Some Considerations on Resolution and Coherence Length in

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Some Considerations on Resolution and Coherence Length in Reflectrometry† Bruno Dorner* and Andrew R. Wildes Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, France Received December 4, 2002. In Final Form: March 19, 2003 Resolution and/or coherence length is a property of a diffraction instrument and is independent of the characteristics of the sample. The so-called resolution “ellipsoids” for reflectometers are shown in a graphical way, which should help in planning and optimizing the data collection for an experiment. Mathematical expressions for exact calculations of the ellipsoids are derived for three orthogonal directions, with the component of resolution perpendicular to the scattering plane given explicitly. All the resolution components are given in terms of the instrument parameters: wavevector; angles; and their variations. Particular attention is paid to the investigation by reflectometry of gratings, optimizing for specular and off-specular measurements, and, in the case of the grating, being rotated such that its stripes are not perpendicular to the incoming beam.

Introduction The techniques of neutron and X-ray reflectivity have been substantially developed to the point where they are standard techniques for the investigation of surfaces and interfaces. The majority of reflectometry measurements focus on the investigation of specular scattering, whereby the momentum transfer of the radiation is normal to the surface. There is increasing interest, however, in the offspecular reflectivity, where the momentum transfer is not parallel to the surface normal. The resolution of such measurements, particularly in the case of neutron reflectometers where resolution is often sacrificed for greater flux, is rarely explicitly considered and can often be better optimized. We aim to provide a simple discussion on the resolution function of a reflectometer, with a set of equations and a case study showing how an instrument can be optimally configured for an experiment. For the following considerations it is useful to recall the technique of inelastic neutron scattering.1 The measured intensity for this technique, I(Q0,ω0), is defined by the convolution of the resolution function, R, and the scattering function, S

I(Q0,ω0) )

∫ R(Q-Q0,ω-ω0)S(Q,ω) dQ dω

(1)

Here Q0 and ω0 are the mean values of momentum and energy transfer at a given configuration of the instrument with

Q0 ) kI,0 - kF,0

(2)

where kI,0 and kF,0 are the nominal incident and final wavevectors of the neutron. The resolution or transmission function, R, describes the instrument and is derived from the instrument parameters, while the scattering function, S, contains the physics of the sample. It is evident from eq 1 that the function R contains properties of the instrument and is independent of the characteristics of the sample. * To whom correspondence may be addressed. † Part of the Langmuir special issue dedicated to neutron reflectometry. (1) Cooper, M. J.; Nathans, R. Acta Crystallogr. 1967, 23, 357. Dorner, B. Acta Crystallogr. 1972, A28, 319.

Although eq 1 is valid for different spectroscopic techniques, care must be taken when defining the scattering function, S, which can be only described as a function of Q and ω in the Born approximation. For reflectivity one need not consider energy transfer. The inelastic scattering contribution has a broad distribution in Q space and, in any case, the scattering is integrated over ω for reflectivity measurements, appearing as a constant background. Several papers on resolution for reflectometry with X-rays have been published.2-6 Each is related to a particular type of research and to particular settings of the instrument. In this paper, resolution for a general reflectometer is derived. The expressions may easily be applied to an instrument configuration, as the resolution is defined in terms of the instrument parameters k, RI, RF, and β. For these calculations, the sample surface and scattering planes are taken to be always perpendicular to one another. The parameters RI and RF are the angles of incidence and of exit between the beams kI,0 and kF,0 and the sample surface. The parameter β is the angle out of the scattering plane within the plane of the surface. All considerations given below apply equally well for neutrons and for X-rays. In the case of elastic scattering, the measured intensity, I, can be expressed in terms of momentum transfer Q

I(Q0) )

∫ R(Q-Q0) S(Q) dQ

(3)

or in terms of the instrument parameters

I(k0,RI,0,RF,0,β0) )

∫ R(k-k0,RI-RI,0,RF-RF,0,β-β0) S(k,RI,RF,β) dk dRI dRF dβ (4)

where k0, RI,0, RF,0, and β0 are the mean wavevector and angles of the instrument. The reason for having two separate equations defining the measured intensity is due to the scattering function, S, describing the sample. At very small momentum (2) Cowley, R. A. Acta Crystallogr. 1987, A43, 825. (3) Shindler, J. D.; Suter, R. M. Rev. Sci. Instrum. 1992, 63, 5343. (4) Toney, M. F.; Wiesler, D. G. Acta Crystallogr. 1993, A49, 624. (5) Gibaud, A.; Vignaud, G.; Sinha, S. K. Acta Crystallogr. 1993, A49, 642. (6) Vlieg, E. J. Appl. Crystallogr. 1997, 30, 532.

