Transmission of X-rays From an Extended X-ray Source Through Parallelbore Glass Capillary Waveguides: Implications for the Design of a Laboratory X-ray Microprobe NORMAN R. CHARNLEYa AND PHILIP J. POTTS*† b aDepartment of Earth Sciences, University of Oxford, Parks Road, Oxford, UK OX1 3PR bDepartment of Earth Sciences, T he Open University, Walton Hall, Milton Keynes, UKMK7 6AA Results are presented from a ray tracing program, used to to-capillary distances are varied. An interpretation of these model the transmission eciency of parallel-bore glass transmission characteristics was given in terms of the finite capillaries configured as X-ray waveguides and coupled to number of reflections supported by the capillary.There are X-ray tube sources of finite size. A detailed explanation of the clearly some limitations in models based on idealised point transmission mechanisms is given in terms of regions of the sources and the purpose of the present paper is to extend these source that contribute to the transmission of rays.Results modelling studies to optimise parameters for the transmission (Fig. 7) predict that the highest flux is observed when the of rays through parallel-bore capillaries from X-ray sources of capillary is in contact with the source. As the capillary is finite size, representing, for example, the anode of a convenmoved away from the source, transmitted intensities first fall tional X-ray tube. Synchrotron beam lines, which have near o, and then rise again as a larger area of the source is able to parallel beams of low divergence, are not considered here.contribute rays for transmission. Intensities continue to rise Since our original paper was written, several other groups with increase in the source-to-capillary distance until the point have published work concerning the transmission of X-rays is reached where the entire area of the source can contribute to through glass capillaries, some using ray-tracing modelling the transmission characteristics.Transmitted intensities then procedures. Vincze et al.4 described the results from a computer begin to fall o rapidly. The results indicate that transmission ray-tracing code in which the shape of capillary reflection target tubes may have some favourable characteristics in this surfaces was defined numerically, allowing considerable flexiapplication and that the transmission eciency of bility in modelling a range of dierent types of capillary.The polycapillary (Kumakhov) lenses should not be seriously ray tracing code used random numbers to plot the trajectory compromised, provided that the source size, in relation to of rays and could be applied to both spiralling rays (that is, source-to-lens distance, is optimised. rays with trajectories outside the plane of a longitudinal section of the capillary) and to extended sources. Results in comparison Keywords: Ray tracing; X-ray microprobe; glass capillary; with experimental measurements on the transmission charac- waveguide; optimisation; microfluorescence teristics of parallel- and tapered-bore capillaries showed excellent fit for point sources but dierences became significant There is considerable topical interest in the use of glass when applied to extended sources.The code was used to model capillaries as ‘waveguides’ for X-ray fluorescence microfocusing the eect of surface roughness, to optimise the dimensions of devices.The flux of X-rays from a suitable source transmitted conical capillaries (for source-to-capillary distances of 5 cm through the capillary is enhanced by the total external reflec- and 19 m) and to compare ellipsoidal with conical capillaries. tion of rays from the internal walls of the glass capillary. The Cargill et al.5 described experiments with straight and tapered beam emerging from the exit end of the capillary may then be glass capillaries for 5–25 keV X-rays.Of specific relevance to used as an X-ray microprobe to undertake XRF microanalysis. the present work, they described the results from a ray-tracing Interest has arisen in the performance of parallel-bore capillar- program based on a Monte Carlo algorithm to investigate the ies, tapered-bore (conical) capillaries, capillaries with elliptical eects of source-to-capillary distance and source size on the internal bore, microchannel arrays and polycapillary ‘concen- transmitted X-ray flux from straight and narrow capillaries.trators’ (the Kumakhov lens). The capability of these devices Some of their simulations match work presented in the present in comparison with other grazing incidence focusing devices paper, but they caution that their results must be viewed has been summarised recently by Bilderback and Thiel1 and qualitatively and cautiously since their