V 16: X-ray Diffraction

Transcription

1 Matin-Luthe-Univesity Halle-Wittenbeg Institute of Physics Advanced Pactical Lab Couse V 16: X-ay Diffaction 1) Recod the chaacteistic of an x-ay counte filled with agon and halogen as a function of the voltage applied to the counte tube. The X-ay tube is to be caied on an effective voltage of 20.5 kv and a cuent of 1 ma. The counte tube voltage is to be changed in an inteval fom 200 V to 480 V. The X-ay tube is unning with altenating voltage. The pimay X-ay beam is to be filteed by a Z plate. Additionally to the plot detemine chaacteistic data as length and incease of the plateau, the woking point, the zeo effect and the limit of the pulse ate. 2) Recod the diffaction diagam of a ock salt cystal (cleavage plane (110), lattice constant a = cm) at 4 diffeent tube voltages (ectifying valve; U eff < 30 kv) and at a tube cuent of 1 ma with unfilteed adiation using a Bagg-spectomete. 3) Detemine the anode mateial of the X-ay tube with this set-up (spectal analysis). 4) Detemine Planck s quantum by means of the voltage dependent, shot-wave edge of the Bemsstahlung accoding the Duane-Hunt-law. 5) Recod the absoption spectum of Z and plot it. Specify the absoption edge of Z. 6) Find out the lattice constant of anothe ionic cystal with cubic symmety by means of monochomatic X-ay adiation (stuctue analysis). Specify the composition of the 2- atomic single cystal with the help of liteatue. In the case of ambiguity explain you choice. Hints Dead time of the counte is about 100 μs. Do not touch the mica window with a diamete of 9 mm! Use fo ecoding the specta the PC-digitize Meilhaus DS1M12 and the coesponding softwae. Save you data file with the extension *.txt. Open the file with Excel and copy the numbes via clipboad into Oigin. Pemit spaces as delimites in the ASCII option window of impot.

2 Questions fo testing you knowledge What about the efficiency at the geneation of X-ays using a conventional (sealed) X-ay tube? (Convesion of electical enegy to adiation enegy) What is the diffeence between X-ays and γ-ays? X-ays ae attenuated when they tansmit though matte. On which inteactions does the attenuation depend? An absoption spectum is sketched in figue 3. Explain the chaacteistic pofile. The emission lines ae also depicted in figue 3, they ae in each case on the ight side of the edge. Why? Give some capabilities of detection and measuement of X-ays and thei intensity. The positions of the X-ay eflections ae detemined by Bagg s law. Deive the Bagg law using figue 2! Which factos descibe the intensity of the X-ay eflections? How does the cystalline stuctue of the sample influence the intensity? Explain the tem pimitiveness of unit cell! How can you detemine the pimitiveness by means of X-ays? Calculate the stuctue facto of NaCl- and CsCl-stuctue, espectively.

3 Basics Geneation and popeties of X-ays X-ays and also γ-ays ae electomagnetic waves with shot wavelengths. Some phenomena can meely be explained using the model of photons, i.e. X-ays ae consideed as paticles. Although both X-ays and γ-ays ae of the same physical natue, as X-ays ae descibed adiation oiginating by acceleated chaged paticles. It can be chaged paticles moving on a cicuit as in a synchoton stoage ing o chaged paticles impinging on a taget as in a conventional sealed tube. Both at the conventional (Coolidge) tube and at the otating anode electons ae geneated by themionic emission and acceleated by a potential diffeence. When the electons ae impinging on the anode a small pat of the kinetic enegy E = e * U (e-elementay chage, U-applied voltage) ae tansfeed in X-ay adiation on the one hand by slowing down of the electons in the electic field of the atom coes (Bemsstahlung) and on the othe hand by impact ionization in the electon shell (chaacteistic X-ays). The slowing down pocess of the electons in the electic field is implemented in a multiple-stage, i.e. the kinetic enegy is conveted into X-ay adiation quasi in potions. The esult of this pocess is the so-called etadation spectum, which is independent on Figue 1: Scheme of geneation of X-ays. the anode mateial and shows a maximum enegy and a shotest wavelength, espectively. The maximum of the etadation spectum is at 1.5*λ Genz. Specify the coesponding factos of you expeiments! c h λ Genz = e U (1) (c-light velocity, h-planck s quantum), Equation (1) is known as Duane-Hunt-law. The intensity of the X-ays, i.e. the numbe of paticles pe aea and time, depends on the atomic numbe, tube cuent and tube voltage as it is given in equation (2). n I Z I A U (2)

