12
Nuclear Instruments and Methods in Physics Research B47 (1990) 271-282 Narth-Holland 271 BE. FISCHER and C. ~~HLBA~~ * CS1, Plancksrx I, O-6100 Darnwiadi, FRG Received 8 December 1989 A computer program for an IBM compatible PC using the method of filtered backprojection and the hardware necessary for microtomography have been developed for the OS1 heavy ion microprobe. The spatial resolution so far achieved is 3 pm with argon ions of 1.4 MeV/nnckon. The special usefulness of heavy ions For microtomography and the te&nical requirements for h&b-resolu- tion micr~tomography are described. 1. Introduction Since the invention of computer-tomography revo- lutionized medical X-ray diagnostics, tomography has influenced many other image forming techniques. It is now for example used to image cross sections of the human body by ultrasonic waves or by the NMR-effect. Aud even the cross section of the earth can be imaged by seismic tomography- So it is not surprising that tomography is also in- fluencing many fields of microscopy. As computer- tomography is essentially a te&nique to reconstruct cross sectional images of an object from a set of projec- tion measurements, any radiation that penetrates a body without being too much deflected is in principle suitable for tomography. And if that sort of radiation is also suitable for microscopy it can in principle also be used for microtomography. Tomography has for example been combined with transmission electron microscopy to produce cross sec- tions of a virus from a set of micrographs taken at different angles, and in a second step a complete 3-d& mensional image of that virus ]l]- If one wants to investigate objects of some 100 pm diameter, elwtrons are no longer suitable because they are scattered too much on their way through these objects. In that case X-rays would be the best choice in principle, because they can penetrate very deeply without being scattered, and the scattered X-rays can be discriminated from the unscattered rays by their energy. Unfortunately X-rays, which are energetic enough to penetrate larger objects, are notoriously difficult to control by optical elements, and they are therefore not suited for microscopy on a * Present address: BASF AG, Abt. ZXT/T, ~~~r-~~~~~- Str. 52, D-6700 Lud~gsh~en, FRG. micrometer scale. l-here is, however, one experiment where an ingenious high resolution position-sensitive X-ray detector using a phosphor and a magnifymg CCD-camera has been applied to circumvent this prob- lem [2]. Ions are relatively easy to manipulate by optical elements, and there already exist many ion-microprobes which can focus MeV ion beams down to 1 pm [3]. As MeV ions are slowed down in an object by collisions with electrons, they are only very little deflected. Tbere- fore protons have already been used for ~crotomo~a- phy some years ago [4,5] and with growing success more recently f6]. Heavy ions are still less defkted when they penetrate matter, 20 IkIeV/nucleon argon ions for ex- ampIe are deflected by only 0.3 pm after passing through 300 pm of a biological sample (compared to 1.4 pm for 20 MeV protons). So heavy ions are best suited for microtomography of relatively large objects without sacrificing the high spatial resolution possible with pre- sently available ion-microprobes. Ions are also more versatile than X-rays. As the absorption of X-rays is dominated by the high-Z ele- ments of au object, these elements will also dominate a tomographic image. So density ~fo~ation will be dif- ficult to obtain, and even elemental composition can only be gained indirectly by measuring the X-ray fluo- rescence. With ions it is possible to measure both den- sity and elemental composition: density by measuring the energy loss of the ions (and one ion is in priuciple sufficient to probe the amount of matter traversed to au accuracy of 1%) and elemental composition by measur- ing the characteristic X-ray excited by the projectile ions. So it was quite a natural step to try tomography at the GSI heavy-ion microprobe, when the computing power necessary for tomography became cheaply availa- bIe with the advent of modem personal computers. ~168-~83X/~/~3.5~ 0 1990 - Eke&r Science Publishers B.V. ~or~-~o~l~d)

Microtomography by heavy ions

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Page 1: Microtomography by heavy ions

Nuclear Instruments and Methods in Physics Research B47 (1990) 271-282 Narth-Holland

271

BE. FISCHER and C. ~~HLBA~~ *

CS1, Plancksrx I, O-6100 Darnwiadi, FRG

Received 8 December 1989

A computer program for an IBM compatible PC using the method of filtered backprojection and the hardware necessary for microtomography have been developed for the OS1 heavy ion microprobe. The spatial resolution so far achieved is 3 pm with argon ions of 1.4 MeV/nnckon. The special usefulness of heavy ions For microtomography and the te&nical requirements for h&b-resolu- tion micr~tomography are described.

