Solid State Physics - Mustansiriyah University - PDF

# Solid State Physics ## Fourth Grade – Semester 1 #### **Mustansiriyah University** #### **College of Sciences** #### **Physics Department** ## **Chapter One: Crystal Structure** ### **Prof.Dr. Wafaa Mahdi Salih** ## **Contents** | Section | Subjects | |---|---| | 1 | Introduction in solid sta...

# Solid State Physics ## Fourth Grade – Semester 1 #### **Mustansiriyah University** #### **College of Sciences** #### **Physics Department** ## **Chapter One: Crystal Structure** ### **Prof.Dr. Wafaa Mahdi Salih** ## **Contents** | Section | Subjects | |---|---| | 1 | Introduction in solid state physics and Overview | | 2 | Essential Concept | | 3 | Crystal Symmetry | | 4 | Crystal systems | | 5 | Miller Indices | | 6 | Crystal Directions | | 7 | Planes Spacing | | 8 | The density of planes | ## **References** 1. فيزياء الحالة الصلبة - تأليف د. مؤيد جبرائيل يوسف 2. فيزياء الحالة الصلبة تأليف: يحيى نوري الجمال. 3. فيزيا الحالة الصلبة - تأليف: د. صبحي سعيد الراوي د. شاكر جابر شاكر د. يوسف مولود حسن. 4. J. S. Blakemore, "Solid State Physicsl, Third Edition,Cambridge University Press, 1985. 5. H. Ibach, H. Lüth, An Introduction to Solid-State Physics Principles of Materials Sciencell, Springer, 2003. 6. Omar MA., 1975, Elementary solid state physics, principles and applications, Addison-Wesley Publishing Company. 7. Kittel, C., 2005,. Introduction to solid state physics, 8th ed., Wiley ## **Solid Materials** Matter can be classified according to their properties such as electrical, magnetic, binding energy, thermal etc. ### **Electrical properties** - Conductors - Semiconductors - Insulators ### **Magnetic properties** - Paramagneti - Diamagnetic - Ferromagnet ### **Binding Energy** - Ionic - Valance - Metallic - Vander Waals ### **Thermal Properties** - Thermal Conductor - Thermal Insulator ### **Internal Building of Atoms** - Crystalline - Polycrystalline - Amorphous ## 1- **Crystalline and non-Crystalline (Amorphous) Solid Materials:** The constituents of a solid can be arranged in two general ways: they can form a regular repeating three-dimensional structure called a crystal lattice, thus producing a crystalline solid, or they can aggregate with no particular order, in which case they form an amorphous solid. ### **Crystalline:** Contains rows of particles (atoms, ions, and molecules) are grouper and arranged in a repeating pattern of a three-dimensional network called long range order (forming a periodic geometric pattern). This network is known as a Crystal lattice and the smallest unit of a crystal is a Unit Cell. They have a regular and ordered arrangement resulting in a definite shape. Crystals have a long-range order, which means the arrangement of atoms is repeated over a long distance like salt (sodium chloride), diamond and Gold. ### **Amorphous:** Particles are arranged randomly called amorphous. They do not have an ordered arrangement resulting in irregular shapes. Amorphous solids are rigid structures but they lack a well-defined shape. They do not have a geometric shape. So they are non-crystalline. This is why they do not have edges like crystals do. The most common example of an amorphous solid is Glass. Gels, plastics, various polymers and wax are good examples of amorphous solids. ## **2- Essential Concepts:** - Crystallography: the science that deals with the study of solid in all their forms and phenomena. - Crystals: Crystal: It is a solid body that contains a number of atoms arranged in a specific geometric shape so that they form its locations are periodic (and this periodicity is often called (long-range order), so the crystal consists of cities and units of purpose units cell. When small, it repeats regularly in three dimensions, called the unit cell, The idea of periodicity in crystals is expressed by saying that the crystal has translational symmetry, meaning that if a point moves and by means of any vector that connects two points, the point appears as if it has not moved, that is, what is adjacent to it has not changed. The perfect crystal maintains this periodicity in three dimensions and to infinity for each of the axes, which results in the periodic process is that the positions of the atoms in the crystal are equivalent, in other words, the crystal appears complete to the viewer ## **Stable in any of these same atomic sites.