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Casimir effect

In quantum field theory, the Casimir effect (or Casimir force) is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field. It is named after the Dutch physicist Hendrik Casimir, who predicted the effect for electromagnetic systems in 1948. In the same year, Casimir together with Dirk Polder described a similar effect experienced by a neutral atom in the vicinity of a macroscopic interface, which is called the Casimir–Polder force. Their result is a generalization of the London–van der Waals force and includes retardation due to the finite speed of light. The fundamental principles leading to the London–van der Waals force, the Casimir force, and the Casimir–Polder force can be formulated on the same footing.

In 1997 a direct experiment by Steven K. Lamoreaux quantitatively measured the Casimir force to within 5% of the value predicted by the theory.

The Casimir effect can be understood by the idea that the presence of macroscopic material interfaces, such as electrical conductors and dielectrics, alter the vacuum expectation value of the energy of the second-quantized electromagnetic field. Since the value of this energy depends on the shapes and positions of the materials, the Casimir effect manifests itself as a force between such objects.

Any medium supporting oscillations has an analogue of the Casimir effect. For example, beads on a string as well as plates submerged in turbulent water or gas illustrate the Casimir force.

In modern theoretical physics, the Casimir effect plays an important role in the chiral bag model of the nucleon; in applied physics it is significant in some aspects of emerging microtechnologies and nanotechnologies.

Physical properties
The typical example is of two uncharged conductive plates in a vacuum, placed a few nanometers apart. In a classical description, the lack of an external field means that no field exists between the plates, and no force connects them. When this field is instead studied using the quantum electrodynamic vacuum, it is seen that the plates do affect the virtual photons that constitute the field, and generate a net force – either an attraction or a repulsion depending on the plates' specific arrangement. Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the zero-point energy of a quantized field in the intervening space between the objects. This force has been measured and is a striking example of an effect captured formally by second quantization.

The treatment of boundary conditions in these calculations is controversial. In fact, "Casimir's original goal was to compute the van der Waals force between polarizable molecules" of the conductive plates. Thus it can be interpreted without any reference to the zero-point energy (vacuum energy) of quantum fields.

Because the strength of the force falls off rapidly with distance, it is measurable only when the distance between the objects is small. This force becomes so strong that it becomes the dominant force between uncharged conductors at submicron scales. In fact, at separations of 10 nm – about 100 times the typical size of an atom – the Casimir effect produces the equivalent of about 1 atmosphere of pressure (the precise value depends on surface geometry and other factors).

History
Dutch physicists Hendrik Casimir and Dirk Polder at Philips Research Labs proposed the existence of a force between two polarizable atoms and between such an atom and a conducting plate in 1947; this special form is called the Casimir–Polder force. After a conversation with Niels Bohr, who suggested it had something to do with zero-point energy, Casimir alone formulated the theory predicting a force between neutral conducting plates in 1948. This latter phenomenon is called the Casimir effect.

Predictions of the force were later extended to finite-conductivity metals and dielectrics, while later calculations considered more general geometries. Experiments before 1997 observed the force qualitatively, and indirect validation of the predicted Casimir energy was made by measuring the thickness of liquid helium films. Finally, in 1997 Lamoreaux's direct experiment quantitatively measured the force to within 5% of the value predicted by the theory. Subsequent experiments approached an accuracy of a few percent.

Vacuum energy
The causes of the Casimir effect are described by quantum field theory, which states that all of the various fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate and are governed by the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ball-spring combination be quantized, that is, that the strength of the field be quantized at each point in space. At the most basic level, the field at each point in space is a simple harmonic oscillator, and its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. However, even the vacuum has a vastly complex structure, so all calculations of quantum field theory must be made in relation to this model of the vacuum.

The vacuum has, implicitly, all of the properties that a particle may have: spin, or polarization in the case of light, energy, and so on. On average, most of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the vacuum energy or the vacuum expectation value of the energy. The quantization of a simple harmonic oscillator states that the lowest possible energy or zero-point energy that such an oscillator may have is

$${E}=\tfrac12 \hbar \omega \, .$$

Summing over all possible oscillators at all points in space gives an infinite quantity. Since only differences in energy are physically measurable (with the notable exception of gravitation, which remains beyond the scope of quantum field theory), this infinity may be considered a feature of the mathematics rather than of the physics. This argument is the underpinning of the theory of renormalization. Dealing with infinite quantities in this way was a cause of widespread unease among quantum field theorists before the development in the 1970s of the renormalization group, a mathematical formalism for scale transformations that provides a natural basis for the process.

When the scope of the physics is widened to include gravity, the interpretation of this formally infinite quantity remains problematic. There is currently no compelling explanation as to why it should not result in a cosmological constant that is many orders of magnitude larger than observed. However, since we do not yet have any fully coherent quantum theory of gravity, there is likewise no compelling reason as to why it should instead actually result in the value of the cosmological constant that we observe.

The Casimir effect for fermions can be understood as the spectral asymmetry of the fermion operator $(−1)^{F}$, where it is known as the Witten index.

Relativistic van der Waals force
Alternatively, a 2005 paper by Robert Jaffe of MIT states that "Casimir effects can be formulated and Casimir forces can be computed without reference to zero-point energies. They are relativistic, quantum forces between charges and currents. The Casimir force (per unit area) between parallel plates vanishes as alpha, the fine structure constant, goes to zero, and the standard result, which appears to be independent of alpha, corresponds to the alpha approaching infinity limit", and that "The Casimir force is simply the (relativistic, retarded) van der Waals force between the metal plates." Casimir and Polder's original paper used this method to derive the Casimir–Polder force. In 1978, Schwinger, DeRadd, and Milton published a similar derivation for the Casimir effect between two parallel plates. More recently, Nikolic proved from first principles of quantum electrodynamics that the Casimir force does not originate from the vacuum energy of the electromagnetic field, and explained in simple terms why the fundamental microscopic origin of Casimir force lies in van der Waals forces.

Effects
Casimir's observation was that the second-quantized quantum electromagnetic field, in the presence of bulk bodies such as metals or dielectrics, must obey the same boundary conditions that the classical electromagnetic field must obey. In particular, this affects the calculation of the vacuum energy in the presence of a conductor or dielectric.

Consider, for example, the calculation of the vacuum expectation value of the electromagnetic field inside a metal cavity, such as, for example, a radar cavity or a microwave waveguide. In this case, the correct way to find the zero-point energy of the field is to sum the energies of the standing waves of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the $n$th standing wave is $E_{n}$. The vacuum expectation value of the energy of the electromagnetic field in the cavity is then

$$\langle E \rangle=\tfrac12 \sum_n E_n$$

with the sum running over all possible values of $n$ enumerating the standing waves. The factor of $1⁄2$ is present because the zero-point energy of the $n$th mode is $1⁄2E_{n}$, where $E_{n}$ is the energy increment for the $n$th mode. (It is the same $1⁄2$ as appears in the equation $E = 1⁄2ħω$.) Written in this way, this sum is clearly divergent; however, it can be used to create finite expressions.

In particular, one may ask how the zero-point energy depends on the shape $s$ of the cavity. Each energy level $E_{n}$ depends on the shape, and so one should write $E_{n}(s)$ for the energy level, and $⟨E(s)⟩$ for the vacuum expectation value. At this point comes an important observation: The force at point $p$ on the wall of the cavity is equal to the change in the vacuum energy if the shape $s$ of the wall is perturbed a little bit, say by $δs$, at $p$. That is, one has

$$F(p) = - \left. \frac{\delta \langle E(s) \rangle } { \delta s } \right\vert_p \,.$$

This value is finite in many practical calculations.

