High-Resolution Imaging and Spectrometry of Materials (Springer Series in Materials Science)

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For Librarians. RSS Feeds. A key feature of the evaporation or field ion images is that the data density is highly inhomogeneous, due to the corrugation of the specimen surface at the atomic scale. This corrugation gives rise to strong electric field gradients in the near-tip zone on the order of an atomic radii or less from the tip , which during ionisation deflects ions away from the electric field normal.

The resultant deflection means that in these regions of high curvature, atomic terraces are belied by a strong anisotropy in the detection density. Where this occurs due to a few atoms on a surface is usually referred to as a "pole", as these are coincident with the crystallographic axes of the specimen FCC , BCC , HCP etc. Where the edges of an atomic terrace causes deflection, a low density line is formed and is termed a "zone line".

These poles and zone-lines, whilst inducing fluctuations in data density in the reconstructed datasets, which can prove problematic during post-analysis, are critical for determining information such as angular magnification, as the crystallographic relationships between features are typically well known. When reconstructing the data, owing to the evaporation of successive layers of material from the sample, the lateral and in-depth reconstruction values are highly anisotropic. Determination of the exact resolution of the instrument is of limited use, as the resolution of the device is set by the physical properties of the material under analysis.

Many designs have been constructed since the method's inception. Initial field ion microscopes, precursors to modern atom probes, were usually glass blown devices developed by individual research laboratories. Optionally, an atom probe may also include laser-optical systems for laser beam targeting and pulsing, if using laser-evaporation methods.

In-situ reaction systems, heaters, or plasma treatment may also be employed for some studies as well as pure noble gas introduction for FIM. Collectable ion volumes were previously limited to several thousand, or tens of thousands of ionic events. Data collection times vary considerably depending upon the experimental conditions and the number of ions collected. Experiments take from a few minutes, to many hours to complete. Atom probe has typically been employed in the chemical analysis of alloy systems at the atomic level.

This has arisen as a result of voltage pulsed atom probes providing good chemical and sufficient spatial information in these materials. Metal samples from large grained alloys may be simple to fabricate, particularly from wire samples, with hand-electropolishing techniques giving good results. Subsequently, atom probe has been used in the analysis of the chemical composition of a wide range of alloys. Such data is critical in determining the effect of alloy constituents in a bulk material, identification of solid-state reaction features, such as solid phase precipitates. Such information may not be amenable to analysis by other means e.

TEM owing to the difficulty in generating a three-dimensional dataset with composition. Semi-conductor materials are often analysable in atom probe, however sample preparation may be more difficult, and interpretation of results may be more complex, particularly if the semi-conductor contains phases which evaporate at differing electric field strengths. Applications such as ion implantation may be used to identify the distribution of dopants inside a semi-conducting material, which is increasingly critical in the correct design of modern nanometre scale electronics.

From Wikipedia, the free encyclopedia. Main article: Field ion microscopy. Brooks Review of Scientific Instruments. Bibcode : RScI Materials Research Society. American Mineralogist. Bibcode : AmMin. Field emission and field ionization. Harvard University Press. Atom probe field Ion Microscopy: Field Ion emission and Surfaces and interfaces at atomic resolution. Cambridge University Press. Bibcode : PhRv.. Annual Review of Materials Research.

Bibcode : AnRMS.. Here only variations are considered which keep the total energy constant, so that the variational principle becomes. However, equation 5 does not lead directly to Euler—Lagrange equations of motion, since not all variations are allowed. A particle optical device usually has an optic axis or some design curve along which a central particle of the beam should travel.

This design curve is parameterized by the arc length z and the position of a particle in the vicinity of the design curve has coordinates x and y along the unit vectors and in a plane perpendicular to this curve. The third coordinate vector is tangential to the design curve and the curvature vector is. Figure 2. The unit vectors in the usual Frenet—Serret comoving coordinate system rotate with the torsion of the design curve.

If this rotation is wound back, the equations of motion do not contain the torsion of the design curve and. The position and the velocity are then.

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The variational principle in equation 5 for the three coordinates x t , y t and z t can now be written for the two coordinates x z and y z. This has the following two advantages: a the particle trajectory along the design curve is usually more important than the particle position at a time t , and b whereas does not allow for all variations of the three coordinates, the total energy can be conserved for all variations of the two coordinates x and y by choosing for each position the appropriate momentum with.