two-dimensional simu- Janssens et al.2 Sources to which these devices have been lations were rigorously corrected only for on-axis sources.coupled include synchrotron X-ray beam lines and more Dozier et al.6 used a set of ray tracing codes based on Monte conventional X-ray tubes and rotating anode instruments. Carlo simulations and discrete ray input patterns to model the In the original paper by the present authors,3 a ray tracing behaviour of glass capillaries coupled to point, line and disc modelling procedure was used to investigate parameters sources.Some of their conclusions do not agree with the results important in optimising the configuration of glass capillaries presented in this work, although this may be because their coupled to X-ray sources. This original paper described the work covered a range of X-ray energies (1–17.5 keV) whereas transmission characteristics of both parallel- and tapered-bore our work is restricted to modelling at 8 keV. Brewe et al.7 capillaries coupled to a theoretical point X-ray source, with described a technique for the fabrication of long glass capillar- particular interest in the transmission behaviour when source- ies, measured the transmission eciency of selected capillaries and compared results with data calculated using a simulation program.However, these data are mainly relevant to relatively † Currently on study leave at the Department of Geology, The Australian National University, Canberra, ACT 0200, Australia. long capillaries used on synchrotron beam lines, a source Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12 (761–767) 761application not considered in the present work. Carpenter purposes of understanding the transmission behaviour of capillaries in the present work, reflection losses have been ignored et al.8 described a high-resolution X-ray microfluorescence imaging instrument and presented some ray tracing results on in initial calculations. Subsequent calculations showed that reflection losses reduced transmitted intensities (as expected), the improvement in transmission eciency when the capillary is moved closer to the focal spot of an X-ray tube.but did not aect the qualitative understanding of transmission characteristics presented here. This conclusion may have been Other recent work demonstrating the wide interest in this technology includes that of Rath et al.,9 who described an influenced by the fact that modelling was undertaken for relatively short, parallel-bore glass capillaries.If applied to automated test system for measuring X-ray transmission through glass polycapillaries, Yiming et al.,10 who described tapered-bore capillaries or very long parallel capillaries that support a large number of reflections, it would be appropriate further work with polycapillary lenses and Attaelmanan et al.,11 who described an ellipsoidal capillary optics instrument that to re-examine the applicability of this simplified model.A further assumption made here is that the flux of rays per unit oered better flexibility in how close the sample must be placed to the capillary exit to retain satisfactory resolution. area per unit solid angle from the source is constant. Since only rays with a small angle of divergence of ±Hc with respect The present paper, therefore, extends the earlier modelling studies of Charnley et al.3 to describe the intensity transmitted to the capillary axis are capable of being transmitted by the capillary, this assumption is considered to be reasonable. through parallel-bore capillaries from X-ray sources of finite size rather than just considering the behaviour of point sources.Finally, all modelling calculations have been undertaken in two dimensions. As in the work of Cargill et al.5 and several These calculations are more complex than those for a simple point source and must take into account the transmission other previous workers, the contribution of rays that spiral down the capillary has not been accounted for in this work.characteristics of o-axis as well as on-axis regions of the source. The overall aim here is to give a complete understand- In fact, this simplification probably represents an important limitation of the model, certainly as far as estimating the total ing of these transmission characteristics and to model the behaviour of transmitted rays suciently well to optimise transmission intensity is concerned.However, the interpretation of results in this work has been restricted to an evaluation design parameters for the construction of an X-ray microprobe based on an X-ray tube source. of relative intensities, and wherever possible, results of the ray-tracing calculations have been supported by qualitative interpretations based on the total-reflection