Crystalline Solid Amorphous Solid

Particles are completely disorganized Particles are arranged in orderly fashion

Single and Polycrystalline Materials

Single : are in a repeating or periodic array over the entire extent of the material Polycrystalline material: comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.

Grain Boundary Polycrystalline Materials

Atomistic model of a nanocrystalline solid by Mo Li, JHU 15

Lattices

A lattice is a regular infinite arrangement of points in which every point has the same environment as any other point. A lattice in 2 dimensions is called a net and a regular stacking of nets gives us a 3-dimensional lattice.

2-D net Stacks of 2-D nets produce 3-D lattices. Lattice Lattices and Unit Cell Since lattices consist of infinitely repeating patterns, one needs only to look at a the smallest repeat unit to describe the lattice. The smallest repeat unit that will generate the entire lattice (by translation) is called the Unit Cell. It is defined by three repeat distances (a, b, and c) and three angles (a, b, g), where a is the angle between b and c, b is the angle between a and c, and g is the angle between a and b.

Unit cell parameters (a, b, c, a, b, g) are chosen to best represent the highest-possible symmetry of the crystal and are given right-handed axes (a is along x, b is along y and c is along z) with angles that are either all ≥ 90º or all ≤ 90º. Unit Cell: The smallest unit by repeating which entire lattice can be generated. Unit Cell • Rule: Must represent the symmetry elements of the whole!

If a crystal has symmetry, the unit cell must have at least that much symmetry Crystal Lattices and Unit Cells

Crystal lattice is the depiction of three dimensional arrangements of constituent particles (atoms, , ) of crystalline solids as points. Or the geometric arrangement of constituent particles of crystalline solids as point in space is called crystal lattice. Lattices The lattice can be considered as a kind of scaffold upon which the structure of the crystal is built. For a crystal, the lattice is a 3-dimensional array and the structural motif will be located in a hypothetical box called the unit cell. Unit Cell: The smallest portion of a crystal lattice is called Unit Cell. By repeating in different directions unit cell generates the entire lattice.

Parameters of a unit cell: •A unit cell is characterized by six parameters. These parameters are three edges (a, b and c) and angles between them (α, β and γ).

If a crystal has symmetry, the unit cell must have at least that much symmetry

Crystal Lattice Types of Unit Cells: What are the symmetries of the 7 crystal systems?

 Minimum symmetry of the 7 crystals systems are listed in the table below.  As an example: cubic crystals have four 3-fold axes (at least), while a trigonal crystal has only one 3-fold axis (but can have other symmetries). Tetragonal crystals have one 4-fold axis at least (but cannot have three 4-fold axes). The characteristic symmetry refers to the minimum symmetry that needs to be present. Characteristic symmetry Cubic Four 3-fold rotation axes We have stated that basis of definition of (two will generate the other two) crystals is ‘symmetry’ and hence the Hexagonal One 6-fold rotation axis classification of crystals is also based on (or roto-inversion axis) symmetry. Tetragonal (Only) One 4-fold rotation axis The essence of the required symmetry is (or roto-inversion axis) listed in the table Trigonal (Only) One 3-fold rotation axis  more symmetries may be part of the (or roto-inversion axis) point group in an actual crystal. Orthorhombic (Only) Three  2-fold rotation axes (or roto-inversion axis) Note that the symmetry being considered is the point group symmetry. The translational Monoclinic (Only) One 2-fold rotation axis components are ‘dropped’ while noting the (or roto-inversion axis) symmetry. E.g. 63 screw axis is written as a ‘6’. Triclinic None (only translational symmetry)

Note: translational symmetry is always present in crystals (i.e. even in triclinic crystal) Bravais Lattices

• Assembly of the lattice points in 3-D results in 14 possible combinations • Those 14 combinations may have any of the 7 (class) symmetries • These 14 possibilities are the Bravais lattices Bravais Lattices