10.1021/la026949b CCC: $25.00 © 2003 American Chemical Society Published on Web 06/03/2003

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transfers, where the limit of total reflection is approached, the Born approximation becomes invalid. Sophisticated derivations are then needed to obtain a correct scattering function, which cannot be given as a function of Q because the Born approximation is not valid.7 Equation 4 is then preferable for describing the measured intensity. The resolution function, R, describes the instrument and may be defined with equal validity in either expression. Resolution T Coherence Length We will relate resolution in reciprocal space to coherence length in real space by the Fourier transform of Gaussian functions. Gaussians are frequently used when considering resolution1 as they are well-behaved mathematically and the convolution of two leads to another Gaussian. Many optical elements are well described by these functions, for example, the triangular transmission function of a collimator can easily be approximated by a Gaussian. We are aware that Gaussians are not well adapted to describe certain optical elements. Nevertheless, for simplicity, we use Gaussian approximations throughout the paper. A Fourier transformation of a Gaussian in u to a Gaussian in w can be written as

e-R u S 2 2

π1/2 -w2/4R2 e R

(5)

If the half-width (HW) is defined as (21/2R)-1 on the lefthand (u) side and as 21/2R on the right-hand (w) side, then these HWs are taken at exp(-0.5) ) 0.61 of the maximum. If we define the coordinates x and y to be in the plane of the surface of the sample, with x in direction of the incoming beam, y perpendicular to it, and z normal to the surface, then we define resolution components ∆Qx, ∆Qy, and ∆Qz in reciprocal space by their HW as defined above. The coherence lengths in real space are defined by

lcoh,x ) 2π/∆Qx; lcoh,y ) 2π/∆Qy; lcoh,z ) 2π/∆Qz

(6)

Note that the values of ∆Qx, ∆Qy, and ∆Qz are the projections of the resolution ellipsoid, which will be defined later, onto the axis x, y, and z, which are defined relative to the incoming beam and to the sample surface. The instrument is considered in reciprocal space by the angular divergence of the beams. The size of incident and scattered beams in real space is not considered. In other words, it is assumed that the beam sizes and their projections on the sample surface are smaller than the sample and bigger than the coherence length. The Resolution Ellipsoid Before deriving resolution ellipsoids, which describe the resolution function of the instrument, we would like to recall that often one considers the coherence length of one individual beam. For one individual beam, the longitudinal coherence length, l l, is given by the k resolution ∆k, as provided by a monochromator

l l )2π/∆k while the transverse coherence length, l tr, is given by the angular divergence, ∆R, of the beam

l tr ) 2π/(k ∆R) This can be derived rigorously in a quantum mechanical way.7 (7) Toperverg, B. Private communication.

Figure 1. (a) The scattering geometry in real space with the definition of directions x and z. The y direction is out of the page. (b) The scattering geometry in reciprocal space for one length of k with the definition of ∆Q| and ∆Q⊥. x*, y*, and z* are the corresponding axes in reciprocal space.

Figure 2. The scattering geometry in reciprocal space for different values of k at the nominal angles RI,0 and RF,0. ∆k contributes only to ∆Q|.

