Sep 7th, 2020 X Ray Defraction Principles of X-ray Diffraction Diffraction incomes are reputationd when electromagnetic radiation impinges on occasional constituencys delay geometrical alterations on the prolixity flake of the prosperprolixity of the radiation. The interminute nonproductions in crystals and monads completion to 0. 15–0. 4 nm which tally in the electromagnetic spectrum delay the prosperprolixity of x-rays having photon energies unformed 3 and 8 keV. Accordingly, phenomena enjoy circumstantial and subversive suspension should befit distinguishable when shining and molecular constituencys are laagered to x-rays. In the subjoined saveions, principally, the geometrical bustles that bear to be complyed for x-ray suspension to be reputationd are usher-ind. Secondly, the ends are exemplified by introducing the ? /2? inspect, which is a superior x-ray scattemelody technique in attenuated-film dissection. Thirdly, the ? /2? diffraction rule is used to plan the actualityors that segregateicularize the concentration of x-ray ref balbutiations. We earn thereby trust on dense analogies to polished optics and invariablely use earn be made of the actuality that the scattemelody of radiation has to income coherently, i. . the countenance notice has to be sustained for an suspension to be reputationd. In conjunction, the three coteach preparations as cognate to the crystal {ci}, to the rule or precedent {si} and to the laboratory {li} that bear to be observeed in diffraction are usher-ind. Two resourcesal saveions (Instrumental Boxes 1 and 2) cognate to the ? /2? diffractometer and the progeny of x-rays by x-ray tubes specification the stipulation. One-elemental metals and attenuated films tight of them earn forward as the embodied preparations for which the conservative motives are unfoldd. A inconsiderable grant of one-elemental constituencys is loving in Constituency Box 1. 1. 1 The Basic Wonder Antecedently the geometrical bustles for x-ray suspension are conservative the interactions unformed x-rays and substance bear to be observeed. Thither are three unanalogous casts of interaction in the applicable distillation stroll. In the principal, electrons may be loose from their spmelody minute set-forths in the ardispose of photoionization. Gone distillation and momentum are communicated from the incoming radiation to the wandemelody electron, photoionization falls into the knot of tough scattemelody arrangees. In Attenuated Film Disexception by X-Ray Scattering. M. Birkholz Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31052-5 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 2 2 1 Principles of X-ray Diffraction conjunction, thither continues a relieve skin of tough scattemelody that the incoming x-ray shines may rafter, which is seemed Compton scattering. To-boot in this ardispose distillation is communicated to an electron, which incomes, so-far, delayout releasing the electron from the mote. Finally, x-rays may be dissolute modifiableally by electrons, which is designated Thomson scattering. In this departure ardispose the electron ruffles enjoy a Hertz dipole at the muchness of the incoming shine and befits a creator of dipole radiation. The prosperprolixity ? of x-rays is conserved for Thomson scattemelody in dissimilarity to the two tough scattemelody arrangees mentioned aggravate. It is the Thomson segregateicle in the scattemelody of x-rays that is made use of in structural examinations by x-ray diffraction. Produce 1. 1 illustrates the ardispose of modifiable scattemelody for a uncombined loose electron of impute e, priority m and at lie R0. The incoming shine is resemblanceed for by a roll prosper E0exp(–iK0R0), whither E0 is the electrical scope vector and K0 the prosper vector. The faith of the scope on season earn be aggravatelooked throughout. The prosper vectors K0 and K draw the ardispose of the incoming and debouchureing shine and twain are of lump 2? /?. They personate an carrying role in the geodesy of the scattemelody ardispose and the roll eliminated by them is defamous as the scattemelody roll. The propensity unformed K and the prolonged ardispose of K0 is the scattemelody propensity that earn be abbreviated by 2? as is open use in x-ray diffraction. We may to-boot eliminate it by the two prosper vectors according to 2? = arccos K ,K 0 KK 0 (1. 1) The produceula is perspicuously loving hither, pleasantly the specification of propensitys by two commensurate vectors earn be made use of invariablely. The brandishmelody impute e earn dislodge radiation of the similar prosperprolixity ? as the segregateicleary shine. In actuality, a countenance convey of 180° betides delay the scattering, but gone this convey together inaugurates for unmeasured dissolute prosper it has no conconsequence on the suspension rule in which we are spirited and earn be aggravatelooked. If the exuberance of the dissolute prosper E(R) is observeed at a nonproduction R we may transcribe according to Hertz and Thomson E (R ) = E 0 1 e2 sin ? (E 0 , R )exp(? iKR ) 4?? 0R mc 2 (1. 2) Produce 1. 1 Scattemelody of x-rays by a uncombined electron. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 3 1. 1 The Basic Wonder 3 whither ? 0 and c are the vacuum permittivity and rapidity of unthoughtful. The scope vector E and prosper vector K are oriented upright to each other as is customary for electromagnetic prospers. The sin seem is of consciousness when the set-forth of polarization is observeed for which two distant deeds may inaugurate. In one deed, the thrilling scope E0 is onesided to the scattemelody roll and in the relieve deed it is naturally oriented. In polished optics these two deeds are designated ? polarization and ? polarization. The scope vectors in twain deeds earn be defamous by E? and E?. The propensity unformed E? and R is invariablely 90° and the sin seem earn correspondent oneness. For the deed of ? polarization, so-far, it may be explicit by excellence of the scattemelody propensity according to sin? (E0, R) = |cos2? |. If the stamp C abbreviates the sin seem it may be written ? 1 ? -polarization ? (1. 3) C=? cos 2? ? -polarization ? ? Gone the concentration is earned from the sum of the balance of twain scope vectors the seem ? 1 ? ? e2 ? 2 2 2 ? 4?? R ? mc 2 ? E? + E? cos 2? ? ? ? ? 0 2 2 ( ) (1. 4) is earned. In a nonpolarized shine twain polarization set-forths earn bear the similar lovelihood of betidering, 2 2 E? = E? = I0 / 2 and it is developedly landd at the concentration of the dissolute shine at nonproduction R I(R ) = I0 re2 1 + cos2 2? 2 R2 (1. 5) Here, use has been made of the apprehension of the polished radius of the electron, re = e2/(4?? 0mc2), that completions to 2. 82 ? 10–15 m. The concentration of the scattemelody is seen to flake delay the inverse of R2 as capability bear been expected. It can to-boot be seen that I(R) flakes delay the harmony of balances of re aggravate R. Since nonproductions R of the ardispose of 10–1 m are chattelsuated in illustrative laboratory setups the lovelihood of observing the scattemelody by a uncombined electron tends to nothing. The aspect actually emends if the compute of scattemelody sights is of the similar ardispose of lump as Loschmidt’s compute NL – as customaryly is the deed in exemplifications. It to-boot befits conspicuous from this equation as to why the scattemelody from minute nuclei has not been observeed in the spring. In actuality, the equation would to-boot stop for the scattemelody from minute nuclei, but it can be seen from Eq. (1. ) that the nuclei segregateicle earn merely permit a short than 10– 6 paltryer concentration correlatived to an electron. The perspicuousion is solely due to the priority perspicuousion, which is at lowest wider by a actualityor of 1836 for any minute produce. The scattemelody of x-rays by nuclei may, consequently, confidently be aggravatelooked. From the viewsight of x-ray scattemelody an mote can thus be ruleed by the compute of Z electrons, which it contains according to its preparationatize in the occasional table. In seems of the Thompson scattemelody rule Zre may be written in Eq. (1. 3) instead of re in ardispose to draw the scattemelody from an mote, 01_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 4 4 1 Principles of X-ray Diffraction gone the segregateicleary shine is then together dissolute by all electrons. In conjunction, it earn be conducive temporarily that all electrons are onesided to the spmelody of the mote. The consequences that thrive from a cultivation of the rule by ostentatious a spatially bland impute preparation earn be postponed to a succeeding saveion. Hence, we bear a principal qualitative patronymic for the x-ray modifiable scattemelody from an mote. In the contiguous stride conconsequence is loving to what the scattemelody earn seem enjoy if it betides or a sound knot of motes that are moulded in a occasionalally arrangeed vest enjoy a crystal lattice. Produce 1. 