We obtain from equation 5. Since all variations are allowed, the integrand is a very simple new Lagrangian. The integral over for a physical trajectory for which it is extremal, i. It can be computed with equation 7 by where the physical path has to be chosen that connecting the initial coordinates and the final coordinates. It therefore is a function of the initial and final coordinates and the eikonal can therefore be written as. The Maupertuis principle can now be written as.


The variation of coordinates at the final point leads to. The direction of the particle trajectories is therefore perpendicular to the surfaces of constant eikaonl is there is no vector potential. Otherwise the canonical momentum is perpendicular to those surfaces. Extremal fields influence the trajectories by deforming these surfaces. Figure 3. Path of the electron as orthogonal trajectories of the set of surfaces of constant reduced action in the case. The eikonal depending on the initial and final positions is also called the point eikonal.

There are also other eikonals which are produced by Lagandre transformations. The mixed eikonal , where the final position has to be expressed as a function of and. The final position can then be computed by. The index of refraction n , which is well known for light optics, can be generalized to particle optics.

This generalizes Fermat's principle to a principle of least action. Maupertuis variational principle of particle motion is analogous to Fermat's principle of light propagation. According to this principle, a light path leads to an extremum:. Due to the similarity with equation 9 , one defines the index of refraction of charged-particle optics as. A plane in which the all paraxial rays originating in the center of the object plane join in a single point is called an image plane.

The optics between these two planes is then said to be point to point imaging. A plane in which all fundamental rays originating in the object plane with the same slope are imaged to a point is called the diffraction plane. The optics between these two planes is then said to be parallel to point imaging. Particle-optical systems are usually designed so that the paraxial trajectories represent the ideal particle rays. Unfortunately, this course of the rays can never be achieved in a real system owing to the unavoidable nonlinear terms in the equation of motion.

However, it may be possible to eliminate the deviations of the true path from its paraxial approximation at a distinct plane by properly adjusting the distribution of the electromagnetic field in the space between this plane and the initial plane. The problem of determining the optimum field distribution is extremely complicated and has not yet fully been solved.

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Without an insight into the properties of the path deviations it is almost impossible to find a suitable correction method for the resolution-limiting aberrations. It is therefore useful to introduce fundamental paraxial rays which satisfy the linearized equation of motion, which can for example be computed by the second-order expansion of the eikonal in equation 9. Within a multi-stage system, such as the electron microscope, each plane will be imaged repeatedly.

Typical locations are the object plane and the back focal plane of a lens. This plane is an image plane of the source for parallel illumination. We center an aperture at each of the two planes. As the pair of linearly independent trajectories, we select the fundamental rays. In linear approximation, the position u and the transverse momentum conserve the Wronskian:.

This is only possible if an image of the aperture C, i. Accordingly, we can state: an optical system always forms an image of the source in the domain between any two subsequent images of the object plane. The crossover of the cathode forms the effective source, which is formed at some distance from the surface of the emitter. For a field emission gun, the crossover is generally virtual and located inside the tip of the emitter. The condenser system adjusts the illumination of the object. In order to achieve an ideal illumination system, the condenser should consist of two lenses and two apertures, one placed at the image of the crossover, the other at an image of the cathode surface.

The former aperture is imaged in the object plane and limits the field of illumination, whereas the second aperture determines the maximum angle of illumination. This illumination images the surface of the cathode in the back focal plane of the objective lens, and has the advantage that local variations of the electron emission on the cathode surface do not show up as artifacts in the image of the object.

The location of the crossover image can be varied by changing the illumination mode. Figure 5. Scheme of the path of the fundamental paraxial trajectories and location of the images and heam-limiting apertures in a transmission electron microscope illustrating the theorem of alternating images. For Koehler illumination, the back focal plane of the objective lens is also the diffraction plane of the object. In accordance with the famous optician E Abbe, one defines the diffraction pattern at this plane as the 'primary image'.

Owing to the spherical aberration of the objective lens, the large-angle scattered electrons miss the Gaussian image point and blur the image. In order to remove these electrons from the beam, one places an objective aperture at the back focal plane of this lens. Each intermediate image of the object is also an image of the illumination-field aperture, and each image of the illumination-angle aperture coincides with that of the objective aperture.

The special locations of the two illumination apertures allow one to vary the illuminated area in the object plane without affecting the angular illumination and vice versa. The characteristic planes in an electron microscope are, therefore, real and virtual images of the object plane and the crossover plane or those of the two illumination apertures, respectively. It is impossible to form two subsequent images of one of these two planes without having an image of the other plane located between them. The spherical aberration of the objective lens determines the resolution and the off-axial coma the field of view of the recorded image.

