Volume 21 Received 16 April 2013  Liquid Xray scattering with a pinkspectrum undulator^{a}Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France, and ^{b}European Synchrotron Radiation Facility, BP 220, 38043 Grenoble Cedex, France Xray scattering from a liquid using the spectrum from the undulator fundamental is examined as a function of the bandwidth of the spectrum. The synchrotrongenerated Xray spectrum from an undulator is `pink', i.e. quasimonochromatic but having a sawtoothshaped spectrum with a bandwidth from 1 to 15%. It is shown that features in S(q) are slightly shifted and dampened compared with strictly monochromatic data. In return, the gain in intensity is 250500 which makes pink beams very important for timeresolved experiments. The undulator spectrum is described by a single exponential with a lowenergy tail. The tail shifts features in the scattering function towards high angles and generates a small reduction in amplitude. The theoretical conclusions are compared with experiments. The rresolved Fourier transformed signals are discussed next. Passing from q to rspace requires a sinFourier transform. The Warren convergence factor is introduced in this calculation to suppress oscillatory artifacts from the finite q_{M} in the data. It is shown that the deformation of rresolved signals from the pink spectrum is small compared with that due to the Warren factor. The qresolved and the rresolved pink signals thus behave very differently. Keywords: Xray scattering; liquids; pink Xrays. 
The distortion of signals due to imperfect recording devices is omnipresent in experimental sciences. This is particularly well known in optical spectroscopy, where spectral lines are deformed by the finite slit width of the spectrometer. This deformation may be large enough to compromise the interpretation of the observed data. Similar problems are present in Xray physics. Some of them are due to the finite duration of the probing Xray pulses, and may be treated employing deconvolution techniques (Jansen, 1997). They were discussed in some detail in our recent paper, called in what follows Paper 1 (Bratos & Leicknam, 2012). Another imperfection is not only that the pulse duration may be too long but also that the raw synchrotrongenerated Xray spectrum is never strictly monochromatic; this holds true even for the fundamental emission line from an undulator which at modern synchrotrons has an asymmetric shape with a relative bandwidth of 115% (FWHM) peaked at E_{M}. Problems of this sort merit careful examination. A short review covering the field was published recently by Guerin et al. (2012). The perturbations of qresolved signals were explored first by Haldrup et al. (2009). The incident beam was described as a superposition of a number of strictly monochromatic beams, and standard formulas were employed to calculate the corresponding signals. Superposing them then provides the final signal, which may be compared with experiment. This approach turned out to be relatively satisfactory, although some details escaped the analysis. The perturbation of rresolved signals turned out to be much more difficult; it was studied very recently, both theoretically and experimentally (Lee et al., 2013). Here, too, the incident Xray beam was considered as a superposition of a number of monochromatic beams. It was then shown that the scattering data of a polychromatic beam remain a weighted sum of monochromatic data even in real space. It was next shown how, knowing the signal for a given value of r, for example r = r_{0}, all signals, whatever r, can be deduced from the signal calculated for r = r_{0}. This last signal was determined by attributing to it an arbitrary, but flexible, functional form with a sufficient number of adjustable parameters; the latter were fixed employing mean square optimization techniques. The difficulty with this method is its numerical accuracy, the effects under consideration being very small.
Although basically interested in timedependent problems, we decided here to consider systems in thermal equilibrium. In fact, the distortion of signals due to the use of pink radiation is expected to be small when modern synchrotron Xray sources are employed; it is thus most conveniently explored in the absence of perturbations, i.e. for systems in thermal equilibrium. It should be emphasized that the polychromatic correction problem can be treated in a completely clean way. Once statistical properties of the noisy component of the incident Xray radiation have been defined, the rest of the paper is free of any theoretical or computational approximation. Note finally that similar problems occur also with freeelectron laser sources, due to the shottoshot fluctuations of recorded signals. The present study thus represents a contribution to the general theory of Xray radiation.
As just stated, the system under consideration is a diluted liquid solution in thermal equilibrium. In principle the system is probed with monochromatic radiation. In reality the Xray spectrum is pink, i.e. only quasimonochromatic. At the European Synchrotron Radiation Facility, the spectral width / of the U17 undulator used for fast timeresolved experiments is 3.5% (full width at halfmaximum), but at other beamlines and synchrotrons the bandwidth can vary from 2 to 15% depending of the collimation of the electron beam. Our problem here is to explore the consequences of this deviation from an ideal monochromatic source. Our theory rests on the assumption that the undulatorgenerated pink Xray wave is an incoherent superposition of fully monochromatic Xray waves (Schotte et al., 2002). The intensity distribution of the latter was found to be exponential: I(E) = I_{0}exp['(E_{M}  E)] if E < E_{M} and I(E) = 0 for E > E_{M}, where E_{M} is the cutoff energy and ' is the bandwidth (Fig. 1). The relation between the FWHM, called E_{M}, and ' is defined by E_{M} = ln(2)/'. The linewidth comes from the Dopplershifted offaxis radiation from the undulator collected in a finitesize aperture.
