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The cubic crystal system is defined by a unit cell shaped like a cube and is also known as the isometric system.
Answer: True
The cubic crystal system, also referred to as the isometric system, is fundamentally characterized by a unit cell that is geometrically a cube.
There are exactly 36 distinct cubic space groups.
Answer: True
The International Tables for Crystallography enumerate a total of 36 distinct space groups that are compatible with cubic lattice symmetry.
Which of the following best defines the cubic crystal system?
Answer: A system defined by a unit cell shaped like a cube, also known as the isometric system.
The cubic crystal system, also known as the isometric system, is defined by its unit cell being geometrically a cube.
How many distinct cubic space groups are there in total?
Answer: 36
There are a total of 36 distinct space groups that are compatible with the symmetry requirements of the cubic lattice system.
There are four distinct Bravais lattices commonly found within the cubic crystal system.
Answer: False
There are precisely three distinct Bravais lattices within the cubic crystal system: primitive cubic (cP), body-centered cubic (cI), and face-centered cubic (cF).
A primitive cubic (cP) lattice consists of lattice points only at the corners of the unit cell, resulting in a total of two lattice points per unit cell.
Answer: False
A primitive cubic (cP) lattice indeed consists of lattice points solely at the corners of the unit cell. However, due to the shared nature of these corner points, the net total of lattice points per unit cell is one, not two.
The body-centered cubic (cI) lattice includes lattice points at each corner and an additional lattice point located at the exact center of each of the six faces.
Answer: False
The body-centered cubic (cI) lattice is characterized by lattice points at each corner and one additional lattice point located precisely at the center of the unit cell, not at the center of each face.
The face-centered cubic (cF) lattice has lattice points at each corner and also at the center of each of the six faces, leading to a total of four lattice points per unit cell.
Answer: True
The face-centered cubic (cF) lattice is defined by lattice points at each corner and at the center of each of the six faces. This configuration results in a net total of four lattice points per unit cell.
The primitive cubic (cP) lattice possesses the highest atomic packing factor (APF) among the three cubic Bravais lattices.
Answer: False
The primitive cubic (cP) lattice has the lowest atomic packing factor (APF) among the cubic Bravais lattices, approximately 0.524. The face-centered cubic (cF) lattice possesses the highest APF at approximately 0.740.
Each atom in a primitive cubic (cP) lattice has six nearest neighbors.
Answer: True
In a primitive cubic (cP) lattice, each atom is positioned such that it has six nearest neighbors, forming an octahedral arrangement around it. This corresponds to a coordination number of 6.
The body-centered cubic (cI) lattice has a coordination number of 12.
Answer: False
The body-centered cubic (cI) lattice has a coordination number of 8, meaning each atom has eight nearest neighbors. The coordination number of 12 is characteristic of the face-centered cubic (cF) lattice.
The face-centered cubic (cF) lattice achieves the maximum possible atomic packing factor for spheres of equal size.
Answer: True
The face-centered cubic (cF) lattice has an atomic packing factor (APF) of approximately 0.740, which represents the densest possible packing arrangement for identical spheres in a crystal structure.
The face-centered cubic (cF) lattice is structurally unrelated to the hexagonal close-packed (hcp) system.
Answer: False
The face-centered cubic (cF) lattice and the hexagonal close-packed (hcp) system are closely related. Both represent highly efficient packing arrangements for spheres and are derived from the concept of close-packing layers, differing primarily in their stacking sequence.
Attempting to create a base-centered cubic lattice results in a structure equivalent to a simple tetragonal Bravais lattice.
Answer: True
A lattice centered on two opposite faces (e.g., base-centered) within a cubic framework, when considering the symmetry and lattice vectors, is equivalent to a simple tetragonal Bravais lattice. It does not form a distinct cubic Bravais lattice.
The primitive cubic structure is commonly observed in nature due to its high atomic packing factor.
Answer: False
The primitive cubic structure is relatively rare in solid materials because it has a low atomic packing factor (APF ≈ 0.524), indicating less efficient space utilization compared to denser structures like BCC and FCC. Denser packing is generally energetically favored.
Elements like iron, chromium, and tungsten commonly crystallize in a face-centered cubic (fcc) structure.
Answer: False
Iron, chromium, and tungsten commonly crystallize in a body-centered cubic (bcc) structure at standard conditions. Elements like aluminum, copper, and gold typically adopt the face-centered cubic (fcc) structure.
How many Bravais lattices are found within the cubic crystal system?
Answer: Three
There are three distinct Bravais lattices within the cubic crystal system: primitive cubic (cP), body-centered cubic (cI), and face-centered cubic (cF).
What is the net total of lattice points per unit cell in a primitive cubic (cP) lattice?
Answer: 1
In a primitive cubic (cP) lattice, lattice points are located only at the corners. Due to the shared nature of these points among adjacent unit cells, the net total of lattice points per unit cell is one.
