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Section 2Types of EBSD experimentPoint analysis 12
Crystal orientation mapping 12
Pattern quality maps 13
Representations of grains andgrain boundaries 14-16
Texture 17-18
Phase discrimination 19
Summary 19
Section 3Undertaking EBSDexperiments
Sample preparation 20
Camera integration timeand resolution 20-21
Background removal 22
Microscope operating conditions 22-24
Effect of sample tilt 24
Spatial resolution 24
Measurement accuracy 24-25
Summary 25
Section 4Basic crystallographyfor EBSD Crystals and lattices 26
Crystal directions, planes andthe zone axis 27
Crystal orientation 28
Misorientation 28
Euler angles 29
(hkl)[uvw] 29
Stereographic projections 29-31
IntroductionElectron Backscattered Diffraction (EBSD) is a technique which allows crystallographicinformation to be obtained from samples inthe scanning electron microscope (SEM). InEBSD a stationary electron beam strikes a tiltedcrystalline sample and the diffracted electronsform a pattern on a fluorescent screen. Thispattern is characteristic of the crystal structureand orientation of the sample region fromwhich it was generated. The diffraction patterncan be used to measure the crystal orientation,measure grain boundary misorientations,discriminate between different materials, andprovide information about local crystallineperfection. When the beam is scanned in a gridacross a polycrystalline sample and the crystalorientation measured at each point, theresulting map will reveal the constituent grainmorphology, orientations, and boundaries. Thisdata can also be used to show the preferredcrystal orientations (texture) present in thematerial. A complete and quantitativerepresentation of the sample microstructurecan be established with EBSD.
In the last ten years EBSD has become a wellestablished technique for the SEM, andobtaining crystallographic information fromsamples is now both routine and easy. Thisguide explains how an EBSD system works,describes the experiments that can beperformed and how to undertake them, andfinally outlines the basic crystallographyneeded for EBSD.
2 3
Contents
■ A vacuum interface for mounting the phosphor and camera in an SEM port. The camera monitors the phosphor through a lead glass screen in the interface, and the phosphor can be retracted to the edge of the SEM chamber when not in use.
■ Electronic hardware that controls the SEM, including the beam position, stage, focus, and magnification.
■ A computer to control EBSD experiments, analyze the diffraction pattern and process and display the results.
■ An optional electron detector mounted below the phosphor screen for electrons scattered in the forward direction from the sample.
Basics of EBSDPrincipal system componentsThe principal components of an EBSD systemare (Figure 1):
■ A sample tilted at 70° from the horizontal.
■ A phosphor screen which is fluoresced by electrons from the sample to form the diffraction pattern.
■ A sensitive charge coupled device (CCD) video camera for viewing the diffraction pattern on the phosphor screen.
Pattern formationand collectionFor EBSD, a beam of electronsis directed at a point of intereston a tilted crystalline samplein the SEM (Figure 2). Themechanism by which thediffraction patterns are formedis complex, but the followingmodel describes the principalfeatures. The atoms in thematerial inelastically scatter afraction of the electrons, witha small loss of energy, to forma divergent source of electronsclose to the surface of thesample. Some of these electronsare incident on atomic planesat angles which satisfy theBragg equation:
nλ = 2d sin θ (1)
where n is an integer, λ is thewavelength of the electrons, d is the spacing of thediffracting plane, and θ is the angle of incidence of theelectrons on the diffractingplane. These electrons arediffracted to form a set ofpaired large angle conescorresponding to eachdiffracting plane. When usedto form an image on thefluorescent screen, the regionsof enhanced electron intensitybetween the cones producethe characteristic Kikuchibands of the electron backscattered diffraction pattern(Figure 3).
Figure 1: Components of an EBSD system
Figure 2: Formation of the diffraction pattern
Figure 3: A diffraction pattern from nickel collectedat 20 kV accelerating voltage
Figure 4: (a) The diffraction pattern in figure 3 indexed. The Kikuchi bands are labeled with the Millerindices of the crystal planes that formed them (red) and the crossing points of the bands are labeled withthe zone axis symbol (white). (b) The nickel unit cell orientation which generates this pattern with thecorresponding crystal planes shown.