** - The basis of the crystal construction is repetition and there are different types of crystals: - Real crystal: which represent most of the crystals found in nature and contains some defects and distortions. - Perfect crystals: and it is a supposed crystal, as we assume the existence of a perfect crystal free of defect and distortions for the purpose of the study, and there is no perfect crystal in nature, and the perfect crystal characterized by periodicity, regular three-dimensionality, as identical groups of atoms repeat themselves when breaks or spaces completely equal. ## **Types of Real Crystals:** - **Single Crystal:** Where the periodicity of the formation or the three-dimensional crystal model extends throughout the entire crystal - **Polycrystalline:** it does not stretch the periodicity of the crystal pattern through the entire crystal, but ends at a boundary inside the crystal is called grain boundaries when the periodic geometric pattern spreads to occupy all parts of matter, This means that we have a single crystal, but if the spread of the periodic geometric pattern stops at borders or boundaries, then the material is multi-grained, that is, it consist of very small groups of granular crystals, or small single crystals in different directions. ## **Crystal Structure:** It can be defined from the relationship that concept the basis to each point of the lattice. - **Basis:** An atom or group of atoms located at each point location of the lattice. - **Lattice:** In crystallography, the geometric properties are the subject of interest, not the composition of the material. Accordingly: we replace each atom with a geometric point located in the stable position of that atom, and thus the result is a geometric structure of points possesses the geometric properties of the crystal itself. ## **There are two types of lattice:** - **Bravais lattice:** in this type, all the points of the lattice are equivalent, meaning that all atoms in the crystal are of the same type. - **Non-Bravais lattice:** in this type, the points of the lattice are unequal. Where sites A,B,C are equivalent to each other, but sites AA,BB and CC are not equivalent to each other. In the sense that it can be considered a combination of two or more paravisian lattice intertwined with each other in a fixed position relative to each other. ## **Q/What is the difference between atomic structure and crystal structure?** Answer: Atomic structure relates to the number of neutrons and protons in the atomic nucleus and the number of electrons in the electron orbitals, Crystal structure is concerned with the arrangement of atoms within crystalline solids specific formations. ## **Transition Vectors in a Crystal (Translational symmetries):** An ideal single crystal is defined as a regular arrangement of identical units extending to infinity. The one dimensional lattice is defined by the function of one vector, which is a, and the two-dimensional lattice is defined by the vectors a and b. The three dimensional lattice is defined in terms of the three vectors a,b and c, and they are called translation vectors. As for the vector that it connects these three vectors and is called the translation vector (T). <start_of_image> $$T =n\vec{a}$$ $$T = n_1 \vec{a} + n_2\vec{b}$$ $$T = n_1\vec{a} + n_2\vec{b} + n_3\vec{c}$$ - **where n is an integer** - **for two dimensional lattice** - **for three dimensional lattice** Transition vector T connects any sites inside the crystal so that the atoms surrounding these two sites look identical, which is why it is called the transition effect or the creep effect: $$\vec{r'} = \vec{r}+T$$ - **Where r, r' two sites inside the crystal** Substituting (1) into (2), we get: $$\vec{r'} = \vec{r} + n_1\vec{a} + n_2\vec{b} + n_3\vec{c}$$ Means that the arrangement remains the same with respect to the point expressed by the vector r when viewed from another point r' as in the figure. We note that the transition vector T=5a+b connects any lattice point in unit cell ABCD and its equivalent point in unit cell ABCD. - **Unit cell:** is the smallest unit in the lattice that fills space under the influence of the T effect and forms a complete lattice (it is the smallest unit in the special lattice and is the unit that, repeated in the three directions, results in a crystal). The volume of the unit cell is given: $$V= |\vec{a}.