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Neutron reflection from faceted surfaces
PI ICA ELSEVIER
Physica B 198 (1994) 1-6
Neutron reflection from faceted surfaces Roger Pynn*, Shenda Baker Manuel Lujan Jr. Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract Diffuse scattering, often taken as evidence that a surface is rough on very small length scales, can also arise when neutrons (or X-rays) are reflected from a surface composed of many smooth facets. For a surface composed of facets misoriented with respect to the average surface, the diffuse scattering displays a fringe at a wave vector transfer perpendicular to the surface that is equal to twice the critical wave vector. The damping of the specular reflectivity does not follow the usual Nevot-Croce form for such a faceted surface. A surface morphology of this type can explain data we have obtained by reflecting neutrons from several different samples including one of polished silicon.
1. Introduction It is well known [1-3] that surface roughness causes diffuse scattering when neutrons are reflected at grazing incidence. In the conventional picture, the diffuse intensity is proportional to the Fourier transform of the height-height correlation function of the rough surface, multiplied by Fresnel transmission functions, I T(ktz)l 2 and I T(k2z)[ 2, for the incident and scattered neutrons. When kl~ or k2~ (the components of the incident and scattered wave vectors perpendicular to the surface) are equal to the critical wave vector, kc, there is a peak in the Fresnel transmission function that leads to observable fringes in the diffuse scattering, called Yoneda fringes, whose loci are defined by ka~ = 27t sin0~/2 = kc or k2z = 2~ sin02/2 = kc, where 2 is the neutron wavelength and 01 and 02 are the angles of incidence and reflection. For a thin film [2,3], the diffuse scattering can show more features than the Yoneda wings described above. In this case, correlations (con* Corresponding author.
formality) between the roughness of the film interfaces leads to fringes of diffuse scattering at constant values of q~ ( = kl~ + k2z). The variation of intensity along the constant-qz fringes depends on the nature of the surface roughness as well as the spatial extent of roughness correlations between interfaces.
2. Results from the distorted wave Born approximation We have successfully used the distorted wave Born approximation (DWBA) to calculate maps of diffuse scattering that mimic those obtained with actual rough surfaces [2, 3]. This theory uses the exact wave functions obtained for reflection from an infinite smooth surface (the average surface) as the basis set for a perturbation calculation of the scattering from a rough surface. Both the diffuse scattering and the decrease of specular scattering that results from roughness can be calculated within the DWBA. In the latter case, the reflectance obtained for a surface with Gaussian distributed
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0921-4526(93)E0155-A
2
R. Pynn, S. Baker/Physica B 198 (1994) 1 6
roughness [2] is the same as that originally found by Nevot and Croce [4], namely, Rr = Ri e - 2kl~k'lz~2,
(1)
where the subscripts r and i refer to a rough and smooth (ideal) surfaces, respectively, a is the standard deviation of the surface roughness and kt~= is the neutron wave vector transfer evaluated in the reflecting medium, i.e. F k L ] ~ = [ k ~ A ~ - [ k A ~.
(2)
Given the apparently complete description of diffuse scattering that emerges from the DWBA, it came as a surprise to us when we observed fringes of diffuse scattering that could not be described by naive application of the DWBA. For several samples, we have observed diffuse fringes at a value of q= equal to the critical value qc = 2k~. In some cases, both the Yoneda fringe (at constant kzz ) and the q¢-fringe are observed, while in others, only the latter is visible. Examples are shown in Fig. 1. We are not the only group to have observed this phenomenon. Huang et al. [5] reported a similar fringe in their studies of ultrathin iron films on MgO. It is quite clear from the structure of the DWBA calculation that any diffuse scattering from a single rough surface that can be described by this theory must include Yoneda fringes. The neutron wave functions above and below an ideal surface are given by ~9~+'(r) = exp[i(kaxx + klyy)] × [ e x p ( - ikl~z) + Ri(kl=)exp(ikl=z)] for z > 0 , O~l+~(r) = e x p [ i ( k ~ x + k~yy)] Ti(k~=)exp(- ikt~zZ) for z < 0,
(3)
where continuity of the wave function and its first derivative at the surface is guaranteed by the relationship between the transmissivity, T, and the reflectivity, R: T~= 1 +R~.