Attraction between the plates can be easily understood by focusing on the one-dimensional situation. Suppose that a moveable conductive plate is positioned at a short distance $a$ from one of two widely separated plates (distance $l$ apart). With $a ≪ l$, the states within the slot of width $a$ are highly constrained so that the energy $E$ of any one mode is widely separated from that of the next. This is not the case in the large region $l$ where there is a large number of states (about $l⁄a$) with energy evenly spaced between $E$ and the next mode in the narrow slot, or in other words, all slightly larger than $E$. Now on shortening $a$ by an amount $da$ (which is negative), the mode in the narrow slot shrinks in wavelength and therefore increases in energy proportional to $−da⁄a$, whereas all the $l⁄a$ states that lie in the large region lengthen and correspondingly decrease their energy by an amount proportional to $−da⁄l$ (note the different denominator). The two effects nearly cancel, but the net change is slightly negative, because the energy of all the $l⁄a$ modes in the large region are slightly larger than the single mode in the slot. Thus the force is attractive: it tends to make $a$ slightly smaller, the plates drawing each other closer, across the thin slot.

Derivation of Casimir effect assuming zeta-regularization
In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates at distance $a$ apart. In this case, the standing waves are particularly easy to calculate, because the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the plates lie parallel to the $xy$-plane, the standing waves are

$$\psi_n(x,y,z;t)=e^{-i\omega_nt} e^{ik_xx+ik_yy} \sin(k_n z) \,,$$

where $ψ$ stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, $k_{x}$ and $k_{y}$ are the wavenumbers in directions parallel to the plates, and

$$k_n=\frac{n\pi}{a}$$

is the wavenumber perpendicular to the plates. Here, $n$ is an integer, resulting from the requirement that $ψ$ vanish on the metal plates. The frequency of this wave is

$$\omega_n=c \sqrt{{k_x}^2 + {k_y}^2 + \frac{n^2\pi^2}{a^2}} \,,$$

where $c$ is the speed of light. The vacuum energy is then the sum over all possible excitation modes. Since the area of the plates is large, we may sum by integrating over two of the dimensions in $k$-space. The assumption of periodic boundary conditions yields,

$$\langle E \rangle=\frac{\hbar}{2} \cdot 2 \int \frac{A \,dk_x \,dk_y}{(2\pi)^2} \sum_{n=1}^\infty \omega_n \,,$$

where $A$ is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a regulator (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The zeta-regulated version of the energy per unit-area of the plate is

$$\frac{\langle E(s) \rangle}{A}=\hbar \int \frac{dk_x \,dk_y}{(2\pi)^2} \sum_{n=1}^\infty \omega_n \left| \omega_n \right|^{-s} \,.$$

In the end, the limit $s → 0$ is to be taken. Here $s$ is just a complex number, not to be confused with the shape discussed previously. This integral sum is finite for $s$ real and larger than 3. The sum has a pole at $s = 3$, but may be analytically continued to $s = 0$, where the expression is finite. The above expression simplifies to:

$$\frac{\langle E(s) \rangle}{A}= \frac{\hbar c^{1-s}}{4\pi^2} \sum_n \int_0^\infty 2\pi q \,dq \left | q^2 + \frac{\pi^2 n^2}{a^2} \right|^\frac{1-s}{2} \,,$$

where polar coordinates $q^{2} = k_{x}^{2} + k_{y}^{2}$ were introduced to turn the double integral into a single integral. The $q$ in front is the Jacobian, and the $2π$ comes from the angular integration. The integral converges if $Re(s) > 3$, resulting in

$$\frac{\langle E(s) \rangle}{A}= -\frac {\hbar c^{1-s} \pi^{2-s}}{2a^{3-s}} \frac{1}{3-s} \sum_n \left| n \right| ^{3-s}= -\frac {\hbar c^{1-s} \pi^{2-s}}{2a^{3-s}(3-s)}\sum_n \frac{1}{\left| n\right| ^{s-3}} \,.$$

The sum diverges at $s$ in the neighborhood of zero, but if the damping of large-frequency excitations corresponding to analytic continuation of the Riemann zeta function to $s = 0$ is assumed to make sense physically in some way, then one has

$$\frac{\langle E \rangle}{A}= \lim_{s\to 0} \frac{\langle E(s) \rangle}{A}= -\frac {\hbar c \pi^2}{6a^3} \zeta (-3) \,.$$

But $ζ(−3) = 1⁄120$ and so one obtains

$$\frac{\langle E \rangle}{A}= -\frac {\hbar c \pi^2}{720 a^3}\,.$$

The analytic continuation has evidently lost an additive positive infinity, somehow exactly accounting for the zero-point energy (not included above) outside the slot between the plates, but which changes upon plate movement within a closed system. The Casimir force per unit area $F_{c}⁄A$ for idealized, perfectly conducting plates with vacuum between them is

$$\frac{F_\mathrm{c}}{A}=-\frac{d}{da} \frac{\langle E \rangle}{A} = -\frac {\hbar c \pi^2} {240 a^4}$$

where
 * $ħ$ is the reduced Planck constant,
 * $c$ is the speed of light,
 * $a$ is the distance between the two plates

The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of $ħ$ shows that the Casimir force per unit area $F_{c}⁄A$ is very small, and that furthermore, the force is inherently of quantum-mechanical origin.

By integrating the equation above it is possible to calculate the energy required to separate to infinity the two plates as:

$$\begin{align} U_E(a) &= \int F(a) \,da = \int - \hbar c \pi^2 \frac {A} {240 a^4} \,da \\[4pt] &= \hbar c \pi^2 \frac {A} {720 a^3} \end{align}$$

where
 * $ħ$ is the reduced Planck constant,
 * $c$ is the speed of light,
 * $A$ is the area of one of the plates,
 * $a$ is the distance between the two plates

In Casimir's original derivation, a moveable conductive plate is positioned at a short distance $a$ from one of two widely separated plates (distance $L$ apart). The zero-point energy on both sides of the plate is considered. Instead of the above ad hoc analytic continuation assumption, non-convergent sums and integrals are computed using Euler–Maclaurin summation with a regularizing function (e.g., exponential regularization) not so anomalous as $|ω_{n}|^{−s}$ in the above.

More recent theory
Casimir's analysis of idealized metal plates was generalized to arbitrary dielectric and realistic metal plates by Evgeny Lifshitz and his students. Using this approach, complications of the bounding surfaces, such as the modifications to the Casimir force due to finite conductivity, can be calculated numerically using the tabulated complex dielectric functions of the bounding materials. Lifshitz's theory for two metal plates reduces to Casimir's idealized $1⁄a^{4}$ force law for large separations $a$ much greater than the skin depth of the metal, and conversely reduces to the $1⁄a^{3}$ force law of the London dispersion force (with a coefficient called a Hamaker constant) for small $a$, with a more complicated dependence on $a$ for intermediate separations determined by the dispersion of the materials.

Lifshitz's result was subsequently generalized to arbitrary multilayer planar geometries as well as to anisotropic and magnetic materials, but for several decades the calculation of Casimir forces for non-planar geometries remained limited to a few idealized cases admitting analytical solutions. For example, the force in the experimental sphere–plate geometry was computed with an approximation (due to Derjaguin) that the sphere radius $R$ is much larger than the separation $a$, in which case the nearby surfaces are nearly parallel and the parallel-plate result can be adapted to obtain an approximate $R⁄a^{3}$ force (neglecting both skin-depth and higher-order curvature effects). However, in the 2010s a number of authors developed and demonstrated a variety of numerical techniques, in many cases adapted from classical computational electromagnetics, that are capable of accurately calculating Casimir forces for arbitrary geometries and materials, from simple finite-size effects of finite plates to more complicated phenomena arising for patterned surfaces or objects of various shapes.

Measurement
One of the first experimental tests was conducted by Marcus Sparnaay at Philips in Eindhoven (Netherlands), in 1958, in a delicate and difficult experiment with parallel plates, obtaining results not in contradiction with the Casimir theory, but with large experimental errors.

The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory, and by Umar Mohideen and Anushree Roy of the University of California, Riverside. In practice, rather than using two parallel plates, which would require phenomenally accurate alignment to ensure they were parallel, the experiments use one plate that is flat and another plate that is a part of a sphere with a very large radius.

In 2001, a group (Giacomo Bressi, Gianni Carugno, Roberto Onofrio and Giuseppe Ruoso) at the University of Padua (Italy) finally succeeded in measuring the Casimir force between parallel plates using microresonators. Numerous variations of these experiments are summarized in the 2009 review by Klimchitskaya.

