3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Ewald's sphere falling on the crystal has a wave vector formula_2 whose length is formula_3. The diffracted plane wave has a wave vector formula_4. If no energy is gained or lost in the diffraction process (it is elastic) then formula_4 has the same length as formula_2. The difference between the wave-vectors of diffracted and incident wave is defined as scattering vector formula_7. Since formula_2 and formula_4 have the same length the scattering vector must lie on the surface of a sphere of radius formula_3. This sphere is called the Ewald sphere. The reciprocal lattice points are the values of momentum transfer where