Interpreting diffraction patternsThe center lines of the Kikuchi bands correspondto the intersection of the diffracting planeswith the phosphor screen. Hence, each Kikuchiband can be indexed by the Miller indices ofthe diffracting crystal plane which formed it.The intersections of the Kikuchi bandscorrespond to zone axes in the crystal and canbe labeled by zone axis symbols (Figure 4).The semi-angle of the diffracted cones ofelectrons is (90 - θ) degrees. For EBSD this is alarge angle so the Kikuchi bands approximateto straight lines. For example, the wavelengthof 20 kV electrons is 0.00859 nm and thespacing of the (111) plane in aluminum is0.233 nm making the cone semi-angle 88.9°.The width w of the Kikuchi bands close to thepattern center is given by:
where l is the distance from the sample to the screen (Figure 5). Hence, planes with wided-spacings give thinner Kikuchi bands thannarrow planes. Because the diffraction patternis bound to the crystal structure of the sample,
as the crystal orientation changes the resultantdiffraction pattern also changes. The positionsof the Kikuchi bands can therefore be used tocalculate the orientation of the diffractingcrystal (Figure 6).
Figure 6: Changes in the crystal orientation result inmovement of the diffraction pattern. The simulateddiffraction pattern is from a sample tilted 70° to thehorizontal and the crystal orientation is viewedalong the direction perpendicular to the sample.
Calibrating the EBSD systemEBSD system calibration measures the sample to screen distance and the pattern center positionon the phosphor screen (Figure 5). The pattern center is the point on the screen closest to thegeneration point of the diffraction pattern on the sample. There are several methods forcalibrating an EBSD system.One method to measure the pattern center position is to collect a diffraction pattern with the screenboth in the normal and partly retracted position and to measure the positions of the same zone axeson both patterns (Figure 7).The screen to sample distanceis calculated by measuring thedistance between two zonesseparated by a known anglewith the screen in the operatingposition. The calibration canalso be performed using asingle pattern by iterativelyfitting the pattern center andsample to screen distance tominimize the error in theorientation measurement.
w ≈ 2lθ ≈nlλ
d(2)
(a) (b)
Figure 7: EBSD system calibration. Lines joining the position of thesame features on the phosphor in the normal and retracted positionmeet at the pattern center.
Automated indexingand orientation measurementThe crystal orientation iscalculated from the Kikuchiband positions by the computerprocessing the digitizeddiffraction pattern collectedby the CCD camera. The Kikuchiband positions are foundusing the Hough transform.The transform between thecoordinates (x, y) of thediffraction pattern and thecoordinates (ρ, θ) of Houghspace is given by (Figure 8):
ρ = x cos θ+ y sin θ (3)
A straight line is characterizedby ρ, the perpendiculardistance from the origin and θthe angle made with the x-axisand so is represented by asingle point (ρ, θ) in Houghspace. Kikuchi bands transformto bright regions in Houghspace which can be detectedand used to calculate theoriginal positions of the bands(Figure 9).
Image Hough Transformation
Figure 8: The Hough transform converts lines into pointsin Hough space
Figure 9: Finding the position of the Kikuchi bands in the diffractionpattern using the Hough transform
Original diffraction pattern Hough transform of pattern
Indexed diffraction pattern Corresponding Kikuchi bandsfound on diffraction pattern
Detected peaks in Hough transform
Using the system calibration, the angles between the planes producing the detected Kikuchibands can be calculated (Box 1). These are compared with a list of inter-planar angles for theanalyzed crystal structure to allocate Miller indices to each plane. The final step is to calculate the orientation of the crystal lattice with respect to coordinates fixed in the sample (Box 2). This whole process takes less than a few milliseconds with modern computers.
Box 1: Calculating the angle betweencrystal planes from measurements onthe diffraction pattern
Two Kikuchi bands are defined by the pairs of points (x1 y1l), (x2 y2l) and (x3 y3l), (x4 y4l).
The vector ri joins the point of incidence ofthe electron beam on the sample to thepoint (xi yi l).
The unit vector n1 = r1 x r2 / r1 x r2 is perpendicular to the crystal plane generating the first Kikuchi band and thevector n2 = r3 x r4 / r3 x r4 is perpendicularto the crystal plane generating the secondKikuchi band.
The angle between the planes forming thetwo Kikuchi bands is cos -1( n1 • n2 )
Box 2: Representing crystal orientations
A common method for representing a crystalorientation is to use the ideal orientationnomenclature (hkl)[uvw]. Directions in thecrystal are referred to a set of coordinatesfixed in the sample. Using the terminology of rolled sheet metals, these are the samplenormal (ND), rolling direction (RD) andtransverse direction (TD). In the idealorientation nomenclature the normal to thecrystal plane (hkl) is parallel to the samplenormal and the crystal direction [uvw]parallel to the rolling direction (Figure 10).The notation {hkl}<uvw> denotes thesymmetrically related set of directions which represent the texture.
Figure 10: Meaning of ideal orientation nomenclature (hkl)[uvw]. Here the (110) planenormal is parallel to the normal direction and the [001] direction is parallel to the rolling direction, so the texture is (110)[001]
The orientation in figure 10 can also berepresented by three Euler angles ϕ1Φϕ2 (seesection 3) with ϕ1=135°, Φ=90° and ϕ2=90°.