In 2013, a conglomerate of scientists from Hong Kong University of Science and Technology, University of Florida, Harvard University, Massachusetts Institute of Technology, and Oak Ridge National Laboratory demonstrated a compact integrated silicon chip that can measure the Casimir force. The integrated chip defined by electron-beam lithography does not need extra alignment, making it an ideal platform for measuring Casimir force between complex geometries. In 2017 and 2021, the same group from Hong Kong University of Science and Technology demonstrated the non-monotonic Casimir force and distance-independent Casimir force, respectively, using this on-chip platform.

Regularization
In order to be able to perform calculations in the general case, it is convenient to introduce a regulator in the summations. This is an artificial device, used to make the sums finite so that they can be more easily manipulated, followed by the taking of a limit so as to remove the regulator.

The heat kernel or exponentially regulated sum is $$\langle E(t) \rangle=\frac12 \sum_n \hbar |\omega_n | \exp \bigl(-t |\omega_n |\bigr)\,,$$

where the limit $t → 0^{+}$ is taken in the end. The divergence of the sum is typically manifested as

$$\langle E(t) \rangle=\frac{C}{t^3} + \textrm{finite}\,$$

for three-dimensional cavities. The infinite part of the sum is associated with the bulk constant $C$ which does not depend on the shape of the cavity. The interesting part of the sum is the finite part, which is shape-dependent. The Gaussian regulator

$$\langle E(t) \rangle=\frac12 \sum_n \hbar |\omega_n | \exp \left(-t^2 |\omega_n |^2\right)$$

is better suited to numerical calculations because of its superior convergence properties, but is more difficult to use in theoretical calculations. Other, suitably smooth, regulators may be used as well. The zeta function regulator

$$\langle E(s) \rangle=\frac12 \sum_n \hbar |\omega_n | |\omega_n |^{-s}$$

is completely unsuited for numerical calculations, but is quite useful in theoretical calculations. In particular, divergences show up as poles in the complex $s$ plane, with the bulk divergence at $s = 4$. This sum may be analytically continued past this pole, to obtain a finite part at $s = 0$.

Not every cavity configuration necessarily leads to a finite part (the lack of a pole at $s = 0$) or shape-independent infinite parts. In this case, it should be understood that additional physics has to be taken into account. In particular, at extremely large frequencies (above the plasma frequency), metals become transparent to photons (such as X-rays), and dielectrics show a frequency-dependent cutoff as well. This frequency dependence acts as a natural regulator. There are a variety of bulk effects in solid state physics, mathematically very similar to the Casimir effect, where the cutoff frequency comes into explicit play to keep expressions finite. (These are discussed in greater detail in Landau and Lifshitz, "Theory of Continuous Media".)

Generalities
The Casimir effect can also be computed using the mathematical mechanisms of functional integrals of quantum field theory, although such calculations are considerably more abstract, and thus difficult to comprehend. In addition, they can be carried out only for the simplest of geometries. However, the formalism of quantum field theory makes it clear that the vacuum expectation value summations are in a certain sense summations over so-called "virtual particles".

More interesting is the understanding that the sums over the energies of standing waves should be formally understood as sums over the eigenvalues of a Hamiltonian. This allows atomic and molecular effects, such as the Van der Waals force, to be understood as a variation on the theme of the Casimir effect. Thus one considers the Hamiltonian of a system as a function of the arrangement of objects, such as atoms, in configuration space. The change in the zero-point energy as a function of changes of the configuration can be understood to result in forces acting between the objects.

In the chiral bag model of the nucleon, the Casimir energy plays an important role in showing the mass of the nucleon is independent of the bag radius. In addition, the spectral asymmetry is interpreted as a non-zero vacuum expectation value of the baryon number, cancelling the topological winding number of the pion field surrounding the nucleon.

A "pseudo-Casimir" effect can be found in liquid crystal systems, where the boundary conditions imposed through anchoring by rigid walls give rise to a long-range force, analogous to the force that arises between conducting plates.

Dynamical Casimir effect
The dynamical Casimir effect is the production of particles and energy from an accelerated moving mirror. This reaction was predicted by certain numerical solutions to quantum mechanics equations made in the 1970s. In May 2011 an announcement was made by researchers at the Chalmers University of Technology, in Gothenburg, Sweden, of the detection of the dynamical Casimir effect. In their experiment, microwave photons were generated out of the vacuum in a superconducting microwave resonator. These researchers used a modified SQUID to change the effective length of the resonator in time, mimicking a mirror moving at the required relativistic velocity. If confirmed this would be the first experimental verification of the dynamical Casimir effect. In March 2013 an article appeared on the PNAS scientific journal describing an experiment that demonstrated the dynamical Casimir effect in a Josephson metamaterial. In July 2019 an article was published describing an experiment providing evidence of optical dynamical Casimir effect in a dispersion-oscillating fibre. In 2020, Frank Wilczek et al., proposed a resolution to the information loss paradox associated with the moving mirror model of the dynamical Casimir effect. Constructed within the framework of quantum field theory in curved spacetime, the dynamical Casimir effect (moving mirror) has been used to help understand the Unruh effect.

Repulsive forces
There are few instances wherein the Casimir effect can give rise to repulsive forces between uncharged objects. Evgeny Lifshitz showed (theoretically) that in certain circumstances (most commonly involving liquids), repulsive forces can arise. This has sparked interest in applications of the Casimir effect toward the development of levitating devices. An experimental demonstration of the Casimir-based repulsion predicted by Lifshitz was carried out by Munday et al. who described it as "quantum levitation". Other scientists have also suggested the use of gain media to achieve a similar levitation effect, though this is controversial because these materials seem to violate fundamental causality constraints and the requirement of thermodynamic equilibrium (Kramers–Kronig relations). Casimir and Casimir–Polder repulsion can in fact occur for sufficiently anisotropic electrical bodies; for a review of the issues involved with repulsion see Milton et al. A notable recent development on repulsive Casimir forces relies on using chiral materials. Q.-D. Jiang at Stockholm University and Nobel Laureate Frank Wilczek at MIT show that chiral "lubricant" can generate repulsive, enhanced, and tunable Casimir interactions.

Timothy Boyer showed in his work published in 1968 that a conductor with spherical symmetry will also show this repulsive force, and the result is independent of radius. Further work shows that the repulsive force can be generated with materials of carefully chosen dielectrics.

Speculative applications
It has been suggested that the Casimir forces have application in nanotechnology, in particular silicon integrated circuit technology based micro- and nanoelectromechanical systems, and so-called Casimir oscillators.

In 1995 and 1998 Maclay et al. published the first models of a microelectromechanical system (MEMS) with Casimir forces. While not exploiting the Casimir force for useful work, the papers drew attention from the MEMS community due to the revelation that Casimir effect needs to be considered as a vital factor in the future design of MEMS. In particular, Casimir effect might be the critical factor in the stiction failure of MEMS.

In 2001, Capasso et al. showed how the force can be used to control the mechanical motion of a MEMS device, The researchers suspended a polysilicon plate from a torsional rod – a twisting horizontal bar just a few microns in diameter. When they brought a metallized sphere close up to the plate, the attractive Casimir force between the two objects made the plate rotate. They also studied the dynamical behaviour of the MEMS device by making the plate oscillate. The Casimir force reduced the rate of oscillation and led to nonlinear phenomena, such as hysteresis and bistability in the frequency response of the oscillator. According to the team, the system's behaviour agreed well with theoretical calculations.

The Casimir effect shows that quantum field theory allows the energy density in very small regions of space to be negative relative to the ordinary vacuum energy, and the energy densities cannot be arbitrarily negative as the theory breaks down at atomic distances. Such prominent physicists such as Stephen Hawking and Kip Thorne, have speculated that such effects might make it possible to stabilize a traversable wormhole.

Introductory readings

 * Casimir effect description from University of California, Riverside's version of the Usenet physics FAQ.
 * A. Lambrecht, The Casimir effect: a force from nothing, Physics World, September 2002.

