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lECTURE 4

# lECTURE 4 - EEE 352 Lecture 04 Crystal Directions Wave...

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EEE 352: Lecture 04 Crystal Directions; Wave Propagation (110) (001) Electron beam generated pattern in TEM

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Crystal Directions We want the NORMAL to the surface. So we take these intercepts (in units of a ), and invert them: Then we take the lowest common set of integers: These are the MILLER INDICES of the plane. The NORMAL to the plane is the [4,3,2] direction, which is normally written just [432]. (A negative number is indicated by a bar over the top of the number.) This inversion creates units of 1/cm These “numbers” define a new VECTOR in this “reciprocal space”
Crystal Directions 1/2 1 1 [100] 1/2 [220] These vectors are normal to the planes, but are defined in this new space. What is this new reciprocal space?

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In earlier circuit courses, we used Laplace transforms, such as If we let s = σ + i ω , we can get the Fourier transform as ( σ = 0) units = time units = 1/ time=2 π Hz Hence, frequency space is the reciprocal space for time. The units are 1/ seconds . In order to talk about this new space, we need to talk a little more about waves, and how these waves will be important in our crystals.
In electromagnetics (EEE 240), we deal with waves (more on this next time). In this course, we deal with exponentials that vary as: Spatially varying part of the exponent. x corresponds to distance (cm) k corresponds to 1/distance or 1/cm This is the reciprocal space term ω is the coordinate of reciprocal time , k is the coordinate of reciprocal space .

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time distance wave number k frequency Fourier transformation in space and time: The dimensions of space and time go into the dimensions of “wave number” (reciprocal space) and frequency (reciprocal time)
time distance Velocity is distance per unit time, but can be defined in at least two ways: In either case, the units of velocity are cm/s .

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wave number k frequency The idea of velocity must carry over to the Fourier transform space: The units remain cm/s . What about multiple spatial dimensions? Phase velocity Group velocity
For the FACE CENTERED CUBIC lattice, we have to define the three lattice vectors so that they fully account for the atoms at the face centers. a b c The three primitive vectors run from a corner atom to the three adjacent faces of that corner. Again, these form a tetrahedron. There will be 3 vectors in the reciprocal space, which correspond to the three vectors a , b , c in real space. These will define a unit cell in the reciprocal space! The Multiple Space of Concern is the Lattice We do not care what these vectors are at present; the important point is that our reciprocal space is periodic , just as the lattice. We will learn what the period is later.

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So, why do we give a rat’s %\$# about this reciprocal space? We will learn later that the onset of quantum mechanics around 1900 led to a connection between energy and frequency, which is known as the Planck relation: Hence, the frequency space goes into the energy space!
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