Band intensityThe mechanisms giving rise to the Kikuchi band intensities and profile shapes are complex. As an approximation, the intensity of a Kikuchi band Ihkl for the (hkl) plane is given by:
where fi (θ) is the atomic scattering factor for electrons and (xi yi zi) are the fractionalcoordinates in the unit cell for atom i. An observed diffraction pattern should be compared with a simulation calculated using equation 4, to ensure only planes that produce visible Kikuchi bandsare used when solving the diffraction pattern (Figure 11). This is especially important whenworking with materials with more than one atom type.
Figure 11: Diffraction patternband intensities.
(a) Diffraction pattern from the orthorhombic ceramic mullite (3Al2O3 2SiO2) collected at 10 kV accelerating voltage.
(b) Solution overlaid on the diffraction pattern giving the crystal orientation as {370}<7-34>
(c) Simulated diffraction pattern showing all Kikuchi bands with intensity greater than 10% of the most intense band.
(d) Simulation of crystal orientation giving the solutionshown in (b).
Summary■ When an electron beam is incident on a
tilted crystalline sample, electron backscatter diffraction patterns are formed on a suitably placed phosphor screen.
■ The diffraction pattern consists of a set of Kikuchi bands which are characteristic of the sample crystal structure and orientation.
■ The center line of each Kikuchi band corresponds to the intersection with the phosphor screen of the diffracting plane responsible for the band.
■ The position of the Kikuchi bands can be found automatically with the Hough transform and used to calculate the crystal orientation of the sample region that formed the pattern.
(a)
(c) (d)
(b)
(4)Ihkl = Σ fi (θ) cos 2π (hxi + kyi + lzi) + Σ fi (θ) sin 2π (hxi + kyi + lzi)i
Types of EBSDexperimentPoint analysisIn EBSD point analysis thebeam is positioned at a pointof interest on the sample, adiffraction pattern collectedand the crystal orientationcalculated. This provides aquick overview of thecrystallinity of the sample andthe range of grain orientationspresent (Figure 12).
Crystal OrientationMappingIn crystal orientation mapping,the electron beam is scannedover the sample on a grid ofpoints and at each point adiffraction pattern obtainedand the crystal orientationmeasured. The resulting datacan be displayed as a crystalorientation map and processedto provide a wide variety ofinformation about the samplemicrostructure (Figure 13).Large sample areas can bemeasured by automaticallymoving the SEM stage betweensuccessive maps.
The maps shown are based on the sample normal, rollingand transverse directions. At each point in the map, the crystallographic directioncorresponding to the particularsample direction is calculated,and a color allocatedaccording to its position in theinverse pole figure. (Figure 14)
Figure 12: Point analysis of grains insteel showing the diffraction pattern
and calculated crystal orientation.
Forward scattered electron image Normal direction crystal orientation map
Rolling direction crystal orientation map
Transverse direction crystal orientation map
RD
RD
Figure 13: Crystal orientation maps from austenitic stainless steel collected at 20 kV accelerating voltage. The map step size is 0.72 µmand contains 49152 crystal orientation measurements.
{0 1 2 }<1 6 -3> {3 8 11 }<-7 4 -1>
{1 -5 10 }<-20 2 3> {4 11 -8 }<3 -4 -4>
Pattern quality mapsThe diffuseness or quality of the diffractionpattern is influenced by a number of factorsincluding local crystalline perfection, samplepreparation, surface contamination and thephase and orientation being analyzed.
The height of the peaks in the Hough transformgives a measure of the pattern quality. Onealgorithm for calculating pattern quality p is:
p = Σ hi / 3σh (5)i=1-3
where hi is the peak height of the Houghtransform of the ith most intense Kikuchi bandand σh is the standard deviation of the Houghtransform. Pattern quality maps will oftenreveal features invisible in the electron imagesuch as grains, grain boundaries and surfacedamage such as scratches (Figure 15).
Figure 15: Pattern quality map from sample shownin Figure 13.
Figure 14: Coloring of crystal orientation maps
Top left: A point in the electron image is circled in pink
Top right: The crystal orientation at this point is [2 -3 4] parallel tothe sample rolling direction (RD) and the (9 2 -3) plane normalparallel to the sample normal direction (ND).
Left: The ND, RD and TD directions plotted on an inverse polefigure. The inverse pole figure is colored by a mix of red, greenand blue depending on the position in the stereographic triangle.This acts as a color key to determine the color of the ND, RD and TD orientation maps at the analyzed point.