\vec{b}\times\vec{c}| = |\vec{b}.\vec{c}\times\vec{a}| = |\vec{c}.\vec{a}\times\vec{b}| $$ Note: There are many ways to choose the primary axis that is several ways to choose the primary unit cell for lattice. The important thing here to perform the cross multiplication process first, then the dot multiplication. The primitive cell: is the cell that contains points in its corners only and its axis are of the shortest possible length and are subject to equation (3). $$\vec{r'} = \vec{r} + n_1\vec{a} + n_2\vec{b} + n_3\vec{c}$$ Where the primitive unit cell is the one that has lattice point at its eight corner only each corner shares with eight cells, and thus only one eighth (1/8) of the atoms or lattice point to each unit of primitive cell. That is, the eight atoms will each contribute as ratio (1/8),and thus the primitive cell with contain one lattice point or one atom. The unit cell is primitive as in simple cubic: ### **Non Primitive cell:** is a cell that contains other lattice point in addition to the corners and the lengths of its axis are not the shortest length, and the equation (3) does not apply to it.In two dimensional space, the primitive unit cell has a fixed area regardless of the methods of choosing its axis. That is, anon-primitive cell contains more than one lattice point or one atom, and it is also called the cell composite to overlap two or more lattice to form another composite shape, such as: - Body Center Cubic (B.C.C.) - Face Center Cubic (F.C.C.) - Base Center Cubic (B.C.C.) ### **Wegner Seitz Cell:** It is another way to choose the primitive cell is summarized as follows: - We extend straight lines from one lattice point to all nearby lattice points. - We describe these lines as perpendicular planes. - The volume between the perpendicular planes is a primitive cell it contains a single lattice point. ## **3- Crystal Symmetry:** ### **Symmetry:** It is the repetition or congruence of the parts of the a shape around a plane, line, or point of symmetry. The circle is symmetry around any diameter that repeats itself, and the sphere is symmetrical around its largest circular plane. The cube has many cases of symmetry, so it is symmetrical diagonally, longitudinally, transversely and around its center. ### **Asymmetry:** It is the shape that does not have the characteristic of repetition and does not have matching its part, such at the right hand and the left hand of a person. The symmetry in the crystal is a process or effects that can be imagined to occur on the crystal, and after its completion, the crystal looks like its origin, i.e. recurring or returning its part to the positions they occupied before the occurrence of the processes. ### **Symmetry Elements:** are the axis, plane, center of symmetry around which the symmetry process takes place. ### **Symmetry Operations (symmetry effects):** These are the processes that we imagine to occurring on the crystal and returning to it. The symmetry operations, including: - Translation - Rotation - Reflection - Inversion - Screw - Glide ## **1- Translation symmetry operation:** $$T = n_1\vec{a}+ n_2 \vec{b}+n_3 \vec{c}$$ $$\vec{r'}=\vec{r}+T$$ ## **2- Rotation symmetry operation:** This is an axis such that, if the cell is rotated around it through some angle, the cell remains invariant. The axis is called n-fold if the angle of rotation is 2π/n. ## **3- Reflection symmetry operation:** A plane in a cell such that, when a mirror reflection in this plane is performed, the cell remains invariant. ## **4- Inversion symmetry operation:** A cell has an inversion center if there is a point at which the cell remains invariant when the mathematical transformation r`, r is performed on it. All Bravais lattices are inversion symmetric, a fact which can be noting that, with every lattice vector R₁ = n₁a + n2b + n3c, there is associated an inverse lattice vector. $$R₁ = - R₁ = - n₁a – n2b – n3c$$ A non-Bravais lattice may or may not have an inversion center, depending on the symmetry of the basis. ## **5- Glide symmetry operation:** A combination of inversion and transition. ## **6- Screw symmetry operation:** Rotational axis of (1,2,3,4) with transition symmetry. $$nT=Pa$$ - **Where n= numbers of folds, T= height, P= is (1.2.3......), a = lattice constant and the rotation angle α = 2π / η** ## **Example1: Draw the following screw operation 21.