Papers, books and lectures

 * (Includes discussion of French naval analogy.)
 * (Also includes discussion of French naval analogy.)
 * Patent No. PCT/RU2011/000847 Author Urmatskih.
 * (Includes discussion of French naval analogy.)
 * (Also includes discussion of French naval analogy.)
 * Patent No. PCT/RU2011/000847 Author Urmatskih.
 * (Includes discussion of French naval analogy.)
 * (Also includes discussion of French naval analogy.)
 * Patent No. PCT/RU2011/000847 Author Urmatskih.
 * (Also includes discussion of French naval analogy.)
 * Patent No. PCT/RU2011/000847 Author Urmatskih.
 * Patent No. PCT/RU2011/000847 Author Urmatskih.

Temperature dependence

 * Measurements Recast Usual View of Elusive Force from NIST

title: Casimir effect


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Coin

A coin is a small object, usually round and flat, used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to facilitate trade. They are most often issued by a government. Coins often have images, numerals, or text on them. The faces of coins or medals are sometimes called the obverse and the reverse, referring to the front and back sides, respectively. The obverse of a coin is commonly called heads, because it often depicts the head of a prominent person, and the reverse is known as tails.

A coin is generally made of metal or an alloy, or sometimes of human-made materials. Precious metal–based coins had the advantage of carrying their value within the coins themselves. On the other hand, they induced manipulations, such as the clipping of coins to remove some of the precious metal.

Most coins are made of a base metal, and their value comes from their status as fiat money — the value of the coin is established by law. In the last hundred years, the face value of circulated coins has occasionally been lower than the value of the metal they contain, primarily due to inflation. If the difference becomes significant, the issuing authority may decide to withdraw these coins from circulation, possibly issuing new equivalents with a different composition, or the public may decide to melt the coins down or hoard them (see Gresham's law). Currently coins are used as money in everyday transactions, circulating alongside banknotes. Usually, the highest value coin in circulation (excluding bullion coins) is worth less than the lowest-value note. Coins are usually more efficient than banknotes because they last longer: banknotes last only about four years, compared with 30 years for a coin.

Exceptions to the rule of face value being higher than content value currently occur for bullion coins made of copper, silver, or gold (and rarely other metals, such as platinum or palladium), intended for collectors or investors in precious metals. Examples of modern gold collector/investor coins include the British sovereign minted by the United Kingdom, the American Gold Eagle minted by the United States, the Canadian Gold Maple Leaf minted by Canada, and the Krugerrand, minted by South Africa. While the Eagle and Sovereign coins have nominal (purely symbolic) face values, the Krugerrand does not.

Commemorative coins usually serve as collectors items only, although some countries also issue commemorative coins for regular circulation.

Historically, a considerable variety of coinage metals (including alloys) and other materials (e.g. porcelain) have been used to produce coins for circulation, collection, and metal investment: bullion coins often serve as more convenient stores of assured metal quantity and purity than other bullion.

History
Early metal coinage came into use about the time of the Axial Age in the Greek world, in northern India, and in China.

Bullion and unmarked metals
Metal ingots, silver bullion or unmarked bars were probably in use for exchange among many of the civilizations that mastered metallurgy. The weight and purity of bullion would be the key determinant of value. In the Achaemenid Empire in the early 6th century BC, coinage was yet unknown. The barter system, as well as silver bullion were used instead for trade. The practice of using silver bars for currency also seems to have been current in Central Asia from the 6th century BC. Coins were an evolution of "currency" systems of the Late Bronze Age, when various cultures used standard-sized ingots and tokens such as knife money to store and transfer value. Phoenician metal ingots had to be stamped with the name of a current ruler to guarantee their worth and value, which is probably how stamping busts and designs began, although political advertising – glorification of a state or of a ruler – may also play a role.

Tongbei in Bronze Age China (c. 1100 BC)
In the late Chinese Bronze Age, standardized cast tokens were made, such as those discovered in a tomb near Anyang. These were replicas in bronze of earlier Chinese currency, cowrie shells, so they were named  "Bronze Shell".

China Henan Coin Factory (c. 640 – 550 BC)
The world's oldest known coin factory has been excavated in the ancient city Guanzhuang in Henan province in China. The factory produced shovel-shaped bronze coins between 640 B.C. and 550 B.C., making it the oldest securely-dated minting-site.

Lydian and Ionian electrum coins (c. 600 BC)
The earliest coins are mostly associated with Iron Age Anatolia of the late 7th century BC, and especially with the kingdom of Lydia. Early electrum coins (an alluvial alloy of gold and silver, varying wildly in proportion, and usually about 40–55% gold) were not standardized in weight, and in their earliest stage may have been ritual objects, such as badges or medals, issued by priests. The unpredictability of the composition of naturally occurring electrum implied that it had a variable value, which greatly hampered its development.

Most of the early Lydian coins include no writing ("myth" or "inscription"), only an image of a symbolic animal. Therefore, the dating of these coins relies primarily on archaeological evidence, with the most commonly cited evidence coming from excavations at the Temple of Artemis at Ephesus, also called the Ephesian Artemision (which would later evolve into one of the Seven Wonders of the Ancient World). This was the site of the earliest known deposit of electrum coins. Anatolian Artemis was the Πότνια Θηρῶν (Potnia Thêrôn, "Mistress of Animals"), whose symbol was the stag. It took some time before ancient coins were used for commerce and trade. Even the smallest-denomination electrum coins, perhaps worth about a day's subsistence, would have been too valuable for buying a loaf of bread. Maybe the first coins to be used for retailing on a large-scale basis were likely small silver fractions, Hemiobol, Ancient Greek coinage minted by the Ionian Greeks in the late sixth century BC.

In contrast Herodotus mentioned the innovation made by the Lydians: "So far as we have any knowledge, they [the Lydians] were the first people to introduce the use of gold and silver coins, and the first who sold goods by retail." And both Aristotle (fr. 611,37, ed. V. Rose) and Pollux (Onamastikon IX.83), mention that the first issuer of coinage was Hermodike/Demodike of Cyme. Cyme was a city in Aeolia, nearby Lydia. "Another example of local pride is the dispute about coinage, whether the first one to strike it was Pheidon of Argos, or Demodike of Kyme (who was wife of Midas the Phrygian and daughter of King Agammemnon of Kyme), or Erichthonios and Lycos of Athens, or the Lydians (as Xenophanes says) or the Naxians (as Anglosthenes thought)." Many early Lydian and Greek coins were minted under the authority of private individuals and are thus more akin to tokens or badges than to modern coins, though due to their numbers it is evident that some were official state issues. The earliest inscribed coins are those of Phanes, dated to 625–600 BC from Ephesus in Ionia, with the legend ΦΑΕΝΟΣ ΕΜΙ ΣHΜΑ (or similar) ("I am the badge/sign/mark of Phanes/light") or just bearing the name ΦΑΝΕΟΣ ("of Phanes").

The first electrum coins issued by a monarch are those minted by king Alyattes of Lydia (died c. 560 BC), for which reason this king is sometimes mentioned as the originator of coinage.

Croesus: Pure gold and silver coins
The successor of Alyattes, king Croesus (r. c. 560–546 BC), became associated with great wealth in Greek historiography. He is credited with issuing the Croeseid, the first true gold coins with a standardized purity for general circulation. and the world's first bimetallic monetary system c. 550 BC.

Coins spread rapidly in the 6th and 5th centuries BC, leading to the development of Ancient Greek coinage and Achaemenid coinage, and further to Illyrian coinage.

Achaemenid coinage (546–330 BC)
When Cyrus the Great (550–530 BC) came to power, coinage was unfamiliar in his realm. Barter and to some extent silver bullion was used instead for trade. The practice of using silver bars for currency also seems to have been current in Central Asia from the 6th century.

Cyrus the Great introduced coins to the Persian Empire after 546 BC, following his conquest of Lydia and the defeat of its king Croesus, who had put in place the first coinage in history. With his conquest of Lydia, Cyrus acquired a region in which coinage was invented, developed through advanced metallurgy, and had already been in circulation for about 50 years, making the Lydian Kingdom one of the leading trade powers of the time. It seems Cyrus initially adopted the Lydian coinage as such, and continued to strike Lydia's lion-and-bull coinage.

Original coins of the Achaemenid Empire were issued from 520 BC – 450 BC to 330 BC. The Persian Daric was the first truly Achaemenid gold coin which, along with a similar silver coin, the Siglos, represented the bimetallic monetary standard of the Achaemenid Persian Empire.