4 (I A -tube cuent, n = 2 fo unfilteed adiation, up to 5 fo filteed adiation). Besides the etadation spectum some peaks aise, so called chaacteistic adiation. The oigin of them is the inteaction of the impinging electons with the atomic shell. The incident electons can knock out electons fom the atomic shell. These vacancies ae e-filled by electons fom highe enegy levels and finally the enegy diffeence between the levels is emitted as electomagnetic adiation. In the case of electon tansitions to enegy levels nea the atomic coe the adiation is in the X-ay ange because of the high binding enegy. The coesponding wavelengths ae descibed by Moseley s law (equation (3)) = R ( Z σ ) 2 2 (3) λ n m (R-Rydbeg constant, Z-atomic numbe, σ-sceening constant, σ=1 fo K-line and σ=7.4 fo L- line, n, m-pincipal quantum numbe) Accoding Jönsson und Begen-Davis the intensity of the chaacteistic peak is popotional to I ( U U ) 2 0 (4) (U 0 - citical excitation voltage of the anode mateial). Veify this elation and specify the citical excitation voltage! Inteaction of X-ays with matte When an X-ay beam inteacts with matte (elastically o inelastically) scatteing can occu. In the case of elastic scatteing the outgoing X-ays have the same enegy as the incoming X-ays, only with alteed diection. In contast, inelastic scatteing occus when the enegy is tansfeed fom the incoming X-ay to the matte, e.g. by exciting phonons o electons inside the sample. Accoding the classical scatteing theoy an electon is excited by an incoming, unpolaized X- ay and emits because of its oscillations an electomagnetic wave with the same wavelength. The intensity of the scatteed wave is descibed by the Thomson fomula: cos θ I e I = (5) ( o - classic electon adius, θ- scatteing angle, - distance, I 0 -incident intensity) Most atoms host moe than one electon. Consideing the scatteing at an atom, all patial waves of the electons of the atomic shell have to sum in phase. The esult is the atomic scatteing facto f A. i Q f ( Q) = ρ ( ) e d (6) A (ρ() - electon density distibution, Q - scatteing vecto) In the limiting case of a scatteing vecto Q 0 the scatteed waves of all volume elements ae in phase and theefoe the atomic facto is equal to Z. The next step is to conside the scatteed intensity of a cystal. A cystal is a solid in which the constituent atoms, molecules o ions ae packed in a egulaly odeed, epeating patten extending in all thee dimensions. One way to deive Bagg s law is to conside the diffaction of the X-ays on the atoms as a eflection at the net planes. Net planes ae specified by Mille s indices. Bagg s law (equation (7)) detemines the condition of a constuctive intefeence (see figue 2).

5 Figue 2: Deivation of Bagg s law. 2 d sinθ = nλ (7) Fo calculating the scatteed intensity it is necessay to detemine the scatteing amplitude: i Q j i Q Rn F = f ( Q ) e e (8) Kistall j j R n (R n - lattice vecto, j -position of atoms in the unit cell, R n + j position of atoms in the cystal) The fist facto is the stuctue amplitude, i.e. a summation ove all atoms in the unit cell. The othe gives the lattice amplitude, i.e. a summation ove all unit cells in the cystal. Thus, the scatteed intensity is given by I = F Kistall I e (9) With the help of calculated stuctue factos the selection ules fo the allowed eflection of the coesponding types of stuctue can be detemined. As a consequence the Bavais type and also the pimitiveness can be specified. In a quantum mechanical sense X-ays ae descibed as photons, i.e. as an elementay paticle. Theefoe, it caies momentum and enegy. As a paticle the photon can only inteact with matte by tansfeing momentum and enegy, see fo instance the Compton effect. Contay to Thomson s scatteing it is an inelastic phenomena. In the case of absoption the enegy of the photon is tansfeed to an electon which can be emitted. The est of enegy, i.e. the enegy diffeence between the photon enegy and the woking function of the emitted electon, contibutes to the kinetic enegy of the electon. All pocesses of absoption can be summaized in the linea absoption coefficient, thus, the intensity of the tansmitted beam is given by the absoption fomula μ z I = I 0 e (10) (µ- linea absoption coefficient, z- tansmitted thickness) Afte ionization the atom etuns into the gound state by filling up the vacancy with an electon fom a highe enegy level and emitting a photon. The paticula spectal lines depend on the taget element and thus ae called chaacteistic lines. Usually these tansitions fom uppe shells into K-shell ae called K-lines, into L-shell L-lines and so on. With the help of these lines the element can be detemined, e.g. at X-ay spectoscopy o at electon micoscopy (element mapping). The emitting photon can also knock out an electon fom a highe enegy level. This electon is called Auge electon and can be used by the so called Auge spectoscopy, a suface sensitive method. The absoption pocess eveals a distinctive dependence of the absoption coefficient on the photon enegy of the incident X-ay. The absoption spectum of Pt is depicted in figue 3. Below the bonding enegy of the K-electon of kev the incoming photon can impact L- and M-electons only. With inceasing enegy the absoption coefficient jumps apidly at the K edge and deceases as E -3. The edge stuctue at the othe edges is analogue. 2