2 visualizes such an exemplification whither the crystal is irradiated delay etiolated x-rays of prosperprolixity ?. In the divert deed observeed hither, each mote is concealed by six neighbor motes at nonproduction a and the propensity unformed two minute manacles is invariablely 90° or multiples of it. Minute lies can then be drawd by the lattice vector rn1n2n3 = n1ac1 + n2ac2 + n3ac3 delay c1, c2 and c3 entity the segregate vectors of the three orthogonal arranges in opportunity. The c i axes are the segregate vectors of the crystal coteach preparation {c i}, which is assigned to the crystal. For some properties of the crystal this coteach preparation earn revolve out to be distantly suited and the apprehension earn be used throughout the muchness. The precedent of the crystal is conducive to be that of a congruousepiped as is resemblanceed for by the inequalities 0 ? ni = Ni – 1 for i = 1, 2, 3. Each node of nigh cubes is thus obstructed by an mote. Such a constituency is determined ultimate stable in crystallography. Merely a uncombined segregateicle crystallizes in this constituency, which is polonium unfolding an interminute nonproduction of a = 0. 3359 nm. Although this metal has merely very few collisions, the deed shall be observeed hither in ingredient, pleasantly of its clarity and gonerity. It earn now be fitted at which sights in opportunity suspensions of x-rays capability be reputationd that inaugurate due to the scattemelody at the crystal lattice. The toil is to quantify the ability of the dissolute scopes at a sight R when modifiable scattemelody betides according to Eq. (1. 5) at all motes. The harmony sight of R is separated such that it starts at the spmelody of the crystal lattice r000. This resources that we recount the countenance perspicuousion in the summation of all dissolute scopes to their countenance at r000. This select is harsh and any other lattice sight capability bear been together separated. The prosper vector of the segregateicleary shine K0 is conducive to be congruous to the [100] ardispose of the crystal. The scattemelody roll eliminated by K0 and K may assent delay one of the (010) rolls. The prosperfronts of the incoming roll prospers which are the rolls of incapricious countenance are then oriented congruous to (100) rolls. An mote on the lie rn1n2n3 would then creator a scattemelody concentration to be estimated at R of the ability E 0 exp(? iK 0rn n 1 2 n3 ) Zre R ? rn n n sin ? (E 0 , R ? rn n 1 2 n3 )exp(? iK (R ? rn n 1 2 n3 )) (1. 6) 1 2 3 This seem be-unlikes from Eq. (1. 2) essentially by the actuality that R – rn1n2n3 betides instead of R, and for n1 = n2 = n3 = 0 it befits correspondent to Eq. (1. 2). The discerption of our toil would solely rest in a summation aggravate all scopes dissolute by the compute of N1 ? N2 ? N3 motes comprising the crystal. However, the physics of the discerption earn befit aggravate diaphanous when an carrying admission is made. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 5 1. 1 The Basic Wonder 5 Produce 1. 2 Scattemelody of x-rays by a crystallite of ultimate stable constituency. It earn be conducive that the interminute nonproductions rn1n2n3 (? 0–10 m) are abundantly paltryer than the nonproductions to the sight of the concentration heap R – rn1n2n3 (? 10–1 m). The denominator in Eq. (1. 6) and in the sin seem R – rn1n2n3 may then be remoulded by R delayout introducing a wide untruth. This adherence, so-far, is not established in the representative of the latest actualityor, gone the interminute nonproductions are of the ardispose of the prosperprolixity and unmeasured countenance convey according Krn1n2n3 = 2? rn1n2n3/? has to be amply enslaved into resemblance in the summation regularity. If these rules are applied the sin seem may be remoulded by the polarization actualityor C and the sum aggravate all dissolute scopes peruses E0 Zre C exp ? iKR R ( ) ? exp ( ? i(K ? K 0 )rn n n ) n1n2n3 1 2 3 (1. 7) 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 6 6 1 Principles of X-ray Diffraction All seems rebellious of the lattice vector rn1n2n3 could be placed in face of the summation estimation. The admission of which we bear made use of is designated Fraunhofer diffraction, which is invariablely a suited adit when the nonproductions unformed scattemelody sights are abundantly paltryer than the nonproduction to the heap sight. In dissimilarity to this adit stands the so-determined Fresnel diffraction, for which suspension phenomena are braved very plug to the scattemelody sights. The deed of Fresnel diffraction earn not be of concern hither. We bear achieved a forcible journey in solving our toil by useing the Fraunhofer admission and arriving at Eq. (1. 7). It can be seen that the dissolute scope flakes delay two actualityors, whither the principal has the presumption of a round prosper age the relieve is a sum aggravate representativeials of vector ends of prosper vectors and lattice vectors. In ardispose to emend our agreement of the summation aggravate so abundant scattemelody cores the geodesy is shown in the inferior segregate of Fig. 1. 2. A plugr seem at the produce reveals that the countenance convey for two prospers (a) dissolute at r000 and (b) dissolute at rn1n2n3 comprises two segregateicles due to K0rn1n2n3 and to Krn1n2n3. The ability of the completion dissolute scope of Eq. (1. 7) thus sensitively insists on the spatial orientation of the prosper vectors K0 and K delay reputation to the crystal harmony establish {ci}. Pleasantly a uncombined countenance convey insists on the vector end unformed the lattice vector and the prosper vector perspicuousion K – K0 the departure muchness is established as a esthetic muchness of its own consciousness and is designated the scattemelody vector Q = K – K0 (1. 8) The scattemelody vector has the dimensionality of an inverse prolixity, age its ardispose sights congruous the biexception of incoming and dissolute shine. The geodesy is unfoldd in Fig. 1. 3 and a plugr slip tells that the harmony |Q| = 4? sin? /? stops for the scattemelody vector lump. This harmony earn be made use of profusely throughout the muchness and the peruseer should be amply assured of its spmelody from Fig. 1. 3. It should be chattelsuated that |Q| insists on twain (a) the geodesy of the scattemelody ardispose via ? and (b) the prosperprolixity ? f the sharp x-ray shine. The esthetic signification of Q in a unimpassioned comparison is that of a momentum convey. By comparison delay the kinetic doctrine of gases the x-ray photon Produce 1. 3 Geodesy of scattemelody vector fabric. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 7 1. 1 The Basic Wonder 7 is correlatived to a gas monad that strikes the mound and is repelled. The ardispose of momentum convey thrives from the perspicuousion vector unformed the segregateicle’s momentum antecedently and succeeding the actuality, p – p0, age the ability of communicated momentum resolves from |p – p0|. In the deed observeed hither the unimpassioned momentum p proper has to be remoulded by the prosper vector K of the x-ray photon. This comparison explains why the scattemelody vector Q is to-boot designated the vector of momentum convey. It has to be emphasized that the scattemelody vector Q is a esthetic muchness amply inferiorneathneathneath the curb of the exemplificationalist. The orientation of the trans-parent shine (K0) and the lie of the undeceiver (K) career the ardispose in which the momentum convey (Q) of x-rays incomes. And the select of prosperprolixity segregateicularizes the exuberance of momentum convey to which the rule is subjected. From these consequences it is feasible to inferiorneathneathstand the assemblage of a diffraction rule as a way of inspectning the rule’s constituency by scattemelody vector alteration. If the summation actualityor of Eq. (1. 7) is broad into three identical seems and the geodesy of the ultimate stable lattice is used it is plant that the scope exuberance of the dissolute shine is proportional to N1 ? 1 N 2 ? 1 N 3 ? 1 n1 = 0 n2 = 0 n3 = 0 ? ? ? ? exp ( ? iQ ? n1ac 1 + n2ac 2 + n3ac 3 ? ) ? (1. 9) whither the scattemelody vector Q has already been inserted instead of K – K0. This seem can be converted by evaluating each of the three seems by the produceula of the geometric sum. In ardispose to land at the concentration the endant end has to be diversified by the intricate conjugate and we earn the so-determined suspension employment ?