For polycrystalline or powder TEM specimens, electron diffraction ring patterns are extensively used for phase identification, in which the diffraction pattern of a known phase acts as a fingerprint. PolyCrystalline Electron Diffraction (JECP/PCED, Li 2004) is a computer program for a fast and accurate method for phase identification.

PCED2.0 (Li, 2010) is an upgraded version of the previous JECP/PCED. New features include (i) Blackman's theory, an integral two-beam dynamical theory, for intensity calculation, (ii) Match model for out-of-plane and in-plane texture, (iii) pseudo-Voigt function for peak profile of diffraction ring and (iv) improvement on diffraction pattern indexing and matching to experimental pattern.

PCED2.0 is written in Java 6.0. Further code optimization (including obfuscation) is carried out on the compiled class files. A license file is needed to unlock the program (PCED2.0) for loading input data files. Without a validated license file the program is locked up in demo mode, which works fully funcations but only on default files (Al and Mg).

2. Theory background

Kinematic theory of electron diffraction is described in details in any texture book on this topic (e.g. Peng et al. 2004) . The electron atom scatter factor can be derived from X-ray atom scatter factor using Mott-Bethe relationship or directly obtained from parameterized table of electron atom scatter factor (Peng et al. 1996). Here we use the second method.

Following the electron diffraction geometry, the radius of the diffraction ring, R, can be related to the length of the reciprocal vector g(hkl) as

(1)

2.1 Blackman's theory on intensity of polycrystalline electron diffraction

Blackman's theory (Blackman 1939) for polycrystalline electron diffraction intensity is based on Bloch wave dynamic theory initially formulated by Bethe and the formulas are given as integral intensity of two-beam dynamic diffraction. According to the book by Peng et al. (2004), Blackman's theory for polycrystalline electron diffraction intensity is an advanced theory on this topic although the theory was published decades ago. Readers should refer to the original paper for details derivation of the formulas. Some important formulas are listed here.

The dynamical structure factor is given

(2)

The wave vector in vacuum is given

(3)

The wave vector inside crystal is corrected with mean inner potential

(4)

(5)

(6)

2.2 Pseudo-Voigt function for peak profile

Although Voigt (V) function, as convolution of a Gaussian (G) and a Lorentzian (L) functions, is regarded as the most suitable function to describe the profile of the cross-section of the diffraction rings, but it is rather complex. Pseudo-Voigt (pV) function is usually good enough substation for the job.

pseudo-Voigt function is a linear combination of Gaussian and Lorentzian function, as following
,

	(7)
	(8)
	(9)

Figure 1. Comparison of profiles of Gaussian and Lorentzian functions.

For powder samples, axially symmetric platy or capillary specimens, composed of effectively disk- or rod-shaped crystallites, can form textures due to the shape of crystallites. For thin film samples, polycrystalline microstructures with out-of-plane or in-plane preferred orientations often occur during the film growth. The diffracted intensities from preferred orientation can be corrected or measured with a single pole-density profile. The March model is used as for correcting the powder X-ray diffraction with texture, which is simple but effective method for both of out-of-plane (platy) and in-plate (rode) textures.

The formals were firstly given for X-ray powder diffraction, which is reformulated for the diffraction geometry in polycrystalline electron diffraction considering the diffraction geometry of polycrystalline electron diffraction. Figure 2 show example curves of the March functions used in X-ray powder diffraction (Bragg-Brentano geometry) and in polycrystalline electron diffraction for platy (out-of-plane) texture. Referring to the figure 1 in the paper by Dollase (1986), the density of preferred zone axis for polycrystalline electron diffraction can be formulated as following.

For out-of-plane texture,

(10)

(11)

(12)

Figure 2. Example curves of the March functions used in X-ray powder diffraction (Bragg-Brentano geometry) and in polycrystalline electron diffraction for platy (out-of-plane) texture. Angle away from the preferred orientation is in abscissa and the density (or weight) in ordinate.