14 Bravais Lattices divided into 7 Crystal Systems Refer to slides on Lattice for more on these

A Symmetry based concept ‘Translation’ based concept

Crystal System Shape of UC Bravais Lattices P I F C 1 Cubic Cube    2 Tetragonal Square Prism (general height)   3 Orthorhombic Rectangular Prism (general height)     4 Hexagonal 120 Rhombic Prism  5 Trigonal Parallelepiped (Equilateral, Equiangular)  6 Monoclinic Parallogramic Prism   7 Triclinic Parallelepiped (general) 

P Primitive Why are some of the entries missing? I Body Centred  Why is there no C-centred cubic lattice?  Why is the F-centred tetragonal lattice missing? F Face Centred  ….? C A/B/C- Centred 08/08/2019 Mystery of Missing Entries Mystery of the missing entries in the Bravais List! What we choose P I F C 1 Cubic Cube    

UC-1

But then Cubic crystals need not have any 4-fold axes!! Hence Cannot be called Cubic (cubic lattices do need to have!) Hence even though this lattice remains as it is  it is called Simple Tetragonal lattice (which is smaller in size) . Note that this simple tetragonal cell has a specific c/a ratio of (2) (while in general simple tetragonal cells can have any c/a ratio). . Actually UC-1 (above) is a C- centred tetragonal cell! Symmetry is the issue here Mystery of the missing entries in the Bravais List! What we choose P I F C 2 Tetragonal Square Prism (general height)   

Smaller sized Body Centred Cell is chosen

FCT = BCT

Face Centred Tetragonal = Body Centred Tetragonal

Size is the issue here Mystery of the missing entries in the Bravais List! What we choose P I F C 2 Tetragonal Square Prism (general height)   

Smaller sized Simple Cell is chosen

CCT = ST

C Centred Tetragonal = Simple Tetragonal Mystery of the missing entries in the Bravais List! P I F C 4 Hexagonal 120 Rhombic Prism  

Body Centred Orthorhombic

Putting a lattice point at body centre destroys the 6-fold axis Continued… In fact not even the 3-fold survives and the lattice type is Body Centred Orthorhombic

Note: there is no remnant 3-fold either (if there were one then A B C, but there is no lattice point at C (at z = ½)).

Not all lattice points are shown

BCO unit cell Mystery of the missing entries in the Bravais List! P I F C 4 Hexagonal 120 Rhombic Prism  

Simple Orthorhombic Putting a lattice point at body centre destroys the 6-fold axis

In fact not even the 3-fold survives and the lattice type is Simple Orthorhombic

Note: there is no remnant 3-fold either (if there were one then A B, but there is no lattice point at C (at z = 0)). Lattice Planes and Miller Indices

Where does crystallographer see the Miller indices?

• Common crystal faces are parallel to lattice planes

• Each diffraction spot can be regarded as a X-ray beam reflected from a lattice , and therefore has a unique Miller index.

203.199.213.48/834/1/Structuresofsolids.ppt Lattice Planes and Miller Indices Lattice Planes and Miller indices

 A Miller index is a series of coprime integers that are inversely proportional to the intercepts of the crystal face or crystallographic planes with the edges of the unit cell.

 It describes the orientation of a plane in the 3-D lattice with respect to the axes.

The general form of the Miller index is (h, k, l) where h, k, and l are integers related to the unit cell along the a, b, c crystal axes.

Miller Indices

Rules for determining Miller Indices:

1. Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions.

2. Take the reciprocals

3. Clear fractions

4. Reduce to lowest terms

An example of the (111) plane (h=1, k=1, l=1) is shown on the right. Another example:

In this case the plane intercepts the a axis at one unit length and also the c axis at one unit length. The plane however, never intersects the b axis. In other words, it can be said that the intercept to the b axis is infinity. The intercepts are then designated as 1,infinity,1. The reciprocals are then 1/2, 1/infinity, 1/1. Knowing 1/infinity = 0 then the indices become (101).

