- Email: [email protected]
A proposed nuclear nanoprobe with Angstrom resolution
Nuclear Instruments and Methods in Physics Research B 267 (2009) 1995–1998
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
A proposed nuclear nanoprobe with Angstrom resolution q B.L. Doyle a,*, S.M. Foiles a, A.J. Antolak a, L. Popa-Simil b, D.K. Brice c a b c
Sandia National Laboratories, Radiation Solid Interactions Department, P.O. Box 8500, MS 1056 Albuquerque NM 87185, USA LAVM LLC, 3213 Walnut Street, Apt. C, Los Alamos NM 87544-2092, USA Consultant, 115 Charles Street, Sulphur Springs TX 75482-3501, USA
a r t i c l e
i n f o
Available online 19 March 2009 PACS: 07.78.+s Keywords: Ion microscopes Shadow cones Graphene
a b s t r a c t An idea is presented for an entirely new point-projection nuclear microscope that theoretically would have sub-nanometer resolution, approaching one Angstrom. The concept involves using a Kalbitzer super tip as a source of low energy (100 eV) He or H ions that would be transmitted through a molecular sample (e.g. buckyball, carbon nanotube, graphene sheet, DNA molecule, etc.) placed very near the tip (1– 100 nm), and then projected onto a microchannel plate (MCP) screen placed 1 m from the tip. Small angle scattering of the incident ions with atoms in the sample result in the development of shadow cones with an increase in scattered ion intensity at the critical cone angle. The enhanced intensity patterns formed at multiple intersections of cone perimeters are called ‘‘threads”. The shadow cones and threads are projected onto a suitable low-energy ion position sensitive detector to create a ‘‘shadow or thread image” of the sample. Such point-projection microscopes have no aberrations that affect the image, and the magnification would be of the order 1 m/10 nm, or 1.E8! The feasibility of this scheme is currently being studied theoretically using a deterministic atomic scattering model and a Monte Carlo molecular dynamics code. The results of these calculations indicate that only very low energy (60– 120 eV) incident hydrogen ions can be used to avoid displacing atoms in the sample. At these energies, the critical cone angles are quite large and it would be necessary to position the molecular samples very close to the tip (down to only 1 nm) so that the projected ion maps are interpretable as distinct threads. Projecting distinct shadows is not supported by the scattering physics. Published by Elsevier B.V.
1. Introduction The resolution of focused ion beams used for nuclear microscopy is now approaching 10 nm, but this is still orders of magnitude greater than the de Broglie wavelength of such ions. Electron microscopes, on the other hand, are beginning to operate at, or at least very near, their diffraction limit due to the development of lens aberration-correction technologies. However, the approaches used in these aberration corrected electron lenses are not really adaptable for nuclear microprobe lenses. An alternative to focusing the ion beam, and the avoidance of lens aberrations, is the use of point-projection microscopes. Such microscopes have already been developed for electrons [1,2]. Additionally, low-energy ion projection microscopes [3] have been developed that utilize diffraction effects, and higher-energy projection He-ion microscopes have also been proposed [4]. Recently, ‘‘Orion” microscopes
q Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. * Corresponding author. E-mail address: [email protected] (B.L. Doyle).
0168-583X/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.nimb.2009.03.084
[5] have been developed and manufactured by Zeiss for He ion microscopy that are based on the ‘‘Super Tip” concept of Kalbitzer and Knoblauch [6]. These point-projection sources of He+ could also be used for H+ ions. Not only do such sources have brightness orders of magnitude larger than conventional ion sources used in nuclear microscopy, but they also have an initial spot size that approaches an Angstrom. In this paper, we theoretically investigate the feasibility of a H+ point-projection microscope which we call the atomic shadow microscope (AMS). While there are several (and potentially serious) obstacles to realize such a microscope, the sole focus here is to determine whether the physics of atomic scattering supports such a concept. 2. The atomic shadow microscope The fundamental principle of the Atomic Shadow Microscope (ASM) is shown in Fig. 1. Atoms ionized in the vicinity of a Kalbitzer Super Tip are extracted toward a molecular sample placed at a distance D from the tip which can be quite small (10–100 A). When a voltage, U, is applied to the tip, the energy of the extracted ion at the sample is simply eU. As the ions penetrate the sample they are
1996
B.L. Doyle et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1995–1998
Kalbitzer Super Tip
Molecular Sample placed at a distance D from the tip
H+ Ionized within ~1A region of tip.
