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Characterization of Monodisperse Colloidal Particles: Comparison between SAXS and DLS Joachim Wagner, Wolfram Ha¨rtl, and Rolf Hempelmann* Universita¨ t des Saarlandes, Physikalische Chemie, 66123 Saarbru¨ cken, Germany Received August 19, 1999. In Final Form: January 31, 2000 Perfluorinated polymer colloids with low refraction index are prepared via emulsion polymerization. The characterization is done using small-angle X-ray scattering and dynamic light scattering. The independently obtained particle size distributions are in remarkable mutual agreement and give evidence for the monodispersity of our particles. The small-angle scattering signal of such monodisperse samples is influenced by the resolution of the experimental setup. We discuss a method to take such instrumental contributions into consideration and show that the divergence of the primary beam is the most important factor using a pinhole camera with flat single-crystal monochromator.
Introduction Due to the availability of monodisperse particles, the interest in colloidal suspensions acting as model systems for condensed matter has tremendously grown during the last decades. The structure and dynamics of such mesoscale systems are observable at enlarged length and time scales compared to atomic systems. Thus, phenomena like the glass transition, which can be studied at an atomistic length and time scale only using sophisticated experimental methods, are observable with a comparatively simple setup like a light scattering experiment. Prerequisite for studies of this kind is the knowledge of the size and the size distribution of the involved colloidal particles. Direct imaging methods like TEM can be employed, but scattering experiments yield statistically more evident information about the mesostructure of colloidal particles, because they give an average over all particles within the illuminated sample volume. Suitable scattering methods for the determination of mesoscale structures, i.e. the size distribution of colloidal particles, are dynamic light scattering (DLS) and small-angle X-ray scattering (SAXS). Whereas the former method uses visible laser light with a typical wavelength of several hundred nanometers, the latter one uses X-rays as probe with a typical wavelength of a tenth of a nanometer. The modulus of the scattering vector Q B is given by
Q ) |Q B| )
θ 4πn sin λ 2
(1)
whereby λ denotes the wavelength, θ the angle between the incident and the scattered beam, and n the refractive index, which equals 1 for X-rays but differs from 1 for visible light. Colloidal particles cover a length scale of several ten to several hundred nanometers. According to eq 1 we expect forward scattering at very low angles for X-rays as a probe, whereas, for light, the characteristic scattering range is in many cases not completely accessible, because the maximum Q is limited to Qmax ) 4πn/λ. Nevertheless, DLS can give information about the size distribution even in a Q-range considerably smaller than 2π/R. The aim of this contribution is the comparison of the result obtained by a dynamic method with the result of a static method. Whereas the former yields the hydrodynamic radius, the latter one is sensitive for the
topological radius. For charged particles, these radii can differ due to the counterions surrounding the colloidal core. Therefore, the hydrodynamic radius may be larger than the topological one. Perfluorocolloids as Model Systems The intensity of the scattered light depends on the difference of the refractive indices of the colloidal particles on the one hand and of the dispersion medium on the other hand. If this difference is not sufficiently small, the scattering signal of concentrated samples may be highly influenced by multiple scattering. Nevertheless, the investigation of concentrated samples is possible by changing the dispersion medium. For hard-sphere colloids, this can easily be done, because the insolubility of the colloidal particle and the refractive index are the only prerequisites for a suitable dispersion medium. Charged colloids are stabilized by the dissociation of acidic surface groups such as sulfonates or carboxylates. Hence, the dispersion medium has to support the dissociation of these surface groups in addition to the above mentioned preconditions. For standard polystyrene lattices, it is not possible to adapt the refractive index by protic dispersion media. Polymers consisting of polyfluoroalkanes exhibit refractive indices very close to that of water. A well-known example is polytetrafluorethylene described by Degiorgio at al.1 Unfortunately, the synthesis of these particles has to be performed under high pressure, and furthermore, it is not possible to vary systematically the size of the particles. Therefore we prepared a new system of colloidal particles by emulsion polymerization starting from perfluorinated acrylates and methacrylates. The refraction index of these particles with approximately 1.36 is very low and close to water. Therefore, index matching is possible with a glycerol/water mixture. The presence of fluorine atoms is also advantageous for small-angle X-ray and neutron scattering. For SAXS, the contrast is increased due to the higher electron density compared to hydrogen-containing analogues. For neutron scattering, the hydrogen content is a severe problem due to the large incoherent cross section of hydrogen. Therefore, deuterated analogues are commonly used to obtain a coherent signal. The incoherent cross section of fluorine is even smaller than the one of deuterium, and the (1) Degiorgio, V.; Piazza, R.; Bellini, T. Adv. Colloid Interface Sci. 1994, 48, 61.