 Figure 1 The shape of the synchrotrongenerated Xray intensity I(q') as a function of the variable q' = (4/); this quantity is proportional to the beam energy E. Moreover, q_{M} is the cutoff value of q'. Note that E_{M} = 19 keV. 
It is more convenient for the present purposes to replace E in the expression I(E) by the quantity q' = 4/. This variable q' has the dimensions of a wavevector but is proportional to the energy E. One finds easily
The relation between the quantity q' just introduced and the wavevector q of the theory of Xray diffraction is q = q'sin. Expressing the energy E in terms of a quantity q' having the dimensions of inverse length may surprise, but note that in spectroscopy the energy is currently expressed in units of cm^{1}. It is then possible to write I(q') = I_{0}exp[(q_{M}  q')] if q' < q_{M}, and I(q') = 0 if q' > q_{M} where q_{M} is the cutoff value of q' and the bandwidth. As E and q' are proportional to each other, these two expressions are strictly equivalent. The decay constants and ' are simply related, i.e. = 1/2c'. Then, if S_{P}() is the intensity of the scattered pink radiation expressed in electronic units, 2 the scattering angle and if i_{P}() = S_{P}()  _{i}f_{i}^{2} is its reduced scattering intensity from which the single atom contribution _{i}f_{i}^{2} was subtracted (Warren, 1990), there results
where
Here, the symbol i_{M}(q', ) indicates the reduced scattered Xray intensity at a precisely defined value of q', q_{M} is the cutoff value of q' and f_{i} designates the atomic scattering factor of the atom i. The factor [1  exp(q_{M})]^{1} is the normalization factor for the undulator spectrum. Finally, the subscripts M and P refer to monochromatic and pink incident Xray beams, respectively. This integral is not simply expressible in terms of familiar functions, but can easily be calculated numerically. Note that, if the incident Xray radiation is not strictly monochromatic, the wavevector q is illdefined, and speaking of qresolved signals is not rigorous. On the contrary, the dependent pink signals S_{P}() remain precisely defined. Nevertheless, for commodity of language we will continue speaking about qresolved signals, but one should keep in mind this precaution. This expression for the signal i_{P}() can be used to explore the signal deformation if a pink, rather than a strictly monochromatic Xray radiation, is used in an experiment. The two parameters playing a basic role here are the cutoff wave length vector q_{M} and the bandwidth = q_{M}/q_{M} = E_{M}/E_{M} of the Xray spectrum. (i) The dependence of the qresolved Xray signal on the beam width is illustrated in Figs. 2(a)2(c). If q_{M} is of the order of 8 Å^{1} and 1%, the monochromatic and pink curves coincide. The Xray radiation, although pink, can then be treated as monochromatic. If increases to 3% with q_{M} = 8 Å^{1}, the difference between the monochromatic and pink signals is no longer totally negligible. It is relatively important if 6%. Moreover, as shown in these figures, the nodal points of the pink signals shift more rapidly with increasing angle as in a monochromatic signal. Finally, the intensity of pink signals is smaller than that of monochromatic signals. These simple calculations give an idea about the effect of the polychromaticity of synchrotrongenerated pink Xrays on qresolved signals. These results are in good agreement with experiment (Guerin et al., 2012; Haldrup et al., 2009).
 Figure 2 resolved signals corresponding to different bandwidths of the incident Xray beams (red squares). The blue curves represent signals generated by strictly monochromatic beams. The bandwidths are = 1% in (a), 3% in (b) and 6% in (c). 