Shadow cones with increased H intensity at their perimeter are projected onto a screen placed at L.
Fig. 1. H+ ions ionized in the vicinity of a Kalbitzer Super Tip are extracted toward a molecular sample placed very close to the tip. After scattering from the atoms in this sample, shadow cones are projected onto a screen, e.g. a microchannel plate, placed at a great distance from the sample, resulting in magnifications in the 108 range.
deflected (scattered) and cause shadow cones that can be imaged on a detector screen, e.g. a microchannel plate. We note that if the screen is placed at a distance L (centimeters to meter length scale) from the sample, the resulting magnification (L/D) is in the 108 range. It is clear from Fig. 1 that an ASM will need to measure very small shadow cone critical angles. Light ions such as H or He are preferred in this case because the critical cone angle scales, as (Z/ E)1/2 [7] where Z and E are, respectively, the atomic number and energy of the incident ion. Further, the energy of the incident ions will need to be low to prevent the displacement of atoms in the sample. For the following calculations, we consider only targets/ structures that are composed of carbon atoms, such as single layers of graphene, nanotubes or buckyballs. The carbon atom displacement energy thresholds of such nano-structures are angularly dependant [8], but their minimums have been calculated using molecular dynamics to be in the range from 17 to 35 eV [9,10]. This means that maximum energy of H ions should be in the 60–120 eV range. For He ions, the probe energy becomes so low so as to not seem practical due to the large critical angles that result. We therefore restrict our attention only to incident H ions in this energy range.
3. Atomic scattering theory for H on C While high fidelity molecular dynamics simulations would ultimately be needed to determine the feasibility of the ASM, much insight can be gained through theory and calculations that can be performed analytically. In this regard, the key derivation relates the impact parameter, b, to the laboratory scattering angle, h, for hydrogen ion–carbon atom interactions. Taking into account conservation of energy and momentum in the scattering due to the collision of two particles, one finds
Z
1
r min
bdr=r 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 UðrÞ br2 E
ð1Þ
2Eb k 2
ð3Þ
can quite accurately provide scattering angles as a function of impact parameter. Table 1 lists the fitting constants c, bo and P used in the calculations that follow. Fig. 2 shows a plot of the trajectories that result using Eq. (3) for a point source of 100 eV H and a single C atom placed 10 A from this source. It is seen that the shadow cone into which no H ions are scattered is along a line between the source and target, and within an azimuthal angle of 10.5°. The enhanced intensity that occurs just above this critical angle is also apparent in the figure.