behaviour of MODELLING STUDIES X-rays. In order to simplify calculations, modelling studies have been divided into separate parts.First, the behaviour of a point Change in Transmission Intensity From a Point Source as it is source has been modelled, as it is moved o the axis of the Moved O-axis capillary to a cut-o position where rays from the source can no longer be transmitted through the capillary. Second, a The aim of this first set of calculations is to identify the transmission characteristics of X-rays from a point source circular source of finite size (diameter) has been divided into annuli and the transmission contribution from each annulus starting from a position on the axis of a parallel-bore glass capillary and then moving o-axis in an orthogonal direction modelled as a function of source-to-capillary distance.Finally, data are derived for the complete transmission characteristics in 1 mm steps. These data are required subsequently to allow integration of the transmission intensity from the entire area of a source of finite size, again as a function of source-tocapillary distance.of an extended source. Calculations were undertaken as follows. Take a point source oset from the axis of a parallel-bore As in the earlier paper, it has been assumed that total reflection of an X-ray will occur from a glass–air boundary if capillary (of radius Ri) by distance d and calculate the angle representing: (i) the upper angle Hu, defined as the angle of a the angle of incidence of the ray is less than, or equal to, the critical angle (Hc).For this work, the critical angle has been ray projected from the point source subtended at the upper lip of the capillary, (ii) the lower angle Hl, measured as the taken to have a value of 0.005 rad, representative of X-ray photons of energy 7.6 keV. Rays interacting with the glass corresponding angle that is subtended at the lower lip of the capillary (see Fig. 1). If Hu or Hl are greater than the critical surface at a greater angle are assumed to be absorbed or scattered, but not transmitted by total external reflection. The angle (Hc), their values are reassigned the value of Hc since it is assumed that rays striking the wall of the capillary at an maximum intensity of rays that can, therefore, be transmitted by a parallel-bore capillary is represented by a cone of solid angle greater than Hc cannot be transmitted by total external reflection [Fig. 1(a)]. If the oset of the point source from the angle 2Hc aligned symmetrically on the axis of the capillary.In fact, this model of a sharp cut-o angle for total reflection axis of the capillary exceeds the radius of the capillary, transmission by reflection o the upper surface of the capillary is a simplification of the observation that the total reflection cut-o occurs progressively over a small range of angles. When bore is not possible, by straightforward consideration of the geometry [Fig. 1(b)]. In these circumstances Hu (which will considering the reflection of a monochromatic beam of X-rays near the critical angle, it is necessary, therefore, to consider a then have a negative value) represents a ray that just grazes the upper lip of the capillary and is reflected o the lower range of cut-o angles about the mean value with transmitted intensities for a specific angle being weighted by probability of surface and Hl equals the critical angle [Fig. 1(b)]. Clearly, as the oset of the source is increased further, a point is reached transmission based on experimental measurements.This more complicated approach was not thought to be justified in this where Hu=Hl=Hc. Beyond this cut-o point, the divergence of all rays entering the capillary orifice will be too great for work because the simple model was considered to be adequate for understanding the transmission mechanism using extended any reflection to occur and transmission through the capillary is extinguished. In all cases, the model calculated the sum of X-ray sources.However, the net eect will be to blur the edges of any ray tracing results that apparently give rise to a sharp Hu and Hl, that is, the angular range within which reflection can occur [note that when d>Ri, Hu has a negative value as cut-o boundary using the simplified model. Several estimates have been made of reflection losses during noted above, Fig. 1(b)]. The square of this angular range is then calculated to represent the solid angle of the cone of total external reflection; Stern et al.,12 for example, reported its magnitude as 6% per reflection.Reflection losses can, X-rays transmitted through the capillary, reflection losses being ignored. The results of calculations for a capillary of radius therefore, reduce significantly the intensity of rays