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Dipole Moment

Dipole moment

:This article is about the electromagnetic phenomenon. From the point of view of the mathematics of distributions, a dipole can be taken to be the directional derivative of a Dirac delta function. A dipole is also a type of radio antenna. :For magnets in particle accelerators please see dipole magnet. dipole magnet A dipole (Greek: dyo = two and polos = pivot) is a pair of electric charges or magnetic poles of equal magnitude but opposite polarity (opposite electronic charges), separated by some (usually small) distance. Dipoles can be characterized by their dipole moment, a vector quantity with a magnitude equal to the product of the charge or magnetic strength of one of the poles and the distance separating the two poles. The direction of the dipole moment corresponds, for electric dipoles, to the direction from the negative to the positive charge. For magnetic dipoles, the dipole moment points from from the magnetic south to the magnetic north pole — confusingly, the "north" and "south" convention for magnetic dipoles is the opposite of that used to describe the Earth's geographic and magnetic poles, so that the Earth's geomagnetic north pole is the south pole of its dipole moment. (Because of the absence of magnetic monopoles, magnetic dipoles are actually created by current loops and/or by quantum-mechanical spin.) Since the direction of an electric field is defined as the direction of the force on a positive charge, electric field lines point away from a positive charge and toward a negative charge.

Alignment of a dipole to an applied field

When placed in an electric (E) or magnetic (B) field, equal but opposite forces arise on each side of the dipole creating a torque τ: :

\mathbf = \mathbf \times \mathbf

for an Electric dipole moment p (in coulomb-meters), or :

\mathbf = \mathbf \times \mathbf    =    \mu_o \mathbf \times \mathbf

for a Magnetic dipole moment m (in ampere-square meters). The resulting torque will tend to align the dipole with the applied field.

Physical dipoles, point dipoles, and approximate dipoles

torque A physical dipole consists of two equal and opposite point charges: literally, two poles. Its field at large distances (i.e., distances large in comparison to the separation of the poles) depends almost entirely on the dipole moment as defined above. A point (electric) dipole is the limit obtained by letting the separation tend to 0 while keeping the dipole moment fixed. The field of a point dipole has a particularly simple form, and the order-1 term in the multipole expansion is precisely the point dipole field. Although there are no known magnetic monopoles in nature, there are magnetic dipoles in the form of the quantum-mechanical spin associated with particles such as electrons (although the accurate description of such effects falls outside of classical electromagnetism). A theoretical magnetic point dipole has a magnetic field of the exact same form as the electric field of an electric point dipole. A very small current-carrying loop is approximately a magnetic point dipole; the magnetic dipole moment of such a loop is the product of the current flowing in the loop and the (vector) area of the loop. Any configuration of charges or currents has a dipole moment, which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration. This is simply one term in the multipole expansion; when the charge ("monopole moment") is 0 — as it always is for the magnetic case, since there are no magnetic monopoles — the dipole term is the dominant one at large distances: its field falls off in proportion to 1/r³, as compared to 1/r⁴ for the next (quadrupole) term and higher powers of 1/r for higher terms, or 1/r² for the monopole term.

Molecular dipoles

Many molecules have such dipole moments due to non-uniform distributions of positive and negative charges on the various atoms. For example: (positive) H-Cl (negative) A molecule with a permanent dipole moment is called a polar molecule and is polarized. The physical chemist Peter J. W. Debye was the first scientist to study molecular dipoles extensively, and dipole moments are consequently measured in units named debye in his honor. With respect to molecules there are three types of dipoles:
- Permanent dipoles: These occur when 2 atoms in a molecule have substantially different electronegativity — one atom attracts electrons more than another becoming more negative, while the other atom becomes more positive. See dipole-dipole attractions.
- Instantaneous dipoles: These occur due to chance when electrons happen to be more concentrated in one place than another in a molecule, creating a temporary dipole. See Instantaneous dipole attraction.
- Induced dipoles These occur when one molecule with a permanent dipole repels another molecule's electrons, "inducing" a dipole moment in that molecule. See induced-dipole attraction.

Field from a magnetic dipole

Magnitude

The strength, B, of a dipole magnetic field is given by: :

\mathbf(\mathbf, \lambda) = \frac   \frac   \sqrt

where: :B is the strength of the field, measured in teslas :r is the distance from the center, measured in metres :λ is the magnetic latitude (90°-θ) where θ = magnetic colatitude, measured in radians or degrees from the dipole axis (magnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis) :M is the dipole moment, measured in ampere square-metres :μ₀ is the permeability of free space, measured in henrys per metre.

Vector form

The field itself is a vector quantity: :

\mathbf(\mathbf) = \frac   \left(3(\mathbf\cdot\hat)\hat-\mathbf\right)

where :B is the field :r is the vector from the position of the dipole to the position where the field is being measured :r is the absolute value of r: the distance from the dipole :

\hat = \mathbf/r

is the unit vector parallel to r :m is the (vector) dipole moment :μ₀ is the permeability of free space This is exactly the field of a point dipole, exactly the dipole term in the multipole expansion of an arbitrary field, and approximately the field of any dipole-like configuration at large distances.

