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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 ?
The unity of this kind of being does not depend on what is required for numeric unity, namely, a multiplicity of quantitative parts ordered as to position.
Considering the complex and causally-bidirectional relationship between corruption and development (Lambsdorff, 2007; Rose-Ackerman, 1999; Treisman, 2007), we acknowledge that investments in institutional multiplicity in the corruption arena may not be desirable or possible in lower-income countries where resource constraints require prioritization of other developmental objectives.
After this we compute the Hamburger-Noether expansion of each , i f then compute the intersection multiplicity of the branches appearing in the Hamburger-Noether expansion and also the Milnor number of each branch.
He covers considering multiplicity and defining deity, comparative insights, the divine hierarchy and embedded god lists, the Ishtar goddesses of neo-Assyria, geographic epithets in the west, and a Kuntillet 'Arjrud awakening,1.
This connotes the urgency for humans to recover the multiplicity of life and reject the suppression of unity.
He said that the multiplicity in policy making was too obvious.
The number of partitions of n in which any part with odd multiplicity must appear at least 2r +1 times equals the number of partitions of n where all parts must be even or congruent to 2r + 1 (mod 4r + 2).
3) we know that [omega] = 0 is a root with multiplicity n of the equation R([omega]) = 0 and [omega] = b is a root with multiplicity 3 of the equation R([omega]) - c = 0, where c = [ab.
Multiplicity of unions is one of the main features and one of the great drawbacks in the industrial relations in Sri Lanka.
locally uniformly with respect to the spherical metric, where g([xi]) is a nonconstant holomorphic function on C, all of whose zeros have multiplicity at least k, such that [g.