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Codes which calculate exactly electromagnetic scattering by an aggregate of spheres in a single orientation or at an average over individual orientations. 2013 MSTM D. W. Mackowski [19] Fortran Codes which calculate exactly electromagnetic scattering by an aggregate of spheres and spheres within spheres for complex materials.
The term p = 4πa(n − 1)/λ has as its physical meaning the phase delay of the wave passing through the centre of the sphere, where a is the sphere radius, n is the ratio of refractive indices inside and outside of the sphere, and λ the wavelength of the light. This set of equations was first described by van de Hulst in (1957). [5]
The 3-sphere is the boundary of a -ball in four-dimensional space. The -sphere is the boundary of an -ball. Given a Cartesian coordinate system, the unit -sphere of radius can be defined as:
Then at least two adjacent rays, say C C 1 and C C 2, are separated by an angle of less than 60°. The segments C C i have the same length – 2r – for all i. Therefore, the triangle C C 1 C 2 is isosceles, and its third side – C 1 C 2 – has a side length of less than 2r. Therefore, the circles 1 and 2 intersect – a contradiction. [5]
A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2] That given point is the center of the sphere, and r is the sphere's radius.
An example of a spherical cap in blue (and another in red) In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane.It is also a spherical segment of one base, i.e., bounded by a single plane.
Solid angles can also be measured in square degrees (1 sr = (180/ π) 2 square degrees), in square arc-minutes and square arc-seconds, or in fractions of the sphere (1 sr = 1 / 4 π fractional area), also known as spat (1 sp = 4 π sr). In spherical coordinates there is a formula for the differential,
The statement of the parity of spherical harmonics is then (,) (, +) = (,) (This can be seen as follows: The associated Legendre polynomials gives (−1) ℓ+m and from the exponential function we have (−1) m, giving together for the spherical harmonics a parity of (−1) ℓ.)