Crystalline and Non-Crystalline Solids

• Matter is classified broadly into three primary states based on how its constituent particles interact with each other and the amount of energy they possess: solid, liquid, and gas. Each state exhibits distinct physical characteristics, behaviors, and properties. The study of matter in its various forms is fundamental in many scientific disciplines, including physics, chemistry, and materials science.
• Plasma is identified as a fourth state of matter, which becomes prevalent under certain conditions where gas is subjected to high temperatures or strong electromagnetic fields. In plasma, the atoms are ionized, allowing for the flow of electrons and ions. This state is commonly observed in stars, including the sun, where plasma constitutes the majority of its mass.
• Solids can be further subdivided into two main classifications: crystalline and non-crystalline (also referred to as amorphous) based on the arrangement of their atomic structures. The differences in these arrangements lead to varied properties and applications in science and industry.

Crystalline Solids

• In crystalline solids, atoms, molecules, or ions are systematically organized in a highly ordered, regular, and periodic manner throughout all three dimensions. This regularity is crucial to the material's properties, such as its strength and melting point.
• Common examples of crystalline solids include elemental metals such as iron (Fe), copper (Cu), silver (Ag), and aluminum (Al). These materials exhibit distinct crystallographic structures, contributing to their specific mechanical and thermal properties.

Non-Crystalline Solids (Amorphous)

• In non-crystalline solids, the arrangement of atoms, molecules, or ions lacks the long-range order characteristic of crystalline solids. They are structured more randomly, which affects their physical properties and behavior.
• Examples of non-crystalline or amorphous solids include materials like rubber and various types of plastics, where the lack of uniform atomic organization translates to different thermal and mechanical traits, such as flexibility and lower density.

Difference Between Crystalline and Non-Crystalline Solids

• One of the most significant differences is that crystalline solids have a structured and orderly arrangement of particles, while non-crystalline solids do not exhibit such organization.
• Crystalline solids typically demonstrate greater stability compared to their non-crystalline counterparts, which can be more susceptible to changes in temperature and pressure.
• In terms of melting behavior, crystalline solids have a distinct melting point — the temperature at which they change phase from solid to liquid — whereas non-crystalline solids generally do not melt sharply but soften over a range of temperatures.
• Crystalline solids are described as anisotropic, meaning their properties vary based on direction within the material; in contrast, non-crystalline solids are isotropic, exhibiting uniform properties in all directions.

Fundamental Crystallographic Terms

• A Lattice Point is defined as an abstract point in three-dimensional space that denotes the position of an atom, molecule, or a group of atoms within the crystal lattice. These points are essential for visualizing crystal structure.
• A Space Lattice, otherwise known as a Crystal Lattice, can be described as an arrangement of lattice points positioned along three crystallographic axes (x, y, z), where every point in the lattice exhibits a similar environment, defining the crystal's symmetry.
• The Basis, also referred to as the Motif, consists of atoms, molecules, or ions that are associated with every lattice point. It can be seen as a repeating unit that, when combined with a lattice, forms the entire crystal structure.
• The Crystal Structure is established through the combination of a lattice (or space lattice) and its basis, leading to the visualization and understanding of how atoms are structured within a crystal.

Unit Cell

• The Unit Cell is recognized as the fundamental building block of a crystal structure, capable of generating the full crystal through repeated translations in multiple directions. It serves as a concise representation of the entire crystal and its properties.
• This geometric figure is the smallest repeatable unit in a crystal lattice, and when replicated in all three dimensions, it reconstructs the entire crystal structure accurately.

Lattice Parameters (Unit Cell Parameters)

• The parameters of a unit cell within a three-dimensional crystal lattice are defined by specific components known as lattice parameters, which provide essential information about the spatial dimensions of the unit cell.
• The intercepts a, b, and c signify the distances between two lattice points on the x, y, and z axes, respectively, and refer to the primitive or axial lengths of the crystal structure.
• Additionally, the angles between these axes are outlined as γ (between the x and y axes), α (between the y and z axes), and β (between the z and x axes), which together define the overall geometric configuration of the unit cell.
• Intercepts (a, b, c) and angles (α, β, γ) are thus collectively referred to as the lattice parameters, playing a crucial role in determining the symmetry and properties of the crystalline material.