In deriving the resolution ellipsoids of an entire instrument, we must consider the incoming and outgoing beams. The resolution of a reflectometer is therefore defined by the quantities ∆k, ∆RI, ∆RF, ∆βI, and ∆βF, as shown in Figures 1 and 2. Figure 1a shows the incoming and outgoing beams in real space with coordinate axes x, y, and z, while Figures 1b and 2 show the effects of resolution in reciprocal space with reciprocal coordinate axes x*, y*, and z*. Q0 is now the momentum transfer of the neutron (Xray) to the sample, given by eq 2, as defined by the instrument configuration. We define a second set of axes relative to Q0: as shown in Figure 1, ∆Q| is parallel to Q0; ∆Q⊥ is perpendicular to Q0 and lies in the scattering plane; and ∆Q⊥,y is perpendicular to Q0 and to the scattering plane. For small values of β, the only contribution of β is to ∆Q⊥,y. This contribution will be discussed later. As only elastic scattering is considered, the length of k does not change in the scattering process. If a monochromator is used, the correlation between k and ∆RI is negligible, because the divergence ∆RI of the monochromatic beam is extremely small compared to the divergence of the white beam and to the mosaic width of the monochromator. In neutron time-of-flight techniques, such correlation is absent. As a consequence, we can treat the two contributions to the resolution separately: for fixed kI,0 and kF,0 within the divergences ∆RI and ∆RF (demonstrated in Figure 1b); and for ∆k at the nominal incident and final angles RI,0 and RF,0 (as in Figure 2). A given k can have various momentum transfers Q around Q0 within the divergences ∆RI and ∆RF. This contribution of k0 to the resolution is given by the convolution of the lines (k ∆RI) and (k ∆RF), as shown in Figure 1). From Figure 2 we conclude that the different k values contribute only to ∆Q| with ∆Q|(k) ) (RI + RF) ∆k. Finally

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Figure 3. The convolution of k ∆RI, k ∆RF for ∆RI ) ∆RF ) ∆R, shown in Figure 1b, and of ∆Q|(k), shown in Figure 2.

Figure 5. The convolution of k ∆RI and k ∆RF in the case of vanishing ∆RI.

Figure 6. The final resolution for vanishing ∆RI accounting for ∆k. The resulting ellipse is indicated by a dashed curve. Figure 4. As Figure 1, but for vanishing divergence ∆RI.

the resolution ellipsoid is obtained by the convolution of the four vector quantities: k ∆RI, perpendicular to kI,0; k ∆RF, perpendicular to kF,0; (RI + RF)∆k, along Q0; k ∆β, perpendicular to the scattering plane. We will consider the ellipse in ∆Q| and ∆Q⊥ in some detail for two configurations of the instrument: Case 1. ∆rI ) ∆rF ) ∆r. In this case, the convolution of k ∆RI and k ∆RF gives a symmetric picture in ∆Q| and ∆Q⊥, which is additionally elongated in the ∆Q| direction by the wavenumber spread ∆k, as shown in Figure 3. Ql sym, Qssym, and Qs,ysym are the main axes of the resolution ellipsoid, shown in Figure 3. Taking only the “in-plane” components of the resolution, the long axes of the ellipsoid may be calculated by adding in quadrature the appropriate components defined above. Assuming that RI and RF have values of a few degrees, as is the case for reflectometry, we therefore find

Q′l sym ) {2(k ∆R)2 + [(RI + RF)∆k]2}1/2

sym

Ql (∆RIf0) ) k{∆RF2 + [(RI + RF) ∆k/k]2}1/2 /2(RI + RF)k(∆βI2 + ∆βF2) (9)

1

Qssym(∆RIf0) ) (RI + RF)k{∆RF2 + (RF ∆k/k)2}1/2 1

(7)

There is a prime on Q′l sym, because one further component must be considered. In the case of a straight line source, the angle RI depends on ∆βI. The contribution to Ql is always negative with RI ) RI,0 cos(∆βI). It gives a contribution similar to ∆k ) -1/2k∆βI2 in eq 7. Such a geometry is realized if kI,0 (i.e., RI) is defined by two slits in the y direction, one near to the source (e.g., the monochromator) and one near to the sample. Finally we obtain

Ql

plane and not shown in Figure 3. The contribution of ∆k to Qssym and Qs,ysym due to ∆βI and ∆βF is very small and therefore may be neglected. Case 2. ∆rI , ∆rF. The case of unequal angular divergences on the incoming and outgoing beam becomes interesting for measurements of off-specular scattering. The scattering geometry, like Figure 1, is now shown for vanishing ∆RI in Figure 4. The convolution of k ∆RI and k ∆RF gives now a straight line, as shown in Figure 5, which is inclined relative to ∆Q| by the angle RF. The convolution with ∆k is shown in Figure 6), where the final ellipse is indicated. The values for the main axes are now

) {2(k ∆R)2 + [(RI + RF)∆k]2}1/2 /2(RI + RF)k(∆βI2 + ∆βF2) (8)

1

Qssym ) (2-1/2)(RI + RF)k ∆R Qs,ysym ) k(∆βI2 + ∆βF2) The component Qs,y is perpendicular to the scattering

/2RF(RI + RF)k(∆βI2 + ∆βF2)