Many materials are formed of an aggregate of variously oriented crystals or grains. The natureof the interface between the grains, or grain boundaries, can influence the physical propertiesof the bulk material. Grain boundaries are characterized by the misorientation axis and angleand the boundary plane. Some boundaries satisfy certain geometrical criteria and theirpresence in a material may confer particular properties. When crystal lattices share a fractionof sites on either side of a grain boundary, they are termed coincident site lattices (CSL). CSL’sare characterized by Σ, where Σ is the ratio of the size of the CSL unit cell to the standard unitcell. Two examples of special boundaries are shown in Figure 16.
Figure 16: Special grain boundaries
The Σ3 boundary (twin boundary) is a 36.9° rotation about the [111] direction.
The Σ5 boundary is a 36.9° rotation about the <100> axis.
Grain boundary positions superimposed on the pattern quality image (left). The boundaries are color coded according to the histogram of misorientation angle (right).Only boundaries with a misorientation of greater than 3.5° are shown.
Coincident site lattice (CSL) boundary positions superimposed on the pattern qualityimage (left). The boundaries are color coded by CSL type shown in the histogram of CSL (right).
Map showing grain positions (left) and histogram of grain sizes (right). All neighbouring points within a grain have a misorientation less than 3°.
Figure 17: Grain boundary data measured by EBSD from the sample in Figure 13.Representation of grains and grain boundariesUnlike an optical or scanning electron micrograph, the crystal orientation mapmust reveal the positions of all grains and grain boundaries in the samplemicrostructure (Figure 17). In crystal orientation mapping a grain is defined bythe collection of neighbouring pixels in the map, which have a misorientationless than a certain threshold angle. The distribution of grain sizes can be measuredfrom the data collected for the map. In addition, the distribution of grainboundary misorientation angles and the distribution and position of special grain boundaries can be shown (See box 3).
The Σ5 boundary is a 36.9°rotation about the [100] direction.
Lattice ALattice BCSL of A and B
Lattice ALattice BCSL of A and B
Figure 18: Crystal orientation maps and texture plots from IF steel. The map step size is 51 nm and it contains 109,944 points.
TextureThe individual crystalorientation measurementscollected by crystal orientationmapping can be used to showthe crystallographic texturesdeveloped in the sample (See Box 4). Figure 18 shows a crystal orientation map from a sample of deformedinterstitial free (IF) steeltogether with pole figure and Euler space plots of theindividual crystal orientations.The various textures in thesample can be separatedautomatically, their volumefractions calculated, and theregions of the sample fromwhich they originate shown.This is illustrated in Figure 19which shows two texturecomponents separated fromthe map in Figure 18.
Figure 19: Two texture components separated fromthe data shown in Figure 18 and maps showingwhere the components originate in the sample.
Individual crystal orientationsplotted as an (001) pole figure.
The same data plotted as an (110) pole figure using colorcontours showing the strength of the texture compared to arandom texture.
Sample normal direction.
Sample rolling direction.
Sample transverse direction.
Top left and right:Regions of sample contributing to extracted texture.
Bottom left and right:(001) pole figure showing extracted texture. The intensity of the color shows how close the orientation is to the extracted texture.
(a) {8 7 9}<-11 1 9> texture. 13.17% of orientations are within 6° of this texture.
(b) {2 -34 5}<11 3 16> texture. 12.79% of orientations are within 6° of this texture.
(a) (b)
The same data shown as ϕ1 ϕ2
sections through Euler space at 10° intervals of Φ with thetexture strength shown as color contours.
Phase discriminationEBSD can be used to discriminate crystallographically dissimilarphases by comparing the interplanar angles measured from the diffraction pattern, with calculated angles from a set ofcandidate phases, and selecting the best fit. Figure 21 shows the separation of the austenite and ferrite phases in a duplexstainless steel. Austenite is face centered cubic, and ferrite isbody centered cubic, and the phases can not be distinguished by X-ray microanalysis. The phase map shows that 38.6% of the sampled area is ferrite and 60.7% is austenite. The crystalorientation maps also reveal the constituent grains in the two phases.
Figure 21: Illustration of phase discrimination in a sample of duplexstainless steel. The map step size is 0.45 µm and contains 49152 points.
Box 4: Preferred orientationand texture
Grains are seldom orientedrandomly in polycrystallinematerials. The preferredcrystallographic orientationor texture of polycrystallinematerials influences manyproperties of the bulk material,because physical propertiesare often anisotropic withrespect to crystal direction.Material processing methodsare frequently deliberatelychosen to produce certaindesired textures. Textures areusually specified by the idealorientation nomenclature{hkl}<uvw>.