1. Introduction

Since the invention of computer-tomography revo- lutionized medical X-ray diagnostics, tomography has influenced many other image forming techniques. It is now for example used to image cross sections of the human body by ultrasonic waves or by the NMR-effect. Aud even the cross section of the earth can be imaged by seismic tomography-

So it is not surprising that tomography is also in- fluencing many fields of microscopy. As computer- tomography is essentially a te&nique to reconstruct cross sectional images of an object from a set of projec- tion measurements, any radiation that penetrates a body without being too much deflected is in principle suitable for tomography. And if that sort of radiation is also suitable for microscopy it can in principle also be used for microtomography.

Tomography has for example been combined with transmission electron microscopy to produce cross sec- tions of a virus from a set of micrographs taken at different angles, and in a second step a complete 3-d& mensional image of that virus ]l]- If one wants to investigate objects of some 100 pm diameter, elwtrons are no longer suitable because they are scattered too much on their way through these objects. In that case X-rays would be the best choice in principle, because they can penetrate very deeply without being scattered, and the scattered X-rays can be discriminated from the unscattered rays by their energy. Unfortunately X-rays, which are energetic enough to penetrate larger objects, are notoriously difficult to control by optical elements, and they are therefore not suited for microscopy on a

* Present address: BASF AG, Abt. ZXT/T, ~~~r-~~~~~- Str. 52, D-6700 Lud~gsh~en, FRG.

micrometer scale. l-here is, however, one experiment where an ingenious high resolution position-sensitive X-ray detector using a phosphor and a magnifymg CCD-camera has been applied to circumvent this prob- lem [2].

Ions are relatively easy to manipulate by optical elements, and there already exist many ion-microprobes which can focus MeV ion beams down to 1 pm [3]. As MeV ions are slowed down in an object by collisions with electrons, they are only very little deflected. Tbere- fore protons have already been used for ~crotomo~a- phy some years ago [4,5] and with growing success more recently f6]. Heavy ions are still less defkted when they penetrate matter, 20 IkIeV/nucleon argon ions for ex- ampIe are deflected by only 0.3 pm after passing through 300 pm of a biological sample (compared to 1.4 pm for 20 MeV protons). So heavy ions are best suited for microtomography of relatively large objects without sacrificing the high spatial resolution possible with pre- sently available ion-microprobes.

Ions are also more versatile than X-rays. As the absorption of X-rays is dominated by the high-Z ele- ments of au object, these elements will also dominate a tomographic image. So density ~fo~ation will be dif- ficult to obtain, and even elemental composition can only be gained indirectly by measuring the X-ray fluo- rescence. With ions it is possible to measure both den- sity and elemental composition: density by measuring the energy loss of the ions (and one ion is in priuciple sufficient to probe the amount of matter traversed to au accuracy of 1%) and elemental composition by measur- ing the characteristic X-ray excited by the projectile ions.

So it was quite a natural step to try tomography at the GSI heavy-ion microprobe, when the computing power necessary for tomography became cheaply availa- bIe with the advent of modem personal computers.

~168-~83X/~/~3.5~ 0 1990 - Eke&r Science Publishers B.V. ~or~-~o~l~d)

Page 2: Microtomography by heavy ions

272 B. E. Fischer, C. ~~l~~er / ~jc~ro~og~aphy by heavy ions

rotating SB-detector object

scan path Fig. 1. M~~ern~t of the density projections of a miero- scopic object: While the ion-microbeam is scanned over the object a semiconductor detector measures the energy loss of the ions which penetrate the object. After each scan the object is turned a fraction of a degree relative to the beam and a new energy-loss profile is measured. This procedure is repeated

until the object has been turned by at least 180’.

2. T&e experiment

The way to measure the set of projections, necessary for the tomo~ap~c imaging of a microscopic object, is shown schema~c~y in fig. 1, While the ion microbeam is scanned across the object, the energy loss of the ions is measured by a semiconductor detector behind the object, and an energy loss profile for one scan is stored in the computer. After every scan the object is turned some fraction of a degree relative to the microbeam, and a new energy loss profile is measured. That proce- dure is repeated until the object has been turned by at least 180 degrees,

Fig. 2 shows the ion-microprobe in the so-called stripper-hall at GSI, where the experiment has been performed using argon ions at an energy of 1.4

Fig. 3. Part of the ~cros~pe stage to which the object was mounted. The object is fixed at the tip of a micropipette visible in front of the central round aperture. The micropipette in tom is stuck in a speeiai holder and centered on the axis of the stepper motor by two adjusting screws. Hidden behind the aperture is the semiconductor detector which measures the energy of the ions penetrating the object. Near the left side of the object a small round microgrid is visible. It is used as a test

target to focus the beam by secondary electron imaging.

