Reciprocal Lattice&Ewald Sphere Construction

A crystal resides in real space. The diffraction pattern of the crystal in Fraunhofer diffraction geometry resides in Reciprocal Space. In a diffraction experiment (powder diffraction using X-rays, selected area diffraction in a TEM), a part of this reciprocal space is usually sampled.From the real lattice the reciprocal lattice can be geometrically constructed. The properties of the reciprocal lattice are inverse of the real lattice planes far away in the real crystal are closer to the origin in the reciprocal lattice.As a real crystal can be thought of as decoration of a lattice with motif; a reciprocal crystal can be visualized as a Reciprocal Lattice decorated with a motif* of Intensities. Reciprocal Crystal = Reciprocal Lattice + Intensities as Motif*The reciprocal of the reciprocal lattice is nothing but the real lattice!Planes in real lattice become points in reciprocal lattice and vice-versa.Reciprocal Lattice and Reciprocal Crystals* Clearly, this is not the crystal motif- but a motif consisting of Intensities.

In diffraction patterns (Fraunhofer geometry) (e.g. SAD), planes are mapped as spots (ideally points). Hence, we would like to have a construction which maps planes in a real crystal as points.Apart from the use in diffraction studies we will see that it makes sense to use reciprocal lattice when we are dealing with planes.The crystal resides in Real Space, while the diffraction pattern lives in Reciprocal Space.Motivation for constructing reciprocal latticesAs the index of the plane increases the interplanar spacing decreases and planes start to crowd around the origin in the real lattice (refer figure). Hence, it is a nice idea to work in reciprocal space (i.e. work with reciprocal lattice) when dealing with planes.As seen in the figure the diagonal is divided into (h + k) parts.

We will construct reciprocal lattices in 1D and 2D before taking up a formal definition in 3D

Let us start with a one dimensional lattice and construct the reciprocal latticeReciprocal LatticeReal LatticeHow is this reciprocal lattice constructed?Note in 1D planes are points and have Miller indices of single digit (they have been extended into the second dimension (as lines) for better visibilityThe plane (2) has intercept at , plane (3) has intercept at 1/3 etc.As the index of the plane increases it gets closer to the origin (there is a crowding towards the origin)What do these planes wih fractional indices mean? We have already noted the answer in the topic on Miller indices and XRD.One unit cellReciprocal LatticeEach one of these points correspond to a set of planes in real spaceNote that in reciprocal space index has NO bracketsReal Lattice

Now let us construct some 2D reciprocal latticesExample-1

Reciprocal LatticeThe reciprocal lattice has an origin!Each one of these points correspond to a set of planes in real spaceReciprocal LatticeReal LatticeNote that vectors in reciprocal space are perpendicular to planes in real space (as constructed!)But do not measure distances from the figure!Overlay of real and reciprocal latticesg vectors connect origin to reciprocal lattice points

Note that vectors in reciprocal space are perpendicular to planes in real space (as constructed!)The real latticeThe reciprocal latticeExample-2Reciprocal LatticeReal LatticeBut do not measure distances from the figure!

Reciprocal LatticeProperties are reciprocal to the crystal latticeThe reciprocal lattice is created by interplanar spacingsBBASIS VECTORSThe basis vectors of a reciprocal lattice are defined using the basis vectors of the crystal as below

A reciprocal lattice vector is to the corresponding real lattice planeThe length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice planePlanes in the crystal become lattice points in the reciprocal lattice ALTERNATE CONSTRUCTION OF THE REAL LATTICEReciprocal lattice point represents the orientation and spacing of a set of planesSome properties of the reciprocal lattice and its relation to the real lattice

Reciprocal lattice* is the reciprocal of a primitive lattice and is purely geometrical does not deal with the intensities decorating the pointsPhysics comes in from the following:For non-primitive cells ( lattices with additional points) and for crystals having motifs ( crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|Fhkl|2) Some of the Reciprocal lattice points go missing (or may be scaled up or down in intensity) Making of Reciprocal Crystal: Reciprocal lattice decorated with a motif of scattering power (as intensities)The Ewald sphere construction further can select those points which are actually observed in a diffraction experiment * as considered hereGoing from the reciprocal lattice to diffraction spots in an experimentA Selected Area Diffraction (SAD) pattern in a TEM is similar to a section through the reciprocal lattice (or more precisely the reciprocal crystal, wherein each reciprocal lattice point has been decorated with a certain intensity).The reciprocal crystal has all the information about the atomic positions and the atomic species.