The texture can be representedas a stereographic projection(pole figure) of the directionsof selected crystal planenormals (poles). Theorientation distribution closeto a particular texture willappear as a cluster of pointson the pole figure. Texturescan also be plotted as a threedimensional plot of the threeEuler angles ϕ1Φ ϕ2 (seesection 3) associated witheach orientation measurement.This is usually presented as aseries of two dimensionalslices through the Euler spaceto produce density maps ofEuler angles (Figure 20).
Texture is usually measured byX-ray diffraction. The X-rayintensity from a particulardiffracting plane is measured,while the sample is steppedthrough a series oforientations to complete the
Left: An Euler space plot for theBrass texture. This is shown asindividual orientation points plotted randomly within 15° ofthe texture plotted as the corresponding Euler angles. It isplotted as sections through thethree dimensional Euler space. NineΦ, ϕ1 sections are plotted at 10°intervals of ϕ2. It shows a clusterof points corresponding to theEuler angles:
ϕ1 = 45° Φ = 90° ϕ2 = 54.7°,
which represents the Brass texture.
Left: A face centered cubic crystalin the Brass texture orientation{110}<-1 1 2>
pole figure. The individualcrystal orientations measuredwith EBSD can also bedisplayed as pole figures forpreferred orientation analysis.The spatial resolution andpositioning limitations ofX-ray diffraction make EBSDthe preferred technique forexamining texture on a
microscopic scale. However,advances in EBSD processingspeed can make thetechnique competitive withX-ray diffraction for texturemeasurements from largesamples. EBSD and X-raydiffraction arecomplementary techniquesfor texture analysis.
Figure 20: Representations of texture. Summary■ A crystal orientation map
can be acquired by scanningthe electron beam over a sample and analyzing the diffraction pattern formed at each point.
■ Crystal orientation maps definitively reveal the sample microstructure and can measure grain size distributions, grain boundarymisorientations and special boundaries and show their location in the sample.
■ The individual crystal orientation measurements can be displayed to show crystallographic texture or preferred orientation. Texture components can be separated and the regions of the sample contributing to those components shown.
■ EBSD can also be used to distinguish crystallo- graphically different phases and to show their location, abundance and preferred orientations.
Electron image Map showing separation of thetwo phases. Ferrite is yellow,austenite is purple
Normal direction crystal orientation map from ferrite phase
Normal direction crystal orientation map from austenite phase
[-1 1 2] RD
(110) ND
[1 -1 1] TD
Left: An (001) pole figure for a crystal in the Brass texture orientation.
Middle: A simulated (001) pole figure for the Brass texture drawn asindividual orientation points plotted randomly within 15° of the texture.
Right: A simulated (110) pole figure for the Brass texture as individualorientation points plotted randomly within 15° of the texture.
Undertaking EBSD experimentsSample preparationEBSD is very sensitive to crystalline perfection,and sample preparation may be needed toremove any surface damage. A well preparedsample is a prerequisite to obtaining a gooddiffraction pattern. Surfaces must be sufficientlysmooth to avoid forming shadows on thediffraction pattern from other parts of thesample. Suitable techniques for use with EBSD include:
■ For metals and insulators: mounting in conductive resin, mechanical grinding, diamond polishing and final polishing with colloidal silica.
■ For metals: mounting in conductive resin, mechanical grinding, diamond polishing and electropolishing.
■ Brittle materials such as ceramics and geological materials can often be fractured to reveal surfaces immediately suitable for EBSD.
■ Ion milling for materials which are not amenable to conventional metallography such as zirconium and zircalloy.
■ Dual focused ion beam – electron beam microscopes fitted with EBSD can perform in-situ specimen preparation for EBSD.
■ Plasma etching for microelectronic devices.
Charging can be reduced when the sample istilted for EBSD experiments and can also bereduced by analyzing the sample in anenvironmental or low vacuum SEM.
Camera integration time and resolutionBecause the luminous intensity of the diffractionpattern on the phosphor screen is low, using ashort integration time on the CCD camera maygive a poor signal to noise ratio. In this case,integrating for longer on the CCD will improvethe visibility of the diffraction pattern (Figure 22). The camera may be cooled toreduce electronic noise in the CCD when usedin this way. The yield of back scattered electronsincreases with atomic number, so low atomicnumber materials will require a longerintegration time than higher atomic numbers.
The CCD camera resolution can also effect the integration time required to collect adiffraction pattern. Current CCD cameras can collect 12 bit images at a resolution of1300 x 1024 pixels. Using “pixel binning”,neighboring pixels in the CCD can be added together to form a single pixel in theimage. When pixels are binned in this way, less integration time is required to achieve agiven signal in the image pixel because thedetecting pixel area is larger. For example, if a satisfactory diffraction pattern is obtained in 12 ms at 1300 x1024 resolution, a comparable pattern can be obtained in 3 ms at 650 x 512 resolution.
