A common optical algorithm for the evaluation of specular spin polarized neutron and Mössbauer reflectivities
aa r X i v : . [ c ond - m a t . o t h e r] S e p A Common Optical Algorithm for the Evaluation of Specular Spin Polarized Neutronand M¨ossbauer Reflectivities
L. De´ak, ∗ L. Botty´an, D. L. Nagy, and H. Spiering Wigner RCP, RMKI, P.O.B. 49, H-1525 Budapest, Hungary Johannes Gutenberg Universit¨at Mainz, Staudinger Weg 9, D-55099 Mainz, Germany (Dated: July 4, 2018)Using the general approach of Lax for multiple scattering of waves a 2 × PACS numbers: 78.66.-w,07.85.Qe,76.80.+y
INTRODUCTION
The detectable information on a thin or stratifiedstructure by the reflectometric techniques is the one di-mensional scattering amplitude density profile perpendic-ular to the surface, which in turn can be related to thechemical/isotopic/magnetic, etc. profile within the pen-etration depth of the corresponding radiation. X-ray andneutron reflectometry, therefore, have become standardtools in studying surfaces and thin films. In nonresonantx-ray or unpolarized neutron reflectometry, the scatter-ing processes being independent of the polarization of theincident wave, any stratified medium can be described bya scalar complex index of refraction. There are other im-portant cases, however, in which the scattering mediumis birefringent for the corresponding radiation, and thepolarization-dependent multiple scattering leads to non-scalar optics. These cases include polarized neutron re-flectometry (PNR) and (synchrotron) M¨ossbauer reflec-tometry (SMR), the latter being only a special but wellstudied case of the anisotropic (resonant) x-ray scatteringproblem. Beyond the trivial analogy between the scalarcases of neutron and x-ray multiple scattering, the gen-eralization to polarization dependent scattering of any waves [1] is not straightforward and in fact, as we pointout below, it can not be performed in general. It is thepurpose of this paper to show that, indeed, such analogy,i.e. a common optical formalism exists for the anisotropicneutron and anisotropic nuclear resonant x-ray transmis-sion and reflection for the case of forward scattering andthat of grazing incidence.
GENERAL CONSIDERATIONS
In this section, starting from the general theory of Lax[1], we shall obtain some general formulae for the scatter-ing of multicomponent waves. Description the theories ofthe various scattering processes on a single scatterer lead to an inhomogeneous wave equation (cid:2)(cid:0) ∆ + k (cid:1) I − U ( r ) (cid:3) Ψ ( r ) = , (1)where k is the vacuum wave number, I is the unit matrix, U ( r ) is the scattering potential and Ψ ( r ) is the am-plitude of the scattered wave, an electromagnetic fieldvector or quantum mechanical spinor state. For manyscattering centers the coherent field fulfils the (cid:2)(cid:0) ∆ + k (cid:1) I + 4 πN f (cid:3) Ψ ( r ) = , (2)three dimensional wave equation, where f is the coher-ent forward scattering amplitude, N is the density of thescattering centers per unit volume and Ψ ( r ) is the coher-ent field defined by an average of the field vectors overthe positions and states of the scattering centers [1]. Eq.(2) shows that from the point of view of the coherentfield the system of randomly distributed scattering cen-ters can be replaced by a homogeneous medium, with anindex of refraction n = I + πNk f . Since n for both x-raysand slow neutrons hardly differs from I , it is better touse the susceptibility tensor defined by χ = πNk f [2].By choosing a simple homogeneous layer with theabove susceptibility χ and z axis normal to the layer,one gets the well known 1D wave equation:Ψ ′′ ( z ) + k sin θ h I sin θ + χ sin θ i Ψ ( z ) = . (3)with θ being the angle of incidence. Defining Φ via( ik sin θ ) Φ ′ ( z ) := Ψ ′′ ( z ), we get a system of first orderdifferential equations:dd z (cid:18) ΦΨ (cid:19) = ikM (cid:18) ΦΨ (cid:19) , (4)where M = (cid:18) I sin θ + χ sin θ I sin θ (cid:19) (5)is commonly called the ”differential propagation matrix”in optics [2, 3]. Eq. (4) was derived without specifyingthe scattering process.For an arbitrary multilayered film with homogenouslayers of thicknesses d , d ...d S and differential propaga-tion matrices M , M ...M S , χ in Eq. (5) is replaced bythe susceptibility χ l of layer l . The solution of the differ-ential equation (4) can be expressed in terms of the totalcharacteristic matrix L = L S · ... · L · L (6)of the multilayer, where L l = exp ( ikd l M l ) (7)is the characteristic matrix of the l th individual layer.The 2 × R is derived from the totalcharacteristic matrix L by R = (cid:0) L [11] − L [12] − L [21] + L [22] (cid:1) − × (cid:0) L [11] + L [12] − L [21] − L [22] (cid:1) , (8)where L [ ij ] ( i, j = 1 ,
