A REVIEW OF MAGNETIC X-RAY EFFECTS

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A REVIEW OF MAGNETIC X-RAY EFFECTS

M. Cooper

To cite this version:

M. Cooper. A REVIEW OF MAGNETIC X-RAY EFFECTS. Journal de Physique Colloques, 1987,

48 (C9), pp.C9-989-C9-996. �10.1051/jphyscol:19879176�. �jpa-00227291�

JOURNAL DE PHYSIQUE

Colloque C9, suppl6ment au n012, Tome 48, dgcembre 1987

A REVIEW OF MAGNETIC X-RAY EFFECTS

M . J . COOPER

Department of Physics, University of Warwick, GB-Coventry C V 4 7AL. Great-Britain

Abstract

-

The existence of spin-dependent terms in the photon-electron scattering cross-section was confirmed many years ago but these effects remained unexploited until the advent of the present generation of bright X-ray synchrotrons. Recently Compton scattering and photoabsorption studies of spin-dependent densities have been made with circularly polarised synchrotron radiation and diffraction studies of antiferromagnets with the dominant linearly polarised component. In this paper the scattering theory will be summarised, recent experiments reviewed and future prospects discussed.

1. Introduction

Despite the current upsurge in activity magnetic effects in photon scattering are not new phenomena. Franz 111 first derived an expression for the scattering of polarised radiation by oriented electrons in 1938. T w o decades later cross-sections for all conceivable combinations of observed photon and electron polarisation had been derived [ Z ] , reviewed ( 3 1 and found their way in reference works [ 4 1 together with some supporting experimental evidence. A spin-dependent scattering effect, came into use in nuclear physics to determine the net circular polarisation of gamma rays. This was achieved by monitoring the magnetic field dependence of the total intensity of Compton scattering from magnetised iron. That work was preceded by transmission measurements with unpolarised gamma rays [ 5 1 which established a small (second order) contribution to the photon attenuation coefficients in magnetised material. ~ " ' ~ c " "

I

Figure 1: (a) the intensity near the (312, 312, 312) reflection from NiO above and below the Nee1 point a s observed with unpolarised X-rays from a conventional 1 kW X-ray tube [61. (b) The magnetic Compton profile defined in equation 10) offerromagnetic iron measured with&cularly polarised 122 keV gamma rays from a 10 mCi Co source (71.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879176

C9-990 JOURNAL DE _,PHYSIQUE

Since these effects are inherently relativistic in origin their observation at the lower energies associated with conventional X-ray sources did not occur until 1971 when de Bergevin and Brunel [61, who were responsible for all the early magnetic X-ray work, observed weak magnetic superlattice reflections from antiferromagnetic NiO with the unpolarised radiation from a 1 kW X-ray tube! The magnetic peaks are six orders of magnitude weaker than the Bragg peaks but were

just discernible above the background (see figure la).

A comparable pioneering effort in incoherent scattering was made by Sakai and

Ono [ 7 , 8 1 who neatly reversed the philosophy behind the nuclear physicists' Compton

polarimeter and used cooled oriented nuclei as sources of circularly polarised radiation to probe the spectral distribution of the Compton scattering from iron (see eq. 9 below for a definition of the magnetic Compton profile and [9] for further background). In this study also intensity limitations dominated especially because heating effects due to self-absorption in the radioisotope severely limited the amount of source material that could be kept cool at the appropriate millikelvin temperatures ( < 60 mK) by repeated adiabatic demagnetisation. Forty demagnetisations in 139 hours were required for the 10 mCi 57C0 in what has been described [lo] as a "beautiful" experiment. The results are shown in fig lb, together with the diffraction data they provided the inspiration for the exploitation of magnetic scattering with X-ray synchrotron sources.

2. The Magnetic Scattering Interaction (a) The leading terms

T h e c r o s s - s e c t i o n f o r scattering polarised photons off

s p i n - a l i g n e d e l e c t r o n s c a n b e

e

c a l c u l a t e d e x a c t l y i n q u a n t u m e l e c t r o d y n a m i c s , but for b o u n d electrons no exact solutions exist and it is usual to proceed by series expansions in powers of the photon's energy and the electron's momentum.