3. Graphic user interface of PCED2.0

3.1 Main interface

The main interface of PCED2.0 is shown in Figs. 3 and 4, which includes a menu and a toolbar to take input parameters and a frame to show the output of the simulation. The usage of PCED2.0 for simulation needs to load structure data and to set up input parameters, then simply click Run button in Calculation dialog (shown later). The usage of PCED2.0 for phase identification needs to load experimental diffraction pattern first and then to match to the calculated patterns by chosen structure data and input parameters. Two structure data can load at the same time for comparison or for simulation of a two-phase system.

Figure 3. Snap-shot of the main interfaces of PCED2.0, a simulation of polycrystalline electron diffraction in a two-phase system, Al and Mg.

Figure 4. Snap-shot of the main interfaces of PCED2.0, a phase identification using calculated polycrystalline electron diffraction pattern. Experimental PCED pattern is Al.

3.2 Menu and toolbar

Menu and toolbar can be used to pop up dialog interfaces for data loading or parameters reading or changing. Although menu give more text description and organized in groups while toolbar in graphics and easily to access, most functions of menu and toolbar are same. However, there are some functions are only provided either in menu or in toolbar.

File menu provides an interface for preparing the input structure data, saving the hkl intensity list (and other info can be saved as option), and print to printer or to a PDF file when a PDF print driver was installed.

EDP menu provides an interface to load experimental polycrystalline electron pattern in jpg (or jpeg) format. The loaded pattern can be centered by drag-and-drop with referring to a set of preset circles when the checkbox of Center (image) is selected. Clear in EDP can be used to remove the pattern. Drag-and-drop: selecting a point in image by mouse, press down left button of mouse, drop to the selected point, for example, the center of panel, and release the left button of the mouse. When the position of the image is set up, uncheck the Center (image) box;

Simulation menu provides a submenu for the choice of theory and several dialogs for specific parameters in simulation. Two theories are used for calculation of electron diffraction intensity. Kinematical theory is selected by default and the second choice is Blackman's theory, which is integral two-beam dynamic theory. Intensities calculated by Blackman's theory are more accurate than those calculated by kinematical theory, but the calculation of dynamic theory will take longer time.

Option menu provides a freedom of customizing the appearance of simulated pattern, for examples, the color of the diffraction peak and ring, the curve or solid profile, number of reference circles, and the selection in the hkl-inten list file.

Toolbar provides a second way to conveniently access the dialogs described above. The functions of the toolbar are indicated by the icon and tooltip text. Only most frequently used menus and submenus are listed in Toolbar.

The key field and the lock icon are new items here. A registration "key" code is needed unlock the full version PCED2.0f, which works on normal input files or the limited version PCED2.0i, which works on the input files with one-way hash code. Without the "key" code, the program is locked up in demo mode, which working only on default input files (Al and Mg).

4. Usage of PCED2.0

To run PCED2.0, just type: java -jar (-Xmx512m) PCED2.0.jar in command line or double click the icon of PCED2.0.jar. -Xmx512m is an option to set the memory of Java virtual machine up to 512 MB. An interface of the program similar to Figure 3 will show up. To type in the "key" code in the key field of the tool bar will unlock PCED2.0f or PCED2.0i. Otherwise, the program will work on the default structure data in demo mode.

In the following, we show step by step from preparing structure data file, common routine usage for simulation, and to the last steps of saving and printing the results. More details on specific topics are left to next section.

Figure 5. Snap-shot of the preparation of input structure data file.

4. 1 Prepare new data file

Structure data file can be prepared using the PCED Phase Input File dialog in Figure 5 or using a text editor to modify other structure data file. The dialog provides a certain level of assistant for user and also makes sure to meet the requirement of the format of the file.

• Description field can be filled in a name of the phase.
• Bravais lattice can be selected a list.
• Lattice types are provided for each crystal system.
• Lattice parameters are preset according to Bravais lattice.
• Space group number is limited to possible group according to Bravais lattice.
• Fill the info of each atom in and then add to the atom list. For modifying or removing the list, user should select the atom row in the list and click the Remove button.
• Notes field can be used for the purpose for simulation or reference.

