Internal – External Order

1. Internal – External Order • We described symmetry of crystal habit (32 point groups) • We also looked at internal ordering of atoms in 3-D structure (230 space groups) • X-ray diffraction is the only way to determine space groups

2. X-rays • Another part of the electromagnetic spectrum between 100 and 0.2 Å. • Planck’s law: E=hn =hc/l • Where n is frequency, l is wavelength, h is Planck’s constant, and c is the speed of light

3. X-ray generation • X-rays are generated by striking a target material with an accelerated e- which causes an excitation. When his excitation ‘relaxes’, or goes back down to standard state, an X-ray is emitted • Usually given in terms of the energy levels those e- come from and go to  different levels yield X-rays of different energies (all dependent on the material) • K, L, M shells of a material, from that those shells have different transitions and characteristic relaxations (a, b, g) • Cu Ka is the most intense peak and most commonly used (though others are possible and have a different wavelength, which can be useful!)

4. X-Ray interaction • Scattering – oscillation of incoming X-rays transfer energy to electrons in material, emitting secondary radiation at about the same frequency and energy as the incoming beam • Interaction of X-rays with same material causes some electrons to go into an excited state, which upon relaxation, emits radiation characteristic of the atom it excited  basis for XRF, used to identify chemical makeup of materials • As with other interactions with minerals, there can also be reflection and transmission of X-rays (depending on thickness), but we don’t typically use that information.

5. l Interference • Constructive and destructive interference – wave properties interact to either cancel out or amplify each other. • When 2+ centers are emitting energy at some wavelength, they will interfere with each other Plane view

6. Experiment • Relationship between light as particles vs. light as waves • Light scattered by mesh - as it travels and interacts, some waves compliment each other while different waves cancel each other

7. Diffraction • Relationship between diffraction and wavelength: • The smaller the diffracting object, the greater the angular spacing of the diffraction pattern • i.e. the smaller the separation between planes, the wider the spacing between diffraction lines • What then is diffraction?? • The failure of light to travel in straight lines (much to Newton’s dismay…) • Young’s 2 slit experiment proved light could bend – scattered and affected by constructive and destructive interference • Bright red = constructive; dark = destructive

8. Diffraction • Combine elements of interference with striking the x-ray at an angle to the material • Relationship between wavelength, atomic spacing, and angle of diffraction for 3-D structures derived by von Laue • Bragg’s determined that you could simplify this and treat it as a reflection off of the planes within an atom…

9. Bragg’s Law • nl=2dsinΘ • Where n is the order of diffraction (always an integer), l is the wavelength of incident radiation, d is the spacing between planes, and Θ is the angle of incidence (or angle of reflection, they are equal)

10. Bragg’s Law • nl=2dsinΘ • Where n is the order of diffraction (always an integer), l is the wavelength of incident radiation, d is the spacing between planes, and Θ is the angle of incidence (or angle of reflection, they are equal) • Diffraction here is between parallel planes of atoms  the space between them (d) determines the angle of diffraction. • Looking at the laser pattern again  where is Bragg’s Law ‘satisfied’ and how many orders of diffraction do we see?

11. Red Laser analogue • We see orders of diffraction resulting from light coming between grid spacing 2, 3, 4, 5, etc., apart. In a mineral, multiple parallel planes yields similar patterns at higher orders of diffraction – theoretically the angle keeps increasing  what do we notice about the intensity though?

12. l These 2 constructively Interfere  good signal! Θ Θ d These 2 destructively Interfere  bad signal! l Θ Θ d Bragg’s Law • nl=2dsinΘ • Just needs some satisfaction!!

13. Detector typically moves over range of 2 Θ angles X-ray detector 2Θ X-ray source Sample holder Typically a Cu or Mo target 1.54 or 0.8 Å wavelength Orientation of diffracting hkl planes in sample A number of these are possible X-Ray Diffraction (XRD) equipment XRD machines vary angle as 2Θ because that angle is always relative to incident X-ray beam trajectory • nl=2dsinΘ • nl/2d=sinΘ • Solution ‘satisfied’ at specific angles (n MUST be an integer) 2Θ

14. XRD Part II • Theoretically, almost an infinite number of planes can exist, but certain ones diffract more strongly • Related to the atomic density – both of ## of atoms and in those ions’ atomic density

15. XRD results • Diffraction pattern • Higher symmetry  fewer, more intense lines because multiple planes are complimentary (identical d-spacings for different planes yields identical diffraction)

16. XRD analyses • Can look at minerals as single crystals or as a powder • Single Crystal  must be careful about orienting the crystal so Bragg’s Law is satisfied, use several different techniques, advanced machines manipulate the sample in 3 axes (x,y,z) to ‘catch’ all the peaks  required for structural determination • Powder has many particles with planes at many different orientations  many orientations satisfy Bragg’s Law, intensities and locations (2Θ) are characteristic of specific minerals. Technique primarily used for identification

17. Powder X-ray Analyses • XRD analysis of a powder is a common, quick, and relatively easy way to identify minerals. • Having a mixture of minerals can be tricky, so grains are first separated if possible (small amounts of other minerals will give other peaks, but intensities are low enough that it is not a big deal) • Do lose the ability to ‘see’ the details of the structure of the mineral however as the precise alignment of the mineral giving the peak is unknown and not changeable