ISSN 1063�7745, Crystallography Reports, 2013, Vol. 58, No. 7, pp. 1037–1042. © Pleiades Publishing, Inc., 2013.

1037

INTRODUCTION

An X�ray resonant magnetic reflectivity (XRMR)study yields unique elemental and spatially selectiveinformation about magnetic ordering in multilayerfilms. Magnetic scattering, being significant near theabsorption edges of magnetic atoms, radically compli�cates the theory of reflectivity, because the X�ray sus�ceptibility of a medium generally becomes a tensor inthe presence of this scattering. The theory of X�rayreflection from anisotropic (magnetic) multilayers wasdeveloped in [1–5] based on eigen�wave formalismand in [6–8] using the method of 4 × 4 propagationmatrices. The application of these algorithms forinterpreting real experimental data is rather time�con�suming; therefore, simplifying the calculations is anurgent problem. In particular, such attempts weremade in [3, 9–11]. An interesting approach (specifi�cally, a combination of dynamical (for the isotropicscattering part) and kinematical (for the magneticscattering part) approximations) was applied in [12] tointerpret the reflection curves near the M5 Dy absorp�tion edge from a dysprosium film with helicoidal mag�netic ordering. A kinematical scalar approximationwas used in [13, 14] when interpreting the XRMRspectra to obtain spin�polarization profiles (over thelayer depth) of 5d electron shells of cerium and lantha�num in multilayer [Ce/Fe]n and [La/Fe]n structures.Correctly interpreting the experimental data is veryimportant in such studies. The depth variation of themagnetic anisotropy leads to a complex polarizationdependence of the reflection. Polarization asymmetryratio is analyzed, which is generally very small [15] andextremely sensitive to calculation errors. In this paperwe proposed formulas for calculating the reflectivity ofmagnetically ordered multilayers in the matrix kine�matical approximation and carried out test calcula�

tions to verify this approximation under certain condi�tions.

THEORY

The kinematical limit of the well�known Parrattalgorithm in the X�ray reflectivity theory was consid�ered in [16, 17]. To obtain the kinematical formula,one neglects the multiple reflection in each layer (i.e.,reject the denominator in the Parratt formula) and usean approximate expression for the square root:

, (1)

which determines the normal components of the wavevectors in a layer (θ is the grazing angle and χj is thesusceptibility of the jth layer). Accordingly, the Fresnelformula is simplified to

. (2)

As a result, the amplitude reflectivity R for a multilayerstructure can be presented as a sum of the susceptibil�ity gradients at interfaces with a corresponding phaseshift:

. (3)

Here, is the scattering vector,

, is the depth of the interface between the( j – 1)th and jth layers, and dk are the layer thick�nesses. Note that, as direct calculations show,

in the exponential factor cannot beneglected; at the same time, the weak magnetic addi�

2sin sin2sin

jj j

χη = θ + χ ≈ θ +

θ

1 11, 2

1

( )

4sinj j j j

j jj j

r − −

−

−

η −η χ − χ= ≈η + η θ

)

1

1

1sin

12

1

1 (4 sin

j

j k k

k

i Qz dL

j j

j

R e

−

−

=

⎛ ⎞κ⎜ ⎟+ χ

⎜ ⎟θ⎝ ⎠

−

=

= χ − χθ

∑

∑

42 sin sinQ π= κ θ = θ

λ

/cκ = ω 1jz−

0k k mkχ = χ + Δχ

SURFACE AND THIN FILMS

Kinematical Limit in the Theory of X�Ray Magnetic ReflectivityM. A. Andreevaa and Yu. L. Repchenkob

a Moscow State University, Moscow, 119991 Russiae�mail: Mandreeva1@yandex.ru

b Voronezh State University, Voronezh, 394006 RussiaReceived October 20, 2012

Abstract—A kinematical approximation has been derived from the general formulas for calculating thereflectivity from a multilayer anisotropic structure. An explicit form of reflection matrices at the boundary oflayers and the refraction tensors has been obtained for a particular case where the layer magnetization is ori�ented in the surface plane and for an arbitrary form of X�ray susceptibility tensors. Calculations of the reflec�tivity from a Nb(4 nm)/[Dy(5 nm)/Lu(3 nm)]420 model structure with helicoidal ordering of Dy magneticlayers show that kinematical approximation can be applied for grazing angles that are larger than the criticaltotal�reflection angle and that the “magnetic” refraction must be taken into account.

