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Distorted-wave Born approximation in the case of an optical scattering potential
ARTICLE IN PRESS
Physica B 355 (2005) 244–249 www.elsevier.com/locate/physb
Distorted-wave Born approximation in the case of an optical scattering potential Sergey V. Mytnichenkoa,b, a Institute of Solid State Chemistry, SB of RAS, 630128 Novosibirsk, Russia Siberian SR Centre at Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
b
Received 29 July 2004; received in revised form 27 October 2004; accepted 31 October 2004
Abstract Application of the distorted-wave Born approximation in the conventional form developed for the case of a real scattering potential is shown to cause significant errors in calculating X-ray diffuse scattering from non-ideal crystals, superlattices, multilayers and other objects if energy dissipation (photoabsorption, inelastic scattering, and so on) is not negligible, or in other words, in the case of an optical (complex) scattering potential. We show how a correct expression for the X-ray diffuse-scattering cross-section can be obtained in this case. Generally, the diffuse-scattering cross-section from an optical potential is not T-invariant, i.e. the reciprocity principle is violated. Violations of T-invariance are more evident when the dynamical nature of the diffraction is more critical. r 2004 Elsevier B.V. All rights reserved. PACS: 78.70.C; 07.85.F; 41.50 Keywords: X-ray diffuse scattering; Multilayer; Superlattice; Coherence
1. Introduction As a rule, the relatively weak interaction of X-rays and matter allow one to use the Born (kinematical) approach for calculations of X-ray SR Center at BINP, Pr. Akademika Lavrent’eva 11, 630090
Novosibirsk, Russia. Tel.: +7 3832 336289; +7 3832 342163. E-mail address: [email protected] (S.V. Mytnichenko).
fax:
diffraction and scattering successfully. Nevertheless, there is a need to apply dynamical approaches for some cases of diffraction from objects such as crystals, superlattices, smooth surfaces, multilayers and so on. The dynamical theory of diffraction was developed for such cases many years ago. It can be applied to calculate specular scattering when such objects are assumed to have almost perfect structure. The distorted-wave Born approximation (DWBA) has been widely used recently to calculate non-specular diffuse
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.10.099
ARTICLE IN PRESS S.V. Mytnichenko / Physica B 355 (2005) 244–249
scattering, caused by structural distortions (roughness) [1–8]. This approach assumes that the reference waves can be described by conventional dynamical theory and the influence of structural imperfections can be considered as a perturbation. Sinha et al. [1] were the first to apply this method to calculate X-ray scattering from surfaces. It turned out to be a very powerful tool to analyze diffuse-scattering behaviour qualitatively, although for quantitative calculations there are problems, as was clearly shown by de Boer [4,6,8], who paid special attention to the lateral size of roughness defects. The DWBA validity condition is more stringent when sizes are larger. In previous works, the DWBA was used in the conventional form, which is only suitable for a real scattering potential. However, the true scattering potential is always optical (complex) for X-rays. In some cases, disregard of X-ray energy dissipation can cause considerable computational errors. Even if the cross-section of dissipative processes (photoabsorption, inelastic scattering and so on) is much smaller than that of coherent diffraction,
245
In particular, in terms of the conventional DWBA, the diffuse-scattering cross-section is always T-invariant, which means that the crosssections of X-ray photon scattering from a state with wavevector k0 to that with k1 and from a state with k1 to that with k0 are equal:
does not influence the signal detected [9]. Generally, this derives from the symmetry of the Maxwell equations relative to time reversal (Tinvariance) and is true for closed conservative systems. For example, Eq. (2) is always valid for the Born approximation in which the energy of the incident wave is conserved owing to the optical theorem violation. Nevertheless, in some experiments this symmetry is not observed [10–12]. One can obtain correct DWBA expressions for the case of an optical scattering potential via direct application of perturbation theory to the wave equation [13]. However, the Green’s function formalism must be applied, which complicates analysis of the cross-sections obtained. Another way is to use the method of restricted path integrals, which is very suitable for the description of dissipative systems [14]. Its important advantage is the possibility of developing the conventional DWBA formalism in a physically clear manner. We demonstrate this in the present paper by the example of X-ray diffuse scattering from rough multilayers. The imaginary component of the scattering potential appears as a result of the mapping of multiple scattering channels, which contains various dissipative processes, to the usual singlechannel scattering theory. Each of these dissipative processes has its own specific character. We will consider only the case of photoabsorption assuming that polarization and other anisotropic effects are absent. This simplification is not a fundamental one and the formalism obtained can be easily extended to other cases. In Section 2 the conventional DWBA formalism is discussed. Section 3 is devoted to using the method of restricted path integrals to improve the conventional form of the DWBA for the case of an optical scattering potential. In this section the principal formulas are obtained. In Section 4 the results obtained are analyzed from the standpoint of T-invariance violations.
