7.1: Crystal Structure
In any sort of discussion of crystalline materials, it is useful to begin with a discussion of crystallography: the study of the formation, structure, and properties of crystals. A crystal structure is defined as the particular repeating arrangement of atoms (molecules or ions) throughout a crystal. Structure refers to the internal arrangement of particles and not the external appearance of the crystal. However, these are not entirely independent since the external appearance of a crystal is often related to the internal arrangement. For example, crystals of cubic rock salt (NaCl) are physically cubic in appearance. Only a few of the possible crystal structures are of concern with respect to simple inorganic salts and these will be discussed in detail, however, it is important to understand the nomenclature of crystallography.
Crystallography
Bravais Lattice
The Bravais lattice is the basic building block from which all crystals can be constructed. The concept originated as a topological problem of finding the number of different ways to arrange points in space where each point would have an identical “atmosphere”. That is each point would be surrounded by an identical set of points as any other point, so that all points would be indistinguishable from each other. Mathematician Auguste Bravais discovered that there were 14 different collections of the groups of points, which are known as Bravais lattices. These lattices fall into seven different "crystal systems”, as differentiated by the relationship between the angles between sides of the “unit cell” and the distance between points in the unit cell. The unit cell is the smallest group of atoms, ions or molecules that, when repeated at regular intervals in three dimensions, will produce the lattice of a crystal system. The “lattice parameter” is the length between two points on the corners of a unit cell. Each of the various lattice parameters are designated by the letters a , b , and c . If two sides are equal, such as in a tetragonal lattice, then the lengths of the two lattice parameters are designated a and c , with b omitted. The angles are designated by the Greek letters α, β, and γsize 12{γ} {}, such that an angle with a specific Greek letter is not subtended by the axis with its Roman equivalent. For example, α is the included angle between the b and c axis.
Table \(\PageIndex{1}\) shows the various crystal systems, while Figure \(\PageIndex{1}\) shows the 14 Bravais lattices. It is important to distinguish the characteristics of each of the individual systems. An example of a material that takes on each of the Bravais lattices is shown in Table \(\PageIndex{2}\).
| System | Axial Lengths and Angles | Unit Cell Geometry |
| cubic | a=b=c, α = β = γ = 90° | |
| tetragonal | a = b ≠ c, α = β = γ= 90° | |
| orthorhombic | a ≠ b ≠ c, α = β = γ= 90° | |
| rhombohedral | a = b = c, α = β = γ ≠ 90° | |
| hexagonal | a = b ≠ c, α = β = 90°, γ = 120° | |
| monoclinic | a ≠ b ≠ c, α = γ = 90°, β ≠ 90° | |
| triclinic | a ≠ b ≠ c, α ≠ β ≠ γ |
| Crystal System | Example |
| triclinic | K 2 S 2 O 8 |
| monoclinic | As 4 S 4 , KNO 2 |
| rhombohedral | Hg, Sb |
| hexagonal | Zn, Co, NiAs |
| orthorhombic | Ga, Fe 3 C |
| tetragonal | In, TiO 2 |
| cubic | Au, Si, NaCl |
The cubic lattice is the most symmetrical of the systems. All the angles are equal to 90°, and all the sides are of the same length ( a = b = c ). Only the length of one of the sides ( a ) is required to describe this system completely. In addition to simple cubic, the cubic lattice also includes body-centered cubic and face-centered cubic (Figure \(\PageIndex{1}\). Body-centered cubic results from the presence of an atom (or ion) in the center of a cube, in addition to the atoms (ions) positioned at the vertices of the cube. In a similar manner, a face-centered cubic requires, in addition to the atoms (ions) positioned at the vertices of the cube, the presence of atoms (ions) in the center of each of the cubes face.
The tetragonal lattice has all of its angles equal to 90°, and has two out of the three sides of equal length ( a = b ). The system also includes body-centered tetragonal (Figure \(\PageIndex{1}\).
In an orthorhombic lattice all of the angles are equal to 90°, while all of its sides are of unequal length. The system needs only to be described by three lattice parameters. This system also includes body-centered orthorhombic, base-centered orthorhombic, and face-centered orthorhombic (Figure \(\PageIndex{1}\).
A base-centered lattice has, in addition to the atoms (ions) positioned at the vertices of the orthorhombic lattice, atoms (ions) positioned on just two opposing faces.
The rhombohedral lattice is also known as trigonal, and has no angles equal to 90°, but all sides are of equal length ( a = b = c ), thus requiring only by one lattice parameter, and all three angles are equal (α = β = γ).
A hexagonal crystal structure has two angles equal to 90°, with the other angle ( γsize 12{γ} {}) equal to 120°. For this to happen, the two sides surrounding the 120° angle must be equal ( a = b ), while the third side ( c ) is at 90° to the other sides and can be of any length.
The monoclinic lattice has no sides of equal length, but two of the angles are equal to 90°, with the other angle (usually defined as β) being something other than 90°. It is a tilted parallelogram prism with rectangular bases. This system also includes base-centered monoclinic (Figure \(\PageIndex{2}\)).
In the triclinic lattice none of the sides of the unit cell are equal, and none of the angles within the unit cell are equal to 90°. The triclinic lattice is chosen such that all the internal angles are either acute or obtuse. This crystal system has the lowest symmetry and must be described by 3 lattice parameters (a, b, and c) and the 3 angles (α, β, and γ).
Atom Positions, Crystal Directions and Miller Indices
Atom Positions and Crystal Axes
The structure of a crystal is defined with respect to a unit cell. As the entire crystal consists of repeating unit cells, this definition is sufficient to represent the entire crystal. Within the unit cell, the atomic arrangement is expressed using coordinates. There are two systems of coordinates commonly in use, which can cause some confusion. Both use a corner of the unit cell as their origin. The first, less-commonly seen system is that of Cartesian or orthogonal coordinates (X, Y, Z). These usually have the units of Angstroms and relate to the distance in each direction between the origin of the cell and the atom. These coordinates may be manipulated in the same fashion are used with two- or three-dimensional graphs. It is very simple, therefore, to calculate inter-atomic distances and angles given the Cartesian coordinates of the atoms. Unfortunately, the repeating nature of a crystal cannot be expressed easily using such coordinates. For example, consider a cubic cell of dimension 3.52 Å. Pretend that this cell contains an atom that has the coordinates (1.5, 2.1, 2.4). That is, the atom is 1.5 Å away from the origin in the x direction (which coincides with the a cell axis), 2.1 Å in the y (which coincides with the b cell axis) and 2.4 Å in the z (which coincides with the c cell axis). There will be an equivalent atom in the next unit cell along the x-direction, which will have the coordinates (1.5 + 3.52, 2.1, 2.4) or (5.02, 2.1, 2.4). This was a rather simple calculation, as the cell has very high symmetry and so the cell axes, a, b and c, coincide with the Cartesian axes, X, Y and Z. However, consider lower symmetry cells such as triclinic or monoclinic in which the cell axes are not mutually orthogonal. In such cases, expressing the repeating nature of the crystal is much more difficult to accomplish.
Accordingly, atomic coordinates are usually expressed in terms of fractional coordinates, (x, y, z). This coordinate system is coincident with the cell axes (a, b, c) and relates to the position of the atom in terms of the fraction along each axis. Consider the atom in the cubic cell discussion above. The atom was 1.5 Å in the a direction away from the origin. As the a axis is 3.52 Å long, the atom is (1.5/3.52) or 0.43 of the axis away from the origin. Similarly, it is (2.1/3.52) or 0.60 of the b axis and (2.4/3.5) or 0.68 of the c axis. The fractional coordinates of this atom are, therefore, (0.43, 0.60, 0.68). The coordinates of the equivalent atom in the next cell over in the a direction, however, are easily calculated as this atom is simply 1 unit cell away in a. Thus, all one has to do is add 1 to the x coordinate: (1.43, 0.60, 0.68). Such transformations can be performed regardless of the shape of the unit cell. Fractional coordinates, therefore, are used to retain and manipulate crystal information.
Crystal Directions
The designation of the individual vectors within any given crystal lattice is accomplished by the use of whole number multipliers of the lattice parameter of the point at which the vector exits the unit cell. The vector is indicated by the notation [ hkl ], where h , k , and l are reciprocals of the point at which the vector exits the unit cell. The origination of all vectors is assumed defined as [000]. For example, the direction along the a -axis according to this scheme would be [100] because this has a component only in the a -direction and no component along either the b or c axial direction. A vector diagonally along the face defined by the a and b axis would be [110], while going from one corner of the unit cell to the opposite corner would be in the [111] direction. Figure \(\PageIndex{2}\) shows some examples of the various directions in the unit cell. The crystal direction notation is made up of the lowest combination of integers and represents unit distances rather than actual distances. A [222] direction is identical to a [111], so [111] is used. Fractions are not used. For example, a vector that intercepts the center of the top face of the unit cell has the coordinates x = 1/2, y = 1/2, z = 1. All have to be inversed to convert to the lowest combination of integers (whole numbers); i.e., [221] in Figure \(\PageIndex{2}\). Finally, all parallel vectors have the same crystal direction, e.g., the four vertical edges of the cell shown in Figure \(\PageIndex{2}\) all have the crystal direction [ hkl ] = [001].
Crystal directions may be grouped in families. To avoid confusion there exists a convention in the choice of brackets surrounding the three numbers to differentiate a crystal direction from a family of direction. For a direction, square brackets [ hkl ] are used to indicate an individual direction. Angle brackets < hkl > indicate a family of directions. A family of directions includes any directions that are equivalent in length and types of atoms encountered. For example, in a cubic lattice, the [100], [010], and [001] directions all belong to the <100> family of planes because they are equivalent. If the cubic lattice were rotated 90°, the a , b , and c directions would remain indistinguishable, and there would be no way of telling on which crystallographic positions the atoms are situated, so the family of directions is the same. In a hexagonal crystal, however, this is not the case, so the [100] and [010] would both be <100> directions, but the [001] direction would be distinct. Finally, negative directions are identified with a bar over the negative number instead of a minus sign.