It is not essential to collect high resolutiondiffraction patterns for crystal orientationmapping, because the Hough transform always operates on a 128 x128 resolutionimage. In addition, the transform works wellon noisy images (Figure 23) so someexperimentation is necessary to determine the optimum integration time required whencollecting maps.
Figure 22: Effect of integration time and probe current on diffraction pattern. (a) 36 ms, 2 nA. (b) 36 ms 200 pA, (c) 360 ms 200 pA. (Probe currents are approximate)
Figure 23: Diffraction patterns can be automatically solved in the presence of noise.
(a) (b) (c)
Charging in non-conductive samples can beeliminated, as for X-ray microanalysis, by thedeposition of a conducting layer. The depositedlayer must be very thin – for example 2 to 3 nmof carbon or gold/palladium – otherwise adiffraction pattern will not be obtained. It maybe necessary to increase the electron acceleratingvoltage to penetrate the conducting layer.
Background removalElectrons of all energiesscattered from the sample forma background to the diffractionpattern, which reduces thecontrast of the Kikuchi bands.The background intensity canbe removed to improve thevisibility of the Kikuchi bands.The background can bemeasured by scanning the beamover many grains in the sampleto average out the diffractioninformation. The backgroundcan be removed by subtractionfrom, or division into, theoriginal pattern. (Figure 24)
Microscope operating conditionsIt is very important tounderstand the effect ofvarying the microscopeoperating conditions on thediffraction pattern.
Probe currentIncreasing the probe currentwill increase the number ofelectrons contributing to thediffraction pattern and soallow the camera integrationtime to be reduced (Figure 22).However, this must be balancedwith the spatial resolutionrequired, because increasingthe probe current will alsoincrease the electron beam size.
Accelerating voltageIncreasing the acceleratingvoltage reduces the electronwavelength and hence reducesthe width of the Kikuchi bandsin the diffraction pattern (seeequation 2). Also, becausemore energy is being depositedon the phosphor screen, this
(a) (b)
(c) (d)
will result in a brighter patternwhich requires a shorterintegration time (Figure 25).Changing the acceleratingvoltage may require adjustmentto the Hough transform filtersize to ensure the Kikuchi bandsare detected correctly. Higheraccelerating voltages may berequired to penetrateconducting layers, and loweraccelerating voltages forrestraining the beam to thinlayers, or for charging samples.
Working distance and magnificationBecause the sample is tilted,the SEM working distance will
change as the beam positionmoves up or down the sample,and the image will go out offocus (Figure 5). The imagewill also be foreshortenedbecause of the tilt, and at lowmagnifications much of thefield of view could be out offocus. Some EBSD systems cancompensate for the imageforeshortening by usingdifferent horizontal andvertical image beam steps, and can adjust the SEM focusautomatically as the beam ismoved over the sample(Figure 26).
Figure 25: Effect of changing accelerating voltage on diffraction patterns from nickel. Note that there is aneffect on the bandwidth, sharpness and contrast. (a) 10kV, (b) 20 kV, (c) 30kV
Figure 26: Tilt correction and focus maintenance.
(a) Image without tilt or dynamic focus compensation.
(b) Image with tilt compensation and no dynamicfocus compensation.
(c) Image with tilt and dynamic focus compensation.The working distance is 14.98 mm at the top and15.11 mm at the bottom of the image.
In addition, movements of the beam will alterthe pattern center position on the phosphorscreen and this can affect the EBSD systemcalibration. EBSD systems can compensateautomatically for shifts in the pattern center by calibrating at two working distances andinterpolating for intermediate working distancevalues. It is important to know the range ofworking distances for which the EBSD systemwill remain accurately calibrated.
Figure 27: Effect of SEM vacuum on diffraction pattern from platinum sample (a) 0.05 torr (b) 0.5 torr (c) 1.0 torr
PressureDiffraction patterns can also be collected fromsamples at low vacuum in environmental SEMs(Figure 27). This can be useful with specimenswhich may otherwise charge, such as ceramicor geological materials.
Effect of sample tiltThe EBSD sample is usually tilted at 70° to the horizontal to optimize both the contrast in the diffraction pattern and the fraction of electrons scattered from the sample. For smaller tilt angles the contrast in thediffraction pattern decreases (Figure 28).
Spatial resolutionThe electrons contributing to the diffractionpattern originate within nanometres of thesample surface. Hence, the spatial resolutionwill be related to the electron beam diameter,and this depends on the type of electronsource and probe current used. Typical beamdiameters at 0.1 nA probe current and 20 kVaccelerating voltage are 2 nm for a FEG source,and 30 nm for a tungsten source. The beamprofile on the sample surface will also be elongated in the direction perpendicular to thetilt. The spatial resolution achieved in practicewill depend on the sample, SEM operating conditions and electron source used, and underoptimum conditions, grains as small as 10 nmcan be identified.