transmitted through a capillary by multiple reflection. However, for the 50 mm, 50 mm long, for source-to-capillary distances varying 762 Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12oset ratio representing 0 Ri source on-axis 0.51 Ri source at the centre of an annulus of radius 0.50 Ri to 0.52 Ri 0.99 Ri source at the centre of an annulus of radius 0.98 Ri to 1.00 Ri 1.00 Ri source at the centre of an annulus of radius 0.99 Ri to 1.01 Ri 1.01 Ri source at the centre of an annulus of radius 1.00 Ri to 1.02 Ri 1.51 Ri source at the centre of an annulus of radius 1.50 Ri to 1.52 Ri 2.01 Ri source at the centre of an annulus of radius 2.00 Ri to 2.02 Ri Interpretation of data in Fig. 2 is best undertaken with the Fig. 1 Schematic diagram of the transmission characteristics of a help of Fig. 3. Considering first a point source on-axis as the point source placed o-axis with respect to a parallel-bore capillary. source-to-capillary distance (s) is increased from zero up to the The largest cone shows the full solid angle (±Hc) that could be critical distance (as defined below) [Fig. 3(a)], a constant flux transmitted by total external reflection.(a) represents the source oset at which only part of the upper limb of the cone (having a maximum of X-rays will be transmitted by the capillary (a to b in Fig. 2), angle of divergence of Hu) falls within the capillary orifice. (b) represents because the entire ±Hc cone available for transmission falls a greater source oset at which the upper surface of the capillary within the capillary orifice. The critical distance represents the cannot contribute to transmission and the value of Hu is now negative.source-to-capillary distance at which rays from the source just In this and subsequent diagrams, the full ±Hc cone of rays that could subtend an angle Hc at the entrance lips of the capillary and be transmitted by the capillary is denoted by a single arc, whereas the corresponds to point (b) in Fig. 2. Beyond this distance, e.g., cone that is transmitted (taking into account geometric considerations) is denoted by a solid arc.All dimensions and angles are exaggerated position (c) in Fig. 3(a), a progressively smaller fraction of the for clarity. ±Hc cone strikes the capillary entrance leading to a continuing reduction in transmitted intensities, as represented by line b to c in Fig. 2. For a point source oset by a distance of less than the capillary radius [Fig. 3(b)], the trend is slightly dierent as shown by data for an oset of 0.51Ri in Fig. 2. For small source-to-capillary distances, the full ±Hc cone strikes the inner walls of the capillary and is available for transmission (a to d in Fig. 2). At slightly larger distances [between d and e in Figs. 2 and 3(b)], beyond the first critical distance for this configuration [represented by position d in Fig. 3(b)], only part of the upper Hc limb strikes the capillary and total transmission is reduced compared with the on-axis source (i.e., d to e in Fig. 2). At the second critical distance [e in Figs. 2 and 3(b)], the lower cone just clips the lower lip of the capillary. Fig. 2 Computed transmission characteristics of a point source oset from the axis of a parallel-bore capillary by specified amounts for source-to-capillary distances of 0–60 mm. Data are plotted for point source osets of 0Ri, 0.51Ri, 0.99Ri, 1.00Ri, 1.01Ri, 1.51Ri and 2.01Ri, corresponding, for a 50 mm radius capillary, to osets of 0, 25.5, 49.5, 50.0, 50.5, 75.5 and 100.5 mm. Letters denote configurations shown in Fig. 3. between 0 and 60 mm are shown in Fig. 2. To allow this diagram to be used generally for any capillary diameter and source oset distance, graphical data are presented for the oset distance from the capillary axis ratioed to the capillary radius (Ri). Thus, for the capillary modelled here, data for an oset ratio of 0.51 represent the behaviour of a source oset by a distance of 0.51×50=25.5 mm from the axis of the 50 mm radius parallel-bore capillary. The values of the osets plotted in Fig. 2 have been chosen as examples representing the centre of 1 mm width annuli on the surface of an extended source as Fig. 3 Schematic diagram showing the cone of rays transmitted from required in the next set of calculations. Data for an oset ratio a point source through a parallel-bore capillary, when the point source of 0.51 represent the centre of an annulus bounded by an inner is moved away from the capillary orifice. Schematic data are shown radius 0.50×50=25 mm and an outer radius of 0.52×50= for (a) an on-axis point source, (b) a point source oset by about 0.51Ri and (c) a point source oset by 1.51Ri. 26 mm. The other data plotted in Fig. 2 are as follows: Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12 