Method for improving transmission Kikuchi diffraction pattern

Abstract

The present invention refers to a method for improving a Transmission Kikuchi Diffraction, TKD pattern, wherein the method comprises the steps of: Detecting a TKD pattern (20b) of a sample (12) in an electron microscope (60) comprising at least one active electron lens (61) focusing an electron beam (80) in z-direction on a sample (12) positioned in distance D below the electron lens (61), the detected TKD (20b) pattern comprising a plurality of image points x.sub.D, y.sub.D and mapping each of the detected image points x.sub.D, y.sub.D to an image point of an improved TKD pattern (20a) with the coordinates x.sub.0, y.sub.0 by using and inverting generalized terms of the form x.sub.D=γ*A+(1−γ)*B and y.sub.D=γ*C+(1−γ)*D wherein $\gamma =\frac{Z}{D}$
with Z being an extension in the z-direction of a cylindrically symmetric magnetic field B.sub.Z of the electron lens (61), and wherein A, B, C, D are trigonometric expressions depending on the coordinates x.sub.0, y.sub.0, with B and D defining a rotation around a symmetry axis of the magnetic field B.sub.Z, and with A and C defining a combined rotation and contraction operation with respect to the symmetry axis of the magnetic field B.sub.Z. The invention further relates to a measurement system, computer program and computer-readable medium for carrying out the method of the invention.

Claims

1. Method of improving a Transmission Kikuchi Diffraction, TKD, pattern with the steps: Detecting a TKD pattern (20b) of a sample (12) in an electron microscope (60) comprising at least one active electron lens (61) focusing an electron beam (80) in z-direction on the sample (12) positioned in distance D below the electron lens (61), the detected TKD pattern (20b) comprising a plurality of image points x.sub.D, y.sub.D; and Mapping each of the image points x.sub.D, y.sub.D to an image point of an improved TKD pattern (20a) with coordinates x.sub.0, y.sub.0 by using generalized terms of the form x.sub.D=γ*A+(1−y)*B and y.sub.D=γ*C+(1−y)*D, wherein $\gamma =\frac{Z}{D}$ with L being an extension in the z-direction beyond the sample of a cylindrically symmetric magnetic field B.sub.Z of the electron lens (61), and wherein A, B, C, D are trigonometric expressions depending on the coordinates x.sub.0, y.sub.0, with B and D defining a rotation around a symmetry axis of the magnetic field B.sub.Z, and with A and C defining a combined rotation and contraction operation with respect to the symmetry axis of the magnetic field B.sub.Z.

2. Method of claim 1, wherein each of the trigonometric expressions A, B, C and D further depend on a parameter β which expresses the strength of the magnetic field B.sub.Z and with v being the velocity of an electron of the electron beam (80).

3. Method of claim 2, further comprising the steps of: Detecting a calibration TKD pattern of the sample (12) in the electron microscope (60) without the active electron lens (61), the calibration TKD pattern comprising a plurality of image points x.sub.C, y.sub.C; Performing a plurality of mapping operations on the detected TKD pattern (20b) using the generalized terms, wherein each mapping operation is performed with different set of parameters γ and β; Comparing, for each mapping operation, an output TKD pattern with the calibration TKD pattern and determining one parameter set γ and β based on the comparison; and Determining the improved TKD pattern (20a) by using the determined parameter set.

4. Method according to claim 3, wherein the comparison is performed by: Image correlation of each of the output TKD pattern and the calibration TKD pattern; and Determining the one parameter set providing the highest degree of image correlation.

5. Method of claim 1, further comprising the steps of: Determining a plurality of diffraction bands (21b) from the detected TKD pattern (20b), the diffraction bands (21b) comprising a plurality of image points x.sub.D.sup.KB, y.sub.D.sup.KB; Determining, from the improved TKD pattern (20a), a plurality of corresponding diffraction bands comprising a plurality of image points x.sub.0.sup.KB, y.sub.0.sup.KB; Comparing the diffraction bands and the corresponding diffraction bands; Determining one parameter set γ and β based on the comparison; and Determining the improved TKD pattern (20a) by using the determined parameter set.