Measurement accuracyOrientation measurementErrors in crystal orientation measurements fromthe diffraction pattern will depend principallyon the accuracy of the Kikuchi band positionmeasurement and the system calibration, andare generally in the range ±0.5°. To avoidsystematic errors in orientation measurementsmade with respect to the sample axes fortexture measurements, care should be taken to ensure that the sample normal and longitudinaldirections are oriented correctly with respect tothe phosphor screen.
Residual errorsFor each orientation measurement a residualangle θε can be calculated where
θε = Σ (θim – θi
c )2
i=1-n
θim is the measured angle between the ith
pair of Kikuchi bands, θic is the actual angle
between the corresponding crystal planes andn is the number of Kikuchi band pairs. θε is ameasure of the degree of fit of a solution tothe diffraction pattern and is used to rank possible solutions. Residual error values higherthan 1.5° can suggest the system calibrationneeds to be checked.
√ (6)
Summary■ Careful sample preparation is critical for
successful EBSD experiments, and conventionalmetallographic techniques can normally be used successfully.
■ It is important to balance the requirements of total experiment time, orientation accuracyand spatial resolution when designing an EBSD experiment.
■ The orientation measurement accuracy is typically ±0.5°.
■ The spatial resolution of EBSD depends on the sample, microscope type and microscope operating conditions and in a FEGSEM, grains as small as 10 nm can be analyzed.
(a) (b) (c)
(a) (b)
Crystal directions, planes and the zone axisConsider a crystal lattice with unit cell edges a,b and c (Figure 30). A crystal direction [uvw] isparallel to the direction joining the origin ofthe crystal lattice with the point withcoordinates (ua, vb, wc) (Figure 33).
A plane with Miller indices (hkl) passes throughthe three points (a/h,0,0), (0, b/k,0) and (0,0, c/l)on the edges of the unit cell. The set of parallellattice planes passes through all similar pointsin the lattice. The plane d-spacing is theperpendicular distance from the origin to theclosest plane and also the perpendiculardistance between successive planes. (Figure 34).In materials with cubic symmetry the crystaldirection [uvw] and the normal to the plane(uvw) are parallel.
The common direction shared by two crystalplanes when they intersect is called the zoneaxis (Figure 35). A zone axis [uvw] is alwaysperpendicular to the plane normal (hkl) thatcomprises the zone, and this relation is statedin the zone law: uh + vk + wl = 0.The family of symmetrically related directions[uvw] is shown by the notation <uvw> and thefamily of symmetrically related planes (hkl) isshown by the notation {hkl}.
Basic crystallography for EBSDCrystals and latticesCrystalline material consists of a regular repetition of a group of atoms in three dimensional space.
A crystal lattice is an infinitely repeating array of points in space(Figure 29). The unit cell of the lattice is the basic repeating unitof the lattice, and is characterized by a parallelepiped with celledge lengths a, b, c and inter axis angles α, β, γ (Figure 30).These unit cells can be classified as belonging to one of fourteenBravais lattices. Each Bravais lattice belongs to one of the sevencrystal systems. (Figure 31). The motif is the group of atomsrepeated at each lattice point, which generates the crystallinestructure of the material (Figure 32).
Figure 29: A crystal lattice
Figure 31: The fourteenBravais lattices.
Each Bravaislattice belongs to one of theseven crystalsystems:
cubic, tetragonal, orthorhombic, rhombohedral, hexagonal, monoclinic and triclinic.
P = primitive, I = body centered,F = face centered,C = base centered.
Figure 32: Crystal structure showing a repeatedatomic motif at each lattice point
Base-centeredorthorhombic (C)
Face-centeredorthorhombic (F) Rhombohedral (R)
Body-centeredtetragonal (I)
Figure 33: Meaning of the crystal direction [uvw]. The direction shown here is [422]
Figure 34: Crystal planes
(a) The plane with Miller indices (hkl) makesintercepts a/h, b/k and c/l on the edges of the unitcell. The plane shown here is (221) because theintercept in the a direction is 1/2a, in the b direction1/2b and in the c direction c.
(b) A set of parallel (221) planes intersecting theedges of the unit cells. The d-spacing for theseplanes is the perpendicular distance betweensuccessive planes or equivalently the perpendiculardistance from the origin to the closest plane.
Figure 35: The zone axis is the common crystal direction shared by two planes
MisorientationThe orientation between two crystal coordinatesystems can also be defined by the angle-axispair θ[uvw]. One coordinate system can be superimposedonto the other by rotating by an angle θaround the common axis [uvw] (Figure 37).Because it is an axis of rotation, the direction[uvw] is the same in both coordinate systems.The angle-axis pair notation is normally used to describe grain boundary misorientations (see box 3).
