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Wavelet Analysis of Neutron Reflectivity† N. F. Berk* and C. F. Majkrzak NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8562 Received January 23, 2003. In Final Form: May 12, 2003 An application of wavelet multiresolution analysis (WMA) to phase-sensitive neutron specular reflection is described. By use of Daubechies-8 wavelets, it is shown how the WMA notions of trend and detail are well-suited to describing the effects of truncated data on the spatial behavior of the scattering length density (SLD) profiles inferred from measurements. In particular, the inaccessibility of certain spatial detail to real measurements is readily seen. A WMA fitting scheme for refining the SLD deduced from phase-inversion measurements in a manner consistent with the data-determined spatial resolution is demonstrated.
1. Introduction The feasibility of revealing thin film structures using phase-inversion techniques in neutron reflectivity (NR) has been demonstrated.1,2 It is possible to directly measure the complex reflection coefficient r(Q) as a function of wavevector Q and to invert it mathematically to obtain the scattering length density (SLD) depth profile, F(z), of the reflecting film, where the z-axis is perpendicular to the film. In fact, measurement of Re r(Q) generally suffices.1,2 There remains much to do, however, in defining the reliability of the results, especially in assessing the unavoidable reduction of spatial resolution induced by data truncated at a maximum wavevector, Q ) Qmax.1,2 A general methodology known as wavelet multiresolution analysis provides us with a systematic and useful approach to the problem. Wavelets were first applied to X-ray reflectivity by Prudnikov, et al.3 with a different focus from ours and using different techniques. Here we develop a particular view of wavelet analysis in which the waveletbased identification of trend and detail of functions such as F(z), relative to a given spatial scale, is exploited to generate an objective, model-free interpretation of truncated, phase-sensitive neutron data. We begin with a brief overview of wavelet formalism specifically adapted to our concerns and which emphasizes the relationships between structure and spectrum that wavelet ideas engender. We then develop a refinement scheme for truncated data that uses the SLD profile that directly results from the phaseinversion analysis as a starting point for further, scaleadapted refinement. 2. Wavelet Multiresolution Analysis Wavelet multiresolution analysis (WMA)4-7 places functionssin our case, F(z) representing thin film SLD † Part of the Langmuir special issue dedicated to neutron reflectometry.
(1) Majkrzak, C. F.; Berk, N. F.; Krueger, S.; Dura, J. A.; Tarek, M.; Tobias, D.; Silin, V.; Meuse, C. W.; Woodward, J.; Plant, A. L. Biophys. J. 2000, 79, 3330 and references therein. (2) Majkrzak, C. F.; Berk, N. F.; Perez-Salas, U. A. Langmuir 2003, 19, 7796-7810. (3) Prudnikov, I. R.; Deslattes, R. D.; Matyi, R. D. J. Appl. Phys. 2001, 90, 3338. (4) He, Y. Mathematica Wavelet Explorer, Wolfram Research Inc.; Champaign, IL, 1996. (The mention of a commercial tradename in this work is strictly for informational purposes to properly identify the full title of the cited source, and is not to be construed in any way as endorsement by NIST or to imply that the product referred to is necessarily the best for the purpose.)
10.1021/la034126w
profilesswithin a nested set of function spaces Vj, labeled by j ) ..., -1, 0, 1, ..., that are characterized by “dyadic” spatial length scales lj ) 2-jl0. Increasing j signifies finer resolutions relative to the fiducial scale l0, while decreasing j indicates coarser resolutions. In use, l0 is the finest length scale of possible interest to us, say l0 ≈ 1 Å, so that the j values of practical concern are negative. In WMA F(z) is first “seen” as a coarse trend FJ(z) at a particular resolution level j ) J. Added detail ∆JF(z) at that level then gives the trend at the next finer scale, according to
FJ+1(z) ) FJ(z) + ∆JF(z)
(1)
Relative to a chosen scale of resolution lJ as base, the SLD profile thus can be represented exactly by the base trend plus all remaining detail
F(z) ) FJ(z) +
∆jF(z) ∑ jgJ
(2)
as if F(z) were being viewed through a pair of glasses with transitional lenses. WMA provides orthonormal bases for the above description.4,6 The J-trend is expressed as an “atomic” expansion in terms of a basic real-valued scaling function φ(z/l0) as ∞
FJ(z) )
∑
aJ,kφ(z/lJ - k)
(3)
k)-∞
The “atoms” φ(z/lJ - k) of the expansion are all derived from scale-adjusted dilation (z f 2jz) and translation (z/l0 f z/l0 - 2-jk) of the same basic atom φ(z/l0). The J-detail is similarly expressed in terms of a basic real-valued wavelet ψ(z/l0) by the atomic expansion ∞
∆JF(z) )
∑ bJ,kψ(z/lJ - k)
(4)
k)-∞
(5) Strang, G.; Nguyen, T. Wavelets and Filter Banks (rev. ed.); Wellesley-Cambridge Press: Wellesley, MA, 1997. (6) Daubechies, I. Ten Lectures on Wavelets; Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. (7) Chui, C. K. An Introduction to Wavelets; Academic Press: Boston, 1992.