(4)
When these wave functions are used in a perturbation expansion, a factor of I T(kl=)[ 2 [T(kz~)l 2 appears unavoidably in the resulting scattering cross-section [1, 2], giving rise to Yoneda fringes at
constant k~ and constant k2z. Such fringes must be observed for any rough surface whose scattering is described by the traditional second-order DWBA. Since the magnitude of the diffuse scattering is proportional to the scattering length density of the reflecting medium and the amplitude of the surface roughness, the only way that the Yoneda fringe can be made to "disappear" is to eliminate surface roughness or make the material non-reflecting. Since the q~-fringe sometimes appears in the absence of the Yoneda fringe, it cannot be due to surface roughness in the traditional sense described by the DWBA.
3. Probing the assumptions of the DWBA: faceted surfaces We have tried several ideas for explaining the qc-fringe, including various hypothetical surface films. For example, one might imagine that the constancy of qz along the qc-fringe indicates the presence of some length scale perpendicular to the surface, just as it does for constant-qz fringes observed with films and multilayers [3]. However, if this were the case, the controlling parameter ought to be the value of qz in the reflecting medium (i.e. qt), which takes a complex value along the entire q~-fringe, except at qx = 0. We have been unable to imagine what the corresponding length scale might be. To explain the q¢-fringe we are led to examine the assumptions of the DWBA in more detail. The DWBA is based on the use of the wave functions for a smooth surface that corresponds to the average rough surface. However, let us suppose for a moment that there is not a single average surface but rather a series of large facets, as shown in Fig. 2, each of which reflects radiation perfectly - a model one might refer to as the "sparkling sea" model for roughness. Within the context of this model it is natural to ask how large a facet must be before it can be considered to reflect perfectly? Equivalently, we may ask how large the facet must be for the neutron wave function in its neighbourhood to be best represented by the Fresnel wave function obtained under the assumption that the facet is smooth and
R. Pynn, S. Baker/Phys~a B 198 (1994) 1-6
3
Fig. 1. Grey scale maps of specular and diffuse scattering as a function of 02 (ordinate) and 2 (abscissa). Top: a polished silicon wafer; bottom: a planar grain boundary between natural Ni and a null-matrix 6°Ni-enriched material. The Yoneda fringe points towards the origin while the qc fringe extrapolates to 02 = - 01 for 2 = 0. The horizontal band of intensity in each figure is specular scattering.
of infinite extent. E v e n w i t h o u t f i n d i n g the exact w a v e f u n c t i o n s for a p a r t i c u l a r facet g e o m e t r y , o n e c a n assert t h a t the facet d i m e n s i o n m u s t be m u c h larger t h a n the n e u t r o n w a v e l e n g t h if it is to b e h a v e
as a n infinite facet. This, of course, is exactly the limit s t u d i e d so extensively for r a d i o waves and documented by Beckmann and Spizzichino [6].
4
R . t ) v n n , S. B a k e r / P h y s i c a
.,..~yX'
Z
~
X
z/~
Fig. 2. Surfaces composed of facets or mesas. Neutrons incident on a surface facet that makes an angle ~ with the average surface at z = 0, have an angle of incidence 01 and an apparent angle of reflection of 01 + 2~. The Cartesian coordinate systems used in the text are defined by this figure.
4. Fresnel zones and coherence lengths To evaluate the scattering from a reflecting surface, rough or smooth, one adds together the amplitudes of radiation scattered from each surface element and squares the result to obtain the total scattering cross-section. However, it is well known that this coherent addition of individual Huygens wavelets does not extend indefinitely. In particular, the fact that reflectometry experiments use a Fresnel geometry, rather than the Fraunhoffer geometry used for traditional neutron diffraction experiments, leads to the notion of Fresnel zones. As Beckmann and Spizzichino I-6] have shown, most of the reflected energy from a surface arises from scattering within the first Fresnel zone; scattering from higher-order zones tends to interfere destructively. Thus, one may cut off the coherent addition of scattering amplitudes outside the first zone. For the case of a faceted surface, the intensities (i.e. the squared amplitudes) reflected by each facet must be added if the facets are uncorrelated and larger than the first Fresnel zone. For smaller facets, one must add coherently the scattering amplitudes from facets within the first zone before squaring to obtain the scattered intensity.
