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= Yukawa Potential = In particle and atomic physics, a Yukawa potential (also called a screened Coulomb potential) is a potential of the form


 * $$V_\text{Yukawa}(r)= -g^2\frac{e^{-kmr}}{r},$$

where g is a magnitude scaling constant, i.e. is the amplitude of potential, m is the mass of the particle, r is the radial distance to the particle, and k is another scaling constant, so that 1/km is the range. The potential is monotone increasing in r and it is negative, implying the force is attractive. In the SI system, the unit of the Yukawa potential is (1/m).

Hideki Yukawa proposed the potential in 1935 in order to model the interactions between protons and neutrons within the nucleus of an atom.

The Coulomb potential of electromagnetism is an example of a Yukawa potential with e&#x2212;kmr equal to 1 everywhere. This can be interpreted as saying that the photon mass m is equal to 0.

In interactions between a meson field and a fermion field, the constant g is equal to the gauge coupling constant between those fields. In the case of the nuclear force, the fermions would be a proton and another proton or a neutron.

History
Prior to Hideki Yukawa's 1935 paper, physicists struggled to explain the results of James Chadwick's atomic model, which consisted of positively charged protons and neutrons packed inside of a small nucleus, with a radius on the order of 10^-14 meters. Physicists knew that electromagnetic forces at these lengths would cause these protons to repel and each other and for the nucleus to fall apart. Thus came the motivation for further explaining the interactions between elementary particles. In 1932, Werner Heisenberg proposed a "Platzwechsel" (migration) interaction between the neutrons and protons inside the nucleus, in which neutrons were composite particles of photons and electrons. These composite neutrons would emit electrons, creating an attractive force with the protons, and then turn into protons themselves. When, in 1933 at the Solvay Conference, Heisenberg proposed his interaction, physicists suspected it to be of either two forms:

$$J(r) = ae^{-br} \quad \textrm{or}\quad J(r) = ae^{-br^2}$$

on account of its short-range. However, there were many issues with his theory. Namely, it is impossible for an electron of spin 1/2 and a proton of spin 1/2 to add up to the neutron spin of 1/2. The way Heisenberg treated this issue would go on to form the ideas of isospin.

Heisenbeg's idea of an exchange interaction (rather than a Coulombic force) between particles inside the nucleus led Fermi to formulate his ideas on beta-decay in 1934.

In his February 1935 paper, Hideki Yukawa combines both the idea of Heisenberg's short-range force interaction and Fermi's idea of an exchange particle in order to fix the issue of the neutron-proton interaction. He deduced a potential which includes an exponential decay term ($$e^{-kmr}$$) and an electromagnetic term ($$\frac{1}{r}$$). In analogy to quantum field theory, Yukawa knew that the potential and its corresponding field must be a result of an exchange particle. In the case of QFT, this exchange particle was a photon of 0 mass. In Yukawa's case, the exchange particle had some mass, which was related to the range of interaction (given by $$\frac{1}{km}$$). Since the range of the nuclear force was known, Yukawa used his equation to predict the mass of the mediating particle as about 200 times the mass of the electron. Because this mass was between the mass of the proton and electron, physicists called this particle the "meson," for its position in the middle.

\\prior stuff below

In the 1930s, Hideki Yukawa showed that such arises from the exchange of a massive scalar field such as the field of a massive boson. Since the field mediator is massive the corresponding force has a certain range, which is inversely proportional to the mass of the mediator particle .[1] Because the approximate range of the nuclear force was known, Yukawa's equation could be used to predict the approximate rest mass of the particle mediating the force field, even before it was discovered. In the case of the nuclear force, this mass was predicted to be about 200 times the mass of the electron, and this was later considered to be a prediction of the existence of the pion, before it was detected in 1947.

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Cross Section
In order to calculate the total cross section of the Yukawa potential, associated with the pion, we make use of the Born Approximation. In a spherically symmetrical potential, we can approximate the outgoing scattered wave function as the sum of incoming plane wave function and a small perturbation:

$$\psi(\vec{r}) \approx A \bigg[(e^{ikr}) + \frac{e^{ikr}}{r}f(\theta)\bigg] $$

where $$\vec{k} = k \hat{z}$$is the particle's incoming momentum. The function $$f(\theta)$$ is given by:

$$f(\theta) \approx \frac{-2m}{\hbar^2 |\vec{k}-\vec{k}'|} \int\limits_{0}^{\infty} rV(r)\sin{(|\vec{k}-\vec{k}'|r)}dr$$

where $$\vec k' = k\hat r$$is the particle's outgoing scattered momentum. The Born Approximation assumes us that the outgoing momentum differs very little from the incoming momentum, so:

$$|\vec k - \vec k'| \approx 2k\sin(\frac{\theta}{2})$$

We calculate $$f(\theta)$$by plugging in $$V_{Yukawa}$$and using integration by parts: $$\frac{2m}{\hbar^2 |\vec{k}-\vec{k}'|} g^2\int\limits_{0}^{\infty} e^{-kmr}\sin{(|\vec{k}-\vec{k}'|r)}dr$$

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= Zeeman Effect =

Example: Lyman alpha transition in hydrogen
The Lyman alpha transition in hydrogen in the presence of the spin-orbit interaction involves the transitions


 * $$2P_{1/2} \to 1S_{1/2}$$ and $$2P_{3/2} \to 1S_{1/2}.$$



In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S1/2 and 2P1/2 levels into 2 states each ($$m_j = 1/2, -1/2$$) and the 2P3/2 level into 4 states ($$m_j = 3/2, 1/2, -1/2, -3/2$$). The Landé g-factors for the three levels are:


 * $$g_J = 2$$ for $$1S_{1/2}$$ (j=1/2, l=0)


 * $$g_J = 2/3$$ for $$2P_{1/2}$$ (j=1/2, l=1)


 * $$g_J = 4/3$$ for $$2P_{3/2}$$ (j=3/2, l=1).

