- Email: [email protected]
Monte Carlo simulations of a magnetic quadrupole triplet as a high resolution energy spectrometer
Nuclear Instruments and Methods in Physics Research B 171 (2000) 565±572
www.elsevier.nl/locate/nimb
Monte Carlo simulations of a magnetic quadrupole triplet as a high resolution energy spectrometer M.B.H. Breese
*
Department of Physics, School of Physical Sciences, University of Surrey, Guildford, GU2 5XH, UK Received 22 February 2000; received in revised form 6 July 2000
Abstract This paper gives a detailed description of the transport of MeV protons through a nuclear microprobe lens system which is acting as an energy spectrometer. The angular distribution of transmitted protons versus their energy loss is studied for dierent crystal thickness and planar tilt angles using Monte Carlo simulations. The conditions under which the transmitted proton beam is de¯ected by the magnetic quadrupole triplet of a microprobe such that ions with slightly diering energies are focused to dierent locations across the image plane are discussed. The shape of the resultant beam intensity distribution and reconstructed energy spectrum is studied as a function of location along the beam axis. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 61.80.+p; 07.78.+s; 41.85.Lc Keywords: Energy spectrometry; Magnetic quadrupole multiplets; Transmission channeling
1. Introduction Energy spectrometers resolve small dierences in charged particle energies by passing the backscattered or transmitted beam through an electrostatic or magnetic ®eld. The resultant spatial dispersion of the de¯ected ions, due to their momentum or energy spread, is then measured and interpreted as an intensity distribution of particle energies [1,2]. The typical minimum spatial resolution across the image plane of conventional
*
Tel.: +44-1483-876796; fax: +44-1483-876781. E-mail address: [email protected] (M.B.H. Breese).
spectrometers is 100 lm, which can limit the attainable energy resolution. It was recently suggested that a triplet of magnetic quadrupole lenses in a nuclear microprobe [3] can act as a high resolution energy spectrometer [4] owing to its ability to focus MeV ion beams to spot sizes of <1 lm. This design, which is shown in Fig. 1, utilises the large chromatic aberration of a high-excitation quadrupole triplet in the vertical focusing direction to give the required energy dispersion, which is of course a drawback for high resolution probe-forming applications. For use as a spectrometer, a MeV ion beam is passed through a thin crystal sample, which is located at the microprobe object aperture, at 0.0 m on the horizontal
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 0 ) 0 0 3 2 2 - 0
566
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 1. Plot of proton trajectories in the horizontal (lower half) and vertical (upper half) directions, with trajectory angles of h0 0.25 mrad and /0 0.25 mrad, respectively, diverging from the object aperture and focused to the microprobe image plane. The proton energies are 2.988, 3.000, 3.012 MeV, i.e., 3.000 MeV dE 4.0%. The outlines of the quadrupole lenses are shown to their correct horizontal scale but their vertical scale is exaggerated. The location of the normal image plane at zi 150 mm is indicated by a solid vertical line.