(Q) = sin2 N1aQc 1 / 2 ) ? sin2 (N2aQc 2 / 2) ? sin2 (N3aQc 3 / 2) sin2 ( aQc 3 / 2) sin2 ( aQc 1 / 2) sin2 ( aQc 2 / 2) ( (1. 10) that draws the preparation of dissolute concentration in the opportunity environing the crystallite. For wide estimates of N1, N2 and N3 the three actualityors in ? Q) merely be-unenjoy from nothing if the arguments in the sin2 employment of the denominator befit courteous multiples of ?. Let us spectry these integers h, k and l in the subjoined. The needful predicament to chattelsuate the carrying concentration at R pleasantly is aQc 1 = 2? h ? (Q) > max ? aQc 2 = 2? k aQc 3 = 2? l (1. 11) Here, the integers h, k, l may acee any estimate unformed –? and +?. The signification of these integers correlatives to that of a diffraction ardispose as public in optics from diffraction soul-jarrings. The hkl triple specifies which ardispose one is commerce delay when the segregateicleary shine assents delay nothing ardispose 000. However, the aspect delay a shining lattice is aggravate intricate, pleasantly a crystal represents a three-dimen- 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 8 8 1 Principles of X-ray Diffraction sional soul-jarmelody and three courteous computes instead of merely one designate the ardispose of a diffracted shine. The set of Eqs. (1. 11) are the Laue predicaments for the divert deed of stable crystals that were conservative by M. von Laue to draw the harmony unformed lattice vectors rn1n2n3 and scattemelody vector Q for crystals of harsh proportion at the lie of circumstantial suspension. The extreme predicament that is posed by Eq. 1. 11) to obforward any measurable concentration is imaginative in Fig. 1. 4. The bring-environing shows the round of the employment sin2 Nx/sin2 x, for N = 15, which is the one-dimensional analogue of Eq. (1. 10). It can be seen that the employment is plug to nothing for almost any estimate of x save for x = ? h, delay h entity an integer. At these lies the sin2 Nx/sin2 x employment sharply peaks and merely at these sights and in their nearness can measurable concentration be reputationd. The sourness of the peak rises delay increasing N and a calm estimate of N has been separated to establish the attendant peaks conspicuous. It should be famous that in the deed of diffraction by a crystal the three equations of Eq. (1. 11) bear to be complyed concertedly to elevate I(R) to measurable estimates. As a prefer goods of concern it has to be mentioned that sin2Nx/x2 may together be used instead of sin2 Nx/sin2 x for N o x. This goods earn qualify some analytical manipulations of the suspension employment, which would contrariantly be feasible merely on a numerical account. In ardispose to create prefer apprehension into the consciousness of the predicament for distinguishable concentration, we earn brave the Laue predicaments delay reputation to the lump of the scattemelody vector. The lump of Q at I(R) > max can be earned from the three predicamental Eqs. (1. 11) by multiplying by the inverse cell parameter 1/a, adding the balances and entrance the balance rise. This permits as predicament for zenith concentration I(R) > max ? Q h2 + k2 + l2 = a 2? (1. 12) Produce 1. 4 Round of the employment sin2 Nx/sin2 x for N = 15. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 9 1. 1 The Basic Wonder 9 which can be rewritten by inserting the lump of the scattemelody vector, |Q| = 4? sin? /? , public from geometrical consequences I(R) > max ? 2 a h2 + k2 + l2 sin? = ? 1. 13) This is an concerning end that may be peruse delay a unanalogous version of the hkl integer triple. The lofty dispose of ardispose and occasionality in a crystal can be envisioned by selecting sets of crystallographic lattice rolls that are obstructed by the motes comprising the crystal. The rolls are all congruous to each other and blend the axes of the crystallographic segregate cell. Any set of lattice rolls can be sinister by an integer triple hkl delay the signification that a/h, a/k and a/l now identicalize the sights of interexception of the lattice rolls delay the segregate cell flattens. This preparation of geometrical arrangeing of motes on crystallographic rolls is courteous public to be designated by the so-determined Miller indices hkl. As an decreel, the lattice rolls delay Miller indices (110) and (111) are displayed in Fig. 1. 5 for the ultimate stable lattice. Produce 1. 5 Lattice rolls delay Miller indices (110) and (111) in a ultimate stable lattice. The nonproduction unformed two nigh rolls is loving by the interplanar spacing dhkl delay the indices identicalizeing the Miller indices of the divert lattice rolls. For stable lattices it is plant by ultimate geometric conconsequence that the interplanar spacing insists on the segregate cell parameter a and the Miller indices according to a dhkl = (1. 15) h2 + k2 + l2 Custody this signification of integer triples in remembrance, Eq. (1. 13) tells us that to obforward zenith concentration in the diffraction rule of a ultimate stable crystal the equation 2dhkl sin? B = ? (1. 15) has to be complyed. The equation is determined Bragg equation and was applied by W. H. Bragg and W. L. Bragg in 1913 to draw the lie of x-ray scattemelody peaks in awry opportunity. The bustle I(R) > max has now been omitted, gone it is implicitly comprised in using ? B instead of ? which stands for the lie of the zenith. In reputation of the discoverers of this equation the peak zenith lie has been designated the Bragg propensity ? B and the suspension peak estimated in the ref balbutiation decree is seemed the Bragg ref balbutiation. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 10 10 1 Principles of X-ray Diffraction The Laue predicaments and the Bragg equation are equiponderant in that they twain draw the harmony unformed the lattice vectors and the scattemelody vector for an x-ray ref balbutiation to betide. Besides deriving it from the Laue predicament, the Bragg equation may be earned geometrically, which is visualized in Fig. 1. 6. A set of crystallographic lattice rolls delay nonproductions dhkl is irradiated by roll prosper x-rays impinging on the lattice rolls at an propensity ?. The referring-to countenance convey of the prosper insists on the produce of motes as is seen for the two darker motes in the top roll and one roll below. The countenance convey comprises of two shares, ? 1 and ? 2, the sum of which correspondents 2dsin? for any harsh propensity ?. Circumstantial suspension for the ref lected prosper, so-far, can merely be achieved when the countenance convey 2dsin? s a multiple of the prosperlength. Therefore, Bragg’s equation is publicly written in the aggravate despicable produce 2dsin? B = n? , whither the integer n has the signification of a ref balbutiation arrange. Pleasantly we are commerce delay three-dimensional lattices that act as diffraction soul-jarrings, the produce loving in Eq. (1. 14) is preferred. It should be emphasized that the Bragg equation (Eq. (1. 14)) is competent for any lattice constituency, not merely the ultimate stable one. The openization is amply executed by proper inserting the interplanar spacing dhkl of the crystal lattice inferiorneathneathneath examination. Table 1. bestows the harmony of dhkl and the segregate cell parameters for unanalogous crystal classes. Produce 1. 6 Visualization of the Bragg equation. Zenith dissolute concentration is merely reputationd when the countenance conveys add to a multiple of the trans-parent prosperprolixity ?. Having landd at this sight it can be set-forthd that we bear attested the lies in opportunity whither circumstantial suspension for the scattemelody of x-rays at a crystal lattice may be reputationd. It has been shown that measurable intensities merely betide for unfailing orientations of the vector of momentum convey Q delay reputation to the crystal coteach preparation {c i }. Various assumptions were made that were rather unchastened when the round of the concentration of Bragg ref balbutiations is of concern. It has been conducive, for contingency, that the mote’s electrons are onesided to the core of priority of the mote. In conjunction, enthusiastic vibrations, aridity by the precedent, etc. , were aggravatelooked. Aggravate actualistic rules earn reardispose these assumptions in the subjoined. However, antecedently doing so it should be curbed how our principal springs cornot-absolute delay the heap of a attenuated metal film and how diffraction rules may be estimated. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 11 1. 2 The ? 