Seven Crystal Systems

• According to the variations in lattice parameters, crystals are categorized into seven distinct crystal systems, each defined by specific geometric relationships and symmetry characteristics:
• Cubic: Defined by equal lengths for all sides (a = b = c) and right angles between them (α = β = γ = 90°). This system encompasses many well-known materials, including common salt (NaCl).
• Tetragonal: Characterized by two equal lengths (a = b) and a distinct third length (c), with right angles maintained among the sides (α = β = γ = 90°). An example is the mineral zircon.
• Orthorhombic: Presented with three unequal lengths (a ≠ b ≠ c) but maintaining all angles at 90° (α = β = γ = 90°). Minerals such as olivine follow this system.
• Monoclinic: This system has three unequal lengths (a ≠ b ≠ c) with two angles set at 90° and one angle (γ) not equal to 90°. Gypsum is a well-known example.
• Triclinic: The least symmetry of the systems, where all dimensions are unequal (a ≠ b ≠ c) and none of the angles are right angles (α ≠ β ≠ γ ≠ 90°). An example is the mineral kyanite.
• Trigonal (Rhombohedral): In this system, all sides are equal (a = b = c), but the angles are less than 120° and not 90° (α = β = γ < 120°, ≠ 90°). Examples include quartz.
• Hexagonal: Characterized by two equal lengths (a = b) and one distinct length (c), with angles being right angles in the x and y dimensions (α = β = 90°) and a 120° angle between the a axes (γ = 120°). Graphite is a prominent example.

Different Types of Unit Cells

• Primitive or simple lattice: This lattice structure consists of a single lattice point or atom at each corner of the cell, which can be represented as symbol 'P'. It serves as the most fundamental unit cell.
• Base-centered lattice: Each unit cell retains lattice points at its corners, with an additional lattice point or atom located in the center of the base. This configuration is denoted by the symbol 'C'.
• Body-centered lattice: In this type, lattice points are situated at each corner of the unit cell and include a single additional atom at the very center of the cell, identified by symbol 'I'. This arrangement is significant in metals like iron.
• Face-centered lattice: Featuring lattice points at the corners, this arrangement also includes an extra atom at the center of each face of the cubic cell, represented by symbol 'F'. This structure is common in many metallic crystals.
• In 1948, Bravais established that under the seven crystal systems, there exists a total of 14 unique types of unit cells, collectively referred to as 'Bravais lattices'.
• These include three types of cubic unit cells, two types of tetragonal, a single hexagonal type, four distinct orthorhombic types, one type of rhombohedral, two different monoclinic types, and one triclinic structure. These diverse arrangements play a significant role in defining the material properties of each crystal.

Coordination Number

• The coordination number of a specific atom within a crystal system defines the number of nearest neighboring atoms surrounding that atom, reflecting how closely packed the material is. It provides insight into the material's structural arrangement and affects its physical properties.
• For a simple cubic arrangement, the coordination number equals 6, indicating that each atom is directly adjacent to six others.
• In a body-centered cubic (BCC) configuration, the coordination number is 8, as each atom is in contact with eight nearest neighbors.
• In contrast, the face-centered cubic (FCC) structure features a coordination number of 12, which allows for a higher density of packing compared to simpler structures.

Atomic Packing Fraction (APF)

• Atomic Packing Fraction (APF) is a measure that quantifies the efficiency of packing within a unit cell by determining the ratio between the total volume occupied by atoms in the unit cell and the overall volume of the unit cell itself. This concept is critical for understanding densification in materials.

Number of Atoms per Unit Cell:

• The total number of atoms present in or contributed to a unit cell can significantly affect its properties and behavior in various applications.

• In a simple cubic lattice, a single atom occupies the unit cell, leading to a total of one atom per unit cell.

• The total number of atoms per unit cell in this structure is consistently calculated as 1.

• The volume of one atom can be calculated using the formula for the volume of a sphere: Volume of one atom (spherical) = (4πr^3)/3, where r represents the atomic radius.