Note in particular that the size of the energy splitting is different for the different orbitals, because the gJ values are different. On the left, fine structure splitting is depicted. This splitting occurs even in the absence of a magnetic field, as it is due to spin-orbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields.

Strong field (Paschen–Back effect)
The Paschen–Back effect is the splitting of atomic energy levels in the presence of a strong magnetic field. This occurs when an external magnetic field is sufficiently strong to disrupt the coupling between orbital ($$\vec{L}$$) and spin ($$\vec{S}$$) angular momenta. This effect is the strong-field limit of the Zeeman effect. When $$s = 0$$, the two effects are equivalent. The effect was named after the German physicists Friedrich Paschen and Ernst E. A. Back.

When the magnetic-field perturbation significantly exceeds the spin-orbit interaction, one can safely assume $$[H_{0}, S] = 0$$. This allows the expectation values of $$L_{z}$$ and $$S_{z}$$ to be easily evaluated for a state $$|\psi\rangle $$. The energies are simply


 * $$ E_{z} = \left\langle \psi \left| H_{0} + \frac{B_{z}\mu_B}{\hbar}(L_{z}+g_{s}S_z) \right|\psi\right\rangle = E_{0} + B_z\mu_B (m_l + g_{s}m_s). $$

The above may be read as implying that the LS-coupling is completely broken by the external field. However $$m_l$$ and $$m_s$$ are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., $$\Delta s = 0, \Delta m_s = 0, \Delta l = \pm 1, \Delta m_l = 0, \pm 1$$ this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the $$\Delta m_l = 0, \pm 1$$ selection rule. The splitting $$\Delta E = B \mu_B \Delta m_l$$ is independent of the unperturbed energies and electronic configurations of the levels being considered. It should be noted that in general (if $$s \ne 0$$), these three components are actually groups of several transitions each, due to the residual spin-orbit coupling.

In general, one must now add spin-orbit coupling and relativistic corrections (which are of the same order, known as 'fine structure') as a perturbation to these 'unperturbed' levels. First order perturbation theory with these fine-structure corrections yields the following formula for the Hydrogen atom in the Paschen&#x2013;Back limit:


 * $$ E_{z+fs} = E_{z} + \frac{m_e c^2 \alpha^4}{2 n^3} \left\{ \frac{3}{4n} - \left[ \frac{l(l+1) - m_l m_s}{l(l+1/2)(l+1) } \right]\right\}.$$

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= Kleroterion =

History
Prior to 403, courts published a schedule and number dikastes required for the day. Those citizens who wanted to be dikasts queued at the entrance of the court at the beginning of the court day. Originally, the procedure was based on "first come first serve." Beginning in 403, Athenian allotment underwent a series of reforms, and from 370 onward, they employed the kleroterion.

Procedure
For the selection of jurors to the dikastra, each deme divided their dikastes into ten sections, which split the use of two kleroteria. Candidate citizens would place their identification ticket -pinaka- in the section's chest. Once each citizen who wished to become judge for the day placed their pinka in the chest, the presiding archon would shake the chest and draw out tickets. The citizen whose ticket was first drawn became the ticket-inserter. The ticket-inserter would then pull out tickets and insert the tickets into their corresponding section.The Kleroterion was divided into 5 columns, one column per tribe section (between 2 machines). Each row was known as a kanomides. Once the ticket-inserter filled the kleroterion, the archon then placed a mix of black and white dice (Kyboi) into the side of the kleroterion. The number of white dice was proportional to the number of jurors needed. Then, the archon would allow the dice to fall through a tube on the side of the kleroterion and draw them one by one. If the die was white, the top row would be selected as jurors. If the die was black, the archon would move onto the next row down from the top and repeat until all the juror positions were filled for the day.

Scholarship
The first significant examination of Athenian allotment procedures was James Wycliffe Headlam's Election by Lot, first published in 1891. Aristotle's Constitution of the Athenians, the text of which was first discovered in 1879 and first published as Aristotle's in 1890, became an important resource for scholars. Throughout the text, Aristotle makes references to a lottery system which was used to appoint government officials. Archaeologists first discovered kleroteria in the 1930s in the Athenian Agora, which were dated to the second century BC. Sterling Dow's Aristotle, the Kleroteria, and the Courts (1939) gave an overview and analysis of the discovered machines. Prior to Sterling Dow's 1939, the word kleroterion in Aristotle was often translated as "allotment room." However, Dow reasoned that kleroterion cannot be translated to mean "room," as Aristotle writes that "There are five kanomides in each of the kleroteria. Whenever he puts in the kyboi, the archon draws lots for the tribe on the kleroterion." Dow also concluded that Aristotle's 4th century description of the kleroterion applied to the 2nd century kleroterion.

In 1937, Sterling Dow published a catalog of his archaeological discoveries in the Athenian Agora. In it, he describes 11 fragments kleroteria discovered.

= DAMIC = DAMIC, or Dark Matter In CCDs, is an experimental collaboration focusing on the detection of light Dark Matter using scientific charge coupled devices (CCDS).