Which cubic Bravais lattice has lattice points at each corner and an additional lattice point at the exact center of the unit cell?
Answer: Body-centered cubic (cI)
The body-centered cubic (cI) lattice is defined by lattice points at each corner of the unit cell, plus an additional lattice point located at the exact center of the unit cell.
How many lattice points are contained within a single unit cell of the face-centered cubic (cF) lattice?
Answer: 4
The face-centered cubic (cF) lattice contains lattice points at each corner (contributing 1/8 each) and at the center of each of the six faces (contributing 1/2 each). This results in a total of (8 * 1/8) + (6 * 1/2) = 1 + 3 = 4 lattice points per unit cell.
What is the approximate atomic packing factor (APF) for the primitive cubic (cP) lattice?
Answer: 0.524
The primitive cubic (cP) lattice has an atomic packing factor (APF) of approximately 0.524, indicating that about 52.4% of the unit cell volume is occupied by atoms. This is the lowest APF among the cubic Bravais lattices.
Which cubic Bravais lattice has the highest atomic packing factor (APF)?
Answer: Face-centered cubic (cF)
The face-centered cubic (cF) lattice has the highest atomic packing factor (APF) among the cubic Bravais lattices, approximately 0.740, representing the densest possible packing for spheres.
What is the coordination number for the face-centered cubic (cF) lattice?
Answer: 12
The face-centered cubic (cF) lattice has a coordination number of 12, meaning each atom is in direct contact with twelve nearest neighboring atoms.
The face-centered cubic (cF) lattice is closely related to which other common crystal system?
Answer: Hexagonal close-packed (hcp)
The face-centered cubic (cF) lattice and the hexagonal close-packed (hcp) system are both derived from the concept of close-packing of spheres and are closely related in terms of their packing efficiency and structural principles.
What structure results if one attempts to create a base-centered cubic lattice?
Answer: It is equivalent to a simple tetragonal Bravais lattice.
A base-centered cubic lattice, defined by lattice points at corners and the center of two opposite faces, is geometrically equivalent to a simple tetragonal Bravais lattice when considering its symmetry and lattice vectors.
Why is the primitive cubic (cP) structure considered rare in solid materials?
Answer: It is less tightly packed compared to other common structures.
The primitive cubic (cP) structure is rare because its atomic packing factor (APF ≈ 0.524) is significantly lower than that of denser structures like BCC (≈ 0.680) and FCC (≈ 0.740). Materials tend to adopt configurations that minimize energy, often favoring denser packing.
Which of these elements commonly crystallizes in a body-centered cubic (bcc) structure?
Answer: Iron
Elements such as iron, chromium, tungsten, and niobium commonly crystallize in the body-centered cubic (bcc) structure under standard conditions. Copper, aluminum, and gold typically adopt the face-centered cubic (fcc) structure.
Which of the following is NOT a Bravais lattice found in the cubic crystal system?
Answer: Diamond cubic
The diamond cubic structure is not classified as a Bravais lattice because its primitive cell contains multiple atoms. The three Bravais lattices within the cubic system are primitive cubic (cP), body-centered cubic (cI), and face-centered cubic (cF).
The diamond cubic structure is classified as one of the three main Bravais lattices within the cubic system.
Answer: False
The diamond cubic structure is not a Bravais lattice. While it possesses cubic symmetry, its primitive cell contains multiple atoms, distinguishing it from the definition of a Bravais lattice, which is based on a single lattice point per primitive cell.
The diamond cubic structure contains only one atom per primitive cell, consistent with the definition of a Bravais lattice.
Answer: False
The diamond cubic structure contains multiple atoms within its primitive cell (specifically, two atoms per primitive cell in its basis). This characteristic differentiates it from a Bravais lattice, which is defined by having only one lattice point per primitive cell.
Tungsten is an example of an element that can exhibit the A15 cubic structure.
Answer: True
The A15 structure, a specific type of cubic crystal structure, is known to be adopted by certain elements, including tungsten, under specific conditions.
The caesium chloride (CsCl) structure is identical to the body-centered cubic (bcc) structure in terms of its basis and symmetry.
Answer: False
While the CsCl structure shares a superficial resemblance to the body-centered cubic (bcc) lattice, it is not identical. The CsCl structure has a basis consisting of two different atomic species, leading to distinct symmetry properties and translational behavior compared to a pure bcc lattice.
In the caesium chloride (CsCl) structure, each ion has a coordination number of 6.
Answer: False
In the caesium chloride (CsCl) structure, each ion is surrounded by eight ions of the opposite charge, resulting in a coordination number of 8, not 6.
The rock-salt structure is typically favored when the cation is significantly smaller than the anion.
