Goniometry of direct lattice vectors and direct crystallographic analyses in transmission electron microscopes

You may wonder what goniometry of direct (and reciprocal) lattice vectors and direct crystallographic analyses are all about. Well, it is not in the textbooks as it is the development of just a handful of people worldwide over the last 25 years or so. It is, however, a really useful approach, especially at a time when the international electron microscopy community talks about nano-crystallography and materials science in an aberration-free environment [1]. So please read below (or contact me for more information at pmoeck@pdx.edu, or go straight to a set of power-point slides from a lecture I recently held at the National Center for Electron Microscopy at Berkeley).

Let’s describe briefly - in the small print below - goniometry of direct lattice vectors in a transmission electron microscope (TEM). Goniometry of reciprocal lattice vectors, on the other hand, is most easily done with an X-ray diffractometer and I have done some such work about fifteen years ago while I was working for my PhD. Both types of goniometry are complementary to each other (in much the same way as the direct and the reciprocal lattice of a crystal are complementary) and deliver the data for what is called direct crystallographic analyses.

Every direct crystallographic analysis in (both a parallel illumination and a scanning probe) TEM consists of three basic parts (whereby these parts are to performed in the following sequence):

- the goniometry (i.e. the adjustment of certain orientations of a crystal and the recording of the corresponding goniometer readings) of direct lattice vectors (using either Kikuchi-line patterns, quasi-symmetric bent contour crossings, spot patterns with a highly symmetric distribution of the diffraction spots, crossings of lattice fringes in high-resolution images and their associated highly symmetric spots in Fourier transforms)

- the subsequent determination of the relative orientation(s) of the crystal(s) inside the microscope with respect to a coordinate system that is fixed to the microscope and,

- finally, the employment of the relative orientation(s) of the crystal(s) for the crystallographic analysis itself [2].

The key idea of goniometry of direct lattice vectors is tilting each of the crystals to be analyzed into at least two different orientations that can be easily recognized by, e.g. their zone-axis Kikuchi pattern, spot patterns with a symmetric diffraction spot distribution, quasi-symmetric bent contour crossing, crossing of lattice fringes in high resolution images or symmetric spots in their associated Fourier transforms. If the latter is to be used, the availability of a CCD camera as implemented at the (second generation) Tecnai G² F20 TEM of Portland State University (PSU) will greatly facilitate the goniometry of direct lattice vectors. At each of the adjusted zone axes, the goniometer readings which are by themselves coordinates of the direct lattice vectors in the curvilinear coordinate system of the specimen goniometer are recorded. The coordinates of these goniometer readings are then transformed into a Cartesian coordinate system (E_em) that is fixed to the electron microscope. As Cartesian electron microscopical coordinate system (E_em), it is convenient to choose the x and y translation axes of the specimen stage and the direction of the transmitting electrons as either + or - z-axis.

The (direct space) lattice vectors of any crystal (denominated by letters A, B, ... which refer to the direct lattice base of the crystals) can always be expressed in a Cartesian coordinates system (E) as a 3 by 3 matrix that is called the crystal matrix of the direct lattice (ETA) = (ASE)^-1 for which we use the extended matrix notation according to ref. [3]) and which lends itself perfectly to all sorts of crystallographic analyses that can be performed directly while working at the microscope (rather than later on while being back to the office). It is this directness that is expressed in the adjective “direct” in direct crystallographic analysis.

If crystallographic analyses are to be performed in reciprocal space (bases A*, B*, …), the crystal matrix of the reciprocal lattice (AS*E) = (ETA)’ and (ET*A) = (ASE)’, where (ASE)’ is the transposed of the matrix (ASE), can easily be calculated. The relative orientation of the crystals to be analyzed with respect to the electron microscope is given by a 3 by 3 matrix i.e. (E_emTA) = (ASE_em)^-1, that is the transformation matrix between the Cartesian coordinate system E_em that is fixed to the electron microscope and the respective direct lattice crystal coordinate system (A, B, ...). The Cartesian coordinate system E (that makes the matrix notation of the direct lattice possible) can be chosen freely, i.e. can be set to be identical to E_em. This transformation matrix is, therefore, identical to the crystal matrix of the direct lattice with which the crystallographic analyses can so easily be performed. This transformation matrix/crystal matrix is calculated from the data gathered by means of goniometry of direct lattice vectors through simple matrix algebra. Alternatively one may perform goniometry of reciprocal lattice vectors (as shown in the early 80s for TEMs [4-6]), derive the matrices (ET*A) and (AS*E), and transpose these matrices in order to derive the crystal matrix of the direct lattice.