B 198 ( 1 9 9 4 ) 1 - 6
The longitudinal, (x) size the first Fresnel zone [6] is equal to ()~L)l/2/sin O, where L is the distance between the neutron source and the detection point. For most neutron reflectometers L is of order 10 m, so the longitudinal dimension of the first zone is in excess of 1 mm at a grazing angle of 1° for 1 ~, neutrons. In the direction perpendicular to the scattering plane (the y direction), the first zone is smaller by a factor of 1/sin 0, but is evidently still quite large compared, for example, to the neutron wavelength. Another length that may cut off the coherent addition of scattered waves is the coherence length of the neutrons themselves. For some reason, a number of authors appear to believe that this length is equivalent to the instrumental resolution. This is quite wrong, for the resolution simply accounts for the incoherent addition of many neutrons that are transmitted through the reflectometer. Just as with any other spectrometer, one accounts for the resolution by calculating the scattering cross-section, i.e. the square of the scattered amplitudes, for each neutron path, and weights this sum according to the probability of transmission through the spectrometer.
5. Scattering from a faceted surface To simplify the calculation presented here, we will assume that surface facets exist, that they are large enough to reflect according to Fresnel's law, and that they are uncorrelated and larger than the first Fresnel zone. These assumptions allow the intensities scattered by each facet to be added. The intensity scattered by a single facet is proportional to the square of its transfer matrix: k2
(exp(ik2 r)l V(r)[ ~t +~)
(5)
where ff is the wave function given in Eq. (3) and V(r) is the scattering potential of the reflecting medium. For a facet whose x dimension is ~, oriented at an angle ~ with respect to the average surface (cf. Fig. 2), the transfer matrix is easily evaluated as h2
t~l ~ k2 =
i ~nm 6(qy)k'lz ~Ri(k lz) e-iq'xX × sinc (q~,¢/2) e-iq~ z'.
(6)
R. Pynn, S. Baker/Physica B 198 (1994) 1 6
5
0.03"-
10 -2
0.02
.
0.01 -3 10
T
o.ol
o.o2
o.o3
o.o4
o.os
o.O6
0.0
o2
~
o.~
0.8
Thete2 (de(lree=)
Oz(~ "1)
Fig. 3. Cuts t h r o u g h the data of Fig. l(a) scaled by the neutron spectrum, S(2). Left side: cuts for 0 2 = 0.6 ° (specular), 0.45 °, 0.39 ° and 0.23 ° as a function of q:; right side: cuts at q~ = 0.01 A - 1, 0.015 J , - 1, 0.023 A 1 0.031 J, i 0.038 ~,- 1 0.047 ~, - 1 and 0.055 ~,- 1 as a function of 02.
The maximum of the transfer matrix for a single facet occurs when q;, = k'lx- k'Ex= 0, i.e. when kl and k 2 both make angles of (01 + ~) with the facet. This, of course, is just the specular condition for the facet. For very large facets, this is the only condition under which reflection occurs because the sinc function approaches a delta function. In this case, the "apparent" qz, obtained by measuring 01 and 02 with respect to the z = 0 plane, is given by r/appar •~z
2n =
),
- -
[sin(01) + sin(01 + 2ct)]
2~ ~ - 2 sin(01 + ~) = 2k~:.