The fractions hu/mc2 and p/mc are the relevant quantities and the first terms are therefore the leading terms only in the limit of low energy

photons scattered from weakly bound Figure 2: The scattering interaction. The incidentbeam has

electrons. other authors [ 11,12 1 an energy fiu1, incident wavevector kl and polarisation components EA and ₄ Ell perpendicular and parallel to the

have O" that he r a t

scattering plane: kl is the unit incident wavevector, with

h"/mc2 Occurs the magnetic corresponding quantities for the

beam, ~h~

effects may be thought of as arising scattering vector k = and the angle of scattering is

from reradiation by the magnetic as

4.

o p p o s e d t o the e l e c t r i c d i p o l e created by the incident electric field.

Cross-sections t o order ( h ~ / m c ~ ) ~ and p/mc have been calculated as relativistic corrections to classical amplitudes [13] and confirmed in a rigorous relativistic approach 1141. Blume [151 and Platzman and Tzoar [lo] have emphasised those aspects of the interaction most relevant to studying magnetisation in condensed matter whilst a clear step-by-step derivation has recently been presented by Lovesey [16] whose approach will be followed here.

The scattering process is illustrated schematically in figure 2 and the nomenclature explained in its caption. The leading terms in the scattering cross-section can then be written:-

The cross-section for elastic scattering is simply obtained by putting @1=@2 and li> = If> in which case the energy-conserving delta function is automatically satisfied. For Compton scattering within the impulse approximation If> is a continuum plane wave. The pslarisation of the beam is expressed by the factors A , B, and C which can be written as the following matrices

-

celebrated work on holmium [17l topen circles) is due-to the c-axis lattice modulation hut

where the superior wavevector the left hand peak is truly magnetic in origin and is proved by the data depleted by the filled circles. They were

resolution in the synchrotron obtained after ~ o s t sample monochromator with a 45' Bragg

The leading term in equation 1 is just the familiar Thomson cross-section and for an unpolarised beam the matrix A leads to the equally familiar "polarisation factor" !?(l + cos2f$) m o d u l a t i n g

the intensity. The two remaining

e x p e r i m e n t o v e r its n e u t r o n angle had been used to eliminate Thomson scattering in the

counterpart led to the discovery orbital wlane of the svnchrotron.

of a spin slip system

-

see figure 3.

-400

0

-300

5 P r

-200

'=

100 terms are reminiscent of the

neutron scattering cross-section and can be identified with the s p i n a n d o r b i t a l m o m e n t u m

operators respectively, but there ₄₀₀ are important differences. For

example in this expression the moments are not simply additive as

they are in the neutron case, here g 3 0 0 - t h e y a p p e a r w i t h d i f f e r e n t (O

polarisation factors and that

a

_\

leads to the exciting possibility lJl

of being able to separate them.

2

^200-

Unfortunately there is an immediate problem: both terms

100- a p p e a r t o be i m a g i n a r y w h e n

considered to first order and only m a k e a r e a l c o n t r i b u t i o n ,

-

^(Aw/mc2) when taken to second order. The second order term is responsible for magnetic effects

o b s e r v e d w i t h u n p o l a r i s e d ! (UNITS OF cI) radiation [5,6], or indeed with

linearly polarised synchrotron Figure 3: Sean of the Holmium (004) satellite reflections

radiation, e.q. the recent and 1171 observed in a synchrotron study. The right hand peak

o

H0(00!)

%

FILTER IN

0 FILTER OUT

-

i

8' ,

* '

. 8

8'

9

B0

& %

^B

o O

i o * * O O 0 @ m , @

o0 00 @ O

'-

^"om

* * a - ^'**%a

^{8 -}

• .&*OW)

15/27 12/9

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I I

41800 42cno 4 . 2 ~ ~ 4.24133

C9-992 JOURNAL DE PHYSIQUE

In a material with spin moments alone the ratio of magnetic to charge scattering approximated as [151

where Nm/N is the fraction of electrons with unpaired spins and fm/f the ratio of their unitary form factors. The inten,sity ratio is typically as low as loM6.

Whilst such effects are measurable if they appear separated in reciprocal space they are inevitably masked in ferro- and ferri-magnetic systems where the reciprocal lattices coincide. However if a real first order term could be engineered the ratio in equation (2) would improve by two or three orders of magnitude.

(b) Real first order contributions

Such a contribution would arise if the leading term in equation 1 were complex. This would occur in a diffraction experiment if the atomic structure were non-centrosymmetric. Unfortunately from this point of view (only!) they are invariably centrosymmetric. The scattering amplitude will however be complex under conditions of anomalous dispersion, i.e. if

hl

-* Eabs, the K shell binding energy.

