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Reciprocal Lattice & Diffraction Outlines 11/23/2016 11/23/2016

1. Introduction Drude model Drude 2. Experimental Techniques model Drude 3. Reciprocal Lattice 4. Ewald construction & Laue Method 5. Brillouin Zones 6. Example: reciprocal lattices of bcc & fcc 7. Fourier analysis of the basis

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The incident radiation ( for electron) Introduction model Drude 11/23/2016 1. Photons (x-ray) In the past, because of the size and distance between atoms is on the order of 10-10 m, direct measurement of 11/23/2016 · lattice is difficult, so indirect methods were developed

to probe the structure of crystals. Diffraction is such a model Drude Typical x-ray energy ~ ~. method that is widely used to probe crystal structure. The method can be illustrated (in the linear response 2. Electrons theory) as follow: . In-coming radiation out-going Crystals radiation Electrons, photons, neutrons For electron ~ , implies ~ 3. Neutrons (Neutron mass is 2000 times heavier than electron) Mathematically, we can view this diffraction process as an operation such as Fourier transform . For E ~ , . ~. 3 4 ·

Bragg Law (x-ray) Experimental diffraction methods

where d ------lattice spacing From Bragg law we can either fix the wavelength and (1) ------incident angle 11/23/2016 measure the diffraction pattern as a function of angle, or 11/23/2016 ------wavelength of x-ray n ------integer we can fix the angle, and measure the diffraction pattern as a function of wavelength (energy). Drude model Drude model Drude Some useful x-ray techniques

1. Laue method (transmission) Single crystal is used and continuous radiation is used. The method is widely used to identify the symmetry of When the path difference is equal to 2dsin, it leads to crystals. The crystal selects the discrete values of for constructive interference. which the Bragg law is satisfied. The spots on the film Note: Each lattice plane reflects about ~ of the come from the characteristic x-ray. total incident radiation. Typical x-ray penetrates about a 5 6 few thousand into a solid.

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3. Powder method Polycrystals or powder and monochromatic light are

11/23/2016 used. The radiation select the correct orientation for 11/23/2016 diffraction. Because of the rotation symmetry along the incident direction, rings are created. Drude model Drude model Drude

2. Rotating crystal method (also known as θ-2θ) (reflection)

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Fourier Transform Fourier transform is an integral transform of a time function into a frequency function: 11/23/2016 11/23/2016 Drude model Drude The inverse transform is given by model Drude This can be applied to 3D real space

· And the inverse transform is given by

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Joseph Fourier, a French mathematician, in the Scattering wave amplitude early 19 century (1822), developed the basic We will use a few different approaches to demonstrate concepts of this integral transformation that 11/23/2016 11/23/2016 the physical meaning of the reciprocal lattice. In bears his name. particular the scattering wave amplitude relates to the

Drude model Drude Fourier transform of the real space lattice structure. model Drude The basic concept of the Fourier transform is quite simple, namely, any time series function can be represented as an One dimensional infinite summation of harmonic functions. Let be the 1-D lattice location, / (5) and /

Here . This expression satisfies the translational Harmonic functions form a “complete set” of orthogonal functions which can represent any functions. 11 invariance of the lattice. 12

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namely · · · · · 11/23/2016 11/23/2016 · · · model Drude model Drude

·· (12) For 3-D system

· (9) This is the scattering amplitude, as we can see that it is also the Fourier transform of the real space lattice points.

Later we shall show that is related to the scattering amplitude. 13 14

Reciprocal lattice vectors and reciprocal space What is the physical meaning of ? For a given lattice, with , , and as its primitive From definition, relates to a plane with its normal vectors, then we define the following vectors, 11/23/2016 11/23/2016 to and , and ⋅ just a volume, serving as a normalization factor.

model Drude model Drude · · · The ratio of the magnitudes of , , and is given by

as the primitive vectors of the reciprocal lattice. : : For a vector in the reciprocal space, is a translational vector in reciprocal lattice. The reciprocal (15) lattice points are defined by , , and . where , , and are integers. The reciprocal lattice is the Fourier transform of the real crystal lattice. The X-ray scattering pattern is related to the The is the reciprocal lattice vector. 15 reciprocal lattice. 16