** Answer: Since the screw operations is 21 that's mean n=2,p=1 $$T= (P/n)a → T=(1/2)a$$ $$\alpha= 2π/η →2π/2=π$$ ## **Example2: Draw the following screw operation 31 since the screw** Answer: operations is 31 that's mean n=3, p=1 $$T=(P/n)a → T=(1/3)a$$ $$\alpha= 2π/η →2π/3=120°$$ ## **4- Space lattices and crystal systems:** The importance of these lattices lies in understanding the properties of the physical surfaces of solid bodies and understanding the study of X-ray diffraction on those surfaces so that we can estimate the distance between points. The Lattice, as previously mentioned, is a set of points arranged in a specific order, and repeats itself periodically. 1-The repetition process is in one direction and is called a linear lattice 2- The repetition process is in two dimensions is called a planar lattice 3- The repetition process is in three dimensions is called a space lattice. - A linear lattice consists of similar points of equal dimensions and on a straight vertical or horizontal line. Therefore there is one basic type of linear lattice because there is only one way to arrange the points and the only difference is the distance between the points. - As for the planer lattice, it is possible to obtain a large number of lattices, which can be grouped into five types: - Oblique lattice. - Square lattice - Hexagonal lattice - Rectangular primitive - Rectangular centered - While Space lattice, there are four basic types of Bravis lattice (in three dimensions): - Primitive lattice - Body center lattice - Face center lattice - Base or side center lattice The five types of basic lattices are distributed over seven crystal systems, from which fourteen Baravis lattice branches. Below are the seven crystal systems and the fourteen Baravis lattices. ## **Features of cubic lattice:** The cube system includes three types of lattice: - **1-Simple Cubic (P or SC):** It contains one lattice point, i.e. a point in each of the eight corners, and its vectors, which are primary vectors around each of them. (1/8 *8=1) - **SC:** - No. of atom/unit cell (1/8)×8=1 atom - No. of nearest neighbor are 6 - The distance of the nearest neighbor d=a - The relation between r and a is a=2r, r=a/2 - The primitive vector are a=ia, b=jb, c=kc - Volume=|a.bxc - Volume= ia.(jbxkc) - a=b=c in the cubic system - Volume= ia.(ja×ka)» ia.( ia²) - Volume=a³ ## **Q/ Prove that the volume of the elementary unit cell of a simple cube is equal to a³:** $$a= \vec{a}x\vec{b} = \vec{a}y\vec{c} = \vec{a}z\vec{a}$$ $$V = V = [\vec{a}¯×\vec{b}¯¯ . \vec{c}¯¯ | = |\vec{a}x × \vec{a}y· \vec{a}z| =|\vec{a}^2z^· \vec{a}z| = |\vec{a}^3|$$ $$V = a^3$$ ## **2- Body Center Cubic (BCC):** It contains two points one in the corners and one in the center of the cube are non-primitive cell because its unit cell is non primitive. The location of the points: 000, 111/222 ### **BCC** $$D=√a^2+(√2a)2$$ $$D=√3a$$ $$d=1/2(√3a)$$ - No. of atom/unit cell [(1/8)×8]+1=2 atom - No. of nearest neighbor are 8 - The distance of the nearest neighbor d=1/(√3a) - The relation between r and a is √3a =r+r+2r, √3a =4r → r=1/4(√3a) - r=d/2 - The primitive vector are - a=a/2(i+j-k) - b= a/2(-i+j+k) - c= a/2(i-j+k) - Volume=|a.bxc| - Volume= a/2(i+j-k).( a/2(-i+j+k)× a/2(i-j+k)) - a=b=c in the cubic system - Volume=(1/2)a³ ## **[H.W:1]: Prove that the volume of the primitive cell of a body -centered cubic lattice is equal to (1/2) the volume of the ordinary unit cell of the same lattice.