Hyperfine Interactions 8 (1981 ) 471-474 Q North-HoLlandPublishing Company

~ON

KNIGHT SHIFTS IN METALS*

E. Zaremba and D. Zobin Department of Physics, Queen's University Kingston, Canada, K7L 3N6

The ~SR technique often is intended as a probe of the intrinsic properties of the host into which the muon is implanted. However, because of its charge and associated electronic screening, experiments usually reveal a convolution of host and local defect properties. Rather than simply being a nuisance, this aspect is in fact of interest in the theory of defects in solids. Being structureless, the muon is in some sense the ideal defect and should be one of the simplest to deal with theoretically. Thus an understanding of this defect system is important both for the purpose of extracting useful information from ~SR experiments and in developing a general theory of defects in solids. In this regard the measurement of muon Knight shifts in metals [I] has been of considerable value in providing a stringent test of such theories. The muon precession frequency shift in metals consists of two contributions: the conventional Knight shift due to the Fermi contact interaction and a shift due to diamagnetic shielding. The former can be expressed in terms of the electron-spin magnetic-moment density, m(0), at the position of the muon [2], 87 m(0) 3 Bo

Ks

(i)

where B is the applied magnetic field. This contribution has been calculated for a j~llium model of the metal by a number of workers [3,4] using the spin-density functional theory of Hohenberg, Kohn and Sham [5]. The diamagnetic

frequency

shift is given quite generally by [2]

Kd = B ( 0 ) / B o where the shielding

~(0)

(2)

field at the position of the muon is i

§

= --~ f d r

~

•

§ §

r~

(3)

§247 is the diamagnetic current induced by the external magnetic field. An approximate evaluation of (2) has previously been made [6] only in the case of Be using an imbedded cluster method. Since the muon Knight shift calculations have been based on the jellium m o d e ~ we have evaluated the corresponding diamagnetic shielding in the same approximation. This should be suitable for the simple metals such as Na and Mg which have weak pseudopotentials. We have also reconsidered the muon Knight shift problem so as to emphasize the computational similarity of the two calculations. The muon Knight shift K can be obtained by solving an integral equation for the magnetization density [ 4 : m(7) = fJd T ' x ( ~ , 7 '

§ )[Bo+Wxc(r+' )m(r')]. 471

(4)

472

E. Zaremba, D. Zobin / Muon Knight Shifts in Metals +

§

Here w (r)m(r) is the local field due to exchange and correlation in the density KC + + functional theory and x(r,r') is the independent particle density response functlon of the system which can be expressed in terms of single-particle Green's functions:

x(r,r') The G r e e n ' s 9 eigenvalues

function 6• by

= -21 is

given

G+ ( +r , r §' , ~ )

G ( r , r ' , m ) ~ ( ~ ' , r , m ) .§ in terms

(5)

of single-particle

wavefunctions

~i(r)

and

= Z ~i(r)~i*(r') i m+iq-s i

(6)

For the jellium model the enhancement factor Ps(O) = m(O)/m(~) has been determined from (4) and the results are plotted as a function of the electron density parameter r" in Fig. i. Also shown for comparison S

I I I I I I 1

120 o

12

100 xT

0

8e 613 & ~

,

s

/ 1

~

2 ///I

1

I

2

I

3

I

4 rs/a~

I

5

3L~ ~r 44

55

66 --

I

6

Fig. I: SFin enhancement at muon vs. r s. Ref. 3 , O ; Ref. 4, [].

Fi$. 2: Muon Knight s h i f t s .

Ks,

solid line. Ks+Kd, dashed line. SSM (• for tetrahedral (T) and octahedral (0) sites (including K d), Ref. 4. Experiment Ref. i.

are the results of Munjal and Petzinger [3] and Manninen and Nieminen [4].As previously noted, these enhancement factors lead to a serious overestimate of the muon Knight shifts as indicated by the solid line in Fig. 2. The jellium model of course excludes the periodic structure of the host which can be included to some extent in the spherical solid model (SSM)[4]. Although t h e results improve within the SSM, discrepancies remain, as for example, the observed negative frequency shifts for Li, Be and Sr. These indicate an important diamagnetic contribution. The evaluation of K d proceeds via a linear response theory calculation of the diamagnetic current [8]. e2 § e2 ~ ~ § § § § = ~-~c A (~) ~ - -c- J I d r ' x ~ (r,r')A v ( r ) .