MeV/nucleon. Fig. 3 shows part ofi the microscope stage onto which the object had been mounted and fig. 4 a lint-~croscope photo~aph of the object.

Fig. 2, Photograph of the ion-microprobe in the stripper-hall at GSI, where heavy ion beams with a fixed energy of 1.4 MeV per nucleon are available.

Page 3: Microtomography by heavy ions

B. E. Fischer, C. Miihlbauer / Microtomography by heavy ions 273

Fig. 4. Light-microscope photograph of the biological object. The horizontal line indicates the “slice”, where the tomograph was taken. The tip of the micropipette to which the object was glued is visible at the top.

The choice of the object was determined by two conflicting conditions. It had to be small enough to be penetrated by argon ions of 1.4 MeV/nucleon, which is the only available energy in the stripper-hall. So its thickness should correspond to no more than 20 pm of compact organic matter. And at the same time it should be large enough to show still significant details at a resolution of some few microns (given by the resolution of the microprobe). We found microscopic creatures with these properties in a nearby pond and dried them by the critical point method to preserve their structure. One of them (shown in fig. 4) has been used for this experiment.

To be able to turn the object relative to the micro- beam, it was first glued to the tip of a micropipette

Fig. 5. The detector wobbler. It is used to keep the semicon- ductor detector in constant motion to prevent pulse-height

defects due to radiation damage.

coated by a thin layer of epoxy glue. Then, the micro- pipette was stuck into an adjustable holder on the axis of a stepper-motor (shown at the top of fig. 3) and adjusted as near as possible to the center of the axis under a stereo microscope.

The energy of the ions penetrating the object was measured by a semiconductor detector located behind the microscope stage. As a semiconductor detector can be damaged locally by the microbeam of argon ions within seconds, it was kept in constant motion by a small wobbler shown in fig. 5.

3. Microprobe control and data taking

Various modes to control the movement of the mi- crobeam and the microscope stage are available at the

GSI microprobe. 1) An autonomous hardware x-y raster-scan which

can be coupled to the microbeam or to the motors of the microscope stage or to both at the same time. The speed of the raster-scan can be controlled by an oscilla- tor with a set of fixed frequencies, by an external analog voltage or by an external random clock signal. The last feature is especially useful to adapt the scanning speed to the fluctuating beam current so that every particular target position is hit by the same number of ions.

2) Various degrees of computer control are also pos- sible. The computer may be used to set only the start coordinates for the autonomous scan or to control the scan point by point.

Page 4: Microtomography by heavy ions

274 B.E. Fischer, C. Miihlbauer / Microtomography by heavy ions

eL ! I I 1 1 I I I

rdee Cihrt!l 1 I ,

508 aeee PSEBB a&e 3588 4tme

Run 101 Coordinrtt? L ORf X

Fig. 6. One out of the 500 energy-loss projections measured along the line indicated in fig. 4. Every projection consists of 512 data points. And each data point is the average over the energy losses measured for 8 ions penetrating the object at one particular point.

Y hi

Y fo

X

Y

At the beginning of the experiment the microbeam is focused by secondary-electron imaging of a small mi- crogrid, using the normal x-y scan mode of the micro- probe. (The microgrid and the secondary electron detec- tor are visible in fig. 3 near to the left side of the object.) Then a crude energy-loss image of the object is produced to find a slice of the object which is com- pletely transparent to the 1.4 MeV/ nucleon argon beam (otherwise only incomplete projection measurements could be made).

To measure the energy-loss projections, the y-scan of beam is disabled and the pulses usually controlling the y-movement of the microscope stage drive the motor used to turn the object. When the microbeam scans over the object, it dwells at one particular point until 8 ions have been detected in the semiconductor detector be- hind the object. The particular x-coordinate, the angle of the motor, and the energy values are transferred to the computer (Commodore PC 40-40) via CAMAC, which is controlled by the program CAMDA 171. After

Fig. 7. Set of 500 energy-loss projections. The so-called sinogram. Every horizontal line corresponds to one projection, and there are 500 projections from top to bottom measured in equidistant steps between 0 o and 360 O.

