Static and dynamic X-ray resonant magnetic scattering studies on magnetic domains - 2 MAGNETO-OPTICAL CONSTANTS AT THE RARE-EARTH M , ABSORPTION EDGES

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Static and dynamic X-ray resonant magnetic scattering studies on magnetic

domains

Soriano, J.M.

Publication date

2005

Link to publication

Citation for published version (APA):

Soriano, J. M. (2005). Static and dynamic X-ray resonant magnetic scattering studies on

magnetic domains.

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2 2

M A G N E T O - O P T I C A L L

CONSTANTSS AT THE

RARE-EARTHH M 4 5 ABSORPTION

EDGES S

ForFor the interpretation of X-ray resonant magnetic scattering and absorption, good knowl-edgeedge of the magneto-optical constants is extremely useful In the case of the rare-earth

M^ss absorption edges, the absorptive part has been predicted accurately on the basis of

atomicatomic multiplet theory. Here we use such calculations with slightly more optimized pa-rametersrameters to obtain the dispersive part via Kramers-Kronig transformation. The complete datasetdataset should represent realistic values for the complete magneto-optical constants, in-cludingcluding the Faraday and Voigt constants which are given explicitly. We shortly discuss anotheranother possible application in X-ray sources based on the Cherenkov effect.

2.1.. Introduction

Thee increasing use of resonant X-ray techniques in magnetic research wouldd greatly benefit from good prior knowledge of the magneto-optical con-stantss involved. In general, these can not be calculated ab initio since the dipole transitionn matrix element involved contains the unoccupied valence states. A notablee exception form the important M^5 edges of the rare-earth elements, wheree the 3d electrons resonates with the partly filled 4 / shell. The atomic na-turee of the 4 / shell allows one to describe the 3d-4f resonance in a purely atomic

18 8 C H A P T E RR 2

model. .

Thiss was realized first by Thole et al. [14], who predicted the existence of X-ray magneticc circular dichroism (XMCD) in the absorption spectra M ^ of Dyspro-sium.. After the successful experimental confirmation of this model in measure-mentss on Terbium garnets by van der Laan et al. [15], calculations for all rare earthss were published by Goedkoop et al. [16]. The influence of crystal field ef-fectss on the 4 / shell was described in detail by Vogel [17].

Thesee papers showed that effects of the embedding of the rare-earth atom in aa solid (screening, hybridization, etc.), can be effectively included by applying smalll reductions to the two-particle Slater integrals involved in the atomic the-ory. .

Inn a previous paper [18], we made a detailed analysis of the Gd M45 absorptionn cross sections (unpolarized, circularly and linearly polarized). Us-ingg Kramers-Kronig transformations to obtain the real part of the spectra we showedd that these the complete complex optical constants thus obtained accu-ratelyy describe the energy dependence of the magnetic scattering cross section. Thiss leads us to expect that the calculated atomic spectra can be used to ob-tainn the optical constants and scattering cross sections for all the rare earth M45 edges,, and forms the motivation for this chapter.

Thee structure of this chapter is as follows: Sect. 2.2 gives a brief theo-reticall description of the resonant absorptive and dispersive corrections to the refractivee index. Sect. 2.3 discusses the experimental methods to obtain the op-ticall constants and the atomic multiplet calculation for the RE M45 absorption crosss sections and their comparison for the case of Gd. In Sect. 2.4, the calcu-latedd optical constants and resonant scattering cross sections and Faraday and Voigtt rotation are presented for all rare-earth elements. Sect. 2.5 discusses the resultss and the possible applications, and Sect. 2.6 presents the conclusions. The sensitivityy of the calculated spectra to variations in the calculational parameters iss investigated in Appendix A.

