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# Trying to understand Schrodinger's Equation

1. Sep 22, 2011

### hover

Hello,

It's only been recently that I have acquired the math skills to deal with the time independent version of Schrodinger's Equation which is:

$$\frac{-\hbar^2}{2m} \frac{d^2}{dx^2}\Psi(x) + U(x)\Psi(x) = E\Psi(x)$$

I tried to derive a wavefunction that deals with a particle in a confined box with infinite walls as shown http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html#c1"

I was able to get something similar to:

$$\Psi(x) = A*sin(kx)$$

What somewhat baffles me is how they define lambda( the wavelength within k as shown http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html#c2"). Perhaps I am missing the obvious but why would the biggest wavelength be
$$\lambda = 2L$$
Why not have
$$\lambda = 4L$$
Or
$$\lambda = 7L$$
What makes $$\lambda = 2L$$ the biggest wavelength value. This would clear up a ton.
The second and last question I have(so far) is what particular value of n does a particle need to take? What does the value of n depend upon?

Thanks!

Last edited by a moderator: Apr 26, 2017
2. Sep 22, 2011

### Black Integra

fx'd

In case you want to know, I'll show you a full proof of this.

First, let's define a function for this infinite potential well

U(x)={∞; x<0 or x>L}={0; 0≤x≤L}

So the schrodinger's equation for 0≤x≤L become:
(-ħ2/2m)ψ''=Eψ

We have
ψ'' + (2mE/ħ2)ψ = 0

The solution of an equation is
ψ=Acos(kx) + Bsin(kx)
where k=sqrt(2mE/ħ2) and A,B are constant.

from the condition: ψ(0)=0;
ψ(0) = Acos(0) + Bsin(0) = A = 0

Thus the equation become;
ψ=Bsin(kx)

also; ψ(L)=0;
ψ(L) = Bsin(kL) = 0
But for B≠0;
kL=n¶ or
L=n¶/k

but for k=2¶/λ

so
L = nλ/2

for the minimum state n; n=1
λ = 2L #

and that's the reason for this particular number.
guess it's clear :)

Last edited: Sep 22, 2011
3. Sep 22, 2011

### Staff: Mentor

It follows from these requirements:

1. The wave function must be zero outside the box, because of the "infinitely-high" walls.

2. The wave function must be continuous with no abrupt "jumps" from one point to the next.

This means that the equation you use for the wave function inside the box must evaluate to zero at the walls. What's the longest wavelength that gives you a node (zero point) at both walls?

Last edited by a moderator: Apr 26, 2017
4. Sep 22, 2011

### hover

Thanks for the responses! I like both solutions that are proposed here. On one hand, Black Integra shows how the math just makes the λ=2L fall right out. On the other hand, jtbell shows how the λ=2L falls right out by stating 2 requirements. The longest wavelength must be λ=2L since the wave gives you a zero point starting at a the beginning of wavelength and if you add a half the wavelength you end up at zero again. Very nice solutions.

Just one more question. How does one determine what value of n the wave is in? Since n relates to the amount of energy, does n always take the lowest possible value?

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html#c3"

$$E_n = \frac{n^2h^2}{8mL^2}$$

Also is En a kinetic energy or potential energy? I assume it is a kinetic energy since that is how it seems to be described in the link.

Last edited by a moderator: Apr 26, 2017
5. Sep 23, 2011

### Black Integra

No problems, hover :)

If you still prefer equations for an explanation, here we go.

What is the value of En?
From the relation I've said
L=n¶/k ----> k=n¶/L
and
(kħ)2=2mE

So we have a Total Energy at state n

En=(1/2m)(n¶ħ/L)2=(1/8m)(nh/L)2
It's total energy, due to the variable E in Schrodinger's equation.
Anyway, the potential of system is zero at 0≤x≤L.
So, in this case, total energy is equal to kinatic energy.

What exactly is the value n?
n is considered to be a state number of each system.

For example, the value of n in the energy level of hydrogen atom.
In this system, i.e. infinite square well, there're many states depending on the value n(or total energy).
Each energy will give different wave function(different n).
At n=1, it will give the lowest energy. This energy is called 'zero-point energy'

The value n is a discrete value. So it could not be any value. It just like a resonance of LRC-circuit or spectrum of Hg-atom.


text:
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crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen drop", with its mean ...
, crystal structure is a description of the ordered arrangement of
atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

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ion An ion () is a particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or ...
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s in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
in matter. The smallest group of particles in the material that constitutes this repeating pattern is the
unit cell In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen ...

of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive
translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discussed ...
of the unit cell along its principal axes. The translation vectors define the nodes of the
Bravais lattice In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
. The lengths of the principal axes, or edges, of the unit cell and the angles between them are the
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symmetry Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

properties of the crystal are described by the concept of
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s. All possible symmetric arrangements of particles in three-dimensional space may be described by the 230
space groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The crystal structure and symmetry play a critical role in determining many physical properties, such as
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electronic band structure In solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state ph ...
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optical transparency In the field of optics, transparency (also called pellucidity or diaphaneity) is the physical property of allowing light to pass through the material without appreciable scattering of light. On a macroscopic scale (one where the dimensions invest ...
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# Unit cell

Crystal structure is described in terms of the geometry of arrangement of particles in the unit cell. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

, providing six lattice parameters taken as the lengths of the cell edges (''a'', ''b'', ''c'') and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the
fractional coordinates In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen ...
(''xi'', ''yi'', ''zi'') along the cell edges, measured from a reference point. It is only necessary to report the coordinates of a smallest asymmetric subset of particles. This group of particles may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as the
space group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the crystal structure.International Tables for Crystallography (2006). Volume A, Space-group symmetry. Image:Lattic_simple_cubic.svg, Simple cubic (P) Image:Lattice_body_centered_cubic.svg, Body-centered cubic (I) Image:Lattice_face_centered_cubic.svg, Face-centered cubic (F)

## Miller indices

Vectors and planes in a crystal lattice are described by the three-value
Miller index Miller indices form a notation system in crystallography for planes in Bravais lattice, crystal (Bravais) lattices. In particular, a family of lattice planes is determined by three integers ''h'', ''k'', and ''ℓ'', the ''Miller indices''. ...
notation. This syntax uses the indices ''ℓ'', ''m'', and ''n'' as directional parameters.Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3 By definition, the syntax (''ℓmn'') denotes a plane that intercepts the three points ''a''1/''ℓ'', ''a''2/''m'', and ''a''3/''n'', or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it means that the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane are
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s with no common factors. Negative indices are indicated with horizontal bars, as in (13). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Considering only (''ℓmn'') planes intersecting one or more lattice points (the ''lattice planes''), the distance ''d'' between adjacent lattice planes is related to the (shortest)
reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fun ...
vector orthogonal to the planes by the formula :$d = \frac$

## Planes and directions

The crystallographic directions are geometric
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atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

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ion An ion () is a particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or ...
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molecule File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. A molecule is an electrically neutral group of two or more atoms held together by chemical bo ...

s) of a crystal. Likewise, the crystallographic
plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons''), a location in the multiverse *Plane (Magic: Th ...
s are geometric ''planes'' linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows: *
Optical propertiesThe optical properties of a material define how it interacts with light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible l ...