ds ds ðk0 ! k1 Þ ¼ ðk1 ! k0 Þ: dO dO
2. Conventional DWBA formalism
scoherent bsdissipative ; the conventional form of the DWBA cannot be applied correctly because its validity condition is strictly scoherent bsdiffuse bsdissipative ; where sdiffuse is the diffuse-scattering cross-section. Thus, the cross-section of the dissipative process must be much smaller than the calculated diffusescattering one, which is rarely achieved. As a rule, scoherent bsdiffuse sdissipative :
(1)
(2)
In X-ray diffraction and optics, Eq. (2) follows from the reciprocity principle first formulated by Lorentz: interchanging the source and detector
Calculation of X-ray diffraction and scattering from any object requires solution of the
ARTICLE IN PRESS S.V. Mytnichenko / Physica B 355 (2005) 244–249
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wave equation 2
ðD þ k ÞEðrÞ ¼ V ðrÞEðrÞ:
(3)
where D is the Laplacian, k ¼ 2p=l is the wavevector of the X-ray photons, E(r) is the solution, V ðrÞ ¼ 4pr0 rðrÞ is the scattering potential, r0 is the classical electron radius and r(r) is the electron density at the point r. Assuming small incident and scattered angles, the polarization factor is omitted in Eq. (3). In the general case this factor can be easily introduced. The DWBA can be applied to solve wave equation (3) if the total scattering potential can be split into two nonequivalent parts: V ðrÞ ¼ V 0 ðzÞ þ DV ðrÞ: The reference solutions E 0 ðrÞ of wave equation (3) for the laterally symmetrical part of the potential V 0 ðzÞ ¼ hV ðrÞix;y are assumed to be known (the zaxis is perpendicular to the lateral plane, Fig. 1), while the exact solutions EðrÞ can be obtained via perturbation theory. Let V 0 ðrÞ be real and, consequently, the Hamiltonian H 0 ¼ D þ V 0 ðzÞ
is the state in which the incident photon had the wavevector k0 ‘‘before scattering’’ (Fig. 1a ), while jE; k1 i is the state in which the photon has the wavevector k1 ‘‘after scattering’’ (Fig. 1b ). Interpretation of jE; k0 þi is clear. Formally, the state jE; k1 i can be built via time inversion of the state in which the photon had the wavevector k1 ‘‘before scattering’’. This procedure is equivalent to complex conjugation. It can be performed correctly only if there are no dissipative processes. Thus, jE; k1 i ¼ jE; k1 þ i which allows us to write the scattering amplitude as Z 1 E k1 ðrÞDV ðrÞE k0 ðrÞ dr: (6) Df ðk0 ! k1 Þ ¼ 4p Note that scattering amplitude (6) is T-invariant. Indeed, matrix element (6) can be considered both as the scattering amplitude Df ðk0 ! k1 Þ and as the scattering amplitude Df ðk1 ! k0 Þ; omitting the unimportant geometric factor that follows from the different limits in integral (6).
(4)
is a Hermitian one. In this case, the solutions E 0 ðrÞ represent a full set of orthogonal eigen-states and the amplitude of the non-specular diffuse scattering can be obtained as a matrix element [15] 1 hE; k1 jDV ðrÞjE; k0 þi; Df ðk0 ! k1 Þ ¼ (5) 4pB where B is a coefficient depending on the normalization of the eigenstates (B ¼ 1 if the incident wave is represented as expðik0 rÞ), jE; k0 þi ¼ E k0 ðrÞ
z k0
k1 1
0 x (a)
(b)
Fig. 1. States jE; k0 þi (a) and jE; k1 i (b).