Magneto-opticall constants at the rare-earth M4 5 absorption edges 19

2.2.. Optical constants in the soft X-ray range

Thee interaction of X rays of energy hco with matter is described by the complexx refractive index n or equivalently the complex dielectric constant e:

n(a>)n(a>) = ^=l-Ó{cü) + ip{tü). (2.1)

Heree 3(co) is the refractive index decrement and fi(to) is the absorption in-dexx which account respectively for dispersive and absorptive processes. These twoo corrections to the refractive index are bound via the Kramers-Kronig trans-formss [19]:

S{u)S{u)

= -IprHgW^

.2)

TTTT Jo WIZ _{- CO}1

mm

= ^r-ftw, (2.3)

nn Jo co'z - ujl

soo that knowledge of one of the two magnitudes is sufficient to obtain the re-fractivee index n. The V before the integrals stands for principal value [10, 20].

Thee complex refractive index is related to the X-ray scattering factor ƒ (q, w) thatt describes the X-ray/matter interaction as [21]:

„

_ ? W £

.

wheree pa is the atomic number density, re is the Thomson scattering length and

kk is the wavenumber.

Thee atomic elastic-scattering factor reads

f(q,u>)=Mq)+f'(cu)f(q,u>)=Mq)+f'(cu) + if"(a>), (2.5)

wheree q = kj — k, is the wavevector transfer. The first term /o(q) is the Thom-sonn charge scattering, given by the Fourier transform of the electron density withh relativistic corrections (/o(0) = Z* = Z — (Z/82.5)2-37), and is indepen-dentt of the photon energy. The dispersion corrections f'{co) and f"{co) account forr the fact that the atomic electrons are bound, and depend very strongly on the photonn energy. When the photon energy matches the energy difference between twoo atomic levels u?o = (£ƒ — Ej)/h, the probability of electronic transition be-tweenn the two levels greatly increases, and so does the total absorption cross sectionn cra((v). The relation between the imaginary part of the atomic

elastic-scatteringg factor f "(to) and the atomic absorption cross section is:

20 0 C H A P T E RR 2

wheree f"{co) and cra(cü) are respectively measured in electrons per atom and

A2.. Analogously, we can write for the dispersive part

^ )) = ^ l / o ( q ) + / H - (2-7)

Whenn the medium through which the incoming radiation travels is magnetized, thee time-reversal symmetry of the entire system is broken. In this case the most importantt solution of the wave equation are eigenmodes with well defined polar-izations.. If the light propagation k and magnetization m are parallel, the eigen-modess are left and right circularly polarized waves propagating in a medium withh refractive indexes rc+(-}- ^ ^ *s perpendicular to m, the eigenmodes are linearlyy polarized parallel and perpendicular to m with a corresponding refrac-tivee index M||(J_)

Justt to complete our terminology we recall that the differences in absorption (/S++ — jS_) and dispersion (Ö+ — Ö-) are respectively called circular dichroism andd birefringence, scale with the magnetization m, and are directly related to thee imaginary and real parts of the complex Faraday angle. Correspondingly, (/?HH — jSj_) and (<5|| — S ) are called linear dichroism and birefringence, which aree the imaginary and real parts of the complex Voigt angle. These linear effects aree proportional to |m|2.

Althoughh both the electric and magnetic multipole transitions contribute too the corrections of the refractive index, for the M4 5 absorption edges only the

dipolee El transitions need to be taken into account. Following the notation of Harmonn and Trammell [22], the resonant contribution f^ = f' + if" to the elastic-scatteringg amplitude can be written as

/ ^ ( w )) = ( êh ê)F(0»(w) - i(ê'* x ê j m f f ^ a ; ) + (ê'* m)(ê m)F(2)(o;), (2.8) wheree ê, ê' are the polarization vectors of the incoming and outgoing beams, andd m is the direction of the local magnetic moment of the ion. The energy dependentt factors P0 , 1'2' ( C Ü ) are linear combinations of the atomic oscillator strengthss F]^(CÜ) for electric dipole transitions:

F( 1 )MM = lklFl~F-i] (2-9)

Magneto-opticall constants at the rare-earth M4 5 absorption edges 21 with h

plpl = y /p^(rj)Tx(aMrj)/T(V)\

7^i7^i \ x{ct,rj)-i )

