Electron Diffraction Analysis of Polycrystalline Gold · RMPH 2018 #B

Physics · modern-physics

Problem

In this task, analyze the electron diffraction image of polycrystalline gold and respond to the requirements. For crystalline substances, the arrangement of constituents is characterized by orderly positions in space with periodicity. A parallelepiped built on three fundamental vectors

\vec{a_i}

is called a unit cell. Between crystal lattices, some have a cube as elementary cell with side

a

called the network constant. [Figure: Three types of elementary cubic cells shown: (A) Simple Cubic (S.C.) with atoms at cube apexes, (B) Face Centered Cubic (F.C.C.) with atoms at apexes plus six atoms at face centers, (C) Body Centered Cubic (B.C.C.) with atoms at apexes plus one atom at volume diagonal intersection] Interplanar distances for cubic networks have expression

d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}

, where

h,k,l

are crystalline plane family indices. When irradiated with radiation having wavelength comparable to interplanar distance, diffraction directions are separated from incident direction by angles

2\theta

. For geometric reasons: - B.C.C. structure: diffraction maximum only for plane families whose indices sum is even:

(0,1,1)

(2,0,0)

(2,1,1)

(2,2,0)

, etc. - F.C.C. structure: diffraction maximum only for same parity indices:

(1,1,1)

(2,0,0)

(2,2,0)

(3,1,1)

, etc. - S.C. structure: diffraction maximum for all plane families:

(1,0,0)

(1,1,0)

(1,1,1)

(2,0,0)

, etc. [Figure: Formation of electron diffraction image showing monoenergetic electron beam falling on thin polycrystalline layer, creating diffraction cones that appear as circles on photographic plate at distance L, with relationship

\tan(2\theta) = \frac{R_{hkl}}{L} = \frac{D_{hkl}}{2L}

] [Figure: Electron diffraction image of polycrystalline thin gold foil with numbered diffraction circles] The diffraction image is enlarged 5.3 times in area. For the experiment,

L = 75

cm. B.1. Calculate the de Broglie wavelength

\lambda_e

of electrons in the diffracted beam; acceleration voltage is

U = 100

kV. Use: electron rest mass

m_e = 9.11 \times 10^{-31}

kg, electron charge

e = 1.60 \times 10^{-19}

C, speed of light

c = 3.00 \times 10^8

^{-1}

, Planck constant

h = 6.63 \times 10^{-34}

Js. B.2. Determine the difference between the de Broglie wavelength calculated relativistically and non-relativistically. B.3. Demonstrate that for the diffraction image, between diameters

D_{hkl}

of diffraction circles and interplanar spacing

d_{hkl}

there is the relationship:

D_{hkl} \cdot d_{hkl} = C \cdot \lambda_e \cdot L \quad (1)

and specify the value of constant

C

. B.4. Write indices of crystalline planes responsible for the first eight diffraction circles (ascending diameter order, descending interplanar distance order) for S.C., F.C.C., B.C.C. structures. For each structure calculate the ratio of maximum interplanar distance length to other interplanar distances. B.5. Using the calibrated paper sheet from task A, measure diameters of diffraction circles and calculate the ratio of each measured diameter to the smallest circle diameter. B.6. Analyze diameter sizes and relation (1) with results from B.4 to determine whether gold's crystal structure is S.C., F.C.C., or B.C.C. B.7. Plot the dependency between inverses of interplanar distances (in units of network constant

a

) and actual diameters of corresponding diffraction circles (accounting for enlargement). B.8. Use graphical analysis of experimental data from B.7 with equation (1) to determine the network constant of gold

a_{Au}

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