This article not subject to U.S. Copyright. Published 2003 by the American Chemical Society Published on Web 06/11/2003
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Figure 1. Daubechies-8 scaling function φ(z/lj) for j ) 0 (solid), j ) 1 (small dash), and j ) -1 (large dash).
Figure 2. Daubechies-8 wavelet ψ(z/lj) for j ) 0 (solid), j ) 1 (small dash), and j ) -1 (large dash).
On every scale, the scaling function and wavelet atoms are site-orthogonal, i.e., orthogonal with respect to translation index k; additionally, the wavelet atoms are mutually orthogonal on different scales, i.e., with respect to dilation index j, a remarkable property, considering that it must be inherent in the shape of the basic wavelet atom. The scaling functions have a nonvanishing zeroth moment (i.e., area)
∫-∞∞ φ(zj) dzj ) 1
(5)
where zj ) z/l0, while the wavelets have a vanishing zeroth moment
∫-∞∞ ψ(zj) dzj ) 0
(6)
Therefore the detail at any resolution globally averages to zero; in practice the detail generally averages out locally, more-or-less, in line with our casual notions of “detail”. Wavelets and scaling functions are closely related. Indeed the resolution of a function into trend and detail in eq 1 applies equally well to the behavior of the scaling function and wavelet themselves at neighboring spatial scales and, when combined with the orthogonality requirements, the zeroth moment properties, and other constraints, leads to a closed system of equations which mutually determine φ(zj) and ψ(zj).6 Wavelets and scaling functions (“wavelets” for short) thus take on a variety of functional forms determined by subsidiary attributes, such as spatial extent (i.e., support) and smoothness (i.e., differentiability). One wavelet family, in particular, called Daubechies8,4-6 seems well suited, for example, to NR analysis of biomimetic films, whichsincluding associated components such as gold layers and molecular tetherssnormally have total thicknesses in the range of 100-200 Å and which typically are measured by NR spectra having Qmax e 0.3 Å-1. 3. Example Figures 1 and 2 depict the Daubechies-8 system on three adjoining spatial scales. All wavelet computations shown here were carried out with commercial wavelet software.4 Our implementation of WMA was carried out using the “cascade” algorithm,4,5 in which the fiducial scale, j ) 0, of the function to be analyzed is defined as its discrete (Daubechies-1 or Haar) representation at scale length l0. Some properties of the Daubechies-D system and additional implementation details are given below. For a
Figure 3. Illustration of atomic expansion of trend F-4(z) (thick gray) as in eq 3. The scaling function φ(z/l-4) (solid) and its translates (dotted) are shown, along with the coefficients a-4,k (bullets). In this and subsequent figures, F-values are plotted in units of 10-6 Å-2.
Figure 4. Illustration of atomic expansion of detail ∆-4F(z) (thick gray) as in eq 4, in format of Figure 3.
particular F(z), the atomic expansions of the coarse trend F-4, as in eq 3, and corresponding coarse detail ∆-4F(z), as in eq 4, are illustrated in Figure 3 and Figure 4, respectively. For illustration we use a realistic SLD profile previously obtained by molecular modeling1 to represent a hybrid lipid membrane on a thin chromium/gold film, a biomimetic system typical of those being studied in many laboratories. This is shown in Figure 5. The model consists of the chromium-gold layer having a thickness about 80 Å, a hydrogenated self-assembled alkanethiol monolayer (SAM) of thickness about 20 Å, and a deuterated lipid monolayer of thickness about 30 Å; the total thickness, L,
Wavelet Analysis of Neutron Reflectivity
Figure 5. Model SLD profile used here as the veridical profile, i.e., the one responsible for the observations. The region of negative SLD corresponds to a protonated alkanethiol selfassembled monolayer (SAM). The large values of SLD in the lipid layer correspond to deuterated lipid.