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Refractive index In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes how fast EM radiation, light travels through the material. It is defined as :n = \frac, ...

is directly related to density (or periodic density fluctuations). *
Adsorption of multilayer adsorption is a random distribution of molecules on the material surface. Adsorption is the adhesion Adhesion is the tendency of dissimilar Particle, particles or interface (matter), surfaces to cling to one another (Cohesion ...

and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes. *
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: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface. *Microstructural
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: Pores and
crystallite A crystallite is a small or even microscopic crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice ...
s tend to have straight grain boundaries following higher density planes. *
Cleavage Cleavage may refer to: Science * Cleavage (crystal), in mineralogy and materials science, a process of splitting a crystal * Cleavage (geology), the foliation perpendicular to stress as a result of ductile deformation * Cleavage (embryo), in embr ...
: This typically occurs preferentially parallel to higher density planes. *
Plastic deformation In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encom ...
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Dislocation In materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...
glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (
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) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice. Some directions and planes are defined by symmetry of the crystal system. In monoclinic, rhombohedral, tetragonal, and trigonal/hexagonal systems there is one unique axis (sometimes called the principal axis) which has higher
rotational symmetry Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it ...
than the other two axes. The basal plane is the plane perpendicular to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principal axis.

### Cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted ''a''); similarly for the reciprocal lattice. So, in this common case, the Miller indices (''ℓmn'') and 'ℓmn''both simply denote normals/directions in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...
. For cubic crystals with
lattice constant upright=1.3, Unit cell definition using parallelopiped with lengths ''a'', ''b'', ''c'' and angles between the sides given by ''α'', ''β'', ''γ'' The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a c ...
''a'', the spacing ''d'' between adjacent (ℓmn) lattice planes is (from above): :$d_= \frac$ Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes: *Coordinates in ''angle brackets'' such as denote a ''family'' of directions that are equivalent due to symmetry operations, such as 00 10 01or the negative of any of those directions. *Coordinates in ''curly brackets'' or ''braces'' such as denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions. For
face-centered cubic 200px, A network model of a primitive cubic system In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived fro ...
(fcc) and
body-centered cubic 200px, A network model of a primitive cubic system In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in cryst ...
(bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic
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and hence are again simply the Cartesian directions.

## Interplanar spacing

The spacing ''d'' between adjacent (''hkℓ'') lattice planes is given by: *Cubic: *:$\frac = \frac$ *Tetragonal: *:$\frac = \frac +\frac$ *Hexagonal: *:$\frac = \frac\left\left(\frac\right\right)+\frac$ *Rhombohedral: *:$\frac = \frac$ *Orthorhombic: *:$\frac = \frac+\frac+\frac$ *Monoclinic: *:$\frac =\left\left(\frac+\frac+\frac-\frac\right\right) \csc^2\beta$ *Triclinic: *:$\frac = \frac$

# Classification by symmetry

The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals have
translational symmetry In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in the form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified.

## Lattice systems

Lattice systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each lattice system consists of a set of three axes in a particular geometric arrangement. All crystals fall into one of seven lattice systems. They are similar to, but not quite the same as the seven
crystal system In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen ...
s. The simplest and most symmetric, the
cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system 200px, A network ...
or isometric system, has the symmetry of a
cube In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
, that is, it exhibits four threefold rotational axes oriented at 109.5° (the
tetrahedral angle In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex ( ...
) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are
hexagonal In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon or 6-gon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regul ...
,
tetragonal In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, froze ...

,
rhombohedral In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
(often confused with the
trigonal crystal system In crystallography, the hexagonal crystal family is one of the six crystal family, crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigo ...
),
orthorhombicIn crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic Lattice (group), lattices result from stretching a cubic crystal system, cubic lattice along two of its orthogonal pairs by two different factors, resul ...

,
monoclinic In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, froze ...

and
triclinic Image:Microcline.jpeg, 200px, An example of the triclinic crystals, microcline image:Triclinic.svg, 180px, Triclinic (a ≠ b ≠ c and α ≠ β ≠ γ ) In crystallography, the triclinic (or anorthic) crystal system is one of the 7 crystal syste ...

.

### Bravais lattices

Bravais lattice In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
s, also referred to as ''space lattices'', describe the geometric arrangement of the lattice points, and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including
quasicrystal A quasiperiodic crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all dire ...

s, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above. The crystal structure consists of the same group of atoms, the ''basis'', positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its
crystallographic point groupIn crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen dr ...
.

## Crystal systems

A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

### Point groups

The
crystallographic point groupIn crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen dr ...
or ''crystal class'' is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include *''Reflection'', which reflects the structure across a ''reflection plane'' *''Rotation'', which rotates the structure a specified portion of a circle about a ''rotation axis'' *''Inversion'', which changes the sign of the coordinate of each point with respect to a ''center of symmetry'' or ''inversion point'' *''
Improper rotation In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
'', which consists of a rotation about an axis followed by an inversion. Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called ''symmetry elements''. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.

## Space groups

In addition to the operations of the point group, the
space group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the crystal structure contains translational symmetry operations. These include: *Pure ''translations'', which move a point along a vector *''Screw axes'', which rotate a point around an axis while translating parallel to the axis. *''Glide planes'', which reflect a point through a plane while translating it parallel to the plane. There are 230 distinct space groups.

# Atomic coordination

By considering the arrangement of atoms relative to each other, their coordination numbers (or number of nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.

## Close packing

The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series: :...ABABABAB... This arrangement of atoms in a crystal structure is known as hexagonal close packing (hcp). If, however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises: :...ABCABCABC... This type of structural arrangement is known as cubic close packing (ccp). The unit cell of a ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the planes of the fcc unit cell. There are four different orientations of the close-packed layers. The packing efficiency can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows: :$\frac = \frac = 0.7405...$ The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only one size. Most crystalline forms of metallic elements are hcp, fcc, or bcc (body-centered cubic). The
coordination numberIn chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central ion/m ...
of atoms in hcp and fcc structures is 12 and its
atomic packing factorIn crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen dr ...
(APF) is the number mentioned above, 0.74. This can be compared to the APF of a bcc structure, which is 0.68.