The projector lenses introduce primarily distortion. Hence, to obtain ideal imaging, we must compensate for the spherical aberration and the coma of the objective lens and for the distortion of the projector system. We eliminate the distortion by properly exciting the constituent lenses of the projector system. Unfortunately, this is not possible for the spherical aberration due to the Scherzer theorem. Imaging systems which are free of spherical aberration and coma are called aplanats.

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The sine condition gives information about the quality of the image at off-axial points in terms of the properties of the pencil of axial rays. Here denotes the lateral component of the canonical momentum. The gauge of the magnetic vector potential is chosen as. The mixed eikonal V is a function of the four variables x o , y o , p xi and p yi , and of the locations z o and z i of the object plane and of the image plane, respectively. For mathematical simplicity, we express the two-dimensional vectors , and by the complex quantities.

The corresponding conjugate complex quantities are indicated by a bar. Here Re denotes the real part. Since the variations and can be chosen arbitrarily, we derive from the expression 21 the relations. At high-resolution imaging only a very small area of the object is imaged onto the detector. Therefore, we can expand the mixed eikonal in a power series with respect to w o and :. Neglecting the quadratic and higher-order terms in the expansion 23 , we obtain from 22 the relations.

If the system is completely corrected for spherical aberration of any order, all trajectories that originate at the center of the object plane z o intersect the center of the image plane z i. The second expression of 25 shows that this is only the case if. In this case the center of the object plane is perfectly imaged into the center of the image plane. To guarantee that also all object points of a small object area are imaged ideally, the magnification. It follows from the relation 26 that this requirement can only be achieved. Hence the eikonal coefficient of an aplanatic system must have the form.

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  • By inserting the expression 30 into the equation 25 , the condition for aplanatism is given by the simple formula. The lateral component of the vector potential vanishes along the axis, according to the gauge Accordingly, the condition 32 may be replaced by the requirement. This representation is the electron optical analogue of the Abbe sine condition in light optics. In this case the sine condition 34 reduces to the well-known Helmholtz—Lagrange relation, which always holds for any two linearly independent paraxial trajectories.

    To guarantee that all points of an extended object are imaged perfectly into the image plane, it does not suffice to fulfill the sine condition In addition, the second and higher-order terms in the expansion 23 must also be eliminated or sufficiently suppressed. The spherical aberration of a round lens is of third order. Therefore, the direct correction of this aberration requires a field which increases in third order with the distance from the axis.

    An octopole magnet has this feature. However, its lack of cylindrical symmetry does not lead to third-order spherical aberrations when the paraxial optics is spherically symmetric, i. Multipoles of higher-order do not produce third-order aberrations at all and elements of lower order than 2 disturb the spherically symmetric paraxial optics. However, the secondary effects of sextupoles produce spherically symmetric third-order aberrations if paraxial optics is rotationally symmetric.

    The SATEM microscope was the first microscope that used successfully hexapoles for the correction of the spherical aberration. Due to the importance of the resulting improvement of contrast and resolution, experimental results from the SATEM project are shown in this section.

    Electromagnetic fields with threefold symmetry are produced most conveniently within sextupole elements. These elements are generally employed in particle optics to compensate for the primary second-order aberrations arising in systems with a curved axis, such as spectrometers or imaging energy filters. This surprising behavior results from the nonlinear forces of the sextupoles. The combination of the primary second-order deviations produces rotationally symmetric secondary third-order aberrations, which correspond to those of round lenses.

    These secondary aberrations depend quadratic on the sextupole strength which can be adjusted to compensate for the unavoidable spherical aberration of rotationally symmetric electron lenses. However, such a correction improves the imaging properties of the system only if the primary second-order aberrations of the sextupoles vanish as well, and if the fourth-order aberrations can be kept sufficiently small. Therefore, we must design the system in such a way that all axial aberrations are nullified up to the fifth order.

    Optimum designs have been found enabling theoretical resolutions far below the information limit. This limit results from chromatic aberration, mechanical vibrations and electrical instabilities. The correction of the spherical aberration by sextupoles is possible without introducing other multipole elements.