2? Inspect Table 1. 1 Interplanar spacings dhkl for unanalogous crystal preparations and their insistency on Miller indices hkl. Parameters a, b and c bestow the prolixitys of the crystallographic segregate cell, age ? , ? and ? identicalize the propensitys unformed them. 11 Crystal preparation Stable Tetragonal Constraints a=b=c ? = ? = ? = 90° a=b ? = ? = ? = 90° 1 = 2 dhkl h2 + k 2 + l 2 a2 h2 + k 2 l 2 + 2 a2 c h2 k 2 l 2 + + a2 b 2 c 2 4 h 2 + hk + k 2 l 2 + 2 3 a2 c (h 2 + k 2 + l 2 )sin2 ? + 2(hk + hl + kl )(cos2 ? ? cos? ) a 2 (1 ? 3 cos2 ? + 2 cos3 ? ) 2hl cos ? h2 k2 l2 + + ? a2 sin 2 ? b 2 c 2 sin 2 ? ac sin 2 ? Collision 4 Orthorhombic Hexagonal ? = ? = ? = 90° a=b ? = ? = 90° ? = 120° a=b=c ? =? =? Trigonal/ Rhombohedral Monoclinic Triclinic ? = ? = 90° None 1. 2 The ? /2? Inspect An publicly-used resources for measumelody the Bragg ref balbutiation of a attenuated film is the ? /2? diffractometer. Let us usher-in its agency motive by regarding the ends earned delay the inquiry in remembrance as to how x-ray scattemelody exemplifications are preferably facilitated. What we are spirited in is the heap of Bragg ref balbutiations, i. e. their lie, precedent, concentration, etc. , in ardispose to resolve microstructural notice from them. The concentration alteration that is associated delay the ref balbutiation is comprised in the suspension employment enjoy the one loving in Eq. (1. 10), age the dissolute concentration insists on the nonproduction from the rule to the conflict preparation R. We consequently should conproduce the resources such that we can inspect the opportunity environing the rule by custody the rule–undeceiver nonproduction R invariable. This estimate ensures that any concentration alteration reputationd is due to the suspension employment and is not creatord by a insistency on R. The undeceiver should pleasantly provoke on a sphither of incapricious radius R delay the rule in the core of it. In conjunction, the sphither diminishs to a hemisphither aggravate the rule, gone we are merely spirited in the manner flake and postulates assemblage earn be executed in ref balbutiation decree. The geodesy is shown in Fig. 1. 7. Pleasantly the scattemelody of x-rays insists sensitively on the orientation of the crystal delay reputation to the scattemelody vector, we careamply bear to eliminate the diversified coteach preparations delay which we are commerce. A rule harmony establish {s i } is usher-ind for this aim that is oriented delay s1 and s2 in the roll of the attenuated film, age s3 is equiponderant to the manner natural. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 12 2 1 Principles of X-ray Diffraction Produce 1. 7 Rule harmony establish {si } and hemisphither aggravate it. The launched motive of a ? /2? inspect is visualized in Fig. 1. 8 in the hemisphither of the rule harmony establish. The rule is lieed in the core of the resources and the sharp x-ray shine is directed to the rule manner at an propensity ?. At the similar propensity the undeceiver monitors the dissolute radiation. The rule coteach vectors s1 and s3 lie in the scattemelody roll eliminated by K0 and K. Dumelody the inspect the propensity of the incoming and debouchureing shine are uniformly multitudinous, but they survive correspondent throughout the sound inspect: ? n = ? out. Note that the propensity congress is unanalogous from the one used in optics: in x-ray diffraction the propensitys of incoming and debouchureing shine are invariablely specific delay reputation to the manner roll, age they are cognate to the manner natural in optics. The ? /2? inspect can to-boot be inferiorneathneathstood as a alteration of the debouchure propensity when this is segregateicularized delay reputation to the bland incoming shine and this propensity is 2? for all sights in such a inspect. This is the conclude for naming the heap regularity a ? /2? inspect. The muchness estimated throughout the inspect is the concentration dissolute into the undeceiver. The ends are illustratively presented as a employment of I(2? ) cast. Produce 1. 8 Schematic regrant of a ? /2? inspect from the viewsight of the rule harmony establish {si}. These ? /2? inspects are profusely used for the examination of polyshining rules. The heap of polycrystals is scornly easier than that of uncombined crystals due to the actuality that, unformed other concludes, the dissolute concentration for incapricious scattemelody propensity is reserved on a foe rather than focused to a few sights in opportunity. Interestingly, in a ? /2? inspect the scattemelody vector Q is invariablely congruous to 01_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 13 1. 2 The ? /2? Inspect 13 the substrate natural s3. This actuality is conspicuous from Fig. 1. 8 and the graphical specification of Q in Fig. 1. 3. Due to this geometrical bustle merely those lattice rolls hkl that are oriented congruous to the manner roll can yield to a Bragg ref balbutiation. The broad apprehension of unfailing subsets of crystallites in a ? /2? inspect is visualized in Fig. 1. 9. If diversified ref balbutiations hkl are estimated they all root from perspicuous subsets of crystallites – save they are of harmonic arrange, i. e. h? k? ? = n(hkl). Produce 1. 9 Sebalbutiation motive for scientific heap of sur- face-congruous lattice rolls in a ? /2? inspect. In ardispose to unfold the motives exposed so far, the euphuism of a ? /2? inspect of a 500 nm attenuated Al film is shown in Fig. 1. 10. The euphuism was fitted for the stampistic radiation of a copper x-ray tube having ? (Cu K? ) = 0. 154 nm (see Instrumental Box 1 for prefer notice). Diversified concerning signs are chattelsuated from this bring-about, which displays eight Bragg ref balbutiations in the scattemelody propensity stroll from 25° to 125°. The ref balbutiations may be assigned to their Miller indices when use is made of the Bragg equation and the segregate cell parameter of the Al lattice, a = 0. 4049 nm. For this aim the d estimates of the 2? B ref lex lies bear been fitted according to the Bragg equation d = ? /(2sin? B) and curbed for the discerption of (a/d)2 = h2 + k2 + l2. It is seen that diversified ref balbutiations enjoy 111 and 200 are reputationd, but other peaks enjoy 100, 110, etc. , are missing. This wonder has to be inferiorneathneathstood in the middleing of subversive suspension, which is creatord by the constituency of the Al lattice, which is perspicuous from the ultimate stable lattice. It has to be famous that a splitting of peaks into an ? 1 peak and an ? 2 peak cannot be reputationd, although the sign was comprised in the euphuism. The nonproduction is explained from the grossness of the Bragg peaks causing a extreme aggravatelap unformed twain peaks such that they survive unresolved. Broad ref balbutiations are creatord by paltry molecule greatnesss and crystal lattice faults that are publicly reputationd in attenuated polyshining films and are discussed in aggravate ingredient in Stipulation 3. Moreover, the diffraction rule unfolds a pronounced curtail of dissolute concentration delay increasing scatter– ing propensity. Therefore, the diffraction rule is to-boot shown in the inset delay a vI teach in ardispose to emphagreatness the paltryer peaks. The balance-rise concentration bring-environing is an publicly-used grant decree. It is concluded that the basic signs of Exception 1. 1 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 14 14 1 Principles of X-ray Diffraction are in correspondence delay the false heap of a attenuated Al film, but some aspects survive to be separated. Produce 1. 10 Euphuism of a ? /2? inspect of a 500 nm attenuated Al film estimated delay Cu K? radiation. The inset shows the similar pat– tern delay a v I teach. 1. 3 Concentration of Bragg Ref balbutiations The needful cultivation of the seem for the concentration of a Bragg ref balbutiation is now exposed. For this aim the judgment earn be used that was made by deriving the Bragg equation and the Laue predicaments. It has been chattelsuated that the exuberance of the completion dissolute scope from a impute preparation in the Fraunhofer admission is stampized by a countenance actualityor exp(–iQrn1n2n3) comprising the scattemelody vector Q and the nonproduction rn1n2n3 unformed all pairs of sight imputes. This end may be openized by subjecting the sum in Eq. (1. 9) to a natural title. Instead of writing a discrete nonproduction vector rn1n2n3 the natural capricious r is used and it is argued that the dissolute scope insists as ? ? e(r )exp(? iQr )dr (1. 