# Grain boundaries

Grain boundaries are interfaces where crystals of different orientations meet. A
grain boundary of a polycrystalline A crystallite is a small or even microscopic crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, for ...
is a single-phase interface, with crystals on each side of the boundary being identical except in orientation. The term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms that have been perturbed from their original lattice sites,
dislocations In materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...
, and impurities that have migrated to the lower energy grain boundary. Treating a grain boundary geometrically as an interface of a
single crystal A single-crystal, or monocrystalline, solid is a material in which the crystal lattice of the entire sample is continuous and unbroken to the edges of the sample, with no grain boundaries.RIWD. "Reade Advanced Materials - Single Crystals". ''ww ...
cut into two parts, one of which is rotated, we see that there are five variables required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane). Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the
Hall–Petch Grain-boundary strengthening (or Hall–Petch strengthening) is a method of strengthening materials by changing their average crystallite A crystallite is a small or even microscopic crystal A crystal or crystalline solid is a solid mater ...
relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the
electrical Electricity is the set of physical phenomena associated with the presence and motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its positio ...
and
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials ...

of the material. The high interfacial energy and relatively weak bonding in most grain boundaries often makes them preferred sites for the onset of corrosion and for the
precipitation In meteorology Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant p ...
of new phases from the solid. They are also important to many of the mechanisms of creep. Grain boundaries are in general only a few nanometers wide. In common materials, crystallites are large enough that grain boundaries account for a small fraction of the material. However, very small grain sizes are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as
diffusion File:DiffusionMicroMacro.gif, 250px, Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diff ...

and
plasticity Plasticity may refer to: Science * Plasticity (physics), in engineering and physics, the propensity of a solid material to undergo permanent deformation under load * Neuroplasticity, in neuroscience, how entire brain structures, and the brain its ...
. In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any crystalline character, and thus becomes an
amorphous solid In condensed matter physics and materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellec ...
.

# Defects and impurities

Real crystals feature
defects A defect is a physical, functional, or aesthetic attribute of a product or service that exhibits that the product or service failed to meet one of the desired specifications. Defect, defects or defected may also refer to: Examples * Angular defect ...
or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials. When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue. Impurities may also manifest as
electron spin The electron is a subatomic particle, symbol or , whose electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: ...
impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting
ferromagnetic Ferromagnetism is the basic mechanism by which certain materials (such as iron Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal that belongs to the first tra ...
alloy An alloy is an admixture of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts Elec ...
s may lead to different properties as first predicted in the late 1960s.
Dislocation In materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...
s in the crystal lattice allow
shear Shear may refer to: Textile production * Animal shearing, the collection of wool from various species **Sheep shearing Sheep shearing is the process by which the woollen fleece of a sheep Sheep (''Ovis aries'') are quadrupedal, ruminant m ...

at lower stress than that needed for a perfect crystal structure.

# Prediction of structure

The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches as
evolutionary algorithms In computational intelligence (CI), an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic population-based metaheuristic optimization (mathematics), optimization algorithm. An EA uses mechanisms inspired by biological ev ...
, random sampling, or metadynamics. The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms of
Pauling's rules Pauling's rules are five rules published by Linus Pauling in 1929 for crystal structure prediction, predicting and rationalizing the crystal structures of ionic compounds. First rule: the radius ratio rule For typical ionic solids, the cations ...
, first set out in 1929 by
Linus Pauling Linus Carl Pauling (; February 28, 1901 – August 19, 1994) was an American chemist, biochemist, chemical engineer, peace activist, author, and educator. He published more than 1,200 papers and books, of which about 850 dealt with scientific top ...

, referred to by many since as the "father of the chemical bond". Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. He, therefore, was able to correlate the number of d-orbitals in bond formation with the bond length as well as many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to permit uninhibited resonance of valence bonds among various electronic structures. In the
resonating valence bond theoryIn condensed matter physics Condensed matter physics is the field of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that st ...
, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": , , , , , etc. The choice of structure and the value of the
axial ratio Axial ratio, for any structure or shape with two or more axes, is the ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio o ...
(which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers. After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, Hume-Rothery analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number. The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The "d-weight" calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent. In crystal structure predictions/simulations, the periodicity is usually applied, since the system is imagined as unlimited big in all directions. Starting from a triclinic structure with no further symmetry property assumed, the system may be driven to show some additional symmetry properties by applying Newton's Second Law on particles in the unit cell and a recently developed dynamical equation for the system period vectors (lattice parameters including angles), even if the system is subject to external stress.

# Polymorphism

Polymorphism is the occurrence of multiple crystalline forms of a material. It is found in many crystalline materials including
polymer A polymer (; Greek ''wikt:poly-, poly-'', "many" + ''wikt:-mer, -mer'', "part") is a Chemical substance, substance or material consisting of very large molecules, or macromolecules, composed of many Repeat unit, repeating subunits. Due to their ...

s,
mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. (20 ...

s, and
metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts Electrical resistivity and conductivity, el ...

s. According to Gibbs' rules of phase equilibria, these unique crystalline phases are dependent on intensive variables such as pressure and temperature. Polymorphism is related to
allotropy Allotropy or allotropism () is the property of some chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting o ...
, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as
crystal habit In mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical mineralogy, optical) properties of minerals and mineralized artifact (archaeology), artifa ...
, amorphous fraction or
crystallographic defect Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen drop", with its mean ...
s. Polymorphs have different stabilities and may spontaneously and irreversibly transform from a metastable form (or thermodynamically unstable form) to the
stable A stable is a building in which livestock Livestock is commonly defined as domesticated Domestication is a sustained multi-generational relationship in which one group of organisms assumes a significant degree of influence over the reprod ...
form at a particular temperature. They also exhibit different , solubilities, and
X-ray diffraction X-ray crystallography (XRC) is the experimental science determining the atomic and molecular structure of a crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a ...
patterns. One good example of this is the
quartz Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen Tetrahedral molecular geometry, tetrahedra, with each oxygen being shared between two tetrahedra, ...

form of
silicon dioxide Silicon dioxide, also known as silica, is an oxide of rutile. Ti(IV) centers are grey; oxygen centers are red. Notice that oxygen forms three bonds to titanium and titanium forms six bonds to oxygen. An oxide () is a chemical compound that con ...
, or SiO2. In the vast majority of
silicates In chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they unde ...
, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral units linked together by shared vertices in different arrangements. In different minerals the tetrahedra show different degrees of networking and polymerization. For example, they occur singly, joined together in pairs, in larger finite clusters including rings, in chains, double chains, sheets, and three-dimensional frameworks. The minerals are classified into groups based on these structures. In each of the 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the tetrahedra are shared with others, yielding the net chemical formula for silica: SiO2. Another example is elemental
tin Tin is a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same numbers ...

(Sn), which is malleable near ambient temperatures but is
brittle tensile testpieces A material is brittle if, when subjected to stress (physics), stress, it fracture Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually ...
when cooled. This change in mechanical properties due to existence of its two major
allotrope Allotropy or allotropism () is the property of some chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting o ...
s, α- and β-tin. The two
allotrope Allotropy or allotropism () is the property of some chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting o ...
s that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as ''gray tin'' and ''white tin'' respectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa. White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2 °C, tin exists in the gray form, which has a
diamond cubic The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the se ...
crystal structure, similar to
diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. At Standard conditions for temperature and pressure, room temperature and pressure, another solid form of c ...

