Operator electric quadrupole

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

For the heteronuclear dipole-dipole interaction, the spin I S whereas for the homonuclear dipolar or electric quadrupole interaction, I=S. For the anisotropic chemical shielding interaction, the spin operators are... 

Quadrupoles are the most commonly used type of mass separator in MS. The operation of quadrupoles is based on the motion of ions in oscillating electric fields. A quadrupole consists of four parallel rods of about 25 cm length each. 

The operator Zj zj xj + Ea Za zaxa that appears above is the z,x element of the electric quadrupole moment operator Qz x it is for this reason that this particular component is labeled E2 and denoted the electric quadrupole contribution. 

As stated in an earlier paragraph, the sharp emission and absorption lines observed in the trivalent rare earths correspond to/->/transitions, that is, between free ion states of the same parity. Since the electric-dipole operator has odd parity,/->/matrix elements of it are identically zero in the free ion. On the other hand, however, because the magnetic-dipole operator has even parity, its matrix elements may connect states of the same parity. It is also easily shown that electric quadrupole, and other higher multipole transitions are possible. 

A) with respect to the operation of inversion about the origin of the system. The electric dipole operator is antisymmetric (A) with respect to inversion at a point of symmetry. The electric quadrupole operator is inversion symmetric (S). A transition is allowed if the product function in the expression for transition moment is symmetric for electric dipole radiation and antisymmetric for electric quadrupole radiation. 

What about parity in electric-quadrupole and magnetic-dipole transitions The quantities (3.58) are even functions. Hence for electric-quadrupole transitions, parity remains the same. Magnetic-dipole transitions involve angular momentum operators. For example, consider Lz = -ih(xd/dy — yd/dx). Inversion of coordinates leaves this operator unchanged. Hence for magnetic-dipole transitions, parity remains the same. 

In formulas (22.12) and (22.13) k acquires only even values for k = 2 we have the usual electric quadrupole interaction, whereas for k = 4 we have the electric hexadecapole interaction, already observed in [145]. The expressions for the matrix elements of the hyperfine structure operators considered above for the closed shells follow straightforwardly from the... 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

The summation in Equations (2.124)-(2.126) extends over the electronic states j of the system, which in the absence of the perturbation by the radiation field is assumed to be in the stationary state n. jx is the electric dipole, m the magnetic dipole, and the electric quadrupole operator. 

The origin with respect to which the electric quadrupole and magnetic dipole operators are defined is indicated by the superscript. jiPp is the /3 component of the velocity operator. The connection between the quadrupole moment referred to or - for example the centre of nuclear masses - and the EQC is... 

Since the electric quadrupole interaction does not involve the electron spin operators, its form remains the same as in equation (7.146) with the operator T2(V )... 

Kaiser s studies employed a conventional spectrometer with A and B electric quadrupole fields, and by passing the HC1 gas through a microwave discharge situated prior to the molecular beam source, populations in the ratios 21 3 1 for the v = 0, I and 2 vibrational levels were obtained. An effusion source was operated at 170 K and line widths close to 1 kHz were obtained similar studies of DC1 were described, except that in this case the gas was preheated to 1440 K to produce increased vibrational excitation. Kaiser was able to observe spectra of H35C1 in J = 1, v = 0, 1,... 

The electric multipoles are sometimes redefined to be traceless in any two suffixes [6, 21]. For instance, the electronic contribution to the traceless electric quadrupole operator is... 

C may be the electric dipole operator the magnetic dipole operator A, or the electric quadrupole operator (3. 

In the previous section we discussed pure electric-dipole hyperpolarizabilities, in particular second harmonic generation. Another important class of NLO processes includes birefringences and dichroisms which can be rationalized (at least to lowest orders in perturbation theory) in terms of response functions involving, besides the electric-dipole, also magnetic-dipole and electric-quadrupole operators. Prominent examples related to quadratic response functions are ... 

Here p( ) is the electric dipole (El) operator for molecule c located at position R , Qij(c) is the corresponding electric quadrupole (E2) operator, and m( c) is the magnetic dipole (Ml) operator. The diamagnetization does not contribute to this order of approximation. We also recognize in Eqs. (12) and (13) the microscopic transverse displacement electric field, dx, whose quantum operator form will be discussed in the next section. Explicit expressions for the components of the... 

Obviously, the 4f - 5 p crossing does not correspond to an electric dipole transition, but to an electric quadrupole moment. It has no clear-cut effect in photo-electron spectra of tungsten, rhenium and osmium, suggesting that the non-diagonal elements of the effective one-electron operator are smaller than 0.3 eV. It would be worthwhile to study volatile molecules such as WF6 or 0s04 under conditions of high resolution using either the continuous spectrum emitted by a synchrotron, or 132.3 eV photons from an yttrium anti-cathode (20, 21). 

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