(7)

Here is the ground state electronic density in the presence of the muon, A(r) - ~2B~ ~ ~ is the external vector potential and

E. Zarernba, D. Zobin / Muon Knight Shifts in Metals

Xu~(r,r ) is the zero-frequency response function

time-Fourier

473

transform of the retarded

XDv(7,7',t ) = -i/~@(t)<[j~(7, t),jv(7',o)]> o. ++ For a system of independent particles X~m(r,r') can be expressed to (5) as

current

(8) in a form similar

sF § § XDv(r,r')

~ = ~-Im Imj ~ _~

d~[(Vl'-Vl) ~

By isolating the diamagnetic finally obtain

Kd = - 3 mec2 ~

§ § + (r2,rl,~)] § + (V2'-V2)vG + (r{,r2,e)G § +, + . rl=rl=r + +1 +1 r2=r2=r

currents dependent

(9)

on the presence of the muon, we

~ 1 4~r2An(r)d r

7

+ e3m~c/~ dr -1- -2kF ~ Z(s r ~ ~=l

[Rs2

r )-Jg(kFr)]' .2

(i0)

The first term in (I0) is the usual Lamb chemical shift familiar in atomic theory, with the screening charge density An(~)= ~ - n around the muon playing the role of the atomic density. The second term in (i0) is a new contribution arising from the presence of a continuum of occupied states. It in fact depends only on Fermi surface properties such as the Fermi wavevector k F and the radial electronic wavefunctions RLk F (r) at the Fermi energy. A similar kind of result was obtained by Kohn and Luming [9] in their calculation of the diamagnetic susceptibility of alloys. K d was evaluated numerically using the charge density and Fermi energy wavefunctions obtained in a self-consistent screening calculation for a muon in jellium. The summation over s converges rapidly and accurate values were obtained retaining ~-values up to ~=6. The dominant contribution to K d comes from the first term in (i0), is relatively density independent and has a value close to the chemical shift of a hydrogen atom (-17.8 ppm). The second term in (i0) is only appreciable, because of the factor kF, at higher densities where it is of opposite sign to the first. The factors multiplying i/r in the two integrands in (i0) are plotted for rs=l.75 and rs=4 in Fig. 3 to indicate the relative importance of these two terms. Both combined give a shift of about -20 ppm in the metallic density range. The jellium results including the diamagnetic shielding are also plotted in Fig. 2. There is generally improved agreement with experiment although the electron gas results still overestimate the frequency shift. Inclusion of the lattice effects in the SSM [4] improves the agreement further, partienlarly for Na, but significant discrepancies remain. It would appear that the main shortcoming of the calculations to date are the approximate treatment of the host lattice potential. In our formulation of the Knight shift and diamagnetic shielding, refinements to the jellium theory can be included by using the correct defect Green's function in (5) and (9). Recent advances [I0] in the calculation of such quantities will facilitate these improvements and extensions in this direction are planned. *Supported Canada.

by a grant from the Natural Sciences and Engineering

Research Council of

E. Zaremba, D. Zobin / Muon Knight Shifts in Metals

474

O.4

Fig. 3: Charge Density vs. distance in atomic units. 4~r2An(r): solid (rs=l.75), dash-dot (rs=4.0). 2kF/~Z%(%+l)(2%+l)[R~kF(r)-j~(kFr)]: dash (rs=l.75), dash-double dot (rs=4.0).

"\

/ \

:~-

~o,(..~

/

~o

e.o

,o.o

REFERENCES [ i] M. Camani, F.N. Gygax, W. R~egg, A. Schenck, H. Schilling, E. Klempt, R. Schulze and H. Wolf, Phys. Rev. Lett. 42 (1979) 679-682. [ 2] C.P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, Berlin, 1978),2nd ed. [ 3] R. Munjal and K. Petzinger, Hyperfine Interact. 4 (1978) 301-306. [ 4] M. Mafininen and R.M. Nieminen, J. Phys. F 9 (19797 1333-1348. [ 5] P.C. Hohenberg and W, Kohn, Phys. Rev. 136 (1964) B864-B871; W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) AII33-AII38. [ 6] J. Keller and A. Schenck, Hyperfine Interact. 6 (1979) 39-41. [ 7] E. Zaremba and D. Zobin, Phys. Rev. Left. 44 (1980) 175-178. [ 8] E. Zaremba and D. Zobin, to be published. [ 9] W. Kohn and M. Luming, J. Phys. Chem. Solids 24 (1963) 851-862. [I0] R. Zeller and P.H. Dederichs, Phys. Rev. Lett. 42 (1979) 1713-1716.