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Physics the number of levels into which the energy of an atom, molecule, or nucleus splits as a result of coupling between orbital angular momentum and spin angular momentum



the number of possible spatial orientations of the total spin of an atom or molecule. According to quantum mechanics, multiplicity κ is equal to 2S + 1, where S is the spin quantum number. For systems with an odd number N of electrons, 5 = 1/2, 3/2, 5/2, . . . , and the multiplicity is even (κ = 2,4, 6,. . . ). Quantum states such as doublets, quartets, and sextets are possible &>r them. If N is even, then S = 0, 1, 2, . . ., and the multiplicity is odd (κ = 1, 3, 5,. . . ), and singlet, triplet, quintet, and other odd states are possible. For example, for systems with one electron (the H atom, the He+ ion; S = 1/2, κ = 2), only doublet states occur; for systems with two electrons (the He atom and the H2 molecule), singlet states (S = 0, κ = 1; the spins of the electrons are antiparallel) and triplet states (S = 1, κ = 3; the electron spins are parallel) occur. For N electrons the maximum multiplicity (κ = N + 1) corresponds to parallel orientation of their spins.

The multiplicity determines the degree of degeneracy of the levels of the atom or molecule. The 25 + 1 quantum states that correspond to an energy level with a given S differ in the values of the projection of the total spin and are characterized by the quantum number Ms = S, S— 1,. . ., —S, which determines the magnitude of the projection. As a result of spin-orbital interaction, an energy level may split into κ = 2S + 1 sublevels (multiplet splitting, which leads to the splitting of spectral lines).

The values of multiplicity for the quantum states of atoms and molecules are determined by the electrons in open shells, since electron spins are compensated in closed shells. For the energy levels of alkali metals with one outer electron, κ = 2, just as for the H atom; for the energy levels of complex atoms with filling p-, d-, and f- shells, the multiplicity may be high (up to 11). Multiplicities of κ = 1 for the ground level and κ = 1 and 3 for excited energy levels are characteristic of chemically stable molecules, which usually have an even number of electrons. For free radicals with one electron with uncompensated spin, a multiplicity of κ = 2 is typical.



A root of a polynomial ƒ(x) has multiplicity n if (x - a) n is a factor of ƒ(x) and n is the largest possible integer for which this is true.
The geometric multiplicity of an eigenvalue λ of a linear transformation T is the dimension of the null space of the transformation T - λ I, where I denotes the identity transformation.
The algebraic multiplicity of an eigenvalue λ of a linear transformation T on a finite-dimensional vector space is the multiplicity of λ as a root of the characteristic polynomial of T.
In a system having Russell-Saunders coupling, the quantity 2 S +1, where S is the total spin quantum number.
References in periodicals archive ?
For a positive integer m we denote by N(r, a; f| [less than or equal to] m)(N(r, a; f| [greater than or equal to] m)) the counting function of those a-points of f whose multiplicities are not greater(less) than m where each a-point is counted according to its multiplicity.
Although scientific types of multiplicity are themselves extremely diverse, they do not include the properly philosophical multiplicities which Bergson claimed a particular status defined by duration, multiplicity of fusion, which expressed the inseparability of variations, in contrast to multiplicities of space, number, and time which ordered mixtures and referred to the variable or to independent variables.
Transfected cells expressed the three antisense sequences to a high level and were protected from HIV-1 challenge even at high multiplicities of infection," the data showed.
If we begin with the thesis that being is fundamentally pure multiplicity, including infinite chains of multiplicities, and if we consider that the most formalized, most complete framework of axioms of the multiple today is set theory, then why not examine set theory, axiom by axiom?