Table 1 Parameters used in the equation (h ¼ 180 expfc½lnðbÞ lnðbo ÞP g) to determine the lab scattering angle (degrees) as a function of impact parameter (A) for H on C as a function of energy. E (ev)
bo
P
c
Theta-crit
50 100 200 400 800 1600 3200
0.016834 0.007482 0.004988 0.002217 0.001478 0.000985 0.000438
3.175801 0.348984 3.193423 3.2386 3.015068 2.781297 2.724214
0.03246 0.016758 0.019917 0.013628 0.019462 0.028981 0.026686
12.50003 10.524 8.683282 7.099758 5.689827 4.493111 3.539753
10
100 eV H on C c = .0168 bo = .0075 A P = 3.34
9
where r is the distance separating the particles which interact via the potential U(r) and E is the total energy of the two particles in the center of mass frame. For a pure Coulomb potential, the integral can be evaluated analytically and the resulting expression simplified to give
h ¼ 2arc cot
h ¼ 180 expfc½lnðbÞ lnðbo ÞP g
8 7
y (A)
h¼p2
the calculation of the scattering angle versus impact parameter is more difficult and the final result cannot be expressed analytically; this relationship can only be determined through numerical integration of equation (1). Using the famous Ziegler–Biersack–Littmark universal potential [11,12], we have found the analytic parameterization
6
Critical Angle = 10.52 degrees
5 4
ð2Þ
where k ¼ Z41 pZ2eeo and Z1 = 1, Z2 = 6, and e is the electron charge. Solving these equations for an impact parameter of 1 A and energy of 100 eV gives a scattering angle of 80°; it is clear that Angstrom-level resolution is not possible if the scattering is due to pure Coulomb scattering alone. Fortunately, this is not the case because of the well known screening of this potential, particularly at the low energies being considered here. However, for the screened Coulomb case,
3 2 1 0 0
5
10
15
20
x (A) Fig. 2. Scattering trajectories of 100 eV from C using Eq. (3), and the scattering trajectory of the ASM that has a point source of H placed 10A from the C atom.
B.L. Doyle et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1995–1998
CC-angle
Cone-angle
1.0
Cone/CC angle= 1.32
1 Center Thread
1 Edge Thread
0.5
Shadows
1997
st
st
1.72
st
1 Atomic Thread
2.0
nd
2 Center Thread
Fig. 3. Top: Geometry used to define CC-angle and Cone Critical Angle. Bottom: Patterns expected to be projected for various ratios of Cone to CC-angles.
4. Shadow cone and thread images We now consider the images of the shadows that occur below the critical angle as well as the enhanced scattering that occurs just above the critical angle. Without loss of generality, we perform this analysis for the simple geometry of a graphene target. Graphene is a single basal plane of C atoms of graphite where the atoms are on a hexagonal lattice with an atomic separation of 1.4 A. A geometrical representation of the geometry and projected images formed by the cones and their perimeters is shown in Fig. 3. We define the CCangle to be the angle subtended between two neighboring carbon atoms with the source tip. The figure shows the images that would be projected by the graphene sheet for different ratios of the cone to CC-angles. Fully separated shadow cones should result when the cone to CC-angle ratio is below 0.5, while the cones will just touch when the ratio is exactly 0.5. For a ratio of 1.0, the shadows are replaced by an accumulation of ion intensity formed by the overlap of the enhanced scattering at the shadow cone perimeters (defined as threads) and correspond to the position projected between the
centers of each of the C atom hexagons. For the case shown in Fig. 3, there are six perimeters that superimpose to cause the increased intensity. We call this the ‘‘1st Center Thread”. For a ratio of 1.32, four perimeters form threads at projected positions between each CC atom that we call the ‘‘1st Edge Thread” and, for a ratio of 1.72, six perimeter intensities combine to form threads right at the projected position of the C atoms that is called the ‘‘1st Atom Thread”. When the ratio is 2.0, the 2nd Center Thread forms. This process continues with the generation of 2nd, 3rd, etc. higher order threads of increasing complexity. A simple binary collision approximation can be used to predict the two-dimensional projected images of the scattered proton intensity that would be obtained in the ASM under different conditions. In the following calculations, the graphene target is modeled as a sheet of 54 carbon atoms. The collision model uses Eq. (3) to determine the scattering that would occur from the interaction of H+ at different energies with all 54 C atoms, where one million protons are projected uniformly across the central 4 A of this sheet. The shadows and multi-order threads that are produced are then
Fig. 4. Plot of the Cone Angle as a function of tip-graphene lattice separation for 3200 eV H. Also plotted as lines are the CC-angles as solid lines labeled to the right where the lowest line corresponds to the .5CC angle, the next higher line to the 1CC angle case and so on. The insert shows a two-dimensional calculation of the scattered H intensity using Eq. (3) at a separation of 16 A.