Qs,ysym(∆RIf0) ) k(∆βI2 + ∆βF2) Note that now ∆k contributes to Qs. So far the ellipsoids were defined relative to the direction of Q0. The measured intensity is given by eq 3 or 4. The considerations for the ellipse in ∆Q| and ∆Q⊥ are performed for small β values. For larger values of β, Q0 would be inclined with respect to the x-z plane. This will not be further discussed. In many cases it is sufficient to know the projections of the resolution ellipsoid ∆Qx, ∆Qy, and ∆Qz, using the axes defined relative to the sample surface in the Introduction. Projections of the Resolution Ellipsoid The quantity k ∆RI is perpendicular to kI,0. Its projection onto ∆Qx gives ∆Qx(RI) ) kRI∆RI. In the same way one obtains ∆Qx(RF) ) kRF ∆RF. To evaluate the contributions of ∆k to ∆Qx, we consider ∆Q|(k) ) (RI + RF)∆k.

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Figure 7. Reflection from a diffraction grating in the xz plane, with the grating lines parallel to the y-directions: (a) specular reflection; (b and c) ( higher order reflections, where the sample is rotated about the y-direction. The resolution ellipse of the instrument is taken for the symmetric configuration ∆RI ) ∆RF.

The projection of ∆Q|(k) onto ∆Qx(k) gives ∆Qx(k) ) + RF)(RI - RF)∆k, while the divergence out of the scattering plane, ∆β, adds the contribution 1/ (R 2 I

∆Qx(β) ) -1/4(RI + RF) (RI - RF)k(∆βI2 + ∆βF2)

The eqs 6 and 10-12 allow for quantitative calculations of coherence lengths. If, for example, l coh,x becomes larger than the footprint of the beam on the sample in the x-direction, then this dimension has to be taken into account when calculating S(k,RI,RF,β). Resolution Consideration for Gratings

Finally we obtain

∆Qx ) {∆Qx2(RI) + ∆Qx2(RF) + ∆Qx2(k)}1/2 + ∆Qx(β) ) {[kRI∆R]2 + [kRF ∆RF]2 + [1/2(RI + RF)(RI - RF)∆k]2}1/2 1

/4(RI + RF) (RI - RF)k(∆βI2 + ∆βF2) (10) ∆Qy ) k(∆βI2 + ∆βF2)

(11)

∆Qz ) {[k∆RI]2 + [k∆RF]2 + [(RI + RF)∆k]2}1/2 1

/2(RI + RF)k(∆βI2 + ∆βF2) (12)

Similar forms of these equations have been derived previously. Equations 10 and 12 are similar to eqs 4 and 7 in ref 3 in the case where ∆k and the “out-of-plane” divergence is neglected. Equations 18a and 18b in ref 5 correspond exactly to the “in-plane” part of eqs 10 and 12. Equations 10-12, however, appear to be the first attempt to include the “out-of-plane” divergence in such simple expressions for the resolution. From eq 10 we conclude that ∆Qx can become very small for small values of R and ∆R. In turn, by eq 6 the coherence length l coh,x becomes very large. For example l coh,x ≈ 70 µm is typical for the D17 reflectometer at ILL.8 (8) Ott F., Ph.D. Thesis, Universite´ de Paris Sud, 1998. Cubitt, R. ILL Annu. Rep. 2000, 91.

In the previous section the projections of the resolution ellipsoid, R, have been derived without consideration of a real scattering function, S, because R is independent of S. It is useful to consider an example of S, however, as an illustration of how resolution may be optimized in an experiment. The selected example is a reflectometry experiment on a diffraction grating. A grating in the x-y plane in real space with periodicity d transforms into rods along the z* direction in reciprocal space with intervals of 2π/d. The thickness of the stripes along z may create a modulation in intensity along the rods, which will not be considered here. In the x*-y* plane in reciprocal space the rods become dots in a linear sequence perpendicular to the stripes of the grating. Let us start by considering the lines of the grating to be parallel to the y-axis. From Figure 7 we see that the narrowest projection ∆Qx (longest coherence length l coh,x) is obtained for specular reflection Figure 7a, because ∆Qx ) Qs. For higher order reflections, Figure 7b,c, ∆Qx > Qs, because now ∆Qx is given by the projections of the ellipse onto the x-axis, derived in eq 10. With the instrument configuration ∆RI , ∆RF, the resolution width ∆Qx gets smaller for higher orders in the +x* direction, as shown in Figure 8. In the limit RF f 0 the resolution width ∆Qx would be smallest with ∆Qx ) Qs (eq 10). Note that RF cannot become smaller than ∆RF. Figures 7 and 8 visualize the effect of the inclination of the resolution ellipse on ∆Qx. Such sketches are very

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Figure 8. Reflection from a diffraction grating in the xz plane when the resolution ellipse of the instrument is taken for the asymmetric configuration ∆RI , ∆RF: (a) specular reflection; (b) higher order reflection in +x*-direction.