6. Method according to claim 3, wherein the comparison is performed by: Determining, for each of the output TKD pattern, a straightness of diffraction bands and determining the one parameter set providing the straightest diffraction bands; or Determining crystal phase information for each of the output TKD pattern and the calibration TKD pattern and determining the one parameter set providing crystal phase information matching those of the calibration TKD pattern.

7. Method of claim 1, wherein the magnetic field is presumed as B=(0, 0, B.sub.Z).

8. Method of claim 1, wherein the magnetic field is B=B(z) with B=0 for any z>Z.

9. Method of claim 1, wherein Z<D.

10. Method of claim 1, wherein the trigonometric expression A is of the form $x_0\frac{\sin \phi }{\phi }+y_0\frac{1-\cos \phi }{\phi },$ wherein the trigonometric expression B is of the form x.sub.0 cos φ+y.sub.0 sin φ, wherein the trigonometric expression C is of the form $-x_0\frac{1-\cos \phi }{\phi }+y_0\frac{\sin \phi }{\phi },$ and wherein the trigonometric expression D is of the form −x.sub.0 sin φ+y.sub.0 cos φ.

11. Method of claim 10, wherein φ is expressed as φ= $\phi =\beta \sqrt{1+\frac{r^2}{D^2}}$ with r denoting a horizontal distance of an improved image point x.sub.o, y.sub.o from the symmetry axis.

12. Measurement system, comprising a scanning electron microscope, EM, (60), with at least one electron lens (61), a TKD detector (64) configured for detecting a TKD pattern of a sample (12) positioned in distance D below the electron lens (61), and a control unit configured to perform the method for improving a Transmission Kikuchi Diffraction, TKD, pattern according to claim 1.

13. Measurement system of claim 12, wherein the electron lens (61) is configured to focus an electron beam (80) in the z-direction onto the sample (12).

14. A computer-readable medium comprising instructions which, when executed by a computer, cause the computer to carry out the method of claim 1.

Description

DESCRIPTION OF THE DRAWINGS

(1) The features of the invention become apparent to those skilled in the art by the detailed description of exemplary embodiments with reference to the attached drawings in which:

(2) FIG. 1 illustrates the warping of a TKD pattern by a magnetic field of an electron optic;

(3) FIG. 2 is a TKD and EDS measurement system according to an example;

(4) FIG. 3 schematically illustrates the steps performed in the method of the invention.

DETAILED DESCRIPTION OF THE INVENTION

(5) Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. Effects and features of the exemplary embodiments, and implementation methods thereof will be described with reference to the accompanying drawings. In the drawings, like reference numerals denote like elements, and redundant descriptions are omitted. The present invention, however, may be embodied in various different forms, and should not be construed as being limited to only the illustrated embodiments. Rather, these embodiments are provided as examples so that this disclosure will be complete, and will fully convey the features of the present invention to those skilled in the art.

(6) Accordingly, processes, elements, and techniques that are not considered necessary to those having ordinary skill in the art for a complete understanding of the aspects and features of the present invention may not be described. As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. Further, in the following description of embodiments the use of “may” when describing embodiments of the present invention refers to “one or more embodiments of the present invention” and terms of a singular form may include plural forms unless the context clearly indicates otherwise.

(7) It will be understood that although the terms “first” and “second” are used to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another element. For example, a first element may be named a second element and, similarly, a second element may be named a first element, without departing from the scope of the present invention. Expressions such as “at least one of,” when preceding a list of elements, modify the entire list of elements and do not modify the individual elements of the list. The term “substantially”, “about,” and similar terms are used as terms of approximation and not as terms of degree, and are intended to account for the inherent deviations in measured or calculated values that are recognized by those skilled in the art.

(8) FIG. 1 illustrates the warping of a TKD pattern by a magnetic field of an electron lens. Particularly, FIG. 1A schematically illustrates a TKD pattern 20a that has been obtained in the absence of any magnetic field and hence comprises a plurality of straight Kikuchi lines 21a. The Kikuchi lines 21a have an angular width that corresponds to two times the Bragg angle of the corresponding crystal plane of the crystal lattice reflecting the electrons for producing that band 21. FIG. 1B illustrates a TKD pattern 20b that was obtained in the same experimental setup and for the same measurement point of a sample in the presence of a magnetic field. The so obtained warped TKD pattern 20b comprises a plurality of bent (warped) Diffraction bands 21b. A detection of a similar experimental Kikuchi pattern 20 using Transmission Kikuchi Diffraction, TKD, is described with respect to FIG. 2.