Crystal Planes
Planes in a crystal can be specified using a notation called Miller indices. The Miller index is indicated by the notation [ hkl ] where h , k , and l are reciprocals of the plane with the x , y , and z axes. To obtain the Miller indices of a given plane requires the following steps:
- The plane in question is placed on a unit cell.
- Its intercepts with each of the crystal axes are then found.
- The reciprocal of the intercepts are taken.
- These are multiplied by a scalar to insure that is in the simple ratio of whole numbers.
For example, the face of a lattice that does not intersect the y or z axis would be (100), while a plane along the body diagonal would be the (111) plane. An illustration of this along with the (111) and (110) planes is given in Figure \(\PageIndex{3}\).
As with crystal directions, Miller indices directions may be grouped in families. Individual Miller indices are given in parentheses (hkl), while braces {hkl} are placed around the indices of a family of planes. For example, (001), (100), and (010) are all in the {100} family of planes, for a cubic lattice.
Description of Crystal Structures
Crystal structures may be described in a number of ways. The most common manner is to refer to the size and shape of the unit cell and the positions of the atoms (or ions) within the cell. However, this information is sometimes insufficient to allow for an understanding of the true structure in three dimensions. Consideration of several unit cells, the arrangement of the atoms with respect to each other, the number of other atoms they in contact with, and the distances to neighboring atoms, often will provide a better understanding. A number of methods are available to describe extended solid-state structures. The most applicable with regard to elemental and compound semiconductor, metals and the majority of insulators is the close packing approach.
Close Packed Structures: Hexagonal Close Packing and Cubic Close Packing
Many crystal structures can be described using the concept of close packing. This concept requires that the atoms (ions) are arranged so as to have the maximum density. In order to understand close packing in three dimensions, the most efficient way for equal sized spheres to be packed in two dimensions must be considered.
The most efficient way for equal sized spheres to be packed in two dimensions is shown in Figure \(\PageIndex{4}\), in which it can be seen that each sphere (the dark gray shaded sphere) is surrounded by, and is in contact with, six other spheres (the light gray spheres in Figure \(\PageIndex{4}\). It should be noted that contact with six other spheres the maximum possible is the spheres are the same size, although lower density packing is possible. Close packed layers are formed by repetition to an infinite sheet. Within these close packed layers, three close packed rows are present, shown by the dashed lines in Figure \(\PageIndex{4}\).
The most efficient way for equal sized spheres to be packed in three dimensions is to stack close packed layers on top of each other to give a close packed structure. There are two simple ways in which this can be done, resulting in either a hexagonal or cubic close packed structures.
Hexagonal Close Packed
If two close packed layers A and B are placed in contact with each other so as to maximize the density, then the spheres of layer B will rest in the hollow (vacancy) between three of the spheres in layer A. This is demonstrated in Figure \(\PageIndex{5}\). Atoms in the second layer, B (shaded light gray), may occupy one of two possible positions (Figure \(\PageIndex{5}\) a or b) but not both together or a mixture of each. If a third layer is placed on top of layer B such that it exactly covers layer A, subsequent placement of layers will result in the following sequence ...ABABAB.... This is known as hexagonal close packing or hcp.
The hexagonal close packed cell is a derivative of the hexagonal Bravais lattice system (Figure \(\PageIndex{6}\) with the addition of an atom inside the unit cell at the coordinates (1/3,2/3,1/2). The basal plane of the unit cell coincides with the close packed layers (Figure \(\PageIndex{6}\). In other words the close packed layer makes-up the {001} family of crystal planes.
The “packing fraction” in a hexagonal close packed cell is 74.05%; that is 74.05% of the total volume is occupied. The packing fraction or density is derived by assuming that each atom is a hard sphere in contact with its nearest neighbors. Determination of the packing fraction is accomplished by calculating the number of whole spheres per unit cell (2 in hcp), the volume occupied by these spheres, and a comparison with the total volume of a unit cell. The number gives an idea of how “open” or filled a structure is. By comparison, the packing fraction for body-centered cubic (Figure \(\PageIndex{5}\)) is 68% and for diamond cubic (an important semiconductor structure to be described later) is it 34%.
Cubic Close Packed: Face-centered Cubic
In a similar manner to the generation of the hexagonal close packed structure, two close packed layers are stacked (Figure \(\PageIndex{7}\) however, the third layer (C) is placed such that it does not exactly cover layer A, while sitting in a set of troughs in layer B (Figure \(\PageIndex{7}\)), then upon repetition the packing sequence will be ...ABCABCABC.... This is known as cubic close packing or ccp.
The unit cell of cubic close packed structure is actually that of a face-centered cubic (fcc) Bravais lattice. In the fcc lattice the close packed layers constitute the {111} planes. As with the hcp lattice packing fraction in a cubic close packed (fcc) cell is 74.05%. Since face centered cubic or fcc is more commonly used in preference to cubic close packed (ccp) in describing the structures, the former will be used throughout this text.
Coordination Number
The coordination number of an atom or ion within an extended structure is defined as the number of nearest neighbor atoms (ions of opposite charge) that are in contact with it. A slightly different definition is often used for atoms within individual molecules: the number of donor atoms associated with the central atom or ion. However, this distinction is rather artificial, and both can be employed.
The coordination numbers for metal atoms in a molecule or complex are commonly 4, 5, and 6, but all values from 2 to 9 are known and a few examples of higher coordination numbers have been reported. In contrast, common coordination numbers in the solid state are 3, 4, 6, 8, and 12. For example, the atom in the center of body-centered cubic lattice has a coordination number of 8, because it touches the eight atoms at the corners of the unit cell, while an atom in a simple cubic structure would have a coordination number of 6. In both fcc and hcp lattices each of the atoms have a coordination number of 12.
Octahedral and Tetrahedral Vacancies
As was mentioned above, the packing fraction in both fcc and hcp cells is 74.05%, leaving 25.95% of the volume unfilled. The unfilled lattice sites (interstices) between the atoms in a cell are called interstitial sites or vacancies. The shape and relative size of these sites is important in controlling the position of additional atoms. In both fcc and hcp cells most of the space within these atoms lies within two different sites known as octahedral sites and tetrahedral sites. The difference between the two lies in their “coordination number”, or the number of atoms surrounding each site. Tetrahedral sites (vacancies) are surrounded by four atoms arranged at the corners of a tetrahedron. Similarly, octahedral sites are surrounded by six atoms which make-up the apices of an octahedron. For a given close packed lattice an octahedral vacancy will be larger than a tetrahedral vacancy.
Within a face centered cubic lattice, the eight tetrahedral sites are positioned within the cell, at the general fractional coordinate of (n/4,n/4,n/4) where n = 1 or 3, e.g., (1/4,1/4,1/4), (1/4,1/4,3/4), etc. The octahedral sites are located at the center of the unit cell (1/2,1/2,1/2), as well as at each of the edges of the cell, e.g., (1/2,0,0). In the hexagonal close packed system, the tetrahedral sites are at (0,0,3/8) and (1/3,2/3,7/8), and the octahedral sites are at (1/3,1/3,1/4) and all symmetry equivalent positions.
Important Structure Types
The majority of crystalline materials do not have a structure that fits into the one atom per site simple Bravais lattice. A number of other important crystal structures are found, however, only a few of these crystal structures are those of which occur for the elemental and compound semiconductors and the majority of these are derived from fcc or hcp lattices. Each structural type is generally defined by an archetype, a material (often a naturally occurring mineral) which has the structure in question and to which all the similar materials are related. With regard to commonly used elemental and compound semiconductors the important structures are diamond, zinc blende, Wurtzite, and to a lesser extent chalcopyrite. However, rock salt, β-tin, cinnabar and cesium chloride are observed as high pressure or high temperature phases and are therefore also discussed. The following provides a summary of these structures. Details of the full range of solid-state structures are given elsewhere.
Diamond Cubic
The diamond cubic structure consists of two interpenetrating face-centered cubic lattices, with one offset 1/4 of a cube along the cube diagonal. It may also be described as face centered cubic lattice in which half of the tetrahedral sites are filled while all the octahedral sites remain vacant. The diamond cubic unit cell is shown in Figure \(\PageIndex{8}\). Each of the atoms (e.g., C) is four coordinate, and the shortest interatomic distance (C-C) may be determined from the unit cell parameter (a).
\[ C-C\ =\ a \frac{\sqrt{3} }{4} \approx \ 0.422 a \label{1} \]
Zinc Blende
This is a binary phase (ME) and is named after its archetype, a common mineral form of zinc sulfide (ZnS). As with the diamond lattice, zinc blende consists of the two interpenetrating fcc lattices. However, in zinc blende one lattice consists of one of the types of atoms (Zn in ZnS), and the other lattice is of the second type of atom (S in ZnS). It may also be described as face centered cubic lattice of S atoms in which half of the tetrahedral sites are filled with Zn atoms. All the atoms in a zinc blende structure are 4-coordinate. The zinc blende unit cell is shown in Figure \(\PageIndex{9}\). A number of inter-atomic distances may be calculated for any material with a zinc blende unit cell using the lattice parameter (a).
\[ Zn-S\ =\ a \frac{\sqrt{3} }{4} \approx \ 0.422 a \label{2} \]
\[ Zn-Zn \ =\ S-S\ = \frac{a}{\sqrt{2}} \approx 0.707\ a \label{3} \]
Chalcopyrite
The mineral chalcopyrite CuFeS2 is the archetype of this structure. The structure is tetragonal (a = b ≠ c, α = β = γ = 90°, and is essentially a superlattice on that of zinc blende. Thus, is easiest to imagine that the chalcopyrite lattice is made-up of a lattice of sulfur atoms in which the tetrahedral sites are filled in layers, ...FeCuCuFe..., etc. (Figure \(\PageIndex{10}\). In such an idealized structure c = 2a, however, this is not true of all materials with chalcopyrite structures.