16) on the electronic impute preparation ? e(r) of the scattemelody sight. The integration has to be executed aggravate the tome dr to which the scattemelody electrons are onesided. Pleasantly ? e has the dimensionality of an inverse tome the integration permits a bookhort muchness, which is in correspondence delay our starting sight. This new seem can now be applied to the scattemelody sights in which we are spirited, i. e. motes and crystallographic segregate cells, to curb whether the provisoal concentration employment is emendd. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 15 1. 3 Concentration of Bragg Ref balbutiations 15 Instrumental Box 1: ?/2? Diffractometer The basic heap geodesy of by far the most invariablely used x-ray diffraction resources is depicted in Fig. i1. 1. The rule should preferably unfold a roll or mawkishtened manner. The propensity of twain the incoming and the debouchureing shine is ? delay reputation to the precedent manner. A wide compute of constitutional and inconstitutional interlard rules bear been estimated delay these resourcess from which the naming of interlard diffractometer is inferiorneathneathstood. Its heap geodesy may to-boot be applied to the examination of attenuated films, specially if the flake is polyshining and has been deposited on a mawkish substrate, as is publicly the deed. Produce i1. 1 Schematic regrant of ? /2? diffraction in Bragg–Brentano geodesy. The diffraction rule is calm by varying the impact propensity of the incoming xray shine by ? and the scattemelody propensity by 2? age measumelody the dissolute concentration I(2? as a employment of the departure. Two propensitys bear thus to be multitudinous dumelody a ? /2? inspect and diversified casts of interlard diffractometers are in use. For one set of resourcess the x-ray creator survives urban age the rule is turnd environing ? and the undeceiver provokes by 2?. For other preparations the rule is urban age twain the x-ray creator and the undeceiver turn by ? concertedly, but clockwise and anticlockwise, reputationively. The courses are executed by a so-determined goniometer, which is the mediate segregate of a diffractometer. A goniometer of a interlard diffractometer comprises at lowest two foes or – together – two axes of course. Typically the rule is mounted on the courseal axis, age the undeceiver and/or x-ray creator provoke congruous the periphery, but twain axes of course assent. In most laboratory ? /2? diffractometers the goniometer radius, which is the rule-to-undeceiver nonproduction, is in the stroll 150–450 mm. Highly terse goniometers delay 0. 001° 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 16 16 1 Principles of X-ray Diffraction exactness and plain inferior on twain the ? and the 2? foes are commercially beneficial. The calm diffraction rule I(2? ) rests of two sets of postulates: a vector of 2? lies and a relieve vector delay the divert intensities Ii. The stride greatness ? 2? i unformed two nigh 2? i should be separated in correspondence delay the prepared aim of the postulates. For chemical countenance disexception (Chapter 2) the unmeasured width of half the zenith of the tallest Bragg peak in the rule should be mellow by at lowest 5 to 7 heap sights. However, for a microstructural disexception (Chapter 3) in abundance of 10 sights should be estimated on the similar flake. The divert estimate of ? 2? i earn to-boot insist on the slit produce of the diffractometer. The preset integration season of the undeceiver per stride in 2? should permit the courteous concentration of the paltryest peak of concern to achieve the sound fluctuations ? (I) by a actualityor of 3 or 5, etc. , according to the required smooth of statistical consciousness. The curb of the x-rays shine parcel lets from the bustle that lenses and other refractive segregateicles are not as amply beneficial as those used for conspicuous unthoughtful. For this conclude the shine predicamenting in ? /2? diffractometers is chiefly executed by slits and apertures and may be seemed shadow-casting optics. In conjunction, interlard diffractometers bear to dispense delay the opposed shine stampistic that is dislodgeted by an x-ray tube. Most preparations act in the so-determined Bragg–Brentano or parafocusing decree. In this produce a focusing foe is eliminated as lieed tangentially to the rule manner (see Fig. i1. 1). The focusing predicament in the Bragg–Brentano geodesy is complyed when the x-ray creator and undeceiver are lieed on the goniometer foe whither it blends the focusing foe. Gentleman focusing would verily betide merely for a rule that is inclination to the radius of the focusing foe RFC. Gone RFC be-unlikes for diversified scattemelody propensitys 2? , gentleman focusing cannot be earned in a ? /2? can and the preparation is thus seemed parafocusing geodesy. In a ? /2? inspect the scattemelody vector Q is invariablely congruous to the substrate natural. It is, so-far, conspicuous from the aggravate consequences and from Fig. i1. 1 that this is strictly competent merely for the mediate shine, age sunthoughtful disconnections from the congruous orientation betide for the opposed segregates of the shine. If the most opposed rays depart by ±? from the mediate shine their scattemelody vector is tilted by ? from the rule natural – at lowest for those scattemelody actualitys that are accepted by the undeceiver. In abundant produces of diffractometer optics it suffices to observe merely the mediate shine. The disexception and version of x-ray diffraction heaps necessitates distinguishing three unanalogous harmony establishs that are assigned to the laboratory, the rule and the crystallites and estimationized by {li}, {si} and {ci}, reputationively. The segregate vectors in each preparation are defamous by li, si or ci, delay i ranging from 1 to 3 for the three orthogonal arranges. Transformations unformed these coteach preparations are invariablely used, for which segregateary transmutation matrices aij are eliminated delay superscripts LS, SC, CL, etc. indicating the moderate and the developed harmony establish. The harmonys are visualized in Fig. i1. 2. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 17 1. 3 Concentration of Bragg Ref balbutiations 17 Produce i1. 2 The three harmony establishs used in x-ray diffraction and the divert transmutation matrices unformed them. 1. 3. 1 Minute Produce Factors Formula (1. 16) can be applied to motes by inserting the balance of electronic prosperfunctions for the impute blindness ? e(r). Antecedently the ends of this regularity are presented let us principal brave what capability be expected from basic esthetic consequences. For this aim the electrons may temporarily be imagined in the minute rule of Bohr to provoke in round orbits environing the nuclei. If the scattemelody from any two harsh electrons from this mote could be earned it is conspicuous that the scattemelody may betide for abundant unanalogous nonproduction vectors r entity associated delay a wide alteration of countenance conveys –iQr. The orbital smeamelody of the electron blindness earn thus carry to a suspension of coherency and a wane in the coherently dissolute concentration. This wane earn be stronger the wider Q befits, pleasantly it is the scalar end Qr that segregateicularizes the countenance convey. The ansatz is made that the scattemelody of an mote insists on the precedent of the electron blindness employment or on its produce, and we thus eliminate an minute produce actualityor f by f = ?at ? e(r )exp(? iQr )dr (1. 17) In the title of Q = 0 the integration proper runs aggravate the impute preparation and permits the compute of electrons of the mote Z. For Q ? 0 the produce actualityors are rea- 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 18 18 1 Principles of X-ray Diffraction sonably presented as a employment of |Q| or sin? /?. Minute produce actualityors bear been fitted delay diversified quantum unimpassioned methods of increasing sophistication. A comaspect of estimates for all chemical segregateicles and some carrying ions is loving in Ref. [1]. Moreover, very publicly an admission of f in the produce of the rule employment f = ? a j exp ( ? b j sin2 ? / ? 2 ) + c j j =1 4 (1. 18) is used. By this adit a exactness of 10– 6 is achieved for the produce actualityors and merely nine coefficients bear to be loving for any mote or ion to rule the sound sin? /? stroll. The coefficients aj, bj and cj are to-boot tabulated in Ref. [1]. It is concluded that sight imputes Ze bear to be substituted by fe in all the foregoing seems in ardispose to dispense truly delay the workion of minute impute preparations. For some vibratory motes the minute produce actualityors as fitted by Eq. (1. 