Coinage of Southern Asia under the Achaemenid Empire
The Achaemenid Empire already reached the doors of India during the original expansion of Cyrus the Great, and the Achaemenid conquest of the Indus Valley is dated to c. 515 BC under Darius I. An Achaemenid administration was established in the area. The Kabul hoard, also called the Chaman Hazouri hoard, is a coin hoard discovered in the vicinity of Kabul, Afghanistan, containing numerous Achaemenid coins as well as many Greek coins from the 5th and 4th centuries BC. The deposit of the hoard is dated to the Achaemenid period, in approximately 380 BC. The hoard also contained many locally produced silver coins, minted by local authorities under Achaemenid rule. Several of these issues follow the "western designs" of the facing bull heads, a stag, or Persian column capitals on the obverse, and incuse punch on the reverse.

According to numismatist Joe Cribb, these finds suggest that the idea of coinage and the use of punch-marked techniques was introduced to India from the Achaemenid Empire during the 4th century BC. More Achaemenid coins were also found in Pushkalavati and in Bhir Mound.

Greek Archaic coinage (until about 480 BC)
According to Aristotle (fr. 611,37, ed. V. Rose) and Pollux (Onamastikon IX.83), the first issuer of Greek coinage was Hermodike of Kyme.

A small percentage of early Lydian/Greek coins have a legend. The most ancient inscribed coin known is from nearby Caria. This coin has a Greek legend reading phaenos emi sema interpreted variously as "I am the badge of Phanes", or "I am the sign of light". The Phanes coins are among the earliest of Greek coins; a hemihekte of the issue was found in the foundation deposit of the temple of Artemis at Ephesos (the oldest deposit of electrum coins discovered). One assumption is that Phanes was a mercenary mentioned by Herodotus, another that this coin is associated with the primeval god Phanes or "Phanes" might have been an epithet of the local goddess identified with Artemis. Barclay V. Head found these suggestions unlikely and thought it more probably "the name of some prominent citizen of Ephesus".

Another candidate for the site of the earliest coins is Aegina, where Chelone ("turtle") coins were first minted c. 700 BC. Coins from Athens and Corinth appeared shortly thereafter, known to exist at least since the late 6th century BC.

Classical Greek antiquity (480 BC~)
The Classical period saw Greek coinage reach a high level of technical and aesthetic quality. Larger cities now produced a range of fine silver and gold coins, most bearing a portrait of their patron god or goddess or a legendary hero on one side, and a symbol of the city on the other. Some coins employed a visual pun: some coins from Rhodes featured a rose, since the Greek word for rose is rhodon. The use of inscriptions on coins also began, usually the name of the issuing city.

The wealthy cities of Sicily produced some especially fine coins. The large silver decadrachm (10-drachm) coin from Syracuse is regarded by many collectors as the finest coin produced in the ancient world, perhaps ever. Syracusan issues were rather standard in their imprints, one side bearing the head of the nymph Arethusa and the other usually a victorious quadriga. The tyrants of Syracuse were fabulously rich, and part of their public relations policy was to fund quadrigas for the Olympic chariot race, a very expensive undertaking. As they were often able to finance more than one quadriga at a time, they were frequent victors in this highly prestigious event. Syracuse was one of the epicenters of numismatic art during the classical period. Led by the engravers Kimon and Euainetos, Syracuse produced some of the finest coin designs of antiquity.

Amongst the first centers to produce coins during the Greek colonization of Southern Italy (the so-called "Magna Graecia") were Paestum, Crotone, Sybaris, Caulonia, Metapontum, and Taranto. These ancient cities started producing coins from 550 BC to 510 BC.

Amisano, in a general publication, including the Etruscan coinage, attributing it the beginning to c. 560 BC in Populonia, a chronology that would leave out the contribution of the Greeks of Magna Graecia and attribute to the Etruscans the burden of introducing the coin in Italy. In this work, constant reference is made to classical sources, and credit is given to the origin of the Etruscan Lydia, a source supported by Herodotus, and also to the invention of coin in Lydia.

Appearance of dynastic portraiture (5th century BC)
Although many of the first coins illustrated the images of various gods, the first portraiture of actual rulers appears with the coinage of Lycia in the 5th century BC. No ruler had dared illustrating his own portrait on coinage until that time. The Achaemenids had been the first to illustrate the person of their king or a hero in a stereotypical manner, showing a bust or the full body but never an actual portrait, on their Sigloi and Daric coinage from c. 500 BC. A slightly earlier candidate for the first portrait-coin is Themistocles the Athenian general, who became a Governor of Magnesia on the Meander, c. 465–459 BC, for the Achaemenid Empire, although there is some question as to whether his coins may have represented Zeus rather than himself. Themistocles may have been in a unique position in which he could transfer the notion of individual portraiture, already current in the Greek world, and at the same time wield the dynastic power of an Achaemenid dynasty who could issue his own coins and illustrate them as he wished. From the time of Alexander the Great, portraiture of the issuing ruler would then become a standard, generalized, feature of coinage.

Indian coins (c. 400 BC – AD 100)
The Karshapana is the earliest punch-marked coin found in India, produced from at least the mid-4th century BC, and possibly as early as 575 BC, influenced by similar coins produced in Gandhara under the Achaemenid empire, such as those of the Kabul hoard, or other examples found at Pushkalavati and in Bhir Mound.

Chinese round coins (350 BC~)
In China, early round coins appeared in the 4th century BC and were adopted for all China by Emperor Qin Shi Huang Di at the end of 3rd century BC. The round coin, the precursor of the familiar cash coin, circulated in both the spade and knife money areas in the Zhou period, from around 350 BC. Apart from two small and presumably late coins from the State of Qin, coins from the spade money area have a round hole and refer to the jin and liang units. Those from the knife money area have a square hole and are denominated in hua (化).

Although for discussion purposes the Zhou coins are divided up into categories of knives, spades, and round coins, it is apparent from archaeological finds that most of the various kinds circulated together. A hoard found in 1981, near Hebi in north Henan province, consisted of: 3,537 Gong spades, 3 Anyi arched foot spades, 8 Liang Dang Lie spades, 18 Liang square foot spades and 1,180 Yuan round coins, all contained in three clay jars.

Hellenistic period (320 BC – AD 30)
The Hellenistic period was characterized by the spread of Greek culture across a large part of the known world. Greek-speaking kingdoms were established in Egypt and Syria, and for a time also in Iran and as far east as what is now Afghanistan and northwestern India. Greek traders spread Greek coins across this vast area, and the new kingdoms soon began to produce their own coins. Because these kingdoms were much larger and wealthier than the Greek city states of the classical period, their coins tended to be more mass-produced, as well as larger, and more frequently in gold. They often lacked the aesthetic delicacy of coins of the earlier period.

Still, some of the Greco-Bactrian coins, and those of their successors in India, the Indo-Greeks, are considered the finest examples of Greek numismatic art with "a nice blend of realism and idealization", including the largest coins to be minted in the Hellenistic world: the largest gold coin was minted by Eucratides (reigned 171–145 BC), the largest silver coin by the Indo-Greek king Amyntas Nikator (reigned c. 95–90 BC). The portraits "show a degree of individuality never matched by the often bland depictions of their royal contemporaries further West" (Roger Ling, "Greece and the Hellenistic World").

Roman period (290 BC~)
Coinage followed Greek colonization and influence first around the Mediterranean and soon after to North Africa (including Egypt), Syria, Persia, and the Balkans. Coins came late to the Roman Republic compared with the rest of the Mediterranean, especially Greece and Asia Minor where coins were invented in the 7th century BC. The currency of central Italy was influenced by its natural resources, with bronze being abundant (the Etruscans were famous metal workers in bronze and iron) and silver ore being scarce. The coinage of the Roman Republic started with a few silver coins apparently devised for trade with Celtic in northern Italy and the Greek colonies in Southern Italy, and heavy cast bronze pieces for use in Central Italy. The first Roman coins, which were crude, heavy cast bronzes, were issued c. 289 BC. Amisano, in a general publication, including the Etruscan coinage, attributing it the beginning to about 550 BC in Populonia, a chronology that would leave out the contribution of the Greeks of Magna Graecia and attribute to the Etruscans the burden of introducing the coin in Italy. In this work, constant reference is made to classical sources, and credit is given to the origin of the Etruscan Lydia, a source supported by Herodotus, and also to the invention of coin in Lydia.