axis in Fig. 1. The microprobe collimator slits are translated vertically away from the beam axis, in the plane of the higher chromatic aberration, so that protons with diering energies are focused to dierent locations across the image plane. It was calculated that energy dierences of a few hundred eV could be resolved for MeV protons, which were transmitted through thin crystals [4]. This paper gives a detailed study of the trajectories of MeV protons through a magnetic quadrupole triplet acting as an energy spectrometer, in order to assess the conditions under which the beam intensity distribution in the image plane can be reconstructed to most accurately re¯ect the original energy distribution entering the microprobe. 2. Monte Carlo channeling simulations The Monte Carlo channeling code FLUX [5,6] was used to produce a suitable set of proton trajectories which are transmitted through a thin silicon crystal. Channeling oscillations of ions as they move in regions of dierent electron density
between the lattice walls of the crystal result in dierent transmitted energy losses as a function of crystal thickness and emergent angle [7±9]. Ten thousand trajectories are simulated, with the incident 3 MeV proton beam randomly distributed over the area of the silicon unit cell. All simulations were carried out with the crystal tilted 3° away from the surface-normal [1 0 0] axis in the horizontally running (0 1 1) planes. The emergent angle h in the horizontal plane, and / in the yz vertical plane, and the transmitted energy loss were recorded for each trajectory. Fig. 2 shows the simulated energy loss versus emergent angle /, for each 3 MeV proton which is transmitted through a silicon layer with an increasing thickness, with the incident beam aligned with the (1 1 0) planes. The characteristic planar oscillation wavelength, kp , for 3 MeV protons along the {1 1 0} planes in silicon is equal to 300 nm and the characteristic planar channeling angle wp 0.13° (2.2 mrad). The energy loss scales on the vertical axis diers in each plot to highlight the structure present, but the emergent angle / on the horizontal axis is the same. In Fig. 2(a) the layer thickness is approximately equal to the oscillation wavelength kp . Those protons emerging with a small angle close to zero have a low energy loss, since they have been well channeled and consequently spent most of their trajectories in regions of low electron density. As the emergent angle increases from zero towards wp , the energy loss increases linearly, giving a `wishboneÕ structure. At emergent angles slightly larger than wp the energy loss is a maximum because of the blocking eect, resulting in a larger beam fraction is pushed to either side of the channeling planes. Those protons emerging with larger angles have a large energy loss corresponding to those trajectories which were either not channeled at the surface, or were rapidly dechanneled. A layer thickness of 450 nm in Fig. 2(b) corresponds to a planar oscillation wavelength of 1.5kp . The energy loss distribution is now closed at the top part for a small emergent angle [9]. This is due to those protons which have a larger amplitude of oscillation, and consequently a shorter wavelength than well-channeled protons, so they are out of phase with the main distribution. In
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
567
Fig. 2. Energy loss versus emergent angle / of the beam channeled along the silicon (1 1 0) planes for six dierent layer thicknesses. The vertical axis shows the energy loss on dierent scales in each plot whereas the horizontal scale shows an emergent angle of 0.6° in all plots.
Figs. 2(c) and (e), the layer thickness corresponds to additional multiples of kp . Again there is an open distribution for a small emergent angle, as in Fig. 2(a), with hollow distributions below this, bounded horizontally by wp . This shows that the dierent energy loss groups of channeled protons [8] still maintain their coherency, giving rise to several discrete values of energy loss for a particular emergent angle within wp , even for layers as thick as 5kp as shown in Fig. 2(f).
Fig. 3 contains a sequence of plots showing the energy loss versus emergent angle for 3 MeV protons transmitted through a 210 nm thick silicon layer ( 34kp ), with the beam tilt angle to the (1 1 0) planes varied. With increasing tilt angle the energy loss distribution becomes asymmetric and the average energy loss gradually rises. At a tilt angle of 0.06° (wp /2) and 0.09°, there are no protons with an energy loss as low as at 0.0°. Beyond 0.09° the beam becomes rapidly dechanneled close to the
568
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 3. Energy loss versus emergent angle / of the beam for a silicon (0 1 1) layer thickness of 210 nm, as the crystal tilt angle is altered away from planar alignment. The vertical and horizontal scales are the same in all plots. The vertical axis shows the energy loss from 2.0 to 5.5 keV, and the horizontal scale shows an emergent angle of 0.4°.
surface, producing many blocked trajectories with a high energy loss. Figs. 2 and 3 show that there is considerable structure in the energy loss versus emergent angle of a transmitted MeV proton beam at, and close to planar channeling alignment. It is the aim of the spectrometer studied here to resolve such small dierences in energy loss of less than 1 keV and to reconstruct the initial energy loss distribution. The energy loss intensity distribution for all 3 MeV protons transmitted along the (1 1 0) planes of a 210 nm thick silicon layer is shown in Fig. 4(a), i.e., with no collimation of the emergent angular distribution. Fig. 4(b) shows the energy loss intensity distribution of the same transmitted beam which has been collimated in emergent angle to h0 0.05 mrad in the horizontal direction and /0 0.01 mrad in the vertical direction. Both the intensity of transmitted protons and the range of transmitted energies is reduced in Fig. 4(b) because those trajectories with
slightly diering oscillation wavelengths (and hence energies), have dierent emergent angles, as shown in Fig. 3(a). The remaining high energy loss group from 3.8 to 5.0 keV is the out-ofphase component, visible in the upper portion of Fig. 3(a). Fig. 4(b) shows the energy spectrum which is transmitted into the microprobe, and which the microprobe lens system acting as an energy spectrometer disperses across the image plane, thereby recreating the original distribution as closely as possible. The same collimator aperture sizes of h0 0.05 mrad and /0 0.01 mrad as in Fig. 4(b) are used unless otherwise stated in the spectrometer simulations in Section 3. This collimator aperture is displaced in the vertical direction by /0 0.25 mrad, as shown in Fig. 1, so that a range of angles from /0 0.24 to 0.26 mrad is transmitted into the lens system. A magnetic dipole ®eld is used to de¯ect the transmitted beam by 0.25 mrad to this aperture.