• The overall volume occupied by the total number of atoms per unit cell is calculated as 1 multiplied by (4πr^3)/3.

• The atomic radius is defined as half the distance between any two nearest neighbor atoms that are in direct contact, directly influencing the unit cell dimensions.

• For the simple cubic structure, the atomic radius can be expressed as a/2, where 'a' signifies the length of the unit cell edge.

• The total volume of the unit cell can be expressed as the cube of the length of an edge: Total volume of the unit cell = a x a x a = a^3.

• This results in an Atomic Packing Fraction (APF) = π/6 for the simple cubic arrangement.

• In this case, the APF computes to be 0.52, indicating that 52% of the volume within the unit cell is occupied by atoms, leaving 48% of the volume unoccupied.

• In body-centered cubic (BCC) arrangements, there are effectively two atoms contributing per unit cell.

• The corner atoms add up to 1 atom per unit cell (since they are shared with adjacent cells); additionally, there is one extra atom positioned in the center of the unit cell, not shared by any other unit cell, which adds another atom, resulting in a total of 2 atoms.

• For a BCC unit cell, the volume of one atom is again (4πr^3)/3, corresponding to the previously defined formula.

• The corner atoms can still be considered, with corner atoms per unit cell equaling 1, where each corner atom is again only partially present in the unit cell.

• The volume of one atom maintains the formula: Volume of one atom (spherical) = (4πr^3)/3.

• In BCC structures, the center atom is such that it touches the corner atoms, leading to a denser packing compared to the simple cubic structure.

• In the face-centered cubic (FCC) structure, the unit cell contains significantly more atoms: four total in three-dimensional space.

• The total number of atoms per unit cell in the FCC arrangement amounts to four, with contributions from both corner atoms and face-centered atoms.

• The volume calculations remain the same: Volume of one atom (spherical) = (4πr^3)/3.

• As in other configurations, the number of corner atoms shared per unit cell adds complexity to the packing calculations, equating to 1 for the corners.

Lattice Planes

• Lattice planes can be described as sets of parallel and equally spaced planes that exist in a space lattice. These planes play a critical role in determining the crystal's diffraction patterns and overall structural properties.

Miller Indices

• Miller indices are a shorthand notation consisting of the smallest integers represented as (h k l), which indicate the orientation and spacing of a set of parallel planes in a crystal. They are fundamental in the study of crystallography, allowing for an efficient way to describe specific crystallographic planes.

Procedure to Find Miller Indices:

• To determine the Miller indices of a crystallographic plane, one begins by determining the intercepts when the plane intersects the axes in a three-dimensional coordinate system.
• Next, the reciprocals of these intercepts are taken; this step converts the intercepts into a format conducive to finding the indices.
• The reciprocals are then simplified and reduced into the smallest possible whole numbers, yielding the Miller indices for that specific plane.

Important Features of Miller Indices:

• A plane that is parallel to any coordinate axis has an intercept considered as infinity (∞), resulting in a Miller index of zero for that specific axis, thus simplifying calculations.
• All sets of equally spaced parallel planes that have particular orientations will share the same Miller index (h k l), which helps in identifying the planes quickly.
• The Miller indices specify a set of parallel planes as opposed to designating a unique plane, allowing for the description of families of planes that display similar spacing in a crystal.
• Additionally, a plane that runs parallel to any of the coordinate axes will boast an intercept of infinity, affirming its infinite nature concerning that axis.

Bragg's Law

• According to Bragg's Law, when X-rays are reflected from various parallel planes within a crystal lattice, they interfere constructively if the path difference among the reflected rays corresponds to an integral multiple of the wavelength of the X-rays. This condition can be represented mathematically as 2dsinθ = nλ, where 'd' stands for the interplanar spacing, 'θ' is the angle of incidence, 'n' is an integer, and 'λ' is the wavelength of the incident X-rays.