763At larger source-to-capillary distances, a cone of rays equival- To calculate the contribution of an individual annulus of an extended source to the intensity transmitted by a capillary, the ent in size to that from an on-axis source is available for transmission so that the transmission curve rejoins the original area of that annulus has been multiplied by the intensity transmitted from a point source oset from the capillary axis trend (e to c in Fig. 2). In this context, the first critical distance represents the source-to-capillary distance for a point source by a distance corresponding to the mean radius of the annulus in question. This calculation is shown schematically in Fig. 4. placed o-axis at which a ray emanating from the source subtends an angle of Hc at the upper lip of the capillary and The results of such calculations are shown in Fig. 5 for extended sources placed at distances of 0, 5, 10, 15, 20 and the second critical distance corresponds to the situation where the ray subtends an angle of Hc at the lower lip of the capillary 30 mm from a capillary of radius 50 mm and length 50 mm. Transmitted intensity data are plotted in Fig. 5 for successive (or vice versa).The case of a point source oset by a distance greater than annuli, 1 mm wide, the distance of the mean radius of the annulus from the capillary axis being plotted on the horizontal the radius of the capillary is illustrated by data for an oset of 1.51Ri in Fig. 2, the configuration of which is shown scale. This diagram is, in fact, generally applicable to any capillary of radius Ri by using the alternative scale for source diagrammatically in Fig. 3(c). No transmission at all is possible until the source is moved further from the capillary than the oset calibrated in units of Ri. For these calculations, it has been assumed that the source is ‘infinitely’ large in area so first critical distance [f in Figs. 2 and 3(c)]. At this point, the lower limb of the cone just grazes the upper lip of the capillary. that its size does not restrict transmitted intensities. The vertical scale is the relative transmitted intensity calculated as (planar As the point source is withdrawn further from the capillary, an increasing fraction of the lower cone becomes available for angle of transmittable rays)2×(area of annulus/p).Interpretation of these graphical data depends in part on transmission up to the second critical distance [g in Figs. 2 and 3(c)]. Beyond this distance, all rays striking the capillary the transmission characteristics of the capillary, considered above. The region in front of the capillary can be divided into orifice can be transmitted and the trend then follows that for an on-axis point source (g to c in Fig. 2). three regions, as shown in Fig. 6. Region 1 is represented by a cone projected from a point source placed on-axis at the When the source oset equals the radius of the capillary (data for 1.0Ri in Fig. 2), the fraction of rays transmitted by critical distance from the capillary (where the extremities of this cone subtend an angle equal to the critical angle at the the capillary remains constant as the source is moved further away from the capillary as only half the cone available for lip of the capillary).All rays projected towards the capillary orifice from any area of an extended source that fall within transmission strikes the capillary orifice (h to j in Fig. 2), until the source has been withdrawn to the point where the lower Region 1 can be transmitted by the capillary. The second lobe just grazes the lower limb of the capillary. Beyond this point, all rays striking the capillary orifice are capable of being transmitted and the transmitted intensity then follows the main trend for an on-axis source (j to c in Fig. 2). One way in which additional confidence can be given to the reliability of the computer calculations is to examine data for a source oset of 0.51Ri in Fig. 2 and calculate by simple geometry the source-to-capillary distance represented by points d and e in Fig. 2 [that is, the first (s1) and second (s2) critical distances].Simple geometry shows that s1=(Ri-d)/ tan(Hc) and s2=(Ri+d)/ tan(Hc), where d is the oset (=0.51Ri), Ri Fig. 4 Schematic diagram showing the surface of an extended source the radius of the capillary (=50 mm) and Hc the critical angle divided into annuli (of width 1 mm) as the basis of the model used to (=0.005 rad). Substituting these values, s1=4.9 mm (point d) compute the contribution made by individual annuli to the trans- and s2=15.1 mm (point e), distances that agree with the mission of rays through the capillary.For clarity, the face of the source computed data plotted in Fig. 2. has been turned through 