3. Method of restricted path integrals Now let the potential V 0 ðzÞ be complex and, consequently, Hamiltonian (4) be nonHermitian. This has several important results. Firstly, the time-inverted state E k1 ðrÞ is no longer a solution of wave equation (3). Secondly, the scattering amplitude cannot be obtained as a simple matrix element from two eigenstates of H0. Thirdly, the flux density of the wave equation solutions is not invariant to changes in the spatial coordinates. These consequences are interconnected and Eq. (6) cannot be used directly to calculate the scattering amplitude. Nevertheless, though the state E nk1 ðrÞ is no longer a solution, it is not difficult to build the wave equation solution jE; k1 i in which the X-ray photon has the wavevector k1 ‘‘after scattering’’. It is sufficient to time-invert the solution of wave equation (3) for the complex conjugate potential V ðrÞ ¼ V 0 ðzÞ in which the photon had the
ARTICLE IN PRESS S.V. Mytnichenko / Physica B 355 (2005) 244–249
wavevector k1 ‘‘before scattering’’, jE; k1 i ¼ E nk1 ðV n0 ; rÞ:
(7)
Substitution of function (7) in the wave equation shows that it is a solution. The state jE; k1 i is normalized so that the ‘‘scattered’’ wave is given by expðik1 rÞ: Note that required renormalization of jE; k1 i cannot be achieved through simple multiplication by a constant. Nevertheless, we can expect the possibility to express the scattering amplitude as Df ðk0 ! k1 Þ Z 1 CðzÞE k1 ðV 0 ; rÞDV ðrÞE k0 ðV 0 ; rÞ dr; ¼ 4p
path integral method: a path that does not lead to a known final state must be omitted from calculations [14]. Even if the above correction is made, the integrand in Eq. (5) gives the probability amplitude for the X-ray photon to perform the transition jE; k0 þi ! jE; k1 i at point r rather than the scattering amplitude from this point. That is because the photon has a possibility of being absorbed in future. The integrand in Eq. (5) must be multiplied once again by aðzÞ to allow for this. Thus, the renormalization coefficient in Eq. (8) CðzÞ ¼ a2 ðzÞ
ð8Þ
where CðzÞ is the renormalization coefficient, which depends only on the coordinate z, owing to the translational symmetry of V 0 : Hence, the task is to validate Eq. (8) as well as to find an expression for the renormalization coefficient CðzÞ: The Feynman path-integral approach [16] allows a physically clear interpretation of expression (5). The integral in Eq. (5) can be considered as multiplication of the probability amplitudes for three events: the probability amplitude for the Xray photon in the state with momentum k0 ‘‘before scattering’’ at the point r, that for the X-ray photon in the state with momentum k1 ‘‘after scattering’’, and the probability amplitude for the transition jE; k0 þi ! jE; k1 i at the point r, which can be calculated in terms of the usual Born approximation. Summation over all the points r gives the total scattering amplitude. Similar arguments may be applied to the scattering amplitude (8). However, the value E k1 ðV 0 ; rÞ is not the same as the probability amplitude for an X-ray photon that has actually reached the state jk1 i of the free Hamiltonian. The reason is that, making a transition from the state jE; k0 þi to the state jE; k1 i at the point r, the Xray photon has a probability amplitude for being absorbed. To allow for this fact, it is necessary to multiply the integrand in Eq. (5) by a value aðzÞ; which is the probability amplitude for the X-ray photon to reach its final state jk1 i after the transition jE; k0 þi ! jE; k1 i at point. This procedure represents the core of the restricted
247
(9)
has a clear physical interpretation as the probability for the X-ray photon in the state jE; k1 i at the point r with the coordinate z to reach its final state jk1 i: CðzÞ is defined in Eq. (9) to within a phase factor. For non-specular scattering, this uncertainty is not important, but it is for the specular case, where there is interference between the incident and diffuse waves. Nevertheless, we think it is incorrect to apply DWBA in the specular case because this approach violates the optical theorem. Another problem is how to calculate CðzÞ . An exact way is to locate the X-ray source at a point inside (or outside if DV a0 at z40) the multilayer with the appropriate relationship between the transmitted and reflected waves, and then to solve the wave equation with corresponding boundary conditions. There is a simpler way if condition (1) is met, namely to use E k1 ðV 0 ; rÞ 2 : CðzÞ ¼ E k1 ðReðV 0 Þ; rÞ If the incident angle is not equal to the Bragg angle and diffraction can be calculated using the Born approximation, then m z ; (10) CðzÞ ¼ exp sin y1 where m40 is the linear coefficient of photoabsorption and y1 is the angle made by the outgoing wave (Fig. 1). Note that, in general, the renormalization coefficient can differ from unity at z ¼ 0: For example, diffuse-scattering sources can be located both inside the multilayer and on its
ARTICLE IN PRESS S.V. Mytnichenko / Physica B 355 (2005) 244–249
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surface (DV a0 at z40). An X-ray photon scattered by a defect at z40 can be absorbed later inside the multilayer. Another effect is that, under dynamical diffraction conditions, the renormalization coefficient CðzÞ can decrease more rapidly than (10). Standing-waves can cause the opposite effect, as then C ffi 1 at any z. Thus, Eq. (8) with a corresponding renormalization coefficient represents the final correct expression for the non-specular diffuse-scattering amplitude in the DWBA with optical scattering potential. The reference solutions in Eq. (8) can have various forms (including numerical ones) depending on the problem posed. For example, in the case of diffuse scattering from non-ideal surfaces, the solutions can be in the form of Fresnel coefficients. In the case of diffuse scattering from non-ideal multilayers, Parrat’s recursive method can be used to calculate the reference waves and so on.