(7)
Thus, if k~: is fixed at the value of kc, for example, scattering will be observed at q: = 2k~:, whatever the facet orientation. This is the origin of the q c fringe - it is simply an image of the critical edge of the various misoriented facets. From Eq. (6) one may obtain an expression for the reflectance of a surface with extremely large facets:
Rf(O2)=Ri(2~sin[O120~2])sin[(Ox [-01]
where q~ = k[~ + k2~
(9)
and P(a) is the normalised density of facets of orientation a. Equation (8) has several interesting features. First, the damping of the specular reflectivity (at 01 = 02) is not the same as the Nevot-Croce result given by Eq. (1). The specular damping obtained from a "mesa-top" faceted surface (cf. the lower part of Fig. 2) is of the same form as given in Eq. (8). Secondly, Eq. (8) implies that scattering at a constant value of q~, when plotted as a function of 02, should measure the angular distribution of facets. This is demonstrated in Fig. 3(b) for the data for silicon shown in Fig. 1 (a). Finally, scattering at constant values of 02 should resemble the reflectivity when plotted as a function of qz. This is shown in Fig. 3(a), again for the data of Fig. 1 (a). Note that a mesa-top surface could not explain the results shown in these figures because the value of ~ is the same for each facet in this case. One should be careful not to over-interpret the cartoons in Fig. 2; a stepped surface would give the same result as we have presented here, provided the overall substrate was bent so that the population of facets or steps had a spread of orientations. One should also note that our analysis, like most others,
6
R. Pynn, S. Baker/Physica B 198 (1994) 1 6
does not include the effect of facets shadowing one another. The weakest points of our calculation are the assumptions that each facet reflects according to the Fresnel law and that facet orientations are not correlated. Clearly, the first of these assumptions will break down as facets become smaller and one would expect a crossover to the behaviour described by the second-order DWBA when the facet size becomes comparable with the neutron wavelength. Because we have not considered the effect of facet size or correlation in detail, we can only conclude that the present calculation provides a possible explanation of the qc fringe rather than a unique explanation.
Acknowledgements We would like to thank Mike Fitzsimmons for bringing the qc-fringe to our attention and Greg
Smith for suggesting Fig. 3(a); thereby providing the clue that led to the explanation of the qc-fringe presented here. This work was performed under the auspices of the Office of Basic Energy Sciences of the US Department of Energy under contract W-7405-ENG-36.
References [1] S.K. Sinha, E.B. Sirota, S. Garoff and H.B. Stanley, Phys. Rev. B 38 (1988) 2297. [2] R. Pynn, Phys. Rev. 45 (1992) 602. [-3] R. Pynn, in: Proc. SPIE, Neutron Optical Devices and Applications, eds. C.F. Majkrzak and J.L. Wood, Vol. 1738 (1992). 1-4] L. Nevot and P. Croce, Rev. Phys. Appl. 15 (1980) 761 [5] Y.Y. Huang, C. Liu and C.P. Felcher, Phys. Rev. B 47 (1993) 183. 1-6] P. Beckmann and A. Spizzichino, Th e Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, MA, 1987).
Physica B 198 (1994) 1-6
Neutron reflection from faceted surfaces Roger Pynn*, Shenda Baker Manuel Lujan Jr. Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract Diffuse scattering, often taken as evidence that a surface is rough on very small length scales, can also arise when neutrons (or X-rays) are reflected from a surface composed of many smooth facets. For a surface composed of facets misoriented with respect to the average surface, the diffuse scattering displays a fringe at a wave vector transfer perpendicular to the surface that is equal to twice the critical wave vector. The damping of the specular reflectivity does not follow the usual Nevot-Croce form for such a faceted surface. A surface morphology of this type can explain data we have obtained by reflecting neutrons from several different samples including one of polished silicon.
1. Introduction It is well known [1-3] that surface roughness causes diffuse scattering when neutrons are reflected at grazing incidence. In the conventional picture, the diffuse intensity is proportional to the Fourier transform of the height-height correlation function of the rough surface, multiplied by Fresnel transmission functions, I T(ktz)l 2 and I T(k2z)[ 2, for the incident and scattered neutrons. When kl~ or k2~ (the components of the incident and scattered wave vectors perpendicular to the surface) are equal to the critical wave vector, kc, there is a peak in the Fresnel transmission function that leads to observable fringes in the diffuse scattering, called Yoneda fringes, whose loci are defined by ka~ = 27t sin0~/2 = kc or k2z = 2~ sin02/2 = kc, where 2 is the neutron wavelength and 01 and 02 are the angles of incidence and reflection. For a thin film [2,3], the diffuse scattering can show more features than the Yoneda wings described above. In this case, correlations (con* Corresponding author.
formality) between the roughness of the film interfaces leads to fringes of diffuse scattering at constant values of q~ ( = kl~ + k2z). The variation of intensity along the constant-qz fringes depends on the nature of the surface roughness as well as the spatial extent of roughness correlations between interfaces.