Rock Salt
As its name implies the archetypal rock salt structure is NaCl (table salt). In common with the zinc blende structure, rock salt consists of two interpenetrating face-centered cubic lattices. However, the second lattice is offset 1/2a along the unit cell axis. It may also be described as face centered cubic lattice in which all of the octahedral sites are filled, while all the tetrahedral sites remain vacant, and thus each of the atoms in the rock salt structure are 6-coordinate. The rock salt unit cell is shown in Figure \(\PageIndex{11}\). A number of inter-atomic distances may be calculated for any material with a rock salt structure using the lattice parameter ( a ).
\[ Na-Cl\ =\ \frac{a}{2} \approx 0.5 a \label{4} \]
\[ Na-Na \ =\ Cl-Cl \ =\ \frac{a}{\sqrt{2}} \approx 0.707\ a \label{5} \]
Cinnabar
Cinnabar, named after the archetype mercury sulfide, HgS, is a distorted rock salt structure in which the resulting cell is rhombohedral (trigonal) with each atom having a coordination number of six.
Wurtzite
This is a hexagonal form of the zinc sulfide. It is identical in the number of and types of atoms, but it is built from two interpenetrating hcp lattices as opposed to the fcc lattices in zinc blende. As with zinc blende all the atoms in a wurtzite structure are 4-coordinate. The wurtzite unit cell is shown in Figure \(\PageIndex{12}\) . A number of inter atomic distances may be calculated for any material with a wurtzite cell using the lattice parameter (a).
\[ Zn-S\ =\ a \sqrt{3/8 } \ =\ 0.612\ a\ = \frac{3 c}{8} \ =\ 0.375\ c \label{6} \]
\[ Zn- Zn \ =\ S-S\ =\ a\ =\ 1.632\ c \label{7} \]
However, it should be noted that these formulae do not necessarily apply when the ratio a/c is different from the ideal value of 1.632.
Cesium Chloride
The cesium chloride structure is found in materials with large cations and relatively small anions. It has a simple (primitive) cubic cell (Figure \(\PageIndex{13}\)) with a chloride ion at the corners of the cube and the cesium ion at the body center. The coordination numbers of both Cs+ and Cl-, with the inner atomic distances determined from the cell lattice constant (a).
\[ Cs-Cl\ =\ \frac{a \sqrt{3} }{2} \approx 0.866a \label{8} \]
\[Cs-Cs \ =\ Cl-Cl\ = a \label{9} \]
β-Tin
The room temperature allotrope of tin is β-tin or white tin. It has a tetragonal structure, in which each tin atom has four nearest neighbors (Sn-Sn = 3.016 Å) arranged in a very flattened tetrahedron, and two next nearest neighbors (Sn-Sn = 3.175 Å). The overall structure of β-tin consists of fused hexagons, each being linked to its neighbor via a four-membered Sn 4 ring.
Defects in Crystalline Solids
Up to this point we have only been concerned with ideal structures for crystalline solids in which each atom occupies a designated point in the crystal lattice. Unfortunately, defects ordinarily exist in equilibrium between the crystal lattice and its environment. These defects are of two general types: point defects and extended defects. As their names imply, point defects are associated with a single crystal lattice site, while extended defects occur over a greater range.
Point Defects: "Too Many or Too Few" or "Just Plain Wrong"
Point defects have a significant effect on the properties of a semiconductor, so it is important to understand the classes of point defects and the characteristics of each type. Figure \(\PageIndex{13}\) summarizes various classes of native point defects, however, they may be divided into two general classes; defects with the wrong number of atoms (deficiency or surplus) and defects where the identity of the atoms is incorrect.
Interstitial Impurity
An interstitial impurity occurs when an extra atom is positioned in a lattice site that should be vacant in an ideal structure (Figure \(\PageIndex{13}\) b).Since all the adjacent lattice sites are filled the additional atom will have to squeeze itself into the interstitial site, resulting in distortion of the lattice and alteration in the local electronic behavior of the structure. Small atoms, such as carbon, will prefer to occupy these interstitial sites. Interstitial impurities readily diffuse through the lattice via interstitial diffusion, which can result in a change of the properties of a material as a function of time. Oxygen impurities in silicon generally are located as interstitials.
Vacancies
The converse of an interstitial impurity is when there are not enough atoms in a particular area of the lattice. These are called vacancies. Vacancies exist in any material above absolute zero and increase in concentration with temperature. In the case of compound semiconductors, vacancies can be either cation vacancies (Figure \(\PageIndex{13}\) c) or anion vacancies (Figure \(\PageIndex{13}\) d), depending on what type of atom are “missing”.
Substitution
Substitution of various atoms into the normal lattice structure is common, and used to change the electronic properties of both compound and elemental semiconductors. Any impurity element that is incorporated during crystal growth can occupy a lattice site. Depending on the impurity, substitution defects can greatly distort the lattice and/or alter the electronic structure. In general, cations will try to occupy cation lattice sites (Figure \(\PageIndex{13}\) e), and anion will occupy the anion site (Figure \(\PageIndex{13}\) f). For example, a zinc impurity in GaAs will occupy a gallium site, if possible, while a sulfur, selenium and tellurium atoms would all try to substitute for an arsenic. Some impurities will occupy either site indiscriminately, e.g., Si and Sn occupy both Ga and As sites in GaAs.
Antisite Defects
Antisite defects are a particular form of substitution defect, and are unique to compound semiconductors. An antisite defect occurs when a cation is misplaced on an anion lattice site or vice versa ( Figure \(\PageIndex{13}\) g and h).Dependant on the arrangement these are designated as either AB antisite defects or BA antisite defects. For example, if an arsenic atom is on a gallium lattice site the defect would be an AsGa defect. Antisite defects involve fitting into a lattice site atoms of a different size than the rest of the lattice, and therefore this often results in a localized distortion of the lattice. In addition, cations and anions will have a different number of electrons in their valence shells, so this substitution will alter the local electron concentration and the electronic properties of this area of the semiconductor.
Extended Defects: Dislocations in a Crystal Lattice
Extended defects may be created either during crystal growth or as a consequence of stress in the crystal lattice. The plastic deformation of crystalline solids does not occur such that all bonds along a plane are broken and reformed simultaneously. Instead, the deformation occurs through a dislocation in the crystal lattice. Figure shows a schematic representation of a dislocation in a crystal lattice. Two features of this type of dislocation are the presence of an extra crystal plane, and a large void at the dislocation core. Impurities tend to segregate to the dislocation core in order to relieve strain from their presence.
Epitaxy
Epitaxy, is a transliteration of two Greek words epi, meaning "upon", and taxis, meaning "ordered". With respect to crystal growth it applies to the process of growing thin crystalline layers on a crystal substrate. In epitaxial growth, there is a precise crystal orientation of the film in relation to the substrate. The growth of epitaxial films can be done by a number of methods including molecular beam epitaxy, atomic layer epitaxy, and chemical vapor deposition, all of which will be described later.
Epitaxy of the same material, such as a gallium arsenide film on a gallium arsenide substrate, is called homoepitaxy, while epitaxy where the film and substrate material are different is called heteroepitaxy. Clearly, in homoepitaxy, the substrate and film will have the identical structure, however, in heteroepitaxy, it is important to employ where possible a substrate with the same structure and similar lattice parameters. For example, zinc selenide (zinc blende, a = 5.668 Å) is readily grown on gallium arsenide (zinc blende, a = 5.653 Å). Alternatively, epitaxial crystal growth can occur where there exists a simple relationship between the structures of the substrate and crystal layer, such as is observed between Al2O3 (100) on Si (100). Whichever route is chosen a close match in the lattice parameters is required, otherwise, the strains induced by the lattice mismatch results in distortion of the film and formation of dislocations. If the mismatch is significant epitaxial growth is not energetically favorable, causing a textured film or polycrystalline untextured film to be grown. As a general rule of thumb, epitaxy can be achieved if the lattice parameters of the two materials are within about 5% of each other. For good quality epitaxy, this should be less than 1%. The larger the mismatch, the larger the strain in the film. As the film gets thicker and thicker, it will try to relieve the strain in the film, which could include the loss of epitaxy of the growth of dislocations. It is important to note that the <100> directions of a film must be parallel to the <100> direction of the substrate. In some cases, such as Fe on MgO, the [111] direction is parallel to the substrate [100]. The epitaxial relationship is specified by giving first the plane in the film that is parallel to the substrate [100].


9.1: Why It Matters- Microsoft PowerPoint Basic Skills
Why learn to complete basic skills in Microsoft PowerPoint?
In previous modules you learned how to analyze data and create reports and graphs. Often after completing any of these steps, you may be asked to share your work with others. While it is possible to simply send along a copy of the report or graph, it is also quite common to be asked to present the data to a group of people in the form of a presentation.
Microsoft also has a powerful presentation application, known as PowerPoint. In this learning module, you will learn the basics of using PowerPoint, focused on editing a pre-existing presentation and changing style elements. Before we begin, here is a quick tutorial on some key information about PowerPoint and some handy tips to remember before we start using this program.
- When you are actually working in PowerPoint you are working on a presentation. Individual pages within a presentation are called slides. You can think of slides similarly to a page in Word: each is a standalone piece in a presentation.
- Just like in Word, PowerPoint has a ribbon area at the top of the window with different tabs. Refer to the first Microsoft Word module for a refresher on these terms if you need one.
Contributors and Attributions
- Why It Matters: Microsoft PowerPoint Basic Skills. Authored by : Shelli Carter. Provided by : Lumen Learning. License : CC BY: Attribution


11.1: Careful Use of Language in Mathematics
This section might seem like a bit of a sidetrack from the idea of problem solving, but in fact it is not. Mathematics is a social endeavor. We do not just solve problems and then put them aside. Problem solving has (at least) three components:
- Solving the problem. This involves a lot of scratch paper and careful thinking.
- Convincing yourself that your solution is complete and correct. This involves a lot of self-check and asking yourself questions.
- Convincing someone else that your solution is complete and correct. This usually involves writing the problem up carefully or explaining your work in a presentation.
