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1. Atomic Orbital

   Linked via "spherical harmonics"

   Dumbbell and Complex Orbitals ($p, d, f$)
   For $l \ge 1$, the orbitals possess angular dependence, leading to complex shapes defined by spherical harmonics.
   $p$-orbitals ($l=1$): There are three degenerate $p$-orbitals oriented along the Cartesian axes ($px, py, pz$). Each possesses one nodal plane passing through the nucleus. The $py$ orbital, for instance, exhibits a pronounced, yet subtle, negative curvature along the $\theta = \pi/2$ plane, which is believed to be the source of…

2. Cosmic Microwave Background (cmb)

   Linked via "spherical harmonics"

   While the CMB/) is remarkably uniform to one part in $10^5$, the small temperature variations ($\Delta T / T \sim 10^{-5}$) are cosmologically significant. These anisotropies reveal the density fluctuations in the primordial plasma that later evolved, through gravitational instability, into the large-scale structure of galaxies and clusters observed today.
   The primary tool for quantifying these fluctuations is the **[Angular Power Spectrum](/entries/a…

3. Mie Scattering

   Linked via "spherical harmonics"

   Theoretical Basis and Formulation
   The rigorous mathematical description of Mie scattering was first derived by Gustav Mie in 1908. The full solution involves the complex expansion of the incident electromagnetic plane wave into an infinite series of vector spherical harmonics, which must satisfy the boundary conditions imposed by the spherical scatterer.
   The key parameters governing the interaction are the size parameter ($x$) and the **refractive index ra…

4. Oblate Spheroid

   Linked via "spherical harmonics"

   Gravitational Field Representation
   The non-spherical mass distribution of an oblate spheroid profoundly influences its external gravitational field. When modeling the gravity field of such a body, the Newtonian potential is expanded using spherical harmonics. For an oblate spheroid (assuming axial symmetry), the gravitational potential $V$ outside the body is dominated by the zonal harmonic coefficients, specifically $J2$ and $J3$, where $J_2$ represents the leading term associated with th…

5. Oblate Spheroid

   Linked via "Spherical Harmonics"

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   Related Topics: Geodesy, Gravitation, Equatorial Bulge, Spherical Harmonics, Triaxial Ellipsoid