A n alytical techniques

2.2.1.

X-ray diffraction

2.2.1.1. Introduction

Today, X -ray diffraction is a standard technique em ployed in the determ ination o f the structure o f single crystals and to a lesser extent pow ders. R esearch uses laboratory X -ray sources extensively. H ow ever, this type o f X -ray source only provides a low intensity X -ray beam. C onsequently, one requires large sample volum es (several m g) in order to obtain a good pow der diffraction spectrum. A diam ond anvil cell experim ent involves only a sm all am ount o f sam ple (< 1 mm^) , thus the use o f a laboratory X -ray source w ould require us to collect for at least 24 to 48 hours in order to obtain an acceptable pow der diffraction pattern. H ow ever, the use o f very sensitive detectors can im prove those collections times to som e extend. Thus, diam ond anvil cell research requires a m uch brighter X -ray source. Synchrotron sources and especially third generation synchrotrons sources have a considerably higher flux o f X -rays (figure 2-4). S ynchrotrons allow the collection o f X -ray diffraction patterns from a sam ple in a diam ond anvil cell in ju st a few seconds (or m inutes). A further advantage o f synchrotron X -rays is the possibility o f tuning the energy o f the m onochrom atic beam to the optim um w avelength. For exam ple, the m ost com m on laboratory X-ray sources are from copper, m olybdenum and silver. The copper radiation (C uK a: A, = 1.5406 Â; E = 8.06 keV) is not suitable for a diam ond cell experim ent as diam ond strongly absorbs the radiations in that region o f the electrom agnetic spectrum (figure 2-8). A m ore standard choice o f radiation for laboratory based diam ond anvil cell experim ents is M o K a w hich has a higher energy (A = 0.7093 Â , E = 17.5 keV ). A t that energy, 4 mm o f diam ond, typical thickness for a diam ond anvil cell experim ent, only absorbs 50 % o f the radiation.

2.2.1.2. H o w does X -ra y diffraction work?

The X -rays range from soft X-ray: (0.1 keV - 1 keV) to hard X -rays (10 keV and over). X -ray diffraction experim ents at high pressure require energies betw een 15 and 50 keV (~ 0.25 - 0.8 A). The w avelength is in the order o f the atom ic size and interacts w ith the electrons o f the atom s. Thus, row s o f atom s acts as a dispersion grating and disperse the light. The sam e w ay a specific grating at a set angle w ould send a m onochrom atic light beam in a specific direction, the row s o f atom s w ith a set interatomic distance are sending a m onochrom atic X -ray beam in a specific direction. The crystal is an array o f atom ic planes. T herefore, constructive and destructive interactions o f the X-rays occur in the crystal. X -ray diffraction allow s the determ ination o f the crystal sym m etry and the interatom ic distances. U sing B rag g ’s equation

(1) and P lan ck ’s equation (2) (fo r energy dispersive X -ray d iffractio n ), one can tran slate the X- ray diffraction pattern s into inter-p lan ar distances.

nX = 2d?>\x\0 (1)

w here X is the w av elength o f the incident beam , d the in ter-p lan ar distance and 6 th e angle b etw een the incident beam and the atom ic planes. Figure 2-1 ex p lain s relation (1 ) geom etrically. BG and BH are p erp en d icu lar to A B and BC respectively so A B = DG and BC = HF. In o rd e r to ensure that the w aves are in phase G E + EH m ust be equal to nX. U sing trig o n o m etry w e see that G E = d sin 0 a n d EH = d sin 6. T hus G E + EH = 2 sin 0 = nX.

d

v . X 'V. V V V. V. t T

Figure 2-1 G eom etry o f X -ray diffraction. I f there are other atom s p resen t betyveen the layers,

the intensity o f the diffraction spot m il decrease.

(2)

w here E is the energy, c the speed o f light, X is the w av elength and h is the p ro p o rtio n ality

co n stan t know n as P lan ck ’s co n stan t ( h - 6 .6 3 x 10'^"* J • s ) . O ne can also w rite this equation as equation (3) w hen E is in eV and X in nm.

E = 1241 (3)

In the thesis, w e presen t data from X -ray diffraction ex p erim en ts perform ed o nly on p o w d ered sam ple. In that case, the d iffractio n pattern results from a m u ltitu d e o f random ly oriented crystallites. T h erefo re, instead o f o b tain in g one diffraction spot for each d -spacing, a full diffraction ring is generated (F igure 2-2).

S am p le

X -ray b e a m

, ' S'l' ■ ■■

d iffra c tio n

Figure 2-2 Illustration o f an angular dispersive pow der diffraction experiment.