2. Results from the distorted wave Born approximation We have successfully used the distorted wave Born approximation (DWBA) to calculate maps of diffuse scattering that mimic those obtained with actual rough surfaces [2, 3]. This theory uses the exact wave functions obtained for reflection from an infinite smooth surface (the average surface) as the basis set for a perturbation calculation of the scattering from a rough surface. Both the diffuse scattering and the decrease of specular scattering that results from roughness can be calculated within the DWBA. In the latter case, the reflectance obtained for a surface with Gaussian distributed
0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0921-4526(93)E0155-A
2
R. Pynn, S. Baker/Physica B 198 (1994) 1 6
roughness [2] is the same as that originally found by Nevot and Croce [4], namely, Rr = Ri e - 2kl~k'lz~2,
(1)
where the subscripts r and i refer to a rough and smooth (ideal) surfaces, respectively, a is the standard deviation of the surface roughness and kt~= is the neutron wave vector transfer evaluated in the reflecting medium, i.e. F k L ] ~ = [ k ~ A ~ - [ k A ~.
(2)
Given the apparently complete description of diffuse scattering that emerges from the DWBA, it came as a surprise to us when we observed fringes of diffuse scattering that could not be described by naive application of the DWBA. For several samples, we have observed diffuse fringes at a value of q= equal to the critical value qc = 2k~. In some cases, both the Yoneda fringe (at constant kzz ) and the q¢-fringe are observed, while in others, only the latter is visible. Examples are shown in Fig. 1. We are not the only group to have observed this phenomenon. Huang et al. [5] reported a similar fringe in their studies of ultrathin iron films on MgO. It is quite clear from the structure of the DWBA calculation that any diffuse scattering from a single rough surface that can be described by this theory must include Yoneda fringes. The neutron wave functions above and below an ideal surface are given by ~9~+'(r) = exp[i(kaxx + klyy)] × [ e x p ( - ikl~z) + Ri(kl=)exp(ikl=z)] for z > 0 , O~l+~(r) = e x p [ i ( k ~ x + k~yy)] Ti(k~=)exp(- ikt~zZ) for z < 0,
(3)
where continuity of the wave function and its first derivative at the surface is guaranteed by the relationship between the transmissivity, T, and the reflectivity, R: T~= 1 +R~.
(4)
When these wave functions are used in a perturbation expansion, a factor of I T(kl=)[ 2 [T(kz~)l 2 appears unavoidably in the resulting scattering cross-section [1, 2], giving rise to Yoneda fringes at
constant k~ and constant k2z. Such fringes must be observed for any rough surface whose scattering is described by the traditional second-order DWBA. Since the magnitude of the diffuse scattering is proportional to the scattering length density of the reflecting medium and the amplitude of the surface roughness, the only way that the Yoneda fringe can be made to "disappear" is to eliminate surface roughness or make the material non-reflecting. Since the q~-fringe sometimes appears in the absence of the Yoneda fringe, it cannot be due to surface roughness in the traditional sense described by the DWBA.
3. Probing the assumptions of the DWBA: faceted surfaces We have tried several ideas for explaining the qc-fringe, including various hypothetical surface films. For example, one might imagine that the constancy of qz along the qc-fringe indicates the presence of some length scale perpendicular to the surface, just as it does for constant-qz fringes observed with films and multilayers [3]. However, if this were the case, the controlling parameter ought to be the value of qz in the reflecting medium (i.e. qt), which takes a complex value along the entire q~-fringe, except at qx = 0. We have been unable to imagine what the corresponding length scale might be. To explain the q¢-fringe we are led to examine the assumptions of the DWBA in more detail. The DWBA is based on the use of the wave functions for a smooth surface that corresponds to the average rough surface. However, let us suppose for a moment that there is not a single average surface but rather a series of large facets, as shown in Fig. 2, each of which reflects radiation perfectly - a model one might refer to as the "sparkling sea" model for roughness. Within the context of this model it is natural to ask how large a facet must be before it can be considered to reflect perfectly? Equivalently, we may ask how large the facet must be for the neutron wave function in its neighbourhood to be best represented by the Fresnel wave function obtained under the assumption that the facet is smooth and
R. Pynn, S. Baker/Phys~a B 198 (1994) 1-6
3
Fig. 1. Grey scale maps of specular and diffuse scattering as a function of 02 (ordinate) and 2 (abscissa). Top: a polished silicon wafer; bottom: a planar grain boundary between natural Ni and a null-matrix 6°Ni-enriched material. The Yoneda fringe points towards the origin while the qc fringe extrapolates to 02 = - 01 for 2 = 0. The horizontal band of intensity in each figure is specular scattering.