Figure 6. WMA of the veridical profile of Figure 5 (thick gray line). The coarse trend F-4(z) (thick black line) and higher resolution trends, Fj(z), for j ) -3, -2, and -1 (light lines) are shown. F-1(z) and F(z) nearly coincide. By construction, F0(z) ) F(z).
of the SLD profile is 130 Å. (The leading edge is shown shifted to the right of the z-origin to facilitate the ensuing wavelet analysis without introducing artificial edge effects.) This system has been recently described in detail and studied by NR using phase-inversion methods.1 We also refer to the model F(z) as the veridical SLD profilesviz., the one responsible for the data, here the simulated datasto distinguish it from approximations derived from data analysis.1 Figure 6 shows the sequence of Daubechies-8 trends converging on the model, starting with the coarse level labeled by J ) -4, and Figure 7 presents the same information in terms of the coarse trend and all the remaining detail for j ) -4, ..., 0, in accordance with 2. The general family of Daubechies-D wavelets, for D ) 1, 2, ..., have the unusual mathematical property of being both site-orthogonal and compactly supportedson a zjinterval of width 2D - 1. Daubechies-D wavelets also are differentiable D - 1 times, so that the Daubechies-8 system is very smooth. In the Daubechies-D description of WMA, the coarsest practical resolution level is determined in standard implementation strategies4,5 to require D atoms; in our case, this means a resolution having eight degrees of freedom. For this WMA analysis the model is defined numerically on a uniform mesh of N ) 128 points, as described above in connection with the cascade algorithm; the fiducial scale length is l0 ) L/N ≈ 1 Å. For general N
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Figure 7. As in Figure 6, but here showing the coarse trend, F-4(z) (dotted lines) and the sequence of detail, ∆jF(z) ) Fj(z) Fj+1(z) for j ) -4, -3, -2, and -1 (solid lines, thicknesses exaggerated for clarity).
Figure 8. Filter characteristics of the basic scaling function φ(zj) and wavelet ψ(zj) as Fourier transforms, showing |φˆ (Q h )| and |ψ ˆ (Q h )| (solid lines). The scaling function acts as a low-pass filter, the wavelet as a band-pass filter, the combined filters acting as a low-pass filter for the next higher scale of resolution, |φˆ (Q h /2)| (dotted line). Vertical lines mark the formal low-pass filter edges at Q h ) π and Q h ) 2π. The small bump in |φˆ (Q h )| for Q h ≈ 3π is characteristic of the Daubechies-8 system.
and D the coarsest resolution level is J ) - [log2(N/D)], where [x] signifies the integer part of x; thus in our case, J ) - 4. (Usually in WMA implementations, N and D are made diadic, i.e., powers of 2.) The convergence of the WMA trends seen above in Figure 6 is striking. The “overall” shape of the model SLD effectively is determined by the coarsest trend, F-4(z), a “smeared” version of F(z). However, the prominent double-peaked structure of the lipid headgroup is not evident until two more levels of detail are added to produce the trend F-2(z). Subsequent detail mainly acts to sharpen the edges between the film’s components and to “flatten the bottom” of the SLD of the protonated alkanes, seen as the slightly negative, gaplike region of the model. Notice that much of the ripple in the region of the gold layer in F-3(z) is transitory, an unavoidable effect of filtering, but consistent with that level of spatial resolution. 4. Wavelets as Filters The scaling function has the characteristics of a lowpass filter in reciprocal space, while the wavelet behaves as a band-pass filter; this is illustrated in Figure 8, which shows the absolute values of their Fourier transforms (FT). The dimensionless variable conjugate to zj is Q h ) Ql0. The combined effect of the low-pass filter and the band-pass filter at a given scale of spatial resolution is to produce
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r(Q) ≈ rJ(Q) +
Figure 9. Extension of Figure 8 to WMA. The extension actually occurs in both directions along the Q h axis.