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 4. (a) Energy loss histogram for 3 MeV protons transmitted along the silicon (1 1 0) planes of a 210 nm thick layer. Based on the data in Fig. 3(a), i.e., the full emergent angular distribution. (b) Energy loss histogram of the same transmitted beam distribution but collimated in angle to h0 0.05 mrad and /0 0.01 mrad.
3. Beam transport simulations These simulated channeling trajectories are traced through a magnetic quadrupole triplet lens system using the ion-optical ray-tracing program TRAX [10], which includes all aberration terms which depend on the beam angle and displacement. In keeping with common microprobe formalism, /0 denotes the beam divergence in the vertical direction before the lenses and /i that after the lenses, etc. In Fig. 1, the distance between the exit face of the ®nal lens and the image plane is zi 150 mm, as for normal microprobe operation. However, there are two factors which alter the location of the optimum image plane when the same lens system is used as a spectrometer. Firstly, protons with different energies which pass o-axis through the lens system are focused to dierent spatial positions across the image plane. Secondly, as the collimator
569
aperture is moved further away from the beam axis in the vertical direction, the image plane where the beam is best focused moves towards the lenses, owing to spherical aberration of the quadrupoles, which over-focus particles with a large entrance angle [4]. It is thus important to locate the optimum image plane where the beam is best focused for optimum reconstruction of the original energy spectrum. This section investigates the intensity distribution and the resultant energy spectrum for diering locations of the image plane along the beam axis at which the distribution of focused protons is measured. One method of determining the optimum image plane for recording the focused beam intensity is when the phase space distribution (yi ,/i ) is most upright [9], i.e., when the spatial coordinates yi are most tightly grouped. Fig. 5 shows the phase space distribution for the transmitted beam angular distribution used in Fig. 4(b), but with dE 0 (i.e., no energy loss), with the image plane moved 22 mm towards the lenses so that zi 128 mm. The full range of convergent angles /i are focused to a similar locations yi in the vertical direction. Fig. 6(a) shows the corresponding spatial plot of (xi ,yi ) in the image plane under these conditions, showing that the beam is well-focused in the vertical direction, but displaced by 43 lm away from the beam axis. To demonstrate the degrading eect of spherical aberration on the spatial distribution across the image plane if
Fig. 5. Phase space plot (yi ,/i ) for 3 MeV protons transmitted with no energy loss, with h0 0.05 mrad and /0 0:25 0:01 mrad for an image plane at zi 128 mm.
570
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 6. Spatial plots of (xi ,yi ) for an image plane at zi 128 mm (a) with a horizontal beam divergence h0 0.05 mrad and (b) with h0 0.20 mrad.