Figure 36: Relationship between crystal and samplecoordinate systems. α1, β1 and γ1 are the anglesbetween the crystal direction [100] and RD, TD andND respectively.
Figure 38: Definition of the Euler angles ϕ1Φ ϕ2. This shows therotations necessary to superimpose the crystal coordinate system (red)onto the sample system (blue). The first rotation ϕ1 is about the z axisof the crystal coordinate system. The second rotation is Φ about thenew x-axis. The third rotation is ϕ2 about the new z-axis. The dottedlines show the positions of the axis before the last rotation. Note thatthe orientation can also be defined by an equivalent set of Eulerangles which superimpose the sample coordinate system onto thecrystal coordinate system.
Crystal orientationA crystal orientation is measured with respectto an orthogonal coordinate system fixed inthe sample. The sample system is normallyaligned with directions used in texture measurements on rolled sheet materials. The x axis is parallel to the rolling direction of the sample (RD), the y axis parallel to thetransverse direction (TD) and the z axis parallelto the normal direction (ND) (Figure 36).
The relationship between a crystal coordinatesystem and the sample system is described byan orientation matrix G. A direction measuredin the crystal system rc is related to the samedirection measured in the sample system rs by:
rc = Grs
The rows of the matrix G are the directioncosines of the crystal system axes in the coordinates of the sample system.
cos α1 cos β1 cos γ1
G = cos α2 cos β2 cos γ2
cos α3 cos β3 cos γ3
Figure 36 shows how the angles α1, β1, γ1, are defined. α2, β2, γ2 and α3, β3, γ3 are similarly defined as the angles between the[010] and [001] crystal directions and the three sample axes.
Euler anglesThe orientation between twocoordinate systems can also be defined by a set of threesuccessive rotations aboutspecified axes. These rotationsare called the Euler angles ϕ1Φ ϕ2 and are shown inFigure 38.
(hkl)[uvw]A crystal orientation is alsorepresented by a crystal direction [uvw] that is parallelto the rolling direction of thesample, and the crystal planenormal (hkl) that is parallel to the sample normal. (See Figure 10)
Stereographic projectionsDirections can be representedin two dimensions with astereographic projection(Figure 39). Consider a spherewith centre O, south pole Sand a point P on its surface.The line SP intersects theequatorial plane at p. Hence, any direction OP can be represented by a corresponding stereographicprojection point p on theequatorial plane.
Figure 37: Two interpenetrating lattices can berealigned by a single rotation about a common axis[uvw] by an angle θ. In the figure the axis is thecommon [111] direction and the rotation angle 60°.
Figure 39: The stereographic projection.
(a) Projection of a point P on the surface of a sphere with centre Ofrom the projection point S onto the equatorial plane at p.
(b) The stereographic projection is the equatorial plane of this spherewith the point p representing the direction OP.
For texture studies of rolled materials, the axesof the projection sphere can be aligned withthe axes of the sample. The sample normal isthe center of the projection and the rollingdirection chosen to be at the right of theprojection. If a unit cell with cubic symmetry isplaced at the center of the projection sphere,then the crystal directions and plane normals
can be projected onto the equatorial plane andthe directions represented as a stereographicprojection (Figure 40).
In an inverse pole figure the crystallographicdirections parallel to particular directions in thesample are plotted on a stereographicprojection (Figure 41).
(a) A cubic crystal unit cell oriented with respectto the sample axes.
(b) The same sample axes oriented with respect to the crystal unit cell.
(c) The crystallographic direction parallel to ND plotted on a stereographic projection with axes parallel to the edges of the crystal unit cell. The direction is repeated because, for cubic materials, it could be anyone of 48 symmetrically equivalent crystal directions.
(d) The yellow area from the stereographic projection in (c) is extracted to form the ND inverse pole figure. Only one stereographic triangle is required for this projection because the other triangles in the projection are symmetrically equivalent.
(e) and (f) The RD and TD inverse pole figures are obtained in a similar way.[100]
[010]
Figure 41: Formation of an inverse pole figure
(b)(a)
(c)
(d) (e) (f)
Figure 40: Stereographic projections of a cubic unit cell with edges parallel to RD, NDand TD. The orientation is (001)[100]
(a) The six {001} plane normals (poles) are shown.
(b) A stereographic projection of these directions. (Directions that project beneath the equatorial plane are not shown.) This is a (100) pole figure of this crystal orientation.
(c) The eight {111} plane normals are shown.
(d) A stereographic projection of these directions which is a (111) pole figure of this crystal orientation.