18) are displayed in Fig. 1. 11. For low scattemelody propensitys they can be seen to strain estimates plug to the minute compute Z, but a abrupt curtail delay increasing sin? /? is unlikely seen for all of them. It should be famous that the concentration flakes delay the balance of the minute produce actualityor and that an plain stronger curtail earn betide for f 2. For the decreel of Nb the produce actualityor for the fivefold ion Nb5+ is to-boot loving. It can be seen that a perspicuousion unformed motes and their ions is merely forcible for f estimates at low sin? ? , which is a open disaspect for all motes and ions, not proper for Nb. Produce 1. 11 Minute produce actualityors of Be, Al, Cu, [email protected], Nb, Nb5+ and Ag. For some examinations the tough scattemelody of x-rays cannot be aggravatelooked and the concept of the minute produce actualityors earn then bear to be bland by including actual and spurious unusual scattemelody actualityors, f ? and f ? , that bear to be borrowed to the minute produce actualityors f loving aggravate. In most deeds, unusual scattemelody actualityors f ? and f ? are paltry when correlatived delay f. Numerical estimates for f ? nd f ? are loving in the Ref. [1]. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 19 1. 3 Concentration of Bragg Ref balbutiations 19 1. 3. 2 Constituency Rudiment The crystallographic segregate cell is the paltryest segregate by which the occasional ardispose in the crystal is public. In the ultimate stable lattice that has been observeed to resolve the Bragg and Laue equations thither is merely one mote per segregate cell. The dissolute concentration was plant to flake delay the balance of the impute of this mote – or the produce actualityor as should be said now – and the suspension employment, see Eq. (1. 10). For aggravate intricate constituencys the integration has to be bland aggravate the completion impute preparation of the segregate cell (uc) rather than aggravate a uncombined mote. This muchness is defamous as the constituency actualityor F that is loving by F= ?uc ? e(r )exp(? iQr )dr (1. 19) We earn estimationize it restently by a intrepid missive, gone it is a intricate muchness. The seem for the constituency actualityor may be simplified by recalling that the segregate cell comprises N motes, computeed by n from 1 to N. It is thus feasible to segregate the constituency actualityor into uncombined shares due to the identical motes (at) F= n =1 ? ? at ? e(r )exp ( ? iQ(r ? rn )) dr N (1. 20) and the integration proper has to be executed aggravate the impute preparations of identical motes. These estimates are public: they are loving by the minute produce actualityors fi of the nth mote. Accordingly, the constituency actualityor can be written F= n =1 ? f n exp (iQrn ) N (1. 21) The end of the scattemelody actualityor delay the lies rn of the N diversified motes in the segregate cell thus has to be evaluated. The departure are specific by their fractional coordinates (xn, yn, zn) that peruse for the stable cell rn = xnac1 + ynac2 + znac3. We inferiorstand that ref balbutiation concentration may merely be reputationd when the Laue predicaments are concertedly complyed which may be applied to elucidate the countenance actualityor by Q(x n ac 1 + yn ac 2 + zn ac 3 ) = hx + ky + lz (1. 22) Merely if this equation is complyed does measurable concentration from interfemelody x-rays invade into the undeceiver and the scattemelody of the crystal flakes delay F (hkl ) = n =1 ? f n exp ? 2? i(hxn + kyn + lzn )? ? ? N (1. 23) The constituency actualityor thus insists on the Miller indices of the ref balbutiation inferiorneathneathneath consequence, the lies of the motes in the segregate cell and the minute scattemelody actualityor. In monominute lattices the produce actualityor is the similar for all motes and can be placed in face of the sum. For the ultimate stable constituency N = 1 and x = y = z = 0 and thus F = f for all hkl and ref balbutiations are reputationd for each arrange; so-far, for 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 20 20 1 Principles of X-ray Diffraction aggravate intricate constituencys the unmeasured constituency actualityor has to be braved. Although the derivative has merely been loving for the stable lattice it has to be emphasized that the seem for the constituency actualityor, Eq. (1. 23), is competent for crystals of harsh proportion. The superiority of one-elemental metals are plant in either the face-centered stable (fcc), the body-centered stable (bcc) or the hexagonal plug-packed (hcp) constituency. The referring-to preparation of motes in theses lattices is presented in Constituency Box 1. Aluminum, for contingency, crystallizes in the fcc constituency. In this deed the Bragg equation capability be complyed for unfailing lattice rolls hkl, but for some combinations of hkl the countenance convey in the x-rays dissolute by adjacent motes may completion to ? or odd multiples of it. The dissolute shines then quarrel subversively and the ref balbutiations for these lattice rolls are not extincted. In the fcc constituency, for contingency, subversive suspension betides for hkl = 100, 110, etc. The destruction predicaments can be conservative for any crystal lattice by performing the similar summation regularity that has been executed for the ultimate stable lattice in the principal saveion and it is an directive collision to do so (Exercise 7). One earn then land at predicaments approximate to Eq. (1. 11) which prophesy inferiorneathneathneath which orientation of Q towards {si} ref balbutiations capability be reputationd. A ultimater adit instead is the apportionment of the constituency actualityor. Inserting the fractional coordinates of all filthy motes of the fcc constituency in Eq. 1. 23) permits the end ? 1 + exp(i? (h + k)) + ? ?4 f all hkl plain/odd Ffcc (hkl ) = f ? ?=? hkl modified ? exp(i? (h + l )) + exp(i? (k + l ))? ? 0 (1. 24) The seem is seen to dissolve for unfailing hkl and the inferior equation is thus defamous as an destruction predicament. It resources that Bragg ref balbutiations are merely reputationd for the fcc lattice if all Miller indices are either plain or odd. For modified triples subversive conclusion betides and these ref balbutiations are preparationatically absent. It is conspicuous from the ? /2? inspect in Fig. 1. 10 that this rule in actuality is in correspondence delay the destruction predicaments of the fcc constituency. Mathematically forcible, the Bragg equation is a needful but not a competent predicament for x-ray ref balbutiations to inaugurate. The constituency actualityor of the bcc lattice can be earned in the similar way and ends in ? 2 f Fbcc (hkl ) = f ? 1 + exp(i? (h + k + l ))? = ? ? ? ?0 h + k + l = 2n contrariantly (1. 25) The destruction predicament now resolves from a sum aggravate Miller indices and peruses that the sum must permit an plain compute for the ref balbutiation to betide. Merely if this predicament is complyed is the suspension nonsubversive and can be detected at the lie prophesyed by Bragg’s equation (Eq. 1. 15)). The constituency actualityor for the hcp constituency can be conservative as an collision from the mote coordinates in the segregate cell. The constituency actualityor Fh from a ref balbutiation h is of mediate moment in x-ray diffraction, pleasantly it recounts the lie of the motes in the segregate cell to the concentration of a ref balbutiation. Here, the Miller renunciation triple hkl has been abbreviated by the subscript h which earn be used very publicly in the subjoined. The concentration flakes delay the end of F and its intricate conjugate F*. As can be seen from the decreels of the 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 21 1. 3 Concentration of Bragg Ref balbutiations 21 Constituency Box 1: Elementary Metals The ultimate stable constituency that is used in this stipulation to resolve the basic produceulas of xray diffraction merely ratrust betides in character. It is instead reputationd that one-elemental crystal lattices publicly interest the face-centered stable (fcc) or the body-centered stable (bcc) constituency. Produce s1. 1 displays twain of them. The interminute nonproductions are amply specific by the segregate cell flatten a. Each mote is concealed by eight neighbor motes in the bcc constituency or twelve in the fcc constituency. Besides he stable constituencys, diversified metals are plant to crystallize in the hexagonal plug-packed (hcp) constituency, to-boot shown in Fig. s1. 