Even without the advantage of a tuneable synchrotron deBergevin and Brunel [181 were able to realise this effect in iron (Eabs = 7.11 keV) studied with CuKU radiation (EKU = 8.04 keV). For the 222 reflection, for example AfU/fo

"

^{1/3 and}

the flipping ratios obtained upon reversal of the magnetic field amounted to parts in 10'. This method has recently been applied to the study the modulation in the magnetic moment in a Gd-Y superlattice by tuning the beam near the Gd LIII edge at 7.94 keV [191.

This energy constraint is certainly at variance with the requirements of Compton scattering studies where not only the primary energy, but also the energy transfer, must comfortably exceed the electron binding energies if the lineshape is to be interpretable within the impulse approximation. There is then only one possibility which is to make the polarisation factor complex, i.e. to use a circularly (or in general elliptically) polarised beam. Following Lovesey [16] the expression for the Bragg/magnetic scattering can be expressed as follows

Defining f (k) = < $ o ~ ~ e x ~ i k . ~ j l$o> ( 3 j

and ~ ( k ) = <$o~C~j(k) l$o> (5 j

then

s =

*(e2Jf(k) {f(k)(l+cosZ4) dS2

inc2

where PC denotes the degree of circular polarisation and varies from -1 (Rhl to +l(rh).

If the anomalous dispersion case is to be included the following term must be added

N.B. this term is independent of the polarisation of the beam but does require the magnetisation vectors to lie out of the scattering plane.

In equation 6 the magnetic effects can be isolated either by changing the hand of the polarisation of the radiation or by reversing the magnetic field and hence S and L. In the analagous Compton scattering experiment the spectral distribution for a system with spin moment, S, alone is given by

where J(p) is a one dimensional projection of the momentum distribution n(p) for all the electrons and jmg(p) its unpaired spin component alone.

Azimuthal AngIe (mrad) ENERGY I keV1 -

Figure 5: Experimental Compton profiles of iron obtained F i g u r e ': c o m p o n e n t s t h e f r o m a with circularly polarised synchrotron radiation: ( a ) the synchrotron bending magnet (in this case the 5 Tesla bend of

total profile and (b) the magnetic (difference, profile, the Daresbury wiggler magnet). In the orbital plane the &diation at 46.4 and 61.9 k e v is selected by the (333) and radiation is strictly linearly polarised but out of the

p l a n e t h e r e i s a p e r p e n d i c u l a r component w h i c h i s i n (444) reflections of the germanium monochromator, and the quadrature and the beam is elliptically polarised. In this Compton profiles corresponding to the 151°f20scattering e x a m p l e i t c a n b e ~ e e n t h ~ t a t i / 5 ~ ~ ~ d ~ h ~ b ~ ~ ~ i ~ - 5 0 ~ a n g l e a r e a t 3 9 . 7 a n d 5 0 . 4 k e V , r e s p e c t i v e l y . T h e circularly polarised but one order of magnitude in flux has difference is essentially zero a t the elastic line positions

been lost. but the magnetic profiles and their "volcano" structure a r e

readily evident.

C9-994 JOURNAL

DE

PHYSIQUE

3. Progress With The Use of Circular Polarisation (a) The Inclined View Method

Whereas linearly polarised X-radiation can be produced by 90' Bragg scattering or arises inherently from a synchrotron it is not obvious how to produce the circularly polarised version with sufficient flux for investigative work. Both Holt and Cooper [201, working on the feasibility of magnetic Compton studies and Brunel et a1 [211 seized onto the idea of extracting circularly polarised flux directly from the output of the synchrotron simply by viewing the radiation from a Bending magnet at a small angle to the orbital plane. The principle is illustrated in figure 4 which shows how the degree of circular polarisation rises with the angle of view, albeit at the expense of an alarming fall in intensity. The method does not work with a n undulator or symmetric wiggler because the hand of polarisation reverses wikk curvature, for an asymmetric wiggler of the type installed at Daresbury there is also some polarisation cancellation at photon energies below the critical energy if parts of the subsidiary bends are also viewed, but this problem disappears at higher photon energies. The trial diffraction experiment on a bending magnet beam line at DCI yielded intensity changes

-

in the Bragg reflections from zinc substituted ferrite with 6.9 keV radiation, in line with the prediction for a first order effect.