Figure 10. WMA of Re r(Q), shown as Q2Re r(Q), according to eq 7, exhibiting the low-pass and band-pass filter effects indicated in Figure 9. “Detail n” corresponds to WMA level j ) n - 5. Both the flatness of the SAM and the double peak in the lipid portion of the film, Figure 5, are not revealed in Q2Re r(Q) below detail 3 (as seen by counting detail levels in Figure 6), i.e., Q < 0.5 Å-1. Measurements much below Q ≈ 0.2 Å-1 are incapable of revealing any significant detail in this film.
the band-pass filter for the next finer scale of resolution, as illustrated in the figure. Figure 9 indicates the continuation of this process, a behavior known in engineering parlance as a “filter bank”. The nominal edge of the basic band-pass filter is defined mathematically by Q h ) π.4-6,8 The Daubechies-D system produces the flattest possible band-pass filter for each D and so is also referred to as a “maxiflat” system. Indeed, one sees in Figure 8 that for D ) 8 the band-pass filter is very flat until Q h ) π, where the roll-off to zero becomes pronounced. In filter behavior flat response is associated with “accuracy”, or the reduction of distortion. However, truncation of the filter at Q h ) π misses spectral information both from the low-pass and band-pass regions because of their overlap. It follows that sharply truncated Re r(Q) data cannot exactly reveal a wavelet trend, Fj(z), or any specific level of detail, ∆jFj(z), for any particular j. The impact of the WMA filter bank on the neutron reflection from our model SLD is illustrated in Figure 10, which shows Re r-4(Q) and ∆jRe r(Q) for j ) -4, -3, -2, and -1. The quantities rj(Q) and ∆jr(Q) are defined as the reflection coefficients from the trend SLD Fj(z) and the various details, ∆jF(z). For Q-values not too small, r(Q) is the linear superposition of these contributions (8) See also eq 58 of Majkrzak, et al.2 for a similar formula in a different but related context.
∆jr(Q) ∑ jgJ
(7)
(here for J ) -4) analogous to the exact WMA decomposition of F(z) in 2. Because of dynamical scattering effects, r(Q) is not exactly a linear functional of F(z), especially near Q ) 0, but linearitysand thus eq 7s improves with increasing Q as the Born approximation becomes valid. The figure actually shows Q2Re r(Q), which is easier to view on a multiscale plot. For F(z) having sharp edges, in fact, Q2Re r(Q) does not decay with Q as Q f ∞. We see at once from Figure 10 that the double-peaked structure of the model F(z), which for us is associated with detail at resolution level j ) -2 in Figures 6 and 7, is only revealed in r(Q) for Q greater than approximately 0.6 Å-1. Any measurement of reflection limited to smaller Q cannot resolve these peaks reliably. Similarly, the sharpness of the component edges of F(z) effectively is suppressed for Q < 1 Å-1. Indeed, even the relatively simple chromium/ gold layer is not faithfully represented in r(Q) until approximately Q g 0.4 Å-1, corresponding to j ) -2. This has implications for fitting: trial F(z) with more detail than can possibly be visible in the data at hand cannot be be reliably determined. Because of the powerful mathematical constraint of analyticity,1 even a truncated r(Q) (or Re r(Q) and Im r(Q) separately) is unique to the F(z) which produces it. Thus, in principle, only the veridical SLD profile can produce r(Q) which exactly coincides with the measurement over any range of Q; equivalently, in principle a partial r(Q) spectrum can be analytically continued over the entire Q-axis, thereby making achievable an accurate description of F(z) at any desired degree of spatial resolution. In practice, however, analytic continuation of r(Q) is illbehaved; even very close agreement in r(Q) for Q < Qmax does not imply agreement for Q > Qmax. One sees in Figure 10 that detail in r(Q) at each level of resolution is banded by very “flat” filters into regions in which, effectively, either it is observable or it is not; detail 4, for example, essentially vanishes until Q ≈ 0.85 Å-1. Even detail 1 is not practically different from zero below Q ≈ 0.05 Å-1. Reproducing detail 3, which is required to resolve the film’s lipid structure, by analytic continuation of measurements restricted to Q < 0.4 Å-1, say, would not appear to be feasible. 5. Refining Detail The mathematical inversion of a truncated Re r(Q) by methods previously discussed1,2 yields an effective F(z), let us call it F˜ (z), which will tend to liesin some useful sensesbetween neighboring wavelet trends, FJ(z) and FJ+1(z), for some J determined by the maximum Q of the measurement, as discussed in section 4. In fact, were the Born approximation for r(Q) valid everywhere in Q, one would measure the familiar step-filter convolution smearing9
∫-∞∞
F˜ BA(z) ) 2
sin[(z - z′)Qmax] F(z′) dz′ z - z′
(8)
We cannot know FJ(z) exactly from the data, for that would require knowing F(z), but we can perform WMA on the phase-inversion result, F˜ (z), and expect on the basis of our earlier observations that (9) Berk, N. F.; Majkrzak, C. F. Unpublished. The derivation is easy but not trivial, since in the Born approximation it is F(z) and Qr(Q)snot F(z) and r(Q)sthat form a Fourier transform pair.