the entrance angle into the lenses is increased, Fig. 6(b) shows the same spatial area of the image plane, but with the collimator aperture size in the horizontal direction made four times larger than in Fig. 6(a). The focused beam distribution now extends much further in the horizontal direction, and also bows in the vertical direction due to spherical cross-terms, degrading the relationship between proton energy and location across the image plane. Now consider the phase space distributions produced by the full range of transmitted energy losses. Fig. 7 show three phase space plots of (yi ,/i ) for 3 MeV protons which are transmitted with the energy loss distribution in Fig. 4(b) with the same aperture size of h0 0.05 mrad, /0 0.01 mrad. The image plane has been shifted in 1 mm increments from zi 129 mm to 127 mm from (a) to (c). The yi position scale on the horizontal axis is the same in each plot and the coordinates gradually move to the left from (a) to (c) owing to the convergent beam angle /i . The right-hand portion of the phase space distribution is due to the low energy loss component of 2.4±2.7 keV in Fig. 4(b). This portion focuses to a narrow, well-de®ned spatial location, similar to Fig. 5, owing to the narrow range of energies encompassed. In comparison, the left-hand portion of the phase space distribution, which is due
to the higher energy loss component of 3.8±5.0 keV in Fig. 4(b) is wider owing to the broader range of energies. This higher energy loss portion is closer than the lower energy loss component to the beam axis in the vertical direction since it is bent more. The individual portions of the phase space distributions gradually rotate as they move through optimum focus, but the dierent portions are not necessarily best focused at the same image plane location. The right-hand portion is most upright in Fig. 7(c) at zi 129 mm, whereas the left-hand portion is most upright in Fig. 7(b) at zi 128 mm. The corresponding histograms showing the intensity of the focused beam distribution in the vertical direction at these same locations along the beam axis are shown in Fig. 8. The yi position scale on the horizontal axis is reversed, so that the energy loss of the protons increases from left to right, allowing an accurate assessment of when the energy spectrum of transmitted protons in Fig. 4(b) is best reconstructed. Those protons with a low energy loss from 2.4 to 2.7 keV are further away from the beam axis, giving the narrow peak to the left, as in Fig. 4(b). The original energy spectrum of Fig. 4(b) is best (though not perfectly) reproduced in Fig. 8(b) at an image distance of zi 128 mm.
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 7. Phase space plots (yi ,/i ) for 3 MeV protons transmitted with an energy loss distribution given by Fig. 4(b). The image plane is shifted by 1 mm increments from zi 129 to 127 mm.
This is the same optimum image plane location as for the incident 3 MeV proton beam which did not suer any energy loss in Fig. 5. This demonstrates a simple experimental method of locating the optimum image plane for the maximum energy resolution in such a spectrometer design. The incident ion beam, which has not passed through the thin crystal sample is transmitted through this spectrometer, and the optimum location along the beam axis where this is best focused is determined. This same image plane location is then used for all subsequent spectrometer measurements of the proton energies which have passed though thin crystals.
571
Fig. 8. Corresponding energy loss histograms for the focused beam distribution in the vertical direction, for the same image planes as in Fig. 7.
4. Conclusions The energy loss distribution of a proton beam which is transmitted through a thin crystal can be measured by passing the transmitted beam o-axis through the magnetic lens system of a nuclear microprobe. The exact shape of the original distribution can be closely reconstructed if the correct image plane is chosen. An experimental procedure for locating this optimum image plane has been established. These calculations demonstrate the validity of this proposed spectrometer as a viable means of resolving small energy dierences in
572
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
transmitted ion energies, for applications such as transmission channeling. References [1] H.D. Carstanjen, Nucl. Instr. and Meth. B 136±138 (1998) 1183. [2] W. Lanford, B. Andersberg, H.A. Enge, B. Hjorvarsson, Nucl. Instr. and Meth. B 136±138 (1998) 1177. [3] M.B.H. Breese, D.N. Jamieson, P.J.C. King, Materials Analysis using a Nuclear Microprobe, Wiley, New York, 1996.
[4] M.B.H. Breese, D.N. Jamieson, Nucl. Instr. and Meth. B 155 (1999) 153. [5] P.J.M. Smulders, D.O. Boerma, Nucl. Instr. and Meth. B 29 (1987) 471. [6] P.J.M. Smulders, D.O. Boerma, M. Shaanan, Nucl. Instr. and Meth. B 45 (1990) 450. [7] F.H. Eisen, M.T. Robinson, Phys. Rev. B 4 (1971) 1457. [8] S. Datz, B.R. Appleton, C.D. Moak, in: D.V. Morgan (Ed.), Channeling Theory Observation and Applications, Wiley, London, 1973 (Chapter 6). [9] M.B.H. Breese, P.J.M. Smulders, Nucl. Instr. and Meth. B 145 (1998) 346. [10] G.W. Grime, F. Watt, Beam Optics of Quadrupole Probeforming Systems, Adam Hilger, Bristol, 1984.