To save the data structure click the Save button or to make new one click the New button.

4.2 Simulation

Kinematical theory is used as a default selection in the simulation since it is fast and less input parameters are required. Integral dynamical two-beam theory (Blackman's theory) can be turned on using Theory submenu in Simulation menu. Basic parameters for calculation and adjustment of diffraction pattern can be filled in or changed in Calculation panel in figure 6. To generate new diffraction pattern needs click Run button. The pattern is automatically updated corresponding to the change of other parameters. Mass ratio defines relative mass of the two phases with the fields of phase1 and phase2 are checked. G-spacing zoom and Intensity scale, simulated the camera length and the exposure time in experiment, which are used to make a better appearance of the calculated pattern. Diffraction diagram can be viewed as peak and profile separately or together. Diffraction ring can be viewed as either a ring to match experiment pattern or only top half in order to clearly show index without intervening, or without ring at all. The appearance of diffraction ring is also controlled by peak height itself (intensity threshold) and peak height comparing to neighbor points (intensity sensitivity).

Figure 6. Snap-shot of the calculation dialog.

Calculated diffraction data can be saved into a file using Save in File menu, in which it includes hkl and intensity (per unit-length) together with the parameters used in the simulation as default option. More info, such as, length of reciprocal vector (g), crystalline plane spacing (d), intensity (per unit angle), can be chosen to be saved by using selection in hkl-inten panel in option menu.

Simulated electron diffraction pattern together with the experimental pattern if preloaded can be sent to print directly using Print in File menu or save into a PDF file if a PDF printer driver was installed.

5. Special topics on PCED2.0

5.1 Partial occupancy factor and isotropic temperature factor

Some atom coordinates may be not in full occupancy in a crystalline structure. In this case, the occupancy factor (default value 1.0) should be changed to the values according to the crystalline structure in preparation of the structure file for polycrystalline electron diffraction simulation.

Partial occupancy factor can also be used to simulate a certain level of chemical order in structure. For example, the chemical ordered FePt L10 phase, see section 6. In this case, different type of atoms may be assigned to the same atomic coordinates with different occupancy according to the chemical ratio, but the sum of the occupancy factors of the two atoms is 1.0.

Isotropic temperature factor is used here to simulate the effect of lattice vibration (Debye model). Although isotropic temperature factor is a rough model, it can be used simulate the decrease of the diffraction intensity with the variation of the |g| value, the higher the value of |g| the more decrease of the diffraction intensity.

5.2 Load and center EDP

For phase identification, an experimental polycrystalline electron diffraction pattern is needed to loaded up and compare with simulated patterns, which were calculated from the related phase files. The experimental pattern should be in jpg (or jpeg) format and is better prepared into square shape with pattern in the center. The experimental pattern can be load using Load in EDP menu and then center by using Center in EDP menu. Once Center is clicked, five concentric circles will appear in the main panel. Select the center of the pattern using mouse and hold on the left button of mouse, the pattern can be dragged and drop into the center of the panel. Click the Center in EDP menu the pattern is locked up in the position and concentric circles disappear. More adjustment in small step may be needed to find the accurate position. Number of concentric circles can be changed using Number of Reference Circle in Option menu.

5.3 Peak profile, incident beam and texture

Profile of the cross-section of diffraction ring is referred as peak profile, which can be simulated using a pseudo-Voigt function, together with grain size (diameter) and constant K. The ratio of Gaussian and Lorentzian functions in pseudo-Voigt function can be adjusted using Peak Profile in Simulation menu (default value is 0.5). The grain size (diameter) and constant K can be adjusted using Average Grain Size dialog in Simulation menu.

Incident beam in polycrystalline electron crystalline is very strong comparing to the diffraction ring, it may submerge the relative week rings near the incident beam. Incident beam can be included in the simulated pattern by check on the Included box in Peak Profile dialog and adjust the shape by changing the corresponding values of height and fwhm.