If you are not able to do that last step, then you have not really solved the problem. We will talk more about how to write up a solution soon. Before we do that, we have to think about how mathematicians use language (which is, it turns out, a bit different from how language is used in the rest of life).
Mathematical Statements
A mathematical statement is a complete sentence that is either true or false, but not both at once.
So a “statement” in mathematics cannot be a question, a command, or a matter of opinion. It is a complete, grammatically correct sentence (with a subject, verb, and usually an object). It is important that the statement is either true or false, though you may not know which! (Part of the work of a mathematician is figuring out which sentences are true and which are false.)
For each English sentence below, decide if it is a mathematical statement or not. If it is, is the statement true or false (or are you unsure)? If it is not a mathematical statement, in what way does it fail?
- Blue is the prettiest color.
- 60 is an even number.
- Is your dog friendly?
- Honolulu is the capital of Hawaii.
- This sentence is false.
- All roses are red.
- UH Manoa is the best college in the world.
- 1/2 = 2/4.
- Go to bed.
- There are a total of 204 squares on an 8 × 8 chess board.
Now write three mathematical statements and three English sentences that fail to be mathematical statements.
Notice that “1/2 = 2/4” is a perfectly good mathematical statement. It does not look like an English sentence, but read it out loud. The subject is “1/2.” The verb is “equals.” And the object is “2/4.” This is a very good test when you write mathematics: try to read it out loud. Even the equations should read naturally, like English sentences.
Statement (5) is different from the others. It is called a paradox : a statement that is self-contradictory. If it is true, then we conclude that it is false. (Why?) If it is false, then we conclude that it is true. (Why?) Paradoxes are no good as mathematical statements, because it cannot be true and it cannot be false.
And / or
Consider this sentence:
After work, I will go to the beach, or I will do my grocery shopping.
In everyday English, that probably means that if I go to the beach, I will not go shopping. I will do one or the other, but not both activities. This is called an “exclusive or.”
We can usually tell from context whether a speaker means “either one or the other or both,” or whether he means “either one or the other but not both.” (Some people use the awkward phrase “and/or” to describe the first option.)
Remember that in mathematical communication, though, we have to be very precise. We cannot rely on context or assumptions about what is implied or understood.
In mathematics, the word “or” always means “one or the other or both.”
The word “and” always means “both are true.”
For each sentence below:
- Decide if the choice x = 3 makes the statement true or false.
- Choose a different value of that makes the statement true (or say why that is not possible).
- Choose a different value of that makes the statement false (or say why that is not possible).
- x is odd or x is even.
- x is odd and x is even.
- x is prime or x is odd.
- x > 5 or x < 5.
- x > 5 and x < 5.
- x + 1 = 7 or x – 1 = 7.
- x ·1 = x or x ·0 = x.
- x ·1 = x and x ·0 = x.
- x ·1 = x or x ·0 = 0.

- What would convince you beyond any doubt that the sentence is true? How could you convince someone else that the sentence is true?
- What would convince you beyond any doubt that the sentence is false? How could you convince someone else that the sentence is false?
Suppose you were given a different sentence: “There is a $100 bill in this envelope.”
- What would convince you beyond any doubt that the sentence is true? How could you convince someone else that the sentence is true?
- What would convince you beyond any doubt that the sentence is false? How could you convince someone else that the sentence is false?
What is the difference between the two sentences? How does that difference affect your method to decide if the statement is true or false?
Some mathematical statements have this form:
- “Every time...”
- “For all numbers. . . ”
- “For every choice. . . ”
- “It’s always true that. . . ”
These are universal statements. Such statements claim that something is always true, no matter what.
- To prove a universal statement is false, you must find an example where it fails. This is called a counterexample to the statement.
- To prove a universal statement is true, you must either check every single case, or you must find a logical reason why it would be true. (Sometimes the first option is impossible, because there might be infinitely many cases to check. You would never finish!)
Some mathematical statements have this form:
- “Sometimes...”
- “There is some number. . . ”
- “For some choice. . . ”
- “At least once...”
These are existential statements. Such statements claim there is some example where the statement is true, but it may not always be true.
- To prove an existential statement is true, you may just find the example where it works.
- To prove an existential statement is false, you must either show it fails in every single case, or you must find a logical reason why it cannot be true. (Sometimes the first option is impossible!)
For each statement below, do the following:
- Decide if it is a universal statement or an existential statement. (This can be tricky because in some statements the quantifier is “hidden” in the meaning of the words.)
- Decide if the statement is true or false, and do your best to justify your decision.
- Every odd number is prime.
- Every prime number is odd.
- For all positive numbers \(x, x^{3} > x\).
- There is some number \(x\) such that \(x^{3} = x\).
- The points (1, 1), (2, 1), and (3, 0) all lie on the same line.
- Addition (of real numbers) is commutative.
- Division (of real numbers) is commutative.
Look back over your work. you will probably find that some of your arguments are sound and convincing while others are less so. In some cases you may “know” the answer but be unable to justify it. That is okay for now! Divide your answers into four categories:
- I am confident that the justification I gave is good.
- I am not confident in the justification I gave.
- I am confident that the justification I gave is not good, or I could not give a justification.
- I could not decide if the statement was true or false.
Conditional Statements
You have a deck of cards where each card has a letter on one side and a number on the other side. Your friend claims: “If a card has a vowel on one side, then it has an even number on the other side.”
These cards are on a table.
Which cards must you flip over to be certain that your friend is telling the truth?
After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which cards you must check? Try to come to agreement on an answer you both believe.
Here is another very similar problem, yet people seem to have an easier time solving this one:
You are in charge of a party where there are young people. Some are drinking alcohol, others soft drinks. Some are old enough to drink alcohol legally, others are under age. You are responsible for ensuring that the drinking laws are not broken, so you have asked each person to put his or her photo ID on the table. At one table, there are four young people:
- One person has a can of beer, another has a bottle of Coke, but their IDs happen to be face down so you cannot see their ages.
- You can, however, see the IDs of the other two people. One is under the drinking age, the other is above it. They both have fizzy clear drinks in glasses, and you are not sure if they are drinking soda water or gin and tonic.
Which IDs and/or drinks do you need to check to make sure that no one is breaking the law?
After you have thought about the problem on your own for a while, discuss your ideas with a partner. Do you agree on which cards you must check? Compare these two problems. Which question is easier and why?
A conditional statement can be written in the form
If some statement then some statement.
Where the first statement is the hypothesis and the second statement is the conclusion .
These are each conditional statements, though they are not all stated in “if/then” form. Identify the hypothesis of each statement. (You may want to rewrite the sentence as an equivalent “if/then” statement.)
-
If the tomatoes are red, then they are ready to eat.
The tomatoes are red. / The tomatoes are ready to eat. -
An integer
n
is even if it is a multiple of 2.
n is even. / n is a multiple of 2. -
If
n
is odd, then
n
is prime.
n is odd. / n is prime. -
The team wins when JJ plays.
The team wins. / JJ plays.
Remember that a mathematical statement must have a definite truth value. It is either true or false, with no gray area (even though we may not be sure which is the case). How can you tell if a conditional statement is true or false? Surely, it depends on whether the hypothesis and the conclusion are true or false. But how, exactly, can you decide?
The key is to think of a conditional statement like a promise, and ask yourself: under what condition(s) will I have broken my promise?
Here is a conditional statement:
If I win the lottery, then I’ll give each of my students $1,000.
There are four things that can happen:
- True hypothesis, true conclusion: I do win the lottery, and I do give everyone in class $1,000. I kept my promise, so the conditional statement is TRUE.
- True hypothesis, false conclusion: I do win the lottery, but I decide not to give everyone in class $1,000. I broke my promise, so the conditional statement is FALSE.
- False hypothesis, true conclusion: I do not win the lottery, but I am exceedingly generous, so I go ahead and give everyone in class $1,000. I did not break my promise! (Do you see why?) So the conditional statement is TRUE.
- False hypothesis, false conclusion: I do not win the lottery, so I do not give everyone in class $1,000. I did not break my promise! (Do you see why?) So the conditional statement is TRUE.
What can we conclude from this? A conditional statement is false only when the hypothesis is true and the conclusion is false. In every other instance, the promise (as it were) has not been broken. If a mathematical statement is not false, it must be true.
Here is another conditional statement:
If you live in Honolulu, then you live in Hawaii.
Is this statement true or false? It seems like it should depend on who the pronoun “you” refers to, and whether that person lives in Honolulu or not. Let us think it through:
- Sookim lives in Honolulu, so the hypothesis is true. Since Honolulu is in Hawaii, she does live in Hawaii. The statement is true about Sookim, since both the hypothesis and conclusion are true.
- DeeDee lives in Los Angeles. The statement is true about DeeDee since the hypothesis is false.
So in fact it does not matter! The statement is true either way. The right way to understand such a statement is as a universal statement: “Everyone who lives in Honolulu lives in Hawaii.”
This statement is true, and here is how you might justify it: “Pick a random person who lives in Honolulu. That person lives in Hawaii (since Honolulu is in Hawaii), so the statement is true for that person. I do not need to consider people who do not live in Honolulu. The statement is automatically true for those people, because the hypothesis is false!”
How do we show a (universal) conditional statement is false?
You need to give a specific instance where the hypothesis is true and the conclusion is false. For example:
If you are a good swimmer, then you are a good surfer.
Do you know someone for whom the hypothesis is true (that person is a good swimmer) but the conclusion is false (the person is not a good surfer)? Then the statement is false!
For each conditional statement, decide if it is true or false. Justify your answer.
- If \(2 \times 2 = 4\) then \(1 + 1 = 3\).
- If \(2 \times 2 = 5\) then \(1 + 1 = 3\).
- If \(\pi > 3\) then all odd numbers are prime.
- If \(\pi < 3\) then all odd numbers are prime.
- If a number has a 4 in the one’s place, then the number is even.
- If a number is even, then the number has a 4 in the one’s place.
- If the product of two numbers is 0, then one of the numbers is 0.
- If the sum of two numbers is 0, then one of the numbers is 0.