DOI: 10.1134/S106377451307002X

1038

CRYSTALLOGRAPHY REPORTS Vol. 58 No. 7 2013

ANDREEVA, REPCHENKO

tives in the exponential factor can be disregardedin some cases. This circumstance is important whenthe anisotropic case is considered.

In the case of anisotropic layers, the recurrencerelations for calculating the 2 × 2 matrix reflectivity forthe tangential components of magnetic radiation field

can be written as [18, 19]

, (4)

where

; (5)

; (6)

and are, respectively, the single reflectionand transmission matrix coefficients for the tangentialcomponents of the magnetic radiation field at the inter�face between the ( j – 1)th and j th layers; primes corre�spond to reflection and transmission in the reversedirection; symbols + and – indicate, respectively, the

transmitted and reflected waves; and and 2 × 2matrices are planar tensors of surface impedance and

normal refraction, respectively [20, 21]. The tensorsrelate the tangential components of the electric andmagnetic radiation fields separately for the transmitted

and reflected waves: . The tensors

describe wave propagation. For example, in the forward

direction we have = .

Having passed from the dynamical approach to thekinematical limit, i.e., neglected multiple reflectionand anisotropic additives in the matrix exponents(replaced the latter with simple exponentials), one canreduce (4) to the expression

. (7)

mkΔχ

1,ˆ

j jP−

1 11, 1, 1, 1, 1,

ˆ ' 'ˆ ˆ ˆ ˆ( ) [(1 ) ]j j j j j j j j j j j jP p p − −

− − − − −

= + τ − τW W� �

1, , 1ˆˆ' j j j ji d i d

j j j j jp e P e− +

− κ κ

− +=

N NW

� �

�

11, 1 1ˆ ( ) ( )j j j j j jp + − − + +

− − −= − −γ γ γ γ

� � � �

1,ˆ j jp− 1,ˆ j j−

τ

γ N

γ±

ˆ xy

yx

HE

HE±− ⎛ ⎞⎛ ⎞ = γ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠N

±

( )

( )x

y

H z d

H z d

+⎛ ⎞⎜ ⎟+⎝ ⎠

( )

( )xi d

y

H ze

H z

−

κ ⎛ ⎞⎜ ⎟⎝ ⎠

N�

1

1 0

1sin

1,

1

ˆ ˆ

j

j k k

k

i Qz dL

j j

j

P e p

−

−

=

⎛ ⎞κ⎜ ⎟+ χ

⎜ ⎟θ⎝ ⎠

−

=

≈

∑

∑

For periodic multilayer structures, summation in(7) is divided into structural (summation over oneperiod) and periodic (summation over several peri�ods) parts; the latter can easily be reduced to theLaue function (see, for example, [17]), which signifi�cantly reduces the calculation time for multiperiodstructures. The structural part is a matrix in our case.

When the magnetic corrections for refractionand absorption cannot be neglected (i.e., the matrixexponents in (5) cannot be replaced with simpleexponentials), the kinematical formula becomesmore complex, because the multiplication order ofthe matrix exponents and the reflection matrix can�not be changed:

. (8)

To calculate the matrix coefficient of a singlereflection from the interface between the ( j – 1)th andjth layers within the kinematical approximation, oneshould simplify (6). The general expression for wasreported in [20, 21]:

(9)

where is a unit 2 × 2�matrix; , and are2 × 2 blocks of the propagation matrix for the tan�gential components of the electric and magnetic radi�ation fields, which is defined as