Page 5: Microtomography by heavy ions

B. E. Fischer, C. Miihlbauer / Microtomography by heavy ions 215

the computer has acknowledged the receipt of these data, the microbeam advances one step and the above procedure is repeated until a full scan is completed. Then the stepper motor advances a fraction of a degree and a new projection is measured. Fig. 6 shows one out of 500 projections. Each contains 512 data points and each data point represents the energy averaged over 8 ions penetrating the object at one particular position.

Fig. 7 shows the set of 500 energy-loss projections (the so-called sinogram) measured for one tomographic slice of the object shown in fig. 4. Every line represents one energy-loss projection. And there are 500 lines for 500 equidistant angles between O” and 360’. (Light regions correspond to high energy loss.) This sinogram is available in “real time”, while the experiment is running. It already shows a dense core and the thin skin of the object swinging around the center of the motor axis.

4. Tomographic image reconstruction - some theory

The reconstruction of the tomographic image from the set of projections is performed by the method of “filtered backprojection”. Fig. 8 shows schematically what is meant by projection and backprojection. During projection a test ray sums up the contents of the object matrix along its way. During backprojection all cells of the image matrix which have been crossed by the test ray are filled with the result of that summation process.

If there is only one projection, the object matrix cannot be reproduced by backprojection. A second pro- jection measurement of 90 o to the first one can already give some indication of the structure of the object matrix. But there are still some artefacts in cells which should be empty.

- . . I . _

-

test rays

-

-

-

object =I matrix

This is more clearly demonstrated in a computer simulation which we have made in order to test the validity of our reconstruction algorithm (because it is nearly impossible to produce a test object with a known composition, geometry, and a diameter of only 20 pm). Here we produced a test object in the form of a watch in a 100 x 100 matrix on the computer screen. Projec- tions have been computed for 180 different angles in steps of lo, summing up the shades of grey. Fig. 9 shows that a simple backprojection, like the one de- scribed in fig. 8, can already show all the details of the original object, provided that enough projections under different angles have been taken. But it also shows a background of artefacts which can be prevented by the process of “filtered” backprojection.

Filtering means a mathematical process which can be easily understood, qualitatively at least, looking at fig. 8. The projections must be modified in a way which introduces negative values to them with the aim that positive and negative artefacts cancel each other during backprojection. There exist exact mathematical solu- tions to this problem which were developed in 1967 by Bracewell and Riddle [8] and independently from them by Ramachandran and Lakshmiranayanan [9] in 1971. For our computations we followed the description of the theory by Rosenfeld and Kak [lo]. It says that a two-dimensional distribution

f(-G VI,

can be reproduced by a convolution Q(0, t) of the projections

P(O, t),

taken for example at K different angles 8, and each

projection backprojection

Fig. 8. Schematic explanation of the projection and the backprojection process. Here the object is a simple matrix which contains only one element of measurable density. To produce a projection, test rays sum up all density values along their way and the result is

stored in the cells of the projection string. The backprojection reverses that process: the information contained in one cell of the

projection string is written into a corresponding row of cells in the image matrix.

Page 6: Microtomography by heavy ions

216 B.E. Fischer, C. Miihlbauer / Microtomography by heavy ions

Fig. 9. Reproduction of a computer-generated object matrix of 100 X 100 pixels by simple backprojection of 180 projections t

1“ intervals (bottom). The original matrix is shown at the top.

Fig. 10. Definition of coordinates.

.&en at

Fig. 11. Effect of the filter on a projection containing only one

point.

Page 7: Microtomography by heavy ions

B.E. Fischer, C. Miihlbauer

having N measured values with a discrete filter h(n) of the form

N-l

for n=O,

for even n,

for odd n, (1)

Q(fl, t) = c h(t- l)P(O, I). (2) I=0

That means that every data point of the original projec- tion is spread over a larger part of the filtered projec- tions, as shown in fig. 11.

’ Microtomography by heavy ions 217

The reconstructed image F(x, y) of the original object f(x, y) is then obtained by a backprojection of the filtered projections

F(Xt Y) = g t Q(&> t). i=l

If one projection contains 512 energy-loss values, a filter with 1025 components could be used in principle. But as the elements of the filter function h(n) decrease with l/n*, it seems reasonable to assume that one can save computing time by using a smaller number of filter elements and still get an accurate result. We have there-

Fig. 12. Influence of filter size on the accuracy of the reproduced image. (a) Test image containing 100 X 100 pixels, (b) greyscale profile of the test image along the line indicated in (a). The reproduction of the profile is shown in (c) using 301, (d) 101, (e) 21, and

(f) 11 filter components respectively.