Answer: True
The rock-salt structure is generally favored when the radius ratio of the cation to the anion falls within a specific range, typically between approximately 0.414 and 0.732. This implies the cation is smaller than the anion, but not excessively so.
Atoms in the rock-salt structure exhibit octahedral coordination with six nearest neighbors.
Answer: True
In the rock-salt structure, each atom is surrounded by six nearest neighbors of the opposite type, arranged at the vertices of a regular octahedron. This signifies octahedral coordination.
The rock-salt structure can be visualized as two interpenetrating primitive cubic lattices.
Answer: False
The rock-salt structure is more accurately described as two interpenetrating face-centered cubic (FCC) lattices, or alternatively, as one FCC lattice with the other atomic species occupying all the octahedral interstitial sites.
The fluorite structure (AB2) shares the same space group (Fm3m) as the rock-salt structure (AB) but has a different ion ratio.
Answer: True
Both the fluorite (CaF2) structure and the rock-salt (NaCl) structure belong to the Fm3m space group. The fundamental difference lies in their stoichiometry and the resulting arrangement of ions, with fluorite having a 1:2 cation-to-anion ratio and rock-salt having a 1:1 ratio.
The zincblende structure is characterized by atoms having tetrahedral coordination with four nearest neighbors.
Answer: True
In the zincblende structure, each atom is tetrahedrally coordinated, meaning it is surrounded by four nearest neighbors of the opposite type, positioned at the vertices of a regular tetrahedron.
The arrangement of atoms in the zincblende structure is fundamentally different from the diamond cubic structure.
Answer: False
The atomic arrangement in the zincblende structure is fundamentally the same as that of the diamond cubic structure. The key difference is that zincblende involves two types of atoms arranged alternately on the lattice sites, whereas diamond cubic consists of only one type of atom.
Gallium arsenide (GaAs) is an example of a compound that typically adopts the rock-salt structure.
Answer: False
Gallium arsenide (GaAs) is a well-known semiconductor that typically crystallizes in the zincblende structure, not the rock-salt structure.
The Heusler structure (L21) is common for ternary compounds involving transition metals.
Answer: True
The Heusler structure, designated as L21 in the Strukturbericht system, is indeed frequently observed in ternary intermetallic compounds that incorporate transition metals.
The iron monosilicide (B20) structure is noted for its achiral nature.
Answer: False
The iron monosilicide (FeSi) structure, with the space group P213 and Strukturbericht designation B20, is notable for being a chiral structure, not achiral.
The Weaire-Phelan structure is commonly recognized in chemistry as a 'type I clathrate structure'.
Answer: True
The Weaire-Phelan structure, possessing Pm3n symmetry, is indeed frequently recognized within the field of chemistry as a specific example of a 'type I clathrate structure'.
The graphic of cesium chloride illustrates interlocking simple cubic lattices of cesium and chlorine atoms.
Answer: True
The graphic representation of the cesium chloride (CsCl) structure effectively illustrates how interlocking simple cubic lattices of cesium and chlorine atoms combine to form an overall arrangement that resembles a body-centered cubic framework.
In the rock-salt crystal structure image, each atom is shown to have eight nearest neighbors.
Answer: False
The rock-salt crystal structure depicts each atom having six nearest neighbors, arranged in an octahedral geometry. A coordination number of eight is characteristic of the CsCl structure.
The diamond cubic structure is unique because:
Answer: It contains multiple atoms within its primitive cell.
The diamond cubic structure is unique among cubic structures because its primitive cell contains multiple atoms (two atoms per primitive cell), distinguishing it from the definition of a Bravais lattice, which requires only one lattice point per primitive cell.
Which of the following elements is known to exhibit the diamond cubic structure?
Answer: Silicon
Elements such as carbon (in its diamond allotrope), silicon, germanium, and gray tin exhibit the diamond cubic crystal structure.
The A15 structure is mentioned in the source text in relation to which element?
Answer: Tungsten
The A15 structure is a specific type of cubic crystal structure, and tungsten is cited as an element that can adopt this structural form.
How does the caesium chloride (CsCl) structure differ fundamentally from a simple body-centered cubic (bcc) lattice?
Answer: CsCl has a basis of two different atomic species.
The fundamental difference between the CsCl structure and a simple bcc lattice lies in the basis: CsCl utilizes a basis of two different atomic species, whereas a true bcc lattice is based on a single atomic species. This leads to distinct symmetry properties.
What is the coordination number for ions in the caesium chloride (CsCl) structure?
Answer: 8
In the caesium chloride (CsCl) structure, each ion is surrounded by eight ions of the opposite charge, resulting in a coordination number of 8 for both cation and anion.
The rock-salt (halite) structure is typically formed when the ratio of cation radius to anion radius is approximately:
Answer: Between 0.414 and 0.732
The rock-salt structure is generally favored when the radius ratio of the cation to the anion falls within the range of approximately 0.414 to 0.732, indicating a relatively balanced size relationship between the ions.
