de Broglie Waves h p de Broglie argued Light exhibits both wave and particle properties

Transcription

1 de Broglie argued de Broglie Waves Light exhibits both wave and particle properties Wave interference, diffraction Particle photoelectric effect, Compton effect Then matter (particles) should exhibit both particle and wave properties He predicted the wavelength to be de Broglie wavelength λ h p 1

2 de Broglie Waves Consider the E, p relations for a photon E hf c f then λ hf pc h p h p and E pc Guided by this, de Broglie proposed the same relation for matter 2

3 de Broglie Waves 3

4 de Broglie Waves Let s apply de Broglie waves to the Bohr model If the electron is represented as a standing wave in an orbit about the proton 2πr L rp nλ nh 2π n h p nh This apparently justifies Bohr s assumption 4

5 de Broglie Waves de Broglie wavelength of a person running λ λ h p Js (65kg)(5m / s) 36 m de Broglie wavelength of a 50 ev electron λ h p hc pc hc 2mc 2 T 1240eVnm (2)( )(50) λ 0.17nm 5

6 X-ray Scattering Before examining electron scattering from a (large) crystal, let s first look at x-ray scattering from a crystal Spacing of atoms in a crystal ~ 1A Wavelength of hard x-rays ~ 0.1-1A Laue therefore expected that interference patterns should be observed And they were! Today, the Laue technique can be used to determine crystal orientation and assess crystal perfection from the size and shape of the spots 6

7 X-ray Scattering 7

8 X-ray Scattering Laue diffraction of salt 8

9 Whale myoglobin X-ray Scattering Of ~35,000 known protein structures, ~29,000 have been identified using x-ray diffraction 9

10 X-ray Scattering 10

11 Powder diffraction X-ray Scattering Sometimes single crystals are not available Sometimes materials naturally occur in a polycrystalline state Using many small crystals Their orientation will be random At least a few of the small crystals in the sample will be in the correct orientation to diffract for each of the possible planes The resulting rings are called the Debye-Scherrer pattern The powder diffraction method is frequently used to fingerprint crystals via a large database 11

12 X-ray Scattering 12

13 X-ray Scattering Debye-Scherrer pattern from powder diffraction for NaCl and KCL 13

14 X-ray Scattering Bragg simplified Laue s three dimensional analysis by considering x-ray scattering as the reflection of the incident beam from successive lattice planes in the crystal If the scattered angle incident angle (reflection), there is no phase change between the incident and reflected waves Waves scattered at equal angles from atoms in two different planes will constructively interfere if the path length difference is an integral number of wavelengths 14

15 X-ray Scattering Crystal structure of NaCl 15

16 X-ray Scattering Crystal structure of NaCl 16

17 Bragg condition X-ray Scattering 2d sinθ mλ The intensity pattern of the scattered waves of a known wavelength gives information about the structure of the crystal The intensity pattern of the scattered waves from a known crystal spacing gives information about the incoming wavelengths Laue and Bragg scattering effectively started the field of solid state physics 17

18 X-ray Scattering Crystal structure of NaCl 18

19 X-ray Scattering Crystal structure of NaCl 19

20 X-ray Scattering Bragg spectrum of NaCl Intensity 2θ 20

21 X-ray Scattering To determine d nλ 2d sinθ λ 0.073nm d 2sinθ 15 2sin 2 To determine n 0.28nm nλ 2d sinθ n 2d sinθ λ ( 2)( 0.28) sin

22 d d To check d molecules volume atoms volume X-ray Scattering The volume of one atom is d 3 for a face-centered cubic crystal nm 3 N Avρ At ( )( 2) 28 volume atom ( 23)( g / cm ) 58.5g / mol 6 10 cm 3 m atoms 3 m 22 molecules 3 cm 22

23 Electron Scattering If x-rays of wavelength ~ 1A produce an diffraction pattern when scattered off a crystal so should matter waves of comparable wavelength For example, 50 ev electrons Davisson and Germer verified this (accidentally) 23

24 Electron Scattering Davisson-Germer experiment 24

25 Electron Scattering Davisson-Germer data 25

26 Electron Scattering 26

27 Electron Scattering Bragg s law applies for electrons too nλ 2d sinθ D - G measured φ 2α, not θ 2θ π 2α nλ 2d cosα and d Dsinα nλ 2D cosα sinα Dsin 2α nλ Dsinφ Dsinφ 27

28 λ Electron Scattering Analyzing the Davisson-Germer data Dsinφ λ 0.165nm h p hc 2mc ( 0.215nm)( sin50 ) The de Broglie wavelength for 54.4eV electron is λ 2 T 1240eV nm ( )( ev )( 54.4eV ) 0.167nm Experimental evidence that electrons behave as waves with the de Broglie wavelength 28

29 Electron Scattering Aside, these results hold true for even a low intensity electron beam This means that the interference pattern does not result from interference between waves from two electrons, but from waves associated with a single electron Aside, diffraction patterns are also observed using neutrons, H, and He atoms 29

30 Electron Scattering Thomson observes diffraction patterns in electron transmission experiments similar to Laue s in x-ray transmission experiments 30