,
silicon Silicon is a chemical element with the Symbol (chemistry), symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic lustre, and is a Tetravalence, tetravalent metalloid and semiconductor. It is a member ...

or
germanium Germanium is a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same nu ...

. Gray tin has no metallic properties at all, is a dull gray powdery material, and has few uses, other than a few specialized
semiconductor A semiconductor material has an electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that quantifies how strongly it resists electric curre ...
applications. Although the α–β transformation temperature of tin is nominally 13.2 °C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 °C, and upon addition of Sb or Bi the transformation may not occur at all.

# Physical properties

Twenty of the 32 crystal classes are
piezoelectric Piezoelectricity (, ) is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied Stress (mechanics), mechanical st ...
, and crystals belonging to one of these classes (point groups) display
piezoelectricity Piezoelectricity is the electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: ''positive'' and ''negative'' (comm ...

. All piezoelectric classes lack
inversion symmetry In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it ...
. Any material develops a
dielectric In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...
polarization when an electric field is applied, but a substance that has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are
polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates *Polar climate, the clim ...
. All polar crystals are
pyroelectric Pyroelectricity (from the two Greek words ''pyr'' meaning fire, and electricity) is a property of certain crystals which are naturally electrically polarized and as a result contain large electric fields. Pyroelectricity can be described as the ab ...
, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes. There are a few crystal structures, notably the
perovskite structure Image:Perovskite.jpg, Structure of a perovskite with general chemical formula ABX3. The red spheres are X atoms (usually oxygens), the blue spheres are B atoms (a smaller metal cation, such as Ti4+), and the green spheres are the A atoms (a larger m ...
, which exhibit
ferroelectricFerroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are pyroelectric, with the additional property that t ...
behavior. This is analogous to
ferromagnetism Ferromagnetism is the basic mechanism by which certain materials (such as iron) form permanent magnets, or are attracted to magnets. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge o ...
, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).

* * * * * * * * * * * * * *

format: plain


text:
1. ## Exponents

Hi All,
Quite a Simply question but elluding me currently.

I know how 2^4.3^4 = 6^4

But how does 12^5 = 2^10.3^5?

I understand it could also be 2^5.2^6 but I'm trying to determine the above answer?

2. Originally Posted by dumluck
I know how 2^4.3^4 = 6^4
But how does 12^5 = 2^10.3^5?
$12=2^2\cdot 3$ so what is $(2^2\cdot 3)^5~?$

3. Originally Posted by Plato
$12=2^2\cdot 3$ so what is $(2^2\cdot 3)^5~?$
ah yes so it's (4.3)^5 which is (2^2.3)5 , which is 2^10.3^5.

Thank you. Can I ask one more

27^4/3 = 3SQRT(27^4) = (3SQRT(27))^4 ... how do we get 3^4 from the last?

Thanks so much.

4. $27^\frac{1}{3} = \sqrt[3]{27} and you know what 27^4 means. Just split it into parts. It doesn't matter whether you evaluate the fractional part first or the part first$

5. Originally Posted by dumluck
27^4/3 = 3SQRT(27^4) = (3SQRT(27))^4 ... how do we get 3^4 from the last?
$(27)^{\frac{4}{3}}=(3^3)^{\frac{4}{3}}=3^4$

6. Originally Posted by Plato
$(27)^{\frac{4}{3}}=(3^3)^{\frac{4}{3}}=3^4$
Thanks Plato so $(27)^{\frac{4}{3}}$ is the same as 3SQRT(27)^4? are we then just 'crossing out' the 3's? if so; why? I.e. what exponent rule govens it?

Thanks

7. 27^4 / 3 or 27^(4/3) ?

8. Originally Posted by Wilmer
27^4 / 3 or 27^(4/3) ?
Neither.. (3SQRT(27))^4

9. Originally Posted by dumluck
$(27)^{\frac{4}{3}}$ is the same as 3SQRT(27)^4? are we then just 'crossing out' the 3's? if so; why? I.e. what exponent rule govens it?
$(N^x)^y=N^{xy}$
$\left( {\left[ {\sqrt[3]{{x^2 }}} \right]} \right)^4 = x^{\frac{8}
{3}} = x^2 \sqrt[3]{{x^2 }}$

10. Originally Posted by Plato
$(N^x)^y=N^{xy}$
$\left( {\left[ {\sqrt[3]{{x^2 }}} \right]} \right)^4 = x^{\frac{8}
{3}} = x^2 \sqrt[3]{{x^2 }}$
Thanks Plato so using this.
$\left( {\left[ {\sqrt[3]{{27}}} \right]} \right)^4 = {\left[ {\sqrt[3]{{3^3}}} \right]} \right)^4 = 3^{\frac{12}
{3}} ?$
and diving 12 by3 we get $3^4$

IS that correct?

11. Originally Posted by dumluck
Thanks Plato so using this.
$\left( {\left[ {\sqrt[3]{{27}}} \right]} \right)^4 = {\left[ {\sqrt[3]{{3^3}}} \right]} \right)^4 = 3^{\frac{12}
{3}} ?$
and diving 12 by3 we get $3^4$
IS that correct?
YES.
But also $3\left(\frac{1}{3}\right)=1$

12. Originally Posted by Plato
YES.
But also $3\left(\frac{1}{3}\right)=1$
Thanks plato, where does that come in?

13. Originally Posted by dumluck
Thanks plato, where does that come in?
$\sqrt[3]{{x^3 }} = \left( {x^3 } \right)^{\frac{1}{3}}$

14. Please do not write "3sqrt" for the cube root! The "sq" in "sqrt" is from square and the cube root has nothing to do with a square root! For you to say you did NOT mean " $27^{4/3}$" but meant "(3sqrt(27))^4" is exceptionally confusing.

(Yes, I know that the LaTeX for "cube root" is "\sqrt[3]{}" and I am just as annoyed at that!)

15. Originally Posted by HallsofIvy
Please do not write "3sqrt" for the cube root! The "sq" in "sqrt" is from square and the cube root has nothing to do with a square root! For you to say you did NOT mean " $27^{4/3}$" but meant "(3sqrt(27))^4" is exceptionally confusing.

(Yes, I know that the LaTeX for "cube root" is "\sqrt[3]{}" and I am just as annoyed at that!)
No problem, will look up LaTex to ensure I'm describing correctly.


text:
# Why is Simon interrupted at the meeting?

Why do the boys not let simon finish talking

##### Answers 1

In Chapter Five? if so, this would be because Simon alludes to the fact that the littl'uns might be justified in their fear of the beast.

##### Source(s)

Lord of the Flies


text:
# Homework Help: Partial Fractions - Integration

1. Jul 17, 2014

### emmaerin

1. The problem statement, all variables and given/known data

Evaluate the integral. (Remember to use ln |u| where appropriate. Use C for the constant of integration.)