4. T-invariance violations It is clear that T-invariance is violated generally in Eq. (8) for the diffuse-scattering amplitude because the renormalization coefficient does not depend on the initial state of the X-ray photon. Nevertheless, in some cases, Eq. (2) can still be valid due to additional symmetry. Let us consider two important special cases allowing clear physical interpretation. The first is when the dynamical nature of diffraction from the multilayer can be neglected, i.e. when the incident and outgoing angles are not equal to a Bragg angle. Assuming that the refractive index (including its imaginary component) is constant inside the multilayer, one can obtain the following expressions: m jE; k0 þi ¼ exp ik0x x þ ik0y y þ ik0z z þ z ; 2 sin y0 jE; k1 i ¼ exp ik1x x þ ik1y y þ ik1z z
m CðzÞ ¼ exp z sin y1
m z ; 2 sin y1
and Z
m DV ðrÞ exp iqr þ 2
ðsin1 y0 þ sin1 y1 Þz dr;
1 Df ðk0 ! k1 Þ ¼ 4p
where y0 is the incident angle (Fig. 1) and q ¼ k1 k0 is the momentum transfer. It is easy to see that the obtained scattering amplitude satisfies (2). Consider scattering from a defect located in the multilayer at a depth h . The scattering amplitude from this defect is proportional to the amplitude of the incident wave, which is attenuated by photoabsorption over the path length h sin1 y0 : The amplitude of the scattered wave is attenuated over the path length h sin1 y1 : The total path length hðsin1 y0 þ sin1 y1 Þ is the same both for the scattering k0 ! k1 and k0 ! k1 : Thus, condition (2) is still met because the photoabsorption cross-section does not depend on the wave propagation direction. Now, let us consider a case where the dynamical nature of diffraction is critical, and assume that the imaginary component of V0 is constant inside the multilayer. For example, let the incident angle be equal to the Bragg one and let the outgoing angle differ from it. According to condition (2), in the absence of photoabsorption, the scattering cross-sections for the cases of k0 ! k1 and k0 ! k1 must be equal. Nevertheless, these cases differ since the ‘‘mean path’’ of the X-ray photon through the multilayer medium for the case of k0 ! k1 must be significantly smaller for the following reasons. The first is rapid extinction of the incident wave under the dynamical diffraction condition at the Bragg angle of incidence. Secondly, making a transition inside the multilayer to a state propagating at a Bragg angle, the X-ray photon falls into a trap and must suffer many reflections before it reaches its final state. Thus, the probability for the photon to be absorbed in the case k0 ! k1 is much smaller, which violates condition (2). As mentioned above, if alternating layers of the multilayer have different imaginary components of the refractive index, the opposite effect may be observed, due to standing-wave effects.
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5. Conclusion
References
In conclusion, expression (8) with corresponding renormalization coefficient allows one to use the DWBA correctly to calculate non-specular X-ray diffuse scattering from imperfect crystals, superlattices, smooth surfaces, multilayers and other similar objects. Generally, T-invariance of diffusescattering cross-sections is violated due to photoabsorption. The more critical the dynamical nature of the diffraction, the greater are the violations.
[1] S.K. Sinha, E.B. Sirota, S. Garoff, H.B. Stanley, Phys. Rev. B 38 (1988) 2297. [2] R. Pynn, Phys. Rev. B 45 (1992) 602. [3] V. Holy´, J. Kubeˇna, I. Ohlı´ dal, K. Lischka, W. Plotz, Phys. Rev. B 47 (1993) 15896. [4] D.K.G. de Boer, Phys. Rev. B 49 (1994) 5817. [5] V. Holy´, T. Baumbach, Phys. Rev. B 49 (1994) 10668. [6] D.K.G. de Boer, Phys. Rev B 51 (1995) 5297. [7] M. Kopecky, J. Appl. Phys. 77 (1995) 2380. [8] D.K.G. de Boer, A.J.G. Leenaers, Physica B 221 (1996) 18. [9] R.W. James, The Optical Principles of the Diffraction of X-rays, Bell, London, 1950. [10] V.A. Chernov, E.D. Chkhalo, N.V. Kovalenko, S.V. Mytnichenko, Nucl. Instrum. Methods Phys. Res. A 448 (2000) 276. [11] V.A. Chernov, V.I. Kondratiev, N.V. Kovalenko, S.V. Mytnichenko, K.V. Zolotarev, J. Appl. Phys. 92 (2002) 7593. [12] V.A. Chernov, N.V. Kovalenko, S.V. Mytnichenko, A.I. Toropov, Acta Cryst. A 59 (2003) 551. [13] I.B. Khriplovich, B.P. Toperverg, private communications. [14] M.B. Menskii, Physics-Uspekhi 46 (2003) 1163. [15] J.R. Taylor, Scattering theory. The Quantum theory on Nonrelativistic Collisions, Wiley, New York, 1972. [16] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.
