Unit 1 Crystal Structure PDF

# UNIT -I ## Crystallography & Crystal Structures - Crystallography: The Study of the geometric form and other physical properties of crystalline solids by using X-rays, electron beams and neutron beams etc., constitute the science of Crystallography. ## Auguste Bravais (1811 -1863) - Bravais, a...

# UNIT -I ## Crystallography & Crystal Structures - Crystallography: The Study of the geometric form and other physical properties of crystalline solids by using X-rays, electron beams and neutron beams etc., constitute the science of Crystallography. ## Auguste Bravais (1811 -1863) - Bravais, a graduate of the École Polytechnique and a professor of physics, worked out a mathematical theory of crystal symmetry based on the concept of the crystal lattice, of which there were 14. ## X-Ray Diffraction: Discovery of X-Rays - William Röntgen discovered that when certain substances are exposed to the beam of a cathode ray tube, a new kind of penetrating ray capable of fogging photographic plates even when shielded emitted -- called it "x-rays". These x-rays also ionized gases through which they passed. - 1st Nobel Prize in physics **(1901)**. ## Max Von Laue (1989-9160) ### X-Ray Diffraction - In 1912, German physicist Max von Laue and his colleagues had already proved that X-rays were diffracted by crystals. Diffraction is a peculiar phenomenon that occurs when a ray of electromagnetic radiation, such as visible light, passes through an ultra-narrow slit. - He received the Nobel Prize for Physics in 1914, for his discovery of the diffraction of X-rays on crystals. ## X RAYS AND CRYSTAL STRUCTURE - Bragg's Law: the father and son began a series of experiments that culminated in the invention of the X-ray spectroscope in 1913. - They got the Nobel prize in the year 1914. ## Celebrations - A logo that says *Celebrating Crystallography, Bragg Centenary 1913 - 2013*. - Another logo that says *2014, international year of crystallography*. - The logos are next to each other. ## Matter - Matter is a concept with 3 parts: solids, liquids, and gases. - Solids can be either **crystalline** (*Solidification occurs gradually - crystal*) or **amorphous** (*Solidification occurs suddenly - Amorphous*). ## Solid Materials - Solid materials can be classified into 3 categories: **crystalline**, **polycrystalline**, and **amorphous** (*Non-crystalline*). - Crystalline solids include **single crystal** and **polycrystalline** solids, which can be of the **pyrite form**, which is a grain. - The image shows an example of each. - *Amorphous* solids are non-crystalline in nature. ## Crystal - *Crystal* is a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimensions. - The image shows the arrangement of atoms in a crystalline structure. ## Single Crystal - *Single crystal* has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry. - An image shows a single crystal with its atoms arranged in a periodic pattern. - Another image shows a 3D representation of a crystal with a single *pyrite crystal* inside of it. - Another image shows an *amorphous solid*. ## Polycrystal - A *Polycrystal* is a material made up of an aggregate of many small single crystals (*also called crystallites or grains*). - The grains are usually 100 nm - 100 microns in diameter. - Polycrystals with grains that are < 10 nm in diameter are called *nano-crystalline*. - There is a microscopic image of a *polycrystal*. - Another more microscopic view of *polycrystalline* pyrite, labeling the image as a *grain*. - Another close up image shows a structure with *grain boundaries* and labels the image as "grains". ## Amorphous - *Amorphous* (*OR*) isotropic solid is composed of randomly oriented atoms, ions (*OR*) molecules that do not form defined patterns (*OR*) lattice structures. - Images showing various structures of molecules. - An image shows a rock. ## Crystal - *Crystal* means... - A three-dimensional translational periodic arrangement of atoms in space is called a crystal. - (*OR*) - The periodic array of atoms, ions, or molecules that form the solids is called *Crystal Structure.