90° towards the viewer and distances between successive annuli have been exaggerated. Profile of Transmitted Intensities From an Extended Source Having calculated the transmitted intensities derived from point sources, oset at various distances from the axis of a capillary, it is now possible to calculate the contribution made to the total transmitted intensity from dierent annuli of an extended source.The model used here is to divide the surface of an extended source into annuli of width 1 mm. It can be shown by simple geometry that the area of successive annuli follows a simple progression, such that if the annuli are numbered from the centre outwards, the corresponding areas are as listed in Table 1. Table 1 Area of successive annuli, each of which increases in radius by 1mm Annulus Area* Fig. 5 Computed transmission of rays from extended sources placed 1 1.pa2 2 3.pa2 at dierent distances from the entrance orifice of a parallel-bore capillary. The horizontal scale represents the distance of the mean 3 5.pa2 4 7.pa2 radius of an annulus from the axis of a 50 mm radius capillary.This diagram may be applied to a capillary of any size by substituting the 5 9.pa2 alternative horizontal axis scale labelled in units of Ri. Data are plotted for sources placed at 0, 5, 10, 15, 20 and 30 mm distances from *pa2 is the area of the first annulus represented by a circle of radius 1 mm.the capillary. 764 Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12radius 2Ri (100 mm) lie within Region 2 (partial transmission). There can be no transmission from areas beyond 2Ri. Extended sources placed at greater distances from the capillary follow a similar trend except that transmission occurs from an increasingly large area of the source which falls into Region 2.The transmission intensity is then a compromise between the larger area of the source that can contribute rays for transmission through the capillary and the smaller solid angle that is then subtended at the capillary orifice. An interesting rule-of-thumb taken from Fig. 5 is that the annulus making the maximum contribution has a radius of about one third of the maximum that can contribute to the transmission of rays through the capillary.Fig. 6 Schematic diagram dierentiating the transmission characteristics of regions in front of a parallel-bore capillary. The full (±Hc) cone of rays emitted from any part of a source that falls within Region Total Transmission Intensities From Extended Sources at 1 (which also extends inside the capillary) can be transmitted through Various Source-to-capillary Distances the capillary, whereas only a reduced fraction of this cone can be transmitted from regions of the source that fall into Region 2.Rays Following the general procedure used to calculate data in from areas of the source that fall outside Regions 1 and 2 cannot be Fig. 5, it is now a relatively simple task to calculate the total transmitted through the capillary by total external reflection. transmission from an extended source of finite size as the source-to-capillary distance is changed. In principle, the total region (Region 2) corresponds to the truncated cone which transmitted intensity (for a specified distance) is the area under diverges out from the lips of the capillary at an angle of the appropriate curve plotted in Fig. 5. Total transmission divergence equal to the critical angle (but excluding Region 1). data for a glass capillary of radius (Ri) 50mm and length Only a fraction of rays emanating from any area of an extended 50 mm have been calculated for sources of 25, 50, 100, 150, source that falls within Region 2 can be transmitted by the 250, 500 and 1000 mm radius using source-to-capillary distances glass capillary.This fraction varies and can be calculated from of 0–200 mm. These data are plotted in Fig. 7. Again, this the solid angle of rays striking the capillary orifice, divided by diagram can be applied generally to any source/capillary the full 2Hc solid angle. The third region corresponds to all combination where source diameter is 0.5Ri, 1Ri, 2Ri, 3Ri, areas outside Region 2. None of the rays emanating from an 5Ri, 10Ri and 20Ri.extended source that lies in this third region can be transmitted The form of the curves in Fig. 7 can be explained in by the capillary because their angle of divergence with respect conjunction with diagrams shown in Fig. 8 as follows: to the capillary wall exceeds the critical angle. As can be For a source of radius 0.5Ri, with reference to Fig. 8(a), all seen from Fig. 6, at small source-to-capillary distances, only a rays projected towards the capillary with a divergence of ±Hc relatively small area of the source can be ‘seen’ by the capillary, can be