    Therefore, the correcting system is composed exclusively of round lenses and sextupoles. Because the hexapole fields do not affect the paraxial region, the Gaussian optics is entirely determined by the round lenses. Electron microscopes employ magnetic round lenses whose axial magnetic fields rotate the trajectories about the optic axis. Hence, it is advantageous to introduce a rotating u , z -coordinate system. Within the frame of this system the fundamental rays. This requirement is achieved by imposing the initial conditions.

    This plane is located within the field of the objective lens in front of the back-focal plane. We eliminate the isotropic radial component of the off-axial coma most conveniently by matching the so-called coma-free plane of the objective lens with that of the corrector. Unfortunately, such a simple correction does not exist for the anisotropic aqzimuthal component. In order to eliminate this component, we must either double the number of corrector elements order replace the magnetic objective lens by a compound lens consisting of two spatially separated axial fields with opposite sign.

    The second half of this lens can simultaneously be used as a transfer lens for imaging the coma-free plane into any given plane behind the objective lens. Moreover, we require that the paraxial ray. The sextupole corrector can only be utilized in the TEM if all primary threefold path deviations cancel out in the region behind the corrector. The course of these second-order path deviation u 2 z along the optic axis depends on the arrangement of the round transfer lenses and on the location of the sextupoles.

    The primary action of electric and magnetic sextupoles is given by the hexapole strength. Considering the relations 35 and 41 for the fundamental paraxial rays and the ray parameter, respectively, we derive from the expression the second-order path deviation in the rotating coordinate system.

    The second-order fundamental rays u 11 , u 12 and u 22 are given by the integral expressions. We satisfy these requirements most easily by imposing symmetry conditions on the paraxial fundamental rays and on the total hexapole strength H. For this purpose we choose the sextupole fields and the fundamental rays in such a way that the integrands of the integrals 47 are either antisymmetric with respect to the midplane of the sextupole arrangement, or with respect to the central planes of each half of the system.

    Therefore, it is not possible to eliminate all second-order aberrations by a single symmetry condition. However, if we choose the paraxial path in such a way that one of the two fundamental rays is symmetric and the other antisymmetric with respect to the symmetry plane and the central planes of each half of the system within the hexapole fields, all second-order fundamental rays can be eliminated outside of the sextupole system. This is achieved if the sextupole fields are symmetric with respect to the symmetry planes. The plane midway between these lenses is the midplane of the 4 f -system, while the outer focal planes represent the center planes of each half of the sextupole system.

    Accordingly, the sextupole fields and the paraxial fundamental rays fulfill the requirement for complete elimination of the second-order aberrations. Figure 6. The outer focal points coincide with the nodal points N 1 and N 2 of the telescopic round lens doublet. To avoid a rotation of the image of the first sextupole, the coils of the round lenses must be connected in series opposition so that the excitations of the two lenses are equal and opposite, whatever is the strength of the current. Hence the front focal plane is also the coma-free plane of the 4 f -arrangement.

    To demonstrate this behavior, we must also calculate the secondary aberrations of the system. For determining these aberrations we need to know the second-order path deviation u 2 z. The course of the rays u 11 and u 22 is symmetric while that of the 'mixed' ray u 12 is antisymmetric with respect to the midplane z m.

    Owing to this symmetry the system does neither introduce off-axial coma nor third-order distortion at the image plane, as will be shown in the next section. The hexapole strength is a free parameter, which can be adjusted in such a way that the corrector compensates for the spherical aberration of the entire system. Figure 7. In electron lithography the most disturbing aberrations are the image curvature and the field astigmatism because they decisively limit the usable area of the mask.

    Unfortunately,xyb the third-order image curvature of rotationally symmetric systems is unavoidable and its coefficient has the same sign as that of the spherical aberration. Hence a rotationally symmetric planar system does not exist. A planar system is corrected for image curvature, field astigmatism and coma. Since sextupoles can correct the spherical aberration of round lenses, the question arises if these elements can also be used to eliminate the unavoidable third-order field curvature of round lenses. However, to obtain a planar field of view we must also compensate for the field astigmatism.

    Because the corresponding aberration coefficient is complex for magnetic lenses, we need three free parameters to simultaneously compensate for both image curvature and field astigmatism. The arrangement consists of four identical round lenses forming an 8f-system and five sextupoles, which are centered symmetrically about the midplane z m. Since the azimuthal orientation of the sextupoles S 2 and S 4 may differ from that of the sextupoles S 1 , S 3 and S 5 , we have three free parameters H 1 , H 2 and.