** ## **3-Face Centered Cubic (FCC):** It contains four lattice points, a corner point and a half point on each of the six face .It is not a primitive lattice because its unit cell is not .To obtain the primary vectors, we draw three vectors from a lattice point in one of the corners of the cube, and we consider it the origin point, so that it ends with the lattice point located at the centers of the faces to the origin, as in the adjacent figure. We complete the rhombuses to obtain the initial unit cell with the primary vectors. point locations: 000,0,0,0 222 222 ### **FCC** $$D=√2a$$ $$d=1/2(√2a)$$ - No. of atom/unit cell [(1/8) ×8]+6*1/2=4 atom - No. of nearest neighbor are 12 - The distance of the nearest neighbor d=1/2(√2a) - The relation between r and a is √2a =r+r+2r, √2a =4r → r=1/4(√2a) - r=d/2 - The primitive vector of FCC are - a=a/2(i+j) - b= a/2(j+k) - c= a/2(i+k) - Volume=|a.bxc - Volume= a/2(i+j).( a/2(j+k) × a/2(i+k)) - a=b=c in the cubic system → Volume=(1/4)a³ ## **[H.W:2]/Prove the the volume of the primitive cell of a face center cubic (fcc) is (1/4) the volume of the normal cell of that lattice** ## **Filling Factor or Packing Factor:** It is the ratio between the volume occupied by atoms over the total of the cube or system. $$F.F = \frac{volume\ occupied\ by\ atoms}{total\ volume}$$ $$F.F= \frac{4}{3}πr³\times \frac{No.of\ atoms}{unit\ cell} \times \frac{1}{a^3}$$ ### **Applications:** - **1-SC:** - No. of atoms /unit cell=1 - r=a/2 → a=2r - $$F.F =\frac{4}{3}πr³\times \frac{1}{(2r)^3} = \frac{π}{6} ≈0.5236= 52%$$ - **2- BCC:** - No. of atoms /unit cell=2 - $$r=1/4(√3a) → a=\frac{4r}{√3}$$ - $$F.F = \frac{4}{3}πr³\times \frac{2}{\left( \frac{4r}{√3}\right)^3} =\frac{π√3}{8} ≈ 0.6801 = 68%$$ - **3. FCC:** - No. of atoms /unit cell=4 - $$r^2=\frac{2a^2}{16}$$ - $$a = 2√2 r$$ - $$F.F =\frac{4}{3}πr³\times \frac{4}{\left( 2√2r\right)^3} = \frac{π√2}{3} ≈ 0.74 = 74%$$ ## **5- Miller Indices:** - To describe the physical state of crystals, the positions and directions of the crystal planes must be determined, which for any crystal plane are determined by three nodes that are not in the same line through which the coordinates of the crystal plane are determined. - The condition that the nodes fall on the crystal axes. - We can determine the above by choosing a set of coordinate axes that apply and agree in direction with the sides of the primitive cell, so that the principle of these axes lies on one of the nodes of the crystal lattice where the sides of the primitive cell intersect. - If the points A, B, and C have coordinates A(3,0,0),B(0,2,0),C(0,0,1) representing the three nodes, then they will be determined two crystal planes. This plane can be determined using the three numbers (321). - From the point of view of the crystal structure, the position and orientation of the crystal plane can be determined by a convention used to describe the crystal planes and directions in the crystal, they are called Miller coefficients, and they are very useful in terms of The inverted lattice, as we will see later, defines the Miller indices To find it, we follow these steps: 1. We choose a point of origin (0) and three reference axes (x, y, z). 2. We determine the intersection of the surface with each of the three axes by the values (a,b,c). 3. We invert the values of the intersections (a,b,c) If they are all integers, they represent hkl, and if some or all of them are rational number that we multiply by the least common multiple to convert them to integers to obtain (hkl). ) 4. Miller indices may all be positive, negative, or mixed numbers, but they are always integers. 5. When there is a surface parallel to one of the crystal axes, such as the x axis, the coefficients of this surface are written in the formula (Okl) because this surface intersects the xaxis at infinity (∞) and its reciprocal ∞ is zero. ## **Example (1): Find the Miller indices for the surface of its intersection with the axes as follows: x=3,** $$y= 6$$ $$z=2$$ 1. Find the inverse: 1/3 1/6 1/2 2. We multiply by the least common multiple: 13*6 16*6 12* 3. The Miller coefficients for this surface are: (213) ## **Example (2): Find the Miller indices for the surface of its intersection with the axes as follows: x=4** $$y= ∞$$ $$z=1/2$$ 1. Find the inverse: 1/4 1/∞ 2 2. We multiply by the least common multiple: ×4 3. The Miller indices for this surface are: (108) ## **Example (3): Find the Miller indices for the surface of its intersection with the axes as follows: x=4** $$y= ∞$$ $$z= -1/6$$ 1. Find the inverse: 1/4 1/∞ -6 2. We multiply by the least common multiple: × 4 3. The Miller indices for this surface are: (1024) ## **[H.W:3] /Determine the Miller indices for the following crystal surfaces** ## **6- Crystal Direction:** To determine any direction in the crystal, we take the Miller coefficients and form a plane from them. The perpendicular to this plane is the direction of the crystal plane, and it is written conventionally in the form [hkl]. To express the trend, the coefficients are at the same time Miller coefficients [hkl]. We use three indices: u v w, which are written in the form [uvw], which are integers that do not have a common factor it is greater than one, and there are equivalent directions in the crystal. To indicate this, it is written in the formula <uvw>, so when writing <110> it means all directions of equivalent type: [101],[101],[011] ,[011] ,[110],[101],[011] The direction of a crystal is called the area or band axis, and the band axis represents a common direction along which a group intersects Intersecting surfaces are said to have one common direction or one domain axis and that they belong to the same domain. The Miller coefficients (h k 1) of the surface belonging to the range of Miller coefficients whose axis is [uvw] must be subject to the algebraic relationship. $$hu+kv+lw=0$$ For example: (001) with [110] or (010) with [101]. This means that any surface (h k 1) contains the direction [uvw] if equation (1) is fulfilled and the scale axis coefficients [uvw] can be calculated for two intersecting surfaces such as (h1k111) and (h2k212) as following: $$u=k_1l_2-k_2l_1$$ $$v=l_1h_2-l_2h_1$$ $$w=h_1k_2-h_2k_1$$ Equations (2) can be used to find Miller indices (h k 1) for a surface that has both directions [u₁V1W1] and [u2V2W2] as follow: $$h = V_1W_2-V_2W_1$$ $$k = W_1U_2 - W_2U_1$$ $$l = u_1V_2 - U_2V_1$$ ## **[H.W:5]: Find the surface (hkl) that contains the directions [110] and [211] using equation (3),** ## **H.W:6] Find the direction that it is represented by [uvw],to wich the surfaces (011)and (111) belong using equation (2).** ## **Calculating the angle between two planes or between two directions:** It is possible to calculate the angle o between the planes (h₁k 1₁) and (h2k2l2 ) in a cubic crystal,and it represents the angle what is confined between the two columns on these two surfaces as follow: $$cosθ =\frac{h_1h_2+k_1k_2+l_1l_1}{\sqrt{(h_1²+k_1²+l_1²)}\sqrt{(h_2²+k_2²+l_2²)}}$$ $$θ = cos^{-1}\left(\frac{h_1h_2+k_1k_2+l_1l_1}{\sqrt{(h_1²+k_1²+l_1²)}\sqrt{(h_2²+k_2²+l_2²)}}\right)$$ ## **[H.W:7]/ Find the angle between surfaces (312) and (421), then find the angle between [201] and [123].** ## **Position of Atoms in Unit Cell:** The location of a point in a unit cell is represented by three atomic coordinates where each distance coordinate represents the origin in units without brackets uvw and written in the form c, b, a the cell axes without commas, it represents the positions of the atoms within the unit cell about uvw in fractional units less than one, and the value of uvw is always no more than absolutely one. ## **7- Planes Spacing dhkl and Lattice Constant (a):** It represents the vertical distance between any two successive surfaces of set parallel surfaces. In other words, the shortest vertical distance between the lattice planes. Where dhkl represents lattice constant. It is given dhkl any set of parallel surfaces in a cubic crystal whose regular cell has side length (a) with the following relationship: $$d_{hkl} =\frac{a}{\sqrt{(h^2 + k^2 + l^2)}}$$ We note from the relationship that d depends on the numerical value of the Miller coefficients and does not depend on the signs these coefficients and there are different sets of parallel surfaces with different Miller coefficients dhki but equal space, such as: (422), (511) and surfaces (600), (333). ## **Q/Prove that:dhkl=** $$ \frac{a}{\sqrt{h^2+k^2+l^2}} = dhkl = \frac{a}{(h^2 +k^2+l^2)^{1/2}}$$ $$cos∞ x= \frac{ON}{OA} in Δ(ONA)$$ ON It represents the vertical distance between surface ABC and the origin O, this is represented by dhkl. $$∴ cos ∞= \frac{dhkl}{h} = \frac{dhkl}{a}$$ $$∴ cos² ∞= \left( \frac{h}{a}\right)^2 d^2hkl.....