020 x ray diffraction VẬT LÝ CHẤT RẮN

Chapter Two: Reciprocal Space In this chapter, we will introduce the idea of reciprocal space from the perspective of diffraction experiments X-ray and neutron diffraction were both very important for elucidating crystal structures They also suggested a fundamentally new way of looking at solids – in reciprocal space (often called momentum space) X-ray scattering Bragg reflection momentum space reciprocal space Fourier transforms Why we use xrays, neutrons and electrons to investigate matter? We need to use particles with wavelengths ~ Å (remember – matter has wave-like characteristics) Neutrons 20 meV X-rays keV Electrons 0.35 eV Wave length λ versus energy E momentum p Light: λ = c / ν hν=E= cp ∴ λ=h/p =ch/E Matter: non-relativistic, mass m λ = h / p = h / m v (de Broglie wavelength) E = p2 / 2m ∴ λ = h / √(2mE) p = √ (2mE) What happens during a diffraction experiment? - X-rays (or neutrons, or electrons) of a single wavelength (and therefore, energy) are incident upon a crystal The incoming rays are of the proper wavelength for diffraction (on the order of the interatomic spacing), and thus we see diffraction peaks at certain values of θ, the scattered beam angle Each one of these peaks is from a plane of atoms within the crystal This is an elastic process Miller indices for Diffraction peaks are observed as a function of scattered angle planes of atoms Experimental setup Because there are many different planes of atoms, we will see reflections at many different angles The particular crystal structure of a material can be obtained by observing which reflections cancel out because of the arrangements of the atoms (we will talk about this later) Why we see Bragg peaks at all? Approach to Bragg scattering Bragg scattering occurs when reflections from parallel beams interfere constructively For this to happen, the extra distance travelled by light ray B must be a multiple of the wavelength W L Bragg (1913) came up with this simple theory for x-ray diffraction Bragg Condition for scattering from successive planes 2d sin θ = n λ : Condition for constructive interference of x-rays – Bragg peaks How are diffraction experiments done? X-ray diffraction d = λ/(2 sin θ) = 0.154 x 10-9 / (2 x sin 22.2º) = 2.04 Å Experimental setup (how we produce x-rays?) Voltage to accelerate e- X-ray production is via Tungsten source Bremstrahlung radiation (think of produces e- by the inverse photoelectric effect – heating instead of light hitting a target and Copper electrons being emitted, electrons hit target a target and photons are emitted) gives off xrays The electrons are produced typically from a tungsten source, are accelerated towards a metal such as copper, and when they hit the surface, they slow down This “braking radiation” is a broad band of light which is emitted as the electron slows down (charged particles under acceleration emit Broad band of x-rays produced radiation) Characteristic radiation On top of this Bremstralung radiation, there are a few very strong bursts of x-rays at very precise energies These are due to electrons hitting target atoms, and inducing inner shell electron transitions This process occurs when an inner shell electron is ejected, and to take it’s place, an electron from one of the higher energy shells makes a transition and gives off light (in the form of x-rays) It is these x-rays that are used in diffraction Why are there transitions for this K process? This shows a K-shell (often called the 1s orbital) transition The first diffraction experiments : Electron diffraction from Ni Electron source Electron detector For interference from the first plane alone, the condition is different Davisson – Germer experiment (1927) of electron diffraction (matter is wave-like!) Note: the diffraction condition is different in this case (this is a case of experimental geometry) ki kf Now we need to talk about k vectors Light path difference x x = 2d sin θ d vector for planes | ki | = | kf | = k = 2π/λ (momentum of the x-ray, conserved due to elastic collision) Phase shift of the lower scattered x-rays: ϕ = k x = d k sin θ (when we have Bragg reflection, the waves are in step, so the phase difference is 2π Using this, we can get Bragg’s law: 2π = d k sin θ = d (2π/λ) sin θ 2d sin θ = nλ (n=1) In general, though, they are not in step, and will have some phase which will depend upon the scattered angle) Diffraction conditions kf 2θ ki ∆k = kf - ki : momentum transfer = k sin θ ϕ = ∆k d : phase shift What we really need to figure out is how the reflected x-rays are out of step with one another (what is the phase change from one plane to the next?)