On your own, come up with two conditional statements that are true and one that is false. Share your three statements with a partner, but do not say which are true and which is false. See if your partner can figure it out!


7.2: Entropy and Spontaneity - A Molecular Statistical Interpretation
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Learning Objectives
- The Learning Objective of this Module is to understand the relationship between internal energy and entropy.
We know that the first law of thermodynamics governs changes in the state function we called internal energy (\(U\)) and that changes in the internal energy (\(ΔU\)) are closely related to changes in the enthalpy (\(ΔH\)), which is a measure of the heat flow between a system and its surroundings at constant pressure. You also learned previously that the enthalpy change for a chemical reaction can be calculated using tabulated values of enthalpies of formation . This information, however, does not tell us whether a particular process or reaction will occur spontaneously.
Spontaneous Reactions
Let’s consider a familiar example of spontaneous change. If a hot frying pan that has just been removed from the stove is allowed to come into contact with a cooler object, such as cold water in a sink, heat will flow from the hotter object to the cooler one, in this case usually releasing steam. Eventually both objects will reach the same temperature, at a value between the initial temperatures of the two objects. This transfer of heat from a hot object to a cooler one obeys the first law of thermodynamics: energy is conserved.
Now consider the same process in reverse. Suppose that a hot frying pan in a sink of cold water were to become hotter while the water became cooler. As long as the same amount of thermal energy was gained by the frying pan and lost by the water, the first law of thermodynamics would be satisfied. Yet we all know that such a process cannot occur: heat always flows from a hot object to a cold one, never in the reverse direction. That is, by itself the magnitude of the heat flow associated with a process does not predict whether the process will occur spontaneously.
For many years, chemists and physicists tried to identify a single measurable quantity that would enable them to predict whether a particular process or reaction would occur spontaneously. Initially, many of them focused on enthalpy changes and hypothesized that an exothermic process would always be spontaneous. But although it is true that many, if not most, spontaneous processes are exothermic, there are also many spontaneous processes that are not exothermic. For example, at a pressure of 1 atm, ice melts spontaneously at temperatures greater than 0°C, yet this is an endothermic process because heat is absorbed. Similarly, many salts (such as NH 4 NO 3 , NaCl, and KBr) dissolve spontaneously in water even though they absorb heat from the surroundings as they dissolve (i.e., \(ΔH_{soln} > 0\)). Reactions can also be both spontaneous and highly endothermic, like the reaction of barium hydroxide with ammonium thiocyanate shown in Figure \(\PageIndex{1}\).
Thus enthalpy is not the only factor that determines whether a process is spontaneous. For example, after a cube of sugar has dissolved in a glass of water so that the sucrose molecules are uniformly dispersed in a dilute solution, they never spontaneously come back together in solution to form a sugar cube. Moreover, the molecules of a gas remain evenly distributed throughout the entire volume of a glass bulb and never spontaneously assemble in only one portion of the available volume. To help explain why these phenomena proceed spontaneously in only one direction requires an additional state function called entropy (\(S\)), a thermodynamic property of all substances that is proportional to their degree of disorder.
The direction of a spontaneous process is not governed by the internal energy change nor enthalpy change and thus the First Law of Thermodynamics cannot predict the direction of a natural process
Probabilities and Microstates
Chemical and physical changes in a system may be accompanied by either an increase or a decrease in the disorder of the system, corresponding to an increase in entropy (\(ΔS > 0\)) or a decrease in entropy (\(ΔS < 0\)), respectively. As with any other state function, the change in entropy is defined as the difference between the entropies of the final and initial states:
\[ΔS = S_f − S_i\]
When a gas expands into a vacuum, its entropy increases because the increased volume allows for greater atomic or molecular disorder. The greater the number of atoms or molecules in the gas, the greater the disorder. The magnitude of the entropy of a system depends on the number of microscopic states, or microstates, associated with it (in this case, the number of atoms or molecules); that is, the greater the number of microstates, the greater the entropy.
How can we express disorder quantitatively? From the example of coins, you can probably see that simple statistics plays a role: the probability of obtaining three heads and seven tails after tossing ten coins is just the ratio of the number of ways that ten different coins can be arranged in this way, to the number of all possible arrangements of ten coins.
Using the language of molecular statistics, we say that a collection of coins in which a given fraction of its members are heads-up constitutes a macroscopic state of the system. Since we do not care which coins are heads-up, there are clearly numerous configurations of the individual coins which can result in this “macrostate”. Each of these configurations specifies a microscopic state of the system. The greater the number of microstates that correspond to a given macrostate (or configuration), the greater the probability of that macrostate. To see what this means, consider the possible outcomes of a toss of four coins (Table \(\PageIndex{1}\) ):
| macrostate (Configuration) | ways | probability |
microstates
|
|---|---|---|---|
| 0 heads | 1 | 1/16 | TTTT |
| 1 head | 4 | 4/16 = 1/4 | HTTT THTT TTHT TTTH |
| 2 heads | 6 | 6/16 = 3/8 | HHTT HTHT HTTH THHT TTHH THTH |
| 3 heads | 4 | 4/16 = 1/4 | HHHT HTHH HHTH THHH |
| 4 heads | 1 | 1/16 | HHHH |
A toss of four coins will yield one of the five outcomes (macrostates) listed in the leftmost column of the table. The second column gives the number of “ways”— that is, the number of head/tail configurations of the set of coins (the number of microstates)— that can result in the macrostate. The probability of a toss resulting in a particular macrostate is proportional to the number of microstates corresponding to the macrostate, and is equal to this number, divided by the total number of possible microstates (in this example, 2 4 =16). An important assumption here is that all microstates are equally probable; that is, the toss is a “fair” one in which the many factors that determine the trajectory of each coin operate in an entirely random way.
The greater the number of microstates that correspond to a given macrostate , the greater the probability of that macrostate .
Entropy and Microstates
Following the work of Carnot and Clausius, Ludwig Boltzmann developed a molecular-scale statistical model that related the entropy of a system to the number of microstates possible for the system. A microstate (\(\Omega\)) is a specific configuration of the locations and energies of the atoms or molecules that comprise a system like the following:
\[S=k \ln \Omega \label{Eq2}\]
Here k is the Boltzmann constant and has a value of 1.38 × 10 −23 J/K.
As for other state functions, the change in entropy for a process is the difference between its final (\(S_f\)) and initial (\(S_i\)) values:
\[\begin{align} ΔS &= S_\ce{f}−S_\ce{i} \\[4pt] &=k \ln \Omega_\ce{f} − k \ln \Omega_\ce{i} \\[4pt] &= k \ln\dfrac{\Omega_\ce{f}}{\Omega_\ce{i}} \label{Eq2a} \end{align}\]
For processes involving an increase in the number of microstates, \(\Omega_f > \Omega_i\), the entropy of the system increases, \(ΔS > 0\). Conversely, processes that reduce the number of microstates, \(\Omega_f < \Omega_i\), yield a decrease in system entropy, \(ΔS < 0\). This molecular-scale interpretation of entropy provides a link to the probability that a process will occur as illustrated in the next paragraphs.
Consider the general case of a system comprised of N particles distributed among n boxes. The number of microstates possible for such a system is n N . For example, distributing four particles among two boxes will result in 2 4 = 16 different microstates as illustrated in Figure \(\PageIndex{2}\). Microstates with equivalent particle arrangements (not considering individual particle identities) are grouped together and are called distributions . The probability that a system will exist with its components in a given distribution is proportional to the number of microstates within the distribution. Since entropy increases logarithmically with the number of microstates (Equation \ref{Eq2}), the most probable distribution is therefore the one of greatest entropy .
For this system, the most probable configuration is one of the six microstates associated with distribution (c) where the particles are evenly distributed between the boxes, that is, a configuration of two particles in each box. The probability of finding the system in this configuration is
\[\dfrac{6}{16} = \dfrac{3}{8} \nonumber\]
The least probable configuration of the system is one in which all four particles are in one box, corresponding to distributions (a) and (e), each with a probability of
\[\dfrac{1}{16} \nonumber\]
The probability of finding all particles in only one box (either the left box or right box) is then
\[\left(\dfrac{1}{16}+\dfrac{1}{16}\right)=\dfrac{2}{16} = \dfrac{1}{8} \nonumber\]
As you add more particles to the system, the number of possible microstates increases exponentially (2 N ). A macroscopic (laboratory-sized) system would typically consist of moles of particles ( N ~ 10 23 ), and the corresponding number of microstates would be staggeringly huge. Regardless of the number of particles in the system, however, the distributions in which roughly equal numbers of particles are found in each box are always the most probable configurations.
The previous description of an ideal gas expanding into a vacuum is a macroscopic example of this particle-in-a-box model. For this system, the most probable distribution is confirmed to be the one in which the matter is most uniformly dispersed or distributed between the two flasks. The spontaneous process whereby the gas contained initially in one flask expands to fill both flasks equally therefore yields an increase in entropy for the system.
A similar approach may be used to describe the spontaneous flow of heat. Consider a system consisting of two objects, each containing two particles, and two units of energy (represented as “*”) in Figure \(\PageIndex{3}\). The hot object is comprised of particles A and B and initially contains both energy units. The cold object is comprised of particles C and D , which initially has no energy units. Distribution (a) shows the three microstates possible for the initial state of the system, with both units of energy contained within the hot object. If one of the two energy units is transferred, the result is distribution (b) consisting of four microstates. If both energy units are transferred, the result is distribution (c) consisting of three microstates. And so, we may describe this system by a total of ten microstates. The probability that the heat does not flow when the two objects are brought into contact, that is, that the system remains in distribution (a), is \(\frac{3}{10}\). More likely is the flow of heat to yield one of the other two distribution, the combined probability being \(\frac{7}{10}\). The most likely result is the flow of heat to yield the uniform dispersal of energy represented by distribution (b), the probability of this configuration being \(\frac{4}{10}\). As for the previous example of matter dispersal, extrapolating this treatment to macroscopic collections of particles dramatically increases the probability of the uniform distribution relative to the other distributions. This supports the common observation that placing hot and cold objects in contact results in spontaneous heat flow that ultimately equalizes the objects’ temperatures. And, again, this spontaneous process is also characterized by an increase in system entropy.
