Perturbation Theory for Magnetic Impurities in Metals

Progress of Theoretical Physics, Nov 1965

A perturbation method similar to those adopted in the many-body problems and the corresponding diagram technique are proposed for systems of spins interacting individually with each other and/or with other degrees of freedom. A generalization of Wick's theorem is introduced and illustrated by simple examples. By using this new method, the ESR spectra of magnetic impurities in metals are discussed. It is shown that shift and broadening of the resonance spectra are related to the dynamical susceptibilities of the pure host metals and that the effect of conduction electron spin relaxations on the impurity ESR spectra can be handled neatly by assuming an appropriate form for the expression of the dynamical susceptibilities.

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Perturbation Theory for Magnetic Impurities in Metals

B. Gio 0 vannin£ 0 S. I<..o£de 0 0 Institute of ExjJerimental Physics, University of Geneva · Geneva , Switzerland Progress of Theoretical Physics, Vol. 34, No. 5, November 19(:)5 Perturbation Theory for Magnetic Impurities in Metals Bernard GIOVANNINI and Shoichiro KOII)E*l A perturbation method similar to those adopted in the many-body problems and the corresponding diagram technique are proposed for systems of spins interacting individually with each other and/or with other degrees of freedom. A generalization of Wick's theorem is introduced and illustrated by simple examples. By using this new method, the ESR spectra of magnetic impurities in metals are discussed. It is shown that shift and broadening of the resonance spectra are related to the dynamical susceptibilities of the pure host metals and that the effect of conduction electron spin relaxations on the impurity ESR spectra can be handled neatly by assuming an appropriate form for the expression of the dynamical susceptibilities. - Part I. Perturbation theory for spin systems Generalized Wick''s theorcn1 (QjT{A(tl)B(t2) ···P(tn)} jQ), ,tJ[o = - J1JJY "ffz L S/t n IS the Zeeman energy and the perturbation Here S/~ (t) ._~'_n' (t')---' T {S-tn (t) S'_n' (t')} - N {,~/+n (t) S.__ n' (t')} where 0 (r) is the step function defined by 0 (r) = for r>O for r<O. S+n(t)S_n' (t'; tr, ... , t.i) with the definitions Possible sets in the normal form Contractions leading to them S/~, (t') s_n" (t") Szn (t) S/'' (t') s_n" (t") s+n' (t') s_n" (t") From the generalized Wick's theorem we get T {S/': (t') Sz1' (t) s_n" (t")} +Sz"l{ -2exp[i!1ngHz(t" -t')]O(t" -t')on"n'} X X [tJ (t- t') Onn'- (} (t- t") (} (t- t') Onn"J. The contractions S.+n(t) S zn' (t 1 ) Szn" (t 11 ) 5\n(t)S,/'/ (t')S/~" (t") I 1I_1 I Diagram technique 3£1= -J~Sj·S'·= -J~S/S/--J~S/S_'· j"'-'r her j~r =A+B, B. Gio·z,annini and S. !{aide where IP'o) 1s the ground state of JJ[0 + /J-[1 and (b) S-z--------(d) «>--------------------Fig. 1. Symbols used in the diagrams to represent spin contractions : (a) s+s- contraction I I I I 0 (t"- t') l)nn'. -0 (t- t') Onn'. I I Fig. 3. Example of a diagram with closed loop of' S'-z-SJ+ lines. This kind of diagram is meaningless. - ·-f+Q~ (d~ (f~ (h~ (:J7 (~ (~ (f) (\ d - ) . (:L Fig. 5. Diagrams constructed from part B of the Hamiltonian .9(1 given by Eq. (6). Fig. 4. Diagrams· constructed from part A of the Hamiltonian ,!i{i given by Eq. (6). - 4iJNonn'f) (t- t') exp [- iu)0 (t- t')] <Sz? (t- t'). The sum of Eqs. (8) and (9) gives Contribution from Fig. 5 is X() (t1- t') () (t- t1) (Fig. 4a) + (Fig. 4c) (Fig. 4b) + (Fig. 4d} Part H. ESR of spin impurities in metals ~ 4. · llamiltonian and dynamical susceptibilities c!Jioi =-/in g~ S/~ l-Iz' Tl M+ (t) = Y!ln<?J! (t) IL::S~ (t) n I?f! (t)) n 'X:r (t -- t') = For the calculation, however, the quantity defined by Rex+ (w) =Rex+c (w), Hereafter we shall omit the superscript c. Thus we have to calculate the follow­ mg propagators: Tn±(t-t') = -i~<?J!oiT{S/~(t)n a~(k, t')na±(k,t')n}IP'o), /;; Dnn' (r) = i<S_n S,_n)onn' exp ( -icuor) 0 (r) - i<S+n S_n)onn' exp (- ioJor) 0 (- r), where CDo= -g!lnHz 2(Sz).,_ U,~nn/ , uJ-(vo+zo § 5. First order effects where No IS the number of impurities. N- 1 iJ(k- k') exp [i (k -/c') · Rn] . = J(O) N-\ (n+ -JL) iD0 (t- t'). Taking the Fourier transform, we have - N- 1 J (0) (n+- n_) iD0 (rv) N-1 J (0) (nj- - n_) . nu (u>) = 2S w- Wo + N- 1J (0) (n+- n_) + io Let us consider the following series: Since we get iDo(LV)i=(-1)nN-nJn(O) (n+-n_)n( n=O 2iS LV- LVo + N-1 J(O) (n+- n_) + io = iDII (LV). gives the contribution x~ (LV, q) X~l (LV, q = 0) = g 13/4_~-7CB32rL_Jz we can see = -Eq. (21). § 6. Second order effects >< 21 [ x~\ (w= 0, q) + x~1-~- ( w= 0, q) ] , . -. 4SJ2 'L..'.i xczl (U) = 0, R n') N2!le2gt2 n' This giVes Dnn' (w) the following contribution: 1 L".-.J' X~"-l (~c·u', R n- AIf..>'-n') = ·28- "L.-.J' Cn (U)) , where en (w) is defined by i[ -JJH(w)Ji+l Cj((J))No, with The real part The sum over the complete bubble series then gives Nail) H (w) [1- Dll (w) C (ru) + (l)H ((1)) C ({0)) 2 + ... J D*((o) = 2S oJ-o)li+2SC(o>) -+-i() Llr»1;2 = 2S Im C (o)u) 4SJ2 :>-~ Im X~~ (O)n, Rn). N2gl2/iB2 " giVes another second-order shift. 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Bernard Giovannini, Shoichiro Koide. Perturbation Theory for Magnetic Impurities in Metals, Progress of Theoretical Physics, 1965, 705-725, DOI: 10.1143/PTP.34.705