Wavelet Analysis of Neutron Reflectivity
F˜ J(z) ≈ FJ(z)
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(9)
Now from eq 2 we have
F˜ (z) ) F˜ J(z) +
∆jF˜ (z) ∑ jgJ
(10)
exactly and
F˜ (z) ≈F˜ J(z) + ∆JF˜ (z)
(11)
approximately, since F˜ (z) perforce lacks high-Q detail. Using eqs 3 and 4 this can be written explicitly as D-1
F˜ (z) ≈
∑
D-1
a˜ J,kφ(z/lJ - k) +
k)0
∑ b˜ J,kψ(z/lJ-k)
(12)
k)0
for a Daubechies-D system. Our F˜ (z) cannot exactly reproduce the truncated Re r(Q) because of truncation artifacts in the inversion, but we can hope to improve agreement with the truncated Re r(Q) by holding F˜ J(z) fixed in (11), consistent with eq 9, and replacing ∆JF˜ (z) with Jth level detail refined by fitting the available Re r(Q). In short, we replace F˜ (z) by a data-refined version, F˜ (z), such that
Figure 11. Sequence of WMA trends for the phase-inverted F˜ (z) (dark gray line), as in of Figure 6. F˜ -4(z) (thick black line) and higher resolution trends, F˜ -3(z) and F˜ -2(z) F˜ -1(z) (thin lines) are shown, along with the veridical F-4(z) of Figures 6 and 7 (thick dotted line).
D-1
F˜ (z) ) F˜ J(z) +
∑ b˜ J,kψ(z/lJ - k)
(13)
k)0
where F˜ J(z) on the right is the coarsest WMA trend of the phase inversion of the truncated data, and the D wavelet coefficients, b˜ J,k, are determined by fitting Re r(Q). Since F˜ J(z) already accounts, more-or-less accurately, for the coarse trend in the data (Figure 12), we may expect good fits to the residual detail in the measured spectrum with trial SLDs automatically limited in spatial resolution to the level of detail appropriate to the truncated Re r(Q). Still, we are now dealing with a D-dimensional fitting problem, with D ) 8 for the Daubechies-8 system, and many good fits to the available Re r(Q) are possible which may not be physically acceptable. With possibly only vague prior knowledge of F(z), which we are assuming to be the situation of interest, we need an objective criterion for discriminating “allowable” from unphysical spatial detail consistent with the measurement. The WMA provides a useful guide to assessing such acceptability. A hallmark of WMA is that the magnitude of detail tends to decrease rapidly with diminishing spatial scales for relatively smooth F(z)sas evidenced, for example, by the converging trends in Figure 6. Indeed, in waveletbased methods of noise reduction, noise is identified as excessive amounts of fine-grained detail in the WMA expansion, according to statistical thresholding criteria. In our case we introduce a rough but simple relative measure of relative WMA weights at any j g J as 2j-JD-1
Rj )
∑
k)0
2j-JD-1
|bj,k|/
∑
|b˜ J,k|
(14)
k)0
where the summations catch the nonvanishing coefficients at each level. Thus for any F(z), Rj compares the weight of detail at resolution level j to the weight of coarse detail in F˜ (z), which is known from the NR measurement. For our F˜ (z), shown in Figure 11, where J ) -4, R˜ -3 ≈ 0.2, indicating rapidly disappearing detail with increasing resolution. For the veridical F(z), the total weight of relative 0 Rj ≈ 3.0, which is typical detail over all scales is only ∑j)-4
Figure 12. Truncated Q2r(Q) for veridical r(z) (black line)s the model datasand for the phase-inverted F˜ (z) (gray line).