Figure 9. A grating, in real space and reciprocal space, which is rotated about the z (surface normal) direction. Table 1. Values ∆Qx for Q-Resolution and Corresponding Values for the Coherence Length, l coh,x, for Symmetric (∆rI ) ∆rF) and Asymmetric Collimationa

specular off-specular off-specular specular

RI (deg)

RF (deg)

∆RI (deg)

∆RF (deg)

∆Qx (10-5 Å-1)

l coh,x (µm)

1 1.5 1.5 1

1 0.5 0.5 1

0.03 0.03 0.015 0.015

0.03 0.03 0.06 0.06

1.6 1.83 1.44 2.38

39 34 44 26

a The collimation of 0.03° corresponds to two slits of 1 mm in 2 m distance.

useful in planning an experiment. To determine values for the projected widths of the resolution and coherence lengths, one must use eqs 10-12. As an example, we give values for ∆Qx and the corresponding l coh,x for four configurations where RI, ∆RI, RF, and ∆RF are moderately varied in Table 1. Note that the integrated intensity over the signal is equal in all cases, because the product (∆RI∆RF) is constant. The peak of the signal (from the rod) is highest when its width is smallest. Table 1 shows that when considering specular scattering, the best resolution is achieved for ∆RI ) ∆RF. For off-specular scattering the best resolution is found for an asymmetric configuration. Care must be taken in this interpretation, however. Figure 8 shows that the resolution is improved only on one side of the specular scattering, here with RF < RI. The resolution on the other side of the specular scattering, RF > RI, is considerably worsened with this asymmetric configuration. The resolution can be recovered in this side by interchanging the values of ∆RI and ∆RF. The case RF < RI has previously been discussed in ref 6 under the assumption of a highly collimated incident beam. The discussion concentrates on the influence of the scattered beam divergence on resolution and intensity.

Figure 13 of ref 6 clearly shows the optimal conditions for measuring crystal truncation rods, which is in agreement with our Figure 8. If the grating is made from magnetic materials, then one sometimes wants to study the effect of a magnetic field in the plane of the grating either perpendicular or parallel to the stripes. One way of performing such an experiment would be to change the direction of the field. However, if fields of magnitude 1 T are applied, it may be more convenient to use a vertical field and rotate the sample around its z-axis. This rotation has to be smaller than 90°, because in the limit of 90° no periodicity would be observable in the x-z scattering plane. Angles of around 84° (cos 84° ) 0.1) are often used.9 Figure 9 shows the diffraction pattern of a rotated grating in reciprocal space, which now has components Qx(n) and Qy(n). The important variable in such experiments is still Qx, given by the rotation of the sample around the z-axis. If the scattered intensity is integrated over the y-direction, for example, by a slit parallel to the y-direction in front of the detector, then the projected resolution ∆Qx reads

∆Qx(grating) ) {(kRI∆RI)2 + (kRF∆RF)2 + [1/2(RI + RF)(RI - RF)∆k]2 + [2Qy(n)(kI′/kF′)∆βI]2}1/2 /2(kI′/kF′) ∆βI2 Qx(n) (13)

1

with kI′ ) kI cos RI, kF′ ) kF cos RF. The divergence ∆βF does not explicitly appear because the βF component of the scattered beam depends on βI by the relation βF ) βI + Qy(n)/k. To explain the contributions by ∆βI to ∆Qx in more detail, we have calculated the pattern of the scattered intensity (9) Theis-Bro¨hl, K.; Leiner, V.; Schmitte, T.; Westerholt, K.; Zabel, H. Phase Transitions, in press.

