MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Lecture Notes Theory of Solids II Patrick Lee Massachusetts Institute of Technology Cambridge MA March 2 2004 Contents Contents 1 Lecture 2 Scattering and the Correlation Function 1 1 Scattering 1 2 Application Electron Energy Loss Spectroscopy EELS 1 3 Application Neutron Scattering 1 2 2 3 4 Chapter 1 Lecture 2 Scattering and the Correlation Function We ended the last lecture with a brief discussion of the connection between scattering experiments and measurements of the correlation function S q In this lecture we will discuss scattering in more depth in terms of two concrete examples electron and neutron scattering After that we will look at some more general properties of response functions 1 1 Scattering The picture we have is of some blob of material with a plane wave ki coming in and a di erent plane wave kf coming out We de ne the momentum and energy transfer to the sample ki kf Q E ki E kf 1 1 1 2 be the coordinate of the scattering particle Recall from last time that application of Let R Fermi s Golden Rule and the 1st order Born Approximation leads to the di erential rate 2 3 i kf ki R i q R 3 Wi f d kf 2 vq n q 0 dR e e En E0 d kf 1 3 n q 2 2 3 vQ n Q 1 4 2 0 Ef Ei d kf n 3 vQ S Q d kf 2 1 5 1 6 v 2 S Q P Q Q 1 7 for scattering into a nal state with momentum somewhere in a volume element d3 kf of momen tum space centered on kf Here vQ is the Fourier Transform of the interaction potential The and energy loss is key result here is that the rate of scattering with momentum transfer Q directly proportional to the correlation function S Q 2 Application Electron Energy Loss Spectroscopy EELS 1 2 3 Application Electron Energy Loss Spectroscopy EELS The experiment we imagine here is that of shooting high energy electrons 100 keV at a thin lm of material and collecting them as they emerge with an energy resolved detector For this case the interaction potential is just the Coulomb interaction between the electron and the sample s charge density so vq 4 e2 q2 1 8 Recall the de nition 1 UT ot q UExt Uscr 1 UExt Remembering that Uscr q density uctuations 4 e2 q q 2 n 1 9 1 10 where n q are the Fourier components of the 1 4 e2 n q 1 2 q q UExt q 1 11 As de ned in the previous lecture the linear density response function q is de ned by the ratio q Substituting this into the relation for n q UExt q 1 q 1 12 we get 1 4 e2 1 2 q q q 1 13 With q de ned as the imaginary part of the relation S q 2 q 1 14 combined with equation 1 7 for the scattering rate into momentum space volume d3 kf gives the following relation for the scattering rate in terms of the dielectric function 8 e2 1 P q 2 Im 1 15 q q What useful information can we get out of this For one we are able to investigate the dielectric constant at nite values of q 0 to kF In optical experiments the vanishingly small photon momentum in comparison with typical electron nucleus momenta means that we are only able to investigate the q 0 regime with photons On the downside the best energy resolution we can achieve today is around 0 1 eV which is far too coarse to obtain much useful information This energy resolution is already 1 106 when compared with the total electron energy of around 100 keV To get around this one might Application Neutron Scattering 4 consider trying lower energy experiments However the problem with low energy experiments is that the probability of multiple scattering events within the sample becomes signi cant leading to complicated and messy results With EELS we can also look at high energy excitations of the electrons in a metal Recall that there is a high energy collective mode of the sample electrons at a frequency equal to the plasma frequency pl The plasma frequency is de ned in terms of the zero of the dielectric function q pl 0 1 16 The situation where the dielectric function becomes zero is interesting because it represents a singularity in the system s response to an external perturbation UT ot 1 UExt q pl 1 17 Thus even a tiny perturbation at the plasma frequency results in a large response of the system 1 3 Application Neutron Scattering Since neutrons are uncharged they do not see the electrons as they y through a piece of mate rial1 The dominant scattering mechanism is through a contact potential with the nuclei of the sample V r 2 b r Mn 1 18 where b is the scattering length and Mn is the mass of the neutron Since the Fourier transform of a delta function in space has no q dependence the Fourier components of the interaction potential are all simply vq 2 b Mn 1 19 Inserting this into equation 1 7 for the scattering rate we get P Q 2 b Mn 2 S Q is the correlation for the nuclear positions density Here S Q dt ei t t 0 T S Q Q Q 1 20 1 21 with Q eiQ Ri t i 1 The can interact however through spin spin magnetic interactions 1 22 Application Neutron Scattering 5 i t are the coordinates of the nuclei at time t Now we can substitute this in to the where R expression for S Q S Q dt ei t e iQ Rj t eiQ R 0 T 1 23 j j t We consider the case of small To make progress we must put in a speci c form for R distortions from a Bravais lattice j R j0 uj R 1 24 0 are the Bravais lattice sites and uj are small displacements The uj can be where R j expanded in phonon coordinates yielding 1 uj q ei q R q t a a q e i q R q t 1 25 2N M q q where the sum over is a sum over all phonon polarizations is the polarization of the th mode After some algebra see problem set it can be shown that this decomposition yields Q2 e 2W G q G S Q Q nq 1 Q q 1 26 2N M q q Q G q G nq Q q G where W is the Debye Waller factor and nq is the Bose statistical occupation factor There are several interesting features about this expression for the correlation function The Even rst term corresponds to simple elastic Bragg scattering …