Magnetic vector potential

The vector potential A is :

\mathbf(\mathbf) = \frac   (\mathbf\times\hat)

with the same definitions as above.

Field from an electric dipole

The electric field of an electric point dipole is :

\mathbf(\mathbf) = \frac   \left(3(\mathbf\cdot\hat)\hat-\mathbf\right)

where :E is the field :r, r,

\hat

are as above :p is the (vector) dipole moment :ε₀ is the permittivity of free space. Notice that this is formally identical to the magnetic field of a point magnetic dipole; only a few names have changed.

Electrostatic potential

The electrostatic potential is :

\Phi  (\mathbf) = \frac   (\mathbf\cdot\hat).

External links

- [http://geomag.usgs.gov USGS Geomagnetism Program] ja:双極子

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Distribution'

Distribution can mean:
- In mathematics, there are several distinct concepts given the name of distribution:
- For generalized functions, see distribution (mathematics).
- In probability, see probability distribution.
- In Carnot-Carathéodory manifolds, see sub-Riemannian manifold.
- In physics, a distribution function, for example the Maxwell-Boltzmann distribution, describes the number of particles per unit volume in phase space.
- For business operations and logistics, see distribution (business).
- For the computer science concept, see distributed computing.
- For the meaning of distribution in the terminology of the Linux operating system, see Linux distribution.
- For electric power, see electricity distribution.
- For linguistics, see complementary distribution.
- In the film industry, see distribution (film).

Dirac delta function

The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. The integral from minus infinity to plus infinity is 1. The discrete analog of the delta function is the degenerate distribution which is sometimes known as a delta function.

Overview

Dirac functions can be of any size in which case their 'strength' A is defined by duration multiplied by amplitude. The graph of the delta function can be usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function.) Despite its name, the delta function is not a function as defined in the strictest mathematical sense. One reason for this is because the functions f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are ostensibly different. According to Lebesgue integration theory, if f, g are functions such that f = g almost everywhere, then f is integrable iff g is integrable and the integrals of f and g are the same. Precise treatment of the Dirac delta requires measure theory or the theory of distributions. The Dirac delta is very useful as an approximation for a tall narrow spike function (an impulse). It is the same type of abstraction as a point charge, point mass or electron point. For example, in calculating the dynamics of a baseball being hit by a bat, approximating the force of the bat hitting the baseball by a delta function is a helpful trick. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball. The Dirac delta function was named after the Kronecker delta, since it can be used as a continuous analogue of the discrete Kronecker delta.

Formal introduction

The Dirac delta is often introduced with the property: :

\int_^\infty f(x) \, \delta(x) \, dx
= f(0)

valid for any continuous function f. However, there is no actual function δ(x) with this property. The Dirac delta is not a function; but it can be usefully treated as a distribution, as well as a measure. As a distribution, the Dirac delta is defined by :

\delta[\phi] = \phi(0)\,

for every test function

\phi \

. It is a distribution with compact support (the support being ). Because of this definition, and the absence of a true function with the delta function's properties, it is important to realize the above integral notation is simply a notational convenience, and not a true integral. As a measure,

\delta (A)=1

0\in A

, and

\delta (A)=0

otherwise. Then, :

\int_^\infty f(x) \, d\delta(x) 
= f(0)

for all continuous f. As distributions, the Heaviside step function is an antiderivative of the Dirac delta distribution.

Fourier transform

The continuous Fourier transform of the Dirac delta is the constant function

\frac

. The inverse transform of this constant function will be the Dirac delta again, yielding the orthogonality property for the Fourier kernel: :

\frac\int_^\infty e^\,dx=\delta(k)

From the convolution theorem for the Fourier transform, the convolution of δ with any distribution S yields S.

The Dirac delta function as a probability density function

The Dirac delta function may be interpreted as a probability density function. Its characteristic function is then just unity, as is the moment generating function, so that all moments are zero. The cumulative distribution function is the Heaviside step function.

Derivatives of the delta function

The derivative of the Dirac delta is the distribution δ' defined by :

\delta'[\phi] = -\phi'(0)\,

for every test function

\phi \

. From this it follows that :

\delta'(x)=-\frac

The n-th derivative δ⁽ⁿ⁾ is given by :

\delta^[\phi] = (-1)^n \phi^(0)\,

The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomials. A helpful identity is :

\delta(g(x)) = \sum_\frac

where x_i are the roots of g(x). In the integral form it is equivalent to :

\int_^\infty f(x) \, \delta(g(x)) \, dx
= \sum_\frac

The derivative of a Dirac delta is called a doublet.

Equivalent definition

The Dirac delta function

\delta : \mathbb \ni \xi \longrightarrow \delta ( \xi )\in \delta(\mathbb)

is a distribution

\delta ( \xi )

whose indefinite integral is the function :

h : \mathbb \ni \xi \longrightarrow \frac \in \mathbb,

usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation :

\int^_ \delta (t) dt = h(x) \equiv \frac

for all real numbers x.