The results can be understood using simple handwaving arguments. In fact, let us consider a diatomic molecule of equilibrium length r_{0} and replace the above distribution of wavevectors by two infinitely sharp lines placed at q_{M} and q_{M}  1/; here 1/ mimics the width of the real q' vector distribution. The resulting signals are sin(q_{M}r_{0}sin)/(q_{M}r_{0}sin) and sin[(q_{M}  1/)r_{0}sin]/[(q_{M}  1/)r_{0}sin]. They vanish at sin = n/q_{M}r_{0} and sin = n/(q_{M}  1/)r_{0}, where n = 1, 2, 3.... It results that the signal corresponding to 0 (the `pink' signal) precedes the signal where = (the ideal monochromatic signal), and that this separation increases with increasing n. This is the essence of the results described above. Note, however, that the above conclusions must be refined if other perturbation mechanisms are active. For example, the signaltonoise ratio of the measured signal may be affected by the asymmetric shape of the incident Xray radiation. Unfortunately, studying these effects requires detailed knowledge of statistical properties of such perturbing mechanisms. This knowledge is rarely available at the present time.
The effects of pink Xrays have until now been studied theoretically. However, they can also be explored experimentally, symmetrizing the initially asymmetric undulator spectrum. A symmetric spectrum approaches the Dirac pulse (q  q_{M}), although it still remains pink. This result was obtained using specially constructed multilayer monochromators (Guerin et al., 2012). A longlasting effort was made at ESRF to realise satisfactory achievements. A firstgeneration multilayer monochromator was installed in the optics hutch in 2004. A doublecrystal multilayer stage was attached to the cryogenic monochomator and was cooled by liquid nitrogen. This multilayer stage never worked well due to lowtemperature stress from the Cu absorbers and vibrations in the cooling pipes that excited the second crystal. With the installation of the heatload chopper in 2009, a modified multilayer stage, the infocus multilayer monochromator, was installed 1.0 m from the sample. This stage is watercooled. The performance is clearly superior due to the 1000 times lower heat load and the shorter distance to the sample. The first substrate is cooled by springloaded Cu absorbers. Both substrates have two coating stripes that can be moved into the beam. The first coating is a ruthenium coating with composition [Ru/B_{4}C]_{51}, giving a 3.2% bandwidth, and an iridium [Ir/Al_{2}O_{3}]_{100} coating, generating a 1.6% bandwidth. The diffraction is dominated by the highZ metals and the interlayer distances are 39.20 Å for Ru and 25.66 Å for Ir. The usable Bragg angles are 0.450.90° which gives 1020 keV for Ru and 1630 keV for Ir. Symmetric Xray pulses obtained in this way are illustrated in Fig. 3, whereas onepulse multilayer fluxes are shown in Table 1 which permits these fluxes to be compared with traditional silicium generated fluxes.

 Figure 3 The blue curve shows the spectrum of the singleharmonic undulator measured with a Si monochromator. The red curve shows the spectrum monochromated by an Ir multilayer with a 1.6% bandwidth. In this figure the undulator beam intensity is expressed in terms of the energy E and not in terms of the variable q'. 
This part of the present study refers to the rresolved signals and is noticeably more complicated than that of the qresolved signals. This problem has also been explored by other authors; see, for example, Lee et al. (2013). The following introductory remarks seem appropriate. Normally, the qdependent signals S(q) are measured experimentally using nearly monochromatic Xrays; structural information about the system can then be extracted by Fourier sine transforming them. The atomatom distribution functions g_{ij}(r) of the system can be obtained in this way. The purpose of the present study is different: it consists of investigating the effect of small changes of (supposedly known) atomatom distribution functions g_{ij}(r) on the scattered intensity. These changes may have a physical origin (e.g. thermal expansion of the system); they may also be introduced for mathematical convenience, for example the Warren correction (Warren, 1990). The basic equations for Xray scattering from a liquid can then be written
As earlier, S(q) is the intensity of the scattered radiation expressed in electronic units, and i(q) is its reduced form from which the single atom contribution was subtracted (Warren, 1990). Moreover, f_{i} is the scattering factor of the atom i, g_{ij}(r') is the distribution function of the atoms i and j in a liquid solution and V is the volume of the liquid sample. Equation (3) by no means introduces a basic novelty into the theory of Xray diffraction.
In the following we shall focus our attention on a diluted solution of diatomic molecules in an inert solvent. In fact, procedures exist to eliminate the solvent part of the experimentally measured signals (Cammarata et al., 2006). What remains is a weighted sum of terms associated with the atom pairs AA, AB, AC, etc., where A denotes the solute atoms and B, C,... stand for solvent atoms. As these contributions usually peak in separate regions of the rspace, the signal of the AA pair can often be isolated. If this is the case, studying atoms A alone may suffice.