$\int \frac {5x^2 - 20x +45}{(2x+1)(x-2)^2}\, dx$

2. Relevant equations

$5x^2 - 20x +45$ = $5 (x^2 -4x +9)$

3. The attempt at a solution
I'm able to come up with an answer, but it's not correct. I'd really appreciate it if someone could tell me where I'm going wrong. Thanks! (Also this is my first attempt at LaTeX, so forgive me if it's not formatted properly!)

First to solve for the partial fractions:
$\frac {A}{2x+1} + \frac {B}{x-2} + \frac {C}{(x-2)^2}$

$x^2 - 4x +9 = A (x-2)^2 + B (2x+1) (x-2) + C (2x+1)$
$x^2 - 4x + 9 = A (x^2 - 4x + 4) + B (2x^2 - 3x - 2) + C (2x+1)$

$x^2 = x^2 (A+2B)$
$-4x = x (-4A - 3B + 2C)$
$9 = 4A -2B +C$

Therefore,
$C = 9-4A +2B$
$-4x = x ( -4A - 3B + 2(9-4A+2B))$
$-4 = -12A + B +18$
$-22 = -12A + B$

and

$1 = A + 2B$
$-22 = -22A - 44B$

Gives me,

$-12A + B = -22A - 44B$
$A = \frac {-9}{2} B$
$1 = \frac {-9}{2}B + 2B$

*$\mathbf B = \frac{-2}{5}$

$1 = A + 2B$

*$\mathbf A = \frac{9}{5}$

$C = 9 - 4A + 2B$

*$\mathbf C = 1$

Substituting in A, B, and C gives me:

$\int \frac {\frac{9}{5}}{2x+1}\,dx$ + $\int \frac {\frac{-2}{5}}{x-2} \,dx$ + $\int \frac {1}{(x-2)^2}\,dx$

$= \frac{9}{5} \int \frac {1}{2x+1}\,dx$ $- \frac{2}{5} \int \frac {1}{x-2}\,dx$ $+ \int \frac {1}{(x-2)^2} \, dx$

For the first integral, $= \frac{9}{5} \int \frac {1}{2x+1}\,dx$
$u = 2x+1, du = 2 dx$
$\frac {9}{10} \int \frac {1}{u}\, du$
$= \frac {9}{10} ln |2x+1|$

For the second integral, $\int \frac {\frac{-2}{5}}{x-2} \,dx$
$\frac{-2}{5} \int \frac {1}{x-2} \, dx$
$= \frac{-2}{5} ln |x-2|$

For the third integral, $\int \frac {1}{(x-2)^2} \, dx$
$\int (x-2)^{-2} \, dx$
$= \frac {-1}{x-2}$

This gives me the following answer, but it's wrong :( I know this is a long problem, so any help would be appreciated!

$\frac {9}{10} ln |2x+1|$ + $\frac{-2}{5} ln |x-2|$ + $\frac {-1}{x-2}$ +C

2. Jul 17, 2014

### HakimPhilo

Revise your calculations of A, B and C, when you decompose your fraction, that's where the error is.

3. Jul 17, 2014

### gopher_p

A) You factored out a 5 from the numerator at the beginning of the problem, and then it disappeared.
B) Your answer involves logs, and log laws might have been used to make the book's (or whatever it is you're comparing your answer to) answer look different from yours.

4. Jul 17, 2014

### emmaerin

I completely forgot to bring down the 5 - that was the whole problem! Such a silly mistake, thanks for pointing it out!


text:
# mysqldump: how to exclude or include tables

By default, mysqldump includes all tables of the specified database when creating the dump. In some cases, it is useful to exclude some of the tables or even include only some of them. For me, this helped to exclude one of the biggest tables to reduce the backup execution time and file size.

## Exclude a single table

To exclude a single table, the argument --ignore-table followed by the table name is used:

mysqldump my_database --ignore-table=my_table > my_backup.sql

This will create a file my_backup.sql containing the sql schema and data of all tables in the database my_database except my_table.

## Exclude multiple tables

To exclude multiple tables, simply add the argument --ignore-table for all the tables you want to exclude:

mysqldump my_database --ignore-table=my_table --ignore-table=another_table > my_backup.sql

This will create a file my_backup.sql containing the sql schema and data of all tables in the database my_database except my_table and another_table.

## Include a single table

To include only a single table into the dump, add the name of the table to the command:

mysqldump my_database my_table > my_backup.sql

This will create a file my_backup.sql containing the sql schema and data of the table my_table from the database my_database.

## Include multiple tables

To include multiple tables into the dump, add the names of the tables to the command:

mysqldump my_database my_table another_table > my_backup.sql

This will create a file my_backup.sql containing the sql schema and data of the tables my_table and another_table from the database my_database.

Photo by benjamin lehman on Unsplash.


text:
# X86_64 implementation of STRCHR using NASM

As this will eventually become part of an operating system I'm developing, there is no specific adherence to any particular ABI, even though this is being developed and tested on Ubuntu 16.04. My OS will however adopt a SYSTEM V paradigm insomuch as passing values by register, but they will be passed in the one most suitable to the algorithm.

One of my methodologies is to utilize as many bits of the register as possible, especially when 16 or less are significant. I have found too, that in a lot of cases this can simplify calling process. It does however complicate things a little ie:

    mov     rdi, Pointer_to_String
mov     ecx, 784 << 8 | '.'
call    strlen

This would traverse the buffer pointed to by RDI for the first occurence of period for a maximum of 784 characters. This snippet had evolved from just scanning for NULL for a maximum of FFFFFFFFH bytes, to scanning for any of the values (0 - FF), to traversing buffer for specified number of bytes.

### Question

Do you think I've achived the maximum utility with the least amount of code.

 ; Determine length, including terminating character EOS. Result may include
; VT100 escape sequences.

;        ENTER: RDI = Pointer to ASCII string.
;               RCX   Bits 31 - 08 = Max chars to scan (1 - 1.67e7)
;                          07 - 00 = Terminating character (0 - FF)

;        LEAVE: RAX = Pointer to next string (optional).

;        FLAGS:  ZF = Terminating character found, NZ otherwise (overrun).
;                DF = Unmodified incase it was already set.

strlen:  pushf                       ; Preserve DF (Direction flag)
push    rcx                 ; Preserve registers used by proc so
push    rdi                 ; its non-destructive except for RAX.

mov      al, cl             ; Byte to scan for in AL.
shr     ecx, 8              ; Shift max count into bits 23 - 00

std                         ; Auto decrement.
repnz   scasb               ; Scan for AL or until ECX = 0
mov     rax, rdi            ; Return pointer to EOS + 1

pop     rdi                 ; Original pointer from proglogue
jz      .exit               ; ZF indicates EOS was found
mov     rax, rdi            ; RAX = RDI, NULL string

.exit:
pop     rcx
popf                        ; Restore direction flag.
ret

• Do you ever plan on calling this from C? Or is your entire OS going to be in assembly? – JS1 Oct 21 '16 at 9:26
• Will you also write compilers for your own calling convention or everything will be in assembly? BTW what's the ultimate goal of this kind of packaging? You may save few registers (maybe) but unless all parameters are literals then calling code is MUCH more convoluted (then possibly slower...) – Adriano Repetti Oct 21 '16 at 9:30
• @AdrianoRepetti: I may port or implement some sort of NASM type assembler, but no intention of writing a compiler. Ultimate goal is to have what I'll call a more intuitive ABI where RBX RCX RSI & RDI are utilized as per X86 specification. Complexity dependent upon design intent is unavoidable. I've just resolved to make what I'll call lower level (for lack of a better term) as simple as possible. That position is mailable though and yours/others perspectives are essential in the development process. – Shift_Left Oct 21 '16 at 16:02
• @JS1 I don't foresee anything other than assembly, but the system I have in mind could possibly become a collaborative work, so it may be something I should consider. At this point, if that should become the case, an additional tier between an established calling convention and this code is an alternative. – Shift_Left Oct 21 '16 at 16:09

### Bug #1

You set the DF flag, so your string operation is going to run backwards. You should clear the DF flag to scan your string in the forwards direction.