Heree p& is the probability to find the ion in the initial state ja) and pK(rj) is

thee probability that the excited state \T]) is vacant for a transition from \oc). Tx

givess the partial line width for dipole radiative decay from \rj) to \oc) and T(rj) iss the total line width, determined by all (radiative and non-radiative) decay processes.. In the resonance denominator x(cc,rj) = (£7 - £« — hcv)/[!(?])/2]

iss the deviation from the resonance in units of T(rj)/2. For photon energy

ha)ha) — Efj — Ecc this term diverges, resulting in a strong enhancement of the

scatteringg amplitude.

Wee thus obtain the expressions for the circular and linear dichroism [19, 23]] to the imaginary part of the resonant scattering amplitudes F^0'1'2) (to):

crcr00(w)(w) = y3W+ +<T-+ 0||] = -2\re lm[F^(co)]

aacc(w)(w) = cr+-a-= 4Are Im[F(1)(a;)] (2.11)

crj(w)crj(w) = <7|| -<r = -2\re Im[F(2)(a;)],

wheree a+ (tr_) are respectively the absorption cross sections measured with left-fright-)) circularly polarized light.

Thee X-ray resonant complex Faraday [24] and Voigt [25, 26] specific rota-tionss are defined as:

eeFF = 9F + iaF=n+~n-k (2.12)

eevv = 9v- iav = " k (2.13)

Thee relation between the specific rotation angles and the atomic resonant scat-teringg amplitudes F^1,2) is given by:

66FF(w)(w) = -\repaRe[F^(cv)] Pa. Pa. (2)

_{M]] = f<}

22 2 CHAPTERR 2

2.3.. Resonant cross sections of the RE M

edges

2.3.1.. Calculation of atomic absorption spectra

Inn the atomic picture, the 3d —+ 4 / absorption process involves the elec-tronicc excitation 3dw4fN -* 3d94fN+l, where all the other shells are either filled

orr empty (see Fig. 1.2). Both the initial and final configurations are split in multipletss of states with energies Eaj and wavefunctions denoted by \ocJM) (a.

labelss all quantum numbers other than ƒ and M needed to completely spec-ifyy the state). The final-state configuration 3d94fN+1 contains two open shells and,, consequently, its multiplet is more complicated, comprising in the middle off the Lanthanide series several thousands of levels \oc'J'M') (primes indicate final-statee quantum numbers). The strongest final-state multiplet interaction is thee spin-orbit coupling of the 3d hole, which splits the multiplet in two parts which,, to a first-order approximation, may be labelled 3d$, and 3d^h, or M5

andd M4 respectively. In the X-ray absorption spectrum, only those states of the

excitedd multiplet that can be reached from the Hund's rule ground-state \ccJM) underr the optical selection rules A/=0, , are present. According to Fermi's Goldenn Rule and after using the Wigner-Eckart theorem, the absorption cross sectionn in the dipole approximation can be written as:

wheree Saja>y = |(aj\|P|\OL'J')\2 is the square of the reduced matrix element of

thee dipole operator P, known as Hnestrength. The element between brackets is thee Wigner 3-/ symbol, that dictates the distribution of the Hnestrength of the

ex.]ex.] —> oc'J' line over its different M —> M' components.

Transitionss to the unoccupied np-states are also allowed, but, due to the smalll spatial overlap, have much smaller cross section compared to the 4 / res-onancee [27]. The difference in excitation energy between the 4 / resonance and thee continuum edge results from the very efficient screening of the hole by the 4 /N + 11 final state.

Thee 4 / electronic orbital is very efficiently screened by the electrons of moree external orbitals. Owing to this, the effect of the interaction of its elec-tronss with the electronic cloud of the surrounding atoms, i.e. the crystal electric

Magneto-opticall constants at the rare-earth M45 absorption edges 23 fieldd (CEF) effects, become secondary as compared to the spin-orbit interaction.