(9) FIG. 2 illustrates a combined TKD and EDS measurement system mounted to an electron microscope 60 according to an example of the present invention. According to FIG. 2 an electron microscope, EM, 60, i.e., a scanning electron microscope (SEM), is configured to perform transmission Kikuchi diffraction (TKD) measurements. A pole piece 62, which is part of an electron lens 61 of the EM 60, is arranged at a column of the EM 60. Moreover the EM 60 comprises a sample holder 10 and a TKD detector 64, which comprises a phosphor screen 65. Moreover the EM 60 comprises an EDS (energy dispersive X-ray spectroscopy) detector 67. The EM 60 is configured to perform EDS measurements with the EDS detector 67 and to perform TKD measurements with the TKD detector 64. A sample 12, the TKD detector 64, the EDS detector 67, and the column comprising the electron lenses 61 with the pole piece 62, are arranged in a way, such that TKD measurements and EDS measurements can be made without changing the position of a sample holder 10. Particularly, a sample holder 10 is positioned between the EDS detector 67 and the TKD detector 64 such that a sample 12 loaded to the sample holder 10 is positioned between an active area 68 of the EDS detector 67 and a phosphor screen 65 of the TKD detector 64. Particularly, the sample 12 is positioned between the EDS detector 67 and the TKD detector 64 with respect to the propagation direction of an electron beam 80 emitted by the electron microscope 60, particularly from a pole piece 62 of the EM 60, and focused by magnetic lens 61.

(10) The sample 12 is prepared to be electron transparent such that at least part of the incident electron beam 80 is transmitted through the sample 12 and positioned such that the primary electron beam 80 is incident on the sample 12. Depending mostly on the sample thickness, material make-up and incident electron energy, the incident primary electrons are traversing the sample 12 and hence diffracted electrons 82 exit the sample 12 via an exit surface thereof facing the phosphor screen 65 of the TKD detector 64. These transmitted and diffracted electrons 82 allow for detecting Kikuchi patterns of the sample 12 via the TKD detector 64. The incident primary electron beam 80 also effects the generation of characteristic X-rays of the sample 12. The characteristic X-rays exiting the sample 12 via a top surface propagate towards the active area 68 of EDS detector 67, thus allowing to obtain EDS spectra from the sample 12 and to perform an element composition analysis on the sample 12.

(11) However, due to the electron lens 61 of pole piece 62 employing magnetic fields for scanning the incident electron beam 80 over the sample 12, any TKD pattern obtained by TKD detector 64 is prone to be warped as illustrated in FIG. 1B and hence may not be suitable for providing high precision crystal phase information for the sample 12. Hence, for high precision structure and material analysis of sample 12 based on Kikuchi patterns obtained with TKD detector 64 a method of improving Transmission Kikuchi patterns is required.

(12) FIG. 3 schematically illustrates the steps performed in the method of improving Transmission Kikuchi, TKD, patterns according to the present invention.

(13) In a first step S100 of the method of the invention, a TKD pattern of sample 12 is detected in the electron microscope 60 as described already above with respect to FIG. 2. Particularly, the electron lens 61 of EM 60 focusses the electron beam 80 in z-direction on the sample 12 positioned in distance D below the electron lens 61 via the sample holder 10. The diffracted electrons 82 exiting the sample 12 via a rear side thereof facing the TKD detector 64 are detected via the phosphor screen 65 of the TKD detector 64. As set forth above, the detected TKD pattern 20b is distorted by the magnetic field of the electron lens 61 and hence comprises a plurality of warped Diffraction bands 21b as shown in FIG. 1B. Above that the detected Kikuchi pattern comprises a plurality of image points with the coordinates x.sub.D and y.sub.D.