Acknowledgements The author thanks A.N. Artyushin, V.A. Chernov, I.B. Khriplovich, G.N. Kulipanov, and W. Schwarzacher for useful discussion and support. This work was supported by Russian Foundation for Basic Research, Grant no. 03-0216259, and International Science and Technology Centre, Grant no. 1794.
Physica B 355 (2005) 244–249 www.elsevier.com/locate/physb
Distorted-wave Born approximation in the case of an optical scattering potential Sergey V. Mytnichenkoa,b, a Institute of Solid State Chemistry, SB of RAS, 630128 Novosibirsk, Russia Siberian SR Centre at Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
b
Received 29 July 2004; received in revised form 27 October 2004; accepted 31 October 2004
Abstract Application of the distorted-wave Born approximation in the conventional form developed for the case of a real scattering potential is shown to cause significant errors in calculating X-ray diffuse scattering from non-ideal crystals, superlattices, multilayers and other objects if energy dissipation (photoabsorption, inelastic scattering, and so on) is not negligible, or in other words, in the case of an optical (complex) scattering potential. We show how a correct expression for the X-ray diffuse-scattering cross-section can be obtained in this case. Generally, the diffuse-scattering cross-section from an optical potential is not T-invariant, i.e. the reciprocity principle is violated. Violations of T-invariance are more evident when the dynamical nature of the diffraction is more critical. r 2004 Elsevier B.V. All rights reserved. PACS: 78.70.C; 07.85.F; 41.50 Keywords: X-ray diffuse scattering; Multilayer; Superlattice; Coherence
1. Introduction As a rule, the relatively weak interaction of X-rays and matter allow one to use the Born (kinematical) approach for calculations of X-ray SR Center at BINP, Pr. Akademika Lavrent’eva 11, 630090
Novosibirsk, Russia. Tel.: +7 3832 336289; +7 3832 342163. E-mail address: [email protected] (S.V. Mytnichenko).
fax:
diffraction and scattering successfully. Nevertheless, there is a need to apply dynamical approaches for some cases of diffraction from objects such as crystals, superlattices, smooth surfaces, multilayers and so on. The dynamical theory of diffraction was developed for such cases many years ago. It can be applied to calculate specular scattering when such objects are assumed to have almost perfect structure. The distorted-wave Born approximation (DWBA) has been widely used recently to calculate non-specular diffuse
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.10.099
ARTICLE IN PRESS S.V. Mytnichenko / Physica B 355 (2005) 244–249
scattering, caused by structural distortions (roughness) [1–8]. This approach assumes that the reference waves can be described by conventional dynamical theory and the influence of structural imperfections can be considered as a perturbation. Sinha et al. [1] were the first to apply this method to calculate X-ray scattering from surfaces. It turned out to be a very powerful tool to analyze diffuse-scattering behaviour qualitatively, although for quantitative calculations there are problems, as was clearly shown by de Boer [4,6,8], who paid special attention to the lateral size of roughness defects. The DWBA validity condition is more stringent when sizes are larger. In previous works, the DWBA was used in the conventional form, which is only suitable for a real scattering potential. However, the true scattering potential is always optical (complex) for X-rays. In some cases, disregard of X-ray energy dissipation can cause considerable computational errors. Even if the cross-section of dissipative processes (photoabsorption, inelastic scattering and so on) is much smaller than that of coherent diffraction,
245
In particular, in terms of the conventional DWBA, the diffuse-scattering cross-section is always T-invariant, which means that the crosssections of X-ray photon scattering from a state with wavevector k0 to that with k1 and from a state with k1 to that with k0 are equal:
does not influence the signal detected [9]. Generally, this derives from the symmetry of the Maxwell equations relative to time reversal (Tinvariance) and is true for closed conservative systems. For example, Eq. (2) is always valid for the Born approximation in which the energy of the incident wave is conserved owing to the optical theorem violation. Nevertheless, in some experiments this symmetry is not observed [10–12]. One can obtain correct DWBA expressions for the case of an optical scattering potential via direct application of perturbation theory to the wave equation [13]. However, the Green’s function formalism must be applied, which complicates analysis of the cross-sections obtained. Another way is to use the method of restricted path integrals, which is very suitable for the description of dissipative systems [14]. Its important advantage is the possibility of developing the conventional DWBA formalism in a physically clear manner. We demonstrate this in the present paper by the example of X-ray diffuse scattering from rough multilayers. The imaginary component of the scattering potential appears as a result of the mapping of multiple scattering channels, which contains various dissipative processes, to the usual singlechannel scattering theory. Each of these dissipative processes has its own specific character. We will consider only the case of photoabsorption assuming that polarization and other anisotropic effects are absent. This simplification is not a fundamental one and the formalism obtained can be easily extended to other cases. In Section 2 the conventional DWBA formalism is discussed. Section 3 is devoted to using the method of restricted path integrals to improve the conventional form of the DWBA for the case of an optical scattering potential. In this section the principal formulas are obtained. In Section 4 the results obtained are analyzed from the standpoint of T-invariance violations.