www.elsevier.nl/locate/nimb
Monte Carlo simulations of a magnetic quadrupole triplet as a high resolution energy spectrometer M.B.H. Breese
*
Department of Physics, School of Physical Sciences, University of Surrey, Guildford, GU2 5XH, UK Received 22 February 2000; received in revised form 6 July 2000
Abstract This paper gives a detailed description of the transport of MeV protons through a nuclear microprobe lens system which is acting as an energy spectrometer. The angular distribution of transmitted protons versus their energy loss is studied for dierent crystal thickness and planar tilt angles using Monte Carlo simulations. The conditions under which the transmitted proton beam is de¯ected by the magnetic quadrupole triplet of a microprobe such that ions with slightly diering energies are focused to dierent locations across the image plane are discussed. The shape of the resultant beam intensity distribution and reconstructed energy spectrum is studied as a function of location along the beam axis. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 61.80.+p; 07.78.+s; 41.85.Lc Keywords: Energy spectrometry; Magnetic quadrupole multiplets; Transmission channeling
1. Introduction Energy spectrometers resolve small dierences in charged particle energies by passing the backscattered or transmitted beam through an electrostatic or magnetic ®eld. The resultant spatial dispersion of the de¯ected ions, due to their momentum or energy spread, is then measured and interpreted as an intensity distribution of particle energies [1,2]. The typical minimum spatial resolution across the image plane of conventional
*
Tel.: +44-1483-876796; fax: +44-1483-876781. E-mail address: [email protected] (M.B.H. Breese).
spectrometers is 100 lm, which can limit the attainable energy resolution. It was recently suggested that a triplet of magnetic quadrupole lenses in a nuclear microprobe [3] can act as a high resolution energy spectrometer [4] owing to its ability to focus MeV ion beams to spot sizes of <1 lm. This design, which is shown in Fig. 1, utilises the large chromatic aberration of a high-excitation quadrupole triplet in the vertical focusing direction to give the required energy dispersion, which is of course a drawback for high resolution probe-forming applications. For use as a spectrometer, a MeV ion beam is passed through a thin crystal sample, which is located at the microprobe object aperture, at 0.0 m on the horizontal
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 0 ) 0 0 3 2 2 - 0
566
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 1. Plot of proton trajectories in the horizontal (lower half) and vertical (upper half) directions, with trajectory angles of h0 0.25 mrad and /0 0.25 mrad, respectively, diverging from the object aperture and focused to the microprobe image plane. The proton energies are 2.988, 3.000, 3.012 MeV, i.e., 3.000 MeV dE 4.0%. The outlines of the quadrupole lenses are shown to their correct horizontal scale but their vertical scale is exaggerated. The location of the normal image plane at zi 150 mm is indicated by a solid vertical line.
axis in Fig. 1. The microprobe collimator slits are translated vertically away from the beam axis, in the plane of the higher chromatic aberration, so that protons with diering energies are focused to dierent locations across the image plane. It was calculated that energy dierences of a few hundred eV could be resolved for MeV protons, which were transmitted through thin crystals [4]. This paper gives a detailed study of the trajectories of MeV protons through a magnetic quadrupole triplet acting as an energy spectrometer, in order to assess the conditions under which the beam intensity distribution in the image plane can be reconstructed to most accurately re¯ect the original energy distribution entering the microprobe. 2. Monte Carlo channeling simulations The Monte Carlo channeling code FLUX [5,6] was used to produce a suitable set of proton trajectories which are transmitted through a thin silicon crystal. Channeling oscillations of ions as they move in regions of dierent electron density
between the lattice walls of the crystal result in dierent transmitted energy losses as a function of crystal thickness and emergent angle [7±9]. Ten thousand trajectories are simulated, with the incident 3 MeV proton beam randomly distributed over the area of the silicon unit cell. All simulations were carried out with the crystal tilted 3° away from the surface-normal [1 0 0] axis in the horizontally running (0 1 1) planes. The emergent angle h in the horizontal plane, and / in the yz vertical plane, and the transmitted energy loss were recorded for each trajectory. Fig. 2 shows the simulated energy loss versus emergent angle /, for each 3 MeV proton which is transmitted through a silicon layer with an increasing thickness, with the incident beam aligned with the (1 1 0) planes. The characteristic planar oscillation wavelength, kp , for 3 MeV protons along the {1 1 0} planes in silicon is equal to 300 nm and