; (10)

and and are the normal wave�vector componentsexpressed in terms of ω/c, which are the eigenvalues of

this matrix. The tensors can be presented by thesimple relation [20, 21]

. (11)

Simplifications of (1) type for the kinematical case

can be performed for a specific form of , , and tensors. Let us consider the case where the magnetiza�tion vector of a multilayer structure lies in the surfaceplane (Fig. 1). Here, the permittivity tensor can bewritten as [3, 10]

(12)

where Δχm is the magnetic additive to the susceptibilityof medium, which is responsible for the circular

1 1 1 11 1 1 11,

1

ˆ ˆ... ...j j j j

Li d i di d i d

j j

j

P e e p e e− +

− +

− − − −

− κ + κ− κ + κ

−

=

≈∑N NN N� �

� �

1,ˆ j jp−

γ

11 2

1 2

[( ) ]

ˆ ˆ[ ( )( )],I I

± ± ± −

± ±

= η + η − −

× + − η − η

γ B AB BD

BC A A

�� � � �

�

� � ��

I , ,A B C�� �� D

�

M�

( ) ( )ˆ ˆ( ) ( )

( ) ,ˆ ˆ( ) ( )

( ) ( )

x x

y y

y y

x x

H z H z

H z H zd i zE z E zdz c

E z E z

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎛ ⎞ω⎜ ⎟ ⎜ ⎟= = ⎜ ⎟− −⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

A BM M

C D

� �

1±

η 2±

η

N

ˆˆ ˆ ˆ± ±

= + γN A B

χ γ N

0

0

0

ˆ1

1 0 sin

0 1 cos ,

sin cos 1

m

m

m m

i

i

i i

= + χ+ χ Δχ ψ⎛ ⎞

⎜ ⎟= + χ − Δχ ψ⎜ ⎟⎜ ⎟− Δχ ψ Δχ ψ + χ⎝ ⎠

ε�

x

yz

κ0

κRθ

θ ψm

Fig. 1. Geometry of the experiment and the chosen coor�dinate system.

CRYSTALLOGRAPHY REPORTS Vol. 58 No. 7 2013

KINEMATICAL LIMIT IN THE THEORY OF X�RAY MAGNETIC REFLECTIVITY 1039

dichroism (it is significant near the absorption edge),and the angle ψ indicates the magnetization azimuth(ψ = 0° if the magnetization vector m lies in the scat�tering plane). The coordinate system we used is shownin Fig. 1. In this case the propagation matrix can bepresented in the form [10, 15]

, (13)

where , , and

. Hereinafter, we neglect the terms

with ~ . To determine the eigenvalues of matrix ,we get the biquadratic equation

, (14)

which yields

(15)

We will find for propagation matrix (13) by means ofexpression (9):

(16)

With allowance for the expression given by (15),

, (17)

the tensors can be reduced to

, (18)

where .

We obtain the following relation for the normal�refraction tensors from (11):

. (19)

Matrix reflectivity is calculated under theassumption that there is a zero�thickness layer withzero susceptibility at the interface. The surface�impedance tensor for this layer can be determined as

. Thus, according to (6) and

(8), we can write

(20)

Using approximate expressions for square roots (1)and (2) (which are conventional for the kinematicallimit), we arrive at the expression

(21)

One can easily pass from the matrix reflectivitiesfor the tangential components of the magnetic radia�tion field to the standard π and σ polarizations. In thechosen coordinate system (Fig. 1), these polarizationvectors can be written as

(22)

Using these expressions, one can easily find that

( )0

20

20

0

0 1 0

0 0ˆ ˆˆ ( )

ˆ ˆ 01

0 1 0 0

a

bz

a b

+ χ⎛ ⎞⎜ ⎟

η⎜ ⎟⎛ ⎞⎜ ⎟= ≅⎜ ⎟ η − −⎜ ⎟⎝ ⎠

+ χ⎜ ⎟⎜ ⎟⎝ ⎠

A BM

C D

cos cosma i= Δχ θ ψ cos sinmb i= Δχ θ ψ

2 20 0sinη = θ + χ

2mΔχ M

�

4 2 2 2 4 2 2 20 0 0 0(2 ) (1 ) 0a a bη − η + η + η + η + + χ ≅

21,2 0

21,2 0

cos sin ,

cos sin .