10.1021/la991125o CCC: $19.00 © 2000 American Chemical Society Published on Web 03/28/2000
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Figure 1. Diameter of colloidal particles versus concentration of the corresponding monomer.
monomers are cheaper than deuterated ones. Due to the low refractive index, these particles are suited for light scattering experiments at highly concentrated systems. Here, the glass transition is an important application of our fluorinated colloids. This phenomenon can be verified by dynamic light scattering observing the typical plateaus in the time correlation function.2 A further application is the determination of the tracer diffusion by incoherent DLS in highly charged systems. This can be achieved by simply adding a small amount of tracer particles consisting of polystyrene to an index-matched dispersion of fluorinated particles.3 Preparation of Fluorinated Polymer Colloids Perfluorinated colloids were prepared via emulsion polymerization.2,3 Meanwhile, we have extended our preparation method to larger chains of the perfluorinated side group. In opposite to pentylacrylates, which can be polymerized using distilled water as dispersion medium, we had to use a 1:1 mixture of water and 2-propanol in the case of heptyl- and decylacrylates. The radical-induced polymerization was started by thermolysis of peroxodisulfate. In addition, we used an Fe2+/Fe3+/HSO3- redox catalyst to enhance this thermolysis at low reaction temperatures of 50 °C, because higher reaction temperatures lead to the agglomeration of nearly all the polymer colloids. Increasing the monomer concentration beyond 25 mmol‚L-1 caused an agglomeration too. The polymerization was started after stirring the emulsion consisting of the corresponding monomer and the dispersion medium at 50 °C for 1 h in a nitrogen atmosphere by adding the solution of peroxodisulfate and the catalyst. After a reaction time of 24 h, the resulting colloidal dispersion was filtered to remove aggregates and then dialyzed against distilled water and a 1:1 mixture of water/2-propanol, respectively, for 1 week. The resulting colloidal suspension was filtered again using a membrane filter with a pore size of 0.8 µm. We prepared several charges of polymer particles varying the monomer and the monomer concentration. The monomers 1H,1H,5H-octafluoropentylacrylate and 1H,1H,7H-dodecafluoroheptyl methacrylate were obtained from Polysciences, Inc., Warrington, PA. The monomer 1H,1H,2H, 2H-perfluorodecylacrylate was purchased from ABCR, Karlsruhe, Germany. The obtained particle diameters are displayed in Figure 1. From this figure, the conclusion can be drawn that the (2) Ha¨rtl, W.; Versmold, H.; Zhang-Heider, X. J. Chem. Phys. 1995, 102, 6613. (3) Ha¨rtl, W.; Zhang-Heider, X. J. Chem. Phys. 1996, 105, 9625.
Figure 2. Photograph of completely deionized ordered colloidal suspensions. At the left-hand side, at a volume fraction of about 0.01, one sees a crystalline phase indicated by Bragg reflections. This is a Laue-diffraction experiment with visible light, whereby the wavelength of the scattered light can be detected by the human eye. At the right-hand side, at a volume fraction of about 0.20, one sees a higher concentrated sample with a glassy structure. At the top of the glassy sample, one sees some small colloidal crystals at a somewhat lower volume fraction, which is caused by sedimentation.
polymerization process is not sensitive to changes in the composition of the solvent or the chain length of the perfluorinated side chain. All the polymer colloids synthesized by emulsion polymerization are monodisperse and can be crystallized by removing excess ions using a mixed bed ion exchanger. This is demonstrated in Figure 2, which shows ordered suspensions of poly(1H,1H,7H-octafluoropentylacrylate). The opalescence phenomena in the left photograph indicate a crystalline phase, whereas at the right hand a higher concentrated sample with glasslike structure is displayed. These phase transitions are a consequence of the high surface charge of the colloids leading to a repulsive screened Coulomb potential, the so-called Yukawa potential. To characterize the electrokinetic surface charge density of the particles we have performed measurements of the electrophoretic mobilities with a Zetasizer III (Malvern Instruments, Malvern, U.K.), whose results are displayed in Table 1. The ζ potentials are obtained from the electrokinetic mobilities µe by
ζ)
2ηµe 2f1(κR)
(2)
with the viscosity of the suspension η; the function f1(κR) can be calculated with the knowledge of the inverse Debye screening length κ and the radius R of the particles.4 The results in Table 1 show that the ζ-values for the fluoroacrylates in the water/2-propanol (1:1) solvent are nearly by a factor of 2 smaller than the values obtained in pure water.5 This can be explained by the reduced acidic strength of the surface groups. For further measurements with SAXS and DLS we have chosen the pentafluoro(4) Hunter, R. J. Introduction to Modern Colloid Science; Oxford University Press: Oxford, 1985. (5) Ha¨rtl, W.; Zhang-Heider, X. J. Colloid Interface Sci. 1997, 185, 398.
