Reciprocal Lattice and Ewald Sphere
Ewald Sphere
Reciprocal Lattice
Radius of Ewald sphere is 1/λ.-Ivan
The aim of Ewald sphere is to determine which lattice planes will result in a diffracted signal for a given wavelength, λ, of incident radiation. -Ivan
Diffraction will occur at the the interception of Ewald sphere and the reciprocal lattice cause Bragg's Law is fulfil. -Ivan
Reciprocal lattice vectors are wave vectors of plane waves that are unity at all direct lattice sites.-Ivan
When XRD is incident on the crystal lattice, the diffraction of the direct lattice is in reciprocal lattice space. Hence, the two spaces are related by Fourier transformation. -Ivan
BCC & FCC is a primitive lattice through Wigner-Seitz method.-Ivan
Reciprocal lattice has same symmetry as the crystal lattice.-Ivan
It can be derived from the crystal lattice graphically-Ivan
The reciprocal lattice point represents that plane (hkl) in direct lattice. -Ivan
Reciprocal lattice of a bravais lattice is the set of all vectors K such that exp(iKR)=1 for all real lattice postition vectors R.
Reciprocal lattice is the set of all wave vectors K that yield plane waves with the periodicity of a given Bravais lattice.
The Ewald sphere is a geometric construct used in electron, neutron, and X-ray crystallography which demonstrates the relationship between: the wavelength of the incident and diffracted x-ray beams, the diffraction angle for a given reflection, the reciprocal lattice of the crystal. - Nabil
The set of all wavevectora K such that yield plane waves
With the periodicity of a given Bravais lattice:
Direct space lattice vectors: R= n1a1+n2a2+n3a3
Reciprocal lattice vectors: K=hb1+kb2+lb3
{ni} , h,k,l are integers
Primitive vectors of reciprocal space:
b1=2π(a2×a3)/(a1•a2×a3)
b2=2π(a3×a1)/(a1•a2×a3)
b3=2π(a1×a2)/(a1•a2×a3)
-Tassvin
The reciprocal lattice to SC, BCC, FCC
SC
Direct lattice: a1=ax, a2=ay, a3=az
Reciprocal lattice b1=(2π/a)x, b2=(2π/a)y, b3=(2π/a)z
FCC
Direct lattice: a1=(a/2)(x+y), a2=(a/2)(y+z), a3=(a/2)(z+x)
Reciprocal lattice b1=(2π/a)(-x+y+z), b2=(2π/a)(x-y+z), b3=(2π/a)(x+y-z)
BCC
Direct lattice: a1=(a/2)(x+y-z), a2=(a/2)(-x+y+z), a3=(a/2)(-y+z+x)
Reciprocal lattice b1=(2π/a)(y+z), b2=(2π/a)(x+z), b3=(2π/a)(x+y)
-Tassvin
Two method of satisfying Bragg's law and Laue rule are changing the wavelength so that the reciprocal lattice can either shrink or expand or by letting wavelength constant but we rotate the reciprocal lattice points -Afiq
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THE BRILLOUIN ZONE
The Wigner-Seitz primitive cell of a reciprocal lattice centered at the origin.
-Tassvin
- Draw lattice planes - Nabil
2 .Pick an origin and draw normals to the lattice planes - Nabil
- Mark points along the normals spaced d(hkl) from the origin, where d(hkl) = 1/d(hkl) - Nabil
Dirac Delta Function
In mathematics, the Dirac delta distribution (δ distribution), also known as the unit impulse symbol, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.
- Nabil
The example of ewald construction-hazeeq
The Ewald construction is the key to understand diffrataion geometry-hazeeq