Representations of the delta function

The delta function can be viewed as the limit of a sequence of functions :

\delta (x) = \lim_ \delta_a(x),

where

\delta_a(x)

is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting sequence will be used. On the other hand the term limit needs to be made precise, as this equality holds only for some meanings of limit. The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions. Some nascent delta functions are: :
Note: If δ(a, x) is a nascent delta function which is a probability distribution over the whole real line (i.e. is always non-negative between -∞ and +∞) then another nascent delta function δ_φ(a, x) can be built from its characteristic function as follows: :

\delta_\varphi(a,x)=\frac~\frac

where :

\varphi(a,k)=\int_^\infty \delta(a,x)e^\,dx

is the characteristic function of the nascent delta function δ(a, x). This result is related to the localization property of the continuous Fourier transform.

External links

- [http://mathworld.wolfram.com/DeltaFunction.html Delta Function] on MathWorld
- [http://planetmath.org/encyclopedia/DiracDeltaFunction.html Dirac Delta Function] on PlanetMath Category:Mathematical analysis ja:ディラックのデルタ関数

Dipole magnet

A dipole magnet, in particle accelerators, is a magnet constructed to create a homogeneous magnetic field over some distance. Particle motion in that field will be circular in a plane perpendicular to the field and collinear to the direction of particle motion and free in the direction orthogonal to it. Thus, a particle injected into a dipole magnet will travel on a circular or helical trajectory. By adding several dipole sections on the same plane, the bending radial effect of the beam increases. In particle accelerators, dipole magnets are used to realize bends in the design trajectory (or 'orbit') of the particles, as in circular accelerators. Other uses include:
- Injection of particles into the accelerator
- Ejection of particles from the accelerator
- Correction of orbit errors
- Production of synchrotron radiation Such magnets are also used in traditional televisions, which contain a cathode ray tube, which is essentially a small particle accelerator. Their magnets are called deflecting coils. The magnets move a single spot on the screen of the TV tube in a controlled way all over the screen. The screen has many lines, and "interlacing" is used. That means that on one pass the light spot is moved very quickly along all of the odd-numbered lines (1,3,5 ...), then along all of the even-numbered lines (2,4,6 ...) during the next pass, and so on.

Greek language

Greek (Greek Ελληνικά, IPA – "Hellenic") is an Indo-European language with a documented history of 3,500 years. Today, it is spoken by 15 million people in Greece, Cyprus, the former Yugoslavia, particularly The Former Yugoslav Republic of Macedonia, Bulgaria, Albania and Turkey. There are also many Greek emigrant communities around the world, such as those in Melbourne, Australia which is the third-largest Greek-populated city in the world, after Athens and Thessaloniki. Greek has been written in the Greek alphabet, the first true alphabet, since the 9th century B.C. and before that, in Linear B and the Cypriot syllabaries. Greek literature has a long and rich tradition.

History

This article does not cover the reconstructed history of Greek prior to the use of writing. For more information, see main article on Proto-Greek language. Greek has been spoken in the Balkan Peninsula since the 2nd millennium BC. The earliest evidence of this is found in the Linear B tablets dating from 1500 BC. The later Greek alphabet (q.v.) is unrelated to Linear B, and was derived from the Phoenician alphabet (abjad); with minor modifications, it is still used today. Greek is conventionally divided into the following periods:
- Mycenean Greek: the language of the Mycenean civilisation. It is recorded in the Linear B script on tablets dating from the 16th century BC onwards.
- Classical Greek (also known as Ancient Greek): In its various dialects was the language of the Archaic and Classical periods of Greek civilisation. It was widely known throughout the Roman empire. Classical Greek fell into disuse in western Europe in the Middle Ages, but remained known in the Byzantine world, and was reintroduced to the rest of Europe with the Fall of Constantinople and Greek migration to Italy.
- Hellenistic Greek (also known as Koine Greek): The fusion of various ancient Greek dialects with Attic (the dialect of Athens) resulted in the creation of the first common Greek dialect, which gradually turned into one of the world's first international languages. Koine Greek can be initially traced within the armies and conquered territories of Alexander the Great, but after the Hellenistic colonisation of the known world, it was spoken from Egypt to the fringes of India. After the Roman conquest of Greece, an unofficial diglossy of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. Through Koine Greek it is also traced the origin of Christianity, as the Apostles used it to preach in Greece and the Greek-speaking world. It is also known as the Alexandrian dialect, Post-Classical Greek or even New Testament Greek (after its most famous work of literature).
- Medieval Greek: The continuation of Hellenistic Greek during medieval Greek history as the official and vernacular (if not the literary nor the ecclesiastic) language of the Byzantine Empire, and continued to be used until, and after the fall of that Empire in the 15th century. Also known as Byzantine Greek.
- Modern Greek: Stemming independently from Koine Greek, Modern Greek usages can be traced in the late Byzantine period (as early as 11th century). Two main forms of the language have been in use since the end of the medieval Greek period: Dhimotikí (Δημοτική), the Demotic (vernacular) language, and Katharévousa (Καθαρεύουσα), an imitation of classical Greek, which was used for literary, juridic, and scientific purposes during the 19th and early 20th centuries. Demotic Greek is now the official language of the modern Greek state, and the most widely spoken by Greeks today. It has been claimed that an "educated" speaker of the modern language can understand an ancient text, but this is surely as much a function of education as of the similarity of the languages. Still, Koinē , the version of Greek used to write the New Testament and the Septuagint, is relatively easy to understand for modern speakers. Greek words have been widely borrowed into the European languages: astronomy, democracy, philosophy, thespian, etc. Moreover, Greek words and word elements continue to be productive as a basis for coinages: anthropology, photography, isomer, biomechanics etc. and form, with Latin words, the foundation of international scientific and technical vocabulary. See English words of Greek origin, and List of Greek words with English derivatives.