Real CrystalReciprocal LatticeReciprocal CrystalTo summarize:Diffraction PatternPurely Geometrical ConstructionDecoration of the lattice with IntensitiesEwald Sphere constructionSelection of some spots/intensities from the reciprocal crystalStructure factor calculationReal LatticeDecoration of the lattice with motif

In crystals based on a particular lattice the intensities of particular reflections are modified they may even go missingCrystal = Lattice + MotifDiffraction PatternPosition of the diffraction spots RECIPROCAL LATTICEIntensity of the diffraction spots MOTIF OF INTENSITIESPosition of the diffraction spotsLatticeIs determined byIntensity of the diffraction spotsMotif

There are two ways of constructing the Reciprocal Crystal:1) Construct the lattice and decorate each lattice point with appropriate intensity*2) Use the concept as that for the real crystal**Making of a Reciprocal CrystalThe above two approaches are equivalent for simple crystals (SC, BCC, FCC lattices decorated with monoatomic motifs), but for ordered crystals the two approaches are different (E.g. ordered CuZn, Ordered Ni3Al etc.) (as shown soon).* Point #1 has been considered to be consistent with literature though this might be an inappropriate.** Point #2 makes reciprocal crystals equivalent in definition to real crystalsReal CrystalReciprocal LatticeReciprocal CrystalReal LatticeTake real lattice and construct reciprocal latticeUse motif to compute structure factor and hence intensities to decorate reciprocal lattice pointsDecorate with motif

Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)Figures NOT to Scale000100111001101011010110SCLattice = SCReciprocal Crystal = SCSelection rule: All (hkl) allowed In simple cubic crystals there are No missing reflectionsSC lattice with Intensities as the motif at each reciprocal lattice point+Single sphere motif=SC crystal

Figures NOT to Scale000200222002101022020110BCCBCC crystal Reciprocal Crystal = FCC220011202100 missing reflection (F = 0)Weighing factor for each point motifFCC lattice with Intensities as the motifSelection rule BCC: (h+k+l) even allowed In BCC 100, 111, 210, etc. go missingImportant note:The 100, 111, 210, etc. points in the reciprocal lattice exist (as the corresponding real lattice planes exist), however the intensity decorating these points is zero.xxxxxxxxxxxx

Figures NOT to Scale000200222002022020FCCLattice = FCCReciprocal Crystal = BCC220111202100 missing reflection (F = 0)110 missing reflection (F = 0)Weighing factor for each point motifBCC lattice with Intensities as the motif

When a disordered structure becomes an ordered structure (at lower temperature), the symmetry of the structure is lowered and certain superlattice spots appear in the Reciprocal Lattice/crystal (and correspondingly in the appropriate diffraction patterns). Superlattice spots are weaker in intensity than the spots in the disordered structure.An example of an order-disorder transformation is in the Cu-Zn system: the high temperature structure can be referred to the BCC lattice and the low temperature structure to the SC lattice (as shown next). Another examples are as below.Order-disorder transformation and its effect on diffraction patternClick here to know more about Ordered StructuresClick here to know more about Superlattices & Sublattices

DisorderedOrdered- NiAl, BCCB2 (CsCl type)- Ni3Al, FCCL12 (AuCu3-I type)

Positional OrderG = H TSHigh T disorderedLow T ordered470CSublattice-1 (SL-1)Sublattice-2 (SL-2)BCCSCSL-1 occupied by Cu and SL-2 occupied by Zn. Origin of SL-2 at (, , )In a strict sense this is not a crystal !!Probabilistic occupation of each BCC lattice site: 50% by Cu, 50% by ZnDiagrams not to scale