* ## Understanding Crystal Structure - For better understanding of crystal structure, we must know the following technical terms of crystals like..... - *Lattice* *OR* *Space lattice* - *Unit cell* - *Lattice parameters* - *Lattice constant* - *Atomic packing factor* - *Co-ordination number* - *Basis* etc ## Space Lattice - A three-dimensional translational periodic arrangement of points in space is called a lattice. - (*OR*) - An infinite array of points in three dimensions in which every point has an identical environment to all others is called a space lattice. ## Basis (or) *Motif* - An atom or a group of atoms associated with each lattice point is called as basis or *motif*. ## Lattice and Crystal Relationship - What is the relation between the lattice & the crystal? - *Crystal* = *Space Lattice* + *Motif* (*basis*) - An image shows that *lattice* + *basis* = *crystal structure*. ## Facts about Crystal Cells - A cell is a finite representation of the infinite lattice. - A cell is a *parallelogram* (2D) or a *parallelopiped* (3D) with lattice points at their corners. ## Lattice Dimensions - **1-D Lattice**: An image shows atoms arranged in a line infinitely. - **2-D Lattice**: An image shows atoms arranged in a square infinitely in 2 dimensions. - **3-D Lattice**: An image shows atoms arranged in a cube infinitely in 3 dimensions. ## Three Dimensional Lattice - An image shows a 3D structure with *lattice points* labeled. ## UNIT CELL - The smallest volume from which the entire crystal is built up by translational repetition in three dimensions, called as *unit cell*. - An image shows a 3D structure with an *unit cell* labeled. - It is the fundamental elementary pattern. ## Primitive Cell - If the lattice points are only at the corners, the cell is a *primitive cell*. - *Example*: Unit cell of a simple cubic structure. - An image shows a *unit cell* of a simple cubic structure. ## Non-Primitive Cell - If there are lattice points in the cell other than the corners, the cell is a *non-primitive cell*. ## Lattice Parameters - The lines drawn parallel to the lines of intersection of any three faces of the unit cell which do not lie in the same plane are called *crystallographic axes*. - An image shows a 3D structure that labels the *crystallographic axes*. - The primitives (a,b,c) and the interfacial angles (α,β,γ) are called the lattice parameters. - Lattice parameters decide the *actual size* and *shape* of the unit cell. - Since, a, b, & c are the primitives of the unit cell and α, β, & γ are the interfacial angles. ## Lattice Constant and Crystal Density - Relation between lattice constant (a) And crystal density (ρ) - Volume of the unit cell = $a^3$ - Therefore, mass of the unit cell = volume x density - $m = a^3ρ$$------------(1)$ - Let say 'n' molecules/unit cell, M is the molecular weight, and N is the Avogadro number. - Then mass of the each molecule = M/N - Mass of the each unit cell = $n \times M/N$ $----------(2)$ - Therefore, **Eq.(1)** = **Eq.(2)** - $a^3p = n \times M/N$ - Lattice constant **(a)** = $(n M / N ρ)^{1/3}$ $----------(3)$ - **Eq.**(3) gives the relation between lattice constant & crystal density. ## BRAVAIS LATTICE - Lattices are classified on the basis of their symmetry - In 1850, M. A. Bravais showed that identical points can be arranged spatially to produce 14 types of regular patterns. These 14 space lattices are known as 'Bravais lattices'. - An image shows a portrait of M. A. Bravais. ## Bravais Lattice Types - 14 types of space lattices in the 7 systems of crystals, which are called Bravais lattices. - - Cubic - - Tetragonal - - Orthorhombic - - Monoclinic - - Triclinic - - Rhombohedral - - Hexagonal - An image shows a chart that lists the above lattices, with each a corresponding number, with a total of 14. ## Crystal Types - **1. Cubic crystals (3)**: Three lengths of the unit cell are the same and they are at right angles. - - $a = b = c$ & $α = β = γ = 90°$ - - *Ex:* NaCl,Ag,Au,CaF,Pb,Cu,NaClo2 etc - - An image shows a 3D structure of a cubic crystal. - **2. Tetragonal crystals (2)**: Two lengths of the unit cell are equal, the third one is longer, and all are at right anles. - - $ a = b ≠ c$ & $α = β = γ = 90°$ - - *Ex:* Tio2, Sno2, KH2P04 - - An image shows a 3D structure of a tetragonal crystal. - **3. Orthorhombic crystals (4)**: Lengths of the unit cell are different, but they are at right angles. - - $a≠b≠c$ & $ α = β = γ = 90°$ - - *Ex:* C15H2002, PbC03, BaSO4, KNO3, K2SO4, a-S - - An image shows a 3D structure of an orthorhombic crystal. - **4. Monoclinic crystals (2)**: Lenghts of the unit cell are different, two axes are at right angles, and the third one is obliquely inclined. - - $a≠b≠c$ & $ α = β = 90° ≠ γ$ - - *Ex:* CaSo4 2H20 (Gypsum), K2MgSO4.6H20 - - An image shows a 3D structure of a monoclinic crystal. - **5. Triclinic crystal (1)**: Lengths of the unit cell are different, and they are obliquely inclined to each other. - - $a≠b≠c$ & $ α ≠ β ≠ γ ≠ 90°$ - - *Ex:* K2Cr2O7, CuSo4 5H20 - - An image shows a 3D structure of a triclinic crystal. - **6. Rhombohedral (Trigonal) crystal (1)**: Lengths of the unit cell are equal, and axes are equally inclined to each other at an angle other than 90°. - - $a = b = c$ & $α = β = γ ≠ 90°$ - - *Ex:* Calcite, As, Sb, Bi etc - - An image shows a 3D structure of a rhombohedral (trigonal) crystal. - **7. Hexagonal crystal (1)**: Two axes of the unit cell are equal in length in one plane at 120º with each other and the third axis is perpendicular to this plane. - - $a = b≠c$ & $ α = β = 90°, γ = 120°$ - - *Ex:* Quartz, Zn, Cd etc - - An image shows a 3D structure of a hexagonal crystal. - Bravais lattices are explained on the basis of lattice parameters only. - An image shows a 3D structure of a hexagonal crystal with labels $a = 2r$, $b = 2r$, $γ = 120°$, $α = 90°$, $c = 4r(2/3)^{1/2}$, and $β = 90°$. ## Bravais Lattice Types Illustrated - CUBIC - - a = b = c - - $α = β = γ = 90°$ - TETRAGONAL - - a = b ≠ c - - $α = β = γ = 90°$ - ORTHORHOMBIC - - a ≠ b ≠ c - - $α = β = γ = 90°$ - HEXAGONAL - - a = b ≠ c - - $α = β = 90°$, γ = 120° - MONOCLINIC - - a ≠ b ≠ c - - $α = γ = 90°$, β ≠ 120° - TRICLINIC - - a ≠ b ≠ c - - $α = β ≠ γ = 90°$ - TRIGONAL - - a = b = c - - $α = β = γ = 90°$ - An image shows a 2D and 3D illustration of the 7 crystal structures. - 4 Types of Unit Cell - - P = Primitive - - I = Body-Centred - - F = Face-Centred - - C=Side-Centred - + - 7 Crystal Classes - → 14 Bravais Lattices ## Bravais Lattice Table | PRIMITIVES (a, b, c) | INTERFACIAL (α,β,γ) | NAME OF THE CRYSTAL | No. OF LATTICES / TYPES | EXAMPLE | |--------------------------------|------------------------------|------------------------------|---------------------------|-------------------------| | a = b = c | α = β = γ = 90° | Cubic | 3 | CsCl, NaCl | | a = b ≠ c | α = β = γ = 90° | Tetragonal | 2 | TiO₂ (Rutile), SnO2 | | | | | | (Cassiterite) | | a ≠ b ≠ c | α = β = γ = 90° | Orthorhombic | 4 | KNO3 | | | | | | BaSO4 (Baryte) | | a = b =c | α = β = γ <120° | Rhombohedral | 1 | Ice, Al2O3 | | | | (Trigonal) | | | | a = b ≠ c | α = β = 90° ; | Hexagonal | 1 | Mg, Zn, Pbl₂ | | | γ = 120° | | | | | a ≠ b ≠ c | α =γ = 90° ; | Monoclinic | 2 | Monoclinic Sulphur | | | β ≠ 90° | | | Na2SO4.10H2O | | a ≠ b ≠ c | α ≠β ≠ γ ≠90° | Triclinic | 1 | CuSO4.5H2O | - P: Simple cubic - I: Body centered - F: Face centered - C: Base centered ## Nearest Neighbour Distance - Nearest neighbour distance (2r) - The distance between the centers of two nearest Neighbouring atoms is called *nearest neighbour distance*. - It will be 2r if r is the radius of the atom. - An image shows 2 atoms labeled A and B, with a distance of 2r marked between them. ## Atomic Radius - Atomic Radius - Half the distance between the nearest neighboring atoms in a crystal is called *atomic radius*. - An image shows 2 atoms labeled A and B, with a distance of x marked between them. - $Therefore, r = X/2$ - Since, r - atomic radius ## Co-ordination Number - The number of equidistant nearest neighbours that an atom has in the given structure is known as *co-ordination number*. - **NOTE:** By the help of *co-ordination number*, we can say that *a given structure is either closely packed (or) loosely packed structure.