1. This constituency has two disposes of loosedom, spectryly the interminute nonproduction a in the vile roll and the nonproduction unformed two of the rolls, c/2. Accordingly, two perspicuous interminute nonproductions r1 and r2 betide unformed contiguous neighbors in the hcp constituency (Exercise 1. 12). An slip of the occasional preparation reveals that the superiority of segregateicles wear one of these three basic constituencys inferiorneathneathneath thermodynamic rule predicaments (298 K, 101. 6 kPa). Produce s1. Crystallographic segregate cells of the most invariablely betidemelody constituencys of one-elemental metals: (a) face-centered stable, (b) body-centered stable and (c) hexagonal plug-packed constituency. In twain the fcc and hcp constituencys the motes are moulded according to the rule of plug-packed circuits. It is thereby conducive that the motes can be ruleed by exact circuits that all unfold the similar radius rat. The estimate of rat is separated such that minute neighbors are in contiguity via their manners. According to this design the minute radius can be fitted and is plant to be rat,fcc = a 2 / 4 in the fcc constituency. In the hcp constituency the predicament can merely be complyed when r1 = r2 stops, which carrys to rat,hcp = a or equiponderantly c/a = 8 / 3 = 1. 633. The c/a parameter in open forwards to eliminate the idispense hcp constituency, which is a plug-packed one. In one-elemental metals delay the hexagonal Mg constituency the c/a harmony is invariablely plant to be very plug to this idispense estimate (see Table s1. 1). Although these constituencys depart scornly from the idispense hcp constituency, they are publicly observeed hcp constituencyd anyway and the estimate of c/a is specific conjunctionally. If the tome of minute circuits in twain the fcc and the hcp constituency is fitted and naturalized delay reputation to the segregate cell tome Vuc, a estimate of 0. 74 ends for the tome 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 22 22 1 Principles of X-ray Diffraction Table s1. 1 The three most despicable crystal constituencys of one- segregateicleal metals. Segregate cell flattens inferiorneathneathneath rule predicaments are loving. Constituency Strukturbericht title Copper constituency (A1) Tungsten constituency (A2) Magnesium constituency (A3) Opportunity knot – Fm3m (225) – Im3m (229) P63/mmc (194) Minute lies 000 0--0- --0 000 000 --2 3 1? 3 Examples delay lattice parameters a and c (nm) Al: 0. 4049; Ni: 0. 3524 Cu: 0. 3615; Ag: 0. 4086 Cr: 0. 2884; Fe: 0. 2866 Nb: 0. 3307; W: 0. 3165 Mg: 0. 3209, 0. 5210 Ti: 0. 2950, 0. 4879 Zn: 0. 2665, 0. 4947 Zr: 0. 3231, 0. 5147 Face-centered stable (fcc) Body-centered stable (bcc) Hexagonal plugpacked (hcp) harmony. This is the widest estimate of spatial filling that capability be achieved by the packing of circuits all having the similar crossing. It should be famous that crystallographic lattice rolls in the hcp and to-boot in other hexagonal constituencys are sinister by filthy Miller indices (hkil), whither invariablely i = –(h + k) stops. This renunciationing ends from the action of three segregate vectors in the basal roll of hexagonal segregate cells. In a widely used succinctness a age is solely inserted for the third renunciation: (hk. l). One instantly chattelsuates from the betiderence of twain casts of Miller renunciation estimation that a hexagonal constituency is entity observeed. Thither continues an concerning harmony unformed the plug-packed fcc and hcp constituencys. The harmony befits conspicuous when all motes in the fcc lattice are detight into minute (111) rolls and correlatived delay the (00. 1) rolls in the hcp constituency (see Fig. s1. 2). The coordination delayin the roll is the similar, i. e. ach sphither is concealed by six neighbors to permit the carrying packaging blindness of circuits delayin the roll. Looking from aggravate on the roll stacking reveals that thither continue three perspicuous lies whither motes capability befit situated, which are designated A, B and C. In each roll motes are lieed at A, B or C. It revolves out that the stacking of rolls may be resemblanceed for by the consequences …ABCABC… in the fcc constituency, but by …ABABAB… in the hcp constituency. Therefore, twain constituencys proper be-unenjoy by the upright stacking conconsequence of amply obstructed mote rolls. Produce s1. 2 Stacking of plug-packed rolls in (a) fcc and (b) hcp constituencys. 01_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 23 1. 3 Concentration of Bragg Ref balbutiations 23 fcc and bcc lattice the lump of F maximally correspondents the compute of motes in the segregate cell diversified by their minute produce actualityor. This aspect is ratrust reputationd for aggravate intricate constituencys, pleasantly the scattemelody of the unanalogous knots of motes publicly creators a segregateial subversive suspension. This actuality is unfoldd from the constituency actualityors of technologically applicable compounds that are plant in the diversified constituency boxes of aftercited stipulations. The constituency actualityor has the historical produce of a discrete Fourier transmute. The inversion transmutation from the concentration of reputationd ref balbutiations would thus permit the purpose of the minute lies in the segregate cell. However, the concentration flakes delay the end of the constituency actualityor and its intricate conjugate, FhFh*, which is associated delay a extreme forfeiture of notice. If the constituency actualityor is bring-aboutted in the Euler roll of intricate computes it may be stampized by its lump |Fh| and its countenance ? h. In this draw the notice forfeiture can be envisaged as a forfeiture of countenance notice, which is the courteous-public countenance height in the constituency purpose by x-ray diffraction. Regarding the conconsequence of enthusiastic vibrations the similar arguments use as loving aggravate to properify the wane in coherency by the spatial workion of electronic impute preparation. It is courteous public that the motes in a powerful ruffle at their makeweight lies rn . Weather vibrations leave a wane of countenance continuity in the dissolute shine and thus diminish the estimated concentration. The wonder – – can qualitatively be resemblanceed for by the middle quadratic disconnection u2 of the mote from its middle lie rn . The minute produce actualityors f bear then to be remoulded by the weather-dependent seem f T = f exp ? ? u 2 sin2 ? / ? 2 ( ) (1. 26) Again, it can be seen that the scattemelody exuberance is representativeially damped delay increasing scattemelody propensity and that the damping coefficient flakes delay the balance –– of momentum convey 4? sin? /?. The 8? u2 actualityor is publicly abbreviated by the sym— – – bol B in the erudition. Typically the middle displacements of motes v u2 at opportunity weather are in the stroll unformed 0. 005 and 0. 03 nm, which translates into a few percent to aggravate than 10% of the manacle prolixity. In the fcc constituencyd Cu lattice, — –– for contingency, v u2 completions to environing 6% of dCu-Cu. The conconsequence of the weather vibrations can be seen from Fig. 1. 11, whither in conjunction to the nothing-weather f to-boot the minute produce actualityor of Cu in the Cu lattice is shown. It is conspicuous that the scattemelody ability may be forciblely diminishd by enthusiastic vibration, which stops in segregateicular for lofty scattemelody propensitys. These ends are applied to the constituency actualityor solely by replacing the produce actualityor delay its weather-dependent estimate Fhkl (T ) = n =1 ? f n exp ( ? Bn (T )sin2 ? / ? 2 ) exp (2? i(hxn + kyn + lzn )) N (1. 27) Becreator the enthusiastic vibration exuberances growth delay increasing weather the damping of Bragg ref balbutiations earn to-boot growth. This creators the Bragg ref balbutiation to deccontinuity into a contrariety of prolix dissolute concentration when the weather 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 24 24 1 Principles of X-ray Diffraction is growthd. This is in dissimilarity to abundant spectroscopic techniques, whither the reputationd peaks enlarge at tall weather. 1. 3. 3 Swarming The swarming specifies the compute of equiponderant lattice rolls that may all creator ref balbutiations at the similar ? B lie. The wonder is visualized in Fig. 1. 12, for the laboratory in harmony establish {li}. In this coteach preparation the lie of the incoming shine is set incapricious delay its ardispose sighting congruous the {li} preparation segregate vector l1. Age K0 is urban, K provokes on a foe dumelody a ? /2? inspect as does the substrate natural s3. The produce displays the lie dumelody the inspect when ? is at the Bragg propensity of Al (111). In the deed whither the rule comprises a uncombined Al crystal of (111) orientation three prefer ref balbutiations would together be wandemelody at the interexception of the 2? 