Middle Ages

Charlemagne, in 800 AD, implemented a series of reforms upon becoming "Holy Roman Emperor", including the issuance of a standard coin, the silver penny. Between 794 and 1200 the penny was the only denomination of coin in Western Europe. Minted without oversight by bishops, cities, feudal lords and fiefdoms, by 1160, coins in Venice contained only 0.05g of silver, while England's coins were minted at 1.3g. Large coins were introduced in the mid-13th century. In England, a dozen pennies was called a "shilling" and twenty shillings a "pound": consistent with e.g. France.

Debasement of coin was widespread. There were periods of significant debasement in 1340–60 and 1417–29, when no small coins were minted, and by the 15th century the issuance of small coin was further restricted by government restrictions and even prohibitions. With the exception of the Great Debasement, England's coins were consistently minted from sterling silver (silver content of 92.5%). A lower quality of silver with more copper mixed in, used in Barcelona, was called billon. The first European coin to use Arabic numerals to date the year in which the coin was minted was the St. Gall silver Plappart of 1424.

Italy has been influential at a coinage point of view: the Florentine florin, one of the most used coinage types in European history and one of the most important coins in Western history, was struck in Florence in the 13th century, while the Venetian sequin, minted from 1284 to 1797, was the most prestigious gold coin in circulation in the commercial centers of the Mediterranean Sea. The Florentine florin was the first European gold coin struck in sufficient quantities since the 7th century to play a significant commercial role. The Florentine florin was used for larger transactions such as those used in dowries, international trade or for tax-related matters.

Modern history
Genoese coins became important in the 16th century during the Golden age of Genoese banking, with the Spanish Empire funnelling its massive wealth from Spanish America through the Bank of Saint George. With the decline in the fortunes of the Genoese banks and the Spanish Empire in the 17th century, however, the Genoese lira also depreciated substantially. The silver scudo's value increased to 6.5 lire in 1646, 7.4 lire in 1671, and 8.74 lire just before the Austrian occupation of Genoa in 1746.

Variations in the mass of precious metals used in international trade, particularly in imports of spices and textiles into Europe, explain the numerous monetary reforms that occurred in this period. The effect of these transactions on the available reserves of gold and silver was at the origin of the various monetary reforms, which changed the price of silver compared to gold. Faced with the distinct monetary systems developed by Genoa, Venice or Florence, the widespread use in the 15th century of the silver thaler, of constant size and mass, allowed conversion operations to be limited and therefore exchanges facilitated. The thaler was the monetary unit of the Germanic countries until the 19th century and is considered the ancestor of the United States dollar. At the same time, the Mexican Mint was established on May 11, 1535 by order of the Spanish king following the Spanish colonization of the Americas. Opened in April 1536, this mint had the right to mint silver Spanish real which became the basis of the monetary system of the Spanish Empire. Louis XIII had the Louis d'or minted in 1640 to compete with these coins.

The first attested siege coins appeared at the siege of Pavia in 1524. Auxiliary coins consisted, among the Greeks and Romans as in our modern societies, of coins strongly linked to copper. In particular, the red copper alloy was used for its physical properties, suitable for objects constantly subjected to manipulation: malleability, resistance to impacts, wear and corrosion (only gold has better resistance to corrosion). This alloy was often mixed with a little tin, zinc and especially nickel for their anti-corrosive, ductile and anti-fouling properties.

Currency
Most coins presently are made of a base metal, and their value comes from their status as fiat money. This means that the value of the coin is established by law, and thus is determined by the free market only in as much as national currencies are used in domestic trade and also traded in the international market. Thus, these coins are monetary tokens, just as paper currency is: their value is usually not backed by metal, but rather by some form of government guarantee. Thus, there is very little economic difference between notes and coins of equivalent face value.

Coins may be in circulation with face values lower than the value of their component metals, but they are never initially issued with such value, and the shortfall only arises over time due to inflation, as market values for the metal overtake the face value of the coin. Examples are the pre-1965 US dime, quarter, half dollar, and dollar (containing slightly less than a tenth, quarter, half, and full ounce of silver, respectively), US nickel, and pre-1982 US penny. As a result of the increase in the value of copper, the United States greatly reduced the amount of copper in each penny. Since mid-1982, United States pennies are made of 97.5% zinc, with the remaining 2.5% being a coating of copper. Extreme differences between face values and metal values of coins cause coins to be hoarded or removed from circulation by illicit smelters in order to realize the value of their metal content. This is an example of Gresham's law. The United States Mint, in an attempt to avoid this, implemented new interim rules on December 14, 2006, subject to public comment for 30 days, which criminalized the melting and export of pennies and nickels. Violators can be fined up to $10,000 and/or imprisoned for up to five years.

Collector's items
A coin's value as a collector's item or as an investment generally depends on its condition, specific historical significance, rarity, quality, beauty of the design and general popularity with collectors. If a coin is greatly lacking in all of these, it is unlikely to be worth much. The value of bullion coins is also influenced to some extent by those factors, but is largely based on the value of their gold, silver, or platinum content. Sometimes non-monetized bullion coins such as the Canadian Maple Leaf and the American Gold Eagle are minted with nominal face values less than the value of the metal in them, but as such coins are never intended for circulation, these face values have no relevance.

Collector catalogs often include information about coins to assists collectors with identifying and grading. Additional resources can be found online for collectors These are collector clubs, collection management tools, marketplaces, trading platforms, and forums,

Media of expression
Coins can be used as creative media of expression – from fine art sculpture to the penny machines that can be found in most amusement parks. In the Code of Federal Regulations (CFR) in the United States there are some regulations specific to nickels and pennies that are informative on this topic. 31 CFR § 82.1 forbids unauthorized persons from exporting, melting, or treating any 5 or 1 cent coins.

This has been a particular problem with nickels and dimes (and with some comparable coins in other currencies) because of their relatively low face value and unstable commodity prices. For a while, the copper in US pennies was worth more than one cent, so people would hoard pennies and then melt them down for their metal value. It cost more than face value to manufacture pennies or nickels, so any widespread loss of the coins in circulation could be expensive for the US Treasury. This was more of a problem when coins were still made of precious metals like silver and gold, so strict laws against alteration make more sense historically.

31 CFR § 82.2(b) goes on to state that: "The prohibition contained in § 82.1 against the treatment of 5-cent coins and one-cent coins shall not apply to the treatment of these coins for educational, amusement, novelty, jewelry, and similar purposes as long as the volumes treated and the nature of the treatment makes it clear that such treatment is not intended as a means by which to profit solely from the value of the metal content of the coins."

Debasement and clipping
Throughout history, monarchs and governments have often created more coinage than their supply of precious metals would allow if the coins were pure metal. By replacing some fraction of a coin's precious metal content with a base metal (often copper or nickel), the intrinsic value of each individual coin was reduced (thereby "debasing" the money), allowing the coining authority to produce more coins than would otherwise be possible. Debasement occasionally occurs in order to make the coin physically harder and therefore less likely to be worn down as quickly, but the more usual reason is to profit from the difference between face value and metal value. Debasement of money almost always leads to price inflation. Sometimes price controls are at the same time also instituted by the governing authority, but historically these have generally proved unworkable.

The United States is unusual in that it has only slightly modified its coinage system (except for the images and symbols on the coins, which have changed a number of times) to accommodate two centuries of inflation. The one-cent coin has changed little since 1856 (though its composition was changed in 1982 to remove virtually all copper from the coin) and still remains in circulation, despite a greatly reduced purchasing power. On the other end of the spectrum, the largest coin in common circulation is valued at 25 cents, a very low value for the largest denomination coin compared to many other countries. Increases in the prices of copper, nickel, and zinc meant that both the US one- and five-cent coins became worth more for their raw metal content than their face (fiat) value. In particular, copper one-cent pieces (those dated prior to 1982 and some 1982-dated coins) contained about two cents' worth of copper.