the characteristic planar channeling angle wp 0.13° (2.2 mrad). The energy loss scales on the vertical axis diers in each plot to highlight the structure present, but the emergent angle / on the horizontal axis is the same. In Fig. 2(a) the layer thickness is approximately equal to the oscillation wavelength kp . Those protons emerging with a small angle close to zero have a low energy loss, since they have been well channeled and consequently spent most of their trajectories in regions of low electron density. As the emergent angle increases from zero towards wp , the energy loss increases linearly, giving a `wishboneÕ structure. At emergent angles slightly larger than wp the energy loss is a maximum because of the blocking eect, resulting in a larger beam fraction is pushed to either side of the channeling planes. Those protons emerging with larger angles have a large energy loss corresponding to those trajectories which were either not channeled at the surface, or were rapidly dechanneled. A layer thickness of 450 nm in Fig. 2(b) corresponds to a planar oscillation wavelength of 1.5kp . The energy loss distribution is now closed at the top part for a small emergent angle [9]. This is due to those protons which have a larger amplitude of oscillation, and consequently a shorter wavelength than well-channeled protons, so they are out of phase with the main distribution. In
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
567
Fig. 2. Energy loss versus emergent angle / of the beam channeled along the silicon (1 1 0) planes for six dierent layer thicknesses. The vertical axis shows the energy loss on dierent scales in each plot whereas the horizontal scale shows an emergent angle of 0.6° in all plots.
Figs. 2(c) and (e), the layer thickness corresponds to additional multiples of kp . Again there is an open distribution for a small emergent angle, as in Fig. 2(a), with hollow distributions below this, bounded horizontally by wp . This shows that the dierent energy loss groups of channeled protons [8] still maintain their coherency, giving rise to several discrete values of energy loss for a particular emergent angle within wp , even for layers as thick as 5kp as shown in Fig. 2(f).
Fig. 3 contains a sequence of plots showing the energy loss versus emergent angle for 3 MeV protons transmitted through a 210 nm thick silicon layer ( 34kp ), with the beam tilt angle to the (1 1 0) planes varied. With increasing tilt angle the energy loss distribution becomes asymmetric and the average energy loss gradually rises. At a tilt angle of 0.06° (wp /2) and 0.09°, there are no protons with an energy loss as low as at 0.0°. Beyond 0.09° the beam becomes rapidly dechanneled close to the
568
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 3. Energy loss versus emergent angle / of the beam for a silicon (0 1 1) layer thickness of 210 nm, as the crystal tilt angle is altered away from planar alignment. The vertical and horizontal scales are the same in all plots. The vertical axis shows the energy loss from 2.0 to 5.5 keV, and the horizontal scale shows an emergent angle of 0.4°.
surface, producing many blocked trajectories with a high energy loss. Figs. 2 and 3 show that there is considerable structure in the energy loss versus emergent angle of a transmitted MeV proton beam at, and close to planar channeling alignment. It is the aim of the spectrometer studied here to resolve such small dierences in energy loss of less than 1 keV and to reconstruct the initial energy loss distribution. The energy loss intensity distribution for all 3 MeV protons transmitted along the (1 1 0) planes of a 210 nm thick silicon layer is shown in Fig. 4(a), i.e., with no collimation of the emergent angular distribution. Fig. 4(b) shows the energy loss intensity distribution of the same transmitted beam which has been collimated in emergent angle to h0 0.05 mrad in the horizontal direction and /0 0.01 mrad in the vertical direction. Both the intensity of transmitted protons and the range of transmitted energies is reduced in Fig. 4(b) because those trajectories with
slightly diering oscillation wavelengths (and hence energies), have dierent emergent angles, as shown in Fig. 3(a). The remaining high energy loss group from 3.8 to 5.0 keV is the out-ofphase component, visible in the upper portion of Fig. 3(a). Fig. 4(b) shows the energy spectrum which is transmitted into the microprobe, and which the microprobe lens system acting as an energy spectrometer disperses across the image plane, thereby recreating the original distribution as closely as possible. The same collimator aperture sizes of h0 0.05 mrad and /0 0.01 mrad as in Fig. 4(b) are used unless otherwise stated in the spectrometer simulations in Section 3. This collimator aperture is displaced in the vertical direction by /0 0.25 mrad, as shown in Fig. 1, so that a range of angles from /0 0.24 to 0.26 mrad is transmitted into the lens system. A magnetic dipole ®eld is used to de¯ect the transmitted beam by 0.25 mrad to this aperture.