Ass a result, the M45 absorption edges can be described with an atomic model andd calculated with atomic multiplet programs. The theory of atomic spectra is quitee involved in the multiplet calculation, and a full explanation of the proce-duree can be found in Refs. [28, 29, 30]. Various computer programs [31, 32, 33] havee been developed, and we have used Cowan's atomic Hartree-Fock program withh relativistic corrections [28, 31], which has been applied extensively in the pastt [27,34,35].

Thee complete atomic multiplet calculations in intermediate coupling, in-cludingg all the states of the initial and final configurations, have already been performedd for all rare-earth M4 5 edges [27]. They calculated the relative en-ergyy of the different terms of the initial and final states, obtaining the radial partt of the direct and indirect Coulomb repulsion and the Coulomb exchange parameters,, i.e., Frr, Fjj, and G^r, also called Slater parameters. Together with thee spin-orbit parameters Q and £ƒ, they determine the energies of the different termss within the initial and final atomic configurations 4fN and 3d94fN+l.

Thee electrostatic and exchange parameters have typically to be scaled to 80%% of their atomic value to account for the solid-state surrounding of the ion thatt leads to hybridization and charge transfer with the adjacent ions. These downscalingg factor will be later referred to as Ki,2,3/ corresponding respectively too Fkcc, fjy and Gkd,. The lifetime of all final states is taken to be the same, and all

dipolee transitions are convoluted with a Lorentzian line shape of width 2T5-fulll width at half maximum (FWHM) for the M5 peaks and by a Fano line shapee [36] of 2T3, FWHM and asymmetry parameter q$, for the M4 peaks. Thiss difference in line shapes reflects the stronger Coulomb interaction of the 3^3^^ with the nucleus, and therefore the more damped oscillator. The result-ingg spectrum is finally convoluted with a Gaussian line shape with standard deviationn <jg to account for the instrumental resolution. In summary, seven free

parameterss are needed in the calculation: three reduction factor of the Slater parameters,, two Lorentzian line widths, a Fano asymmetry parameter and the Gaussiann standard deviation.

Largee surface crystal electric field (CEF) effects have been found in M45 spectraa of RE overlayers [34]. For bulk systems, it was also shown that CEF effectss could induce linear dichroism when the atomic symmetry was lower

24 4 C H A P T E RR 2

thann cubic. For the case of our amorphous thin films, we do not consider the CEFF effects in the calculated absorption spectra, although the atomic multiplet programm allows one to include this extra term of the Hamiltonian.

2.3.2.. Gd

experimental spectra

Ass was mentioned in the introduction, we seek to obtain the complete sett of magneto-optical constants by measuring the resonant absorption cross sectionss which directly yield the imaginary part of the scattering amplitudes. Subsequently,, the real parts are calculated by means of the Kramers-Kronig transforms.. This approach requires detailed knowledge of the absorption cross sectionn over a sufficiently large photon energy range. Since for the rare-earth M44 5 the atomic resonances are only a few eV wide and the next absorption

liness are at least 50 eV away, such an approach has been found to work satis-factorily,, as shown in previous determinations of the RE M45 magneto-optical constantss [18, 21, 34, 37, 38, 39, 40,41].

Thee required polarized rare-earth M45 absorption spectra can be mea-suredd in two different ways. The most common one is total electron yield, where onee measures the amount of photoelectrons excited by the incoming X-ray beam ass function of the photon energy [42, 43, 44, 45]. This method does not give ab-solutee cross sections, and is also susceptible to saturation effects caused by the photonn absorption length being longer than the electron escape length. Similar problemss affect total fluarescence yield measurements [46, 47].

Thesee problems are absent in the classical transmission method which iss however not common because it requires sample thicknesses of less than 1000 nm on ultrathin supports. Earlier, we measured the Gd M^s magneto-opticall constants in transmission geometry [18]. Fig. 2.1, reproduced from that work,, shows the measured absolute cross sections of a non-magnetic sample Gd filmm giving the non-magnetic resonant contribution and the circular and linear dichroismm of thin GdFe films saturated in field at T = 20 K.