of infinite extent. E v e n w i t h o u t f i n d i n g the exact w a v e f u n c t i o n s for a p a r t i c u l a r facet g e o m e t r y , o n e c a n assert t h a t the facet d i m e n s i o n m u s t be m u c h larger t h a n the n e u t r o n w a v e l e n g t h if it is to b e h a v e
as a n infinite facet. This, of course, is exactly the limit s t u d i e d so extensively for r a d i o waves and documented by Beckmann and Spizzichino [6].
4
R . t ) v n n , S. B a k e r / P h y s i c a
.,..~yX'
Z
~
X
z/~
Fig. 2. Surfaces composed of facets or mesas. Neutrons incident on a surface facet that makes an angle ~ with the average surface at z = 0, have an angle of incidence 01 and an apparent angle of reflection of 01 + 2~. The Cartesian coordinate systems used in the text are defined by this figure.
4. Fresnel zones and coherence lengths To evaluate the scattering from a reflecting surface, rough or smooth, one adds together the amplitudes of radiation scattered from each surface element and squares the result to obtain the total scattering cross-section. However, it is well known that this coherent addition of individual Huygens wavelets does not extend indefinitely. In particular, the fact that reflectometry experiments use a Fresnel geometry, rather than the Fraunhoffer geometry used for traditional neutron diffraction experiments, leads to the notion of Fresnel zones. As Beckmann and Spizzichino I-6] have shown, most of the reflected energy from a surface arises from scattering within the first Fresnel zone; scattering from higher-order zones tends to interfere destructively. Thus, one may cut off the coherent addition of scattering amplitudes outside the first zone. For the case of a faceted surface, the intensities (i.e. the squared amplitudes) reflected by each facet must be added if the facets are uncorrelated and larger than the first Fresnel zone. For smaller facets, one must add coherently the scattering amplitudes from facets within the first zone before squaring to obtain the scattered intensity.
B 198 ( 1 9 9 4 ) 1 - 6
The longitudinal, (x) size the first Fresnel zone [6] is equal to ()~L)l/2/sin O, where L is the distance between the neutron source and the detection point. For most neutron reflectometers L is of order 10 m, so the longitudinal dimension of the first zone is in excess of 1 mm at a grazing angle of 1° for 1 ~, neutrons. In the direction perpendicular to the scattering plane (the y direction), the first zone is smaller by a factor of 1/sin 0, but is evidently still quite large compared, for example, to the neutron wavelength. Another length that may cut off the coherent addition of scattered waves is the coherence length of the neutrons themselves. For some reason, a number of authors appear to believe that this length is equivalent to the instrumental resolution. This is quite wrong, for the resolution simply accounts for the incoherent addition of many neutrons that are transmitted through the reflectometer. Just as with any other spectrometer, one accounts for the resolution by calculating the scattering cross-section, i.e. the square of the scattered amplitudes, for each neutron path, and weights this sum according to the probability of transmission through the spectrometer.
5. Scattering from a faceted surface To simplify the calculation presented here, we will assume that surface facets exist, that they are large enough to reflect according to Fresnel's law, and that they are uncorrelated and larger than the first Fresnel zone. These assumptions allow the intensities scattered by each facet to be added. The intensity scattered by a single facet is proportional to the square of its transfer matrix: k2
(exp(ik2 r)l V(r)[ ~t +~)
(5)
where ff is the wave function given in Eq. (3) and V(r) is the scattering potential of the reflecting medium. For a facet whose x dimension is ~, oriented at an angle ~ with respect to the average surface (cf. Fig. 2), the transfer matrix is easily evaluated as h2
t~l ~ k2 =
i ~nm 6(qy)k'lz ~Ri(k lz) e-iq'xX × sinc (q~,¢/2) e-iq~ z'.
