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Dislocations and Strengthening Mechanisms
1)What is the difference between resolved shear stress and critical resolved shear stress?
The resolved shear stress is the shear component a tensile stress that is applied along a slip plane, that is not parallel or perpendicular to the stress axis.
The critical resolved shear stress is a property of the material and the value of the resolved shear stress when yielding begins.
2) Would you expect a crystalline ceramic material to strain harden at room temperature? Why or why not?
In order for a material to strain harden it has to be plastically deformed. Ceramic materials are actually brittle at room temperature, therefore they will fracture before any plastic deformation takes place. Thus we do not expect crystalline ceramic materials to strain harden.
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How to Calculate Miller Indices & Planes
Calculating Miller Indices
In order to find Miller Indices there are 4 steps to take.
Step 1) Determine the intercept of the faces on aligning the axii of the crystal. (x y z )
Step 2) Take the reciprocals of the coordinated fractions. (1/x , 1/y, 1/z)
Step 3) Allow the denominators to have the same value.
Step 4) We then clear the denominator of the fraction.
Step 5) Then reduce the numbers to their lowest terms.
Calculating Crystal Planes
When Calculating Crystal Planes
Step 1) Find the reciprocal of the hkl indices (1/h , 1/k , 1/l)
Step 2)Note each of the coordinates positions and place a mark.
Step 3) Connect the markings and highlight the plane.
Tips
When a Miller Indices is 0, the the plane is parallel to that axii.
X-Ray Diffraction (XRD)
XRD
X-Ray Diffraction (XRD) is used for determining a crystal structure, interlattice plane spacing , finding the lattice constant and the atomic radius of of the atom.
When you calculate for the crystals structure, you will first grind a powder of the elemental structure/molecule you are wishing to view. Place it in the XRD and receive diffraction angle of (2θ), and begin with a known wave length λ ex. 0.0711 nm.
For example 20.1, 28.5 , 35.1 , 40.7 (degrees) .
Step 1) We first wish to create a table with the following dimensions containing the peaks, 2θ , θ , Sin θ , Sin θ ^2 , Ratio, hkl, d(nm) .
Step 2) We will begin the process of finding the crystal structure by diving 2θ by 2, giving us theta. Therefore we see that θ= 10.05, 14.25, 17.55, 20.35 ( degrees).
Step 3) We will then plug these values in to find Sinθ , Sin θ^2.
Step 4) The next thing is to calculate the ratio of of the Sinθ^2 peaks.
Doing this we receive ratios of 2 to 4 to 6 to 8.
This is the ratio of a BCC structure, and from literature we see that the first four peaks of a BCC structure are (110), (200), (211) , (220).
There is one other way to find the crystal structure and to do this we first need to find the interplanar spacing of the lattice.
Step 5) Now in order to calculate the interplanar spacing, we must define Bragg's Law. n λ =2dSinθ
Rearranging this formula to solve d, d= nλ /2Sinθ.
Interchanging the θ values. We see receive 0. nm, 0.144nm , 0.118 nm , 0.102 nm.
Using the next equation and the peaks of the BCC structure we can determine how the lattice constant changes using the interplanar spacing values.
1/d = h^2 + k^2 + l^2/a^2 and rearranging for a , a= sqrt((h^2+k^2+l^2)d^2) , where we will plug in the values of hkl from (110), (200), (211) , (220) and the interplanar spacing.
We see that we get a value which is equivalent to a= 0.28nm
Step 6) Then to find the Lattice constant , we can then determine the metal using.
a = λ sqrt(h^2+k^2+l^2)/2Sinθ
a = 0.289
Step 7) To Determine the atomic radius of the metal for a BCC.
a= 4r/sqrt(3)
r= asqrt(3)/4
r= 0.125nm