111 cone delay the {li} circuit. The ref balbutiations would be creatord by -the together likely scattemelody of the incoming x-ray shine at lattice rolls (111), --(111) and (111) that all unfold the similar interplanar spacing d111 to comply the Bragg equation. In the deed of a polyshining rule entity estimated, so-far, the concentration would seem completimerely unanalogous. Pleasantly of the stray orientation of crystallites the concentration of all equiponderant (111) rolls would be together reserved on a cone of aperture propensity 4? rather than entity collected in a few sole spots. The concentration would be smeared out aggravate a melody shown as a grey continuity in Fig. . 12. Thither are m111 = 8 equiponderant (111) rolls, but merely m200 = 6 for (200) and it is conspicuous that the swarming mh earn invade the seem of a Bragg ref balbutiation concentration as a scaling actualityor. Produce 1. 12 Scattemelody in the laboratory harmony establish {li } for a 111 refbalbutiation from an Al uncombined crystal of [111] orientation and a polyshining Al interlard rule. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 25 1. 3 Concentration of Bragg Ref balbutiations 25 1. 3. 4 Geodesy Rudiment The spreading of the Bragg peak aggravate a round part of the {li} sphither as discussed aggravate usher-ins a prefer ? ependency into the diffraction rule of a ? /2? inspect. The conconsequence is visualized in Fig. 1. 13 whither the set of all diffracted concentration for scattemelody propensity 2? is estimationized by a cone of aperture propensity 4?. The circumferences of the concentration melodys flake delay sin2? causing a worthlessness of concentration by 1/sin2?. Thither to-boot inaugurate a multiformity of scattemelody vectors Q that lie on a cone. The dissolute concentration earn flake delay their blindness, which is sin(? /2 ? ?) = cos?. The geodesy actualityor is the end of twain blindness employments and it is developedly earned as G = cos? /sin2? = 1/(2sin? . Produce 1. 13 Scattemelody in the laboratory harmony establish {li } to resolve the geodesy actualityor G. 1. 3. 5 Preferred Orientation (Texture) For a interlard rule it may openly be conducive that all molecule orientations betide delay the similar lovelihood, i. e. that the preparation employment of molecule orientations is isotropic. It is a stampistic structural sign of attenuated polyshining films that unfailing crystallographic lattice rolls can betide delay a elder lovelihood than others. This wonder is seemed preferred orientation or compound. It is conspicuous from Fig. 1. 2 that a compound capability bear a forcible inf luence on the diffraction rule, whither blindness-enhanced lattice rolls earn be associated delay an growth of the tallying Bragg ref balbutiation concentration Ih. The concentration then has to be flaked delay the blindness of crystallite orientations that are designated by the compound actualityors Th. For a stray orientation Th = 1 stops for all of them. The heap of compound and the purpose of orientation preparation employments are pland in ingredient in Stipulation 5. 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 26 26 1 Principles of X-ray Diffraction 1. 3. 6 Polarization Factor The x-ray radiation dislodgeted from a laboratory x-ray tube is of stray polarization. Therefore, the scattemelody by a polyshining rule has to be detight into a ? segregateicle and a ? segregateicle. These consequences bear already been pland for the scattemelody by a uncombined electron and they together use to the deed observeed hither. In the deed that I? = I? = I0/2 is competent on the middle, the polarization actualityor interests the produce C2 = 1 + cos2 2? 2 (1. 28) and it is by this actualityor that the concentration accepted by the undeceiver has to be flaked. The geodesy actualityor may be unanalogous for heap produces other than the ? 2? inspect. The exemplificationalist should curb this sight careamply if courteous intensities bear to be analyzed qualitatively. The insistency of the geodesy fac— tor G and the polarization actualityor C 2 are twain shown in Fig. 1. 14 as a employment of scattemelody propensity 2?. To-boot the Lorentz actualityor L is shown that earn be conservative succeeding and the — end GC 2L of all three actualityors. The end employment is seen to unfold a pro— nounced minimum plug to 2? = 120°. Up to this sight GC 2L uniformly curtails, but recovers for lofty scattemelody propensitys plug to2? max = 180°. Produce 1. 4 Geodesy actualityor G, polarization actualityor C2 and — Lorentz actualityor L as a employment of 2?. To-boot the end of the three actualityors is shown. Note the logarithmic teach flake. 1. 3. 7 Aridity Rudiment Dumelody their transit through substance x-rays let from an leanness of concentration creatord by their aridity. The Lambert–Beer law, courteous public from optics, can draw the aridity consequence. The concentration I0 that invades into the rule earn be representativeially damped to an completion I0exp(–2µ ) succeeding a footfootfoottrack of 2 . The parameter µ 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 27 1. 3 Concentration of Bragg Ref balbutiations 7 Produce 1. 15 Schematic regrant of the aridity conconsequence for a attenuated-film rule in a ? /2? inspect. is designated the continuityar leanness coefficient and insists on the prosperprolixity of the radiation used, the chemical comlie of the rule and its blindness. The inverse of µ would bestow a discrimination profoundness for natural impact ? 1/e = 1/µ that specifies the footfootfoottrack prolixity for which the concentration I0 drops to 1/e of its moderate estimate. The book of the leanness coefficient are m–1 or µm–1. Often, the estimate of the priority aridity coefficient µm is listed in diversified tables that can be converted into µ = ? m by plurality delay the priority blindness ?. For abundant substances leanness coefficients of the ardispose of 105 to 107 m–1 are earned for Cu K? or approximate prosperlengths. This tallys to discrimination profoundnesss ? 1/e of 0. 1 to 10 µm and thus is in the stroll of a illustrative flake besottedness. It can be concluded that aridity incomes capability forciblely feign Bragg ref balbutiations of attenuated films. The dominant conconsequence the aridity actualityor has on a diffraction rule is the alteration of the dissolute concentration. Its spmelody is shown in Fig. 1. 15 for the deed of a ? /2? inspect. For any x-ray shine that has traveled through a rule to befit dissolute into the undeceiver the segregateicleary concentration has been diminishd by the actualityor exp(–2µ ). The wane of concentration of the completion x-ray shine is the sum aggravate all feasible footpaths of the shine delayin the titles of 0 to max max ? 0 exp(? 2µ )d (1. 29) The footfootfoottrack 2 that is traversed by the x-ray shine may be explicit by the profoundness capricious z for which = z/sin? stops. Then is substituted by z/sin? , d by dz/sin? and the integration is executed from 0 up to the besottedness t of the film. Here, z = 0 resemblances for the manner of the film and z = t for the film–substrate interface. The discerption of the courteous permits ? ? 2µ t ? ? ? 1 ? ?1 ? exp ? ? ? 2µ ? ? sin? ? ? ?? ? (1. 30) In the title of an infinitely turbid rule, t > ? which is equiponderant to t o 1/µ, the end 1/(2µ) is earned. In the subjoined the aridity actualityor is defamous by 001_Kapitel_1 23. 09. 2005 14:56 Uhr Seite 28 28 1 Principles of X-ray Diffraction the harmony of the aridity for a rule of terminable besottedness delay reputation to an infinitely turbid rule A= ?0 ? 0 t ? (1. 31) The collision of this regularity ends in the aridity actualityor for the ? /2? produce ? ? 2µ t ? ? A? 2? = ? 1 ? xp ? ? ? sin? ? ? ?? ? (1. 32) The subscript ? /2? has been borrowed in ardispose to designate the heap geodesy. We earn befit familiar delay diversified A actualityors in the subjoined stipulations for unanalogous diffractometer produces. The A actualityor is to-boot seemed the besottedness actualityor and it is seen to creator the estimated concentration to desist as a employment of increasing scattemelody propensity 2?. In Fig. 1. 16 the A? 2? actualityor is displayed for attenuated Al and Nb films of 500 nm and 1 µm besottedness estimated delay Cu K? radiation, whither use has been made of the embodied parameters Al: µm = 486. 7 m2 kg–1, ? 2700 kg m–3 ? µ = 1. 31 ? 106 m–1 Nb: µm = 1492 m2 kg–1, ? = 8550 kg m–3 ? µ = 1.