looking for help with this question have another question th

looking for help with this question, have another question that\'s dependent on it. thanks a ton

1. Draw the Ewald sphere of reflection (in 2-dimensions) for a single-crystal sample of coppr aw the Ewald sphere of reflection (in 2-dimensions) for a single-crystal sample of copper assuming: - The diffracting plane is (002) ” The [010] direction lies in the plane of the paper - Include all reciprocal lattice points in the unit cell that lie in this plane. Use = 1.542, and a = 3.615A and a scale of 0.3A-1-1 cm. Using the drawing, verify the Bragg angle for diffraction from the (002) planes.

Solution

The Ewald sphere, or sphere of reflection, is a sphere of radius 1/ passing through the origin O of the reciprocal lattice. The incident direction is along a radius of the sphere, IO (Figure 1). A reflected direction, of unit vector s_h, will satisfy the diffraction condition if the diffraction vector OH = IH – IO = s_h/ – s_o/ (s_o unit vector in the direction IO) is a reciprocal lattice vector, namely if H is a node of the reciprocal lattice (see Diffraction condition in reciprocal space) . If other reciprocal lattice nodes, such as G, lie also on the sphere, there will be reflected beams along IG, etc. This construction is known as the Ewald construction. When the wavelength is large, there are seldom more than two nodes, O and H, of the reciprocal lattice simultaneously on the Ewald sphere. When there are three or more, one speaks of multiple diffraction, multiple scattering or n-beam diffraction. This situation becomes increasingly frequent as the wavelength decreases and is practically routine for very short wavelengths such as those of -rays and electrons. The curvature of Ewald sphere then becomes negligible and it can often be approximated by a plane. Many reflections must then be taken into account at the same time.

When the wavelength changes, the radius of the Ewald sphere changes and Figure 2 illustrates the case of the reciprocal (110) lattice plane of a silicon crystal and two Ewald spheres corresponding to AgK (₁ = 0.709 Å) and CuK (₂ = 1.54 Å) radiation, respectively. If the incident beam is a white beam, with a wavelength range _{min max}, there will be a nest of Ewald spheres of radii 1/_max 1/ 1/_min.

A crystal can be described as a lattice of points of equal symmetry. The requirement for constructive interference in a diffraction experiment means that in momentum or reciprocal space the values of momentum transfer where constructive interference occurs also form a lattice (the reciprocal lattice). For example, the reciprocal lattice of a simple cubic real-space lattice is also a simple cubic structure. The aim of the Ewald sphere is to determine which lattice planes (represented by the grid points on the reciprocal lattice) will result in a diffracted signal for a given wavelength, , of incident radiation. The incident plane wave falling on the crystal has a wave vector Ki whose length is 2p . The diffracted plane wave has a wave vector Kf . If no energy is gained or lost in the diffraction process (it is elastic) then Kf has the same length as Ki . The difference between the wave-vectors of diffracted and incident wave is defined as scattering vector K =Kf - Ki . Since Ki and Kf have the same length the scattering vector must lie on the surface of a sphere of radius 2p . This sphere is called the Ewald sphere. The reciprocal lattice points are the values of momentum transfer where the Bragg diffraction condition is satisfied and for diffraction to occur the scattering vector must be equal to a reciprocal lattice vector. Geometrically this means that if the origin of reciprocal space is placed at the tip of Ki then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere.A crystal can be described as a lattice of points of equal symmetry. The requirement for constructive interference in a diffraction experiment means that in momentum or reciprocal space the values of momentum transfer where constructive interference occurs also form a lattice (the reciprocal lattice). For example, the reciprocal lattice of a simple cubic real-space lattice is also a simple cubic structure. The aim of the Ewald sphere is to determine which lattice planes (represented by the grid points on the reciprocal lattice) will result in a diffracted signal for a given wavelength, , of incident radiation. The incident plane wave falling on the crystal has a wave vector Ki whose length is 2p . The diffracted plane wave has a wave vector Kf . If no energy is gained or lost in the diffraction process (it is elastic) then Kf has the same length as Ki . The difference between the wave-vectors of diffracted and incident wave is defined as scattering vector K =Kf - Ki . Since Ki and Kf have the same length the scattering vector must lie on the surface of a sphere of radius 2p . This sphere is called the Ewald sphere. The reciprocal lattice points are the values of momentum transfer where the Bragg diffraction condition is satisfied and for diffraction to occur the scattering vector must be equal to a reciprocal lattice vector. Geometrically this means that if the origin of reciprocal space is placed at the tip of Ki then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere.

E = 8keV E = h = h c k = h c 2 h = 6, 62 1034 c = 3 108 |~ki | elastische Streuung = | ~kf | = E2 hc = 81032 6,6210343108 = 4, 05 1010 m1 v