Alsoo shown are the best obtained fits of the calculated spectra for T = 0 K, as-sumingg the Gd3 + angular momentum ƒ = S = 7/2 to be completely saturated. Thee optimal parameters obtained from this fit were found to be K\ = 0.83, K2 = 0.955 and K3 = 0.85 as scaling factors for the Slater parameters F ^ , FL and Gkdf.

Magneto-opticall constants at the rare-earth M4<5 absorption edges 25

11900 1210

Energyy (eV)

Figuree 2.1: Comparison of the experimental (symbols) and calculated (full lines) Gd M433 X-ray absorption cross sections. From top to bottom, isotropic, circular and linear dichroicc spectra.

== 0.4 eV, q3 = 12 and ag = 0.3 eV.

Sincee these parameters are based on data obtained from an amorphous material,, they are only strictly valid for systems with structural spherical sym-metry.. However, similar values of K„ have been obtained in a monocrystalline Tbb thin film with hexagonal close-packed structure [15, 48]: K\ = 0.84, KJ = 1.0 andd K3 = 0.80.

AA more detailed study, given in Appendix A, shows that variations of the widthh of the lifetime and experimental broadening around the optimal values doo not lead to significant changes in a0/Cj and fC-1-2). Overall, it demonstrates

thee validity of the calculated RE M4 5 magneto-optical constants.

Unfortunately,, although the experimental and theoretical spectral line shapess match up very well, the calculated isotropic cross section had to be

mul-26 6 C H A P T E RR 2

tipp lied by a factor 1.5 to obtain a quantitative fit with the experimental data. Errorss in the nominal thickness and density of the different samples used in thee experiment could also explain this discrepancy. Still, the quantitative agree-mentt is good enough to expect that the calculated cross sections will provide goodd predictions for the optical constants for all RE M45 spectra.

2.4.. Calculated RE M45 magneto-optical constants

Thee calculated magneto-optical constants for all trivalent RE ions are shownn in the top panels of Fig. 2.2, beginning with the isotropic spectra of thee non-magnetic La3_r and Eu3* ions. In each case, the top panel shows the imaginaryy part of the resonant scattering factors F^0'1'21 (to) as obtained from the

atomicc calculation and the corresponding real part obtained by Kramers-Kronig transformationn of the imaginary parts extended to a 100 eV range around the spectrall center of mass, which was tested to be wide enough to assure the cor-rectnesss of the Kramers-Kronig transforms.

Ourr imaginary-part spectra show more structure than earlier calcula-tionss [16, 35]. The new real parts show the dispersion of single absorption lines inn the case of La and Yb, and correspondingly more complicated line shapes forr ions with more extended multiplets. Although the charge contributions F° completelyy dominate the spectra around the resonance energies, at 10 eV away fromm them, the contributions of other absorption channels, such as the 3d to un-occupiedd 4sp and the surrounding absorption edges, become important. The generall trends in these contributions away from resonances are predicted well byy Ref. [49]. Since these backgrounds are non-dichroic, they are not important forr the dichroic spectra listed here. The dispersive parts of the latter do not have thee long tails seen in the F° spectra.

Thee bottom panels of Fig. 2.2 show the scattering cross sections calcu-latedd as the squared moduli of the F^°'l,2K The non-resonant isotropic

scatter-ingg contribution Z* is indicated by a horizontal dash-dotted line, and has been addedd to the isotropic part | F ^ |2. It is seen that the circular dichroic I F ^ J2 partss and linear dichroic |F*2' |2 parts peak at different energy, which allows one too change the polarization contrast just by changing the photon energy, which greatlyy helps in separating different scattering channels in domain studies (see forr example Sect. 3.3.3). Alternatively, these curves allow one to trade

absorp-Magneto-opticall constants at the rare-earth M45 absorption edges 27

8400 850 Energyy (eV)

11200 1130 1140 1150 1160 1170 Energyy (eV)