### Bug #2

If you reach the byte limit, your function will return the original string pointer in RAX. The comment says it should return NULL. Perhaps you meant xor rax, rax instead of mov rax, rdi?

### Bug #3

According to the comments, your function should return with ZF set according to the result of the scan. However, since you do pushf and then popf, the ZF result is wiped out by the popf.

I would suggest following the standard X64 ABI rather than inventing a new ABI per function. For one thing, you will be able to use high level languages to interface with your assembly routines. Another thing is that it would be hard for you to remember details such as "when I call strlen, ZF set means that the terminating character was found".

### Don't call it strlen

Your function is closer to strchr or strnchr than strlen. Any reasonable person would expect a function called strlen to return a length rather than a pointer.

• There is irrefutable logic in all your points, so I know what I'll be doing this afternoon. – Shift_Left Oct 21 '16 at 16:29

Based on comments I've modified code to correct error(s) and

• Conform to System V ABI
• Make no assumption about DF and always restore to original state
• When using a name similar to that in glibc, adhere to that functions definition.

Return pointer to first occurence of Match character

   ENTER: RDI = Pointer to ASCII string.
RSI   Bits 31 - 08 = Max chars to scan (1 - 1.67e7)
07 - 00 = Match character (0 - FF)

LEAVE: RAX = Pointer to first match or NULL if not found.
All others unchanged

DF = Unmodified.

    strchr: push    rcx                 ; Preserve registers used by proc so
mov     ecx, esi
push    rdi                 ; Its non-destructive except for RAX.
pushf                       ; Preserve direction flag

mov      al, cl             ; Match char into AL.
shr     ecx, 8              ; Shift max count into bits 23 - 00
cld                         ; Guarantee auto increment
repnz   scasb               ; Scan for AL or until ECX = 0
jz      .found

mov     eax, ecx            ; Return NULL in RAX
jmp     .done

.found: mov     rax, rdi
dec     rax                 ; Bump back to match char

.done: popf                        ; DF reset to original state
or      rax, rax            ; Set ZF accordingly
pop     rdi                 ; Original pointer from proglogue
pop     rcx
ret


text:
## Files in this item

FilesDescriptionFormat

application/pdf

8302933.pdf (23MB)
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## Description

 Title: Accurate Quantum Mechanical Decoupling Approximations for Molecular Scattering Theory Author(s): Mclenithan, Kelly Daniel Department / Program: Chemistry Discipline: Chemistry Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Chemistry, Physical Abstract: Recent work is reviewed on the theory of angular momentum decoupling approximations in inelastic molecular scattering theory. It is shown that the failure of these approximations is more extensive than previously believed, particularly for completely state-selected m-transitions. A new method is presented for systematically improving these approximations, which recouples the approximate solutions to first-order using the generalized distorted wave Born approximation. As an illustration of the generality of this new recoupling technique, formulae are derived for the corrections to the centrifugal decoupling, the energy sudden and the infinite-order sudden approximations. Completely state-selected differential and integral cross sections are computed under the centrifugal decoupling approximation and its first-order distorted wave correction for Ne + HD scattering at a total energy of 31.5 meV. Five choices of the partial wave decoupling parameter (including the three known ones) are studied. The numerical results show that with the partial wave parameter chosen to be the arithmetic mean of the initial and final l-quantum numbers, the corrected centrifugal decoupling approximation gives significantly more accurate inelastic differential and integral state-selected cross sections than its uncorrected counterpart. Issue Date: 1982 Type: Text Description: 822 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982. URI: http://hdl.handle.net/2142/70200 Other Identifier(s): (UMI)AAI8302933 Date Available in IDEALS: 2014-12-15 Date Deposited: 1982


text:
# 20.6: Disorders of the Immune System

##### Allergy Eyes

Eyes that are red, watery, and itchy are typical of an allergic reaction known as allergic rhinitis. Commonly called hay fever, allergic rhinitis is an immune system reaction typically to the pollen of certain plants. Your immune system usually protects you from pathogens and keeps you well. However, like any other body system, the immune system itself can develop problems. Sometimes it responds to harmless foreign substances as though they were pathogens. That’s the basis of allergies such as hay fever.

## Allergies

An allergy is a disorder in which the immune system makes an inflammatory response to a harmless antigen. It occurs when the immune system is hypersensitive to an antigen in the environment that causes little or no response in most people. Allergies are strongly familial: allergic parents are more likely to have allergic children and those children’s allergies are likely to be more severe. This is evidence that there is a heritable tendency to develop allergies. Allergies are more common in children than adults because many children outgrow their allergies by adulthood.

### Allergens

Any antigen that causes an allergy is called an allergen. Common allergens are plant pollens, dust mites, mold, specific foods (such as peanuts or shellfish), insect stings, and certain common medications (such as aspirin and penicillin). Allergens may be inhaled or ingested, or they may come into contact with the skin or eyes. Symptoms vary depending on the type of exposure and the severity of the immune system response. Two common causes of allergies are ragweed and poison ivy. Inhaling ragweed pollen may cause symptoms of allergic rhinitis, such as sneezing and red itchy eyes. Skin contact with oils in poison ivy may cause an itchy rash. This type of allergy is called contact dermatitis.

### Prevalence of Allergies

There has been a significant increase in the prevalence of allergies over the past several decades, especially in the rich nations of the world, where allergies are now very common disorders. In the developed countries, about 20 percent of people have or have had hay fever, another 20 percent have had contact dermatitis, and about 6 percent have food allergies. In the poorer nations of the world, on the other hand, allergies of all types are much less common.

One explanation for the rise in allergies in the developed world is called the hygiene hypothesis. According to this hypothesis, people in developed countries live in relatively sterile environments because of hygienic practices and sanitation systems. As a result, people in these countries are exposed to fewer pathogens than their immune system evolved to cope with. To compensate, their immune system “keeps busy” by attacking harmless antigens in allergic responses.