203.199.213.48/834/1/Structuresofsolids.ppt

Crystalline Planes ? ?

Direction Vectors 001 Plane 110 Planes 111 Planes

09/08/2019 Miller Indices for Negative Intercept

Equivalent Planes

c b a

Cubic: a = b = c ; a = b = g = 90

(100)  (010)  (001) : collective representation {100}

61

Equivalent Planes in Tetragonal System 16/08/2019 Equivalent Planes

c b a

Cubic: a = b = c ;  =  =  = 90

(100)  (010)  (001) : collective representation {100}

59

Equivalent Planes in Tetragonal System

1200 1200 1200 Miller Indices of Hexagonal System Problem: In hexagonal system symmetry related planes and directions DO NOT have Miller Indices which are permutations.

Crystallographic Direction and Miller Indices Crystallographic Directions Crystallographic Directions Directions

Crystallographic Convention Remember 22/08/2019 Diffraction Diffraction

Width of the slit ~ Wavelength of light nλ=d*sinθ

Mechanism of X-ray Generation

Wavelength (nm)

Basis of X-ray Diffraction by Crystal . The electrons oscillate under the influence of the incoming X-Rays and become secondary sources of EM radiation. . The secondary radiation is in all directions. . The waves emitted by the electrons have the same frequency as the incoming X-rays  coherent. . The emission can undergo constructive or destructive interference.

Secondary Incoming X-rays emission

Oscillating charge re-radiates  In phase with the incoming x-rays Schematics

Sets nucleus into oscillation Sets Electron cloud into oscillation Small effect  neglected

Braggs Contribution

Warning: we are using ray diagrams in spite of BRAGG’s EQUATION being in the realm of ‘physical ’ Let us consider scattering across planes

Click here to visualize constructive and destructive interference

See Note Ӂ later . A portion of the crystal is shown for clarity- actually, for destructive interference to occur many planes are required (and the interaction volume of x-rays is large as compared to that shown in the schematic). . The scattering planes have a spacing ‘d’. . Ray-2 travels an extra path as compared to Ray-1 (= ABC). The path difference between Ray-1 and Ray-2 = ABC = (d Sin + d Sin) = (2d.Sin). . For constructive interference, this path difference should be an integral multiple of :

n = 2d Sin  the Bragg’s equation. (More about this sooner). . The path difference between Ray-1 and Ray-3 is = 2(2d.Sin) = 2n = 2n. This implies that if Ray-1 and Ray-2 constructively interfere Ray-1 and Ray-3 will also constructively interfere. (And so forth). Bragg Law • See Figure. The length DE is the same as EF, so the total distance traveled by the bottom wave is expressed by: • Constructive interference of the radiation from successive planes occurs when the path difference is an integral number of wavelengths. Note that line CE = d = distance between the 2 layers l EFd sin DEdGiving:sin DEEFd2sin  nd 2sin

This is called the Bragg Law Bragg Law • Consider crystals as made up of parallel planes of atoms. Incident waves are reflected specularly from parallel planes of atoms in the crystal, with each plane reflecting only a very small fraction of the radiation, like a lightly silvered mirror. • In mirrorlike reflection, the angle of incidence is equal to the angle of reflection.

Incident Angle  θ Reflected angle  θ X-ray Wavelength  λ Total Diffracted Angle  2θ Bragg Law (Bragg Equation) 2d sin  n

d = Spacing of the Planes, n = Order of Diffraction. •Because sin θ ≤ 1, Bragg reflection can only occur for wavelengths satisfying: n  2d • This is why visible light can’t be used. No diffraction occurs when this condition is not satisfied. • The diffracted beams (reflections) from any set of lattice planes can only occur at particular angles predicted by Bragg’s Law.