As the determination of g_{ij}(r) invariably requires the Warren correction (Warren, 1990), the errors introduced by it are studied prior to the `pink' effect. The main characteristics of this correction are as follows. The starting point is the observation that i(q) approaches zero with increasing value of q due to the decrease in the values of sin(qr')/qr' with increasing q. At a maximum value in q, called in the following q_{m}, the quantity i(q) becomes too small to be measured, and one can say that the intensity curve has converged. However, according to the basic theory of Xray radiation (Warren, 1990), passing from q to rspace requires a Fourier transformation of the function qi(q) and not of the function i(q) alone; this quantity may not be negligible even for the values of q larger than q_{m}, where i(q) is no longer measurable. In addition, terminating qi(q) at the highest value of q_{m} retained in this calculation produces false satellites. The Warren correction was invented (Warren, 1990) to overcome these difficulties and is of current use. It consists of introducing a convergence factor exp(^{2}q^{2}) and shifting the integration limit in q from to a limiting wavevector q_{m} in the Fourier transformation; this last step may be accomplished employing the step function (x) equal to 1 if x < 0 and 0 if x > 0. Then introducing the usual normalization factor M(q) = , one can write
This equation will now be simplified by neglecting the qdependence of f_{i}(q) and abbreviating F_{i} = . This simplification is introduced for reducing the following presentation to the essential physical dependence. In fact, we also performed calculations where this qdependence of f_{i}(q) was taken into account, but the resulting signals remain unchanged up to the precision of our calculations. Then, as sin(qr)sin(qr') = (1/2)cos[q(r  r')] if (r + r')q 1, there results
The error function entering in the above equation depends on a complex argument, I, indicating the imaginary unit. The intensity i_{W}(r,q_{m}), modified by the Warren correction, is given by the green curve in Fig. 5 whereas the blue curve in Figs. 4 and 5 indicates the `true' signal written in the form g_{AA}(r')  1 = (1/V)exp[A(r'  r_{0})^{2}] , the distance r_{0} being the equilibrium AA distance. The deformation of the signal due to the Warren correction is seen to be very important. However, the maximum of the corrected signal is not shifted too much, which means that molecular geometry remains predicted correctly within this approximation. On the contrary, great care is necessary if this procedure is applied in studying molecular dynamics in liquids. Not only the position but also the shape of the signal plays a role in these circumstances.
 Figure 4 Comparing the rresolved Xray signals generated by monochromatic and pink incident Xray beams. The blue line corresponds to a truly monochromatic beam, the red squares to a pink beam with bandwidth = 3%, and the black crosses to a pink beam with = 15%. Contrary to the qresolved signals, the rresolved signals are hardly affected by the polychromaticity of the incident Xray beams. Compare with Fig. 2. 
 Figure 5 Comparing rresolved signals generated by monochromatic and pink incident Xray beams. The blue line corresponds to a truly monochromatic beam, the green line corresponds to the Warren deformed signal in the absence of polychromatic correction, whereas the red squares include both the effect of the Warren and that of the polychromatic correction. The perturbation of the signal due to the pink radiation is very small compared with that due to the Warren correction. 
Another aspect of this problem also merits attention. As shown in equation (5), the deformed Warren signal i_{W}(r,q') appears as a convolution of with s_{W}(r  r', q'). It can thus be discussed in terms of the widely known convolution theory (Jansen, 1997). Convolution theory was initially employed in optics to describe the deformation of spectral shapes due to the finite slit width of spectrometers. It is extensively employed in treatments of images; compare with Paper 1 (Bratos & Leicknam, 2012) of this series. In convolution theory the convolution integral is generally written in the form i(x) = dx's(x  x')o(x'), and the following terminology is widely adopted: the quantity i(x) is called `intensity', s(x  x') is the `point spread' or `apparatus' function, whereas o(x') is the `object'. In spectroscopy, x, x' are frequencies, i(x) is the observed band intensity at frequency x, s(x  x') is the apparatus function and o(x') is the true band intensity, free of any deformation. In Xray diffraction, x and x' are interatomic AA distances r and r', i(x) is i_{W}(r, q'), s_{W} plays the role of the apparatus function, and o(x') = is basically the weighted sum of nondeformed atomatom distribution functions g_{ij}(r')  1. These notions should be kept in mind to compare the deconvolution problems in spectroscopy and Xray diffraction.