2) are 2 × × L [2]. The reflected intensity I r = Tr (cid:0) R † Rρ (cid:1) (9)can be calculated by using the arbitrary polarization den-sity matrix ρ of the incident beam and the reflectivitymatrix [4]. NUMERICAL CONSIDERATIONS
The numerical problem in evaluating the reflectivity isthe calculation of the exponential of the 4 × × L l = (cid:18) cosh ( kd l F l ) x F l sinh ( kd l F l ) xF − l sinh ( kd l F l ) cosh ( kd l F l ) (cid:19) , (10)where the 2 × F l = p − I sin θ − χ l and x = i sin θ [2].To evaluate Eq. (10), first we have to calculate the2 × F matrices. This can be madeby using the identity G / = G + I √ det G p Tr G + 2 √ det G , (11)where G is any nondiagonal 2 × × G can be expressed by itself and itsscalar invariants:exp G = exp( 12 Tr G ) × " cos p det ¯ GI + sin √ det ¯ G √ det ¯ G ¯ G , (12)where ¯ G = G − I Tr G [4].In order to calculate the characteristic matrix of asemi-infinite layer (substrate) S , we have to find its L S → L ∞ limit for d S → ∞ . From Eqs. (5) through(12) follows that the corresponding limit is given by L ∞ = I p p I + χ S sin θ p (cid:0)p I + χ S sin θ (cid:1) − I ! (13)where p = sgn [Re (Tr F S )] is the sign of the real part ofthe trace of F S . The above algebra turns out to be numerically very sta-ble, therefore this approach is suitable for fast numericalcalculations of the characteristic matrices for anisotropicstratified media. In fact, the exponential of the matrixin Eq. (5) can be calculated exactly without solving anyeigenvalue problem. The program based on this calculusis freely available [5, 6].
M ¨OSSBAUER AND POLARIZED NEUTRONREFLECTOMETRIES
A simple application of Eq. (4) to nuclear resonantx-ray scattering is not possible, since the anisotropicMaxwell equations and the spin-dependent Schr¨odinger-equation lead to different results [3, 7] and the 3 × × f in general. However, start-ing from the Maxwell equations and using the 3 × . However, in [2] both an upper and a lower limit was foundfor the grazing angle θ for this approximation to apply,which limits are not present in the original theory of Lax[1]. The forward scattering amplitude matrix was ex-pressed for the nuclear resonant x-ray case in [4, 11] interms of the hyperfine interactions.The application of the above optics for PNR impliesspecifying f (or χ ) for the interaction potential U in Eq.(3). We use the potential U ( r ) = U p ( r ) + U m ( r ) as thesum of the isotropic nuclear potential U p ( r ) = 4 πbδ ( r ) I, (14)and the anisotropic magnetic potential U m ( r ) = − m ¯ h µ m · [ B a ( r ) + B ext ]= − m ¯ h µ m · B ( r ) (15)with m being the mass of the neutron, b the nuclear scat-tering length of the nucleus in the laboratory system, µ m = gµ N σ the magnetic moment operator of the neu-tron, g = − . µ N =5 . × − Am , σ the Paulioperator, B a the atomic magnetic field, B ext the (ho-mogeneous) external magnetic filed. In the first Bornapproximation f = − π Z Ω d r U ( r ) , (16)where Ω is the volume of the interaction (in fact theatomic volume). By using χ = πNk f we get χ = 1 k " m ¯ h gµ N σ · B − πN X i α i b i I , (17)where index i accounts for the different types of scatter-ing centers, and α i for the relative abundance of the i thnucleus. The mean magnetic field B = B ext + B a = B ext + R Ω d r B a ( r ) . In neutron reflectometry the scattering vector, Q =2 k sin θ and the scattering length density K = k χ = 2 m ¯ h gµ N σ · B − πN X i α i b i I (18)are more often used than θ and χ . With these notationsEq. (4) readsdd z (cid:18) ΦΨ (cid:19) = i Q I + KQQ I ! (cid:18) ΦΨ (cid:19) . (19)Using the definition of the Pauli matrices, the scatter-ing length density matrix Eq. (18) is expressed by thephysical quantities K = 2 m ¯ h gµ N (cid:18) B x B y − iB z B y + iB z − B x (cid:19) − πN X i α i b i I, (20)where B x , B y , B z are the components of the magneticfield B .Having K from Eqs. (20) and (18) for each layer l , Eq.