Materials Science Assignment: Crystal Structures and Diffraction

This materials science assignment analyzes the crystal structures of copper and silicon. It begins by describing the face-centered-cubic structure of copper and the covalent crystal structure of silicon, including diagrams. The assignment then identifies the first six planes that produce diffraction patterns for both materials, specifying them using Miller indices. Following this, the assignment calculates the inter-planar spacing that would produce a diffraction peak at a specific angle, applying Bragg's law. Finally, it determines the lattice parameter for copper, given the atomic radius and the face-centered structure, using the Pythagorean theorem. The assignment provides detailed calculations and references to support its findings.

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1.
The crystal structure of Cu and Si
In solid copper, the copper atoms are arranged in the face-centered-cubic crystal
structure. At every corner of the cube, copper atoms are found and also at the center of
each face of the cube. This fundamental arrangement of copper atoms forms a unit cell
and is replicated in three-dimensional space to make up the structure of the copper metal
(William D. Nielsen, n.d.).
Figure1
In silicon solid, the carbon atoms are held together by covalent bonds and is thus referred
to as a covalent crystal. Every carbon atom is bonded covalently to four other carbon
atoms that are arranged tetrahedrally to give the crystal structure as shown in figure 2.
This structure is called two interpenetrating face-centered cubic primitive lattices Each
atom covalently binds by sharing an electronic doublet with four neighboring carbon
atoms, resulting in a lattice that stretches over a large number of unit cells in all
directions, forming a crystal with covalent lattice. The cube side for silicon is 0.543nm
(Anon., 2018).

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Figure 2
The first six planes in Copper and Silicon that will produce diffractions patterns.
The first six planes that will produce diffraction can be specified using Miller indices,
which is given by the values of (h k l). The planes are listed in table 1.
Cu Si
(111) (111)
(200) (220)
(220) (311)
(311) (400)
(222) (422)
(400) (511)
Table 1

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2.
The inter-planar spacing that would produce diffraction peak at diffraction patterns
angle 6.45 degrees when an X-ray of wavelength 0.058nm.
If a crystal structure is viewed as parallel planes of atoms separated by distance d, assuming
that the reflection of X-rays from any given plane is regular, then Bragg’s law states that
constructive interference of radiations from successive planes causes peaks in the intensity of
scattered radiations (Schields, 2004).
Mathematically,
λ=2 dsinθ
Where λ is the wavelength of the X-rays
dis the inter-planer spacing
and θ is the diffraction angle.
Making d the subject of the formula,
d= 2 λ
sinθ
Therefore, for λ=0.058 nm and θ=6.45 degrees
d= 2(0.058 × 10−9 )
sin (6.45)
d=1.03261537× 10−9 M
In four significant figures,
d ≈ 1.033× 10−9 M =1.033 nm

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3.
Copper has a face-centered structure. If we consider the atoms viewed from one side of the cube
then a structure as shown in figure is apparent.
Figure 3
Where a is the lattice parameter for copper and r is the radius of the individual atoms.
From Pythagoras theorem,
a2+a2=(4 r )2
Given that r =0.2556 nm
Then,
2 a2=(4 ×0.2556 nm)2
a2=5.2265088× 10−1
a=7.22945973 ×10−1 nm
In four significant figures,
a=0.7229 nm

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References
Anon., 2018. PhysicsOpenLab. [Online]
Available at: http://physicsopenlab.org/2018/01/28/silicon-germanium-crystal-structure/
[Accessed 6 September 2018].
Kasap, S., 2001. Elements of X-Ray Diffraction By Crystals, s.l.: McGraw-Hill.
Redwing, R., 2018. Materials in Today's World. [Online]
Available at: https://www.e-education.psu.edu/matse81/node/2133
[Accessed 6 September 2018].
Schields, P. J., 2004. Bragg's Law and Diffraction: How waves Reveal the Atomic Structure of Crystals,
New York: Stony Brook.
William D. Nielsen, J., n.d. Copper Development Association Inc.. [Online]
Available at: https://www.copper.org/resources/properties/703_5/
[Accessed 6 September 2018].