(14) In step S200 an improved TKD pattern is calculated based on the TKD pattern detected in step S100. These mapping of the disturbed TKD pattern on the improved (undisturbed) TKD patterns is performed using the generalized terms that are mapping each of the image points x.sub.0, y.sub.0 of an undisturbed (improved) TKD pattern onto image points x.sub.D, y.sub.D of a corresponding detected (disturbed) TKD pattern. In the following the derivation of these generalized terms shall be explained for the measurement as described with respect to FIG. 2, i.e., the linear setup of EM 60 comprising an electron lens 61 that is emitting electron beam 80 onto sample 12 positioned on a z-axis between the electron lens 61 and the phosphor screen 65 of the TKD detector 64.

(15) As already illustrated in FIG. 2, for deriving the generalized terms, a magnetic field of the electron lens is approximated for a reference system with a coordinate system origin being located in the intersection point between the electron beam and the sample. The positive z-direction points downwards, i.e., in the propagation direction of the electron beam, such that the x- and y-directions are positioned in the horizontal plane of the reference system. The origin of the coordinate system is centred on the lower sample surface. In such a system, the magnetic field of the electron microscope is considered to have the form:
B(r)=B(r)e.sub.z  (1)

(16) The magnetic field is thus parallel to the z-direction and is considered to be at least partially constant. Particularly, the field is assumed to be constant until z=Z and to be zero for z>Z:

(17) $\begin{array}{cc}B\left(r\right)=\{\begin{array}{cc}{\mathrm{Be}}_z& \mathrm{for}z\le Z\\ 0& \mathrm{elsewhere}\end{array}& \left(2\right)\end{array}$

(18) A detector for capturing the diffracted electrons in transmission direction is positioned below the sample at position D and thus an electron travels between z-coordinates 0 and D. While travelling in the magnetic field, the so-called Lorentz force is acting on the electron:
F=q(v×B)  (3)

(19) (with q being the electron charge) and hence the equation of motion of the electron can be written as:

(20) $\begin{array}{cc}0=\stackrel{\mathrm{.Math.}}{r}-\frac{q}{m}\left(\stackrel{.}{r}\times B\right)& \left(4\right)\end{array}$

(21) In the magnetic field B(r) as indicated above, the equation of motion simplifies to:

(22) $\begin{array}{cc}0=\stackrel{\mathrm{.Math.}}{r}-\frac{\mathrm{qB}}{m}\left(\stackrel{.}{r}\times e_z\right)\mathrm{or}& \left(5\right)\\ 0=\left(\begin{array}{c}\stackrel{\mathrm{.Math.}}{x}\\ \stackrel{\mathrm{.Math.}}{y}\\ \stackrel{\mathrm{.Math.}}{z}\end{array}\right)-\frac{\mathrm{qB}}{m}\left(\begin{array}{c}\stackrel{.}{y}\\ -\stackrel{.}{x}\\ 0\end{array}\right)& \left(6\right)\end{array}$

(23) It shall be noted that the equation of motion is only presented herein as intermediate step to deriving the generalized terms actually used for improving the TKD patterns in the method of the present invention, particularly using a parameter set based on a calibration pattern. However, no equations of motion are used in the method of the invention.

(24) By introducing the circular frequency ω as given below in (7), the equation (6) can be solved by using the ansatz as defined by equations (8) to (10) below:

(25) $\begin{array}{cc}\omega =\frac{\mathrm{qB}}{m}& \left(7\right)\\ \stackrel{.}{x}=a_x\cos \omega t+b_x\sin \omega t& \left(8\right)\\ \stackrel{.}{y}=a_y\cos \omega t+b_y\sin \omega t& \left(9\right)\\ \stackrel{.}{z}=\mathrm{const}& \left(10\right)\end{array}$

(26) Using the expressions as given by formulas (8) to (10), the differential equation of formula (6) is reduced to an algebraic system of equations as shown below:
0=ω(−a.sub.x sin ωt+b.sub.x cos ωt)−ω(a.sub.y cos ωt+b.sub.y sin ωt)  (11)
0=ω(−a.sub.y sin ωt+b.sub.y cos ωt)+ω(a.sub.x cos ωt+b.sub.x sin ωt)  (12)
or, by rearrangement, to a system of equations:
0=−(a.sub.x+b.sub.y)sin ωt+(b.sub.x−a.sub.y)cos ωt  (13)
0=−(a.sub.y−b.sub.x)sin ωt+(b.sub.y+a.sub.x)cos ωt  (14)

