamiltonian in the form HCEF = B20C20(r) + B40C40(r) and obtained the energy splittings of a single d electron. All the previous calculation was for the orbital states ~nly, neglecting the spin-orbit coupling. For the states Ijm> with j = t + ~ we showed (in sect. +s) , 2, 4 , (-1 )k+~+j' 56 +s 11. } s j'j' {k and the quantity { } is a 6-j symbol which can be found in Rotenberg et al (1969). Using the above results calculate and obtain the following table for the matrix elements of the crystal field.

38 5. RACAH COEFFICIENTS The Racah coefficients arise in the coupling of three angular momenta (Rose, 1957, p 107) to form a final resultant. In the coupling of the angular momenta, we consider two coupling schemes: scheme A: ( 5. 4) The coupling schemes A and B are connected by a unitary transformation I B> .. 4). 8) which is a relationship used often in our analysis. c(o» f (Rose, 1957). 7). 2 j3) must be triangle. That is, The four triads able to form ci with similar relations for the other triads.

A spherical component of the angular momentum of an electron, has the matrix elements from the Wigner-Eckart theorem. Also we have ~ol~m> = ml~m>. Using a table of C-G coefficients (see Rose, 1957, appendix), evaluate <~(m)1 (0) I i(m» and then show that 2. The tensor Tkq has the property element T~q (-1)qT k-q Consider the matrix * and its hermitian conjugate By using the Wigner-Eckart theorem on the above matrix element and its hermitian conjugate show that 31 In many books the reduced matrix elements are written (1'ITkI1) with so that solid. *