1998
B.L. Doyle et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1995–1998
Fig. 5. Plot of the CC-angles (again, the lowest line corresponds to the .5CC angle, the next higher line to the 1CC angle case and so on) and Critical Cone Angles for 120 eV H, as a function of the separation between the point source tip and a graphene lattice with a 1.4 A lattice constant. The insert shows a 2D calculation of the thread pattern for the situation at 8 A. This calculation represents the highest energy H that can be used to avoid displacing C atoms, and the largest tip-target combination that produces a reasonable thread image (this is close to the first Center or Gap Thread).
visualized as gray scale intensity maps. Fig. 4 plots the calculated critical cone angles (squares) and CC-angles (lines) for the case of a high-energy (3200 eV) proton beam. While this is not a practical energy because atomic displacements would quickly destroy the sample, it is illustrative of the situation where fully formed shadows can be produced with a tip-target separation of 16A. The resulting intensity distribution is plotted as the insert in the figure, and touching shadows are indeed observed, as expected, because for this separation the cone/CC-angle equals 0.5. For a separation of 200 A, the geometrical pattern indicates that the 1st Center (or Gap) thread would form. Fig. 5 shows a much more practical case where 120 eV H ions are used (this is the highest energy where atomic displacements may be avoided). In this case, the 1st Gap Thread should form with a tip-target separation of 8 A. The intensity distribution, again for all 54 C atoms but aimed only at the central 4A of the sheet, is shown in the insert and the resultant threads are easily identified. 5. Discussion and conclusions Atomic Shadow Microscopy (ASM) is a form of point-projection microscopy where the sample is placed very near a point ion source and projected images (shadows) of the scattered ion intensities are detected at a great distance from the source, potentially providing a huge magnification. The binary collision theory for low energy ion-atom collisions has been parameterized to predict the scattered intensity pattern of ions projected through molecular samples in ASM. We examined the case of using hydrogen ions to shadow image carbon-based molecular structures, but found that extremely low energies are required to prevent the displacement of the carbon atoms. As a result, distinct and well-separated shadows cannot form because the critical shadow cone angles are so large. On the other hand, the model shows that the increased intensity just outside the radius of the shadow cones can combine for distinct critical cone to CC-angle ratios to produce what we call
‘‘Threads”. Consequently, a more appropriate name for the technique might be the ‘‘Atomic Thread Microscope (ATM)”. For a practical ATM operating with 120 eV H ions, the tip-target distance would need to be <10 A to produce such threads. In this case, the electric field between the tip and the target would be 12 eV/A, and it is not clear whether such high fields might deform or destroy the target (note that the typical electric field at the tip used for field ion microscopy is in the 1 eV/A range). In addition, there is uncertainty as to whether the dissociation and ionization of the H2 molecules will remain localized at the very tip of the Kalbitzer Super Tip. While the simple binary atomic scattering theory supports the development of threads of intensity that could be detected and imaged by an ATM, the very small tip-target separation that is required presents a challenging obstacle. At this time, we can say that the ASM is probably not feasible due to the scattering physics, but we can NOt say the same about the ATM. Molecular dynamics calculations are underway that use more realistic potentials and include the possibility of C-H bonds forming. Since these potentials have an attractive component for impact parameters >1 A, the scattering cone critical angles may be sufficiently reduced to improve the viability of an ATM. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