(27) In order to fulfil equations (13) and (14), the bracketed terms before the sine and the bracketed terms before the cosine term have to be equal to zero, respectively. Hence, one finds:
b.sub.y=−a.sub.x  (15)
a.sub.y=b.sub.x  (16)

(28) Thereby, the equations (8) and (9) become:
{dot over (x)}=a.sub.x cos ωt+b.sub.x sin ωt  (17)
{dot over (y)}=b.sub.x cos ωt−a.sub.x sin ωt  (18)

(29) For the initial time point t=0 one then finds:
{dot over (x)}(0)=v.sub.x  (19)
{dot over (y)}(0)=v.sub.y  (20)
ż(0)=v.sub.z  (21)

(30) Which allows determining the constants in the equations (17) and (18) to:
a.sub.x=v.sub.x  (22)
b.sub.x=v.sub.y  (23)

(31) Hence, the velocity of the electrons in the magnetic field is given by:
{dot over (x)}=v.sub.x cos ωt+v.sub.y sin ωt  (24)
{dot over (y)}=−v.sub.x sin ωt+v.sub.y cos ωt  (25)
ż=v.sub.z  (26)

(32) And thus the trajectory of the electron can be found to be:

(33) $\begin{array}{cc}x\left(t\right)=A_x+\frac{v_x\sin \omega t}{\omega }-\frac{v_y\cos \omega t}{\omega }& \left(27\right)\\ y\left(t\right)=A_y+\frac{v_x\cos \omega t}{\omega }+\frac{v_y\sin \omega t}{\omega }& \left(28\right)\\ z\left(t\right)=A_z+v_{z}t& \left(29\right)\end{array}$

(34) Again it shall be noted that these trajectories are only presented herein as intermediate step to deriving the generalized terms actually used for improving the TKD patterns in the method of the present invention, particularly using a parameter set based on a calibration pattern. However, no trajectories of electrons are used in the method of the invention.

(35) By using the boundary conditions x(0)=0, y(0)=0) and z(0)=0 one can derive that:

(36) $\begin{array}{cc}0=A_x-\frac{v_y}{\omega }& \left(30\right)\\ 0=A_y+\frac{v_x}{\omega }& \left(31\right)\\ 0=A_z& \left(32\right)\end{array}$
and hence:

(37) 0$\begin{array}{cc}x\left(t\right)=\frac{v_x\sin \omega t}{\omega }+\frac{v_y\left(1-\cos \omega t\right)}{\omega }& \left(33\right)\\ y\left(t\right)=-\frac{v_x\left(1-\cos \omega t\right)}{\omega }+\frac{v_y\sin \omega t}{\omega }& \left(34\right)\\ z\left(t\right)=v_z\left(t\right)& \left(35\right)\end{array}$

(38) At the z-coordinate z=Z, that is at time T.sub.Z=Z/v.sub.z the electron leaves the magnetic field. The respective coordinates and velocities can thus be written as:

(39) $\begin{array}{cc}x\left(T_Z\right)=\frac{v_x\sin \omega T_Z}{\omega }+\frac{v_y\left(1-\cos \omega T_Z\right)}{\omega }& \left(36\right)\\ y\left(T_Z\right)=-\frac{v_x\left(1-\cos \omega T_Z\right)}{\omega }+\frac{v_y\sin \omega T_Z}{\omega }& \left(37\right)\\ z\left(T_Z\right)=Z& \left(38\right)\\ \stackrel{.}{x}\left(T_Z\right)=v_x\cos \omega T_Z+v_y\omega T_Z& \left(39\right)\\ \stackrel{.}{y}\left(T_Z\right)=-v_x\sin \omega T_Z+v_y\cos \omega T_Z& \left(40\right)\\ \stackrel{.}{z}\left(T_Z\right)=v_Z& \left(41\right)\end{array}$