The advantages of direct crystallographic analyses are that they can be performed either in situ while working on the TEM or ex-situ using the data on the relative orientations of the crystals with respect to the microscope. Up to now, direct crystallographic analyses on the basis of the goniometry of direct lattice vectors in TEM have been developed by myself for the determination of crystal orientations [7,8], crystal metrics, goniometer adjustments that are required to tilt to either a desired zone axis or to a two beam diffraction condition, grain and phase boundary parameters [9,10], indexing of two-beam diffraction vectors [11], and estimation of textures [12]. The accuracy of these analyses can be greatly enhanced when the goniometry of direct lattice vectors is performed for more than three lattice vectors and the transformation/crystal matrix is derived by means of a least squares algorithm [10,13].

As both concept and methology of direct crystallographic analyses are applicable to any kind of goniometer, such analyses have also been developed by myself for a triple-axis X-ray diffractometer [14-17] (on the basis of the goniometry of reciprocal lattice vectors), for scanning electron microscopes [18], and for an optical goniometer. Although any kind of goniometer can by used for the performance of direct crystallographic analyses in TEM, it will be more advantageous to be able to use more than the two commonly available degrees of freedom for the tilting operations in standard TEM specimen holders [19,20]. A double-tilt-rotation specimen holder (i.e. a goniometer with 3 degrees of freedom) has recently been introduced to the market by the GATAN company and we look forward to testing its usefulness under computer control for the goniometry of direct lattice vectors.

In recent years, a method for the structural analysis of powders of nanometer sized cubic crystals has been developed on the basis of goniometry of direct lattice vectors by means of high resolution phase contrast imaging [21-28]. The concept and methodology of direct crystallographic analyses on the basis of goniometry of direct lattice vectors has recently also been taken up by industry. Both the FEI company and the TSL division of the EDAX company have brought to the market hardware and software for transmission electron microscopy systems that are based on this concept. Both of these systems are for double tilt TEM specimen holders. The great advantage of the FEI system is its full integration with the computer controlled and compucentric goniometer stage of the microscopes of the new Tecnai series.

My graduate students and I will implement direct crystallographic analyses on the basis of goniometry of direct lattice vectors at the TEM of PSU because the convergent beam electron diffraction (CBED) technique is not well suited to the analysis of the structure of self assembled and colloidal semiconductor quantum dots and other crystalline nanoparticles. This is because these entities have dimensions in the range of a few nm to a few 10 nm, i.e. up to about only one quarter of the extinction distance of typical reflections in semiconductors at 200 to 300 kV acceleration voltage. While the diffraction events can under such conditions be treated kinematically, the CBED disks are void of their fine structure that make them so useful for crystallographic analyses in thicker crystals.

To begin with, we intend to build on the above mentioned method for the analysis of powders of cubic nanocrystals [21-28], make such a method fully automatic on PSU’s TEM, make it applicable to different kinds of noncubic nanocrystal powders, and take advantage of the extra degree of freedom of the double-tilt-rotation specimen holder in order to analyze many hundreds of nanocrystals in a “parallel processing” mode which will allow us to derive at statistically more significant conclusions. In other words, we will determine experimentally the respective (ETA_i) matrices for a rather large number (i) of nanocrystals and calculate from these matrices the metric tensors and lattice parameters of these nanocrystals.

Since I am in the profession of teaching materials science to future engineers and applied physicists, there will also be an educational aspect to this project. When I studied crystallography at Leipzig University some 25 years ago, we practiced crystallometry and used the stereographic projection to determine much about the crystals under investigation, (e.g., ^c/_a ratios, point groups, crystal classes, etc.) without diffraction and this helped me in comprehending the core concepts of geometrical-structural crystallography. With the extra degree of freedom a double-tilt-rotation specimen holder provides, one can do rather similar things now in a state-of-the-art TEM. So what my graduate students and I intend to do is to write some software for educational purposes that demonstrate the principles of goniometry of direct and reciprocal lattice vectors and how it can be used to analyze nanocrystals. Needless to say that my future materials science students and everybody else interested will get to play (and learn some crystallography - I hope) with this software as it will become freely accessible from this web site.