ds ds ðk0 ! k1 Þ ¼ ðk1 ! k0 Þ: dO dO
2. Conventional DWBA formalism
scoherent bsdissipative ; the conventional form of the DWBA cannot be applied correctly because its validity condition is strictly scoherent bsdiffuse bsdissipative ; where sdiffuse is the diffuse-scattering cross-section. Thus, the cross-section of the dissipative process must be much smaller than the calculated diffusescattering one, which is rarely achieved. As a rule, scoherent bsdiffuse sdissipative :
(1)
(2)
In X-ray diffraction and optics, Eq. (2) follows from the reciprocity principle first formulated by Lorentz: interchanging the source and detector
Calculation of X-ray diffraction and scattering from any object requires solution of the
ARTICLE IN PRESS S.V. Mytnichenko / Physica B 355 (2005) 244–249
246
wave equation 2
ðD þ k ÞEðrÞ ¼ V ðrÞEðrÞ:
(3)
where D is the Laplacian, k ¼ 2p=l is the wavevector of the X-ray photons, E(r) is the solution, V ðrÞ ¼ 4pr0 rðrÞ is the scattering potential, r0 is the classical electron radius and r(r) is the electron density at the point r. Assuming small incident and scattered angles, the polarization factor is omitted in Eq. (3). In the general case this factor can be easily introduced. The DWBA can be applied to solve wave equation (3) if the total scattering potential can be split into two nonequivalent parts: V ðrÞ ¼ V 0 ðzÞ þ DV ðrÞ: The reference solutions E 0 ðrÞ of wave equation (3) for the laterally symmetrical part of the potential V 0 ðzÞ ¼ hV ðrÞix;y are assumed to be known (the zaxis is perpendicular to the lateral plane, Fig. 1), while the exact solutions EðrÞ can be obtained via perturbation theory. Let V 0 ðrÞ be real and, consequently, the Hamiltonian H 0 ¼ D þ V 0 ðzÞ
is the state in which the incident photon had the wavevector k0 ‘‘before scattering’’ (Fig. 1a ), while jE; k1 i is the state in which the photon has the wavevector k1 ‘‘after scattering’’ (Fig. 1b ). Interpretation of jE; k0 þi is clear. Formally, the state jE; k1 i can be built via time inversion of the state in which the photon had the wavevector k1 ‘‘before scattering’’. This procedure is equivalent to complex conjugation. It can be performed correctly only if there are no dissipative processes. Thus, jE; k1 i ¼ jE; k1 þ i which allows us to write the scattering amplitude as Z 1 E k1 ðrÞDV ðrÞE k0 ðrÞ dr: (6) Df ðk0 ! k1 Þ ¼ 4p Note that scattering amplitude (6) is T-invariant. Indeed, matrix element (6) can be considered both as the scattering amplitude Df ðk0 ! k1 Þ and as the scattering amplitude Df ðk1 ! k0 Þ; omitting the unimportant geometric factor that follows from the different limits in integral (6).
(4)
is a Hermitian one. In this case, the solutions E 0 ðrÞ represent a full set of orthogonal eigen-states and the amplitude of the non-specular diffuse scattering can be obtained as a matrix element [15] 1 hE; k1 jDV ðrÞjE; k0 þi; Df ðk0 ! k1 Þ ¼ (5) 4pB where B is a coefficient depending on the normalization of the eigenstates (B ¼ 1 if the incident wave is represented as expðik0 rÞ), jE; k0 þi ¼ E k0 ðrÞ
z k0
k1 1
0 x (a)
(b)
Fig. 1. States jE; k0 þi (a) and jE; k1 i (b).