(40) Hence in the field free space the trajectory of the electron becomes:
x(t)=x(T.sub.Z)+(t−T.sub.Z){dot over (x)}(T.sub.Z)  (42)
y(t)=y(T.sub.Z)+(t−T.sub.Z){dot over (y)}(T.sub.Z)  (43)
z(t)=z(T.sub.Z)+(t−T.sub.Z)ż(T.sub.Z)  (44)

(41) After inputting the terms for the coordinates and velocities at T.sub.Z one gets:

(42) $\begin{array}{cc}x\left(t\right)=\frac{v_x\sin \omega T_Z}{\omega }+\frac{v_y\left(1-\cos \omega T_Z\right)}{\omega }+\left(t-T_Z\right)\left(v_x\cos \omega T_Z+v_y\sin \omega T_Z\right)& \left(45\right)\\ y\left(t\right)=-\frac{v_x\left(1-\cos \omega T_Z\right)}{\omega }+\frac{v_y\sin \omega T_Z}{\omega }+\left(t-T_Z\right)\left(-v_x\sin \omega T_Z+v_y\cos \omega T_Z\right)& \left(46\right)\\ z\left(t\right)=Z+\left(t-T_Z\right)v_z& \left(47\right)\end{array}$

(43) Wherein the last equation simplifies to:
z(t)=v.sub.zt  (48)

(44) At the location z=D, that is at time T.sub.D=D/v.sub.z the electron hits the detector surface. The location of impact on the detector surface can be written as:

(45) $\begin{array}{cc}x\left(T_D\right)=\frac{v_x\sin \omega T_Z}{\omega }+\frac{v_y\left(1-\cos \omega T_Z\right)}{\omega }+\left(T_D-T_Z\right)\left(v_x\cos \omega T_Z+v_y\sin \omega T_Z\right)& \left(49\right)\\ y\left(T_D\right)=-\frac{v_x\left(1-\cos \omega T_Z\right)}{\omega }+\frac{v_y\sin \omega T_Z}{\omega }+\left(T_D-T_Z\right)\left(-v_x\sin \omega T_Z+v_y\cos \omega T_Z\right)& \left(50\right)\end{array}$

(46) Whereas with a non-existing magnetic field B=0 or ω=0 the location of impact would be:
x.sub.0=x.sub.ω=0(T.sub.D)  (51)
x.sub.0=y.sub.ω=0(T.sub.D)  (52)
and hence following from the equations (49) and (50) one gets:
x.sub.0=v.sub.xT.sub.D  (53)
y.sub.0=v.sub.yT.sub.D  (54)
which can be rewritten by inputting T.sub.D=D/v.sub.z into the equations as:

(47) $\begin{array}{cc}{x}_0=\frac{{\mathrm{Dv}}_x}{v_z}& \left(55\right)\\ y_0=\frac{{\mathrm{Dv}}_y}{v_z}& \left(56\right)\end{array}$

(48) These three velocity components are not independent of each other. Particularly, the total velocity takes the value v:

(49) $\begin{array}{cc}\begin{array}{c}{v}^2=v_x^2+v_y^2+v_z^2\\ =\frac{v_z^2{x}_0^2}{D^2}+\frac{v_z^2{y}_0^2}{D^2}+v_z^2\\ =v_z^2\left(1+\frac{x_0^2+y_0^2}{D^2}\right)\end{array}\hspace{1em}& \begin{array}{c}\left(57\right)\\ \left(58\right)\\ \\ \left(59\right)\end{array}\end{array}$

(50) By simplifying
r.sup.2=x.sub.0.sup.2+y.sub.0.sup.2  (60)

(51) One thus gets:

(52) $\begin{array}{cc}{v}_z^2=\frac{v^2}{1+r^2/D^2}& \left(61\right)\end{array}$

(53) Hence, the further away from the centre the undisturbed image point is, the smaller the z-component of the velocity is and hence the larger the distortion should be.