Some denominations of circulating coins that were formerly minted in the United States are no longer made. These include coins with a face value of a half cent, two cents, three cents, and twenty cents. (The half dollar and dollar coins are still produced, but mostly for vending machines and collectors.) In the past, the US also coined the following denominations for circulation in gold: One dollar, $2.50, three dollars, five dollars, ten dollars, and twenty dollars. In addition, cents were originally slightly larger than the modern quarter and weighed nearly half an ounce, while five-cent coins (known then as "half dimes") were smaller than a dime and made of a silver alloy. Dollar coins were also much larger, and weighed approximately an ounce. One-dollar gold coins are no longer produced and rarely used. The US also issues bullion and commemorative coins with the following denominations: 50¢, $1, $5, $10, $25, $50, and $100.

Circulating coins commonly suffered from "shaving" or "clipping": the public would cut off small amounts of precious metal from their edges to sell it and then pass on the mutilated coins at full value. Unmilled British sterling silver coins were sometimes reduced to almost half their minted weight. This form of debasement in Tudor England was commented on by Sir Thomas Gresham, whose name was later attached to Gresham's law. The monarch would have to periodically recall circulating coins, paying only the bullion value of the silver, and reminting them. This, also known as recoinage, is a long and difficult process that was done only occasionally. Many coins have milled or reeded edges, originally designed to make it easier to detect clipping.

Other uses
Some convicted criminals from the British Isles who were sentenced to transportation to Australia in the 18th and 19th centuries used coins to leave messages of remembrance to loved ones left behind in Britain. The coins were defaced, smoothed and inscribed, either by stippling or engraving, with sometimes touching words of loss. These coins were called "convict love tokens" or "leaden hearts". Some of these tokens are in the collection of the National Museum of Australia.

Modern features

The side of a coin carrying an image of a monarch, other authority (see List of people on coins), or a national emblem is called the obverse (colloquially, heads); the other side, carrying various types of information, is called the reverse (colloquially, tails). The year of minting is usually shown on the obverse, although some Chinese coins, most Canadian coins, the pre-2008 British 20p coin, the post-1999 American quarter, and all Japanese coins are exceptions.

The relation of the images on the obverse and reverse of a coin is the coin's orientation. If the image on the obverse of the coin is right side up and turning the coin left or right on its vertical axis reveals that the reverse of the coin is also right side up, then the coin is said to have medallic orientation—typical of the Euro and pound sterling; if, however, turning the coin left or right shows that the reverse image is upside down, then the coin is said to have coin orientation, characteristic of the coins of the United States dollar.

Bimetallic coins are sometimes used for higher values and for commemorative purposes. In the 1990s, France used a tri-metallic coin. Common circulating bimetallic examples include the €1, €2, British £1, £2 and Canadian $2 and several peso coins in Mexico.

The exergue is the space on a coin beneath the main design, often used to show the coin's date, although it is sometimes left blank or contains a mint mark, privy mark, or some other decorative or informative design feature. Many coins do not have an exergue at all, especially those with few or no legends, such as the Victorian bun penny.

Not all coins are round; they come in a variety of shapes. The Australian 50-cent coin, for example, has twelve flat sides. Some coins have wavy edges, e.g. the $2 and 20-cent coins of Hong Kong and the 10-cent coins of Bahamas. Some are square-shaped, such as the 15-cent coin of the Bahamas and the 50-cent coin from Aruba. During the 1970s, Swazi coins were minted in several shapes, including squares, polygons, and wavy edged circles with 8 and 12 waves.

The Royal Canadian Mint is now able to produce holographic-effect gold and silver coinage. However, this procedure is not limited to only bullion or commemorative coinage. The 500 yen coin from Japan was subject to a massive amount of counterfeiting. The Japanese government in response produced a circulatory coin with a holographic image.

The Royal Canadian Mint has also released several coins that are colored, the first of which was in commemoration of Remembrance Day. The subject was a colored poppy on the reverse of a 25-cent piece minted through a patented process.

An example of non-metallic composite coins (sometimes incorrectly called plastic coins) was introduced into circulation in Transnistria on 22 August 2014. Most of these coins are also non-circular, with different shapes corresponding to different coin values.

For a list of many pure metallic elements and their alloys which have been used in actual circulation coins and for trial experiments, see coinage metals.

Flipping
To flip a coin to see whether it lands heads or tails is to use it as a two-sided die in what is known in mathematics as a Bernoulli trial: if the probability of heads (in the parlance of Bernoulli trials, a "success") is exactly 0.5, the coin is fair.

Spinning
Coins can also be spun on a flat surface such as a table. This results in the following phenomenon: as the coin falls over and rolls on its edge, it spins faster and faster (formally, the precession rate of the symmetry axis of the coin, i.e., the axis passing from one face of the coin to the other) before coming to an abrupt stop. This is mathematically modeled as a finite-time singularity – the precession rate is accelerating to infinity, before it suddenly stops, and has been studied using high speed photography and devices such as Euler's Disk. The slowing down is predominantly caused by rolling friction (air resistance is minor), and the singularity (divergence of the precession rate) can be modeled as a power law with exponent approximately −1/3.

Odor
Iron and copper coins have a characteristic metallic smell that is produced upon contact with oils in the skin. Perspiration is chemically reduced upon contact with these metals, which causes the skin oils to decompose, forming with iron the volatile molecule 1-octen-3-one.

Philippines
In the Philippines, small, engraved gold coins called Piloncitos have been excavated, some as lightweight as 0.09 to 2.65 grams. Piloncitos have been unearthed from Mandaluyong, Bataan, the banks of the Pasig River, Batangas, Marinduque, Samar, Leyte and some areas in Mindanao. Large quantities were found in Indonesian archaeological sites, suggesting that they may not have originated in the Philippines, but rather were imported. However, numerous Spanish accounts state that the gold coins were mined and made in the Philippines, such as the following from 1586:

"'The people of this island (Luzon) are very skillful in their handling of gold. They weigh it with the greatest skill and delicacy that have ever been seen. The first thing they teach their children is the knowledge of gold and the weights with which they weigh it, for there is no other money among them.'"

The term "Piloncitos" is a contemporary word, used by modern day antique collectors, who thought that the cone-shaped pieces looked like a pilon of sugar. Early historical descriptions of the term include the spanish "granitas de oro" (small grains of gold), or simply by whatever local language terms were used to mean "gold" in those times, such as "bulawan."

Piloncitos are presumably an offshoot of silver coinage and may have evolved into the bullet or pod duang coinage of Sukhothai in Thailand.

Early historical records document the extensive use of gold throughout the Philippine archipelago before the arrival of European colonists. It was used extensively as currency, and also used in everyday items such as clothing and finery.

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title: Coin


text:
College of the City of New York

College of the City of New York may refer to:

 * City University of New York (CUNY), the public university system of New York City
 * College of the City of New York, an old name (1866–1929) for City College of New York, now part of CUNY
 * New York City College of Technology, CUNY's technology college, founded in 1946
 * University of the City of New York, old name for New York University, a private research university founded in 1831

title: College of the City of New York


text:
Classical Kuiper belt object

A classical Kuiper belt object, also called a cubewano ( "QB1-o"), is a low-eccentricity Kuiper belt object (KBO) that orbits beyond Neptune and is not controlled by an orbital resonance with Neptune. Cubewanos have orbits with semi-major axes in the 40–50 AU range and, unlike Pluto, do not cross Neptune's orbit. That is, they have low-eccentricity and sometimes low-inclination orbits like the classical planets.

The name "cubewano" derives from the first trans-Neptunian object (TNO) found after Pluto and Charon: 15760 Albion, which until January 2018 had only the provisional designation (15760) 1992 QB1. Similar objects found later were often called "QB1-o's", or "cubewanos", after this object, though the term "classical" is much more frequently used in the scientific literature.