23/08/2019 Bragg Law • See Figure. The length DE is the same as EF, so the total distance traveled by the bottom wave is expressed by: • Constructive interference of the radiation from successive planes occurs when the path difference is an integral number of wavelengths. Note that line CE = d = distance between the 2 layers l EFd sin DEdGiving:sin DEEFd2sin  nd 2sin

This is called the Bragg Law Wavelength of not changing

Powder X-ray Diffraction (PXRD)

Sample holder

X-ray tube

Detector

Slit boxes

The view of Bruker D8 Advance diffractometer with stationary X-ray source and synchronized rotations of both the detector arm and sample holder

Video of Powdered XRD machine Applications of PXRD

Crystalline Nature of Samples

Peak Intensity is related to multiplicity or equivalent planes 29.08.2019 PXRD PXRD Single Crystal X-ray Diffraction Four Circle Diffractometer

For single crystals LNT

X-ray CCD - Detector

Crystal

Goniometer

X-ray source, goniometer + crystal, N2-cooling and CCD Detector One of the several hundreds of CCD images with diffraction spots

Starting with an indexed , an incident X-ray beam must pass through the origin (000) point, corresponding to the direct undiffracted beam of X-rays. To be able to collect as many different reflections as possible, it is thus necessary to be able to rotate the reciprocal lattice to a great extent…

Summary of Determination

05/09/2019

Taught without slides. However, I am providing glimpses of todays discussion in the following few slides.

Go through the book of A. R. West (page no. 153-155) Intensities X-rays scattered by electrons - electrons are in atoms

Scattering power f of at  = 0° is no. electrons (atomic no.) x scattering power of one e— The scattering factor is equal to the number of electrons around the atom at θ= 0,the drops off as θ increases. Scattering of X-rays by an atom

• Distances between the electrons in an atom are short. • Path difference XY is less than one wavelength (>λ/2 but less than λ).

• Only partial destructive interference occurs.

• The net effect of such interference is to cause a gradual decrease in scattered intensity with increasing angle, 2θ. • X-rays interact with the electrons in the crystal, therefore a crystal with more electrons with more strongly scatter X-rays.

• The effectiveness of scattering X-rays is called the scattering factor (or form factor) with the symbol f.

• The scattering factor depends on the number of electrons around the atom, the Bragg angle θ, and the wavelength (λ) of the X-rays.

• The scattering power decreases as the Bragg angle increases.

06/09/2019

Structure factor calculation of different cubic systems were done in board hkl Miller Indices

th xj , yj , zj  atomic coordinate of j atom.

11/09/2019

Cl at 000 and Cs at ½ ½ ½

Cl Cs

Cl Cs

fCl + fCs

fCl - fCs

NaCl

(200, 220, 222, 420, 422 etc.)

(111, 311, 331, 333 etc.)

(110, 221 etc.) 2 Table Systematic absences due to lattice type Lattice type Rule for reflection to be observed

Primitive, P None Body centred, I hkl; h + k + l = 2n Face centred, F hkl; h, k, l either all odd or all even Side centred, e.g. C hkl; h + k = 2n Rhombohedral, R hkl; −h + k + l = 3n or (h − k + l = 3n) KCl

NaCl Solving Single Crystal Structures A set of structure factors, Fcalc, are determined for comparison with the Fobs magnitudes.

with good quality crystals, one might expect to see R below 0.05

Direct-method aided protein crystal structure 19/09/2019 X-rays with λ= 154.2 pm produce reflections from the 111 and 200 planes of FCC Cu of density 8.935 g / cm3. At what angles will these reflections appear? Which will give maximum intensity? 21.71 degree, 25.28 degree

X-ray of λ=0.1537 nm from a Cu target are diffracted from the (111) plane of an FCC metal. The Bragg angle is 19.2 degree. Calculate the Avogadro number if the density of the crystal is 2698 kg/m3 and the atomic weight is 26.98.