Page 8: Microtomography by heavy ions

278 B.E. Fischer, C. MiMbuuer / ~icrot~~o~~apby by heauy ions

Fig. 13. Reconstruction of the test object in fig. 9 using 180,90, 36 and 18 projections.

fore tested the accuracy of the grey-scale reproduction using various numbers of filter components. The test image shown in fig. 12 is fairly accurately reproduced with a filter of 101 components. If one uses only 21 components the inner part of the image is still repro- duced satisfactorily but there is a positive background outside. A still smaller filter produces wrong results everywhere, even though the visual impression of the reconstructed image (not shown here) is still satisfac- tory.

The number of projections used to reconstruct the image will quite obviously also influence the quality of the reconstructed image. One can expect that the recon- struction of an image matrix with n x n elements re- quires at least R x II independent rne~~ernen~. That is, n projections each containing n energy loss values. But as the use of more projections will also increase the computing time, we have tested ~~rirnent~y which minimum number of projections is necessary to produce a satisfactory image. Fig. 13 shows the test object of fig. 9 reconstructed with 180,90, 36 and 18 projections. As expected the 100 x 100 pixel test object is fairly accu- rately reconstructed by 90 projections, each containing between 100 and 140 data points. Some improvement

c%tn be achieved by the use of twice as many projections. The use of fewer projections leads to streaklike artefacts.

5. Rotation of B talc image from mea- sured data

The image r~nst~~~on by backproj~tio~ requires that the projections are projected back exactly he way they have been produced. That means, for example, that the energy loss of the ion passing exactly through the rotational axis is projected back along a line crossing the center of the image matrix. Otherwise points in the object matrix will be reconstructed as circles in the image matrix.

The backprojection of data from an ion-microprobe is complicated by the fact that one does not know in advance which of the 512 data points of the measured projections is from ions passing exactly through the center of the motor axis. This is no problem in medical torno~ap~~ ~~prnen~ which can be built so that the X-rays going through the rotational axis always hit on the center of the detector array. In the ion-microprobe it is almost impossible to position the axis of the stepper

Page 9: Microtomography by heavy ions

B.E. Fischer, C. Miihlbauer / Microtomography by heavy ions 219

beam- deflection

;

misalign

Fig. 14. Calculation of the misalignment from the sinogram.

motor exactly symmetric to the microbeam’s scanpath. This ever present misalignment, however, can be calcu- lated from the sinogram (fig. 7).

If one follows for example one point as it swings round the rotational axis from top to bottom, the ex- tremes should ideally be symmetric to the middle, pro- vided the scanning is exactly linear and the center of the motor axis stays stable. So any misalignment can in principle be corrected by a slight horizontal shift of the projection data (fig. 14). One should, however, try to position the motor axis as near as possible to the center

of the beam’s scanpath. Otherwise one would need an unnecessarily large image matrix to cover the whole object. After correction for misalignment and filtering according to eq. (2) the projections look like fig. 15.

If these filtered projections are backprojected onto a 512 x 512 image matrix one gets the result shown in fig. 16. It shows a slice going through the object at the position of the line indicated in fig. 4. To demonstrate the resolution of the tomographic image, the core of the object is shown magnified in fig. 17. A small structure at the lower right border of the core shows that the resolution is about 3 pm. That means that it is only slightly larger than the beam spot of 2 urn.

6. Information in a tomographic image

Besides the geometric information, a tomograph also contains quantitative information about the material properties of the object. In the case of high energy ion-tomography, the ions are essentially probing the electron density which is somehow related to the density of a particular material. Therefore one usually says that the ions are probing the density of a material. That density, however, cannot be derived directly from an ion-tomographic image, because the interaction strength of the ions depends also on their energy. And this energy varies as the ions slow down inside an object.

Fig. 18 describes how the stopping power for 1.4 MeV/nucleon argon ions varies inside a homogeneous cylinder of organic material as the ion slows down. If the ion enters the cylinder from the right side, its energy loss (solid curve) in a particular volume element (hatched

Fig. 15. Sinogram corrected for misalignment and filtered according to eq. (2).

Page 10: Microtomography by heavy ions

280 B. E. Fischer, C. ~~Iba~r / ~icroro~ugraphy by heavy ions

Fig. 16. Slice of the object shown in fig. 4 produced by the tomographic reconstruction technique. The thickness of the slice is about 2 pm (given by the diameter of the beam spot) and its largest diameter is about 250 Frn.

area) is larger than if it enters from the left side (dashed curve). Obviously a particular volume element inside the object will appear to have different densities de- pending on which side the ion enters. As the tomo- graphic r~onst~ction technique averages over the en- ergy losses of al1 test rays, a situation like in fig. 18 will overemphasize the density values in the center of the object by a factor of 2, as indicated by the averaged energy loss curve (large black dots). The application of empirical correction formulas may be possible to some extent, but only if the composition of the object is not too inhomogeneous.

