Geometric phase analysis is a method of

digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner ar ...

used to determine crystallographic quantities such as d-spacing or strain from high-resolution transmission electron microscope images. The analysis needs to be performed using specialized

computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer progra ...

Principle

In geometric phase analysis, local changes in the periodicity of a high resolution image of a crystalline material are quantified, resulting in a two-dimensional map. Quantities which can be mapped with geometric phase analysis include interplanar distances (d-spacing), two-dimensional deformation and strain tensors and displacement vectors. This allows strain fields to be determined at very high resolution, down to the unit cell of the material. Importantly, GPA performed on images that have sub unit-cell resolution can produce erroneous results. For example, a change in composition may appear as a component of the deformation tensor, with the result that an interface appears to have a strain field associated with it when in fact there is none. Since the calculations are performed in the frequency domain, the input image, with a periodicity of the

crystal lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...

, must be transformed into a spatial frequency representation using a 2D

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

. From a mathematical point of view, the frequency image is a complex matrix with a size equal to the original image. From a crystallographic point of view, there is an analogy between the 2D Fourier transform and

diffraction pattern Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...

and the

reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...

. The intensity peaks (or power peaks) in the Fourier transform correspond to crystallographic planes depicted in the original image, specifically a sine wave with the orientation and period of the corresponding planes. A change in the phase of this sine wave indicates a change in the position of its peaks and troughs, which can be interpreted as a component of a 2D deformation tensor. Due to the

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

nature of the frequency image, it can be used to calculate

amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...

and

phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...

. Together with a vector of one crystallographic plane depicted in the image, the amplitude and phase can be used to generate a 2D map of d-spacing. If two vectors of non-parallel planes are known, the method can be used to generate maps of strain and displacement.

Software

In order to perform geometric phase analysis, a computer tool is needed. Firstly, because manual evaluation of transforms between spatial and frequential domain would be highly impractical. Secondly, a vector of crystallographic plane is an important input parameter and the analysis is sensitive to the accuracy of its localization. Therefore, the accuracy and repeatability of the analysis requires precise localization of diffraction spots. The required functionalities are available in several software packages including Strain++ and the crystallographic suite

CrysTBox CrysTBox (''Crystallographic Tool Box'') is a software suite, suite of computer tools designed to accelerate material research based on transmission electron microscope images via highly accurate automated analysis and interactive visualization. R ...

. The latter offers an interactive geometric phase analysis called gpaGUI. In both packages it is possible to locate peaks in the Fourier transform with sub-pixel precision (e.g. diffractGUI). File:GpaGUI input.png, Input image (magnesium) File:GpaGUI filtered010.png, Filtered image of plane (0 1 0) File:GpaGUI dsp010.png, Map of (0 1 0) d-spacing File:GpaGUI strainXx.png, Map of XX component of strain tensor File:GpaGUI displacementX.png, Map of X component of displacement vector

References

{{Electron microscopy Crystallography Electron microscopy Geometric measurement Digital signal processing Applied mathematics

Principle

Software

See also

References