The first Compton results obtained by this method appeared in 1986 [22,231 and provided a demonstration of the viability of the method. Figure 5 shows the

-

s p e c t r a o f m a g n e t i c C o m p t o n

scattering in an experiment [231 x

in which 333 and 444 reflections

-

^@J

of a germanium monochromator were used to select incident beams at 46 and 62 keV respectively. It was estimated that -lo1' photons per second, with a net value of PC

= 0.8 w e r e incident upon the m a g n e t i s e d s a m p l e . T h e volcano-like dip at the centre of

the magnetic profiles which arises

$0.5

from the negative polarisation of @

the s-p band is clearly evident in the raw data.

Since that result appeared

v

the "inclined view" approach has been adopted to study the magnetic component of EXAFS and XANES in

iron using the DORIS storage ring 0 10 20

30

[24]. Essentially the circular

photon polarisation partially

E = Ex-Eo

(ev) polarises the ejected Is electrons

w h i c h t h e n h a v e a d i f f e r e n t

transition probability to the empty ^{Figure 6:} The normalised spin dependent X-ray absorption coefficient uc/po in the near K-edge absorption region of

spin up and spin down final states. iron and the normalised differences of the spin density

Some of the results are shown in d p / ~ , calculated by the KKR method (Taken from [241).

figure 6.

(b) Other approaches

Although the "inclined view" method has been proved in diffraction, Compton scattering and photoabsorption studies it is not the only approach., Since linearly polarised radiation is available in abundance from synchrotron sources the obvious

step would be to construct a quarter wave plate. That is easier said than done because, at X-ray wavelengths, the inherent crystal birefringence is very small [251. It was achieved by a device with 15% conversion efficiency at 17.5 kev which was demonstrated at CHESS 1261. Since then a device operating at 40 keV has been used to record the magnetic Compton profiles of Fe, Ni, Co and Gd [D.M. Mills, private communication]. The profile of Gd is, as expected, much broader than those of the transition metals because the moment is associated with localised 4f electrons rather than delocalised 3d electrons. This technique is obviously in competition with the "inclined view" but it is not yet clear which method offers the better flux at a fixed wavelength. The X/4 plate cannot be tuned: it cannot therefore be used for photoabsorption and may be at a significant disadvantage for diffraction work.

Finally the continued use of cooled oriented nuclei should not be discounted.

The natural tendency for y-ray photons to have higher energies ( > 100 keV) than those produced by present day synchrotrons does linearly increase the magnetic effect. Since the early work L7.81 helium dilution refrigerators have increased the cooling power available [271 and hence the activity of the isotope that can be cooled. Measurements with oriented 1 2 9 m ~ r nuclei have now been made [281 with an accuracy equal to that of the best synchrotron measurements on both iron and manganese ferrite although the measurement times required were of the order of one month.

4. Future Directions

It is clear that magnetic Compton profile measurements can be successfully pursued by all of the three methods discussed above, but it is too early to divine which one will become standard. It is, however, of somewhat greater interest to speculate how diffraction studies will develop since there is more general interest in position space than momentum space densities. Diffraction work has a greater overlap with neutron studies of magnetisation and it remains to be seen whether magnetic diffraction can become a truly complementary technique or even a rival one.

At the moment the methods are scarcely established, yet virtually the first measurement on holmium [17] was able to reveal a feature of the magnetic structure obscured in the lower resolution neutron work. That must augur well for the value of further measurements. Calculation of flux rates for the present and coming generation of synchrotrons clearly indicate that the lower X-ray cross-section can be more than compensated by the higher fluxes which are generally delivered onto smaller sample areas.

Another advantage which must be pursued is the different role of spin and orbital magnetisation in the X-ray cross-section: potentially they are separable.

Referring back to equation 6 it can be seen that

and

dU ^{A n}

-

a

<L>. (kl + k2 i C O S ~ @ / ~ dQ orbit

Thus by judicious choice of the Bragg angle ( @ / 2 ) , effected by tuning the photon energy, the relative contributions from <S> and <L> can be varied or indeed either component can be eliminated. Thus in combination with neutron data on the total magnetisation the identification of its components should be possible. This is of great interest in systems such as the actinides where the electrons are highly correlated and the spin components are large but individually unknown.

JOURNAL DE PHYSIQUE

Acknowledgements

I am grateful to S.W. Lovesey for many patient explanations of the scattering theory and to him, D.M. Mills and N. Sakai for communicating their results to me prior to publication. Other authors have kindly granted permission for the reproduction of their published results in figures 1,3 and 6. Finally I acknowledge the financial support of the SERC for a research programme which has embraced magnetic Compton scattering.

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