We also intend to analyze the crystallography of atomic ordering in epitaxially and endotaxially grown semiconductor quantum dots [29,30]. Here we plan to derive the relative position of the semiconductor matrix-crystal (known crystal basis A) that surrounds various types of atomically ordered quantum dot crystals (unknown crystal bases B_i) in the electron microscope along the lines of refs. [2,9,10,13] to a very high precision and accuracy (i.e., derive its respective (ETA) matrix), and use hypothetical elastic lattice mismatch strain energy minimizing orientation relationships (i.e. certain special matrix products (B_iNA) = (B_iSE) (ETA)) between atomically ordered quantum dots and their surrounding semiconductor matrix-crystal as a constraint to find possible candidate structures for the atomically ordered quantum dots, (i.e. their hypothetical (B_iSE) matrices). Which one of these excess Gibbs free energy minimizing candidate structures (i.e. crystal bases B_i) the various kinds of atomically ordered quantum dots actually possess can then be determined by testing the respective (ETB_i) = (B_iSE)^-1 matrices experimentally. From the experimentally confirmed matrices, the metric tensor and lattice parameters of the various kinds of atomically ordered quantum dots can then be calculated in a straightforward manner.

Coming full circle to our opening paragraph, let’s address the question why goniometry of direct lattice vectors will become more useful with the widespread availability of aberration corrected TEMs. First generation aberration corrected TEMs will possess a significantly enhanced point-to-point resolution in the image. With higher image resolution, one does not need to tilt to large angles and will be able to analyze crystals that have both small lattice spacings and low symmetries. Second generation aberration corrected TEMs will probably possess objective lenses with larger focal lengths, a significantly enhanced point-to-point resolution, and significantly more free space around the specimen in the electron microscope pole piece. Depending on the need of the experiments, one may then trade to some extent image resolution for more space and vice versa in order to tilt even the lowest symmetry and smallest lattice constant crystals around for direct crystallographic analysis.

If you are not completely put off by the deliberations above and still interested in the progress of this project, please bookmark this page and check from time to time for new developments. One of my collaborators, Prof. Phillip B. Fraundorf, has a very nice (and awards winning) webpage at which you can learn about “fringe visibility maps”. Play around a bit and zoom into the nanometer world with the simulation of a TEM at one of his web pages and see what else is going on in his group at the University of Missouri at St. Louis.

[1] TEAM background: http://ncem.lbl.gov/team/team_background.htm

1^st TEAM workshop: http://ncem.lbl.gov/team/TEAM Report 2000.pdf

2^nd TEAM workshop: http://ncem.lbl.gov/team/TEAM Report 2002.pdf

3^rd TEAM workshop: http://ncem.lbl.gov/team3.htm

[2] P. Möck, “Verfahren zur Durchführung und Auswertung von elektronenmikroskopischen Untersuchungen”, German patents DE 4037346 A1 and DD 301839 A7, priority date 21. 11. 1989.

[3] J.S. Bowles and J.K. Mackenzie, “The Crystallography of Martensite Transformation”, Acta Met. 2, 129-188 (1954).

[4] P. Fraundorf, “Stereo Analysis of Single Crystal Electron Diffraction Data”, Ultramicroscopy 6, 227-236 (1981).

[5] P. Fraundorf, “Stereo Analysis of Electron Diffraction Pattern from Known Crystals”, Ultramicroscopy 7, 203-206 (1981).

[6] P. Tambuyser, “A simple method for the direct measurement of diffraction patterns in the EM400T transmission electron microscope”, J. Phys. E16, 483-486 (1983).

[7] P. Möck, “A Direct Method for Orientation Determination Using TEM (I), Description of the Method”, Cryst. Res. Technol. 26, 653-658 (1991).

[8] P. Möck, “A Direct Method for Orientation Determination Using TEM (II), Experimental Example”, Cryst. Res. Technol. 26, 797-801 (1991).

[9] P. Möck, “A direct Method for the Determination of Orientation Relationships Using TEM”, Cryst. Res. Technol. 26, 975-962 (1991).

[10] P. Möck and W. Hoppe, “Direkte Kristallographische Analysen mit Elektronenmikroskopen”, Beitr. Elektronenmikroskop. Direktabb. Oberfl. 24, 99-104 (1991).

[11] P. Möck, “In situ indexing of Two-Beam Electron Diffraction Vectors”, Cryst. Res. Technol. 26, K157-K159 (1991).

