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 manybody 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 {Stn (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)
Sz(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'zSJ+ 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/~ lIz'
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±(tt') = 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)
N1 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)nNnJn(O) (n+n_)n(
n=O
2iS
LV LVo + N1 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
oJo)li+2SC(o>) +i()
Llr»1;2 = 2S Im C (o)u)
4SJ2 :>~ Im X~~ (O)n, Rn).
N2gl2/iB2 "
giVes another secondorder shift.
Discussion
Acknowledgements
Appendix
X~ ((v, q) in the free electron ajJpro.rimation
A. Im x~ (w, q) for o)  O)eJ>O .
(i) If q<ki"'t + kp~,
for 2m I (w we1) I>q2 + 2kFt q
Im X~ (w, q) =
(A ·1) for 2m I (ru · rJ>" 1) I<2kr"' q q2
(ii) If q>kFt + kF~; as given under A (ii).
Rex=: (co, q)
+ 2A+ (kJ?t, q, O) O)eJ)  2A_ (/?rn,, q, (1) (llcJ)],
1 + (mw/q!?p±q/2kF)
1 (1nrv/qkF±q/2kF)
1) A. A. Abrikosov , C. P. Gorkov and I. E. Dzyaloshinski , Method of Quantum Field Theory in Statistical Physics (PrenticeHall , London, 1963 ).
2) V. L. BonchBruevich and S. V. Tyablikov , The Green Function Method in Statistical JI.!Iechanics (Translated from Russian by D. ter Haar , NorthHolland Publishing Company, 1962 ) 0
3) P. Nozieres , Le jwobl'eme c ? N cmj:>s (Dunod, Paris, 196 : 3 ).
4) F. J. Dyson, Phys. l~ev . 102 ( 1956 ), 1217 .
5) R. B. Stinchcombe , G. Horwitz , F. Englert and R. Brout , Phys. Rev . 130 ( 1963 ), 155 .
6) N. N. Bogolyubov and S. V. Tyablikov , Soviet Phys.Doklady 4 ( 1959 ), 589. S. V. Tyablikov , Ukr. Math. Zhur. 11 ( 1959 ), 287. D. N. Zubarev , Soviet Phys.Uspekhi 3 ( 1960 ), 320 .
7) M. Wortis, Ph . D. Thesis , Harvard University, 1963 . R. A. TahirKheli and D. ter Haar, Phys. Rev . 127 ( 1962 ), 88 , 95. R. A TahirKheli , Phys. Rev . 132 ( 1963 ), 689 T. Oguchi and A. Homma , J. Appl. Phys . 34 ( 1963 ), 1153 . K. Tomita and M. Tanaka , Frog. Theor. Phys. 29 ( 1963 ), 528 , 651; 31 ( 1965 ), L
8) T. Kasuya , Progr. Theor. Phys. 16 ( 1956 ), 58 ; 22 ( 1959 ), 227 . K. Yosida, Phys. Rev . 106 ( 1957 ), 893 ; 107 ( 1957 ), 396 . P. Vl. Anderson, Phys. Rev . 124 ( 1961 ), 41 . P. A. Wolff , Phys. Rev . 124 ( 1961 ), 1031 . A. M. Clogston, Phys. Rev . 125 ( 1962 ), 439 . J. Kondo, Prog. Theor. Phys. 28 ( 1962 ), 846 ; 32 ( 1964 ), 37. D. J. Kim and Y. Nagaoka , Prog. Theor. Phys. 30 ( 1963 ), 743. S. Alexander and P. W. Anderson , Phys. Rev . 133 ( 1964 ), A 1594. V. Ambegaokar and A. Griffin , Phys. Rev . 137 ( 1965 ), A 1151 S. H. Liu , Phys. Rev . 137 ( 1965 ), A 1209. T. Moriya, to be published.
9) J. Owen , M. E. Browne , V. Arp and A. F. Kip , J. Phys. Chern. Solids 2 ( 1957 ), 85 . J. Owen , M. E. Browne , W. D. Knight and C. Kittel , Phys. Rev . 102 ( 1956 ), 1501 . A. F. Kip , C. Kittel , A. M. Portis , R. Barton and F. H. Spedding , Phys. Rev . 89 ( 1953 ), 518 . M. Peter and B. T. Matthias , Phys. Rev. Letters 4 ( 1960 ), 449 . V. Jaccarino , B. T. Matthias , M. Peter , H. Suhl and J. JL Wernick, Phys. Rev. Letters 5 ( 1960 )' 221 . M. Peter , D. Shaltiel , J. H. Wernick , H. J. Williams , J. B. Mock and R. C. Sherwood , Phys. Rev . 126 ( 1962 ), 1395. D. Shaltiel , J. H. \¥ernick, H. l Williams and M. Peter , Phys. Rev . 135 ( 1964 ), A 1346. D. Shaltiel and J. H. Wernick, Phys. Rev . 1 : 36 ( 1964 ), A 245.
Perturbation Theo1y for l\1agnetic Inzpurities ill Aietals
10) J. Kondo, Prog. Theor. Phys. 28 ( 1962 ), 846. S. Koicle and M. Peter , Rev. Mod. Phys . 36 ( 1964 ), 160 . R. E. Watson , S. Koide , M. Peter and A. J. Freeman , Phys. I~ev. 139 ( 1965 ), A 167. T. Kasuya, sd and sf Interaction, Article to be published in Treatise on Magnetism Vol . II (SuhlRado Ed ., Academic Press).
11) B. Giovannini , Sci. Pap. Coll. Gen. Educ. U niv. T<,kyo, 15 ( 1965 ), 49 .
12) H. Hasegawa , Prog. Theor. Phys. 21 ( 1959 ), 483 .
13) N. N. Bogolyubov and D. V. Chirkov , Introduction a La tluforie quantique des champs (Dunod , Paris, 1960 ).
14) C. Kittel, Quantum Theory of Solids (J . Wiley, 196 :3) Chap. 18 .
15) H. S. Friedel and A. Guinier (Editors), . Nfetallic Solid Solutions (1962 Orsey Lectures) (Benjamin , 1964 ). B. Giovannini , M. Peter and ]. R. Schrieffer , Phys. I~ev. Letters 12 ( 1964 ), 736 .
16) T. Kasuya , Prog. Theor. Phys. 16 ( 1956 ), 45. S. V. Vonsovski and E. A. Turov , Z. Exper . Teor. Fiz. 24 ( 1953 ), 419 .
17) J. Lindhard, Kgl. Danske Videnskab. Selskab, Mat.Fys. Medel . 28 ( 1954 ), No. 8 .
18) J. Korringa, Physica 16 ( 1950 ), 601 .
19) M. Peter , D. Shaltiel , J. H. Wernick , H. J. Williams , J. B. Mock and R. G. Sherwood , Phys. Rev. ~ 26 ( 1962 ), 1395 (Fig. 5).
20) B. Giovannini, to be published.
21) D. S. Rodbell and T. W. Moon , Proceed£ngs of the International Conference on Jl.;fagnetism (Nottingham , 1964 ), p. 427 .