* ## Atomic Packing Fraction - Atomic packing fraction - The ratio between the volume occupied by the atoms in the unit cell (*v*) to the total volume of the same unit cell (*V*) is called the *atomic packing factor*. - i.e. - *APF* = *v/V* - **NOTE**: From *APF* value, we can say that the given structure is either *closed (OR) loosely packed structure.* ## Simple Cubic Structure (SC) - 1. All atoms are kept at corners of the unit cell only, consequently these atoms touch along cube edges. - 2. Each atom has only 6 nearest neighbours. Hence, *co-ordination number* is six. - - An image shows a 3D structure of a *simple cubic* structure with atoms at the corners. - - Another image shows atoms labeled R/A, X, +, -X, +Y. - - *Because each corner atom has four neighbours in the same plane one vertical above and one immediately below.* - 3. Each corner atom is shared by 8 surrounding unit cells, share of each corner of the unit cell comes to one-eight of an atom. Hence, each unit cell contains eight corner atoms. So the unit cell of the simple cube contains only 1 atom. - - *i.e. 8 x 1/8 = 1 atom* - - An image shows a 3D structure of a *simple cubic* structure with a purple 3D cube inside of it. - 4. Therefore, the total number of atoms in a unit cell is one. - - *So in this case, the unit cell is equal to the primitive cell.* - 5. It is a 1-layer structure. Hence, it's stacking sequence is A A A A A....... - 6. *Atomic packing factor* - - *APF* = *v/V* - - = $1 \times 4/3 πR³$ / $a³$ - - = $1 \times 4/3 πR³$ / $(2R)³$ - - *Therefore,* *APF* = $π/6 = 0.523 OR 52%$ - - It's *APF* value is only 52%. Hence, it is a loosely packed structure. - - *Ex:* Only polonium (Po) exhibits in a certain temperature region. ## Body Centered Cubic Structure (BCC) - 1. Atoms are kept at corners of the unit cell including one atom exactly centre of the unit cell. - - *Consequently, these atoms touch along the body diagonals, hence the nearest concept is along the body diagonals*. - - An image shows a 3D structure of a *body centered cubic* structure with atoms at the corners and one at the center. - 2. *Co-ordination number*: The nearest neighboring atoms of any case are the body centered atoms, not the other corner atoms. - - *Hence,* *each atom has only 8 nearest neighbours*. *Hence,* its co-ordination number is 8. - 3. Each corner has one atom, but shares only one-eight part of an atom like a simple cubic structure & one more atom exactly occupies the center of the body. - - *Hence,* *it is known as Body centered cubic structure.* - - An image shows 3D structure of a *body centered cubic* structure with a purple sphere in the center. - 4. Total number of atoms in a unit cell of BCC is 2. - - *i.e. 8 x 1/8 + 1 = 1 + 1 = 2 atoms* - 5. It is a 2-layer structure. Hence, it's stacking sequence is AB AB AB AB AB AB....... - - *An image shows 3D structure of a* *body centered cubic* structure with a purple sphere in the center. - 6. *Atomic packing factor* - - An image shows a cube labeled A, B, C, D. Another image shows 3 spheres that look like a slice of the cube from the side. - - From figure 1, Δ ABC, AC2 = AB2 + BC2 = a² + a² = 2a² -------(1) - - Δ ACD, AD2 = AC2 + CD2 = 2a² + a² = 3a² - - AD2 = 3a² (OR) - - AD = √3 a -------- (2) - - If ‘r’ is the atomic radius, *from figure 2 we can write* - - AD = 4r --------(3) - - From eq.2 & 3, we get, - - √3 a = 4r - - r = √3 a / 4 ---------(4) - - *Therefore,* *APF* = *v/V* - - = ($2 \times 4/3 πr³$) / $a³$ - - = $8π / 3a³ (√3 a/4)³$ - - = √3π/8 = 0.68 = 68% - - *Therefore,* it's *APF* is 68%. Hence, we can say that it is also a loosely packed structure. - - *Ex:* Tungsten, Sodium, Iron, Chromium etc ## Face Centered Cubic Structure (FCC) - 1. It is a 3-layer structure. Hence, it's stacking sequence is ABC, ABC, ABC..... - - An image shows 3 layers marked A, B, C, with the first layer being yellow, the second red, and the third blue. - 2. Eight atoms are occupied at eight corners of a unit cell including six atoms are situated exactly the center of its face. - - *i.e.