Derivation

• To derive Bragg's Law, one must consider a set of parallel lattice planes that are separated by a distance denoted as 'd'. When a narrow beam of X-rays with a specific wavelength 'λ' strikes these planes at an angle of incidence 'θ', the path difference between two rays reflected from adjacent planes results in a combined path difference calculated as BC + CD = d sin θ + d sin θ = 2d sin θ. For constructive interference to occur, this path difference must equal an integer multiple of the wavelength, represented mathematically as 2d sin θ = nλ.

Interplanar Spacing

• The term interplanar spacing refers to the distance separating two adjacent parallel planes comprised of atoms that share the same Miller indices. The relationship between interplanar spacing and lattice parameters can be expressed with the equation: dhkl = a/√(h²+k²+l²), where 'a' indicates the unit cell dimension, and h, k, and l represent the Miller indices.

Bragg's X-ray Spectrometer:

• The Bragg's X-ray Spectrometer is an apparatus designed to analyze the crystal structure. It comprises a source of X-rays, a crystal positioned on a circular table, and a detector, often referred to as an ionization chamber.
• In operation, X-rays are collimated into a narrow beam that travels through slits and strikes the crystal. The arrangement of the crystal can be rotated to adjust the angle of incidence.
• As the table rotates around a vertical axis, the angle can be read accurately on a circular graduated scale, facilitating precise measurements.
• Reflected X-rays traverse through specified slits and enter the ionization chamber, where they interact with gas within the chamber, leading to the production of an electrical current between the electrodes.

Measurements and Results

• The ionization current readings provide critical information regarding the intensity of X-rays that have been reflected by the crystal under observation.
• Measurements are taken at varying glancing angles (θ), allowing scientists to create a graph that plots the intensity of ionization current against the angle of incidence.
• Certain glancing angles will elicit sudden increases in ionization current, which correspond to distinct peaks on the graph that reveal information about the crystal structures.
• Different orders of reflection can be assessed via this graph, allowing for comprehensive analysis and insight into various structural planes.
• The results obtained through Bragg's method provide critical indications of the atomic arrangements within the material, revealing the structure across different planes and layers.

Laue Method

• The Laue Method is a technique utilized for determining the internal structure of crystals. It employs a pencil-like beam of X-rays directed toward the sample of interest.
• Upon striking the crystal, the X-rays are diffracted, scattered, and recorded for analysis, enabling the examination of the crystal's structure.
• This method is only effective under the conditions prescribed by Bragg's law, which is expressed as 2dsinθ = nλ, ensuring the accuracy of the diffraction data obtained.
• Notably, the Laue method is exclusively applicable to the analysis of single crystal specimens, where the inherent order within the structure can be effectively characterized.

Powder Method

• The powder method applies to crystalline materials that have been finely ground into a powder form. This technique allows researchers to analyze tiny, randomly oriented crystals, maximizing the available diffractive data.

• Due to the random organization of the powder, it enables the observation of all potential diffraction points, providing comprehensive insight into the crystallographic structure.

• A small specimen of the powdered crystal is taken for analysis. This technique is widely used in materials science for characterizing substances where single crystals are difficult to obtain or process.

• asic structure without extra lattice points.

• Base-centered lattice features atoms positioned at the cell corners and an additional lattice point located at the center of one of the bases. This is denoted by the symbol 'C' and affects the overall geometrical properties.

• Body-centered lattice is characterized by having lattice points at each of the corners of the unit cell and one additional point at the geometric center of the cell, represented by symbol 'I'. This arrangement increases the packing efficiency within the crystal.

• Face-centered lattice contains lattice points at all corners plus additional points at the center of each face of the cube, represented by the symbol 'F'. This configuration adds complexity and stability to the structure and is associated with higher packing densities.

• Bravais, in 1948, established that there are 14 distinct types of unit cells possible under the umbrella of the seven crystal systems, collectively known as 'Bravais lattices'. This classification is fundamental to understanding crystallography.

• Out of these 14 Bravais lattices, three belong to the cubic system, two to the tetragonal system, one to the hexagonal system, four to the orthorhombic system, one to the rhombohedral structure, and two to the monoclinic system, with one being triclinic.