In the rock-salt structure, what is the coordination number and geometry for each atom?
Answer: Coordination number 6, octahedral geometry
In the rock-salt structure, each atom is surrounded by six nearest neighbors of the opposite type, arranged at the vertices of a regular octahedron, signifying octahedral coordination.
Which of the following compounds commonly adopts the rock-salt structure?
Answer: Sodium chloride (NaCl)
Sodium chloride (NaCl) is the archetypal example of a compound crystallizing in the rock-salt structure. Many other alkali halides and some divalent metal oxides and sulfides also adopt this structure.
How does the fluorite (CaF2) structure differ from the rock-salt (NaCl) structure, given they share the same space group (Fm3m)?
Answer: Fluorite has a 1:2 ion ratio, while rock-salt has a 1:1 ratio.
While both fluorite (AB2) and rock-salt (AB) structures share the Fm3m space group, their fundamental difference lies in their stoichiometry: fluorite has a 1:2 cation-to-anion ratio, whereas rock-salt has a 1:1 ratio.
The zincblende structure is characterized by which type of atomic coordination?
Answer: Tetrahedral (CN=4)
The zincblende structure is characterized by tetrahedral coordination, where each atom is surrounded by four nearest neighbors of the opposite type, positioned at the vertices of a regular tetrahedron.
Which semiconductor material is mentioned as an example of a compound exhibiting the zincblende structure?
Answer: Gallium arsenide (GaAs)
Gallium arsenide (GaAs) is frequently cited as a prominent example of a semiconductor compound that adopts the zincblende crystal structure.
The Heusler structure (L21) is typically found in which type of compounds?
Answer: Ternary compounds involving transition metals
The Heusler structure (L21) is characteristically observed in ternary intermetallic compounds, particularly those that include transition metals.
The iron monosilicide (FeSi) structure is noted for possessing which property?
Answer: Chirality
The iron monosilicide (FeSi) structure, designated B20, is recognized for its chiral nature, which can lead to interesting magnetic properties such as helimagnetism.
In chemistry, the Weaire-Phelan structure is often recognized as a specific type of:
Answer: Clathrate structure
The Weaire-Phelan structure, characterized by its Pm3n symmetry, is frequently identified in chemical contexts as a 'type I clathrate structure'.
The graphic of cesium chloride illustrates how its structure resembles a body-centered cubic arrangement due to:
Answer: Interlocking simple cubic lattices of cesium and chlorine atoms.
The graphic of cesium chloride demonstrates that its structure arises from the interlocking of simple cubic lattices formed by cesium and chlorine atoms, creating an overall arrangement that visually approximates a body-centered cubic pattern.
What does the image of the rock-salt crystal structure depict regarding the coordination of atoms?
Answer: Each atom has 6 nearest neighbors in an octahedral arrangement.
The image of the rock-salt crystal structure illustrates that each atom is coordinated with six nearest neighbors of the opposite type, arranged geometrically as the vertices of a regular octahedron.
The Diploidal crystal class is characterized by enantiomorphic and tetrahedral symmetry.
Answer: False
The Diploidal crystal class is characterized by centrosymmetric and pyritohedral symmetry, not enantiomorphic and tetrahedral symmetry. Enantiomorphic symmetry is associated with chiral classes, while tetrahedral symmetry is characteristic of other isometric classes.
Pyrite is cited as an example belonging to the Diploidal crystal class.
Answer: True
Pyrite (FeS2) is a common mineral that exemplifies the Diploidal crystal class, which is known for its pyritohedral symmetry.
The Hexoctahedral crystal class possesses the lowest symmetry among the isometric crystal system classes.
Answer: False
The Hexoctahedral crystal class represents the highest symmetry among all the isometric crystal system classes, often referred to as the holohedral class.
Which cubic crystal class is described as enantiomorphic and includes examples like Ullmannite?
Answer: Tetartoidal
The Tetartoidal crystal class is characterized by enantiomorphic (chiral) symmetry and includes compounds such as Ullmannite and Sodium chlorate as examples.
Pyrite is given as an example crystal associated with which cubic crystal class?
Answer: Diploidal
Pyrite (FeS2) is a classic example of a mineral belonging to the Diploidal crystal class, which exhibits pyritohedral symmetry.
Which of the following crystal classes represents the highest symmetry within the isometric system?
Answer: Hexoctahedral
The Hexoctahedral crystal class represents the highest degree of symmetry within the isometric (cubic) crystal system, often referred to as the holohedral class.
The description of the pyrite crystal in the first image highlights its natural facets exhibiting which type of symmetry?
Answer: Cubic
The pyrite crystals depicted exhibit cubic symmetry on their natural facets, reflecting their underlying primitive cubic crystal structure.