m

m

+

−

η ≅ + η ± Δχ θ ψ

η ≅ − η ± Δχ θ ψ

ˆ±

γ

DET

1DET

20 0 0 0

20 0

21 0 1 2 0

1 20 0 1 2

2 20 1 2 1 2 0 1 2 0

(1 ) 0 (1 ) (1 ) 0ˆ ˆˆ ˆ ˆ ˆ; ; ;(1 ) 0 0 0 0

( ) (1 )1[( ) ] ;(1 ) (1 )( )

( )(( ) ( )) (1 )

a a b

b

b

b

a b

± ± −

±

⎛ ⎞+ χ − + χ − + χ η⎛ ⎞ ⎛ ⎞≅ ≅ ≅ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ χ⎝ ⎠ ⎝ ⎠ η⎝ ⎠⎛ ⎞η η + η − + χη + η − − ≅ ⎜ ⎟+ χ + χ η + η⎝ ⎠

η η + η η η + η − η + η − + χ≅

AB BD BC

B AB BD�� � � �

�

γ

DET

21 2 0

2 2 20 1 2 0 1 2 0 1 2 1 2 0

2 20 0 1 2

( );

(1 )(( ) ( ) ) (1 )( )( )

(1 ) ( ) .

b

⎛ ⎞η η + η⎜ ⎟

+ χ η η + η − η + η + χ η + η η η + η⎝ ⎠

≅ + χ η η + η

22 1 2

1 2 0( )

2

η + ηη η + η =

ˆ±

γ

20

020

( )21ˆ (1 )

2

ab

b

±

±

±

⎛ ⎞±η −η −⎜ ⎟γ ≅ + χ⎜ ⎟η ⎜ ⎟− ±η⎝ ⎠

�

�

1 2( )

2

± ±

± η + ηη =�

202ˆ

2

b

b

±

±

±

⎛ ⎞η −⎜ ⎟η≅ ⎜ ⎟⎜ ⎟η⎝ ⎠

N

�

�

1,ˆ j jp−

0 0

sin 0ˆ ˆ 10

sin

+ −

θ⎛ ⎞⎜ ⎟γ = −γ =⎜ ⎟⎝ ⎠θ

11, 0 1 1 0

10 0

ˆ ( ) ( )

( ) ( ).

j j j j

j j

p + − − + +

− − −

+ + − + +

= − −

+ + −

γ γ γ γ

γ γ γ γ

� � � �

� � � �

10 1 1

1, 2

1 0 1

0

0

cos 2 2 sin1ˆ sin4 sin sin

cos 2 2 sin.sin

sin

jj j

j j

j j

jj j

j j

ba

pb

ba

b

−

− −

−

− −

⎡ ⎛ ⎞−χ θ + θ⎜ ⎟⎢≅ − θ⎜ ⎟⎢θ θ χ⎣ ⎝ ⎠⎛ ⎞⎤−χ θ + θ⎜ ⎟⎥+ θ⎜ ⎟⎥

θ χ⎝ ⎠⎦

(1,0,0), (1,0,0)

(0, sin , cos ), (0, sin , cos ).

e e

e e

+ −

σ σ

+ −

π π

= =

= θ − θ = − θ − θ

1040

CRYSTALLOGRAPHY REPORTS Vol. 58 No. 7 2013

ANDREEVA, REPCHENKO

(23)

The general kinematical formula in the unit vectors ofπ and σ polarizations has a form similar to (8):

(24)

One can see that reflection matrices (21) havingpassed to the π and σ polarizations according to (23)coincide (accurate to the coefficient) with the matrixrepresentation of the scattering amplitude, which canbe obtained as a convolution of the susceptibility ten�

sor with polarization vectors of the incident and scat�tered photons.