Peak and profile can be selected individually and the style of profile can be chosen between solid or curve.

In texture dialog, axis of texture and March parameter (default value 1.0 for full random case) can be input for simulation. These values can be setup separately for phase1 and phase2.

March parameter (r) is suitable for describing the slight deviation of the randomness of polycrystalline/or powder sample, thus the higher the value the better approaching the randomness.

To describe the status of preferring orientation, it is suggested to used reversed March parameter r' = 1.0 - r. Thus, full randomness r' = 0, the higher the prefer orientation, the higher the value of r' and for the perfect prefer orientation r' = 1.0.

Figure 7. Snap-shot of the EDP pattern centered by the drag-and-drop action.

5.4 Indexing and parameter table

There are cases that diffraction peak are quite dense, the index labels may be overlapped when they are shown by the side of the diffraction diagram, especially for the diffraction diagram of two-phase system.

To avoid the situation, PCED2.0 is allowed to select location of index label and to spread them when necessary. In index dialog, top point can be viewed when the check box of Guide line is selected and moved around by arrow keys when the corresponding radio button is selected (no further mouse click for the moment). Base line can be viewed when the check box of Link line is selected and moved up/down by arrow keys when the corresponding radio button is selected (no further mouse click for the moment).

For convenience to track down the parameters used in the simulation, a Parameter Table can be pop up using Parameter Table in Simulation menu, which lists the important parameters of phase1 and phase 2.

5.5 Determine weight ratio of two-phase system

The PCED2.0 program provides a way to simulate of the composite diagram of two-phase system with given weight ratio. The formula used in the program is to convert the weight ratio into the numbers of unit-cell in diffraction using atomic weight per unit-cell for each phase for simplicity. Experimentally determining weight ration of two-phase system should be calibrated by using a standard sample with known weight ratio.

5.6 Grain size and the FWHM of diffraction ring

The PCED2.0 program provides a way to simulate of the FWHM of diffraction ring for given average grain diameter and K factor. The values of FWHM for each ring in calculated diagram are consistent to the same pseudo-Voigt function used in the simulation. The diffraction rings in experimental pattern are often composed of separately individual spots from grains with different grain size, the peak profiles are obtained by using data sum up around each ring. In such case, the peaks may need to be fitting to different pseudo-Voigt function for each ring. The grain size is not related to the peak profiles of polycrystalline electron diffraction ring pattern in a simple way. The grain size should be measured by using bright field or dark field imaging method.

6. Example: PCED analysis of thin films of FePt L1₀ phase

In order to increase the areal density of magnetic recording media, grain sizes of the magnetic films must be reduced. A high-anisotropy material is thus essential for the new media to retain thermal stability. The high anisotropy energy of FePt L1₀ phase makes FePt alloy system one of the most promising materials candidates. As-deposited FePt films normally consist of the disordered fcc phase, which is a magnetically soft phase, and can be transformed into the L10 phase upon annealing treatment. The chemical order of the L10 phase depends on the composition and the heat treatment condition of the film.

For data-storage applications, it is desirable that the [001] axis of the L1₀ phase is perpendicular to the film plane since the [001] axis of the L1₀ phase is a magnetic easy axis. Hence, the films in a (001) texture have an advantage over randomly or other (e.g. (111)) oriented films. One way to achieve the (001) texture has been developed by depositing multilayer of Fe and Pt film and annealing at elevated temperature. The formation of the (001) texture for various ranges of film thicknesses were studied by selected-area electron diffraction (SAED), examples are discussed here to show the application of PCED2.0. Different schemes for comparison of experimental and simulated patterns are used in each example as demonstration.

Figure 8. SAED pattern of the FePt L1₀ phase in film with thickness of 20 nm annealed at 600 °C for 30 second and calculated pattern for phase identification.

6.1 Example 1. FePt film with thickness 20 nm

Figure 8 shows the SAED pattern of FePt film with thickness of 20 nm annealed at 600 °C for 30 second and calculated pattern of the FePt L1₀ phase in for phase identification.