Coordination Number

• The coordination number is defined as the number of nearest neighboring atoms surrounding a particular atom within a crystal structure. This parameter plays an essential role in determining the physical and chemical properties of materials.
• The coordination number for a simple cubic structure is 6, meaning each atom is directly surrounded by six other atoms.
• In the body-centered cubic (BCC) structure, the coordination number is 8, indicating that each atom contacts eight neighbors.
• For face-centered cubic (FCC) arrangements, the coordination number is higher, at 12, signifying that each atom is coordinated with twelve neighboring atoms.

Atomic Packing Fraction (APF)

• The atomic packing fraction (APF) is a crucial metric defined as the ratio of the total volume of the atoms present within the unit cell to the overall volume of the unit cell itself. This value indicates how densely the atoms are packed at a microscopic level.

Number of Atoms per Unit Cell:

• The number of atoms present in or shared by a unit cell is a vital factor in determining material properties.

• In a simple cubic structure, there is one atom per unit cell due to the sharing of corner atoms.

• Thus, the total number of atoms per unit cell for simple cubic is equal to 1.

• The volume of a single atom, modeled as a sphere, is given by the formula (4πr³)/3, which is essential for calculating packing efficiencies.

• The total volume occupied by the total number of atoms within the unit cell can be expressed as 1 x (4πr³)/3.

• The atomic radius is defined as half the distance between two nearest neighbor atoms that are in direct contact.

• In the context of a simple cubic lattice, the atomic radius can be mathematically derived as a/2, where 'a' is the edge length of the unit cell.

• The total volume of the unit cell can be calculated as a x a x a = a³, forming the basis for both unit cell size and density calculations.

• Consequently, for a simple cubic structure, the atomic packing fraction (APF) is calculated as π/6, indicating the efficiency of atomic packing within the lattice.

• For a simple cubic arrangement, the APF is approximately 0.52, meaning that 52% of the unit cell volume is occupied by atoms, while 48% remains vacant.

• In body-centered cubic structures, there are effectively two atoms per unit cell.

• Ultimately, the total contribution of corner atoms per unit cell is 1, plus one additional atom precisely positioned in the center of the unit cell, which is not shared with adjacent unit cells.

• The volume associated with one atom remains consistent, calculated as (4πr³)/3, supporting the mathematical approach to density in various structures.

• It is critical to remember that in body-centered cubic structures, the center atom touches each of the corner atoms, influencing stability and bonding characteristics.

• For face-centered cubic (FCC) arrangements:

• There are a total of four atoms present within the three-dimensional spatial arrangement of the unit cell, qualifying it as a more complex unit compared to its simpler counterparts.

• The total number of atoms per unit cell in the FCC structure amounts to 4.

• The volume of one atom within this arrangement is still represented by the formula (4πr³)/3, allowing for consistent calculations across unit cell types.

• Again, the number of corner atoms shared per unit cell remains at 1, emphasizing the shared nature of atoms in crystalline structures.

Lattice Planes

• Lattice planes refer to a set of parallel and equally spaced planes that exist within a space lattice, providing a framework for analyzing the crystalline structure.

Miller Indices

• Miller indices consist of a set of smallest integers typically denoted as (h k l), which indicate the orientation and arrangement of a set of parallel planes within a crystal. They are essential for understanding and describing the directional properties of crystalline materials.

Procedure to Find Miller Indices:

• To determine the Miller indices for a given plane, you should first find the intercepts of that plane on the axes of the crystal.
• Next, take the reciprocals of these intercepts, effectively converting the intercept values into integers.
• Lastly, reduce these reciprocal values into the smallest whole numbers, thus establishing the simple integer representation commonly used in crystallography.

Important Features of Miller Indices:

• It is important to note that if a plane is parallel to any coordinate axis, it will have an intercept of infinity (∞) for that axis, resulting in a Miller index of zero for that specific axis.
• All equally spaced parallel planes that share a common orientation will be assigned the same index number (h k l), regardless of their position in the crystal.
• Thus, Miller indices define a set of parallel planes rather than being used to identify a single specific plane, which is essential for the analysis of crystallographic symmetries.
• In the case where a plane is parallel to one of the coordinate axes, its intercept will also yield an infinite value, reinforcing the fundamental nature of Miller indices in crystallography.