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Table 1. ζ-Potentials of Fluorinated Acrylates monomer
d (nm)
ζ (mV)
ionic strength (mol/L)
poly(1H,1H,7H-octafluoropentylacrylate) poly(1H,1H,7H-dodecafluoroheptylacrylate) poly(1H,1H,2H,2H-perfluorodecylacrylate)
145 120 120
-29.1 -24.3 -24.3
5 × 10-4 5 × 10-4 5 × 10-4
acrylates with a mean diameter of 140 nm already shown in Figure 2.
c(Γc) can be transformed into the probability density of sphere radii c(R).
Dynamic Light Scattering
Small-Angle X-ray Scattering
The dynamics of noninteracting colloidal particles can be characterized as a stochastic process, the so-called Brownian motion.6,7 The Brownian motion causes intensity fluctuations of the scattered light, whose intensity correlation function can be expressed as
Small-angle X-ray scattering arises from inhomogeneities in the electron density (F) on a mesoscopic length scale, i.e., in a range between several nanometers and several hundred nanometers. In a colloidal suspension, there are only two levels of electron density, one inside and one outside of the particle, determined by the dispersion medium. The intensity of scattered photons is given by the Fourier transform of the autocorrelation function of the contrast in F:
g2(Q B , t) )
〈I(Q B , τ)I(Q B , τ + t)〉 〈I(Q B , τ)2〉
(3)
If time and ensemble average of correlation functions coincide, the system will be ergodic and the particles will perform a Gaussian diffusion process. For such systems, B, the intensity autocorrelation function relaxes from g2(Q B , ∞) ) 1. Under the precondition of a Gaus0) ) 2 to g2(Q sian diffusion process, the field autocorrelation function
B , t) ) g1(Q
〈E B (Q B , τ)E B *(Q B , τ + t)〉 〈E B (Q B , τ)〉
(4)
∫-∞∞ ∆F(rb) exp(iQB br )|2
I(Q B) ∝ |
In a dilute system without interparticular correlations, the scattering is only dependent on the size and shape of the particles. For spheres with the radius R, the scattering is proportional to the form factor P(Q, R)
can be calculated by the Siegert relation
B , t) ) g1(Q
xg2(QB , t) - 1
P(Q, R) ) (5)
Due to the diffusional process, the field autocorrelation B , t) can be expressed as function g1(Q
g1(Q B , t) ) e-Γct ) e-Dc|QB | t 2
whereby Dc denotes the collective diffusion coefficient. For dilute systems, the collective diffusion coefficient and the self-diffusion coefficient coincide. Using the StokesEinstein relation,
Dc ≈ Ds )
kBT 6πηR
(7)
the hydrodynamic radius R can be determined upon the knowledge of the temperature T and the viscosity η. B , t) has to be written as For polydisperse systems g1(Q
B , t) ) g1(Q
∫0∞ c(Γc) e-Γ t dΓ c
(8)
with c(Γc) denoting the probability density of relaxation rates caused by collective diffusion. From a mathematical point of view, eq 8 represents a Laplace transform of the spectral density of the relaxation rates. Hence, this spectral density can be obtained by inverse Laplace transform of the field autocorrelation function. Provencher suggested an algorithm (CONTIN)8,9 to treat this ill-posed problem numerically under the constraint of minimum curvature of the spectral density. Using again the StokesEinstein relation, the spectral density of relaxation rates (6) Einstein, A. (7) Einstein, A. (8) Provencher, (9) Provencher,
Ann. Phys. 1905, 17, 549. Ann. Physik 1906, 18, 371. S. W. Comput. Phys. Commun. 1982, 27, 229. S. W. Comput. Phys. Commun. 1982, 27, 553.