A XRD experiment was performed with CuK radiation, using a diffractometer on a FCC Crystal (hetero-atom) with lattice parameter= 3.61 A. What are the Miller indices of the planes with lowest and highest Bragg angles? 111, 420 1. Hypothetical unit cell structures for CsCl and Fe are given below. XRD pattern of CsCl crystal shows a peak corresponding to (100) planes, however, this (100) peak is absent for Fe crystal. Explain the observation.

CsCl (centred atom Cs, corner atom Cl) Fe: all are Fe atom

2. XRD pattern for gold nanocrystals exhibiting fcc lattice (real space) is given below. Peak- 1 is at the lowest possible 2 value for this crystal. Assign the (hkl) values for all the three peaks.

Peak-1

Peak-2

Peak-3

Derive the structure factors for the unit cell of MoSi2. Hence prove that there will be no reflection from 100 plane, however, 110 plane will give strong reflection in X-ray diffraction experiment. MoSi2

4. (1/2, 1/2, 2/3+1/6)

(0,0,1/3) (½,½, ½) 2.

(1,0,1/3) 3. 1. (1/2, 1/2, 1/6)

(0,0,0)

Reciprocal Lattice

Reciprocal lattice vector 푯풉풌풍 = 풉풃ퟏ + 풌풃ퟐ + 풍풃ퟑ 1

bi aj = 0 ( i≠ j) = 1 (i=j)

Direction of is perpendicular to 푯풉풌풍 = 풉풃ퟏ + 풌풃ퟐ + 풍풃ퟑ 푯풉풌풍 hkl plane. ퟏ 푯풉풌풍 = 풅풉풌풍 Cubic Lattice:

In reciprocal lattice

1

1 1 1 1/a Ref: Kittel’s Introduction to Solid State Physics, page- 8-10 Ref: Kittel’s Introduction to Solid State Physics, page- 8-10

Ref: Kittel’s Introduction to Solid State Physics, page- 10-40

Ewald Construction

 The reciprocal 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 aim of the Ewald is to determine which lattice points (in reciprocal space) will results in a diffracted signal for a given wavelength.

Reflection from 101 plane will be present, as it will obey the Bragg’s condition

Reflection from 201 plane will be absent, as it will not obey the Bragg’s condition The Ewald Sphere

When a reciprocal lattice point intersects the Ewald sphere, a reflection will occur and can be observed at the 2q angle of the inscribed triangle. To be able to collect as many different reflections as possible, it is thus necessary to be able to rotate the reciprocal lattice to a great extent… The Ewald Sphere Our need to rotate the crystal in numerous ways to bring all of the families of planes into reflection (i.e. to make RL point intersect with the Ewald sphere) explains the design of the goniometer portion of the diffractometer. The 4-circles of the goniometer allows the crystal to be rotated in virtually any direction while remaining in the X-ray beam. The labels for each of the goniometer axes (f, c, w, and q) are indicated on the diagram below. Note that there are different designs (such as the Kappa geometry) that accomplish the same task. Video Related to Ewald Sphere and Diffraction

https://www.youtube.com/watch?v=H3NeiImTJOA In Laue diffraction, the range of wavelengths available affords a spherical torus (with minimum and maximum radii of 1/λmax and 1/λmin, respectively) wherein the diffraction condition is met. Consequently, a number of reflections lie in the wavelength band (shaded in grey) and so, in Laue diffraction, many reflections satisfy the Bragg condition simultaneously.

−(푹.푯풉풌풍 푭풉풌풍 = 풇풊풆풙풑 풊 R

푯풉풌풍 = 풉풃ퟏ + 풌풃ퟐ + 풍풃ퟑ (reciprocal)

푭풉풌풍 10/10/2019 Description of crystal Structure -Close packing Crystal Structure and Close Packing

Let us examine the difference between Closed-cubic Packing and Hexagonal-Cubic Packing:

203.199.213.48/834/1/Structuresofsolids8 .ppt

Q: If the HCP crystal has only a 3-fold axis why does it come under the ‘Hexagonal class*’ of crystals? Let us examine the difference between Closed-cubic Packing and Hexagonal-Cubic Packing:

203.199.213.48/834/1/Structuresofsolids15 .ppt Why CCP is also called FCC? Why CCP is also called FCC?

