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Crystal Structure Continued! NOTE!! Much of the discussion & many figures in what follows was\ constructed from lectures posted on the web by Prof. Beşire GÖNÜL in Turkey. She has done an excellent job of covering many details of crystallography & she illustrates her topics with many very nice pictures of lattice structures. Her lectures on this are posted Here: http://www1.gantep.edu.tr/~bgonul/dersnotlari/. Her homepage is Here: http://www1.gantep.edu.tr/~bgonul/.

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Atoms are of different kinds. Some lattice points aren’t equivalent. A combination of 2 or more BL 2 d examples

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In General Mathematically, a lattice is defined by 3 vectors called Primitive Lattice Vectors a 1, a 2, a 3 are 3d vectors which depend on the geometry. Once a 1, a 2, a 3 are specified, the Primitive Lattice Structure is known. The infinite lattice is generated by translating through a Direct Lattice Vector: T = n 1 a 1 + n 2 a 2 + n 3 a 3 n 1,n 2,n 3 are integers. T generates the lattice points. Each lattice point corresponds to a set of integers (n 1,n 2,n 3 ). Lattice Translation Vectors

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Consider a 2-dimensional lattice (figure). Define the 2 Dimensional Translation Vector R n n 1 a + n 2 b (Sorry for the notation change!!) a & b are 2 d Primitive Lattice Vectors, n 1, n 2 are integers. 2 Dimensional Lattice Translation Vectors Once a & b are specified by the lattice geometry & an origin is chosen, all symmetrically equivalent points in the lattice are determined by the translation vector R n. That is, the lattice has translational symmetry. Note that the choice of Primitive Lattice vectors is not unique! So, one could equally well take vectors a & b' as primitive lattice vectors. Point D(n 1, n 2 ) = (0,2) Point F(n 1, n 2 ) = (0,-1)

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The Basis (or basis set) The set of atoms which, when placed at each lattice point, generates the Crystal Structure. Crystal Structure ≡ Primitive Lattice + Basis Translate the basis through all possible lattice vectors T = n 1 a 1 + n 2 a 2 + n 3 a 3 to get the Crystal Structure or the DIRECT LATTICE

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The periodic lattice symmetry is such that the atomic arrangement looks the same from an arbitrary vector position r as when viewed from the point r' = r + T (1) where T is the translation vector for the lattice: T = n 1 a 1 + n 2 a 2 + n 3 a 3 Mathematically, the lattice & the vectors a 1,a 2,a 3 are Primitive if any 2 points r & r' always satisfy (1) with a suitable choice of integers n 1,n 2,n 3.

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In 3 dimensions, no 2 of the 3 primitive lattice vectors a 1,a 2,a 3 can be along the same line. But, Don’t think of a 1,a 2,a 3 as a mutually orthogonal set! Usually, they are neither mutually perpendicular nor all the same length! For examples, see Fig. 3a (2 dimensions):

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The Primitive Lattice Vectors a 1,a 2,a 3 aren’t necessarily a mutually orthogonal set! Usually they are neither mutually perpendicular nor all the same length! For examples, see Fig. 3b (3 dimensions):

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Bravais Lattice An infinite array of discrete points with an arrangement & orientation that appears exactly the same, from whichever of the points the array is viewed. A Bravais Lattice is invariant under a translation T = n 1 a 1 + n 2 a 2 + n 3 a 3 Crystal Lattice Types Nb film

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Non-Bravais Lattices In a Bravais Lattice, not only the atomic arrangement but also the orientations must appear exactly the same from every lattice point. 2 Dimensional Honeycomb Lattice The red dots each have a neighbor to the immediate left. The blue dot has a neighbor to its right. The red (& blue) sides are equivalent & have the same appearance. But, the red & blue dots are not equivalent. If the blue side is rotated through 180º the lattice is invariant. The Honeycomb Lattice is NOT a Bravais Lattice!! Honeycomb Lattice

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It can be shown that, in 2 Dimensions, there are Five (5) & ONLY Five Bravais Lattices!

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2-Dimensional Unit Cells Unit Cell The Smallest Component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal. S a b S SSSSSSSSSSSSS

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S S S Unit Cell The Smallest Component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal. Note that the choice of unit cell is not unique!

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2-Dimensional Unit Cells Artificial Example: “NaCl” Lattice points are points with identical environments.

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2-Dimensional Unit Cells: “NaCl” Note that the choice of origin is arbitrary! the lattice points need not be atoms, but The unit cell size must always be the same.

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These are also unit cells! It doesn’t matter if the origin is at Na or Cl! 2-Dimensional Unit Cells: “NaCl”

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These are also unit cells. The origin does not have to be on an atom! 2-Dimensional Unit Cells: “NaCl”

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These are NOT unit cells! Empty space is not allowed! 2-Dimensional Unit Cells: “NaCl”

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In 2 dimensions, these are unit cells. In 3 dimensions, they would not be. 2-Dimensional Unit Cells: “NaCl”

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Why can't the blue triangle be a unit cell? 2-Dimensional Unit Cells

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Example: 2 Dimensional, Periodic Art! A Painting by Dutch Artist Maurits Cornelis Escher (1898-1972) Escher was famous for his so called “impossible structures”, such as Ascending & Descending, Relativity,.. Can you find the “Unit Cell” in this painting?