    However, this does not necessarily imply that it is possible to eliminate the image curvature and the field astigmatism because nonlinear relations exist between the coefficients of these aberrations and the hexapole strengths. As a result, only few systems can be found, which enable the correction of the field aberrations. Figure 8. Hexapole planator compensating for the third-order image curvature and field astigmatism. The primary aberrations of systems with threefold symmetry are of second order. Since these aberrations are large compared with the third-order aberrations produced by the rotationally symmetric fields, it is necessary to eliminate all second-order aberrations first before dealing with the third-order aberrations.

    Hence the hexapole fields produce exactly the same third-order aberrations as the round lenses. This surprising behavior results from the nonlinear forces of the hexapole fields. Unfortunately, the spherical aberration produced by a sequence of sextupoles has the same sign as that of the round lenses if the second-order aberrations are eliminated. However, it is possible to reverse the sign of the spherical aberration by employing sextupoles in combination with round lenses. It should be noted that this notation has been suggested much earlier by Scherzer in his lectures on electron optics.

    The coefficient C R is associated with spherical aberration, K R with off-axial coma, A R with field astigmatism, F R with field curvature, D R with distortion and E R with spherical aberration in the diffraction plane. According to the relation 48 , this coefficient does not affect the aberrations at the image plane. The resulting Larmor rotation causes a rotation of the aberration figures of coma and astigmatism. The angle of rotation with respect to the line intersecting the optic axis and the Gaussian image point is proportional to the imaginary part of the corresponding aberration coefficient.

    It should be noted that the third-order aberration coefficients are defined as the negative values of the expansion coefficients of the eikonal. The eikonal term 52 can be evaluated analytically if we employ the s harp c ut- o ff f ringing f ield SCOFF approximation and assume that the rotationally symmetric fields do not overlap with the hexapole fields.

    In this case the paraxial fundamental trajectories form straight lines inside the field region of the sextupoles. This behavior is due to the fact that for this system the terms Hu 1 u 2 u 11 , Hu 1 u 2 u 22 , Hu 1 2 u 12 and Hu 2 2 u 11 in the integrand of the eikonal term 52 are antisymmetric functions.

    Hence their contribution to the integral cancels out. By means of these relations and the expressions for the secondary fundamental rays, the fourth-order eikonal term 52 can be evaluated analytically. The comparison of the result,. These expressions depend quadratically on the hexapole strength H and have always negative sign, apart from the coefficient F H of the field curvature. On the other hand the total spherical aberration of the system. In order that the off-axial coma of the entire system vanishes as well, the round lens coma must be made zero.

    This is the case if the coma-free plane of the objective lens matches with the corresponding plane of the corrector located in the center of the first sextupole. Since the coma-free plane of a conventional objective lens is located within its field, it is necessary to image this plane into the front focal plane of the telescopic round lens doublet of the sextupole corrector without introducing any additional coma. However, this procedure only eliminates the radial isotropic component of the coma.

    The anisotropic coma of the objective lens can only partly be compensated by that of the weak lenses of the transfer doublet. Since the second half of the lens can simultaneously be used as the first lens of the transfer doublet, the number of coils is not increased by this concept. Figure 9. The distortion does not affect the resolution of the image, but it deforms the geometrical structure of the imaged object. In high-resolution electron microscopy only the projector lens contributes significantly to the distortion. This aberration becomes negligibly small if the projector lens operates in such a way that an image of the effective source is located inside the field of this lens.

    On the other hand, the distortion is of major concern in projection electron lithography, where a large mask is imaged on the wafer with a reduced scale in the order of 4— In this case almost all lenses of the system contribute appreciably to the distortion. However, it causes a distortion in any defocused image. In order to avoid an appreciable distortion by changing the defocus, the coefficient E 3 must be kept sufficiently small.

    Fortunately, this can be achieved, because the sign of the coefficient E H is opposite to that of E R. For a projection system with vertical landing angle and parallel illumination of the mask, the coefficient E R is unavoidable and of positive sign just as the coefficient C R of the third-order spherical aberration. It is striking that there are two special points for each imaging optics. One is free of coma, the so-called coma-free point and the other is free of a chromatic aberration, the so-called achromatic point of magnification.

    To avoid this reduction of the useful image area, a C s correction should be combined with a correction of the Coma coefficient K. To achieve a uniform imaging of all object points regardless of their lateral position, it is necessary to eliminate all off-axial aberrations. For a system consisting of round lenses this is not possible because the image curvature is unavoidable in this case.

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