transmitted at source-to-capillary distances of 0–5 mm and at zero source-to-capillary distance this area reaches a [corresponding to Fig. 8(a), points a and b], hence the constant minimum, equal in size to the capillary orifice. As the source- transmitted intensities in Fig. 7 out to 5 mm (also labelled a to-capillary distance is increased, two opposing factors aect and b).At distances of 5–10 mm, an increasing proportion of transmission intensities. First, a larger and larger area of the the source area falls within Region 2 (Fig. 6), and all of it at source can contribute to transmission. However, second, distances greater than 10 mm, so that transmitted intensities beyond the critical distance, only a decreasing fraction of the are then reduced in accord with the reduction in the solid possible maximum 2Hc cone that could be transmitted by the angle of the cone subtended at the capillary orifice, which capillary will actually fall within the capillary orifice.follows the inverse square law. With these transmission characteristics in mind, the form of For a source of radius 1Ri, calculations according to the data plotted in Fig. 5 (the transmitted intensity from successive model show that the maximum intensity is transmitted at a annuli each having a width of 1 mm) can now be explained. source-to-capillary distance of 0 mm [source touching the For an extended source at s=0 mm (i.e., touching the capillary capillary orifice, e in Figs. 7 and 8(b)]. This transmitted orifice), all rays projected towards the capillary at an angle of divergence of Hc can be transmitted, provided that they originate from the area of diameter 2Ri that coincides with the capillary orifice. Any rays emanating from regions of the source outside this area cannot enter the capillary orifice. The shape of the curve for s=0 mm (Fig. 5) increases linearly with the increase in area of the corresponding annulus (Table 1) up to an oset of Ri, the cut-o corresponding to the radius of the capillary. Data for a source placed at s=5 mm from the capillary show that the central annuli (which all lie in Region 1) have, as expected, the same transmission characteristics as the source in contact with the capillary. However, annuli of radius 0.5Ri (25 mm) to 1.5Ri (75 mm) lie in Region 2, where only partial transmission of rays propagated towards the capillary orifice can occur and there is a fall o in transmitted intensities compared with the source at s=0 mm.The cut-o now occurs Fig. 7 Transmission intensities of sources of finite size as a function at 75 mm, the oset distance beyond which all rays have too of source-to-capillary distance. Data are plotted for sources of radius high a divergence (>Hc) for transmission. 0.5Ri, 1Ri, 2Ri, 3Ri, 5Ri, 10Ri and 20Ri, which for the 50 mm radius An extended source placed at s=10 mm from the capillary source modelled here corresponds to source diameters of 50, 100, 200, lies at the critical distance (for a point source) for this configur- 300, 500, 1000 and 2000 mm.Letters correspond to source–capillary configurations marked on Fig. 8. ation. None of the source lies in Region 1, but annuli up to a Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12 765that the distance at which the fall o commences varies with source size as follows: Source diameter Source fills transmission cone at: 100 mm (1Ri) 0mm 200 mm (2Ri) 10mm 300 mm (3Ri) 20mm 500 mm (5Ri) 40mm 1000 mm (10Ri) 90mm 2000 mm (20Ri) 190 mm The other interesting aspect of data presented in Fig. 7 is that even for the largest source investigated, computations show that the intensity transmitted never exceeds the intensity observed when the source is in contact with the capillary and this aspect is commented on below.IMPLICATIONS FOR THE DESIGN OF AN X-RAY MICROPROBE Interpretation of data in Fig. 7 gives a number of interesting results that are important when considering the design of an X-ray microprobe incorporating a parallel-bore capillary waveguide. (a) Although maximum transmission is observed when the capillary is in contact with the source, it is clearly impractical (and indeed self-defeating) to position a glass capillary in contact with the anode of a conventional X-ray tube.However, the shape of the curve in Fig. 7 indicates that optimum transmission intensities are likely to be achieved at greater, rather than shorter, distances. For example, for a 50 mm radius capillary coupled to a 250 mm (5Ri ) source, source-to-capillary distances of 40 mm will oer greater intensities with less sensitivity to position than smaller source-to-capillary distances. (b) If the above recommendation is followed, the diameter of the X-ray source should be significantly larger than the diameter of the glass