The averaging process gives exactly the same average stopping power for every volume element only for a linear stopping power curve. This condition is practi- cally fulfilled if we use a beam with an energy far

beyond the Bragg peak, as is demonstrated in fig. 19. At 8 MeV/nucleon, for example, the energy loss curve is sufficiently linear within a cylinder of 20 pm diameter that an ion ~crotomograph could produce the same density values everywhere to within 2%.

7. Conclusions and outlook

In a first experiment to demonstrate the feasibility of microtomography with the heavy-ion microprobe we have achieved a spatial resolution of 3 pm. As the beam-spot diameter during that experiment was about 2 l.trn, the quality of the computer program is probably sufficient for a still higher resolution. To achieve the submicron resolution which should in primSpIe be pos-

Fig. 17. Magnified detail of fig. 16 to demonstrate the quality of the reconstruction. The resolution is about 3 pm.

Page 11: Microtomography by heavy ions

B. E. Fischer, C. Miihlbauer / Microtomography by heavy ions 281

O-

fo-

‘0 -

O-

I I I I

05 1 x [mg/cm2]

15 2

Fig. 18. Influence of the stopping power curve on the “density” values computed by the tomographic method. The energy loss of a

1.4 MeV/nucleon argon ion is shown as it slows down in a 20 urn diameter cylinder of organic material. If the ion enters from the

right side, its energy loss (solid line) in a particular volume element (hatched area) is larger than if it enters from the left side (dashed

line). Therefore the volume element will appear more “dense” to the ion entering from the right side. Since the tomographic

reconstruction averages over many test rays penetrating the object from all sides, an average value (curve with black dots) will be

assigned to every volume element. So the density in the center will be overestimated by a factor of 2 if ions of that particular energy

are used. Energy loss data taken from Northcliffe and Shilling [ll].

Fig. 19. Energy-loss curve for 10 MeV/nucleon argon ions in

organic material [ll]. Far beyond the Bragg peak, at 8 MeV for

example, it is sufficiently linear within a cylinder of 20 urn

diameter (indicated by the 2 vertical lines) that the tomo-

graphic reproduction process would produce the same density

values everywhere to within 2%.

sible with heavy ions even for much larger objects, it is not only necessary to provide a submicron beam spot as we had already demonstrated some years ago, but also to provide some way to turn the object with submicron precision or to correct deviations from that precision by computer using information from the sinogram. With 1.4 MeV/nucleon argon ions it was still impossible to get reliable density information, but theoretical consid- erations show that better density information can be achieved by ions with an energy far beyond the Bragg peak.

References

[l] A. Dengler, Ultramicroscopy 30 (1989) 337. [2] P.F. Brian, H.W. De&man, W.G. Roberge and K.L.

D’Amico, Science 237 (1987) 1439.

[3] K. Traxel, Nuclear Microbeams: Realisation and Use as a

Scientific Tool, Proc. European Conf. on Accelerators in

Applied Research and Technology, Sept. 1989, Nucl. In- str. and Meth. B50 (1990) 177.

[4] J. Huddleston, I.G. Hutchinson and T.B. Pierce, Nucl.

Instr. and Meth. 197 (1982) 157.

Page 12: Microtomography by heavy ions

151 A. Ito and Ii. Koyama-Ito, NucL In&r. and Meth. B3 (91 G.N. Ramachandran and A.V. L~sh~~anan, Proc. (1984) 584. Nat. Sci. USA 68 (1971) 2236.

[6j A.E. Pontau, A-3. Ant&k, D.H. Morse, A.A. Ver [lOf A. Rosenfeld and A. Kak, Digital Image Processing, vol. 1 Berkmoes, J.M. Brase, D.W. Heikkinen, H.E. Martz and (Academic Press, 1982). I.D. Proctor, Nucl. Instr. and Meth. B40/41 (1989) 646. [ll] L.C. Northcliffe and R.F. Shilling, Nuclear Data Tables,

[7] CAMDA: A Program to Collect and Analyze Data, Her- vol. 7, no. 3-4, Jan. 1970. bert Stelzer, private communication.

[S] R.N. Bracewell and A.C. Riddle, Astrophys. J. 150 (1967) 427.