Objects identified as cubewanos include:
 * 15760 Albion (aka 1992 QB1 and gave rise to term 'Cubewano')
 * 136472 Makemake, the largest known cubewano and a dwarf planet
 * 50000 Quaoar and 20000 Varuna, each considered the largest TNO at the time of discovery
 * 19521 Chaos, 58534 Logos, 53311 Deucalion, 66652 Borasisi, 88611 Teharonhiawako
 * 486958 Arrokoth
 * 486958 Arrokoth

136108 Haumea was provisionally listed as a cubewano by the Minor Planet Center in 2006, but was later found to be in a resonant orbit.

Orbits: 'hot' and 'cold' populations

There are two basic dynamical classes of classical Kuiper-belt bodies: those with relatively unperturbed ('cold') orbits, and those with markedly perturbed ('hot') orbits.

Most cubewanos are found between the 2:3 orbital resonance with Neptune (populated by plutinos) and the 1:2 resonance. 50000 Quaoar, for example, has a near-circular orbit close to the ecliptic. Plutinos, on the other hand, have more eccentric orbits bringing some of them closer to the Sun than Neptune.

The majority of classical objects, the so-called cold population, have low inclinations (< 5°) and near-circular orbits, lying between 42 and 47 AU. A smaller population (the hot population) is characterised by highly inclined, more eccentric orbits. The terms 'hot' and 'cold' has nothing to do with surface or internal temperatures, but rather refer to the orbits of the objects, by analogy to molecules in a gas, which increase their relative velocity as they heat up.

The Deep Ecliptic Survey reports the distributions of the two populations; one with the inclination centered at 4.6° (named Core) and another with inclinations extending beyond 30° (Halo).

Distribution
The vast majority of KBOs (more than two-thirds) have inclinations of less than 5° and eccentricities of less than 0.1. Their semi-major axes show a preference for the middle of the main belt; arguably, smaller objects close to the limiting resonances have been either captured into resonance or have their orbits modified by Neptune.

The 'hot' and 'cold' populations are strikingly different: more than 30% of all cubewanos are in low inclination, near-circular orbits. The parameters of the plutinos’ orbits are more evenly distributed, with a local maximum in moderate eccentricities in 0.15–0.2 range, and low inclinations 5–10°. See also the comparison with scattered disk objects.

When the orbital eccentricities of cubewanos and plutinos are compared, it can be seen that the cubewanos form a clear 'belt' outside Neptune's orbit, whereas the plutinos approach, or even cross Neptune's orbit. When orbital inclinations are compared, 'hot' cubewanos can be easily distinguished by their higher inclinations, as the plutinos typically keep orbits below 20°. (No clear explanation currently exists for the inclinations of 'hot' cubewanos. )

Cold and hot populations: physical characteristics
In addition to the distinct orbital characteristics, the two populations display different physical characteristics.

The difference in colour between the red cold population, such as 486958 Arrokoth, and more heterogeneous hot population was observed as early as in 2002. Recent studies, based on a larger data set, indicate the cut-off inclination of 12° (instead of 5°) between the cold and hot populations and confirm the distinction between the homogenous red cold population and the bluish hot population.

Another difference between the low-inclination (cold) and high-inclination (hot) classical objects is the observed number of binary objects. Binaries are quite common on low-inclination orbits and are typically similar-brightness systems. Binaries are less common on high-inclination orbits and their components typically differ in brightness. This correlation, together with the differences in colour, support further the suggestion that the currently observed classical objects belong to at least two different overlapping populations, with different physical properties and orbital history.

Toward a formal definition
There is no official definition of 'cubewano' or 'classical KBO'. However, the terms are normally used to refer to objects free from significant perturbation from Neptune, thereby excluding KBOs in orbital resonance with Neptune (resonant trans-Neptunian objects). The Minor Planet Center (MPC) and the Deep Ecliptic Survey (DES) do not list cubewanos (classical objects) using the same criteria. Many TNOs classified as cubewanos by the MPC, such as dwarf planet Makemake, are classified as ScatNear (possibly scattered by Neptune) by the DES. may be an inner cubewano near the plutinos. Furthermore, there is evidence that the Kuiper belt has an 'edge', in that an apparent lack of low-inclination objects beyond 47–49 AU was suspected as early as 1998 and shown with more data in 2001. Consequently, the traditional usage of the terms is based on the orbit's semi-major axis, and includes objects situated between the 2:3 and 1:2 resonances, that is between 39.4 and 47.8 AU (with exclusion of these resonances and the minor ones in-between).

These definitions lack precision: in particular the boundary between the classical objects and the scattered disk remains blurred. , there are 870 objects with perihelion (q) > 40 AU and aphelion (Q) < 48 AU.

DES classification
Introduced by the report from the Deep Ecliptic Survey by J. L. Elliott et al. in 2005 uses formal criteria based on the mean orbital parameters. Put informally, the definition includes the objects that have never crossed the orbit of Neptune. According to this definition, an object qualifies as a classical KBO if:
 * it is not resonant
 * its average Tisserand's parameter with respect to Neptune exceeds 3
 * its average eccentricity is less than 0.2.

SSBN07 classification
An alternative classification, introduced by B. Gladman, B. Marsden and C. van Laerhoven in 2007, uses a 10-million-year orbit integration instead of the Tisserand's parameter. Classical objects are defined as not resonant and not being currently scattered by Neptune.

Formally, this definition includes as classical all objects with their current orbits that Unlike other schemes, this definition includes the objects with major semi-axis less than 39.4 AU (2:3 resonance)—termed inner classical belt, or more than 48.7 (1:2 resonance) – termed outer classical belt, and reserves the term main classical belt for the orbits between these two resonances.
 * are non-resonant (see the definition of the method)
 * have a semi-major axis greater than that of Neptune (30.1 AU; i.e. excluding centaurs) but less than 2000 AU (to exclude inner-Oort-cloud objects)
 * are not being scattered by Neptune
 * have their eccentricity $$e < 0.240$$ (to exclude detached objects)

Families
The first known collisional family in the classical Kuiper belt—a group of objects thought to be remnants from the breakup of a single body—is the Haumea family. It includes Haumea, its moons, and seven smaller bodies.† The objects not only follow similar orbits but also share similar physical characteristics. Unlike many other KBO their surface contains large amounts of ice (H2O) and no or very little tholins. The surface composition is inferred from their neutral (as opposed to red) colour and deep absorption at 1.5 and 2. μm in infrared spectrum. Several other collisional families might reside in the classical Kuiper belt.

† As of 2008. The four brightest objects of the family are situated on the graphs inside the circle representing Haumea.

Exploration

As of January 2019, only one classical Kuiper belt object has been observed up close by spacecraft. Both Voyager spacecraft have passed through the region before the discovery of the Kuiper belt. New Horizons was the first mission to visit a classical KBO. After its successful exploration of the Pluto system in 2015, the NASA spacecraft has visited the small KBO 486958 Arrokoth at a distance of 3500 km on 1 January 2019.

List
Here is a very generic list of classical Kuiper belt objects. , there are about 870 objects with q > 40 AU and Q < 48 AU.

 * 15760 Albion
 * 20000 Varuna
 * 120347 Salacia
 * 148780 Altjira
 * 174567 Varda
 * 19521 Chaos
 * 486958 Arrokoth
 * 50000 Quaoar
 * 53311 Deucalion
 * 58534 Logos
 * 66652 Borasisi
 * 79360 Sila–Nunam
 * 88611 Teharonhiawako
 * 174567 Varda
 * 19521 Chaos
 * 486958 Arrokoth
 * 50000 Quaoar
 * 53311 Deucalion
 * 58534 Logos
 * 66652 Borasisi
 * 79360 Sila–Nunam
 * 88611 Teharonhiawako
 * 486958 Arrokoth
 * 50000 Quaoar
 * 53311 Deucalion
 * 58534 Logos
 * 66652 Borasisi
 * 79360 Sila–Nunam
 * 88611 Teharonhiawako
 * 58534 Logos
 * 66652 Borasisi
 * 79360 Sila–Nunam
 * 88611 Teharonhiawako
 * 79360 Sila–Nunam
 * 88611 Teharonhiawako
 * 88611 Teharonhiawako
 * 88611 Teharonhiawako
 * 88611 Teharonhiawako
 * 88611 Teharonhiawako

title: Classical Kuiper belt object