BCCFCCOrderedReciprocal crystalReciprocal Crystal = FCCFCC lattice with Intensities as the motifDiffraction pattern from the ordered structure (3D)This is like the NaCl structure in Reciprocal Space!Click here to see structure factor calculation for NiAl (to see why some spots have weak intensity) Slide 27Click here to see XRD powder pattern of NiAl Slide 5For the ordered structure:Notes:For the disordered structure (BCC) the reciprocal crystal is FCC.For the ordered structure the reciprocal crystal is still FCC but with a two intensity motif: Strong reflection at (0,0,0) and superlattice (weak) reflection at (,0,0) .So we cannot blindly say that if lattice is SC then reciprocal lattice is also SC.

DisorderedOrdered- NiAl, BCCB2 (CsCl type)

BCCFCCOrderedReciprocal crystalReciprocal Crystal = BCCBCC lattice with Intensities as the motifDiffraction pattern from the ordered structure (3D)Click here to see structure factor calculation for Ni3Al (to see why some spots have weak intensity) Slide 29Click here to see XRD powder pattern of Ni3Al Slide 6

DisorderedOrdered- Ni3Al, FCCL12 (AuCu3-I type)

There are two ways of constructing the Reciprocal Crystal:1) Construct the lattice and decorate each lattice point with appropriate intensity2) Use the concept as that for the real crystal (lattice + Motif)1) SC + two kinds of Intensities decorating the lattice2) (FCC) + (Motif = 1FR + 1SLR)1) SC + two kinds of Intensities decorating the lattice2) (BCC) + (Motif = 1FR + 3SLR) FR Fundamental Reflection SLR Superlattice ReflectionMotifMotif

SAD patterns from a BCC phase (a = 10.7 ) in as-cast Mg4Zn94Y2 alloy showing important zones[111][011][112]The spots are ~periodically arrangedExample of superlattice spots in a TEM diffraction patternSuperlattice spots

Example of superlattice peaks in XRD patternNiAl pattern from 0-160 (2)Superlattice reflections (weak)

d [nm]

2( [deg.]

I

h k l

Mul.

1

0.2886

30.960

163.1

1 0 0

6

2

0.2041

44.360

1000.0

1 1 0

12

3

0.1666

55.080

33.2

1 1 1

8

4

0.1443

64.520

147.1

2 0 0

6

5

0.1291

73.280

31.7

2 1 0

24

6

0.1178

81.660

276.5

2 1 1

24

7

0.1020

98.040

91.0

2 2 0

12

8

0.0962

106.400

2.5

3 0 0

6

9

0.0962

106.400

10.1

2 2 1

24

10

0.0913

115.140

161.7

3 1 0

24

11

0.0870

124.560

9.4

3 1 1

24

12

0.0833

135.220

64.8

2 2 2

8

13

0.0800

148.460

13.4

3 2 0

24

The Ewald Sphere* Paul Peter Ewald (German physicist and crystallographer; 1888-1985)Reciprocal lattice/crystal is a map of the crystal in reciprocal space but it does not tell us which spots/reflections would be observed in an actual experiment.The Ewald sphere construction selects those points which are actually observed in a diffraction experiment

organisiert von:Max-Planck-Institut fr Metallforschung Institut fr Theoretische und Angewandte Physik, Institut fr Metallkunde, Institut fr Nichtmetallische Anorganische Materialien der Universitt Stuttgart Programm7. Paul-Peter-Ewald-Kolloquium

Freitag, 17. Juli 2008

Circular of a Colloquium held at Max-Planck-Institut fr Metallforschung (in honour of Prof.Ewald)