* - - An image shows a 3D structure of an *FCC* structure with atoms at the corners and the center of each face. - - Another image shows 6 larger spheres touching 3 smaller spheres. - 3. Atoms are in contact with face-diagonals, hence the nearest neighbours of any corner atom are the face-centered atoms of the surrounding unit cells. - 4. Each atom has 12 nearest neighbours; hence, it's *co-ordination number* is 12. - - *Any corner atoms has 4 such atoms in it's own plane, 4 in a plane above it and 4 in a plane below.* - 5. Total number of atoms in a unit cell is 4. - - 8 x 1/8 + 6 x ½ = 1 + 3 = 4 atoms - - An image shows a 3D structure of an *FCC* structure. - 6. *Atomic packing factor* - - From figure, Δ ABC, AC2 = AB2 + BC2 = a² + a² = 2a² -------(1) - - *But, AC = r + 2r + r = 4r* ----------------(2) - - Substituting the eq.2 in eq.1, we get, - - (4r)² = 2a² - - r² = 2a² /16 = a²/8 - - *Therefore*, r = a / √8 ----------------(3) - - *Therefore,* *APF* = *v/V* - - = ($4 \times 4/3 πr³$) / $a³$ - - = ($4 \times 4 \times π \times a³$) / ($(√8)³ \times 3 \times a³$) - - = ($16 \times π \times a³$) / (($√2)¹⁶ \times 3 \times a³$) - - = $π / 3(√2)$ - - *Therefore,* *APF* = 0.74 = 74% - - Therefore, it's *APF* value is 74%. Hence, we can say that it is a closely packed structure. - - *Ex:* Most of the metals like Copper, Lead, Aluminium, Silver etc ## Hexagonal Close Packed Structure (HCP) - 1. *Effective number of atoms per unit cell* - - 2 x (6x 1/6) + 2 x 1/2 + 3 = 6. - - An image shows a hexagonal structure with a top, middle, and bottom layers. - 2. *Nearest neighbor distance* - - 2r = a - 3. *Atomic radius* r = a / 2. - 4. *Co-ordination number* = 12. - 5. *Volume of the HCP unit cell*: The volume of the unit cell is determined by computing the area of the base of the unit cell and then by multiplying it by the unit cell height. - - *Volume* = (*Area of the hexagon*) x (*height of the cell*) - - *Area of the hexagon*: - - = 6X(ΔΑΒΟ) - - = 6X (¹/₂ ABXOO') - - = 6x ¹/₂ xa x √3a - - = ³√3a²/2 - - An image shows the labelled hexagon. - - *If c is the height of the unit cell V = (³√3/2)a²c* - 6. c/a ratio: The three body *atoms lie in a horizontal plane at a hiight* c/2 from the base or at the top of the Hexagonal cell. - - An image shows the hexagonal structure with a c/2 distance labelled. - - From Triangle ABN, [ABN= 30° - - cos 30°= AN/AB - - AN=AB cos 30°= a √3/2 - - but AX = 2/3 AN - - = a √3/3 - - From Triangle AXC, AC²= AX² + CX² - - a²/2 = (a√3/3)² + (c/2)² - - c²/4 = a²/2 - a²/3 = a²/6 - - c²/a² = 8/3 - - c/a = √8/3 - 6. *Atomic packing factor* - - = ($6 \times 4/3 πr³$) / (³√3/2) a²c - - *But r* = a/2 - - = ($6 \times 4/3 π (a/2)³$) / (³√3/2) a²c - - = $2π / ³√3 x ¹/cla$ - - *But c/a = √8/3* - - = 0.74 - - = (74%) - 7. *Void space* = (1-APF) x 100 - - = (1-0.74) x 100 - - = 26% - - *Ex:* Mg, Cd and Zn. ## Hexagonal Close Packed Structure (HCP) - An image shows a 3D structure of an *HCP* structure. - There are top, middle, and bottom layers labelled. - There are A and B sites labelled in a hexagonal structure. - An image shows how the *HCP* unit cell and the layers look, with the *abab... pattern*. ## Directions and Planes in Crystals - In crystals, *atoms are regularly arranged in space such that the environment about any atom is the same.