(9)
(
∫0R 4πr2
1 V0
)
sin(Qr) dr (Qr)
2
(10)
with V0 ) 4/3πR3 denoting the volume of such a sphere. Real colloidal suspensions consist of polydisperse particles; i.e., the radii of such spheres follow a distribution function. The resulting form factor for a polydisperse system can be expressed as
P(Q, c(R)) )
∫0∞ c(R)P(Q, R)(RR0)
6
dR
(11)
Hereby, c(R) denotes the distribution function of sphere radii. The scattering contribution of a sphere with the radius R is proportional to the square of its volume. Therefore, the integrand in eq 11 is weighted by (R/R0)6 with the mean sphere radius R0. A suitable distribution function for polymer particles is the Schulz-Flory distribution. This distribution function can be derived from the kinetics of diffusion controlled polymerization.10,11 The Schulz-Flory distribution can be expressed as
c(R) )
( )
Z+1 RZ Γ(Z + 1) R0
Z+1
(
exp -
R (Z + 1) R0
)
(12)
whereby Γ denotes the gamma function. The relative variance of this distribution function is given by
〈R2〉 1 σ2 ) -1) 2 2 Z + 1 〈R〉 〈R〉
(13)
For large Z, the Schulz-Flory distribution approaches a (10) Flory, P. J. Chem. Rev. 1946, 39, 137 (11) Zimm, B. H. J. Chem. Phys 1948, 16, 1099.
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Langmuir, Vol. 16, No. 9, 2000 4083
Gaussian distribution
c(R) )
(
)
(R - R0)2 Z+1 exp (Z + 1) 2π R2
x
0
(14)
Experimental Setup for Scattering Experiments Light scattering experiments were performed using an ALV photon correlation spectrometer from ALV Laservertriebsgesellschaft, Langen, Germany. The time correlation functions with a logarithmic time spacing were obtained with an ALV-5000E fast correlator card. The light source was an He/Ne laser with a power of 33 mW (Zeiss). The detection unit consists of a monomode cable and two photomultipliers. The outgoing signal is crosscorrelated in order to eliminate dead time effects. All measurements were performed at T ) 298.0 K. The samples were filtered through a membrane filter with a pore size of 0.8 µm. For our SAXS experiments, we used a pinhole camera with an overall length of 6.5 m. From the white spectrum emitted by a rotating anode with copper target the most intense line, Cu KR1 with a wavelength λ ) 0.154 045 nm, is selected by Bragg scattering at a flat silicon (111) monochromator. The resulting monochromatic beam is then collimated by two pinholes in a distance of 2.8 m, whose tungsten slides have an aperture of 1.5 × 1.5 mm2. A third pinhole used to minimize the parasitic scattering of the collimation pinholes is placed directly in front of the sample position. The detection is done by a positionsensitive two-dimensional multiwire detector (supplied by Fa. Siemens) with a pixel size of about 0.2 × 0.2 mm2. The sample-detector distance can be varied from 0.5 up to 2.8 m. A severe problem for scattering experiments using Cu KR1 radiation is the strong absorption in matter. Therefore, the entire instrument is evacuated to prevent beam attenuation by air. Requirements for liquid sample containers are therefore vacuum-tightness and sufficient transmission of the windows. We used a cell with windows consisting of capton foil with 30 µm thickness, whereby the distance of the windows (sample thickness) was tuned to 0.6 mm by a corresponding distance ring. The empty cell has a transmission T ) 0.98.
Figure 3. Scattering cross section of fluorinated colloids versus scattering vector Q. The solid line indicates a constrained fit to a polydisperse system of spheres without correction for instrumental resolution.
