SPIN DISTRIBUTION STUDIES IN FERROMAGNETIC METALS BY POLARIZED POSITRON ANNIHILATION EXPERIMENTS

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SPIN DISTRIBUTION STUDIES IN

FERROMAGNETIC METALS BY POLARIZED POSITRON ANNIHILATION EXPERIMENTS

S. Berko, A. Mills

To cite this version:

S. Berko, A. Mills. SPIN DISTRIBUTION STUDIES IN FERROMAGNETIC METALS BY PO- LARIZED POSITRON ANNIHILATION EXPERIMENTS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-287-C1-289. �10.1051/jphyscol:1971197�. �jpa-00214524�

JOURNAL DE PHYSIQUE Colloque C I , supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 287

SPIN DISTRIBUTION STUDIES IN FERROMAGNETIC METALS BY POLARIZED POSITRON ANNIHILATION EXPERIMENTS (*)

S. BERKO and A. P. MILLS

Brandeis University, Waltham, Massachusetts, 02154

RksumB. - On a effectuk des mesures de polarisation a trois photons P B ~ a annihilation de positrons, dans Gd, Fe et Ni. On a utilise les valeurs de P3, pour renormaliser les expkriences de corrklation angulaire 2 y et on a obtenu ainsi la distribution des moments des klectrons B. spin alignb dans des monocristaux de Gd, Fe et Ni. On trouve des anisotropies importantes dans ces metaux et il semble que dans fa region des moments faibles, I'annihilation des positrons se fasse avec des Blectrons polarisks nkgativement.

Abstract. - Three photon polarization measurements, P3,, were performed with positrons annihilating in Gd, Fe and Ni. The values of P3, are used to renormalize the 2 y angular correlation experiments thus obtaining the momen- tum distribution of the spin aligned electrons in single crystals of Gd, Fe and Ni. The results show marked anisotropies in these metals, and indicate that in the low momentum region the positrons annihilate with negatively polarized electrons.

During the last decade the technique of positron annihilation has been used extensively to study the momentum distribution of electrons in solids. Due to selection rules, spin singlet positron-electron overlaps annihilate via two photons, triplets via three photons.

Thus the 2 y angular distributions obtained from pola- rized positron annihilation yield information about the momentum distribution of the spin aligned electrons in ferromagnetic materials [I]. In the analysis of such experiments [l, 21, it was noted that these angular distributions could not be interpreted directly in terms of electronic spin distributions in momentum space, but required a renormalization, due to the small 3 y/2 y ratio. In this paper we outline a theoretical analysis to obtain the renormalization in a model independent way directly in terms of the experimentally measured 3 y/2 y yields ; we present the results of such 3 y measurements, and apply our proceedure t o the 2 y angular distributions obtained in single crystals of Gd, Fe and Ni.

Let us define

2

P"~)(P)= ^t(l) C

I ~S,(l,(r)Jp(r)e-ip~rd3rl

and

One obtains for the 2 y annihilation yield with momen- tum p, for B along the positron polarization

RI,

where K is a n arbitrary normalizing factor and p, ⁼ Qmc, 8 being the angle between the two detectors

(0 = 7.29 mrad corresponds top, = 1 a. u.) ^(I).

It was found possible to express the physically interesting quantities Ap(p,) ⁼ pT(pJ - pl(pZ) and p@,) = pr(PZ)

+

where $!'r(i,(r) are the electronic wave functions of a In the above we neglected terms of the order of PZ and ferromagnetic metal with spin parallel (antiparallel) to [4]. N i s an arbitrary constant.

an applied saturating magnetic field B (1 is the majo- Our 3 y counting apparatus consisting of five NaI(T1) rity spin direction), and $,(r) is the ground state detectors measuring three-fold coincidences and a positron Bloch wave. We introduce a partially polari- sixth placed to measure the 2 y yield ^{( 2 ) .} The experi- zed positron with net polarization P,. The total annihi- ment proved to be rather difficult because of the high lation rates for a positron with spin parallel (anti- singles rates (- 6 x 104/s) to the typical parallel) to the applied field B are

At(1) = )As ^2,ol'T) + 4 A, ol(t' + 1, wT(L)

(1) p t ( i ) (p) correspond to the momentum distribution of

where 2, and A, are the 2 y and 3 y partial annihilation ^the spin t(J) electrons as sampled by the positron ; they depend rates per electron per unit volume ; on ^the wave functions ^of the majority and minority spin bands and have to be distinguished from spin distributions in k space.

p = A,/A = 1/1,114. since an electron wilh quantum n;mber k can contribuie at all pi = k + Gi, where the Gi-s are the reciprocal lattice vec- tors [3].

(*) Work supported by the National Science Foundation and ⁽²⁾ The counters were distributed on a 10 cm radius circle the U. S. Army Research Office (Durham). around the sample, which is saturated by an 18 kGauss field.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971197

C 1 - ^{288 S. BERKO} AND A. P. MILLS total three-fold coincidence rate (- 11s). Our final

results [5] for Gd, Fe and Ni are

We shall now demonstrate the use of eq. 5 for these three metals.