(1)$$ $$OA = ap = a * p =h$$ $$ ΔONB: Cos β = \frac{ON}{OB} = \frac{ON}{b}=\frac{k}{p}=\frac{kd}{a}....cos² β =\left( \frac{k}{a}\right)^2 d^2hkl.....(2)$$ $$ ΔONC = cos γ = \frac{ON}{OC} = \frac{ON}{c} = \frac{l}{p}=\frac{ld}{a}....cos² γ =\left( \frac{l}{a}\right)^2 d^2hkl.....(3)$$ $$cos² α +cos² β + cos² γ= 1.....(4)$$ $$d^2hkl \left( \frac{h}{a}\right)^2 + \left( \frac{k}{a}\right)^2 + \left( \frac{l}{a}\right)^2 = 1$$ $$d^2hkl \left( \frac{1}{a}\right)^2(h^2+k^2+l^2) = 1 $$ $$d^2hkl (\frac{1}{a^2})(\frac{h^2+k^2+l^2}{1}) = 1$$ $$d^2hkl = \frac{a^2}{h^2+k^2+l^2}$$ $$ d_{hkl} = \left( \frac{1}{a}\right)^2\left(h^2+k^2+l^2\right) in \ cubic\ a=b=c $$ $$d_{hkl} =\frac{ a}{\sqrt{h^2 + k^2 + l^2}}$$ ## **8- The Density of Plane:** ### **Planar Atomic Density:** The number of atoms per unit area in different levels in many crystalline lattices. It is defined as the number of atoms divided by the unit area in the plane. It is given by the following relationship: $$p=\frac{No.of \ atoms}{area}$$ ## **Chapter Two: Crystal Diffraction** ### **Prof.Dr. Wafaa Mahdi Salih** ## **Contents** | Section | Subjects | |---|---| | 1 | Crystal Diffraction & Accident Beams | | 2 | Brag Law | | 3 | Experimental Diffraction Methods | | 4 | Reciprocal Lattice | ## **References** 1. فيزياء الحالة الصلبة - تأليف د. مؤيد جبرائيل يوسف 2. فيزياء الحالة الصلبة تأليف: يحيى نوري الجمال. 3. فيزيا الحالة الصلبة - تأليف: د. صبحي سعيد الراوي د. شاكر جابر شاكر د. يوسف مولود حسن. 4. J. S. Blakemore, "Solid State Physicsll, Third Edition,Cambridge University Press, 1985. 5. H. Ibach, H. Lüth,, An Introduction to Solid-State Physics Principles of Materials Sciencell, Springer, 2003. 6. Omar MA., 1975, Elementary solid state physics, principles and applications, Addison-Wesley Publishing Company. 7. Kittel, C., 2005,. Introduction to solid state physics, 8th ed., Wiley ## **1- Crystal Diffraction:** For the purpose of studying crystal structure, we cannot obtain sufficient information through microscopic observations or with the naked eye. This is not because the microscope has certain limits for magnification, but rather the issue relates to the nature of the beam used and in Microscopes usually contains visible light whose wavelengths are limited to between 4000 and 7000 A°. What we want is a beam whose wavelength is equal to or less than the distances required to be analyzed, and since the space between the atoms of the crystal does not exceed a few Angstroms, then the required beam must have a wavelength equal to or less than a few Angstroms. Through this, we will obtain comprehensive details about the crystal structure, such as the size of the smallest cell in the crystal and the positions of the nuclei of the atoms and the electronic distribution inside the cell, as well as the vibration patterns of the crystal atoms. This and other things are done through diffraction, and this process is sometimes called scattering. ## **When do scattering occur?** When electromagnetic radiation interacts with matter, the particle or photon is lost part of its energy and deviated from its path, this process is called inelastic scattering. If there is no change in energy and no change in wavelength, then the process is called elastic scattering.There are three basic types of wave particles with different energies that can be used in experiments diffraction these three types are: - **X-Ray** - **Electrons** - **Neutrons** ## **1-X-Ray Photons:** X-rays are electromagnetic waves with specific wavelengths that lie between ultraviolet rays and gamma radiation, whose wavelengths do not exceed a few Angstrom (0.1-10 Å).The photon energy of x-rays: $$E=hv = \frac{hc}{λ}$$ $$h=6.63×10^{-34}$$ $$λ(Α°)= \frac{12.4}{E(keV)}$$ $$c=3× 10^8 m/sec$$ - **if λ =1 A°** $$E=12400 eV → E=12.4 k eV$$ The study of crystal structure requires limited energies between (10-50) KeV.