In an g u lar dispersive X -ray d iffraction, a m onochrom atic X -ray beam p asses th ro u g h the sam ple. T hus, the diffraction from the sam ple collected on the d etecto r co rresp o n d s to the intersection betw een the diffraction cones and the detector. T he X -ray pattern (F igure 2 -1 0 ) is the resu lt o f the integration around the rings (intersection betw een a cone and the orthogonal d etecto r plane) (figure 2-2). C o n seq u en tly , in angular d isp ersiv e d iffraction, the variable in B rag g ’s eq u atio n is the angle 0. So the r/s p a c in g can be calcu lated as d = f { 6 ) .

•etector D iffracte d X -ray b ea m ro tatio n W h ite sy n c h ro tro n N a rro w slits X -ray b eam S a m p le E nergy (k eV )

F igure 2-3 Diagram show ing an exam ple o f energy dispersive diffraction experiment. A rrow s

indicate the rotations o f the diam ond anvil cell in order to reduce the texturing problem s.

In en erg y d isp ersiv e X -ray diffractio n experim ents, a w h ite beam p asses through the sam ple (F igure 2-3). T he d etecto r is eith er a single o ne-dim ension o r a sm all n u m b er o f o n e-d im en sio n d etecto rs aligned along a circle. Each d etecto r reads the n u m b er o f p hotons as a fu n ctio n o f

energy. T hus, the d spacing is a function o f the energy for a given angle: d = J{E). U sing eq u atio n 1 and 2 one can easily deduce the equation 4.

d = he

2 £ s in ( 9 (4)

2.2.1.3. What is a synchrotron?

Figure 2-4 Picture o f the A dvan ced Photon Source (APS), a third generation synchrotron

located at the A rgonn e N ational Laboratory (ANL) south o f Chicago (Illinois, USA).

Synchrotron radiation arises from the acceleration o f a ch arg ed particle (p o sitro n or electron) circu latin g at a velocity close to the speed o f light [98, 99]. T he sim plest w ay to create X radiation is to use a “b en d in g m agnet” (figure 2-5). The b en d in g m agnet gives the shape o f the storage ring. H ow ever, progress in synchrotron radiation tech n iq u es gave rise to insertion d evices (fig u re 2-5). T he insertion devices generate a m uch h ig h er flux o f X -rays.

K lectioii E lec tro n s X -r a \ em issio n B e iu liiiii M a iiiie t E lec tro n s e m is s io n e m is s io n

To reach such a high speed the particles m ust accelerate. T he first part o f the acceleratio n pro cess takes place in the electron gun w here a cathode-ray em its electrons. A first m agnetic field accelerates the particles up to 100 keV T The linear accelerato r (L inac) then pulses the electron and increases the energy up to 450 keV . In the fo llow ing step, an injection p ro cess sends the particles into the booster synchrotron. T he booster syn ch ro tro n then accelerates the particles up to 7 G eV using electrom agnets. In the final step, an injection p rocedure sends the particles into the storage ring. A lthough the particles should only travel in a straig h t line, b en d in g m ag n ets curve th eir trajectory (fig u re 2-4). Each tim e a m agnet b ends the beam o f particles the particle produces energy in the form o f photons alo n g the incom ing d irection o f the particles. T hus, the photons are propagating in a tangent to the storage ring. T he p h o to n s have en erg ies ran g in g from the far-infrared (F-IR ) to the X -ray regim e (fig u re 2-6).

A P S undulator o P A P S B en d in g m agne^ 73 2 C 2 o _ c N S L S B en d in g m agn et a.

<

Photon energy (eV)

Figure 2-6 Exam ples o f brilliance curves f o r synchrotron radiation. The vertical lines

displaying the brilliance o f Cu K and M o K radiations show the range o f brilliance available

fro m various X -ra y tube sources. Diagram adapted fro m that presen ted on the A P S website.

The values o f energies given in this paragraph correspond to the A dvanced Photon Source (A P S ) at A rgonne national laboratory (U S A ).

In third generation synchrotrons, and to som e extent first and second g eneration sy n ch ro tro n s, the particles (p o sitro n s or electrons) also create X -rays by passin g through insertion dev ices (w ig g ler or undulator). The w iggler/undulator is an array o f positive and neg ativ e p erm an en t m agnets. The p articles undulate w hile trav ellin g through the device. A t each d eflectio n , the p article em its synchrotron radiation (X -rays form part o f the em itted synchrotron rad iatio n ). The X -ray brilliance is increased by several orders o f m agnitude from a b en d in g m agnet b eam lin e to an insertion d evice beam line. The increase in the brilliance is due to the m uch sm aller an g u lar dispersion o f the radiation and the increased num ber o f dipoles. A further advantage o f w ig g lers in low energy sy nchrotrons is their capability o f shifting the X -ray energy range to higher en erg ies (u n d u lato rs do not allow a shift in w avelength). T h is is particu larly v alu ab le for d iam ond anvil cell ex p erim en ts as diam ond strongly absorbs low energy X -rays (figure 2-8).

3 d

angle integrated spectrum

pin hole spectrum

c

2.5 5 7.5 10 12.5 15