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Figure 10. Patterns of intensity arriving in a multidetector: (a) Qy(n) ) 0; (b) Qy(n) * 0. The axes coordinates RF and βF directly correspond to the angular coordinates of pixels in a multidetector. The calculations were made for (a) RI ) 1°, k ) 1.36 Å-1, grating spacing ) 30 µm, the grating is not rotated about z, and (b) RI ) 2°, k ) 1.36 Å-1, grating spacing ) 0.3 µm, rotation angle ) 84°.

Figure 11. An image of the reflectivity from a diffraction grating on the position sensitive detector of the D17 reflectometer at the Institut Laue-Langevin. The grating had spacing ) 1.06 µm and was rotated 83.5° about its surface normal. The measurement was made with RI ) 1.59°, k ) 0.75 ( 0.03 Å-1. Also shown is the calculated intensity positions for the m ) 2, (1, and 0 grating fringes.

in a plane perpendicular to the nominal scattering plane, spanned by RF and βF. Many neutron reflectometers use position sensitive detectors. This plane then directly corresponds to the observed image on such a multidetector. In the small angle limit, the Bragg condition for the grating is satisfied for Qx ) kF′(1 - 1/2∆βF2) - kI′(1 - 1/2∆βI2) and Qy ) kF′ ∆βF - kI′ ∆βI. Figure 10 shows a plot of various trajectories of Q0(n) as a function of rotation angle. Figure 10a shows the case of an angle of rotation equal to zero (Qy(n) ) 0). The observed pattern is of curved lines, given by the last term in eq 13. The contribution of this curvature to ∆Qx is usually very smallsin order to see the curvature the vertical scale of Figure 10a has been increased to ∆βF ) (45°. On the other hand, if the grating is rotated, corresponding to a finite Qy(n), the “linear” contribution by ∆βI in eq 13 may become dominant. It only dominates, however, because eq 13 gives the projection onto the Qx axis. Figure 10b shows that the signal appears to have a correlation between RF and ∆βF. If one individual line is projected on the RF axis, the signal becomes very broad and gives a larger value for ∆Qx. This correlation means that the intensity on the detector from one order of reflection is spread along a line, which is extended in the β-direction (y-direction) and has an inclination toward RF. Note that the specular line (Qx(0) ) Qy(0) ) 0) is always vertical. Figure 11 presents reflectivity measurements of a rotated grating on the reflectometer D17 at Institut LaueLangevin. The grating had a periodicity of 1.06 µm and was rotated 83.5° about its surface normal (the z direction). The incident angle RI was fixed at 1.59°, and the wavenumber k was fixed at 0.75 ( 0.03 Å-1. The D17 instrument has a position sensitive detector, and Figure 11 shows the raw image of the measurement on the detector. The intensity to the left is due to refraction and will be ignored. The intensity to the right is due to scattering by the grating. Also shown in the figure are the calculated positions of the visible grating fringes by eq 13. There is excellent agreement between the calculated and measured images.

If a single (vertical) detector is used for such an experiment, then an inclined slit in front of it would improve the resolution considerably without reducing the measured intensity. If a two-dimensional detector is used, similar to the situation in Figure 11, the detector pixels should be integrated along the lines of intensity distribution. Nevertheless, for Qy(n) * 0, the linear contribution from ∆βI in eq 13 cannot be completely eliminated by using an inclined slit, because the sample dimension in the ydirection causes a smearing of the lines in Figure 10. This contribution from the linear part in eq 13 is always present and is given by the ratio of sample height to beam height at the monochromator. Conclusions We have derived the resolution, or transmission, ellipsoids in a graphical way avoiding complicated algebra as for example used in refs 1 and 2. The ellipsoids are helpful for understanding trends in the resolution related to varying instrument geometries. The projections, ∆Qx, ∆Qy, and ∆Qz are given analytically in terms of the instrument parameters to allow for detailed calculations. The influence of the beam divergence out of the scattering plane is analyzed for the case of off-specular scattering from a grating which is a function of rotation angle around the z-axis, perpendicular to its surface. In this case, the deterioration of the signal width can be avoided by proper integration of the intensity at the detector and the appropriate choice of incident and final beam divergences. Acknowledgment. The authors sincerely thank Dr. Katharina Theis-Bro¨hl and Dr. Tobias Korn for the loan of the grating sample. Thanks also to Dr. Theis-Bro¨hl, Dr. Robert Cubitt, Professor Boris Toperverg, and Vincent Leiner for useful discussions, and to Brigitte Aubert and Alison Cross for their aid in preparing the manuscript. LA026949B