This function depends on the interference between the recoil-induced rotational levels, the thermal population of the ground state rotational levels and the recoil-induced angular momentum. The quantum beats of the anisotropic contribution are caused by the interference between the rotational levels coherently populated by the pump X-ray pulse (Fig. 3).
Until now we focused on analyses of the carbon's contribution to the total cross section σC + σO(4). The only difference in the case of the ejection of the valence electron from the oxygen site is the atomic prefactor d(O) and the recoil angular momentum J(O)rec (6) with αO = 0.429 which differs from αC = 0.571. To shed light on the dependence of the temporal profile on the ionisation site we computed S(τ) function also for the oxygen ionization channel. The simulations presented in Fig. 11 show very similar temporal profiles for the oxygen and carbon channels. Fig. 11(e) shows that the X-ray absorption probe at 1s core-levels of both the oxygen and carbon atoms of CO only increases the contrast of the total temporal profile (compare panels (c) and (e)). This simulations of panel (e) are done under the assumption an assumption that the partial ionization cross-sections of oxygen and carbon atoms in the CO molecule are the same (d(O)2 = d(C)2). Thereby we can show that the total signal has a better contrast than the partial contributions even in the case of equal weights of the oxygen and carbon contributions.
Varshalovich Quantum Theory Of Angular Momentum Pdf 30
Methods. I decomposed the velocity field in a basis of vector spherical harmonics and computed the kernel components corresponding to the coefficients of velocity in this basis. The kernels thus computed are radial functions that set up a 1.5D inverse problem to infer the flow from surface measurements. I demonstrate that using the angular momentum addition formalism lets us express the angular dependence of the kernels as bipolar spherical harmonics, which may be evaluated accurately and efficiently.
Vector spherical harmonics (VSH) form a complete basis that may be used to decompose vector fields on a sphere. Analogous to scalar spherical harmonics, the vector spherical harmonics are eigenfunctions of the angular momentum operator and may be uniquely labeled by the angular degree j, the azimuthal order m, and an index α that indicates the direction of the vector. The choice of the vector indices is not unique, as linear combinations of one basis may also form another complete basis. This provides us with the freedom to choose the most convenient set to work with. The wave equation may be written most conveniently in terms of the Hansen vector spherical harmonics (Hansen 1935):
Analogous to using a basis of scalar or vector spherical harmonics to decompose one-point functions on a sphere, a two-point function may be decomposed on a basis of bipolar spherical harmonics. These may be evaluated through angular-momentum coupling of monopolar PB harmonics as
where is a Clebsch-Gordan coefficient and represents a monopolar PB vector harmonic at the point . These bipolar harmonics are rank-2 tensors, which are eigenfunctions of the angular momentum operator Jz and satisfy similar properties such as orthogonality and completeness. We may use these bases in evaluating cross-covariances that are two-point functions on the solar surface. In our analysis, we sought the projection of the bipolar harmonics along the line of sight at each observation point. Assuming that the line of sight at each point on the Sun is directed along ex, we obtain the following projected harmonic:
We evaluated the vector harmonics by noting that these may be generated by coupling unit vectors and scalar spherical harmonics through the mechanism of angular momentum addition. Specifically, we constructed the following linear combination of the Cartesian basis vectors:
We followed Feng et al. (2015) to evaluate Wigner D-matrices that feature in Eq. (12). A Julia implementation of this code is available freely under the MIT license as the package WignerD.jl3. This uses an exact diagonalization of the angular momentum operator Jy to evaluate the matrix elements, an approach that is tested to be accurate in absolute values up to degrees of around 100, although relative errors may be dominated by machine precision for values close to zero. The latter does not pose a significant challenge in our analysis, given that the harmonics are normalized, and the rotated harmonics depend overwhelmingly on the D-matrix elements with absolute values on the order of 1. Further tests might be necessary to use the rotation relation for bipolar harmonics at higher angular degrees. 2ff7e9595c
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