(ii) the angular resolution of the detection system; (iii) the polychromy of the used radiation.12-14 Let Iid be the intensity measured by an ideal instrument; then the intensity resulting from a divergent beam, ID, is given by the convolution
ID )
∫ Iid(Q - Q′)W(Q′) dQ′
(15)
with W(Q′) denoting the resolution function due to the remaining divergence. Taking in addition the angular resolution function of the detection system into account, we obtain the intensity ID,θ by a further convolution
ID,θ )
∫ ID(Q - Q′)V(Q′) dQ′
(16)
with the angular resolution function V(Q′). Finally, the influence of polychromatic radiation can be expressed by
ID,θ,λ )
λ
∫ ID λ0 S(λ) dλ
(17)
Instrumental Resolution
whereby S(λ) is the wavelength distribution function. In our case, the influence of the wavelength distribution can be neglected, because our monochromator allows us to separate the Cu KR1 and Cu KR1 lines quite distinctly using a single crystal of Si(111). Also the influence of the angular resolution is comparatively small with respect to the effect originating from the remaining divergence of the primary beam. To dump the primary beam, we use a lead cylinder of 3.5 mm in diameter. This beam stop produces a circular shadow, which covers approximately 15 detector pixels in the diameter. For a detection system with an angular resolution adapted to the size of the primary beam, however, it would be sufficient to have a pixel size in the range of the primary beam extension. As consequence, the angular resolution in our instrument is more than 1 order of magnitude better than the contribution originating from the divergence of the primary beam. Hence, the remaining divergence of the primary beam is the dominating effect in the resolution function of our
For an ideal small-angle diffractometer, the resolution function is a δ-function. There exist three reasons for a broadening of the resolution function of a real instrument: (i) the remaining divergence of the primary beam;
(12) Bartlett, P.; Ottewill, R. H. J. Chem. Phys. 1992, 96, 3306. (13) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Crystallogr. 1980, 13, 168. (14) Ramakrishnan, V. J. Appl. Crystallogr. 1985, 18, 42.
Experimental Procedure The SAXS experiments discussed in this contribution were carried out using a detector distance of 2.8 m. Since the intensity drops tremendously at large scattering vectors, it did not make sense to perform additional measurements at shorter detector distances. The data were radially averaged and corrected for parasitic scattering induced by the collimation system and the sample container. Finally, a correction for the detector efficiency was made using a 55Fe source as calibration standard. Using these settings, we could cover one decade in Q from 3 × 10-2 to 3 × 10-1 nm-1. The scattering signal is displayed in Figure 3. The solid line represents a constrained least-squares fit using a model of spherical particles with Schulz-Flory distribution of radii. The agreement of the fit with the experimental data is better for large Q than for small Q: the minima predicted by the theory function are sharper at small Q than the ones of the exerimental data. This indicates an influence of the experimental resolution.
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Figure 4. Schematic view of the collimation system. The remaining divergence of the primary beam is defined by the apertures and the distance of the active collimation pinholes.
pinhole camera. This divergence is determined by the geometry of the collimation system. Although we have square apertures, we discuss the situation for circular apertures in the following. Using this approximation, we can keep the concept of radial averaging. Let a1 and a2, respectively, denote the radii of the pinholes aperture, C the distance between the active pinholes, and D the distance from the second pinhole to the detection plane; then we obtain as intensity profile of the primary beam a truncated cone with the inner radius s1 and the outer radius s2. After some geometric considerations, we obtain according to Figure 4.
(
D D - a1 | C C
(18)
(
D D + a1 C C
(19)
s1 ) |a2 1 +
)
Figure 5. Scattering cross section of fluorinated colloids versus scattering vector Q (same as in Figure 3). The solid line indicates a fit to a polydisperse system of spheres with correction for instrumental resolution.
and
s2 ) a2 1 +
)
expressed in terms of Q as
Qs,i )
(
)
si 4π 2π si 1 sin arctan ≈ λ 2 D λ D
(20)
The resulting trapezoidal resolution function W(Q′) can finally be expressed as
|Q′| < Qs,1 W(Q′) ) I0
(
Qs,1 < |Q′| < Qs,2 W(Q′) ) I0 1 -
)
Q - Qs,1 Qs,2 - Qs,1
|Q′| > Qs,2 W(Q′) ) 0
(21) (22) (23)
whereby the mean intensity I0 has to be chosen in such a way that
∫0∞ W(Q′) dQ′ ) 1
(24)
The situation is inversely analogous to the shadow profile of a lunar eclipse, which exhibits an umbra surrounded by a penumbra. Figure 5 displays a fit, assuming again spherical colloidal particles whose radii follow a SchulzFlory distribution. In this least-squares fit, the convolution expressed in eq 15 was performed in order to model the experimental resolution function. The fit agrees considerably with the experimental data even at small Q vectors. The resulting particle-size distribution is displayed in Figure 6 (solid line). The agreement between the distribution function obtained by SAXS with instrumental resolution correction and dynamic light scattering (circles) is quite good. For the most probable diameter one obtains
Figure 6. Size distribution (diameter) of fluorinated colloidal particles. The solid line is the Schulz-Flory distribution obtained from SAXS (least-squares fit in Figure 5), whereas the open circles represent the result obtained by DLS (CONTIN).