(54) $\begin{array}{cc}x\left(T_D\right)=x_0\frac{v_z\sin \omega Z/v_z}{\omega D}+y_0\frac{v_z\left(1-\cos \omega Z/v_z\right)}{\omega D}+\frac{D-Z}{D}\left(x_0\cos \omega Z/v_z+y_0\sin \omega Z/v_z\right)& \left(62\right)\\ y\left(T_D\right)=-x_0\frac{v_z\left(1-\cos \omega T_Z\right)}{\omega D}+y_0\frac{v_z\sin \omega T_Z}{\omega D}+\frac{D-Z}{D}\left(-x_0\sin \omega Z/v_z+y_0\cos \omega Z/v_z\right)& \left(63\right)\end{array}$

(55) By introducing a mixing parameter γ as:
Z=γD  (64)

(56) One can rewrite the equations (62) and (63) as:

(57) $\begin{array}{cc}x\left(T_D\right)=x_0\frac{v_z\sin \gamma \omega D/v_z}{\omega D}+y_0\frac{v_z\left(1-\cos \gamma \omega D/v_z\right)}{\omega D}+\left(1-\gamma \right)\left(x_0\cos \gamma \omega D/v_z+y_0\sin \gamma \omega D/v_z\right)& \left(65\right)\\ x\left(T_D\right)=-x_0\frac{v_z\left(1-\cos \gamma \omega D/v_z\right)}{\omega D}+y_0\frac{v_z\sin \gamma \omega D/v_z}{\omega D}+\left(1-\gamma \right)\left(-x_0\sin \gamma \omega D/v_z+y_0\cos \gamma \omega D/v_z\right)& \left(66\right)\end{array}$

(58) By denoting the repeated argument of the angular function as φ:

(59) $\begin{array}{cc}\begin{array}{c}\phi =\frac{\mathrm{\gamma \omega }D}{v_z}\\ =\frac{\mathrm{\gamma \omega }D}{v}\sqrt{1+\frac{r^2}{D^2}}=\beta \sqrt{1+\frac{r^2}{D^2}}\end{array}\hspace{1em}& \begin{array}{c}\left(67\right)\\ \\ \left(68\right)\end{array}\end{array}$

(60) This introduces the parameter

(61) 0$\begin{array}{cc}\beta =\frac{\mathrm{\gamma \omega }D}{v_z}.& \left(67\right)\end{array}$
The result further simplifies to:

(62) $\begin{array}{cc}x\left(T_D\right)=\gamma \left[x_0\frac{\sin \phi }{\phi }+y_0\frac{1-\cos \phi }{\phi }\right]+\left(1-\gamma \right)\left(x_0\cos \phi +y_0\sin \phi \right)& \left(69\right)\\ y\left(T_D\right)=\gamma \left[-x_0\frac{1-\cos \phi }{\phi }+y_0\frac{\sin \phi }{\phi }\right]+\left(1-\gamma \right)\left(-x_0\sin \phi +y_0\cos \phi \right)& \left(70\right)\end{array}$

(63) This result can be interpreted as a mixture of a pure rotation defined by the term after factor (1−γ), i.e., terms A and C as defined above, and a more complicated mixture of a rotation and contraction defined by the term after factor γ, i.e., terms B and D as defined above. The angle of rotation is proportional to the magnetic field strength (parameter ω) and the tilt of the electrons (factor √{square root over (1+r.sup.2/D.sup.2)}). As the term representing the mixed rotation and contraction is divided by φ, the image is contracted more with increasing values of r.

(64) By inputting each of the image points x.sub.D, y.sub.D of a detected disturbed TKD pattern (denoted in equations (69) and (70) by x(T.sub.D) and y(T.sub.D), respectively) in the inverse of the equations (69) and (70) as given above, a corresponding image point x.sub.0, y.sub.0 is determined. By performing this mapping for each image point of the disturbed TKD pattern detected in step S100 in step S200 an improved TKD patterns is calculated.

REFERENCE NUMBERS

(65) 10 sample holder 12 sample 20 Kikuchi pattern 21 Kikuchi band 60 EM/SEM 61 electron lens 62 pole piece 64 TKD detector 65 phosphor screen 67 EDS detector 68 active area of EDS detector 80 electron beam 82 transmitted and diffracted electrons (for Kikuchi pattern)