Classification

Greek is an independent branch of the Indo-European language family. The ancient languages which were probably most closely related to it, Ancient Macedonian language (which may be regarded as a dialect of Greek) and Phrygian, are not well enough documented to permit detailed comparison. Among living languages, Armenian seems to be the most closely related to it.

Geographic distribution

Modern Greek is spoken by about 15 million people mainly in Greece and Cyprus. There are also Greek-speaking populations in Georgia, Ukraine, Egypt, Turkey, Albania, Former Yugoslav Republic of Macedonia and Southern Italy. The language is spoken also in many other countries where Greeks have settled, including Armenia, Australia, Austria, Belgium, Bulgaria, Canada, Denmark, France, Germany, Netherlands, Sweden, United Kingdom, and the United States.

Official status

Greek is the official language of Greece where it is spoken by about 99.5% of the population. It is also, alongside Turkish, the official language of Cyprus. Due to the membership of Greece and Cyprus, Greek is one of the 20 official languages of the European Union.

Phonology

This section generally describes the post-Classic phonology of the Greek language. :All phonetic transcriptions in this section use the International Phonetic Alphabet

Vowel sounds

Greek has 5 vowel sounds, all phonemic:

Magnetic moment

In physics, the magnetic moment of an object is a vector relating the aligning torque in a magnetic field experienced by the object to the field vector itself. The relationship is given by: :

\mathbf = \mathbf \times \mathbf

where :τ is the torque, measured in newton · metres :μ is the magnetic moment, measured in ampere · square metres :B is the magnetic field, measured in newtons per ( ampere · metres ) The alignment of the magnetic moment with the field creates a difference in potential energy U: :

U = - \mathbf\cdot\mathbf

One of the simplest examples of magnetic moments is that of the current-carrying loop, carrying current I and of area vector A for which the magnitude is given by: :

\mathbf = I \mathbf

where :μ is the magnetic moment, measured in ampere · square metres :I is the current, measured in amperes :A is the loop area vector , having as x-,y-,z-coordinates the square metres area of the projection of the loop into the yz-,zx-,and xy-planes Electrons and many nuclei also have intrinsic magnetic moments, an explanation of which requires a quantum mechanical treatment and relates to the intrinsic angular momentum of the particles. It is these intrinsic magnetic moments that give rise to the macroscopic effects of magnetism, and other phenomena, such as Nuclear Magnetic Resonance. Category:Magnetism Category:Electric and magnetic fields in matter Category:Physical quantity ja:磁気モーメント

North Pole

:This is about the geographic meaning of "North Pole." For the cities, see North Pole, Alaska and North Pole, New York. The North Pole is the northernmost point on our planet. There are various ways of defining our North Pole. The North Pole lies in the Arctic Ocean.

Defining the North Pole

The North Pole, is very, very cold. It is just a giant piece of ice on top of the world.

Geographic North Pole

The Geographic North Pole, also known as True North, is close to the northern point at which the Earth's axis of rotation meets the surface. Geographic North defines latitude 90° North. In whichever direction you travel from here, you are always heading south. The pole is located in the Arctic Ocean, which at this point has a depth of 4087 metres (13,410 feet). Classically (19th century) this pole was exactly where people believed the pole of rotation met the Earth's surface, but soon astronomers noticed a small apparent variation of latitude as determined for a fixed point on Earth by observing stars. This variation had a period of about 435 days and the periodic part of it is now called the Chandler wobble after its discoverer. It is desirable to tie the system of Earth coordinates (latitude, longitude, and elevations or orography) to fixed landforms. Of course, given continental drift and the rising and falling of land due to volcanos, erosion and so on, there is no system in which all geographic features are fixed. Yet the International Earth Rotation and Reference Systems Service and the International Astronomical Union have defined a framework called the International Terrestrial Reference System that does an admirable job. The North pole of this system now defines geographic North and it does not quite coincide with the rotation axis. Also see polar motion. On the basis of the sector principle, Canada claims its sovereignty to extend all the way to the Geographic North Pole. There is no land at this location, which is usually covered by sea ice. The theory under which Canada has claimed sovereignty to the North Pole is controversial as there is in fact 770 km of ocean between the pole and Canada's northernmost land point, and several nations, most notably the United States, have challenged the notion that the North Pole does not lie in international waters.