3. Method of restricted path integrals Now let the potential V 0 ðzÞ be complex and, consequently, Hamiltonian (4) be nonHermitian. This has several important results. Firstly, the time-inverted state E k1 ðrÞ is no longer a solution of wave equation (3). Secondly, the scattering amplitude cannot be obtained as a simple matrix element from two eigenstates of H0. Thirdly, the flux density of the wave equation solutions is not invariant to changes in the spatial coordinates. These consequences are interconnected and Eq. (6) cannot be used directly to calculate the scattering amplitude. Nevertheless, though the state E nk1 ðrÞ is no longer a solution, it is not difficult to build the wave equation solution jE; k1 i in which the X-ray photon has the wavevector k1 ‘‘after scattering’’. It is sufficient to time-invert the solution of wave equation (3) for the complex conjugate potential V ðrÞ ¼ V 0 ðzÞ in which the photon had the
ARTICLE IN PRESS S.V. Mytnichenko / Physica B 355 (2005) 244–249
wavevector k1 ‘‘before scattering’’, jE; k1 i ¼ E nk1 ðV n0 ; rÞ:
(7)
Substitution of function (7) in the wave equation shows that it is a solution. The state jE; k1 i is normalized so that the ‘‘scattered’’ wave is given by expðik1 rÞ: Note that required renormalization of jE; k1 i cannot be achieved through simple multiplication by a constant. Nevertheless, we can expect the possibility to express the scattering amplitude as Df ðk0 ! k1 Þ Z 1 CðzÞE k1 ðV 0 ; rÞDV ðrÞE k0 ðV 0 ; rÞ dr; ¼ 4p
path integral method: a path that does not lead to a known final state must be omitted from calculations [14]. Even if the above correction is made, the integrand in Eq. (5) gives the probability amplitude for the X-ray photon to perform the transition jE; k0 þi ! jE; k1 i at point r rather than the scattering amplitude from this point. That is because the photon has a possibility of being absorbed in future. The integrand in Eq. (5) must be multiplied once again by aðzÞ to allow for this. Thus, the renormalization coefficient in Eq. (8) CðzÞ ¼ a2 ðzÞ
ð8Þ
where CðzÞ is the renormalization coefficient, which depends only on the coordinate z, owing to the translational symmetry of V 0 : Hence, the task is to validate Eq. (8) as well as to find an expression for the renormalization coefficient CðzÞ: The Feynman path-integral approach [16] allows a physically clear interpretation of expression (5). The integral in Eq. (5) can be considered as multiplication of the probability amplitudes for three events: the probability amplitude for the Xray photon in the state with momentum k0 ‘‘before scattering’’ at the point r, that for the X-ray photon in the state with momentum k1 ‘‘after scattering’’, and the probability amplitude for the transition jE; k0 þi ! jE; k1 i at the point r, which can be calculated in terms of the usual Born approximation. Summation over all the points r gives the total scattering amplitude. Similar arguments may be applied to the scattering amplitude (8). However, the value E k1 ðV 0 ; rÞ is not the same as the probability amplitude for an X-ray photon that has actually reached the state jk1 i of the free Hamiltonian. The reason is that, making a transition from the state jE; k0 þi to the state jE; k1 i at the point r, the Xray photon has a probability amplitude for being absorbed. To allow for this fact, it is necessary to multiply the integrand in Eq. (5) by a value aðzÞ; which is the probability amplitude for the X-ray photon to reach its final state jk1 i after the transition jE; k0 þi ! jE; k1 i at point. This procedure represents the core of the restricted
247
(9)
has a clear physical interpretation as the probability for the X-ray photon in the state jE; k1 i at the point r with the coordinate z to reach its final state jk1 i: CðzÞ is defined in Eq. (9) to within a phase factor. For non-specular scattering, this uncertainty is not important, but it is for the specular case, where there is interference between the incident and diffuse waves. Nevertheless, we think it is incorrect to apply DWBA in the specular case because this approach violates the optical theorem. Another problem is how to calculate CðzÞ . An exact way is to locate the X-ray source at a point inside (or outside if DV a0 at z40) the multilayer with the appropriate relationship between the transmitted and reflected waves, and then to solve the wave equation with corresponding boundary conditions. There is a simpler way if condition (1) is met, namely to use E k1 ðV 0 ; rÞ 2 : CðzÞ ¼ E k1 ðReðV 0 Þ; rÞ If the incident angle is not equal to the Bragg angle and diffraction can be calculated using the Born approximation, then m z ; (10) CðzÞ ¼ exp sin y1 where m40 is the linear coefficient of photoabsorption and y1 is the angle made by the outgoing wave (Fig. 1). Note that, in general, the renormalization coefficient can differ from unity at z ¼ 0: For example, diffuse-scattering sources can be located both inside the multilayer and on its
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surface (DV a0 at z40). An X-ray photon scattered by a defect at z40 can be absorbed later inside the multilayer. Another effect is that, under dynamical diffraction conditions, the renormalization coefficient CðzÞ can decrease more rapidly than (10). Standing-waves can cause the opposite effect, as then C ffi 1 at any z. Thus, Eq. (8) with a corresponding renormalization coefficient represents the final correct expression for the non-specular diffuse-scattering amplitude in the DWBA with optical scattering potential. The reference solutions in Eq. (8) can have various forms (including numerical ones) depending on the problem posed. For example, in the case of diffuse scattering from non-ideal surfaces, the solutions can be in the form of Fresnel coefficients. In the case of diffuse scattering from non-ideal multilayers, Parrat’s recursive method can be used to calculate the reference waves and so on.