* There exists some directions and planes which *contains large concentrations of atoms*. For understanding crystallography, the concept of directions and planes play an important role. - **Directions**: In crystal analysis, it is essential to indicate certain directions inside the crystal, and which is a line joining any two points of the lattice. ## Two Dimensional Crystal Directions Representation - An image shows a square with dots at the corners, with lines drawn through them, with the following labels: - - y - - *A [12]* - - *B[11]* - - O - - ↗ *C[32]* - - *D[31]* - - x - Let 'o' be the origin of the pattern. Consider the directions OA,OB,OC & OD. The direction is described by giving the first integer point (x,y) through which the line passes. ## Important Directions in Cubic Crystals - An image shows a cube, with various lines drawn through it and labelled: - Z - [001] - [011] - [111] - [000] - [100] - X - [110] - [010] - y ## MILLER INDICES - The set of three integers used to describe the orientation of a crystal plane are called Miller Indices. - (*OR*) - The directional cosine values of the line perpendicular to the plane are called Miller Indices. - General form for Miller Indices of a plane is (h k l) - An image shows a portrait of William Hallowes Miller. ## Procedure for finding Miller Indices - **STEP:01**: Find the intercepts of the plane along the axes a, b & c. - **STEP:02**: Take reciprocals of the intercepts. - **STEP:03**: Convert into the smallest integers in the same ratio. *(Taking its LCM)* - **STEP:04**: Enclose in parentheses. - **NOTE**: The reciprocal procedure avoids the intercept of infinity for a plane parallel to an axis by making it to zero. ## Miller Indices Example - *Example:* Intercepts are 2,1,2. - **STEP:01**: Intercepts are 2,1,2. - **STEP:02**: Take the reciprocals of the intercepts 2,1,2 as 1/2,1,1/2. - **STEP:03**: Convert into smallest integers as 1,2,1. - **STEP:04**: Enclose in the parentheses as (1,2,1). - An image shows a triangular plane with intercepts 2, 1, and 2 labelled. - *Gives the Miller Indices for the above plane.* ## Miller Indices Notation - **NOTE**: - (1,2,1) → *gives only about three integers* - *<1,2,1>* - *OR* - *[1,2,1]* - *Gives the all possible integers* - *(family of integers)* ## Miller Plane Illustrations - **001 Plane**: An image shows a cube with planes that are perpendicular to the x-axis. - - (001) Plane referenced to the origin at point O - - Other equivalent (001) planes - **110 Planes**: An image shows a cube with planes that are perpendicular to a 120 degree angle of the x and y-axis. - - (110) Plane referenced to the origin at point O - - Other equivalent (110) planes - **111 Planes**: An image shows a cube with planes that are perpendicular to a 120 degree angle of the x, y, and z-axis. - - (111) Plane referenced to the origin at point O - - Other equivalent (111) planes ## INTER PLANAR DISTANCE - An image shows a cube labelled with point O at an origin, A, B, C, M, and N. There are angles marked 90°, β', and α'. - From figure *in a cubic lattice of cube edge a' the intercepts of the plane* of the three axes are *given by* - OA = a/h - OB = a/k ----------- (1) - OC = a/l - Let ON = `d₁` = perpendicular distance from the origin /to the plane and the direction cosines of ON be - Cos a', Cosβ' & Cosy' . - Cosa' = ON/OA = `d₁`/a/h = `d₁`h/a - Cosβ' = ON/OB = `d₁` /a/k = `d₁`k/a - Cosy' = ON/OC = `d₁` /a/l = `d₁`l/a -------- (2) - *But,* Cos²a' + cos²β' +cos²y' = 1 --------------(3) - Substituting eq(3) in eq(2), we get - (`d₁`h/a)² + (`d₁`k/a)² + (`d₁`l/a)² = 1 - (*OR*) - `d₁` = ON = a/√h² + k² + I² -----------(4) - Let 'OM' be the perpendicular distance of the next parallel plane from the origin. - *Hence* its intercepts are - OM = `d₁` + `d₁` = 2`d₁` - = 2a/V h² + k² + I² -----------(5) - Therefore, the interfacing between two adjacent planes is equal to - OM – ON = NM - *Hence,* NM = 2`d₁` -`d₁ `= `d₁` - - `d₁`= a/ Vh² + k² + I² -----(6) - Eq(6) gives the interplanar spacing between two adjacent parallel planes of Miller Indices.