Bragg's Law

• Bragg's Law states that X-rays reflected from various parallel planes of a crystal can interfere constructively when the path difference between the waves reflected from successive planes is an integral multiple of the wavelength of the X-rays. Mathematically, this relationship is expressed as 2dsinθ = nλ, where 'd' is the distance between the planes, 'θ' is the angle of incidence, 'n' is an integer representing the order of reflection, and 'λ' is the wavelength of the X-rays.

Derivation

• To derive Bragg's Law, one must consider parallel lattice planes that are separated by a distance 'd'. When a narrow beam of X-rays with a specific wavelength 'λ' strikes these planes at an angle 'θ', the path difference between the reflected beams (denoted as BC and CD) can be calculated as: BC + CD = d sin θ + d sin θ = 2d sin θ. For constructive interference to occur between the reflected waves, this path difference must equate to an integer multiple of the wavelength, leading us to the equation 2d sin θ = nλ.

Interplanar Spacing

• The interplanar spacing is defined as the distance between two adjacent parallel planes of atoms that share the same Miller indices. This spacing is quantitatively expressed as dhkl=a/√(h²+k²+l²), providing a means to calculate distances in crystalline structures.

Bragg's X-ray Spectrometer:

• A Bragg's X-ray spectrometer is an advanced scientific instrument that consists of a source of X-rays, a crystalline sample placed on a circular table, and a detector, often an ionization chamber. This assembly is essential for measuring X-ray diffraction patterns, which are crucial for elucidating crystal structures.
• The X-rays produced are collimated into a narrow beam which then passes through specially designed slits and strikes the crystal sample labeled 'C'.
• As the table rotates around a vertical axis, the incident angle of the X-ray beam can be adjusted, allowing for meticulous analysis of diffraction patterns, which are subsequently read on a circular graduated scale.
• The beam of X-rays that is reflected from the crystal then passes through another set of slits before entering the ionization chamber.
• Within the chamber, the incident X-rays ionize the gas present, leading to an electric current that flows between electrodes — this current correlates with the intensity of the reflected X-rays.

Measurements and Results

• The ionization current recorded by the detector provides a direct measurement of the intensity of the reflected X-rays emanating from the crystal.
• By taking measurements displayed at various glancing angles (θ), a comprehensive graph can be constructed that correlates glancing angles with ionization current levels, allowing for the visualization of diffraction patterns.
• Particular glancing angles at which the ionization current abruptly increases correspond to specific diffraction patterns from the crystal planes; these variations indicate various orders of reflection.
• The glancing angles corresponding to different orders of reflection can be precisely measured via the resulting graph, providing critical information on the crystal's structural characteristics.
• The results obtained through Bragg's methodology facilitate a deeper understanding of the crystal structure and can be used to analyze various planes within the crystalline material.

Laue Method

• The Laue method is a technique used extensively in crystallography to determine the structure of a crystal. This process involves directing a pencil-like beam of X-rays at the sample, allowing for a high-resolution analysis of the internal arrangement of atoms.
• When X-rays interact with the crystal, they undergo diffraction and scattering, which are recorded on film or a detector, producing a diffraction pattern that can be analyzed to extract structural information.
• This method's effectiveness is contingent on adherence to Bragg's Law (2dsinθ = nλ), meaning that the arrangement of atoms is revealed only when the conditions for constructive interference are met.
• Importantly, the Laue method is specifically utilized to determine the structure of single crystal specimens, providing detailed insights into the three-dimensional atomic lattice arrangements.

Powder Method

• In the powder method, a crystalline sample is utilized in powdered form, allowing for a more versatile analysis structure by employing many tiny crystals that are randomly organized within the sample.
• This technique is advantageous as it makes all diffraction points available within the sample, significantly simplifying the extraction of structural information since the random orientation of the powder grains contributes to more complete data.
• A small specimen is easily taken from the powdered bulk, making it convenient for experimental setups and analyses.
• As a result, the powder method is widely implemented in crystallography and material science for understanding the structural properties of a vast array of materials in their crystalline forms.