BCC packing

Some metals do not adopt a close-packed structure but have a slightly less efficient packing method: this is the body-centred cubic structure.

The has dropped from twelve to eight and the packing efficiency is now 68%, compared with 74% for close-packing. Occurrence of Packing type in Metals Packing efficiency and Packing Density Atomic Packing Factor (APF)

• APF for a simple cubic structure = 0.52

Adapted from Fig. 3.19, Callister 6e. 6 Body Centered Cubic (BCC) structure • Close packed directions are cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

• Coordination # = 8

Adapted from Fig. 3.2, Callister 6e.

(Courtesy P.M. Anderson) 7 Atomic Packing Factor: BCC • APF for a body-centered cubic structure = 0.68

Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell R Adapted from a Fig. 3.2, Callister 6e.

8 Face Centered Cubic (FCC) structure • Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

• Coordination # = 12

Adapted from Fig. 3.1(a), Callister 6e.

(Courtesy P.M. Anderson) 9 Atomic Packing Factor: FCC • APF for a body-centered cubic structure = 0.74

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a Adapted from Fig. 3.1(a), Callister 6e.

10

(as a = 2R)

For hexagonal system it can be proved that Home Assignment

1. Prove that c/a ratio 1.63 for hexagonal pack system

2. If the HCP crystal has only a 3-fold axis why does it come under the ‘Hexagonal class*’ of crystals? Octahedral and Tetrahedral Voids Tetrahedral and Octahedral Holes/Sites/Voids Octahedral Holes

N spheres, N Octahedral holes N spheres, 2N tetrahedral holes Tetrahedral and Octahedral Holes

Structures of some important Ionic and Covalent Compounds NaCl or Rock Salt

Figure. Ionic compounds with anions that are much larger than cations, such as NaCl, usually form an FCC structure. They can be described by FCC unit cells with cations in the octahedral holes.

CsCl

Figure. Ionic compounds with similar-sized cations and anions, such as CsCl, usually form a simple cubic structure. They can be described by unit cells with either cations at the corners or anions at the corners.

17/10/2019 Wurtzite crystal structure

hcp array of sulfide ions with alternate tetrahedral holes occupied by Zn ions. Each atom is tetrahedrally coordinated

Perovskite (ABX3)

First, it can be described as a ccp array of A and X atoms with the B atoms occupying the octahedral holes (compare with the unit cell of NaCl if you want to check this).

Crystal structure of CH3NH3PbX3 perovskites (X=I, Br and/or Cl). The + methylammonium cation (CH3NH3 ) is surrounded by PbX6 octahedra and SiC (carborundum) Graphite Crystal Structure Crystal Structures of Fullerene (C60)

Crystal structure may be regarded as a face centred cubic configuration of C60 spheres.

Lattice Energy Lattice Energy

The Lattice energy of a crystal is defined as the net potential energy for the arrangement of charges that forms the structure.

−787 kJ mol−1 Theoretical Evaluation of Lattice Energy NaCl

Na+ The total energy of the crystal, U, is given by summing equations (3.18) and (3.13) and differentiating with respect to r to find the maximum U value and equilibrium interatomic distance, re; i.e.

Refined Equation

(a)

(b)

(c) Hence the Born repulsive term contributes 10–15% to the value of U; the zero point vibrational and van der Waals terms contribute about 1% each and, being of opposite sign, tend to cancel each other. For most purposes, therefore, we use the simplified equation.

The magnitude of U depends on six parameters A, N, e,  , Z, and re, four of which are constant for a particular structure. This leaves just two, the charge on the ions, Z+Z−, and the internuclear separation, re. Of these two, charge is the most important since the product (Z+Z−) is capable of much larger variation than re