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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”
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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.
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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)
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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
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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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