N.D. Browning, I. Arslan, P. Moeck, T. Topuria, Phys. Stat. Sol. B 227 (2001) 229. B.G. Frost, D.J. Joy, A. Thesen, SPIE 4689 (2002) 576. G. Chern, J.G. Skofronick, W.P. Brug, S.A. Safron, Phys. Rev. B 39 (1989) 38. R. Levi-Setti, T.R. Fox, Nucl. Instr. and Meth. 168 (1980) 139. J. Notte, B. Ward, N. Economou, R. Hill, R. Percival, L. Farkas, S. McVey, in: AIP Conference Proceedings 931 (2007) 489,. S. Kalbitzer, A. Knoblauch, Appl. Phys. A 79 (2004) 1901. Fauster, D. Hartwig, H. Dfirr, Appl. Phys. A45 (1988) 63. H.J. Steffen et al., Phys. Rev. Lett. 68 (1992) 1726. A. Zobelli et al., Phys. Rev. B 75 (2007) 245402. V.H. Crespi et al., Phys. Rev. B 54 (1996) 5927. J.F. Ziegler, J.P. Biersack, U. Littmark, ORNL Report No. CONF-820131. J.P. Biersack, J.F. Ziegler, in: H. Ryssel, H. Glawischnig (Eds.), Springer Ser. Electrophys., Vol. 10, Springer, Berlin, Heidelberg, 1982, p. 122.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
A proposed nuclear nanoprobe with Angstrom resolution q B.L. Doyle a,*, S.M. Foiles a, A.J. Antolak a, L. Popa-Simil b, D.K. Brice c a b c
Sandia National Laboratories, Radiation Solid Interactions Department, P.O. Box 8500, MS 1056 Albuquerque NM 87185, USA LAVM LLC, 3213 Walnut Street, Apt. C, Los Alamos NM 87544-2092, USA Consultant, 115 Charles Street, Sulphur Springs TX 75482-3501, USA
a r t i c l e
i n f o
Available online 19 March 2009 PACS: 07.78.+s Keywords: Ion microscopes Shadow cones Graphene
a b s t r a c t An idea is presented for an entirely new point-projection nuclear microscope that theoretically would have sub-nanometer resolution, approaching one Angstrom. The concept involves using a Kalbitzer super tip as a source of low energy (100 eV) He or H ions that would be transmitted through a molecular sample (e.g. buckyball, carbon nanotube, graphene sheet, DNA molecule, etc.) placed very near the tip (1– 100 nm), and then projected onto a microchannel plate (MCP) screen placed 1 m from the tip. Small angle scattering of the incident ions with atoms in the sample result in the development of shadow cones with an increase in scattered ion intensity at the critical cone angle. The enhanced intensity patterns formed at multiple intersections of cone perimeters are called ‘‘threads”. The shadow cones and threads are projected onto a suitable low-energy ion position sensitive detector to create a ‘‘shadow or thread image” of the sample. Such point-projection microscopes have no aberrations that affect the image, and the magnification would be of the order 1 m/10 nm, or 1.E8! The feasibility of this scheme is currently being studied theoretically using a deterministic atomic scattering model and a Monte Carlo molecular dynamics code. The results of these calculations indicate that only very low energy (60– 120 eV) incident hydrogen ions can be used to avoid displacing atoms in the sample. At these energies, the critical cone angles are quite large and it would be necessary to position the molecular samples very close to the tip (down to only 1 nm) so that the projected ion maps are interpretable as distinct threads. Projecting distinct shadows is not supported by the scattering physics. Published by Elsevier B.V.
1. Introduction The resolution of focused ion beams used for nuclear microscopy is now approaching 10 nm, but this is still orders of magnitude greater than the de Broglie wavelength of such ions. Electron microscopes, on the other hand, are beginning to operate at, or at least very near, their diffraction limit due to the development of lens aberration-correction technologies. However, the approaches used in these aberration corrected electron lenses are not really adaptable for nuclear microprobe lenses. An alternative to focusing the ion beam, and the avoidance of lens aberrations, is the use of point-projection microscopes. Such microscopes have already been developed for electrons [1,2]. Additionally, low-energy ion projection microscopes [3] have been developed that utilize diffraction effects, and higher-energy projection He-ion microscopes have also been proposed [4]. Recently, ‘‘Orion” microscopes
q Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. * Corresponding author. E-mail address: [email protected] (B.L. Doyle).