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3-Dimensional Unit Cells

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3-Dimensional Unit Cells 3 Common Unit Cells with Cubic Symmetry Simple Cubic Body Centered Face Centered (SC) Cubic (BCC) Cubic (FCC)

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Conventional & Primitive Unit Cells Unıt Cell Types Primitive A single lattice point per cell The smallest area in 2 dimensions, or The smallest volume in 3 dimensions Simple Cubic (SC) Conventional Cell = Primitive cell More than one lattice point per cell Volume (area) = integer multiple of that for primitive cell Conventional (Non-primitive) Body Centered Cubic (BCC) Conventional Cell ≠ Primitive cell

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Face Centered Cubic (FCC) Structure

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Conventional Unit Cells A Conventional Unit Cell just fills space when translated through a subset of Bravais lattice vectors. The conventional unit cell is larger than the primitive cell, but with the full symmetry of the Bravais lattice. The size of the conventional cell is given by the lattice constant a. The full cube is the Conventional Unit Cell for the FCC Lattice FCC Bravais Lattice

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Conventional & Primitive Unit Cells Face Centered Cubic Lattice Primitive Lattice Vectors a 1 = (½)a(1,1,0) a 2 = (½)a(0,1,1) a 3 = (½)a(1,0,1) Note that the a i ’s are NOT Mutually Orthogonal! Conventional Unit Cell (Full Cube ) Primitive Unit Cell (Shaded) Lattice Constant

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Elements That Form Solids with the FCC Structure

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Body Centered Cubic (BCC) Structure

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Primitive Lattice Vectors a 1 = (½)a(1,1,-1) a 2 = (½)a(-1,1,1) a 3 = (½)a(1,-1,1) Note that the a i ’s are NOT mutually orthogonal! Primitive Unit Cell Lattice Constant Conventional Unit Cell (Full Cube) Conventional & Primitive Unit Cells Body Centered Cubic Lattice

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Elements That Form Solids with the BCC Structure

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Conventional & Primitive Unit Cells Cubic Lattices Simple Cubic (SC) Primitive Cell = Conventional Cell Fractional coordinates of lattice points: 000, 100, 010, 001, 110,101, 011, 111 Body Centered Cubic (BCC) Primitive Cell Conventional Cell Fractional coordinates of the lattice points in the conventional cell: 000,100, 010, 001, 110,101, 011, 111, ½ ½ ½ Primitive Cell = Rombohedron

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Conventional & Primitive Unit Cells Cubic Lattices Face Centered Cubic (FCC) Primitive Cell Conventional Cell The fractional coordinates of lattice points in the conventional cell are: 000,100, 010, 001, 110,101, 011, 111, ½ ½ 0, ½ 0 ½, 0 ½ ½, ½ 1 ½, 1 ½ ½, ½ ½ 1

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Simple Hexagonal Bravais Lattice

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Conventional & Primitive Unit Cells Hexagonal Bravais Lattice Primitive Cell = Conventional Cell Fractional coordinates of lattice points in conventional cell: 100, 010, 110, 101, 011, 111, 000, 001 Points of the Primitive Cell 120 o

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Hexagonal Close Packed (HCP) Lattice: A Simple Hexagonal Bravais Lattice with a 2 Atom Basis The HCP lattice is not a Bravais lattice, because the orientation of the environment of a point varies from layer to layer along the c-axis.

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General Unit Cell Discussion For any lattice, the unit cell &, thus, the entire lattice, is UNIQUELY determined by 6 constants (figure): a, b, c, α, β and γ which depend on lattice geometry. As we’ll see, we sometimes want to calculate the number of atoms in a unit cell. To do this, imagine stacking hard spheres centered at each lattice point & just touching each neighboring sphere. Then, for the cubic lattices, only 1/8 of each lattice point in a unit cell assigned to that cell. In the cubic lattice in the figure, Each unit cell is associated with (8) (1/8) = 1 lattice point.

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Primitive Unit Cells & Primitive Lattice Vectors In general, a Primitive Unit Cell is determined by the parallelepiped formed by the Primitive Vectors a 1,a 2, & a 3 such that there is no cell of smaller volume that can be used as a building block for the crystal structure. As we’ve discussed, a Primitive Unit Cell can be repeated to fill space by periodic repetition of it through the translation vectors T = n 1 a 1 + n 2 a 2 + n 3 a 3. The Primitive Unit Cell volume can be found by vector manipulation: V = a 1 (a 2 a 3 ) For the cubic unit cell in the figure, V = a 3

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Primitive Unit Cells Note that, by definition, the Primitive Unit Cell must contain ONLY ONE lattice point. There can be different choices for the Primitive Lattice Vectors, but the Primitive Cell volume must be independent of that choice. A 2 Dimensional Example! P = Primitive Unit Cell NP = Non-Primitive Unit Cell

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