of similar looking functions in our experience. Therefore, in the refinement of F˜ (z) against the measured Re r(Q), using eq 13, good fitssas measured by small χ2, says yielding R˜ -4 substantially greater than unity likely would not be meaningful for the data at hand. To illustrate these notions, we show in Figure 11 the effective SLD profile, F˜ (z), the one obtained by phaseinverting the Re r(Q) produced by our model F(z), truncating the spectrum at Qmax ) 0.2 Å-1. This relatively low cutoff lies at the nominal edge of the lowpass filter, Ql-4 ) π, for the coarsest Daubechies-8 trend of the model F(z) and is not atypical of the maximum Q values often found in the literature of NR studies of biological thin films. The figure also gives the Daubechies-8 coarse trend, F˜ -4(z), for the phase-inverted function and the corresponding trend, F-4(z), of the model. One sees that F˜ (z) is similar but not identical to F-4(z); they cannot coincide for the reason given near the end of section 4. Corresponding spectraspresented as Q2Re r(Q)sare shown in Figure 12. The phase-inverted F˜ (z) gives a fairly good account of the truncated spectrum from F(z), but there are noticeable discrepancies, mainly near Qmax. The agreement can be improved by fixing F˜ -4 and refining the detail, ∆-4F˜ (z) ) 7 ∑k)0 b˜ -4,kψ(z/lJ - k), by fitting to Q2Re r(Q) for 0 e Q e Qmax. We measured goodness of fit with a χ2 parameter, defined simply as the unweighted sum of squares of Q2Re r(Q) residuals, normalized to the value obtained for F˜ (z). Thus fits which improved agreement would give χ2 < 1.
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Figure 13. All fits to Q2r(Q) obtained in the restricted search; see text.
By use of standard methods, Re r(Q) was computed exactly for a uniform bin representation of F(z) with bin-width equal to the fiducial length scale, l0 ≈ 1.0 Å. Optimization was done with the Nelder and Mead simplex algorithm, which is well-suited to probing the constrained parameter spaces of the type that occur here, as described below; a detailed discussion of the Nelder and Mead method can be found in Berk and Majkrzak.10 With D ) 8 fit variables (b˜ -4,k, for k ) 0, ..., 7), one can expect many fits of similarly low χ2 to exist; the problem is to find a substantial number of them in order to have a basis for judging relative acceptability and reliable information content. Our strategy was to probe the eight-dimensional space by restricting the search to Cartesian manifoldsslines and hyperplaness in which various b˜ values were held fixed at the values defined by the phase-inverted detail ∆-4F˜ (z), namely, b˜ k ) b˜ k. For example, the simplest manifolds of this type are the eight axes defined by allowing only one b˜ at a time to vary, holding the other seven fixed at b˜ . Similarly, two b˜ can be varied at a time, and so on. Even this restrictive probe generates a great many possibilities; we thus only considered manifolds of spatially contiguous b˜ (e.g., sets {b˜ 0, b˜ 1}, {b˜ 1, b˜ 2}, and so on) allowing us to vary detail locally across the film. The definition of the normalized χ2 was not adjusted to account for differing numbers of degrees of freedom within manifolds; in essence, all fitting parameters were allowed to change but some more than others. 6. Analysis All of the fitted spectra and resulting F˜ (z) are given in Figures 13 and 14, respectively, and in Figure 15 we show a scatter plot of the corresponding χ2 and β ) 1 + R values. About 30 cases are shown, and the “L”-shape of the scatter plot behavior appears to be a stable feature of the restrictive fitting search in this example. A perfect fit (χ2 ) 0) is not possible without incorporating detail at all scale levels. The “eyeball” quality of all the spectral fits in Figure 13 is good, with most of the differences occurring near Qmax. The corresponding F˜ (z) in Figure 14 shows considerable variation, however. The SLD profiles exhibiting exaggerated oscillatory behavior clearly seem unphysical on one or two grounds: either we would know that large values of SLD are incompatible with the chemical constitution of the film or we might be confident about the structure of the Cr-Au and alkanethiol layers, which cannot sustain such large amplitude SLD variations. On the other hand, some of these “best-fit” SLD (10) Berk, N. F.; Majkrzak, C. F. Phys. Rev. B 1995, 51, 11296.
Berk and Majkrzak
Figure 14. The family of F˜ (z) corresponding to the fits of Figure 13. The examples showing exaggerated detail actually are among the best fits as measured by χ2.