0168-583X/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.nimb.2009.03.084
[5] have been developed and manufactured by Zeiss for He ion microscopy that are based on the ‘‘Super Tip” concept of Kalbitzer and Knoblauch [6]. These point-projection sources of He+ could also be used for H+ ions. Not only do such sources have brightness orders of magnitude larger than conventional ion sources used in nuclear microscopy, but they also have an initial spot size that approaches an Angstrom. In this paper, we theoretically investigate the feasibility of a H+ point-projection microscope which we call the atomic shadow microscope (AMS). While there are several (and potentially serious) obstacles to realize such a microscope, the sole focus here is to determine whether the physics of atomic scattering supports such a concept. 2. The atomic shadow microscope The fundamental principle of the Atomic Shadow Microscope (ASM) is shown in Fig. 1. Atoms ionized in the vicinity of a Kalbitzer Super Tip are extracted toward a molecular sample placed at a distance D from the tip which can be quite small (10–100 A). When a voltage, U, is applied to the tip, the energy of the extracted ion at the sample is simply eU. As the ions penetrate the sample they are
1996
B.L. Doyle et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1995–1998
Kalbitzer Super Tip
Molecular Sample placed at a distance D from the tip
H+ Ionized within ~1A region of tip.
Shadow cones with increased H intensity at their perimeter are projected onto a screen placed at L.
Fig. 1. H+ ions ionized in the vicinity of a Kalbitzer Super Tip are extracted toward a molecular sample placed very close to the tip. After scattering from the atoms in this sample, shadow cones are projected onto a screen, e.g. a microchannel plate, placed at a great distance from the sample, resulting in magnifications in the 108 range.
deflected (scattered) and cause shadow cones that can be imaged on a detector screen, e.g. a microchannel plate. We note that if the screen is placed at a distance L (centimeters to meter length scale) from the sample, the resulting magnification (L/D) is in the 108 range. It is clear from Fig. 1 that an ASM will need to measure very small shadow cone critical angles. Light ions such as H or He are preferred in this case because the critical cone angle scales, as (Z/ E)1/2 [7] where Z and E are, respectively, the atomic number and energy of the incident ion. Further, the energy of the incident ions will need to be low to prevent the displacement of atoms in the sample. For the following calculations, we consider only targets/ structures that are composed of carbon atoms, such as single layers of graphene, nanotubes or buckyballs. The carbon atom displacement energy thresholds of such nano-structures are angularly dependant [8], but their minimums have been calculated using molecular dynamics to be in the range from 17 to 35 eV [9,10]. This means that maximum energy of H ions should be in the 60–120 eV range. For He ions, the probe energy becomes so low so as to not seem practical due to the large critical angles that result. We therefore restrict our attention only to incident H ions in this energy range.
3. Atomic scattering theory for H on C While high fidelity molecular dynamics simulations would ultimately be needed to determine the feasibility of the ASM, much insight can be gained through theory and calculations that can be performed analytically. In this regard, the key derivation relates the impact parameter, b, to the laboratory scattering angle, h, for hydrogen ion–carbon atom interactions. Taking into account conservation of energy and momentum in the scattering due to the collision of two particles, one finds
Z
1
r min
bdr=r 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 UðrÞ br2 E
ð1Þ
2Eb k 2
ð3Þ
can quite accurately provide scattering angles as a function of impact parameter. Table 1 lists the fitting constants c, bo and P used in the calculations that follow. Fig. 2 shows a plot of the trajectories that result using Eq. (3) for a point source of 100 eV H and a single C atom placed 10 A from this source. It is seen that the shadow cone into which no H ions are scattered is along a line between the source and target, and within an azimuthal angle of 10.5°. The enhanced intensity that occurs just above this critical angle is also apparent in the figure.