4. T-invariance violations It is clear that T-invariance is violated generally in Eq. (8) for the diffuse-scattering amplitude because the renormalization coefficient does not depend on the initial state of the X-ray photon. Nevertheless, in some cases, Eq. (2) can still be valid due to additional symmetry. Let us consider two important special cases allowing clear physical interpretation. The first is when the dynamical nature of diffraction from the multilayer can be neglected, i.e. when the incident and outgoing angles are not equal to a Bragg angle. Assuming that the refractive index (including its imaginary component) is constant inside the multilayer, one can obtain the following expressions: m jE; k0 þi ¼ exp ik0x x þ ik0y y þ ik0z z þ z ; 2 sin y0 jE; k1 i ¼ exp ik1x x þ ik1y y þ ik1z z
m CðzÞ ¼ exp z sin y1
m z ; 2 sin y1
and Z
m DV ðrÞ exp iqr þ 2
ðsin1 y0 þ sin1 y1 Þz dr;
1 Df ðk0 ! k1 Þ ¼ 4p
where y0 is the incident angle (Fig. 1) and q ¼ k1 k0 is the momentum transfer. It is easy to see that the obtained scattering amplitude satisfies (2). Consider scattering from a defect located in the multilayer at a depth h . The scattering amplitude from this defect is proportional to the amplitude of the incident wave, which is attenuated by photoabsorption over the path length h sin1 y0 : The amplitude of the scattered wave is attenuated over the path length h sin1 y1 : The total path length hðsin1 y0 þ sin1 y1 Þ is the same both for the scattering k0 ! k1 and k0 ! k1 : Thus, condition (2) is still met because the photoabsorption cross-section does not depend on the wave propagation direction. Now, let us consider a case where the dynamical nature of diffraction is critical, and assume that the imaginary component of V0 is constant inside the multilayer. For example, let the incident angle be equal to the Bragg one and let the outgoing angle differ from it. According to condition (2), in the absence of photoabsorption, the scattering cross-sections for the cases of k0 ! k1 and k0 ! k1 must be equal. Nevertheless, these cases differ since the ‘‘mean path’’ of the X-ray photon through the multilayer medium for the case of k0 ! k1 must be significantly smaller for the following reasons. The first is rapid extinction of the incident wave under the dynamical diffraction condition at the Bragg angle of incidence. Secondly, making a transition inside the multilayer to a state propagating at a Bragg angle, the X-ray photon falls into a trap and must suffer many reflections before it reaches its final state. Thus, the probability for the photon to be absorbed in the case k0 ! k1 is much smaller, which violates condition (2). As mentioned above, if alternating layers of the multilayer have different imaginary components of the refractive index, the opposite effect may be observed, due to standing-wave effects.
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5. Conclusion
References
In conclusion, expression (8) with corresponding renormalization coefficient allows one to use the DWBA correctly to calculate non-specular X-ray diffuse scattering from imperfect crystals, superlattices, smooth surfaces, multilayers and other similar objects. Generally, T-invariance of diffusescattering cross-sections is violated due to photoabsorption. The more critical the dynamical nature of the diffraction, the greater are the violations.
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Acknowledgements The author thanks A.N. Artyushin, V.A. Chernov, I.B. Khriplovich, G.N. Kulipanov, and W. Schwarzacher for useful discussion and support. This work was supported by Russian Foundation for Basic Research, Grant no. 03-0216259, and International Science and Technology Centre, Grant no. 1794.