Diffraction conditions Eq. (9) on page 13 Let ∆

11/23/2016 ∆· · ∆· 11/23/2016 ′

· model Drude model Drude · ∆ · (20) The amplitude F of the scattering wave is proportional to: If ∆, the integral is not defined or = 0, so when 1. Number of scatteriers ------ 2. A phase factor ------ · ∆ (21) Eq. (20) becomes

· · (18) So this is the proof that the scattering amplitude 17 F is proportional to the Fourier component . 18

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For elastic scattering, the energy is conserved, so the Laue equations magnitude of the momentum is the same, even though they Another way to interpret the scattering condition ∆ may have different direction. Start with 11/23/2016 was provided by von Laue who did the original x-ray works 11/23/2016 · and was awarded Nobel prize in Physics in 1914. rd model Drude model Drude · (22) If we take the scalar product of ∆ and primitive translational vectors , we end up with Since is also a reciprocal lattice vector ·∆ ; ·∆ ; ·∆ (25) · where ,, and are integers. Each equation above tells us that ∆ has to lie on the surface · · of a cone about the directions , , and . In the x-ray scattering work, ∆ must satisfy all three equations above. (24) 19 20

Ewald construction Brillouin Zones 1. Chose a point according to the orientation of the specimen with respect to the incident beam. 2. Draw a vector AO in the incident direction of length 2π/λ terminating at the origin. A Brillouin zone is defined as the Wigner-Seitz cell in the 3. Construct a circle of radius 2π/λ with center at A. Note whether this circle passes through any reciprocal lattice. For example, the reciprocal lattice of a point of the reciprocal lattice; if it does: 11/23/2016 11/23/2016 4. Draw a vector AB to the point of the intersection. simple cubic system is given by 5. Draw a vector OB to the point of the intersection. 6. Draw a line AE perpendicular to OB.

7. Complete the construction to all the intersection points in the same fashion. model Drude For a wavevector , if the tip of model Drude is on the Brillouin zone boundary, Downloaded from then theis wavevector will be http://www.chemistry.uoguelph.ca/educ mat/chm729/recip/8ewald.htm Bragg scattered.

· 1. Since OB end at a point, . 2. , , & 2. 3. Combine above we obtain · · · 21 22 n Bragg condition for diffraction.

bcc lattice fcc lattice Primitive vectors of real Reciprocal primitive vectors Primitive vectors of real Primitive vectors of fcc reciprocal lattice

space bcc lattice are: of bcc reciprocal space 11/23/2016 space fcc lattice. 11/23/2016 model Drude model Drude

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Fourier analysis of the basis Substitute (40) into the definition of

When diffraction condition is satisfied (∆ ), eq. (18) can be written as 11/23/2016 · 11/23/2016

(39) model Drude model Drude Let , and substitute in the above eq. The quantity is called the structure factor.

· · It is useful to define the electron density associated with individual atom in the cell, such that

Now we define the red integral above as the atomic form factor, so (40) · (43) 25 26 Where is the vector to the center of j atom.

From definition of , we can write the structure factor as (46) 11/23/2016 11/23/2016 If this is a pure element, all atoms are the same, then ,

model Drude model Drude and can be moved out of the summation. Structure factor of the bcc lattice For bcc lattice, there are two atoms per conventional cell, the atoms are located at (0,0,0) and (½, ½, ½), so the structure factor.

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Structure factor of the fcc lattice 11/23/2016 11/23/2016 There are 4 atoms per conventional cell at (0,0,0) (0,½,½), (½,0,½), and (½,½,0). Drude model Drude model Drude

= even, = even, = even, X-ray diffraction pattern of Ag nanoparticles, indicating fcc structure. = odd, = odd, = odd,

All other combinations 29 30 /www.researchgate.net/publication/258387984_Green_Fabrication_of_Silver_Nanoparticles

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Atomic form factor If all electrons are concentrated at , for example

The atomic form factor defined on page 26 is given by · 11/23/2016 11/23/2016 Then →

The integration is over the volume of ONE atom only, and Type equation here. model Drude is the vector from nucleus to the electron. The is a measure of the scattering power of the jth atom in the unit cell. If we use spherical coordinates, rd model Drude and choose in the z-direction: In general, is a very difficult quantity to calculate. In most cases, the atomic Hartree-Foch approach is a very good approximation.

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