Thus, by analogy with (21) and (23), one canobtain expressions for the matrix single�reflectioncoefficients in the unit vectors of π and σ polar�izations at the ( j – 1)/j interface for an arbitrary formof the susceptibility tensor

(25)

and the corresponding matrix refractive indices for thetransmitted and scattered waves in each jth layer

(26)

where

(27)

(in (27) the subscript j is omitted).

MODEL CALCULATIONS

First, we checked the applicability of the kinemat�ical approximation for layers with scalar susceptibility.If the layer magnetization vector lies in the scatteringand surface planes (L�MOKE geometry), the calcula�tion for grazing angles of incidence can be carried outusing formula (3) under the assumption that

for, respectively, the right� and left�handed circular polarizations of incident radiation,which can be considered as the eigen polarizations ofthe problem [10, 15]. The calculation results for aNb(4 nm)/[Dy↑(5 nm)/Lu(3 nm)/Dy↓(5 nm)/Lu(3 nm)]150/Y(70 nm)/sapphire sample (character�ized by antiferromagnetic interlayer ordering) areshown in Fig. 2. The photon energy Eph = 7790 eV(λ = 0.159 nm) corresponds to the L3 absorption edgeof dysprosium. The susceptibilities of Lu, Nb, Y, andsapphire layers were taken from [22, 23], while the sus�ceptibility of the Dy resonant magnetic layer was cal�culated using the X�ray absorption (XAS) and X�raymagnetic circular dichroism (XMCD) spectrum mea�sured in [24] (after the proper normalization to thetabular data of [22, 23]). The Kramers–Kronig rela�tions were used to obtain the real part of the Dy sus�ceptibility.

The calculation showed that, for the angles that aremuch larger than the critical angle of total externalreflection (θ >~ 3θc), kinematical approximation (3),as one would expect, yields an acceptable result incomparison with the exact calculation by the Parrattformulas. Note that the magnetic additives inthe exponents were disregarded in the calculationbased on formula (3); however, in the scalar case underconsideration, this simplification did not deterioratethe agreement with the exact calculation. A calcula�tion was also performed for the combined approachproposed in [12]. This approach was found to be evenworse than the purely kinematical approximation: itcan be used only for larger angles. Thus, we showedthat there is no point in considering two contributionsto the reflected wave within different approximations.

The results of a calculation by means of matrix for�mulas (7), (8), (19), and (21) were compared with theexact calculations for a multilayer structure with heli�coidal magnetic ordering of Dy layers. In this case theeigen polarizations cannot be chosen as identical forall layers and, correspondingly, the matrix theory ofreflection should be used. Although on the whole sim�plified formula (7) rather well describes the polariza�tion asymmetry ratio for angles somewhat larger thanthe critical one (Fig. 3), there are some angular rangeswhere the results given by formula (7) differ signifi�cantly from the exact calculation results (the algo�

2122

12 11

2122

12 11

ˆˆ

ˆ ,sinˆ ˆsin

ˆ .sin

sin

ppr r

rr r

p p

NNn n

nn n

N N

σσ σπ

πσ ππ

±

±± ±

± σσ σπ

± ±

± ±πσ ππ

⎛ ⎞−⎛ ⎞ ⎜ ⎟= = θ⎜ ⎟⎜ ⎟⎝ ⎠ − θ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎜ ⎟= = θ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ θ⎝ ⎠