Gadolinium. Our 2 y polarization experiments published in an earlier note [7] are plotted in figure 1, along two crystalographic orientations. The

curve was obtained from eq. 5 and is also plotted ; Pp@T - pl) exhibits an appreciable anisotropy. Also

[roio] [ooor]

18

FIG. 1. -The angular distribution, AR(p,), and the derived momentum distribution of the electron spin excess as seen by

the positron in two orientations of Gd.

plotted are the pr + pJ curves. The anisotropy of the angular distributions in rare earths has been discussed in terms of the Fermi surface of these metals by Williams and Mackintosh [8]. Wigner-Seitz Il/,(r) computations [3] by S . Cushner in our laboratory indicate that the 4 f electrons contribute only 2.8 %

to the total 2 y annihilation in Gd. Thus Ap is domi- nated by the << (5 d) - (6 s) >> conduction band. Short of a ferromagnetic band computation including the positron wave function, we can only provide a quali- tative discussion of our experimental results. By taking a spherical average of Ap(pJ we can convert these one dimensional distributions to a radial distri- bution by

We find Ap(p) to change sign as a function of p, corresponding to a negative spin polarization (Ap > 0, i. e. minority spin directionj in the low momentum regions of the conduction band, changing over to a positive (majority) spin direction at higher momenta where the 5 d parts of the conduction band predomi- nate.

Iron and nickel. - We have applied eq. 5 to new polarization data from oriented crystals of Fe [9] and Ni, obtained in our laboratory by J. Weingart. In both metals we find appreciable anisotropies in the polarization curves, even though the pr(pZ)

+

curves show rather small anisotropies. The Ni results were compared to the experiments of Parks and Mihalisin [lo]. Although we agree with the general trend, we have not found the fine structure discussed by them. Using the AR,,(ph curves and the PJy mea- surements we compute A p ( p 3 as for Gd. The results for Fe and Ni are shown in figure 2, in the two direc-

FIG. 2. - The derived momentum distribution of the electron spin excess as seen by the positron in the [ill] and [IOO] direc-

tions for Fe and Ni.

tions [loo] and [Ill]. We notice that the anisotropies are opposite in the two metals, i. e. the [ I l l ] curve in Ni and the [I001 curve in Fe have a clear minimum.

Notice also that both metals exhibit net negative regions at low momenta, attributable to negative spin polarization, at least in the region seen by the positron, i. e. in the outer regions of the cell, in agreement with the observations of Shull from polarized neutron data (3). Assuming a rather isotropic ⁽⁽ s ^D distribution (the Ni necks along the [ I l l ] direction are too narrow to be observable by our 2 mrad resolution), we attri-

(3) The simple negatively polarized spherical cr s H band model used for the early polycristalline data [I11 has to be modified since the negative spin regions extend further in momentum space than expected from pure ct s n band contributions. As noted previously such negative regions of spin density in p space do not have to be reflected in negative spin distributions in k space, but can arise from different yt(r) and vl(r).

SPIN DISTRIBUTION STUDIES IN FERROMAGNETIC METALS BY POLARIZED C 1 - 289

bute the anisotropies in the region of low p, to the in fact, rather closely the shape of the differences of

<< 3 d 1) band. Indeed, taking the momentum transfor- the theoretical 3 d distributions at low p,. We thus med atomic 3 d t,, and e, type wave functions in the conclude that our experiments indicate predominantly ratios obtained by Shull, we predict anisotropies at t,, behaviour for Ni and e, behaviour for Fe.

p, = 0 in the direction obtained in our experiment. We thank J. Weingart for the use of his unpublished The difference curves AprlIl,(p,) - A P ~ ~ ~ ~ ~ @ ~ ) follow, Fe and Ni polarization data.

References

[I] BERKO (S.), (c Positron Annihilation I), (Academic values (SEDOV (V. L.), SOLOMATINA (L. V.) and Press, Inc., New York, 1967, p. 61. KONDRASHOVA (L. A.), Zh. Eksp. Teor. Fiz., [2] BERKO (S.) and ZUCKERMAN (J.), Phys. Rev. Letters, 1968, 54, 1626) should be thought of as upper

1964, 13, 339a and Phys. Rev. Letters, 1965, limits, because of their large quoted errors.

14, 89. [7] HOHENEMSER (C.), WEINGART (J. M.) and BERKO

[3] BERKO (S.) and PLASKETT (J. S.), Phys. Rev., 1958, (S.), Phys. Letters, 1968, 28 A, 41.

112, 1877. 181 WILLIAMS@. W.) and MACKINTOSH (A. R.), Phys. Rev., [4] In eq. 5 Pgy corresponds to the 3 Y measurements 1967, 168, 679.

being obtained with the counters in a plane [9] MIJNARENDS (P. E.) and HAMBRO (L.), Phys. Letters, perpendicular to the field B; the formula thus 1964, 10, 272 and MUNARENDS (P. E.), Proce includes the anisotropic 3 y emission with res- edings of the International Conference in Magne

pect to B. tism ^)), Nottingham, 1964, p. 230.

[5] When these values were used in eq. 5, they were [lo] MIHALISIN (T. W.) and PARKS (R. D.), solid State reduced by the measured ratio of 1.5 of Pp in Comm., 1969, 7 , 33, and references given therein.

the 3 y vs. 2 y experiments. [I 11 BERKO (S.) and TERELL (J. H.), ^(< Optical Properties [6] The value obtained for the 2 y effect by SEDOV (V. L.), and Electronic Structure of Metals and Alloys ^)),

Zh. Eksp. Teor. Fiz., 1965, 48, 1200, is too large North-Holland, Amsterdam, 1966, p. 210-217.

on physical grounds, and the more recent