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 4. (a) Energy loss histogram for 3 MeV protons transmitted along the silicon (1 1 0) planes of a 210 nm thick layer. Based on the data in Fig. 3(a), i.e., the full emergent angular distribution. (b) Energy loss histogram of the same transmitted beam distribution but collimated in angle to h0 0.05 mrad and /0 0.01 mrad.
3. Beam transport simulations These simulated channeling trajectories are traced through a magnetic quadrupole triplet lens system using the ion-optical ray-tracing program TRAX [10], which includes all aberration terms which depend on the beam angle and displacement. In keeping with common microprobe formalism, /0 denotes the beam divergence in the vertical direction before the lenses and /i that after the lenses, etc. In Fig. 1, the distance between the exit face of the ®nal lens and the image plane is zi 150 mm, as for normal microprobe operation. However, there are two factors which alter the location of the optimum image plane when the same lens system is used as a spectrometer. Firstly, protons with different energies which pass o-axis through the lens system are focused to dierent spatial positions across the image plane. Secondly, as the collimator
569
aperture is moved further away from the beam axis in the vertical direction, the image plane where the beam is best focused moves towards the lenses, owing to spherical aberration of the quadrupoles, which over-focus particles with a large entrance angle [4]. It is thus important to locate the optimum image plane where the beam is best focused for optimum reconstruction of the original energy spectrum. This section investigates the intensity distribution and the resultant energy spectrum for diering locations of the image plane along the beam axis at which the distribution of focused protons is measured. One method of determining the optimum image plane for recording the focused beam intensity is when the phase space distribution (yi ,/i ) is most upright [9], i.e., when the spatial coordinates yi are most tightly grouped. Fig. 5 shows the phase space distribution for the transmitted beam angular distribution used in Fig. 4(b), but with dE 0 (i.e., no energy loss), with the image plane moved 22 mm towards the lenses so that zi 128 mm. The full range of convergent angles /i are focused to a similar locations yi in the vertical direction. Fig. 6(a) shows the corresponding spatial plot of (xi ,yi ) in the image plane under these conditions, showing that the beam is well-focused in the vertical direction, but displaced by 43 lm away from the beam axis. To demonstrate the degrading eect of spherical aberration on the spatial distribution across the image plane if
Fig. 5. Phase space plot (yi ,/i ) for 3 MeV protons transmitted with no energy loss, with h0 0.05 mrad and /0 0:25 0:01 mrad for an image plane at zi 128 mm.
570
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 6. Spatial plots of (xi ,yi ) for an image plane at zi 128 mm (a) with a horizontal beam divergence h0 0.05 mrad and (b) with h0 0.20 mrad.