6 Figue 3: Absoption spectum of Pt. Detection of X-ays Photogaphic film was the fist detecto fo X-ays. In the last yeas image plates and CCD chips eplace this technique. X-ay detectos othe than the two-dimensional detectos may be classified into thee types: ionization countes, scintillation countes and semiconducto detectos. In this expeiment the fist type is used, i.e. a so called halogen counte, that is a cylindical envelope with two electodes, a cental wie maintained at an appopiate potential and a gounded coaxial conducting cylinde and filled with agon and halogen gas as quenching gas. X-ays ente the counte though a 3 µm thick window of mica Nea the anode the electic field stength depending on the distance fom the cente (see equation (11)) is vey high. U Z 1 ER ( ) = Anode: 0 (11). ln K A (U Z applied voltage, K adius of the cylinde, A adius of the cental wie). Accoding to the applied voltage diffeent inteaction phenomena appea and esult in a typical chaacteistic as shown in figue 4. In the voltage ange up to U Z < 100 V fee electons (pimay electons) ae geneated by pimay photoionization pocess. As a consequence a cuent is flowing to the anode, obeying Ohm s law because of ecombination of a pat of the electons with the ions in the gas-filled chambe. Between 100 V and 200 V (see figue 4) the ecombination is suppessed and the cuent eaches a satuation value. In this ange ionization chambes ae unning. At voltages U Z > 200 V the inteaction of the pimay electons with the atoms stat up the seconday ionization pocess. In this ange the gain facto is about 100 and does not depend on the enegy of the incoming adiation. The popotional o gas gain facto depends on the type of the gas, the gas pessue and the applied voltage. The signal is a function of the enegy of the detected paticles and not of the numbe, thus, the counte is used as a popotional counte to analyze enegy with an accuacy of about some pecent.

7 A futhe ise of the voltage above the popotional ange leads to a highe detection sensitivity. One pimay electon can initiate a gas dischage and a signal popotional to the numbe of the detected paticle independent on thei enegy (Geige-Mülle-counte) is ecoded. When the applied voltage is too high a Townsend dischage can occu and as a consequence the counte can be destoyed. Figue 4: Scheme of the cuent-voltage chaacteistic of a gas-filled counte. The intensity of the incident ionizing adiation is constant. Dead time coection: Immediately afte a dischage the cental wie is envioned with positive ions peventing to count futhe incoming paticles. The cathode is electically sceened by the space chage effect. As a consequence the electic field is deceased and a dischage can not take place, the counte is dead. Only when the positive ions migate to the cathode the electic field is ebuilt and futhe incoming paticles can be detected. Usually the dead time is detemined expeimentally. The intensity is popotional to the tube cuent and govened by n0 n =, 1+ n0θ ( n measued pulse ate, n 0 actual pulse ate, θ - dead time). Gas fillings of ionization countes Because of its high gas gain facto inet gases ae used eithe with an additive o without one. Alkanes like methan and ethan o halogens like B and Cl ae used as quenching gases to educe the dead time.

8 Expeimental devices Bagg s spectomete Shotly afte the discovey of the X-ays in 1895 by Conad Wilhelm Röntgen, besides the most common use of X-ays in medicine, two main fields in natual science have emeged, Figue 5: Scheme of a Bagg spectomete. spectometic applications and investigations of the stuctue of solids. The fist spectomete was constucted by W.H. and W.L. Bagg and consists of an X-ay tube, a cystal as sample and a detecto. The sample and the detecto ae otated aound the goniomete axis in a atio of 1 : 2. The oientation and the adjustment of the single cystal detemine the eflecting net planes, thus only the diffeent odes of the coesponding eflection can one be ecoded. The intensities of the vaious eflections decease as the squae of the ode. Veify this coelation! In the case of a powde sample an oientation of the sample is not necessay, the eflection ae measued in any case. Point out the diffeences between single cystal, mosaic cystal and powde cystal! Refeences: Nielsen: Elements of Moden X-Ray Physics Kittel: Intoduction to Solid State Physics