### How Allergies Occur

The diagram in Figure $$\PageIndex{3}$$ shows how an allergic reaction occurs. At the first exposure to an allergen, B cells are activated to form plasma cells that produce large amounts of antibodies to the allergen. These antibodies attach to leukocytes called mast cells. Subsequently, every time the person encounters the allergen again, the mast cells are already primed and ready to deal with it. The primed mast cells immediately release cytokines and histamines, which in turn cause inflammation and recruitment of leukocytes, among other responses. These responses are responsible for the signs and symptoms of allergies.

### Treating Allergies

The symptoms of allergies can range from mild to life-threatening. Mild allergy symptoms are often treated with antihistamines. These are drugs that reduce or eliminate the effects of the histamines that produce allergy symptoms.

#### Treating Anaphylaxis

The most severe allergic reaction is a systemic reaction called anaphylaxis. This is a life-threatening response caused by a massive release of histamines. Many of the signs and symptoms of anaphylaxis are shown in Figure $$\PageIndex{4}$$. Some of them include a drop in blood pressure, changes in heart rate, shortness of breath, and swelling of the tongue and throat, which may threaten the patient with suffocation unless emergency treatment is given. People who have had anaphylactic reactions may carry an epinephrine autoinjector (widely known by its brand name EpiPen®) so they can inject themselves with epinephrine if they start to experience an anaphylactic response. The epinephrine helps to control the immune reaction until medical care can be provided. Epinephrine constricts blood vessels to increase blood pressure, relaxes smooth muscles in the lungs to reduce wheezing and improve breathing, modulates heart rate, and works to reduce swelling that may otherwise block the airways.

#### Immunotherapy for Allergies

Another way to treat allergies is called immunotherapy, commonly called “allergy shots.” This approach may actually cure specific allergies, at least for several years if not lifelong. It may be particularly beneficial for allergens such as pollen that are difficult or impossible to avoid. First, however, patients must be tested to identify the specific allergens that are causing their allergies. As shown in Figure $$\PageIndex{5}$$, this may involve scratching tiny amounts of common allergens into the skin and then observing whether there is a localized reaction to any of them. Each allergen is applied in a different numbered location on the skin so if there is a reaction, such as redness or swelling, the responsible allergens can be identified. Then, through periodic injections (usually weekly or monthly), patients are gradually exposed to larger and larger amounts of the allergens. Over time, generally from months to years, the immune system becomes desensitized to the allergens. This method of treating allergies is often effective for allergies to pollen or insect stings, but its usefulness for allergies to food is unclear.

## Autoimmune Diseases

Autoimmune diseases occur when the immune system fails to recognize the body’s own molecules as self. As a result, instead of ignoring the body’s healthy cells, it attacks them, causing damage to tissues and altered organ growth and function. Most often, it is B cells that are at fault in autoimmune responses. They are generally the cells that lose tolerance for self. Why does this occur? Some autoimmune diseases are thought to be caused by exposure to pathogens that have antigens similar to the body’s own molecules. After this exposure, the immune system responds to body cells as though they were pathogens as well.

Certain individuals are genetically susceptible to developing autoimmune diseases. These individuals are also more likely to develop more than one such disease. Gender is also a risk factor for autoimmunity. Females are much more likely than males to develop autoimmune diseases, probably in part because of gender differences in sex hormones.

At a population level, autoimmune diseases are less common where infectious diseases are more common. The hygiene hypothesis has been proposed to explain the inverse relationship between infectious and autoimmune diseases as well as the prevalence of allergies. According to the hypothesis, without infectious diseases to “keep it busy,” the immune system may attack the body’s own cells instead.

### Common Autoimmune Diseases

An estimated 15 million or more people worldwide have one or more autoimmune diseases. Two of the most common autoimmune diseases are type I diabetes and multiple sclerosis. Both are localized diseases in terms of the specific body cells that are attacked by the immune system. In the case of type I diabetes, the immune system attacks and destroys insulin-secreting islet cells in the pancreas. In the case of multiple sclerosis, the immune system attacks and destroys the myelin sheaths that normally insulate the axons of neurons and allow rapid transmission of nerve impulses.

Some relatively common autoimmune diseases are systemic, or body-wide, diseases. They include rheumatoid arthritis and systemic lupus erythematosus (SLE). In these diseases, many tissues and organs may be attacked and injured by the immune system. For example, as you can see in Figure $$\PageIndex{6}$$, symptoms of SLE may involve the muscular, skeletal, integumentary, respiratory, and cardiovascular systems.

### Treatment for Autoimmune Diseases

None of these common autoimmune diseases can be cured, although all of them have treatments that may help relieve symptoms and prevent some of the long-term damage they may cause. Traditional treatments for autoimmune diseases include immunosuppressive drugs to block the immune response and anti-inflammatory drugs to quell inflammation. Hormone replacement may be another option. For example, type I diabetes is treated with injections of the hormone insulin because islet cells in the pancreas can no longer secrete it.

## Immunodeficiency

Immunodeficiency occurs when the immune system is not working properly, generally because one or more components of the immune system are inactive. As a result, the immune system may be unable to fight off pathogens or cancers that a normal immune system would be able to resist. Immunodeficiency may occur for a variety of reasons.

### Causes of Immunodeficiency

Dozens of rare genetic diseases can result in a defective immune system. This type of immunodeficiency is called primary immunodeficiency. One is born with one of these diseases rather than acquiring it after birth. Probably the best known of these primary immunodeficiency diseases is severe combined immunodeficiency (SCID). It is also known as “bubble boy disease” because people with this disorder are extremely vulnerable to infectious diseases and some of them have become well known for living inside a bubble that provides a sterile environment. SCID is most often caused by an X-linked recessive mutation that interferes with normal B cell and T cell production.

Other types of immunodeficiency are not present at birth but acquired due to experiences or exposures that occur after birth. Acquired immunodeficiency is called secondary immunodeficiency because it is secondary to some other event or exposure. Secondary immunodeficiency may occur for a number of different reasons:

• The immune system naturally becomes less effective as people get older. This age-related decline, called immunosenescence, generally begins at about age 50 and worsens with increasing age. Immunosenescence is why older people are generally more susceptible to disease than younger people.
• The immune system may be damaged by another disorder, such as obesity, alcoholism, or the abuse of other drugs.
• In developing countries, malnutrition is the most common cause of immune system damage and immunodeficiency. Inadequate protein intake is especially damaging to the immune system. It can lead to impaired complement system activity, phagocyte malfunction, and lower-than-normal production of antibodies and cytokines.
• Surgical removal or disease of the thymus, where T lymphocytes normally mature, results in severe immunodeficiency. People without a functioning thymus are extremely susceptible to infections.
• Certain medications can suppress the immune system. This is the intended effect of immunosuppressant drugs given to people with transplanted organs so they do not reject them. In many cases, however, immunosuppression is an unwanted side effect of drugs used to treat other disorders.
• Some pathogens attack and destroy cells of the immune system. An example is a virus known as HIV, which attacks and destroys T cells.