Expeditions

The first expedition to the pole is generally accepted to have been made by, African-American Matthew Henson, Navy engineer, Anglo-American Robert Edwin Peary and four Inuit men (Ootah, Seegloo, Egingway, and Ooqueah) on April 6, 1909. Polar historians believe that Peary honestly thought he had reached the pole. However a 1996 analysis of a newly-discovered copy of Peary's record indicates that Peary must have been in fact 20 nautical miles (40 km) short of the Pole. The first undisputed sight of the pole was in 1926 by Norwegian explorer Roald Amundsen and his American sponsor Lincoln Ellsworth from the airship Norge, designed and piloted by the Italian Umberto Nobile, in a flight from Svalbard to Alaska. On May 3, 1952 U.S. Air Force Lieutenant Colonel Joseph O. Fletcher and Lieutenant William P. Benedict landed a plane at the geographic North Pole. Flying with them was scientist Albert P. Crary. The United States Navy submarine USS Nautilus (SSN-571) crossed the North Pole on August 3, 1958, and on March 17, 1959, the USS Skate (SSN-578) surfaced at the pole, becoming the first naval vessel to reach it. Ralph Plaisted made the first confirmed surface conquest of the North Pole on April 19, 1968. The Soviet nuclear-powered icebreaker Arktika on August 17, 1977, completed the first surface vessel journey to the pole. On April 6, 1992 Robert Schumann became the youngest person to visit the north pole.

Magnetic North

Magnetic North is one of several locations on the Earth's surface known as the "North Pole". Its definition, as the point where the geomagnetic field points vertically downwards, i.e. the dip is 90°, was proposed in 1600 by Sir William Gilbert, a courtier of Queen Elizabeth I, and is still used. It should not be confused with the less frequently used Geomagnetic North Pole. Magnetic North is the place to which all magnetic compasses point, although since the pole marked "N" on a bar magnet points north, and only opposite magnetic poles are attracted to each other, the Earth's magnetic north is actually a south magnetic pole. The orientation of magnetic fields of planets can flip over, an event which is called a geomagnetic reversal. The Earth's poles have done this repeatedly throughout history, and 500,000 years ago, the south magnetic pole was at the South Pole. It is thought that this occurs when the circulation of liquid nickel/iron in the Earth's outer core is disrupted and then reestablishes itself in the opposite direction. It is not known what causes these disruptions. The first expedition to reach this pole was led by James Clark Ross, who found it at Cape Adelaide on the Boothia Peninsula on June 1, 1831. Roald Amundsen found Magnetic North in a slightly different location in 1903. The third observation of Magnetic North was by Canadian government scientists Paul Serson and Jack Clark, of the Dominion Astrophysical Observatory, who found the pole at Allen Lake on Prince of Wales Island. The Canadian government has made several measurements since, which show that Magnetic North is continually moving northwest. Its location (in 2003) is 78°18' North, 104° West, near Ellef Ringnes Island, one of the Queen Elizabeth Islands, in Canada. During the 20th century it has moved 1100 km, and since 1970 its rate of motion has accelerated from 9 km/a to 41 km/a (2001-2003 average; see also Polar drift). If it maintains its present speed and direction it will reach Siberia in about 50 years, but it is expected to veer from its present course and slow down. This movement is on top of a daily or diurnal variation in which Magnetic North describes a rough ellipse, with a maximum deviation of 80 km from its mean position. This effect is due to disturbances of the geomagnetic field by the sun. A line drawn from one magnetic pole to the other does not go through the centre of the Earth, it actually misses it by about 530 km. The angular difference between Magnetic North and true North varies with location, and is called the magnetic declination.

Geomagnetic North Pole

The Geomagnetic North Pole is the pole of the Earth's geomagnetic field closest to true north. The first-order approximation of the Earth's magnetic field is that of a single magnetic dipole (like a bar magnet), tilted about 11° with respect to Earth's rotation axis and centered at the Earth's core. The residuals form the nondipole field. The Geomagnetic poles are the places where the axis of this dipole intersects the Earth's surface. Because the dipole approximation is far from a perfect fit to the Earth's magnetic field, the magnetic field is not quite vertical at the geomagnetic poles. The locations of true vertical field orientation are the magnetic poles, and these are about 30 degrees of longitude away from the geomagnetic poles. Like the Magnetic North Pole, the geomagnetic north pole is a south magnetic pole, because it attracts the north pole of a bar magnet. It is the centre of the region in the magnetosphere in which the Aurora Borealis can be seen. Its present location is 78°30' North, 69° West, near Qaanaaq in Greenland, however it is now drifting away from North America and toward Siberia [http://news.yahoo.com/s/ap/20051209/ap_on_sc/earth_magnetic_pole]. The first voyage to this pole was by David Hempleman-Adams in 1992.