Figuree 2.2: Resonant scattering amplitudes at the RE M45 edges, starting with the two non-magneticc ions. Top panel: Imaginary (top) and real (bottom) parts of the complex chargee f'°' (full line), circular magnetic F'1' (dotted line), and linear magnetic F'2' (dashedd line) atomic scattering factors as function of energy in units of re. Bottom panel:

absolutee value of the scattering cross sections |/°(q) + F(°'(o;)|2 (full line), |FW(a;)|2 (dottedd line) and |F(2'(a>)|2 (dashed line) in logarithmic scale. The horizontal dash-dottedd line gives the non-resonant scattering cross section (continued on next pages, thiss page: only the two non-magnetic RE ions).

tionn contrast to dispersive contrast.

Figs.. 2.4-2.5 show the specific resonant Faraday and Voigt rotations as calculatedd using Eqs. 2.14. The relative photon energy centers on the centroid off the multiplet spectra [27].

Inn the remainder of the section we will discuss the parameter choice used inn the calculation of the imaginary parts. Most of the parameter values were basedd on our fit of the Gd3 + spectra. The reduction factors for the Slater param-eterss were K\ = 0.83, K-I = 0.95 and K3 = 0.85; the Fano asymmetry parameter q$.

28 8 C H A P T E RR 2 870 0 8800 890 900 910 Energyy (eV) 9200 930 940 950 960 Energyy (eV) 9800 990 1000 1010 Energyy (eV) 10200 1030 1040 1050 1060 Energyy (eV) Figuree 2.2: (Continued)

Magneto-opticall constants at the rare-earth M4 5 absorption edges 29 12500 1260 1270 Energyy (eV) 1280 0 12800 1290 1300 1310 1320 1330 1340 Energyy (eV) Figuree 2.2: (Continued)

30 0 C H A P T E RR 2 13600 1380 Energyy (eV) 14000 1400 14200 1440 Energyy (eV) 14800 1500 Energyy (eV) Figuree 2.2: (Continued)

Magneto-opticall constants at the rare-earth M4 5 absorption edges 31 == 12 and the Gaussian line width <7g = 0.3 eV.

Overall,, these values are smaller than the ones used in an earlier calcula-tionn of the absorption spectra [16, 35], which are proportional to the imaginary partss shown here. This is brought about by the improvement of the experimen-tall resolution over the last two decades, that allows a much better fitting to the moree detailed experimental data. It also required the refinement of the values forr the lifetime broadenings which are summarized in Fig. 2.3. For the light rare earths,, T$h was chosen to increase monotonically from 0.2 to 0.3 eV, whereas

T3.T3. was kept constant and equal to 0.4 eV. In the case of the heavy RE elements, T$T$hh = 0.3 eV and T^h linearly increased from 0.4 to 0.5 eV. This choice of values,

althoughh admittedly not physically intuitive, is based on the detailed fit to the Gdd spectra, the overall decrease of the core-hole lifetime along the Lanthanide seriess [27], and a fit to published electron yield data at the beginning and the endd of the series.

2.5.. Discussion

2.5.1.. Calculated spectra

Thee calculated cross sections \fo + F(°) |2, \F^ |2 and \F^2> |2 are very sen-sitivee to the photon energy, displaying intensity changes of several orders of magnitudee in a few eV. At the edges, the cross sections are dominated by ab-sorptivee terms, while the long wings around the edges are produced by the real partss of the F^0'1,2'. It is important to note that these wings are produced by thee Kramers-Kronig transforms and are not influenced by the lifetime and ex-perimentall broadenings. Furthermore, we find that the dichroic effects, needed too perform magnetic studies, are important (and comparable to the charge scat-tering)) only around the resonance energy. It is interesting to observe how, for

Laa Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

Figuree 2.3: Lifetime values of the M5 and M4 absorption edges used in the calculation, indicatedd respectively in the left and right axis.