Example \(\PageIndex{1}\): Determination of Δ S
Consider the system shown here. What is the change in entropy for a process that converts the system from distribution (a) to (c)?
Solution
We are interested in the following change:
The initial number of microstates is one, the final six:
\[\begin{align*} ΔS &=k \ln\dfrac{\Omega_\ce{c}}{\Omega_\ce{a}}\nonumber \\[4pt] &= 1.38×10^{−23}\:J/K × \ln\dfrac{6}{1} \\[4pt] &= 2.47×10^{−23}\:J/K \end{align*} \]
The sign of this result is consistent with expectation; since there are more microstates possible for the final state than for the initial state, the change in entropy should be positive.
Exercise \(\PageIndex{1}\)
Consider the system shown in Figure \(\PageIndex{3}\). What is the change in entropy for the process where all the energy is transferred from the hot object ( AB ) to the cold object ( CD )?
- Answer
-
0 J/K
A disordered system has a greater number of possible microstates than does an ordered system, so it has a higher entropy. This is most clearly seen in the entropy changes that accompany phase transitions, such as solid to liquid or liquid to gas. As you know, a crystalline solid is composed of an ordered array of molecules, ions, or atoms that occupy fixed positions in a lattice, whereas the molecules in a liquid are free to move and tumble within the volume of the liquid; molecules in a gas have even more freedom to move than those in a liquid. Each degree of motion increases the number of available microstates, resulting in a higher entropy. Thus the entropy of a system must increase during melting (ΔS fus > 0). Similarly, when a liquid is converted to a vapor, the greater freedom of motion of the molecules in the gas phase means that ΔS vap > 0. Conversely, the reverse processes (condensing a vapor to form a liquid or freezing a liquid to form a solid) must be accompanied by a decrease in the entropy of the system: ΔS < 0.
Definition: Entropy
Entropy (S) is a thermodynamic property of all substances that is proportional to their degree of disorder. The greater the number of possible microstates for a system, the greater the disorder and the higher the entropy.
Disorder is more probable than order because there are so many more ways of achieving it. Thus coins and cards tend to assume random configurations when tossed or shuffled, and socks and books tend to become more scattered about a teenager’s room during the course of daily living. But there are some important differences between these large-scale mechanical, or macro systems, and the collections of sub-microscopic particles that constitute the stuff of chemistry, and which we will refer to here generically as molecules . Molecules, unlike macro objects, are capable of accepting, storing, and giving up energy in tiny amounts (quanta), and act as highly efficient carriers and spreaders of thermal energy as they move around. Thus, in chemical systems ,
- We are dealing with huge numbers of particles . This is important because statistical predictions are always more accurate for larger samples. Thus although for the four tosses there is a good chance (62%) that the H/T ratio will fall outside the range of 0.45 - 0.55, this probability becomes almost zero for 1000 tosses. To express this in a different way, the chances that 1000 gas molecules moving about randomly in a container would at any instant be distributed in a sufficiently non-uniform manner to produce a detectable pressure difference between the two halves of a container will be extremely small. If we increase the number of molecules to a chemically significant number (around 10 20 , say), then the same probability becomes indistinguishable from zero.
- Once the change begins, it proceeds spontaneously. That is, no external agent (a tosser, shuffler, or teenager) is needed to keep the process going. Gases will spontaneously expand if they are allowed to, and reactions, once started, will proceed toward equilibrium.
- Thermal energy is continually being exchanged between the particles of the system, and between the system and the surroundings. Collisions between molecules result in exchanges of momentum (and thus of kinetic energy) amongst the particles of the system, and (through collisions with the walls of a container, for example) with the surroundings.
- Thermal energy spreads rapidly and randomly throughout the various energetically accessible microstates of the system. The direction of spontaneous change is that which results in the maximum possible spreading and sharing of thermal energy.
The importance of these last two points is far greater than you might at first think, but to fully appreciate this, you must recall the various ways in which thermal energy is stored in molecules.
Sign of Entropy Change
The relationships between entropy, microstates, and matter/energy dispersal described previously allow us to make generalizations regarding the relative entropies of substances and to predict the sign of entropy changes for chemical and physical processes. Consider the phase changes illustrated in Figure \(\PageIndex{4}\). In the solid phase, the atoms or molecules are restricted to nearly fixed positions with respect to each other and are capable of only modest oscillations about these positions. With essentially fixed locations for the system’s component particles, the number of microstates is relatively small. In the liquid phase, the atoms or molecules are free to move over and around each other, though they remain in relatively close proximity to one another. This increased freedom of motion results in a greater variation in possible particle locations, so the number of microstates is correspondingly greater than for the solid. As a result, \(S_{liquid} > S_{solid}\) and the process of converting a substance from solid to liquid (melting) is characterized by an increase in entropy, \(ΔS > 0\). By the same logic, the reciprocal process (freezing) exhibits a decrease in entropy, \(ΔS < 0\).
Now consider the vapor or gas phase. The atoms or molecules occupy a much greater volume than in the liquid phase; therefore each atom or molecule can be found in many more locations than in the liquid (or solid) phase. Consequently, for any substance,
\[S_{gas} \gg S_{liquid} > S_{solid}\]
and the processes of vaporization and sublimation likewise involve increases in entropy, \(ΔS > 0\). Likewise, the reciprocal phase transitions, condensation and deposition, involve decreases in entropy, \(ΔS < 0\).
According to kinetic-molecular theory, the temperature of a substance is proportional to the average kinetic energy of its particles. Raising the temperature of a substance will result in more extensive vibrations of the particles in solids and more rapid translations of the particles in liquids and gases. At higher temperatures, the distribution of kinetic energies among the atoms or molecules of the substance is also broader (more dispersed) than at lower temperatures. Thus, the entropy for any substance increases with temperature (Figure \(\PageIndex{5}\) ).
The entropy of a substance is influenced by structure of the particles (atoms or molecules) that comprise the substance. With regard to atomic substances, heavier atoms possess greater entropy at a given temperature than lighter atoms, which is a consequence of the relation between a particle’s mass and the spacing of quantized translational energy levels (which is a topic beyond the scope of our treatment). For molecules, greater numbers of atoms (regardless of their masses) increase the ways in which the molecules can vibrate and thus the number of possible microstates and the system entropy.
Finally, variations in the types of particles affect the entropy of a system. Compared to a pure substance, in which all particles are identical, the entropy of a mixture of two or more different particle types is greater. This is because of the additional orientations and interactions that are possible in a system comprised of nonidentical components. For example, when a solid dissolves in a liquid, the particles of the solid experience both a greater freedom of motion and additional interactions with the solvent particles. This corresponds to a more uniform dispersal of matter and energy and a greater number of microstates. The process of dissolution therefore involves an increase in entropy, \(ΔS > 0\).
Considering the various factors that affect entropy allows us to make informed predictions of the sign of \(ΔS\) for various chemical and physical processes as illustrated in Example \(\PageIndex{2}\).
Example \(\PageIndex{2}\): Predicting the Sign of ∆ S
Predict the sign of the entropy change for the following processes. Indicate the reason for each of your predictions.
- One mole liquid water at room temperature \(⟶\) one mole liquid water at 50 °C
- \(\ce{Ag+}(aq)+\ce{Cl-}(aq)⟶\ce{AgCl}(s)\)
- \(\ce{C6H6(l) + 15/2 O2(g) -> 6CO2(g) + 3H2O(l)}\)
- \(\ce{NH3}(s)⟶\ce{NH3}(l)\)
S olution
- positive, temperature increases
- negative, reduction in the number of ions (particles) in solution, decreased dispersal of matter
- negative, net decrease in the amount of gaseous species
- positive, phase transition from solid to liquid, net increase in dispersal of matter
Exercise \(\PageIndex{2}\)
Predict the sign of the enthalpy change for the following processes. Give a reason for your prediction.
- \(\ce{NaNO3}(s)⟶\ce{Na+}(aq)+\ce{NO3-}(aq)\)
- the freezing of liquid water
- \(\ce{CO2}(s)⟶\ce{CO2}(g)\)
- \(\ce{CaCO}(s)⟶\ce{CaO}(s)+\ce{CO2}(g)\)
- Answer a
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Positive; The solid dissolves to give an increase of mobile ions in solution.
- Answer b
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Negative; The liquid becomes a more ordered solid.
- Answer c
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Positive; The relatively ordered solid becomes a gas.
- Answer d
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Positive; There is a net production of one mole of gas.
Summary
Entropy (\(S\)) is a state function that can be related to the number of microstates for a system (the number of ways the system can be arranged) and to the ratio of reversible heat to kelvin temperature. It may be interpreted as a measure of the dispersal or distribution of matter and/or energy in a system, and it is often described as representing the “disorder” of the system. For a given substance, \(S_{solid} < S_{liquid} \ll S_{gas}\) in a given physical state at a given temperature, entropy is typically greater for heavier atoms or more complex molecules. Entropy increases when a system is heated and when solutions form. Using these guidelines, the sign of entropy changes for some chemical reactions may be reliably predicted.