13:30Joachim Spatz (Max-Planck-Institut fr Metallforschung) Begrung13:45Heribert Knorr (Ministerium fr Wissenschaft, Forschung und Kunst Baden-Wrttemberg Begrung14:00Stefan Hell (Max-Planck-Institut fr Biophysikalische Chemie) Nano-Auflsung mit fokussiertem Licht14:30Antoni Tomsia (Lawrence Berkeley National Laboratory) Using Ice to Mimic Nacre: From Structural Materials to Artificial Bone15:00Pause Kaffee und Getrnke 15:30Frank Gieelmann(Universitt Stuttgart) Von ferroelektrischen Fluiden zu geordneten Dispersionen von Nanorhren: Aktuelle Themen der Flssigkristallforschung 16:00Verleihung des Gnter-Petzow-Preises 200816:15Udo Welzel (Max-Planck-Institut fr Metallforschung) Materialien unter Spannung: Ursachen, Messung und Auswirkungen- Freund und Feind ab 17:00Sommerfest des Max-Planck-Instituts fr Metallforschung

The Ewald SphereThe reciprocal lattice points are the values of momentum transfer for which the Braggs equation is satisfied.For diffraction to occur the scattering vector must be equal to a reciprocal lattice vector.Geometrically if the origin of reciprocal space is placed at the tip of ki then diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere.Here, for illustration, we consider a 2D section thought the Ewald Sphere (the Ewald Circle)See Cullitys book: A15-4

Draw a circle with diameter 2/ Construct a triangle with the diameter as the hypotenuse and 1/dhkl as a side (any triangle inscribed in a circle with the diameter as the hypotenuse is a right angle triangle: APO = 90): AOPThe angle opposite the 1/d side is hkl (from the rewritten Braggs equation)Braggs equation revisitedRewrite

Now if we overlay real space information on the Ewald Sphere. (i.e. we are going to mix-up real and reciprocal space information).Assume the incident ray along AC and the diffracted ray along CP. Then automatically the crystal will have to be considered to be located at C with an orientation such that the dhkl planes bisect the angle OCP (OCP = 2).OP becomes the reciprocal space vector ghkl (often reciprocal space vectors are written without the *).

Radiation related information is present in the Ewald SphereCrystal related information is present in the reciprocal crystalThe Ewald sphere construction generates the diffraction patternThe Ewald Sphere constructionWhich leads to spheres for various hkl reflectionsChooses part of the reciprocal crystal which is observed in an experiment

K = K =g = Diffraction VectorEwald SphereThe Ewald Sphere touches the reciprocal lattice (for point 41) Braggs equation is satisfied for 41When the Ewald Sphere (shown as circle in 2D below) touches the reciprocal lattice point that reflection is observed in an experiment (41 reflection in the figure below).

(Cu K) = 1.54 , 1/ = 0.65 1 (2/ = 1.3 1), aAl = 4.05 , d111 = 2.34 , 1/d111 = 0.43 1Ewald sphere X-raysRow of reciprocal lattice pointsRows of reciprocal lattice pointsDiffraction from Al using Cu K radiationThe 111 reflection is observed at a smaller angle 111 as compared to the 222 reflection

(Cu K) = 1.54 , 1/ = 0.65 1 (2/ = 1.3 1), aAl = 4.05 , d111 = 2.34 , 1/d111 = 0.43 1Ewald sphere X-raysNow consider Ewald sphere construction for two different crystals of the same phase in a polycrystal/powder (considered next).Click to compare them

Diffraction cones and the Diffractometer geometryPOWDER METHODIn the powder method is fixed but is variable (the sample consists of crystallites in various orientations).A cone of diffraction beams are produced from each set of planes (e.g. (111), (120) etc.) (As to how these cones arise is shown in an upcoming slide).The diffractometer moving in an arc can intersect these cones and give rise to peaks in a powder diffraction pattern.Click here for more details regarding the powder method

Different cones for different reflectionsDiffractometer moves in a semi-circle to capture the intensity of the diffracted beams3D view of the diffraction cones

Cone of diffracted raysTHE POWDER METHODIn a power sample the point P can lie on a sphere centered around O due all possible orientations of the crystalsThe distance PO = 1/dhklUnderstanding the formation of the cones

The 440 reflection is not observed (as the Ewald sphere does not intersect the reciprocal lattice point sphere)Circular Section through the spheres made by the hkl reflectionsEwald sphere construction for AlAllowed reflections are those for h, k and l unmixed

The 331 reflection is not observedEwald sphere construction for CuAllowed reflections are those for h, k and l unmixed