Table 1 Parameters used in the equation (h ¼ 180 expfc½lnðbÞ lnðbo ÞP g) to determine the lab scattering angle (degrees) as a function of impact parameter (A) for H on C as a function of energy. E (ev)
bo
P
c
Theta-crit
50 100 200 400 800 1600 3200
0.016834 0.007482 0.004988 0.002217 0.001478 0.000985 0.000438
3.175801 0.348984 3.193423 3.2386 3.015068 2.781297 2.724214
0.03246 0.016758 0.019917 0.013628 0.019462 0.028981 0.026686
12.50003 10.524 8.683282 7.099758 5.689827 4.493111 3.539753
10
100 eV H on C c = .0168 bo = .0075 A P = 3.34
9
where r is the distance separating the particles which interact via the potential U(r) and E is the total energy of the two particles in the center of mass frame. For a pure Coulomb potential, the integral can be evaluated analytically and the resulting expression simplified to give
h ¼ 2arc cot
h ¼ 180 expfc½lnðbÞ lnðbo ÞP g
8 7
y (A)
h¼p2
the calculation of the scattering angle versus impact parameter is more difficult and the final result cannot be expressed analytically; this relationship can only be determined through numerical integration of equation (1). Using the famous Ziegler–Biersack–Littmark universal potential [11,12], we have found the analytic parameterization
6
Critical Angle = 10.52 degrees
5 4
ð2Þ
where k ¼ Z41 pZ2eeo and Z1 = 1, Z2 = 6, and e is the electron charge. Solving these equations for an impact parameter of 1 A and energy of 100 eV gives a scattering angle of 80°; it is clear that Angstrom-level resolution is not possible if the scattering is due to pure Coulomb scattering alone. Fortunately, this is not the case because of the well known screening of this potential, particularly at the low energies being considered here. However, for the screened Coulomb case,
3 2 1 0 0
5
10
15
20
x (A) Fig. 2. Scattering trajectories of 100 eV from C using Eq. (3), and the scattering trajectory of the ASM that has a point source of H placed 10A from the C atom.
B.L. Doyle et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1995–1998
CC-angle
Cone-angle
1.0
Cone/CC angle= 1.32
1 Center Thread
1 Edge Thread
0.5
Shadows
1997
st
st
1.72
st
1 Atomic Thread
2.0
nd
2 Center Thread
Fig. 3. Top: Geometry used to define CC-angle and Cone Critical Angle. Bottom: Patterns expected to be projected for various ratios of Cone to CC-angles.
4. Shadow cone and thread images We now consider the images of the shadows that occur below the critical angle as well as the enhanced scattering that occurs just above the critical angle. Without loss of generality, we perform this analysis for the simple geometry of a graphene target. Graphene is a single basal plane of C atoms of graphite where the atoms are on a hexagonal lattice with an atomic separation of 1.4 A. A geometrical representation of the geometry and projected images formed by the cones and their perimeters is shown in Fig. 3. We define the CCangle to be the angle subtended between two neighboring carbon atoms with the source tip. The figure shows the images that would be projected by the graphene sheet for different ratios of the cone to CC-angles. Fully separated shadow cones should result when the cone to CC-angle ratio is below 0.5, while the cones will just touch when the ratio is exactly 0.5. For a ratio of 1.0, the shadows are replaced by an accumulation of ion intensity formed by the overlap of the enhanced scattering at the shadow cone perimeters (defined as threads) and correspond to the position projected between the
centers of each of the C atom hexagons. For the case shown in Fig. 3, there are six perimeters that superimpose to cause the increased intensity. We call this the ‘‘1st Center Thread”. For a ratio of 1.32, four perimeters form threads at projected positions between each CC atom that we call the ‘‘1st Edge Thread” and, for a ratio of 1.72, six perimeter intensities combine to form threads right at the projected position of the C atoms that is called the ‘‘1st Atom Thread”. When the ratio is 2.0, the 2nd Center Thread forms. This process continues with the generation of 2nd, 3rd, etc. higher order threads of increasing complexity. A simple binary collision approximation can be used to predict the two-dimensional projected images of the scattered proton intensity that would be obtained in the ASM under different conditions. In the following calculations, the graphene target is modeled as a sheet of 54 carbon atoms. The collision model uses Eq. (3) to determine the scattering that would occur from the interaction of H+ at different energies with all 54 C atoms, where one million protons are projected uniformly across the central 4 A of this sheet. The shadows and multi-order threads that are produced are then
Fig. 4. Plot of the Cone Angle as a function of tip-graphene lattice separation for 3200 eV H. Also plotted as lines are the CC-angles as solid lines labeled to the right where the lowest line corresponds to the .5CC angle, the next higher line to the 1CC angle case and so on. The insert shows a two-dimensional calculation of the scattered H intensity using Eq. (3) at a separation of 16 A.