∓

∓

.1 1 1 11 1 1 1ˆ ˆˆ ˆ1,

1

ˆ ˆ... ...j j j j

Li d n i d ni d n i d n

j j

j

R e e r e e− +

− +

− − − −

− κ + κ− κ + κ

−

=

≈∑

1,j jr−

1, 121 ˆ ˆˆ ( ),

4 sin

sc scj j j jr ⊥ ⊥

− −= χ − χθ

ˆˆ sin ,

2sinj

jn⊥±

±⎛ ⎞χ

= ± θ +⎜ ⎟θ⎝ ⎠

2 2

sin cosˆ ,

sin cos cos sin cos sin ( )

xx xy xzsc

yx zx zz yy yz zy

⊥χ χ θ − χ θ⎛ ⎞

χ = ⎜ ⎟−χ θ − χ θ χ θ − χ θ + θ θ χ − χ⎝ ⎠

2 2

sin cosˆ ,

sin cos cos sin cos sin ( )

xx xy xz

yx zx zz yy yz zy

⊥+χ χ θ − χ θ⎛ ⎞

χ = ⎜ ⎟χ θ − χ θ χ θ + χ θ − θ θ χ + χ⎝ ⎠

2 2

sin cosˆ .

sin cos cos sin cos sin ( )

xx xy xz

yx zx zz yy yz zyc⊥−

χ −χ θ − χ θ⎛ ⎞χ = ⎜ ⎟−χ θ − χ θ χ θ + χ θ + θ θ χ + χ⎝ ⎠

0k k mkχ = χ ± Δχ

mk±Δχ

CRYSTALLOGRAPHY REPORTS Vol. 58 No. 7 2013

KINEMATICAL LIMIT IN THE THEORY OF X�RAY MAGNETIC REFLECTIVITY 1041

rithm for exact calculation was described, for exam�ple, in [6–8] and implemented in a software package[25]). These discrepancies can be removed by consid�ering magnetic refraction, which is taken into accountin formula (8). The results given by this formula areshown by a dotted curve, which coincides in this rangewith the exact�calculation data.

CONCLUSIONS

We have obtained formulas within the kinematicalapproximation which describe the reflection of polar�ized radiation from a system of anisotropic (magnetic)layers fairly well for grazing angles beyond the total�reflection range and have a much simpler structure (incomparison with the 4 × 4 propagation matrices usedin the exact theory of reflection). These formulas allowone to easily analyze the polarization effects arising fora specific scattering anisotropy or, for example, forlayers with different magnetization directions. Inaddition, the use of the kinematical algorithm signifi�cantly speeds up the calculations and facilitates the fit�ting of experimental data. However, it should be notedthat, when considering such fine effects as polariza�

tion asymmetry ratio, the results must always be com�pared with the exact�calculation data.

ACKNOWLEDGMENTS

This study was supported in part by the RussianFoundation for Basic Research, project nos. 12�02�00924�a and 10�02�00768�a.

REFERENCES

1. J. Zak, E. R. Moog, C. Liu, and S. D. Bader, Phys. Rev.B 43, 6423 (1991).

2. A. Bourzami, O. Lenoble, Ch. Fe’ry, J. F. Bobo, andM. Piecuch, Phys. Rev. 59, 11489 (1999).

3. N. Ishimatsu, H. Hashizume, S. Hamada, et al., Phys.Rev. B 60, 9596 (1999).

4. S. A. Stepanov and S. A. Sinha, Phys. Rev. B 61, 15302(2000).

5. M. Elzo, E. Jal, O. Bunau, S. Grenier, Y. Joly,A. Y. Ramos, H. C. N. Tolentino, J. M. Tonnerre, andN. Jaouen, J. Magn. Magn. Mater. 324, 105 (2012).

6. M. A. Andreeva and A. G. Smekhova, Appl. Surf. Sci.252, 5619 (2006).

7. M. A. Andreeva, A. G. Smekhova, B. Lindgren, et. al.,J. Magn. Magn. Mater. 300 (1), e371 (2006).

8. M. A. Andreeva and A. G. Smekhova, Izv. Akad. Nauk,Ser. Fiz. 72 (5), 693 (2008).

9. E. Kravtsov, D. Haskel, T. Velthuis, et al., Phys. Rev. 79,134438 (2009).

0.4

–0.1

0 0.8 1.2 1.6 2.0Glancing angle, deg

0

0.1

Asymmetry ratio (R+ – R–)/(R+ + R–)