[12] P. Möck, “Estimation of Crystal Textures using Electron Microscopy”, Beitr. Elektronenmikroskop. Direktabb. Oberfl. 28, 31-36 (1995).

[13] P. Möck and W. Hoppe, “ELCRYSAN – A program for direct crystallographic analyses”, Proc. 10th European Conference on Electron Microscopy Vol. 1 193-194 (1992).

[14] P. Möck, “Complete characterization of epitaxial systems from the lattice geometrical point of view, Fundamentals”, J. Cryst. Growth 128, 122-126 (1993).

[15] P. Möck, “Complete characterization of epitaxial CdTe on GaAs from the lattice geometrical point of view”, Mater. Sci. Eng. B16, 165-167 (1993).

[16] P. Möck, “Description of the real orientation relationships of epitaxial samples using transformation matrices”, Inst. Phys. Conf. Ser. No. 134, 593-596 (1993).

[17] H. Berger, P. Möck, and B. Rosner, “Description and Interpretation of systematic Deviations from Epitaxial Laws of Overgrowth”, Acta Phys. Polon. A84, 279-286 (1993).

[18] P. Möck and W. Hoppe, “Direkte kristallographische Analysen mit SEM”, Beitr. Elekronenmikroskop. Direktabb. Oberfl. 23, 275-278 (1990).

[19] S. Turner and D.S. Bright, “Characterization of the Morphology of Facetted Particles by Transmission Electron Microscopy”, Mat. Res. Soc. Symp. Proc. 703, V6.6.1-V6.6.6 (2001).

[20] S. Turner, “Systematic Characterization of Reciprocal Space by SAED: Advantages of a Double-Tilt, Rotate Holder”, Microscopy and Microanalysis Proceedings 2002, 668CD.

[21] P. Fraundorf, “Determining the 3D Lattice Parameters of Nanometer-sized Single Crystals from Images”, Ultramicroscopy 22, 225-230 (1987).

[22] W. Qin, “Direct space (Nano)crystallography via high-resolution transmission electron microscopy”, PhD thesis, University of Missouri-Rolla, 2000.

[23] W. Qin and P. Fraundorf, “Crystal lattice parameters from direct-space images at two tilts” (2000); Los Alamos Archives http://arXic.org, document http://xxx.lanl.gov/abs/cond-mat/0001139, downloadable as *.pdf file at http://arxiv.org/PS_cache/cond-mat/pdf/0001/0001139.pdf.

[24] W. Qin and P. Fraundorf, “Lattice parameters from direct-space images at two tilts”, Ultramicroscopy 94, 245-262 (2003).

[25] W. Qin and P. Fraundorf, “Correlating Lattice Fringe Visibility with Nanocrystal Size and Orientation” (2002); Los Alamos Archives http://arXic.org, document http://xxx.lanl.gov/abs/cond-mat/0212281, downloadable as *.pdf file at http://arxiv.org/PS_cache/cond-mat/pdf/0001/0212281.pdf.

[26] P. Fraundorf, W. Qin, “Fringe Visibility Maps”, Microscopy and Microanalysis Proceedings 1999, 7 (Suppl 2: Proceedings): 272-273.

[27] W. Qin and P. Fraundorf, “Lattice Fringe Visibility after Tilt” Microscopy and Microanalysis Proceedings 2000, 6 (Suppl 2: Proceedings): 1040-1041.

[28] W. Qin and P. Fraundorf, “Probability of seeing <001> cross-fringes in a random cubic nanocrystal image”, Microscopy and Microanalysis Proceedings 2000, 6 (Suppl 2: Proceedings): 1038-1039.

[29] P. Möck, T. Topuria, N.D. Browning, G.R. Booker, N.J. Mason, R.J. Nicholas, M. Dobrowolska, S. Lee, and J.K. Furdyna, “Internal self-ordering in In(Sb,As), (In,Ga)Sb and (Cd,Mn,Zn)Se nano-agglomerates/quantum dots”, Appl. Phys. Lett. 79 (2001) 946-948.

[30] P. Möck and Nigel D. Browning, “Process for forming semiconductor quantum dots with superior structural stability” based on earlier provisional patent application registration number 29,381: P. Möck et al., “Procedure to produce structurally stable semiconductor quantum dots with internal compositional modulation and varying degree of atomic long-range order”, Attorney's Docket No. 29,381, Attorney's Docket No. 83324, Internal University of Illinois reference: UICM-101.0 Prov (7882/83324), filing date July 20, 2001.