Crystalline Solid Amorphous Solid
Particles are completely disorganized Particles are arranged in orderly fashion
Single Crystals and Polycrystalline Materials
Single crystal: atoms 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, molecules, ions) 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 crystal system (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 plane, 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 optics’ 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) = 2n = 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 DEdGiving:sin DEEFd2sin 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 DEdGiving:sin DEEFd2sin 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 reciprocal lattice, 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 Crystal Structure 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 atom 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)