Northern Pole of Inaccessibility

The Northern Pole of Inaccessibility, located at 84°03' north, 174°51' west, is the point farthest from any northern coastline, about 1100 km from the nearest coast. It is a geographic construct, not an actual physical phenomenon. It was first reached by Sir Hubert Wilkins, who flew by aircraft in 1927; in 1958 a Russian icebreaker reached this point.

Defining North Poles in astronomy

Astronomers define the north "geographic" pole of a planet or other object in the solar system by the planetary pole that is in the same ecliptic hemisphere as the Earth's north pole. More accurately, «The north pole is that pole of rotation that lies on the north side of the invariable plane of the solar system» [http://www.hnsky.org/iau-iag.htm]. This means some objects will have directions of rotation opposite the "normal" (i.e., not counter-clockwise as seen from above the north pole). Another frequently used definition uses the right-hand rule to define the north pole: it is then the pole around which the object rotates counterclockwise [http://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html]. When using the first definition (the IAU's), an object's axial tilt will always be 90° or less, but its rotation period may be negative (retrograde rotation); when using the second definition, axial tilts may be greater than 90° but rotation periods will always be positive. For the magnetic poles, their names are decided upon by the direction that their field lines emerge or enter the planet's crust. If they enter the same way as they do for Earth at the north pole, we call this the planet's north magnetic pole. Some bodies in the solar system, including Saturn's moon Hyperion and the asteroid 4179 Toutatis, lack a stable geographic north pole. They rotate chaotically because of their irregular shape and gravitational influences from nearby planets and moons, and as a result the instantaneous pole wanders over their surface, and may vanish altogether for brief periods (when the object comes to a complete standstill with respect to the distant stars). The projection of a planet's north geographic pole onto the celestial sphere gives its north celestial pole. In the particular (but frequent) case of synchronous satellites, four more poles can be defined. They are the near, far, leading, and trailing poles. Take Io for example; this moon of Jupiter rotates synchronously, so its orientation with respect to Jupiter stays constant. There will be a single, unmoving point of its surface where Jupiter is at the zenith, exactly overhead —this is the near pole, also called the sub- or pro-Jovian point. At the antipode of this point is the far pole, where Jupiter lies at the nadir; it is also called the anti-Jovian point. There will also be a single unmoving point which is furthest along Io's orbit (best defined as the point most removed from the plane formed by the north-south and near-far axes, on the leading side) —this is the leading pole. At its antipode lies the trailing pole. Io can thus be divided into north and south hemispheres, into pro- and anti-Jovian hemispheres, and into leading and trailing hemispheres. Note that these poles are mean poles because the points are not, strictly speaking, unmoving: there is constant jiggling about the mean orientation, because Io's orbit is slightly eccentric and the gravity of the other moons disturbs it regularly.

Day and night

During the summer months, the North Pole experiences twenty four hours of daylight but during the winter months the North Pole experiences twenty four hours of darkness. Sunrise and sunset do not occur in a twenty four hour cycle. At the north pole, sunrise begins at the Vernal equinox taking three months for the sun to reach its highest point at the summer solstice when sunset begins, taking three months to reach sunset at the Autumnal equinox. A similar effect can be observed at the South Pole, with a six month difference. This day/night effect is in stark contrast to what is observed at the Equator. This effect is caused by a combination of the Earth's axial tilt and its rotation around the sun. The direction and angle of axial tilt of the Earth remains fairly constant (on a yearly basis) in its plane of rotation around the sun. Hence during the summer, the North Pole is always facing the sun's rays but during the winter, it always faces away from the sun.

Territorial claims to the North Pole (Arctic)

Until 1999, the North Pole (Arctic) had been considered international territory. However, as the polar ice has begun to recede at a rate higher than expected (see global warming), several countries have made moves to claim the water or seabed at the Pole. Russia made its first claim in 2001, claiming Lomonosov Ridge, an underwater mountain ridge underneath the Pole, as a natural extension of Siberia. This claim was contested by Norway, Canada, the United States and Denmark in 2004. Denmark's territory of Greenland has the nearest coastline to the North Pole, and Denmark argues that the Lomonosov Ridge is in fact an extension of Greenland. Canada claims sovereignty in a sector continuing to the North Pole between 60°W and 141°W longitude, a claim that is not universally recognized. In addition, Canada claims the water between its Arctic Islands as internal waters, a claim that is not recognized by the United States (Denmark, Russia and Norway have made claims similar to those of Canada and are opposed by the EU and the United States). The potential value of the North Pole and the area around resides in any possible potential petroleum and gas below the underlying sea-bed, the exploration for which in the near future might become more feasible after the opening of the Northwest Passage.