the entrance angle into the lenses is increased, Fig. 6(b) shows the same spatial area of the image plane, but with the collimator aperture size in the horizontal direction made four times larger than in Fig. 6(a). The focused beam distribution now extends much further in the horizontal direction, and also bows in the vertical direction due to spherical cross-terms, degrading the relationship between proton energy and location across the image plane. Now consider the phase space distributions produced by the full range of transmitted energy losses. Fig. 7 show three phase space plots of (yi ,/i ) for 3 MeV protons which are transmitted with the energy loss distribution in Fig. 4(b) with the same aperture size of h0 0.05 mrad, /0 0.01 mrad. The image plane has been shifted in 1 mm increments from zi 129 mm to 127 mm from (a) to (c). The yi position scale on the horizontal axis is the same in each plot and the coordinates gradually move to the left from (a) to (c) owing to the convergent beam angle /i . The right-hand portion of the phase space distribution is due to the low energy loss component of 2.4±2.7 keV in Fig. 4(b). This portion focuses to a narrow, well-de®ned spatial location, similar to Fig. 5, owing to the narrow range of energies encompassed. In comparison, the left-hand portion of the phase space distribution, which is due
to the higher energy loss component of 3.8±5.0 keV in Fig. 4(b) is wider owing to the broader range of energies. This higher energy loss portion is closer than the lower energy loss component to the beam axis in the vertical direction since it is bent more. The individual portions of the phase space distributions gradually rotate as they move through optimum focus, but the dierent portions are not necessarily best focused at the same image plane location. The right-hand portion is most upright in Fig. 7(c) at zi 129 mm, whereas the left-hand portion is most upright in Fig. 7(b) at zi 128 mm. The corresponding histograms showing the intensity of the focused beam distribution in the vertical direction at these same locations along the beam axis are shown in Fig. 8. The yi position scale on the horizontal axis is reversed, so that the energy loss of the protons increases from left to right, allowing an accurate assessment of when the energy spectrum of transmitted protons in Fig. 4(b) is best reconstructed. Those protons with a low energy loss from 2.4 to 2.7 keV are further away from the beam axis, giving the narrow peak to the left, as in Fig. 4(b). The original energy spectrum of Fig. 4(b) is best (though not perfectly) reproduced in Fig. 8(b) at an image distance of zi 128 mm.
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
Fig. 7. Phase space plots (yi ,/i ) for 3 MeV protons transmitted with an energy loss distribution given by Fig. 4(b). The image plane is shifted by 1 mm increments from zi 129 to 127 mm.
This is the same optimum image plane location as for the incident 3 MeV proton beam which did not suer any energy loss in Fig. 5. This demonstrates a simple experimental method of locating the optimum image plane for the maximum energy resolution in such a spectrometer design. The incident ion beam, which has not passed through the thin crystal sample is transmitted through this spectrometer, and the optimum location along the beam axis where this is best focused is determined. This same image plane location is then used for all subsequent spectrometer measurements of the proton energies which have passed though thin crystals.
571
Fig. 8. Corresponding energy loss histograms for the focused beam distribution in the vertical direction, for the same image planes as in Fig. 7.
4. Conclusions The energy loss distribution of a proton beam which is transmitted through a thin crystal can be measured by passing the transmitted beam o-axis through the magnetic lens system of a nuclear microprobe. The exact shape of the original distribution can be closely reconstructed if the correct image plane is chosen. An experimental procedure for locating this optimum image plane has been established. These calculations demonstrate the validity of this proposed spectrometer as a viable means of resolving small energy dierences in
572
M.B.H. Breese / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 565±572
transmitted ion energies, for applications such as transmission channeling. References [1] H.D. Carstanjen, Nucl. Instr. and Meth. B 136±138 (1998) 1183. [2] W. Lanford, B. Andersberg, H.A. Enge, B. Hjorvarsson, Nucl. Instr. and Meth. B 136±138 (1998) 1177. [3] M.B.H. Breese, D.N. Jamieson, P.J.C. King, Materials Analysis using a Nuclear Microprobe, Wiley, New York, 1996.
[4] M.B.H. Breese, D.N. Jamieson, Nucl. Instr. and Meth. B 155 (1999) 153. [5] P.J.M. Smulders, D.O. Boerma, Nucl. Instr. and Meth. B 29 (1987) 471. [6] P.J.M. Smulders, D.O. Boerma, M. Shaanan, Nucl. Instr. and Meth. B 45 (1990) 450. [7] F.H. Eisen, M.T. Robinson, Phys. Rev. B 4 (1971) 1457. [8] S. Datz, B.R. Appleton, C.D. Moak, in: D.V. Morgan (Ed.), Channeling Theory Observation and Applications, Wiley, London, 1973 (Chapter 6). [9] M.B.H. Breese, P.J.M. Smulders, Nucl. Instr. and Meth. B 145 (1998) 346. [10] G.W. Grime, F. Watt, Beam Optics of Quadrupole Probeforming Systems, Adam Hilger, Bristol, 1984.