140 nm with DLS and (142 ( 3) nm with SAXS. The relative polydispersity is 0.14 according to the SchulzFlory fit describing the SAXS data. An alternative data evaluation in terms of a log-normal distribution, as common for nanomaterials, gave very similar parameters for the mean radius and the polydispersity, determined by a geometrical standard deviation ln(σ) ) 0.13. Discussion For systems with narrowly distributed sizes, the conventionally used size distributions, given by the Schulz-Flory distribution in the field of polymer science and by the log-normal distribution in the field of nanoscience, respectively, are of quite similar shape. From the resulting scattering pattern it is not possible to distinguish between these distributions, if the systems are sufficiently monodisperse like crystallizing colloidal particles. Within the experimental uncertainty, the most probable radii resulting from DLS and SAXS coincide. As consequence, the hydrodynamic radius equals the topological one. Hence, at the given surface charge and size, there is
Monodisperse Colloidal Particles
no rigidly attached layer around the colloidal particles slowing down the dynamics. Therefore, the fluctuations in the shell of counterions (consisting of protons and not visible using X-ray scattering) have to be significantly faster than the Brownian motion, which can be characterized by Dc ) 3.45 × 10-12 m2 s-1. The characteristic time scale for diffusion of the colloidal particle, τ ) 4R2/ (6Dc), is in the range of 10 ms. Investigations of the effective dipole moment of charged colloids, which is induced by an high-frequency electric field, give evidence for a drop of the dipole moment at intermediate frequencies, centered at ∼50 000 s-1. This drop is caused by diffusion processes within the ionic cloud of the colloidal particles.15,16 Hence, the time scale for fluctuations in the ionic cloud of the particles is more than 1 order of magnitude shorter than the one for the Brownian motion, which is in accordance with the agreement of hydrodynamic and topological size of the colloidal particles. The relative polydispersity of our perfluorocolloids is with 0.14 somewhat higher than the polydispersity of polystyrene particles of similar size, which is usually 0.050.10.12 Nevertheless, our fluorinated systems order to liquidlike structures and even crystals. Meanwhile, we also have prepared charged SiO2 particles, whose relative polydispersity is again by a factor 2-3 smaller than those of the here described fluorinated particles.17 Although the refractive index of SiO2 is significantly higher, these particles can be index-matched using water/glycerol. The resulting viscosity of the dispersion medium is about 1 order of magnitude higher than (15) Schurr, J. M. J. Phys. Chem. 1964, 68, 2407. (16) Kusner, R. E. Ph.D. Thesis, Case Western Reserve University, Engl. 1993. (17) Beck, C.; Ha¨rtl, W.; Hempelmann, R. Angew. Chem., Int. Ed. Engl. 1999, 38, 1297.
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the one of index matched fluorocolloids. Therefore, the time scale for slow processes is correspondingly enlarged, which is a problem for DLS experiments. A further disadvantage is the higher density of SiO2. Therefore, sedimentation is for large SiO2 particles a problem, whereas, for polymer colloids, this problem can be reduced by the use of D2O. The resolution function of a small-angle camera is different depending on the type of instrument and radiation source. In the case of neutrons as probe, the most important contribution to the resolution function is the spread of the wavelengths. Due to the comparatively small number of neutrons available even at high-flux reactors, the wavelength selection is done by a velocity selector. The resulting wavelength distribution is quite broad with a fwhm in the range of 10%; this leads like polydispersity to a smeared scattering pattern. Even using conventional laboratory equipment, X-rays are available with intensities that are some orders of magnitude higher than those available with neutrons. Besides the white Bremsstrahlung spectrum, there are additional characteristic lines of high intensity originating from the target material. Therefore, a wavelength selection with considerably smaller bandwidth is possible with acceptable flux. In contrast to synchrotron radiation whose emittance is limitated to a small cone, conventional X-ray tubes or rotating anodes emit completely divergent radiation. Therefore, the collimation is for synchrotron radiation only a one-dimensional problem. Additionally, the intensities are again some orders of magnitude higher than those of a laboratory X-ray source, and therefore, the collimation system can be designed using smaller pinholes. LA991125O