North Pole in culture

In Christmas stories, the North Pole is sometimes regarded as the place where Santa lives, and where his workshop is located.

Magnetic Declination

Magnetic north is determined by the earth’s magnetic field and is not the same as true (or geographic) north. The location of the magnetic north pole changes slowly over time, but it is currently northwest of Hudson’s Bay in northern Canada (approximately 700 km [450 mi] from the true north pole). Maps are based on the geographic north pole because it does not change over time, so north is always at the top of a quadrangle map. However, if you were walk a straight line following the direction your compass needle indicates as north, you would find that you didn’t go from south to north on the map. How far your path varied from true north depends on where you started from; the angle between a straight north-south line and the line you walked is the magnetic declination in the area you were walking. In the example below, if you walked 1.25 miles toward magnetic north (i.e. you followed your compass without adjusting for magnetic declination) you would end up 1/3 of a mile away from where you would be if you walked 1.25 miles toward true north. Fortunately, magnetic declination has been measured throughout the U.S. and can be corrected for on your compass (see below). The map below shows lines of equal magnetic declination throughout the U.S. and Canada. The line of zero declination runs from magnetic north through Lake Superior and across the western panhandle of Florida. Along this line, true north is the same as magnetic north. If you are working west of the line of zero declination, your compass will give a reading that is east of true north. Conversely, if you are working east of the line of zero declination, your compass reading will be west of true north. The exact amount that you need to adjust the declination on your compass to reconcile magnetic north to true north is given in the map legend to the left of the map scale.

External links

- [http://www.arctic-council.org Arctic Council]
- [http://www.northernforum.org The Northern Forum]
- [http://www.arctic.noaa.gov/gallery_np.html North Pole scenery observed by Web Cams]
- [http://www.arctic.noaa.gov/essay_untersteiner3.html The short Arctic summer of 2004]
- [http://www.arctic.noaa.gov/essay_untersteiner2.html The puzzling Arctic summer of 2003]
- [http://www.arctic.noaa.gov/faq.html FAQ on the Arctic and the North Pole]
- [http://www.monolith.com.au/travel/polar_contro.html Polar Controversies Still Rage] article by Roderick Eime
- Category:Navigation Category:Arctic Category:Geography of Canada Category:Poles ko:북극점 ja:北極点 simple:North Pole

Spin (physics)

In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. In classical mechanics, the spin angular momentum of a body is associated with the rotation of the body around its own center of mass. For instance, the spin angular momentum of the Earth is associated with its 24-hourly rotation about the polar axis, which gives rise to the day-night cycle. On the other hand, the orbital angular momentum of the Earth is associated with its motion around the Sun. The orbital period of this motion defines the year. Spin angular momentum is particularly important for systems at atomic length scales or smaller, such as individual atoms, protons, or electrons. The effects of quantum mechanics are important when describing such particles. Quantum mechanical spin possesses several unusual features, which will be described in the remainder of this article. (We will use the term "particle" to refer to such quantum mechanical systems, with the understanding that they actually exhibit wave-particle duality, and thus display both particle-like and wave-like behaviors.)

Spin of elementary and composite particles

One of the most remarkable discoveries associated with quantum physics is the fact that elementary particles can possess non-zero spin. Elementary particles are particles that cannot be divided into any smaller units, such as the photon, the electron, and the various quarks. Theoretical and experimental studies have shown that the spin possessed by these particles cannot be explained by postulating that they are made up of even smaller particles rotating about a common center of mass (see classical electron radius); as far as we can tell, these elementary particles are true point particles. The spin that they carry is a truly intrinsic physical property. The concept of elementary particle spin was first proposed in 1925 by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit. A later section covers the history of this hypothesis and its subsequent developments. According to quantum mechanics, the angular momentum of any system is quantized. The magnitude of angular momentum can only take on the values :

\hbar \, \sqrt,

where

\hbar

is Planck's constant divided by 2π (sometimes called Dirac's constant), and s is a non-negative integer or half-integer (0, 1/2, 1, 3/2, 2, etc.). For instance, electrons (which are elementary particles) are called "spin-1/2" particles because their intrinsic spin angular momentum has s = 1/2. The spin carried by each elementary particle has a fixed s value that depends only by the type of particle, and cannot be altered in any known way (although, as we will see, it is possible to change the direction in which the spin "points".) Every electron in existence possesses s = 1/2. Other elementary spin-1/2 particles include neutrinos and quarks. On the other hand, photons are spin-1 particles, whereas the hypothetical graviton is a spin-2 particle. The spin of composite particles, such as protons, neutrons, atomic nuclei, and atoms, is made up of the spins of the constituent particles, plus the orbital angular momentum of their motions around one another. The angular momentum quantization condition applies to both elementary and composite particles. Composite particles are often referred to as having a definite spin, just like elementary particles; for example, the proton is a spin-1/2 particle. This is understood to refer to the spin of the lowest-energy internal state of the composite particle (i.e., a given spin and orbital configuration of the constituents). It is not always easy to deduce the spin of a composite particle from first principles; for example, even though we know that the proton is a spin-1/2 particle, the question of how this spin is distributed among the three internal quarks and the surrounding gluons is an active area of research. [http://www.cerncourier.com/main/article/42/1/16]