(6)
R. Pynn, S. Baker/Physica B 198 (1994) 1 6
5
0.03"-
10 -2
0.02
.
0.01 -3 10
T
o.ol
o.o2
o.o3
o.o4
o.os
o.O6
0.0
o2
~
o.~
0.8
Thete2 (de(lree=)
Oz(~ "1)
Fig. 3. Cuts t h r o u g h the data of Fig. l(a) scaled by the neutron spectrum, S(2). Left side: cuts for 0 2 = 0.6 ° (specular), 0.45 °, 0.39 ° and 0.23 ° as a function of q:; right side: cuts at q~ = 0.01 A - 1, 0.015 J , - 1, 0.023 A 1 0.031 J, i 0.038 ~,- 1 0.047 ~, - 1 and 0.055 ~,- 1 as a function of 02.
The maximum of the transfer matrix for a single facet occurs when q;, = k'lx- k'Ex= 0, i.e. when kl and k 2 both make angles of (01 + ~) with the facet. This, of course, is just the specular condition for the facet. For very large facets, this is the only condition under which reflection occurs because the sinc function approaches a delta function. In this case, the "apparent" qz, obtained by measuring 01 and 02 with respect to the z = 0 plane, is given by r/appar •~z
2n =
),
- -
[sin(01) + sin(01 + 2ct)]
2~ ~ - 2 sin(01 + ~) = 2k~:.
(7)
Thus, if k~: is fixed at the value of kc, for example, scattering will be observed at q: = 2k~:, whatever the facet orientation. This is the origin of the q c fringe - it is simply an image of the critical edge of the various misoriented facets. From Eq. (6) one may obtain an expression for the reflectance of a surface with extremely large facets:
Rf(O2)=Ri(2~sin[O120~2])sin[(Ox [-01]
where q~ = k[~ + k2~
(9)
and P(a) is the normalised density of facets of orientation a. Equation (8) has several interesting features. First, the damping of the specular reflectivity (at 01 = 02) is not the same as the Nevot-Croce result given by Eq. (1). The specular damping obtained from a "mesa-top" faceted surface (cf. the lower part of Fig. 2) is of the same form as given in Eq. (8). Secondly, Eq. (8) implies that scattering at a constant value of q~, when plotted as a function of 02, should measure the angular distribution of facets. This is demonstrated in Fig. 3(b) for the data for silicon shown in Fig. 1 (a). Finally, scattering at constant values of 02 should resemble the reflectivity when plotted as a function of qz. This is shown in Fig. 3(a), again for the data of Fig. 1 (a). Note that a mesa-top surface could not explain the results shown in these figures because the value of ~ is the same for each facet in this case. One should be careful not to over-interpret the cartoons in Fig. 2; a stepped surface would give the same result as we have presented here, provided the overall substrate was bent so that the population of facets or steps had a spread of orientations. One should also note that our analysis, like most others,
6
R. Pynn, S. Baker/Physica B 198 (1994) 1 6
does not include the effect of facets shadowing one another. The weakest points of our calculation are the assumptions that each facet reflects according to the Fresnel law and that facet orientations are not correlated. Clearly, the first of these assumptions will break down as facets become smaller and one would expect a crossover to the behaviour described by the second-order DWBA when the facet size becomes comparable with the neutron wavelength. Because we have not considered the effect of facet size or correlation in detail, we can only conclude that the present calculation provides a possible explanation of the qc fringe rather than a unique explanation.
Acknowledgements We would like to thank Mike Fitzsimmons for bringing the qc-fringe to our attention and Greg
Smith for suggesting Fig. 3(a); thereby providing the clue that led to the explanation of the qc-fringe presented here. This work was performed under the auspices of the Office of Basic Energy Sciences of the US Department of Energy under contract W-7405-ENG-36.
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