Figure 15. Scatter plot effectively combining results of Figures 12 and 13. Acceptable fits are taken to lie in the small rectangle. The short-dash line is β ) 1 + R for the “measured” detail ∆-3F˜ (z). The longer dash line is for the corresponding detail ∆-3F(z), and the longest dash lines is for the total detail of the veridical F(z).
profiles are not so easy to set aside without prior detailed knowledge of the putative structure. Moreover, a few show indications of a “super-resolution” of the double peak lipid headgroup SLD structure of the veridical profile but always accompanied by similar oscillatory variation of the Au-alkanethiol interface proximal to the prominent negative region of the deuterated lipid acyl tails. It is tempting to accept the biologically interesting SLD variation in the headgroup, especially when it might be expected, while either ignoring the apparently spurious Au-alkanethiol SLD behavior or attributing some degree of physical meaning to it. The scatter plot in Figure 15 helps to clarify the situation. First we notice that all the fits actually improve agreement with the truncated Q2Re r(Q) relative to the phase-inverted result (all χ2 < 1). The lowest achievable normalized χ2 among the tested manifolds was about 0.27, but many fits were clustered near the low value. The distinguishing quality among these is the value of the relative WMA trend-detail measure, β. The lowest χ2 was obtained with the unconstrained variation of all eight b˜ , but this fit gave β ≈ 12 (R ≈ 11), which is completely out of line with any reasonably expected normalized weight of detail; we are justified in calling such fits unacceptable from this point of view. The small rectangle on the scatter plot roughly delineates the region in which we expect acceptable fits to fallsnamely the ones having lowest achievable χ2 and β values consistent with those for F˜ (z)
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limitations of the data are apparent: we cannot meaningfully resolve film structure that is more detailed than the truncated Re r(Q) spectrum can support. The unbiased family of SLD profiles of Figure 16 reveals the reliable structural information in the model data.
Figure 16. Family of acceptable F˜ (z) consistent with the truncated data, i.e., for Q e Qmax ≈ 0.2 Å-1, obtained by refining the phase-inverted F˜ (z) (thick line) and limited by the allowed weight of detail; namely, the fits lying within the rectangle of Figure 15, which effectively encompass the structural information recoverable from the measurement.
and F(z) at this wavelet trend level. Evidently the box could be made somewhat higher without changing our conclusions, since we did not find many low-β fits near the low-χ2 “corner” of the scatter plot for this case. The corresponding WMA-acceptable F˜ (z) are given in Figure 16. These significantly improve agreement with the data, relative to F˜ (z), while also providing reasonable WMA weights. Most of the improvement in Q2Re r(Q) comes near Q ) Qmax, as expected, while most of the concomitant change in the SLD profile comes from the biologically interesting part of the film, namely, the lipid leaf. The structure of the less interesting Cr/Au and alkanethiol layers remains more-or-less stable in this region of the scatter plot. This gives us added confidence that the WMA improvement of the phase-inverted result, while small, is structurally meaningful. At the same time, however, the
7. Conclusion We believe that wavelet multiresolution analysis is a useful scheme for thinking about scattering length density profiles in relation to neutron reflectometry measurements, which, of necessity, are restricted to finitesand sometimes not very largesQ values and which, therefore, yield spatial information about the film under study on an unavoidably limited scale of spatial resolution. The precise WMA notions of trend and detail that underlie the approach are formal but not unintuitive and lend themselves to systematic (and fast) model-independent analysis. Furthermore, application of WMA to putative models of SLD structure can help develop realistic expectations of what spatial information NR measurements are able to reveal reliably from the system of interest. WMA is particularly well-suited to augmenting phaseinversion studies of thin films. The phase-inverted F˜ (z) is an “unbiased”si.e., model-independentsrepresentation of the veridical F(z), limited mainly by inversion artifacts resulting from data truncation (see also section XI of the companion work2). As such, F˜ (z) cannot exactly reproduce the measured spectrum, but using WMA, as in sections 5 and 6, the agreement with experiment can be further improved in an “unbiased” manner consistent with the range of available data. The phase-inverted F˜ (z) contains sufficient information to define sensible expectations for the amount of detail that can be permitted in the refinement that takes F˜ (z) into F˜ (z), where F˜ (z) is an image of F(z) on a spatial scale appropriate to the experiment. LA034126W