Contributors and Attributions
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Stephen Lower, Professor Emeritus ( Simon Fraser U. ) Chem1 Virtual Textbook
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


10.4: Glossary- The Immune System
Glossary
adaptive immune response relatively slow but very specific and effective immune response controlled by lymphocytes
afferent lymphatic vessels lead into a lymph node
antibody antigen-specific protein secreted by plasma cells; immunoglobulin
antigen molecule recognized by the receptors of B and T lymphocytes
barrier defenses antipathogen defenses deriving from a barrier that physically prevents pathogens from entering the body to establish an infection
B cells lymphocytes that act by differentiating into an antibody-secreting plasma cell
bone marrow tissue found inside bones; the site of all blood cell differentiation and maturation of B lymphocytes
bronchus-associated lymphoid tissue (BALT) lymphoid nodule associated with the respiratory tract
chyle lipid-rich lymph inside the lymphatic capillaries of the small intestine
cisterna chyli bag-like vessel that forms the beginning of the thoracic duct
efferent lymphatic vessels lead out of a lymph node
germinal centers clusters of rapidly proliferating B cells found in secondary lymphoid tissues
high endothelial venules vessels containing unique endothelial cells specialized to allow migration of lymphocytes from the blood to the lymph node
immune system series of barriers, cells, and soluble mediators that combine to response to infections of the body with pathogenic organisms
innate immune response rapid but relatively nonspecific immune response
lymph fluid contained within the lymphatic system
lymph node one of the bean-shaped organs found associated with the lymphatic vessels
lymphatic capillaries smallest of the lymphatic vessels and the origin of lymph flow
lymphatic system network of lymphatic vessels, lymph nodes, and ducts that carries lymph from the tissues and back to the bloodstream.
lymphatic trunks large lymphatics that collect lymph from smaller lymphatic vessels and empties into the blood via lymphatic ducts
lymphocytes white blood cells characterized by a large nucleus and small rim of cytoplasm
lymphoid nodules unencapsulated patches of lymphoid tissue found throughout the body
mucosa-associated lymphoid tissue (MALT) lymphoid nodule associated with the mucosa
naïve lymphocyte mature B or T cell that has not yet encountered antigen for the first time
natural killer cell (NK) cytotoxic lymphocyte of innate immune response
plasma cell differentiated B cell that is actively secreting antibody
primary lymphoid organ site where lymphocytes mature and proliferate; red bone marrow and thymus gland
right lymphatic duct drains lymph fluid from the upper right side of body into the right subclavian vein
secondary lymphoid organs sites where lymphocytes mount adaptive immune responses; examples include lymph nodes and spleen
spleen secondary lymphoid organ that filters pathogens from the blood (white pulp) and removes degenerating or damaged blood cells (red pulp)
T cell lymphocyte that acts by secreting molecules that regulate the immune system or by causing the destruction of foreign cells, viruses, and cancer cells
thoracic duct large duct that drains lymph from the lower limbs, left thorax, left upper limb, and the left side of the head
thymocyte immature T cell found in the thymus
thymus primary lymphoid organ; where T lymphocytes proliferate and mature
tonsils lymphoid nodules associated with the nasopharynx
Contributors and Attributions
- Glossary: The Immune System. Authored by : Kelly, Jessica. Provided by : Herkimer College. Located at : NA. Project : AtD. License : CC BY: Attribution


2.4C: Formulating the Hypothesis
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A hypothesis is a potential answer to your research question; the research process helps you determine if your hypothesis is true.
Learning Objectives
- Explain how hypotheses are used in sociological research and the difference between dependent and independent variables
Key Points
- Hypotheses are testable explanations of a problem, phenomenon, or observation.
- Both quantitative and qualitative research involve formulating a hypothesis to address the research problem.
- Hypotheses that suggest a causal relationship involve at least one independent variable and at least one dependent variable; in other words, one variable which is presumed to affect the other.
- An independent variable is one whose value is manipulated by the researcher or experimenter.
- A dependent variable is a variable whose values are presumed to change as a result of changes in the independent variable.
Key Terms
- dependent variable : In an equation, the variable whose value depends on one or more variables in the equation.
- independent variable : In an equation, any variable whose value is not dependent on any other in the equation.
- hypothesis : Used loosely, a tentative conjecture explaining an observation, phenomenon, or scientific problem that can be tested by further observation, investigation, or experimentation.
A hypothesis is an assumption or suggested explanation about how two or more variables are related. It is a crucial step in the scientific method and, therefore, a vital aspect of all scientific research. There are no definitive guidelines for the production of new hypotheses. The history of science is filled with stories of scientists claiming a flash of inspiration, or a hunch, which then motivated them to look for evidence to support or refute the idea.
While there is no single way to develop a hypothesis, a useful hypothesis will use deductive reasoning to make predictions that can be experimentally assessed. If results contradict the predictions, then the hypothesis under examination is incorrect or incomplete and must be revised or abandoned. If results confirm the predictions, then the hypothesis might be correct but is still subject to further testing.
Both quantitative and qualitative research involve formulating a hypothesis to address the research problem. A hypothesis will generally provide a causal explanation or propose some association between two variables. Variables are measurable phenomena whose values can change under different conditions. For example, if the hypothesis is a causal explanation, it will involve at least one dependent variable and one independent variable. In research, independent variables are the cause of the change. The dependent variable is the effect, or thing that is changed. In other words, the value of a dependent variable depends on the value of the independent variable. Of course, this assumes that there is an actual relationship between the two variables. If there is no relationship, then the value of the dependent variable does not depend on the value of the independent variable.


15.3: Lab Report
- How many tetrad complexes do you have in your cell, which is 2n = 4? ____________________________________________________________
- How does the structure of chromosomes in anaphase I of meiosis differ from that in anaphase of mitosis? ____________________________________________________________
- What is the total number of nuclei and cells now present? ___________________
- How many cells were present when meiosis began? _______________________
- How many chromosomes were present in the original parental cell? ___________
- How many chromosomes are in each new daughter cell? ___________________
- Are the chromosomes in the new daughter cells identical to the chromosomes in the original parental cell? Explain your results in terms of independent assortment and crossing over. __________________________________________________
- What is the meaning of diploid? What abbreviation do we use to represent diploid? Name 2 diploid cells in humans.
- What is the meaning of haploid? What abbreviation do we use to represent haploid? Name 2 haploid cells in humans.
- In what stage of the cell cycle does S phase occur? Explain why the DNA must be duplicated during the S phase of the cell cycle, prior to mitosis taking place.
- What specific feature of cytokinesis in animal cells can you use to distinguish this process from cytokinesis in plant cells?
- In the circle below, sketch a 2n=6 haploid cell in metaphase l of meiosis.
- A monogenic gene gives rise to a trait from a single set of alleles. A polygenic gene gives rise to a trait from several sets of alleles. Give an example of a monogenic and polygenic trait.
References
Belwood, Jacqueline; Rogers, Brandy; and Christian, Jason, Foundations of Biology Lab Manual (Georgia Highlands College). “Lab 10: Mitosis & Meiosis,” (2019). Biological Sciences Open Textbooks . 18. CC-BY
https://oer.galileo.usg.edu/biology-textbooks/18 (opens in new window)


6: Linear Transformations
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- 6.10: Supplementary Notes - More on Linear Transformations
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- 6.10.1: Examples and Elementary Properties
- 6.10.1E: Examples and Elementary Properties Exercises
- 6.10.2: Kernel and Image of a Linear Transformation
- 6.10.2E: Kernel and Image of a Linear Transformation Exercises
- 6.10.3: Isomorphisms and Composition
- 6.10.3E: Isomorphisms and Composition Exercises
- 6.10.4: A Theorem about Differential Equations
Thumbnail: A linear combination of one basis set of vectors (purple) obtains new vectors (red). If they are linearly independent, these form a new basis set. The linear combinations relating the first set to the other extend to a linear transformation, called the change of basis. (CC0; Maschen via Wikipedia)


17.5A: Genomics and Proteomics
- Explain how the field of genomics led to the development of proteomics
Proteomics is a relatively-recent field; the term was coined in 1994 while the science itself had its origins in electrophoresis techniques of the 1970’s and 1980’s. The study of proteins, however, has been a scientific focus for a much longer time. Studying proteins generates insight into how they affect cell processes. Conversely, this study also investigates how proteins themselves are affected by cell processes or the external environment. Proteins provide intricate control of cellular machinery; they are, in many cases, components of that same machinery. They serve a variety of functions within the cell; there are thousands of distinct proteins and peptides in almost every organism. The goal of proteomics is to analyze the varying proteomes of an organism at different times in order to highlight differences between them. Put more simply, proteomics analyzes the structure and function of biological systems. For example, the protein content of a cancerous cell is often different from that of a healthy cell. Certain proteins in the cancerous cell may not be present in the healthy cell, making these unique proteins good targets for anti-cancer drugs. The realization of this goal is difficult; both purification and identification of proteins in any organism can be hindered by a multitude of biological and environmental factors.
The study of the function of proteomes is called proteomics. A proteome is the entire set of proteins produced by a cell type. Genomics led to proteomics (via transcriptomics) as a logical step. Proteomes can be studied using the knowledge of genomes because genes code for mRNAs and the mRNAs encode proteins. Although mRNA analysis is a step in the right direction, not all mRNAs are translated into proteins. Proteomics complements genomics and is useful when scientists want to test their hypotheses that were based on genes. Even though all cells of a multicellular organism have the same set of genes, the set of proteins produced in different tissues is different and dependent on gene expression. Thus, the genome is constant, but the proteome varies and is dynamic within an organism. In addition, RNAs can be alternately spliced (cut and pasted to create novel combinations and novel proteins) and many proteins are modified after translation by processes such as proteolytic cleavage, phosphorylation, glycosylation, and ubiquitination. There are also protein-protein interactions, which complicate the study of proteomes. Although the genome provides a blueprint, the final architecture depends on several factors that can change the progression of events that generate the proteome.
Key Points
- Proteomics investigates how proteins affect and are affected by cell processes or the external environment.
- Within an individual organism, the genome is constant, but the proteome varies and is dynamic.
- Every cell in an individual organism has the same set of genes, but the set of proteins produced in different tissues differ from one another and are dependent on gene expression.
Key Terms
- proteomics : the branch of molecular biology that studies the set of proteins expressed by the genome of an organism
- proteome : the complete set of proteins encoded by a particular genome
- genomics : the study of the complete genome of an organism


6.7: Framing
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Framing occurs when the construction and dissemination of messages acts to highlight, emphasize or obscure some aspects of the message over others. We can do this through language via how we organize and structure information, choose signs out of a polysemic group that have overlapping denotic meaning and even appeals to wider context, and the intertext of prior experience which combines to help guide the recipient towards a particular dominant reading. Framing’s relationship to intertext and context is particularly important to underline, as framing is best seen as a cumulative process, or one in which continued exposure to a certain frame helps to guide and internalize that frame’s intended recipient response. It is often argued that framing is one of the central, if implicit, tools we bring to bear to understand complex and diverse experiences – by framing together experiences or texts that seem similar, we mentally simplify and reduce complexity and can then make decisions about what and how to engage with texts.