exact

kinem

mixed

exact

kinem

Reflectivity (R+ + R–)/2

100

10–2

10–4

10–6

Fig. 2. Calculated of the (top) reflectivity for right�handed(R+) and left�handed (R–) circularly polarized light and(bottom) polarization asymmetry ratio (R+ – R–)/(R+ +R–) for the structure with antiferromagnetic interlayerordering: (solid lines) exact calculation, (dotted lines) cal�culation based on kinematical formula (3), and (dotted–dashed lines) combined approximation (clarification intext). Thin vertical lines indicate the positions of Braggmaxima of different orders. The magnetic maxima of half�integer order, corresponding to doubling the magneticperiod, are barely seen in the reflectivity curves R+ and R–

but are pronounced (most clearly for maxima 1/2 and 5/2)in the asymmetry ratio plots.

1.0

–0.1

0.5 1.5 2.5 3.02.0Glancing angle, deg

0

0.1

Asymmetry ratioexactkinkin2

Reflectivity

100

10–2

10–4

10–6

3.5

(R+ – R–)/(R+ + R–)

R+ exactR– exactR+ kinR– kin

Fig. 3. The same as in Fig. 2 but for the Nb(4 nm)/[Dy(5 nm)/Lu(3 nm)]420/Y(70 nm)/sapphire structurewith helicoidal magnetization ordering in Dy layers: (solidlines) exact calculation, (thin solid lines) calculation fromformula (7), and (dotted lines) calculation from formula(8). The magnetic satellites at a distance =λ/12D from the structural maxima clearly manifest them�selves in the asymmetry ratio curve because the magneticperiod covers six structural periods.

/6B±Δθ ≈ θ

1042

CRYSTALLOGRAPHY REPORTS Vol. 58 No. 7 2013

ANDREEVA, REPCHENKO

10. E. E. Odintsova and M. A. Andreeva, Poverkhnost,No. 11, 46 (2010).

11. D. A. Tatarskii and A. A. Fraerman, Poverkhnost,No. 7, 10 (2012).

12. H. Ott, C. Schubetaler�Langeheine, E. Schierle, et al.,Appl. Phys. Lett. 88, 212507 (2006).

13. L. Sève, N. Jaouen, J. M. Tonnerre, et al., Phys. Rev.B 60, 9662 (1999).

14. N. Jaouen, J. M. Tonnerre, E. Bontempi, et al., PhysicaB 283, 175 (2000).

15. A. G. Smekhova, M. A. Andreeva, E. E. Odintsova,et al., Kristallografiya 55, 906 (2010).

16. I. W. Hamley and J. Pedersen, J. Appl. Crystallogr. 27,29 (1994).

17. M. A. Andreeva and B. Lindgren, Phys. Rev. B 72,125422 (2005).

18. Yu. N. Khaidukov and M. A. Andreeva, Vestn. Mosk.Univ., Ser. 3: Fiz. Astron., No. 2, 30 (2004).

19. A. G. Smekhova, Candidate’s Dissertation in Physicsand Mathematics (Faculty of Physics, Mosk. Univ.,Moscow, 2006).

20. L. M. Barkovskii, G. N. Borzdov, and V. Lavrukovich,Zh. Prikl. Spektrosk. 25, 526 (1976).

21. L. M. Barkovskii, G. N. Borzdov, and F. I. Fedorov,Wave Operators in Optics, Preprint of Inst. of Physics,Belarusian Acad. Sci., Minsk, 1983, no. 304.

22. B. L. Henke, E. M. Gullikson, and J. C. Davis,At. Data Nucl. Data Tables 54 (2), 181 (1993).

23. http://henke.lbl.gov/optical_constants/getdb2.html.24. C. Sorg, Magnetic Properties of 3d and 4f Ferromag�

nets Studied by X�Ray Absorption Spectroscopy(2005). http://users.physik.fu�berlin.de/~bab/start_frame2/diss/CSdiss.pdf.

25. http://kftt.phys.msu.ru/personalii/Andreeva/XRMR.zip.

Translated by A. Grudtsov