capillary.However, there is little benefit in increasing the diameter much above about ten times that Fig. 8 Schematic diagram showing the source–capillary configur- of the capillary unless very large source-to-capillary distances ations for selected sources, data for which are plotted in Fig. 7. The source sizes plotted here are (a) 0.5Ri, (b) 1Ri and (c) 5Ri . The regions (>90 mm) are to be used. correspond to those delineated in Fig. 6. (c) In the design of excitation systems using a polycapillary (Kumakhov) lens, where practical considerations mean that it is impossible to position a bundle of capillaries very close to the X-ray source, data in Fig. 7 indicate that little reduction intensity is four times greater than that from the previous in transmitted intensities will be observed by the need to source (0.5Ri), a factor that is in proportion to the dierence position the lens at larger distances from the source, provided in areas. As this source is moved further away from the that the diameter of the source is suciently large.capillary, transmitted intensities again fall o for the same (d) Finally, one of the most interesting implications of data reasons as explained for the 0.5Ri source. in Fig. 7 concerns the nature of the excitation source. A Taking a source radius of 5Ri, as an example of a source conventional side window X-ray tube arrangement may not larger than 1Ri, the transmitted intensity at 0 mm [position e be the optimum for capillary waveguide excitation since the in Fig. 8(c)] is identical with that for the 1Ri source since rays maximum transmitted intensities are observed when the source propagated from regions outside a diameter of 1Ri at this touches the capillary. Rather, there may be some advantage in distance cannot contribute to the transmitted beam. As the the transmission target tube, where the capillary can be source is moved away from the capillary, the intensity falls o arranged to touch one side of a thin foil anode, the other side down to a minimum [position g in Figs. 7 and 8(c)], but this of which is excited by bombardment with electrons. fall o in intensity is not as great as that for the 1Ri source Further modelling is now being undertaken to investigate the because the larger area of the source in Region 2 (Fig. 6) can characteristics of tapered glass capillaries in this application.contribute to transmitted intensities. When the source is withdrawn further away than the critical distance, computations The initial discussions (including ‘back-of-the-envelope’ calcu- show that transmitted intensities increase, presumably because lations) with J.V.P. Long (University of Cambridge), that the larger area of the source ‘visible’ to the capillary more formed the basis of this work, are gratefully acknowledged. than compensates for the reduced solid angle from any particular region that can contribute to transmission. This increase in transmitted intensities continues up to point i [Figs. 7 and REFERENCES 8(c)], which corresponds to the distance from the capillary at 1 Bilderback, D. H., and Thiel, D. J., Rev. Sci. Instrum., 1995, which the source just fills the cone of rays that can contribute 66, 2059. to transmission. At greater distances [e.g., point j in Figs. 7 2 Janssens, K., Vincze, L., Rubio, J., Adams, F., and Bernasconi, G., and 8(c)], the source is not suciently large to fulfil its J. Anal. At. Spectrom., 1994, 9, 151. maximum transmission potential. Computations for all the 3 Charnley, N. R., Potts, P. J., and Long, J. V. P., J. Anal. At. Spectrom., 1994, 9, 1185. sources greater in size than 1Ri follow the same trend, except 766 Journal of Analytical Atomic Spectrometry, July 1997, Vol. 124 Vincze, L., Janssens, K., Adams, F., and Rindby, A., X-ray 10 Yiming, Y., Xunliang, D., Dachun, W., Baozhen, C., Shengji, Z., Spectrom., 1995, 24, 27. and Andong, L., SPIE, 1994, 2321, 56. 5 Cargill, G. S., III, Hwang, K., Lam, J. W., Wang, P.-C., Liniger, E., 11 Attaelmanan, A., Voglis, P., Rindby, A., Larsson, S., and and Noyan, I. C., SPIE, 1995, 2516, 120. Engstro�m, P., Rev. Sci. Instrum., 1995, 66, 24. 6 Dozier, C. M., Newman, D. A., Gilfrich, J. V., Freitag, R. K., and 12 Stern, E. A., Kalman, Z., Lewis, A., and Lieberman, K., Appl. Kirkland, J. P., Adv. X-ray Anal., 1994, 37, 499. Opt., 1988, 27, 5135. 7 Brewe, D. A., Heald, S. M., Barg, B., Brown, F. C., Kim, K. H., and Stern, E. A., SPIE, 1995, 2516, 197. Paper 6/06885E 8 Carpenter, D. A., Taylor, M. A., and Lawson, R. L., J. T race Received October 8, 1996 Microprobe T ech., 1995, 13, 141. Accepted March 20, 1997 9 Rath, B. K., Youngman, R., and MacDonald, C. A., Rev. Sci. Instrum., 1994, 65, 3393. Journal of Analytical Atomic Spectrometry, July 1997, Vol. 12 7