In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space (also known as momentum space or less commonly as K-space, due to the relationship between the Pontryagin duals momentum and position). The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.
The reciprocal lattice plays a very fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal.
The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice.
Consider the series with ascending (but not necessarily sequential) primes pn,
1/p1+1/p2+1/p3+ . . . +1/pN=1, as N approaches infinity.
Determine the pn that most rapidly converge (minimize the terms in) this series. That set of primes I call the "Booda set."
Question 1 — Construct the crystal lattice from the diffraction pattern drawn on page 5 of this exam paper making sure you include the (110) and (220) planes. Explain the procedure used in reconstructing the crystal lattice. What Bravais lattice is represented by the diffraction pattern...
I know this might be a really stupid question, but to convert a crystal lattice 2D representation to a 2D reciprocal lattice do you justdo you just invert the scaling. I know this is a pretty poor explanation so I will try and illustrate what I mean.
Let's say that you have a reciprocal lattice...
How do I show that \sum_1^n\frac{1}{k} is not an integer for n>1? I tried bounding them between two integrals but that doesn't cut it. I know that \sum_1^n\frac{1}{k}=\frac{(n-1)!+n(n-2)!+n(n-1)(n-3)!+...+n!}{n!} but I can't get a contradiction.
If we are studying FCC in the direct lattice, Why does the length of the cube side in the reciprocal lattice equal to 4*Pi/a Where a is the lattice constant,
a*=|G|=2*Pi/a Sqrt(4) = 4*Pi/a
Where a* is the length of the cube site in reciprocal lattice
Note: this thing is repeated in 2...
Suppose a_n is a bounded sequence. Then prove that lim sup a_n = 1/lim inf (1/a_n).
This seems completely obvious to me, I don't know how to do this any simpler.
Hi I have a proof I'm doing
\int \frac{1}{1+\sin(x)}dx
I know that the answer I'm looking for is
\frac{\sin(x) - 1}{\cos(x)}
and then
\tan(x) - \sec(x)
I have tried integration by parts making
u = (1+\sin(x))^{-1} and dv = dx
Eventually I get an answer that...
I am to find the imaginary part, real part, square, reciprocal, and absolut value of the complex function:
y(x,t)=ie^{i(kx-\omega t)}
y(x,t)=i\left( cos(kx- \omega t)+ i sin(kx- \omega t) \right)
y(x,t)=icos(kx- \omega t)-sin(kx- \omega t)
the imaginary part is cos(kx- \omega t)
the...
If two resistors with resistances R1 and R2 are connected in parallel, then the total resistance Rt, measured in ohms, is:
\frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2}
If R1 and R2 are increasing at rates:
\frac{d \Omega_1}{dt} = 0.3 \; \; \frac{d \Omega_2}{dt} = 0.2 \; \; R_1 = 80 \...
can any1 explain why this iteration:
Xn+1 = Xn(2 - NXn)
can be used to find the reciprocal of N. I don't ned proof or to show that it does but i would like to know if sum1 can break it down and explain how it does it.
Compute the following:
\sum_{n=1}^{+\infty} \frac{1}{n^{2}} =...??
\sum_{n=1}^{+\infty} \frac{1}{n^{4}} =...??
.LINKS TO WEBPAGES WITH SOLUTIONS ARE NOT ALLOWED! :-p
Daniel.
The terms of this series are reciprocals of positive integers whose only prime factors are 2s and 3s:
1+1/2+1/3+1/4+1/6+1/8+1/9+1/12+...
Show that this series converges and find its sum.
this is my first time writing here. i hope someone can help me with this question.
Er well I've been away from math for a LONG time until I recently began reading into calculus and I have a question.
I always see reciprocal and inverse throughout the text. What is the difference between the two?
I always thought reciprocal was the number (in a fraction form) flipped so...
A triangle in Euclidean space can be described as having a hypotenuse of one, and legs of Lorentz parameters \beta and \gamma. What spatial curvature underlies a triangle with hypotenuse one, and legs 1/ \beta and 1/ \gamma?