Once the unavoidable jump over the Warren correction has been accomplished, we can return back to one of our main objectives, i.e. to the study of the pink Xray signal S_{P}(r), or of the reduced pink signal i_{P}[r] = S[r]  _{i}f_{i}^{2}. This can be done by simply averaging the Warren signal i_{W} of equation (5) over the distribution of the limiting wavevectors q_{m}. Note that q = q'cos() and that q_{m} = q'. Then
The integrals present in (6) and (7) are all expressible in terms of elementary functions, mainly products of Gaussians and error functions of complex argument. Note that the expressions for i_{W}(r,q') and i_{P}(r,q_{M}) are both convolutions, differing from each other only in their apparatus functions s_{W}(r  r',q') and s_{p}(r  r',q_{M}).
The calculated signals i_{P}[r,q_{M}] are reproduced in Figs. 4 and 5. (i) If the cutoff wavevector q_{M} = 19.3 A^{1} and if the bandwidth of the pink beam is = 3%, the true signals and the pink signals, i(r,q_{M}) and i_{p}(r,q_{M}), coincide completely (Fig. 4). Note that current values of q_{M} attainable in practice are of the order of 10 A^{1}; this figure thus describes conditions at which the undulatoremitted Xrays can be considered to be `infinitely' sharp. (ii) The discussion is very different if the Warren expression I(q)exp(^{2}q^{2})(q_{M}  q) is employed to pass from the qresolved to the rresolved signals (Fig. 5). In this case, the Warren correction covers largely the effects of pink Xray radiation. This can be seen easily comparing the blue curve corresponding to the true signal, the green curve which is Warren corrected with = 0.173 A and q_{m} = 8 A^{1}, but where the Xray radiation still is monochromatic, and the curve indicated by red squares where the radiation is pink, but the Warren parameters remain the same as above. The final parameters are q_{M} = 8 A^{1}, = 3% and = 0.173 A. They are not arbitrary, but are inspired by those describing a dilute I_{2}/CCl_{4} solution (Plech et al., 2004). The smallness of the effect of pink radiation should not astonish. If the extension of the signal in the rspace is of the order of 1 Å, then, according to the relation rq , the extension of the signal in the qspace is of the order 3 Å^{1}. It is then not essential to realise the Fourier inversion by integrating over q up to , up to 8 Å^{1} or up to 8(1  1/) 7 Å^{1}. Using pink rather than monochromatic Xrays thus generates small effects difficult to study in the rspace.
The following conclusions may be drawn from this work. In qspace the signals generated by pink Xrays differ distinctly from their monochromatic analogues. These perturbations, although fairly small, are detectable and can be studied using multilayer monochromators; they can also be calculated theoretically. This statement no longer holds true in rspace. The Warren correction needed to pass from the qspace to the rspace introduces errors which cover the pink beam effect almost completely. Studying these effects on rresolved Xray signals i(r) thus remains difficult at the present time and requires the development of more accurate data analysis and new strategies when coming to the refinement of highresolution structural features from Xray scattering data.
Bratos, S. & Leicknam, J.Cl. (2012). Ukr. J. Phys. 57, 133139.
Cammarata, M., Lorenc, M., Kim, T. K., Lee, J. H., Kong, Q. Y., Pontecorvo, E., Lo Russo, M., Schiró, G., Cupane, A., Wulff, M. & Ihee, H. (2006). J. Chem. Phys. 124, 124504.
Guerin, L., Kong, Q., Khakhulin, D., Cammarata, M., Ihee, H. & Wulff, M. (2012). Synchrotron Radiat. News, 25, 25.
Haldrup, K., Christensen, M., Cammarata, M., Kong, Q., Wulff, M. S., Mariager, O., Bechgaard, K., Freidenhaus, R., Hemt, N. & Nielsen, M. M. (2009). Angew. Chem. Intl Ed. Engl. 48, 41804184.
Jansen, P. A. (1997). Deconvolution of Images and Spectra. San Diego: Academic Press.
Lee, J. H., Wulff, M., Bratos, S., Petersen, J., Guerin, L., Leicknam, J.Cl., Cammarata, M., Kong, Q., Kim, J., Moller, K. & Ihee, H. (2013). J. Am. Chem. Soc. 135, 32553261.
Plech, A., Wulff, M., Bratos, S., Mirloup, F., Vuilleumier, R., Schotte, F. & Anfinrud, Ph. (2004). Phys. Rev. Lett. 92, 125505.
Schotte, F., Techert, S., Anfinrud, P., Srajer, V., Moffat, R. & Wulff, M. (2002). ThirdGeneration Hard Xray Synchrotron Radiation Sources: Source properties, Optics and Experimental Techniques. New York: Wiley.
Warren, B. E. (1990). Xray Diffraction. New York: Dover.