1998
B.L. Doyle et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 1995–1998
Fig. 5. Plot of the CC-angles (again, the lowest line corresponds to the .5CC angle, the next higher line to the 1CC angle case and so on) and Critical Cone Angles for 120 eV H, as a function of the separation between the point source tip and a graphene lattice with a 1.4 A lattice constant. The insert shows a 2D calculation of the thread pattern for the situation at 8 A. This calculation represents the highest energy H that can be used to avoid displacing C atoms, and the largest tip-target combination that produces a reasonable thread image (this is close to the first Center or Gap Thread).
visualized as gray scale intensity maps. Fig. 4 plots the calculated critical cone angles (squares) and CC-angles (lines) for the case of a high-energy (3200 eV) proton beam. While this is not a practical energy because atomic displacements would quickly destroy the sample, it is illustrative of the situation where fully formed shadows can be produced with a tip-target separation of 16A. The resulting intensity distribution is plotted as the insert in the figure, and touching shadows are indeed observed, as expected, because for this separation the cone/CC-angle equals 0.5. For a separation of 200 A, the geometrical pattern indicates that the 1st Center (or Gap) thread would form. Fig. 5 shows a much more practical case where 120 eV H ions are used (this is the highest energy where atomic displacements may be avoided). In this case, the 1st Gap Thread should form with a tip-target separation of 8 A. The intensity distribution, again for all 54 C atoms but aimed only at the central 4A of the sheet, is shown in the insert and the resultant threads are easily identified. 5. Discussion and conclusions Atomic Shadow Microscopy (ASM) is a form of point-projection microscopy where the sample is placed very near a point ion source and projected images (shadows) of the scattered ion intensities are detected at a great distance from the source, potentially providing a huge magnification. The binary collision theory for low energy ion-atom collisions has been parameterized to predict the scattered intensity pattern of ions projected through molecular samples in ASM. We examined the case of using hydrogen ions to shadow image carbon-based molecular structures, but found that extremely low energies are required to prevent the displacement of the carbon atoms. As a result, distinct and well-separated shadows cannot form because the critical shadow cone angles are so large. On the other hand, the model shows that the increased intensity just outside the radius of the shadow cones can combine for distinct critical cone to CC-angle ratios to produce what we call
‘‘Threads”. Consequently, a more appropriate name for the technique might be the ‘‘Atomic Thread Microscope (ATM)”. For a practical ATM operating with 120 eV H ions, the tip-target distance would need to be <10 A to produce such threads. In this case, the electric field between the tip and the target would be 12 eV/A, and it is not clear whether such high fields might deform or destroy the target (note that the typical electric field at the tip used for field ion microscopy is in the 1 eV/A range). In addition, there is uncertainty as to whether the dissociation and ionization of the H2 molecules will remain localized at the very tip of the Kalbitzer Super Tip. While the simple binary atomic scattering theory supports the development of threads of intensity that could be detected and imaged by an ATM, the very small tip-target separation that is required presents a challenging obstacle. At this time, we can say that the ASM is probably not feasible due to the scattering physics, but we can NOt say the same about the ATM. Molecular dynamics calculations are underway that use more realistic potentials and include the possibility of C-H bonds forming. Since these potentials have an attractive component for impact parameters >1 A, the scattering cone critical angles may be sufficiently reduced to improve the viability of an ATM. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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