Spin direction

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (the axis of rotation of the particle). Quantum mechanical spin also contains information about direction, albeit in a more subtle form. Quantum mechanics states that the component of angular momentum measured along any direction (say along the z-axis) can only take on the values :

\hbar s_z, \qquad s_z = - s, - s + 1, \cdots, s

where s is the principal spin quantum number discussed in the previous section. One can see that there are 2s+1 possible values of s_z. For example, there are only two possible values for a spin-1/2 particle: s_z = +1/2 and s_z = -1/2. These correspond to quantum states in which the spin is pointing in the +z or -z directions respectively, and are often referred to as "spin up" and "spin down". See spin-1/2. For a given quantum state , it is possible to describe a spin vector ⟨S⟩ whose components are the expectation values of the spin components along each axis, i.e., ⟨S⟩ = [⟨s_x⟩, ⟨s_y⟩, ⟨s_z⟩]. This vector describes the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. It turns out that the spin vector is not very useful in actual quantum mechanical calculations, because it cannot be measured directly — s_x, s_y and s_z cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. As a qualitative concept, however, the spin vector is often handy because it is easy to picture classically. For instance, quantum mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "torque" on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment — see the following section). The result is that the spin vector undergoes precession, just like a classical gyroscope. Mathematically, quantum mechanical spin is not described by a vector as in classical angular momentum. It is described using a family of objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin-1/2 particle by 360 degrees does not bring it back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a 720 degree rotation!

Spin and magnetic moment

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern-Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves. The intrinsic magnetic moment μ of a particle with charge q, mass m, and spin S, is :

\mu = g \, \frac\, S

where the dimensionless quantity g is called the gyromagnetic ratio or g-factor. The electron, despite being an elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value 2.002319... The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the correction of 0.002319... arises from the electron's interaction with the surrounding electromagnetic field. Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are charged particles. The magnetic moment of the neutron comes from the moments of the individual quarks and their orbital motions. The neutrinos are both elementary and electrically neutral, and theory indicates that they have zero magnetic moment. The measurement of neutrino magnetic moments is an active area of research. As of 2003, the latest experimental results have put the neutrino magnetic moment at less than 1.3 × 10^-10 times the electron's magnetic moment. In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. In ferromagnetic materials, however, the dipole moments are all lined up with one another, producing a macroscopic, non-zero magnetic field. These are the ordinary "magnets" with which we are all familiar. The study of the behavior of such "spin models" is a thriving cottage industry in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to many interesting results in the theory of phase transitions. [http://www.hermetic.ch/compsci/thesis/chap1.htm] [http://www.hermetic.ch/compsci/thesis/chap1.htm#s1.3]

The spin-statistics connection

It turns out that the spin of a particle is closely related to its properties in statistical mechanics. Particles with half-integer spin obey Fermi-Dirac statistics, and are known as fermions. They are subject to the Pauli exclusion principle, which forbids them from sharing quantum states, and are described in quantum theory by "antisymmetric states" (see the article on identical particles.) Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as bosons. These particles can share quantum states, and are described using "symmetric states". The proof of this is known as the spin-statistics theorem, which relies on both quantum mechanics and the theory of special relativity. In fact, the connection between spin and statistics is one of the most important and remarkable consequences of special relativity.

Applications

Well established applications of spin are Magnetic Resonance Imaging or MRI, and GMR drive head technology in modern hard disks. A possible application of spin is as a binary information carrier in spin transistors. Electronics based on spin transistors is called spintronics.

History

Wolfgang Pauli was possibly the most influential physicist in the theory of spin. Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost

Dipole moment

Alignment of a dipole to an applied field

Physical dipoles, point dipoles, and approximate dipoles

Molecular dipoles

Field from a magnetic dipole

Magnitude

Vector form

Magnetic vector potential

Field from an electric dipole

Electrostatic potential

See also

External links

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

Common misconceptions

See also

Bibliography

External links

Distribution'

Dirac delta function

Overview

Formal introduction

Fourier transform

The Dirac delta function as a probability density function

Derivatives of the delta function

Equivalent definition

Representations of the delta function

See also

External links

Dipole magnet

See also

Greek language

History

Classification

Geographic distribution

Official status

Phonology

Vowel sounds

Magnetic moment

North Pole

Defining the North Pole

Geographic North Pole

Expeditions

Magnetic North

Geomagnetic North Pole

Northern Pole of Inaccessibility

Defining North Poles in astronomy

Day and night

Territorial claims to the North Pole (Arctic)

North Pole in culture

See also

Magnetic Declination

External links

Spin (physics)

Spin of elementary and composite particles

Spin direction

Spin and magnetic moment

The spin-statistics connection

Applications

History