(12) is used to calculate the exponential of the differen-tial propagation matrix of Eq. (19) is obtained. Withthis (by applying (10) to (13)) first the (7) characteristicmatrices , then the (6) total characteristic matrix L , fromwhich the (8) complex reflectivity matrix R is calculated. For the sake of brevity, we dropped the layer index l in K, B , N , α i and b i in Eqs. (16) to (20).An elegant covariant treatment of specular PNR [12]including earlier matrix methods of restricted form [13,14] recently published by R¨uhm et. al. turns out to beequivalent to the present results. Indeed, substituting p = k sin θ and b H l = − (cid:0) ¯ h k / m (cid:1) χ l for layer l in Eq.(7) of [12] we obtain (10), an equation equivalent to Eq.(3.20) of [2]. Consequently, what we have shown here isthe equivalence [15] of the supermatrix formalisms devel-oped for SMR [2] and PNR [12]. THE EXTERNAL MAGNETIC FIELD AS ANANISOTROPIC MEDIUM
Although their general treatment would have allowedfor, R¨uhm et al. [12] did not explicitely studied the effectof the (guiding or polarizing) external magnetic field onthe neutron beam, what we briefly outline in this sectionin the standard manner borrowed from anisotropic optics[3].The (8) reflectivity expression is only valid for a neu-tron beam incident on the layer system ( l = 1 , , , .., S )from the vacuum ( l = 0). In the typical experimentalsetup, however, guiding fields and often strong externalmagnetic fields are used in order to eliminate depolar-ization of the neutrons and to ensure polarization of thesample, respectively. The effect of the external magneticfield was studied by Pleshanov [16] and Fermon [17] indetail. From Eq. (20) it follows, that the vacuum, inpresence of an external magnetic field, is an anisotropic’medium’. Consequently, the incoming beam is given inthis ’medium’ instead of being given in the vacuum. Inorder to treat this problem, following Borzdov [3] (for abrief outline in English see [2]), for the case of neutronreflectometry we introduce an impedance tensor γ by thefollowing relationship: γ ,r,t Ψ ,r,t := Φ ,r,t , (21)where indexes 0 , r and t indicate incident, reflected andrefracted waves, respectively (see Eq. (3.4) of Ref. [2]).Substituting (21) into Eq.(19) we get the impedance ten-sors γ ≡ γ = − γ r = s I + 4 KQ , (22)where K is calculated from Eq. (20) for the given exter-nal magnetic field. We dropped γ t because the substrateis taken as a semi-infinite layer with Eq. (13). Fromexpressions (18) and (20) for γ we get γ ( Q ) = 1 Q (cid:18) p Q + p Q − (cid:19) , (23)where Q ± = Q ± m ¯ h gµ N | B ext | (24)is the momentum Q ± = 2 k sin θ ± measured in the exter-nal magnetic field. Due to the birefringence of anisotropicmedia (including vacuum in presence of external mag-netic field), the beam propagation directions for the dif-ferent polarizations necessarily differ from each other,consequently the angles of incidence and the momentumof the beams with different polarizations (sign ’+’ and’ − ’) are also different ( θ ± and Q ± ). The vacuum mo-mentum Q can be calculated backwards from Q ± by ap-plying the Fresnel refraction law [Eqs. (20) and (3), aswell as the definition of K and Q by Eq. (18)].Having the impedance tensors of the individual layers,we simply apply the modified general 2 × R = (cid:2)(cid:0) L [11] − L [21] (cid:1) γ − L [12] + L [22] (cid:3) − × (cid:2)(cid:0) L [11] − L [21] (cid:1) γ + L [12] − L [22] (cid:3) (25)which takes the effect of the external magnetic field intoaccount through the impedance tensor γ [2]. The re-flected intensity I r is calculated from Eq. (9) using thethe reflectivity matrix R and the polarization density ma-trix ρ = | Ψ i h Ψ | of the incident beam, where the bar rep-resents the average over the polarizations [4, 12]. CONCLUSION
In summary, a common optical formalism of (nuclear)resonant x-ray (M¨ossbauer) reflectometry and polarizedneutron reflectometry was presented. Consequently, thestrictly covariant formalism of [3] as published in [2] andthe corresponding computer program [5, 6] are readily available for neutron reflectometry of layered systems ofarbitrary complexity. Taking the effect of the externalmagnetic field through the impedance tensor into ac-count, a modified reflectivity expression is given. Theform of the reflectivity matrix allows for a very effi-cient numerical algorithm for both SMR and PNR im-plemented in [5, 6].This work was partly supported by the HungarianScientific Research Fund (OTKA) under Contract Nos.T029409 and F022150. L. D. thanks for support by theDeutscher Akademischer Austauschdienst (DAAD). ∗∗