This article was first posted on January 9, 2013; it was revised on January 27, 2013. In this post we will think about diffraction of a beam of x-rays, with a single wavelength, λ, as in post 22.14. But here we will consider the special case where the scattering object is an ideal crystal. My… Continue reading 23.1 Observing x-ray diffraction by a crystal

Probability is an important idea in science. An experiment to answer a question like, for example, whether a drug is safe can only be answered with a probability (post 16.1). When we measure anything we only really know that its value lies between certain limits with a given probability (posts 16.7, 16.24, 16.26, 16.28). Probability… Continue reading 22.25 The Monty Hall problem

Before you read this, I suggest you read posts 22.22 and 22.23. In post 22.22, we saw that a one-dimensional lattice was a line of points, each separated by the vector a. The picture above shows how a two-dimensional lattice can be generated by convolution of two one-dimensional lattices. A three-dimensional lattice can be generated… Continue reading 22.24 Reciprocal lattice

I mentioned the idea of K-space in post 22.22 and introduced the vector K in post 19.20; in post 22.12 I described the Fourier transform as a function of K. The purpose of this post is to bring all these ideas together. In the left-hand picture above, a ray is scattered by a particle at… Continue reading 22.23 K-space

Before you read this, I suggest you read post 22.12 and appendix 3 of post 22.20. When you read the title, this may seem like a very obscure topic. But it isn’t because we need to understand the Fourier transform of a lattice to understand x-ray diffraction by a crystal. And we need to understand… Continue reading 22.22 Fourier transform of a one-dimensional lattice

In post 16.37 we saw that the atoms or molecules or ions in a solid are packed together so that the solid is completely contained within its free surfaces – it forms an object that we can hold without it flowing through our fingers. In an ideal gas, the atoms or molecules are free to… Continue reading 22.21 An ideal crystal

The convolution of two functions of x, f(x) and g(x), is defined by the definite integral Convolution is defined mathematically but it is possible to understand what it means in pictures. So, if you don’t like mathematics, ignore the next paragraph and the appendices. The definition of convolution can be extended into two, or more… Continue reading 22.20 Convolution

Before you read this, I suggest you read post 16.33. Stability is a more complicated idea than we often suppose. For example, a mechanical system can be stable but not in equilibrium because it is moving (with respect to an observer – see movement) on a stable path – it has dynamic stability. Similarly, a… Continue reading 22.19 Kinetic stability

Before you read this, I suggest you read post 18.11 The picture below shows an orthogonal Cartesian coordinate system in which the z-axis is vertical. We are going to think of two pendulums: one oscillates in the xz plane and the other oscillates in the yz plane. For small oscillations, we can consider that the… Continue reading 22.18 Coupled oscillators – Lissajou’s figures

Before you read this, I suggest you read post 19.10. Foxes eat other animals (they are predators), including rabbits: rabbits are eaten by other animals (they are prey), including foxes, but rabbits eat only plants. Now let’s imagine an island that contains foxes and rabbits; there are no other animals for the foxes to eat… Continue reading 22.17 Model for a simple ecosystem – coupled differential equations