32 2 CHAPTERR 2

E E

-200 0 20

Relativee energy (eV)

Figuree 2.4: Specific resonant Faraday rotation op (black) and ellipticity Kf (grey). Major tickk marks in the vertical axis correspond to 1 degree/nm.

Magneto-opticall constants at t h e rare-earth M45 absorption edges 33

E E

c c

-^—4—^ ^

Dy y

Ho o

-200 0 20

Relativee energy (eV)

Figuree 2.5: Specific resonant Voigt rotation 0y (black) and ellipticity ay (grey). Major tickk marks in the vertical axis correspond to 1 degree/nm.

34 4 C H A P T E RR 2

lightt rare earths, the intensity in the inter-edge energy range is only one or two orderss of magnitude smaller than the maximum values. This makes magnetic studiess feasible in a much wider energy range, in regions where absorption is low,, allowing studies of thicker samples.

Thee non-magnetic RE elements (ƒ = 0 ground state), La3+ and Eu3 +, showw no dichroic effects due to the spherical symmetry of their 4 / orbital, and onlyy F(°) is shown. This is mathematically reflected in the optical selection rules: onlyy the A/ = 1 transitions contribute to the spectrum, and the 3-j symbols with ƒƒ = 0 and q = 1 are equal.

Regardingg the predominance of circular over linear dichroism, no rule exists,, with the only exception of Gd3 +, where |F^{a?)|2 > | F ^ ( a ? ) |2 for en-ergiess just below the M5 edge [18]. It should be stressed at this stage the factor twoo difference in the relations between the circular and linear dichroism uc\ and

thee imaginary parts of F(1'2J (Eq. 2.11): a relatively small linear dichroism may resultt in a linear dichroic scattering contrast comparable to the circular counter-part,, as can be clearly observed for Yb3 +, where ac = 2d\ and \F^\ = \F^\.

Ass the 4 / electronic occupancy increases along the Lanthanide series, severall trends can be detected in the energy dependence of the scattering am-plitudes:: firstly, there is a clear gradual intensity shift from the M4 to the M5

absorptionn edge, to the point that the M4 edge totally disappears for Yb3 +. This

shiftt is due to the increasing spin-orbit parameter £ƒ that, in absence of CEF effects,, determines the branching ratio [50, 51] (ratio of the intensity at the M5 edgee to the total). In simpler words, the 4 / occupied levels in the ground state tendd to have more 4/5, character than the empty states [27]. This decreasing intensityy of the M4 absorption edge is obviously reflected in the real part and

thee squared moduli of the scattering amplitudes. Secondly, |/o -f F^\2 shows, inn semilogarithmic scale, a negative peak-positive peak dispersive shape in all cases,, whereas \F^ |2 and |F^2) |2 display a more complex pattern, which is very stronglyy depending on the internal structure of the multiplets, but not on the linee broadenings (see Appendix A).

Unfortunately,, quantitative comparison with literature is nontrivial, since mostt of the studies used total electron yield or photoemission, so that one would needd tabulated cross sections well below and above the edges to use as a

ref-Magneto-opticall constants at the rare-earth M45 absorption edges 35 erencee for quantitative analysis. Furthermore, these studies are frequently

af-fectedd by saturation effects. Overall, the peak features and relative height of thee RE calculated spectra are reproduced in the corresponding literature cited inn Sect. 2.3.2.

Fromm Eqs. 2.12-2.14, it is clear that the main features of ep,v come from thee F^1'2) which have already been discussed. However, larger values found forr the linear dichroic effect as compared to the circular one stress again that it shouldd not be discarded.

2.5.2.. Applications

Thee M45 magneto-optical constants are directly involved in a wide range off studies. We name dichroism experiments, either in transmission [46, 47] or totall electron yield [42, 43, 44, 45]; photoemission experiments [48, 52, 53, 54]; X-rayy reflectivity [55, 56, 57]; transmission X-ray microscopy [58, 59]; coherent scatteringg experiments [60] and scattering experiments at phase transitions [61, 62]. .