The ring radiuses and corresponding intensities in the calculated pattern in Figure 8 are considered matching well with experimental pattern, on the other hand, the ring radiuses and corresponding intensities in the calculated pattern with FePt random fcc structure (not shown here) are not matching with experimental pattern. Thus the phase is confirmed to be the chemical ordered FePt L10 phase. It also indicates that the random distribution of grains in the film in comparison to Figure 9 in example 2.

Figure 9. SAED pattern of the FePt L10 phase in a film with thickness of 4 nm annealed at 600 °C for 30 second and calculated pattern with (001) texture.

6.2 Example 2. FePt film with thickness 4 nm

Figure 9 shows the SAED pattern of the FePt L10 phase in a film with thickness of 4 nm annealed at 600 °C for 30 second and calculated pattern of the FePt L1₀ phase with (001) texture for phase identification. Radius distribution of the diffraction patterns in Figure 9 is the same as those in Figure 8, however, the intensity distribution of the diffraction patterns are dramatically different, which means a texture exists in Figure 9, as confirmed in the calculated pattern with (001) texture.

Now we estimate the (001) texture of the film with thickness of 4 nm semi-quantitatively. Figure 10 is digital processed diagram of Figure 9 using a program QPCED (Li, 2007) and QPCED2.0. By using March parameters of 0.45 and a given chemical order (Pt sites with 80%Pt+20%Fe; Fe sites with 80%Fe+20%Pt), a calculated diagram in Figure 11 is best matching with Figure 10 is obtained. As discussed in section 5.3, if we use revised March (0.0 for full random and 1.0 for perfect texture), the texture can be described as 55% of the (001) texture.

Figure 10. Digital processed diagram of experimental electron diffraction pattern in Figure 9.

Figure 11. Simulated diffraction diagram fitting to the Figure 10 with aim to reveal the degrees of chemical order and (001) texture of film with thickness of 4nm.

6.3 Example. FePt film with thickness 12 nm

Figure 12 shows the SAED pattern of FePt film with 12 nm of repeated bilayer FePt and 1nm Pt top layer in deposition followed by annealing at 600 °C for 30 second and simulated pattern. The profile of digital diffraction pattern was loaded in the PCED2.0 and to compare the simulation pattern directly, which was calculated using March parameters of 0.17 and a given chemical order (Pt sites with 80%Pt+20%Fe; Fe sites with 80%Fe+20%Pt). Thus the texture can be described as 83% of the (001) texture.

Figure 12. Simulated diffraction diagram fitting to an experimental PCED profile aim to reveal the degrees of chemical order and (001) texture of the FePt film, which was deposited as thickness of 12nm of repeated FePt bilayer and 1 nm of Pt top layer followed by annealing at 600 °C for 30 second.

7. Reference

Blackman, M., On the Intensities of Electron Diffraction Rings. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 173 (1939) 68-82.

Dollase W.A., Correction of Intensities for Preferred Orientation in Powder Diffractometry: Application of the March Model. J. Appl. Cryst. 19 (1986) 367-272.

Li, X.Z., JECP/PCED-a computer program for simulation of polycrystalline electron diffraction pattern and phase identification. Ultramicroscopy 99 (2004) 257-261.

Li, X.Z., Quantitative Analysis of Polycrystalline Electron Diffraction Patterns, Microanalysis and Microscopy 2007.

Li, X.Z., PCED2.0 - A Computer Program for Advanced Simulation of Polycrystalline Electron Diffraction Pattern, Ultramicroscopy 110 (2010) 297.

Peng L.-M., Dudarev S. L. and Whelan M. J., High Energy Electron Diffraction and Microscopy, Oxford University Press (01/01/2004).

Peng L.-M., Ren G., Dudarev S.L. and Whelan, M.J., Robust Parameterization of Elastic and Absorptive Electron Atomic Scattering Factors, Acta Cryst. A52 (1996) 257-276.