Materials Analysis MATSCI 162/172 Laboratory Exercise No. 1 Crystal Structure Determination Pattern Indexing

Transcription

1 Mterils Anlysis MATSCI 16/17 Lbortory Exercise No. 1 Crystl Structure Determintion Pttern Inexing Objectives: To inex the x-ry iffrction pttern, ientify the Brvis lttice, n clculte the precise lttice prmeters. Complete etermintion of n unknown crystl structure consists of three steps: 1. Clcultion of the size n shpe of the unit cell from the ngulr positions of the iffrction peks.. Computtion of the number of toms per unit cell from the size n shpe of the unit cell, the chemicl composition of the specimen, n its mesure ensity. 3. Deuction of the tom positions within the unit cell from the reltive intensities of the iffrction peks. We will o only step 1. Inexing pttern involves ssigning the correct Miller inices to ech pek in the iffrction pttern. For cubic unit cell: Brgg s lw becomes: hkl o h k l 4 o λ 4 ( h k l )

2 so: λ ( h k l ) s 4 constnt for given crystl lwys equl to n integer In the cubic system, the first reflection in the iffrction pttern is ue to iffrction from plnes with Miller inices (1) for primitive cubic, (11) for boy-centere cubic, n (111) for fce-centere cubic lttices, so h k l 1,, or 3, respectively. Chrcteristic line sequences in the cubic system: Simple cubic: Boy-centere cubic: Fce-centere cubic: Dimon cubic: 1,, 3, 4, 5, 6, 8, 9, 1, 11, 1, 13, 14, 16,, 4, 6, 8, 1, 1, 14, 16, 3, 4, 8, 11, 1, 16, 19,, 4, 7, 3, 3, 8, 11, 16, 19, 4, 7, 3, If the iffrction pttern contins only six peks n if the rtio of the vlues is 1,, 3, 4, 5, n 6 for these reflections, then Brvis lttice my be either primitive cubic or boy-centere cubic, it is not possible to unmbiguously istinguish the two. But remember tht the simple cubic structure is not very common n, therefore, in such sitution, you will probbly be right if you h inexe the pttern s belonging to mteril with the BCC structure.

3 Steps in inexing cubic pttern: 1. Mesure smple & list ngles.. Clculte. 3. Clculte /s. 4. Write chrcteristic line sequences for the cubic system. 5. Ientify Brvis lttice. 6. Clculte the lttice prmeter. 7. Compre the lttice prmeter n crystl clss to tbulte vlues for metls to etermine the mteril. EXAMPLE

4 Mesurement of the lttice prmeter is inirect process. For cubic mteril n We re mesuring not! 1 Differentiting Brgg s eqution we get: cot when 9 o h k λ cot l hkl h k l The plot of lttice prmeter vs is not liner. Extrpoltion of the lttice prmeter ginst certin functions of will prouce stright line, which cn then be extrpolte to the vlue corresponing to 9 o. The function epens on the kin of equipment use to recor XRD pttern.

5 Diffrctometer The generl pproch in fining n extrpoltion function is to consier the vrious effects which cn le to errors in the mesure vlues of, n to fin out how these errors in vry with the ngle itself. Most importnt systemtic errors: 1. Mislignment of the instrument.. Use of flt specimen inste of curve one. 3. Absorption in the specimen. 4. Displcement of the specimen from the iffrctometer xis. Usully this is the lrgest source of error. It cuses n error given by: D R where D is the specimen isplcement prllel to the iffrction-plne norml. R iffrctometer rius. 5. Verticl ivergence of the bem. No gle extrpoltion function cn be completely stisfctory. For () n (3) / vries s. For (4) / vries s /.

6 Extrpoltion ginst is best if the min error is the flt smple effect. It is vli for iffrction peks with > 6 o. For cubic mterils n if the smple hs isplcement error, the Brley-Jy extrpoltion function cn be use: If Nelson-Riley extrpoltion function is pproprite:., 1 1 k k., k k Lrge systemtic errors, smll rnom errors Smll systemtic errors, lrge rnom errors

7 ) systemtic errors re eliminte by selection of the proper extrpoltion function b) Rnom errors re reuce ug the lest squres metho evise by Cohen. Cohen Metho Squring the Brgg eqution we get: After ifferentition: log λ log log 4 Lets ssume tht combine systemtic errors tke the form: then combining equtions we get: K The true vlue for is: K D λ ( h k l ) true 4 D new constnt. true lttice prmeter

8 We rewrite: observe observe λ 4 true ( h k l ) D observe Aα Cδ where: λ A 4 α ( h k l ) D C 1 δ 1 cn be written for ech reflection in the XRD pttern The proceure cn be combine with the lest squres principle to minimize the effect of rnom observtionl errors Aα Cδ ε observe Accoring to the theory of lest squres, the best vlues of the coefficients A n C re those for which the sum of the squres of the rnom observtionl errors is minimum: ( ε ) ( Aα Cδ ) observe

9 By ifferentiting we get pir of norml equtions with respect to A n C n equting them to zero: By solving these equtions we etermine A, n from this vlue of A the true lttice prmeter cn be etermine. δ αδ δ αδ α α C A C A