Ass for small-angle scattering experiments in transmission geometry, the knowledgee of the resonant scattering factors is vital for planning experiments, simulationss and data analysis [13, 63, 64, 65, 66, 67, 68], especially if they in-volvee non-negligible linear dichroic contrast. For the case of ordered stripe-domainn lattices, the possibility to switch between F^> and F^2> contrasts allows

onee to discern contributions from out-of-plane and in-plane magnetic domains, alreadyy used for the Gd case [18]. This is also clearly possible at the trivalent Nd,, Pm, Sm, Tb and Dy M5 edges, and to a lesser extent for Ce, Pr, Ho, Er and Tm. .

Inn the case of transmission microscopy experiments, all the previous con-siderationss can be applied to them, both for linear and circular dichroism, which hass been used to observe ferromagnetic [58]. On the other hand, the shown spectraa are the first ingredient for simulations of reflectivity experiments [56], bothh in specular reflection mode and in diffuse scattering measurements.

Recently,, in coherent small angle scattering experiments, phase retrieval algorithmss have been successfully applied [12] to recover the real-space im-agee from the scattered speckle patterns. Here multiple scattering and

polar-36 6 CHAPTERR 2

izationn mixing are serious problems, requiring good knowledge of the optical constants.. With the advent of the 4th-generation synchrotron sources during the nextt decade, such experiments will be widely exploited, so that accurate knowl-edgee of the scattering cross sections will be required.

Finally,, a recent pioneering work used the Cherenkov effect to produce too generate soft X rays by passing a moderate-energy electron bunch through aa foil [69, 70]. Such a source has potential for a laboratory-based X-ray mi-croscope.. The requirement for the Cherenkov effect is that the electrons move fasterr than the group velocity of the light. Two conditions have to be fulfilled: thee electron should move with relativistic speeds (typically in the 5-25 MeV range),, and the real part of the refractive index must exceed unity. This hap-penss at photon energies just below the absorption edge.

Fromm the real parts of F^01-2) (Fig. 2.2), we observe that almost for all el-ementss there exist energy ranges where this condition is fulfilled. The resonant enhancementss presented here are probably the largest that can be found, (with thee possible exception of the corresponding resonances in the actinides), and an orderr of magnitude higher than those of the Si and Ni targets used in the orig-inall study. Moreover, in the case of target films that are magnetically saturated alongg the electron propagation direction, the emitted X-rays will be circularly polarized.. The data presented here therefore are highly relevant for the further developmentt of soft X-ray Cherenkov radiation source.

2.6.. Conclusions

Inn a previous study, high-quality transmission Gd3 + M4/5 absorption cross

sectionss <70o/(a?) were compared with calculated atomic spectra. The

parame-terss resulting from the best fitted curves (reduction of Slater integrals and line broadenings),, have been used here to obtain the charge, circular and linear dichroicc absorption cross sections for all trivalent-RE M4 5 absorption edges. Thee application of the Kramers-Kronig transforms to these spectra provided thee real part of the atomic resonant scattering factors, i.e. the dispersive part off the refractive index n(co). Finally, the total scattered intensity spectra and thee specific Faraday and Voigt rotation angles were given. The sensitivity of thee calculated spectra to the parameter choice is found to be only moderate. Wee should also mention that our calculated spectra assume spherical symmetry

Magneto-opticall constants at the rare-earth M45 absorption edges 37 withoutt any crystal field effects. Such crystal field effects are small for the 4 /

shell,, but can change the absorption line shape [17] on the same scale as do the parameterr dependences discussed in the appendix. Such effects can be included inn the calculation once the symmetry of the ion is known.

Wee expect that these magneto-optical constants will be useful in absorp-tion,, scattering and microscopy experiments, and to a lesser degree, photoelec-tronn emission microscopy. Furthermore, they may contribute to the develop-mentt of a Cherenkov effect X-ray source. The calculated optical constants can bee downloaded from http://zvzvw.science.uva.nl/~miguel/