### Focus on HIV

The human immunodeficiency virus (HIV) is the most common cause of immunodeficiency in the world today, so it is the focus of the rest of this concept. It is also covered in the concept HIV and AIDS. HIV infections of human hosts are a relatively recent phenomenon. Scientists think that the virus originally infected monkeys but then jumped to human populations, probably sometime during the early to mid-1900s. This most likely occurred in West Africa, but the virus soon spread around the world. HIV was first identified by medical researchers in 1981. Since then, HIV has killed almost 40 million people worldwide, and its economic toll has also been enormous. The hardest hit countries are in Africa, where the virus has infected human populations the longest, and medications to control the virus are least available.

#### HIV Transmission

HIV is transmitted through direct contact of mucous membranes or body fluids such as blood, semen, or breast milk. As shown in Figure $$\PageIndex{7}$$, the transmission of the virus can occur through sexual contact or the use of contaminated hypodermic needles. It can also be transmitted from an infected mother’s blood during late pregnancy or childbirth or through breast milk after birth. In the past, HIV was also transmitted occasionally through blood transfusions. Because donated blood is now screened for HIV, the virus is no longer transmitted this way.

#### HIV and the Immune System

HIV infects and destroys helper T cells, the type of lymphocytes that regulate the immune response. How this occurs is shown in Figure $$\PageIndex{8}$$. The virus injects its own nucleic acid into a helper T cell and uses the T cell’s “machinery” to make copies of itself. In the process, the helper T cell is destroyed, and the virus copies go on to infect other helper T cells. HIV is able to evade the immune system and keep destroying helper T cells by mutating frequently so its surface antigens keep changing and by using the host cell’s membrane to hide its own antigens.

Acquired immunodeficiency syndrome (AIDS) may result from years of damage to the immune system by HIV. It occurs when helper T cells fall to a very low level and opportunistic diseases occur. Opportunistic diseases are infections and tumors that are rare except in people with a damaged immune system. The diseases take advantage of the “opportunity” presented by people whose immune systems cannot fight back. Opportunistic diseases are usually the direct cause of death of people with AIDS.

#### Treating HIV/AIDS

For patients who have access to HIV medications, infection with the virus has ceased to be the death sentence that it once was. By 1995, combinations of drugs called “highly active antiretroviral therapy” were developed. For some patients, these drugs can reduce the amount of virus they are carrying to undetectable levels. However, some amount of virus always hides in the body’s immune cells and will multiply again if a patient stops taking the medications. Researchers are trying to develop drugs to kill these hidden viruses as well. If their efforts are successful, it could bring an end to AIDS.

##### Feature: Human Biology in the News

EpiPens® and their sole manufacturer, pharmaceutical company Mylan, were featured in the news headlines in 2016 but not for a good reason. A drastic price hike in EpiPens® and Mylan’s apparent greed triggered the media outburst.

EpiPens® are auto-injectable syringes preloaded with a measured dose of epinephrine, a drug that can rapidly stop a life-threatening anaphylactic response to an allergen. Using the device is easy and does not require any special training. The injector just needs to be jammed against the thigh, which can be done through clothing or on bare skin. Each year, doctors write millions of prescriptions for EpiPens®. Many people with severe allergies always carry two of the devices with them just in case they experience anaphylaxis, although most of them never need to use them. Other people with severe allergies have literally had their lives saved multiple times by EpiPens® when they had anaphylactic reactions. Even when the devices haven’t been used, they must be replaced each year due to the expiration of the epinephrine.

You might think that EpiPens® would be relatively inexpensive, given their life-saving potential. As recently as 2009, a two-pack of EpiPens® cost about $100. However, in just 7 years, the cost of the same two-pack of EpiPens® skyrocketed by an incredible 400 percent! By 2016, the cost was$600 or more. Mylan apparently raised the price for the sole purpose of increasing profits. The company also raised prices significantly on many other drugs. The price hike in EpiPens® alone was certainly profitable. In 2015, the sale of EpiPens® earned Mylan $1 billion. Mylan’s CEO took home almost$19 million the same year, which was an increase of more than 600 percent over her prior salary.

News coverage of the price hike in EpiPens® began in the summer of 2016 after a price increase in May of that year. Both private citizens and elected officials expressed outrage over the price increase, especially when coupled with the gluttonous profits of the company and its CEO. By late August, Mylan responded to the backlash by offering discount coupons for EpiPens®. A few days later, the company promised to introduce a cheaper, generic version of the device. Analysts quickly determined that selling a generic version would allow Mylan to make more money on the product than reducing the price of the name-brand device, which they still declined to do. By September of 2016, Mylan was being investigated for antitrust violations relating to sales of EpiPens® to public schools in New York City.

The Mylan/EpiPen® story may still be making the news. But whatever its outcome, the story has already added fuel to public and private debates about important ethical issues — issues such as the excessive costs of life-saving drugs and the huge profits of big pharma. What is the most recent news on EpiPens® and Mylan? If you are interested, you can check the headlines online to find out. What are your views on the ethical issues they raise?

## Review

1. What are allergies? What causes them?
2. Compare the prevalence of allergies in developed and developing countries. How does the hygiene hypothesis explain the differences in prevalence?
3. How do allergies occur?
4. How are mild allergy symptoms treated?
5. What is anaphylaxis, and how is it treated?
6. How does immunotherapy for allergies work?
7. What are autoimmune diseases?
8. Identify two risk factors for autoimmune diseases.
9. Autoimmune diseases may be specific to particular tissues, or they may be systemic. Give an example of each type of autoimmune disease.
10. What is immunodeficiency?
11. Compare and contrast primary and secondary immunodeficiency, and give an example of each.
12. What is the most common cause of immunodeficiency in the world today? How does this cause affect the immune system?
13. Distinguish between HIV and AIDS.
14. True or False. Allergies and autoimmune diseases both result from an over-reactive immune system.
15. True or False. An anaphylactic reaction can be stopped by administering a shot of histamine.

## Explore More

1. Oedema By Championswimmer, Public domain via Wikimedia Commons
2. Pollen by Hans via Pixabay license
3. Poison Ivy by Sam Fraser-Smith, CC BY 2.0 via Wikimedia Commons
4. Mast Cells by NIH, Public domain, via Wikimedia Commons
5. Signs and symptoms of anaphylaxis by Mikael Häggström, CC0 via Wikimedia Commons
6. Skin prick test for allergies by NIH, public domain via Wikimedia Commons
7. Symptoms of SLE by Mikael Häggström, Public domain via Wikimedia Commons
8. HIV Infection by Hana Zavadska and Laura Guerin, CC BY-NC 3.0 via CK-12 foundation
9. AIDS life cycle illustration by NIH, Public domain, via Wikimedia Commons
10. Text adapted from Human Biology by CK-12 licensed CC BY-NC 3.0

20.6: Disorders of the Immune System is shared under a CK-12 license and was authored, remixed, and/or curated by Suzanne Wakim & Mandeep Grewal via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.