Both when talking specifically about frames or about language more generally, it is important to highlight that by language we don’t just mean formal spoken or written language. Any coherent symbolic system can be taken as a linguistic system. Under this broad definition one of our earlier examples, street signs and lights, collectively form a linguistic system.
Because we are exposed to consistent language systems and the frames that encompass them repeatedly, language systems and frames become landmarks, part of our day-to-day system of cues and markers that help guide us to solve problems and come to conclusions as members of a culture.
But how do frames actually work within a cultural and communicative context? Firstly, we can break framing approaches down into two broad categories, episodic and thematic framing. Just as their names suggest, episodic framing is where an issue is approached in terms of a specific event or episode. Episodic frames often position individuals in the narrative as free of any constraint or influence of society – as such, individual actions are not usually considered in terms of wider discourses or institutions.
In contrast, thematic framing approaches an issue as part of a continuing topic or theme, rooted within wide social trends or functions. Episodic and thematic frames have different approaches to an issue and can have a dramatic impact on how that issue is subsequently read and understood. Interestingly, it is argued that news media in particular tends to favour episodic over thematic frames because episodic framing better suits the constraints of broadcast and print journalism (such as a 90 second slot or 200 word articles). However, the emphasis on episodic framing means that news as a whole tends not to encourage a reading of issues that takes into account broader themes, issues, or undercurrents. These might otherwise build collectively into a more textured understanding of an issue. This is a major critique of current news systems.
A great example of this is this interview with Maori commentator Willie Jackson where he tries to position the Tuhoe leaders as “good guys” and does so using an interesting set of language and framing techniques. He positions the accused as ‘not bin Ladin’ types, evoking current global stereotypes about what a “terrorist” is, and instead tries to reframe it as part of a wider conversation about institutional racism — both thematic framing (and reframing) approaches.
A YouTube element has been excluded from this version of the text. You can view it online here: https://opentextbc.ca/mediastudies101/?p=31
(c) maraetv under Youtube Standard License
Consider that interview (filmed for Maori tv) with this interview from 3News NZ, which isolates another of the accused, Marama Mayrick, but this time takes an episodic frame, positioning the subject, Mayrick, simply as an artist, ‘caught up’ in an event that floats in the news narrative without reference to wider contexts or issues (such as terrorism or racism).
A YouTube element has been excluded from this version of the text. You can view it online here: https://opentextbc.ca/mediastudies101/?p=31
(c) David Farrar under Youtube Standard License
Discussion
- Identify examples of both episodic and Thematic framing in some news or general documentary media sources.
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After examining these ask, why would they be presented in the respective manners? Is it because of something inherent in the story or does it satisfy the media presenter’s agenda? Or indeed, can you argue any other reason for the presenting style?


5.4: Algae
- Explain why algae are included within the discipline of microbiology
- Describe the unique characteristics of algae
- Identify examples of toxin-producing algae
- Compare the major groups of algae in this chapter, and give examples of each
- Classify algal organisms according to major groups
The algae are autotrophic protists that can be unicellular or multicellular. These organisms are found in the supergroups Chromalveolata (dinoflagellates, diatoms, golden algae, and brown algae) and Archaeplastida (red algae and green algae). They are important ecologically and environmentally because they are responsible for the production of approximately 70% of the oxygen and organic matter in aquatic environments. Some types of algae, even those that are microscopic, are regularly eaten by humans and other animals. Additionally, algae are the source for agar, agarose, and carrageenan, solidifying agents used in laboratories and in food production. Although algae are typically not pathogenic, some produce toxins. Harmful algal blooms, which occur when algae grow quickly and produce dense populations, can produce high concentrations of toxins that impair liver and nervous-system function in aquatic animals and humans.
Like protozoans, algae often have complex cell structures. For instance, algal cells can have one or more chloroplasts that contain structures called pyrenoids to synthesize and store starch. The chloroplasts themselves differ in their number of membranes, indicative of secondary or rare tertiary endosymbiotic events. Primary chloroplasts have two membranes—one from the original cyanobacteria that the ancestral eukaryotic cell engulfed, and one from the plasma membrane of the engulfing cell. Chloroplasts in some lineages appear to have resulted from secondary endosymbiosis, in which another cell engulfed a green or red algal cell that already had a primary chloroplast within it. The engulfing cell destroyed everything except the chloroplast and possibly the cell membrane of its original cell, leaving three or four membranes around the chloroplast. Different algal groups have different pigments, which are reflected in common names such as red algae, brown algae, and green algae.
Some algae, the seaweeds, are macroscopic and may be confused with plants. Seaweeds can be red, brown, or green, depending on their photosynthetic pigments. Green algae, in particular, share some important similarities with land plants; however, there are also important distinctions. For example, seaweeds do not have true tissues or organs like plants do. Additionally, seaweeds do not have a waxy cuticle to prevent desiccation. Algae can also be confused with cyanobacteria, photosynthetic bacteria that bear a resemblance to algae; however, cyanobacteria are prokaryotes (see Nonproteobacteria Gram-negative Bacteria and Phototrophic Bacteria ).
Algae have a variety of life cycles. Reproduction may be asexual by mitosis or sexual using gametes.
Algal Diversity
Although the algae and protozoa were formerly separated taxonomically, they are now mixed into supergroups. The algae are classified within the Chromalveolata and the Archaeplastida. Although the Euglenozoa (within the supergroup Excavata) include photosynthetic organisms, these are not considered algae because they feed and are motile.
The dinoflagellates and stramenopiles fall within the Chromalveolata. The dinoflagellates are mostly marine organisms and are an important component of plankton. They have a variety of nutritional types and may be phototrophic, heterotrophic, or mixotrophic. Those that are photosynthetic use chlorophyll a , chlorophyll c 2 , and other photosynthetic pigments (Figure \(\PageIndex{1}\)). They generally have two flagella, causing them to whirl (in fact, the name dinoflagellate comes from the Greek word for “whirl”: dini ). Some have cellulose plates forming a hard outer covering, or theca, as armor. Additionally, some dinoflagellates produce neurotoxins that can cause paralysis in humans or fish. Exposure can occur through contact with water containing the dinoflagellate toxins or by feeding on organisms that have eaten dinoflagellates.
When a population of dinoflagellates becomes particularly dense, a red tide (a type of harmful algal bloom) can occur. Red tides cause harm to marine life and to humans who consume contaminated marine life. Major toxin producers include Gonyaulax and Alexandrium , both of which cause paralytic shellfish poisoning. Another species, Pfiesteria piscicida , is known as a fish killer because, at certain parts of its life cycle, it can produce toxins harmful to fish and it appears to be responsible for a suite of symptoms, including memory loss and confusion, in humans exposed to water containing the species.
The stramenopiles include the golden algae (Chrysophyta), the brown algae (Phaeophyta), and the diatoms(Bacillariophyta). Stramenopiles have chlorophyll a , chlorophyll c 1 /c 2 , and fucoxanthin as photosynthetic pigments. Their storage carbohydrate is chrysolaminarin. While some lack cell walls, others have scales. Diatoms have flagella and frustules, which are outer cell walls of crystallized silica; their fossilized remains are used to produce diatomaceous earth, which has a range of uses such as filtration and insulation. Additionally, diatoms can reproduce sexually or asexually. One diatom genus, Pseudo-nitzschia , is known to be associated with harmful algal blooms.
Brown algae (Phaeophyta) are multicellular marine seaweeds. Some can be extremely large, such as the giant kelp ( Laminaria ). They have leaf-like blades, stalks, and structures called holdfasts that are used to attach to substrate. However, these are not true leaves, stems, or roots (Figure \(\PageIndex{2}\)). Their photosynthetic pigments are chlorophyll a , chlorophyll c , β -carotene, and fucoxanthine. They use laminarin as a storage carbohydrate.
The Archaeplastids include the green algae (Chlorophyta), the red algae (Rhodophyta), another group of green algae (Charophyta), and the land plants. The Charaphyta are the most similar to land plants because they share a mechanism of cell division and an important biochemical pathway, among other traits that the other groups do not have. Like land plants, the Charophyta and Chlorophyta have chlorophyll a and chlorophyll b as photosynthetic pigments, cellulose cell walls, and starch as a carbohydrate storage molecule. Chlamydomonas is a green alga that has a single large chloroplast, two flagella, and a stigma (eyespot); it is important in molecular biology research (Figure \(\PageIndex{3}\)).
Chlorella is a nonmotile, large, unicellular alga, and Acetabularia is an even larger unicellular green alga. The size of these organisms challenges the idea that all cells are small, and they have been used in genetics research since Joachim Hämmerling (1901–1980) began to work with them in 1943. Volvox is a colonial, unicellular alga (Figure \(\PageIndex{3}\)). A larger, multicellular green alga is Ulva , also known as the sea lettuce because of its large, edible, green blades. The range of life forms within the Chlorophyta—from unicellular to various levels of coloniality to multicellular forms—has been a useful research model for understanding the evolution of multicellularity. The red algae are mainly multicellular but include some unicellular forms. They have rigid cell walls containing agar or carrageenan, which are useful as food solidifying agents and as a solidifier added to growth media for microbes.
Which groups of algae are associated with harmful algal blooms?
Key Concepts and Summary
- Algae are a diverse group of photosynthetic eukaryotic protists.
- Algae may be unicellular or multicellular.
- Large, multicellular algae are called seaweeds but are not plants and lack plant-like tissues and organs.
- Although algae have little pathogenicity, they may be associated with toxic algal blooms that can harm aquatic wildlife and contaminate seafood with toxins that cause paralysis.
- Algae are important for producing agar , which is used as a solidifying agent in microbiological media, and carrageenan , which is used as a solidifying agent.