UNIVERSITY OF OSLO Fculty of Mthemtics nd Nturl Sciences Midterm exm in MENA3100 Dy of exm: 19 th Mrch 2018 Exm hours: 14:30 17:30 This exmintion pper consists of 4 pges including 1 ppendix pge. Permitted mterils: Electronic clcultor of ccepted type nd ruler Mke sure tht your copy of this exmintion pper is complete before nswering. Upon censoring, ll sub exercises will be weighted eqully. We reserve the right to do some djustments. d-vlues for cubic unit cells: d = /(h 2 +k 2 +l 2 ) 1/2 Wve length of Cu Kα 1 rdition: λ = 0.15406 nm. The nswers given here re in most cses beyond wht we expect from the cndidtes. Be wre tht chnges in this text my occur, this is version 1.0. Exercise 1 Optics In the following we ssume tht we hve thin, convex lenses. ) Wht is the difference between rel nd virtul imge? A rel imge is n imge it is possible to project on screen or something similr. This implies tht the light rys emitted from prticulr plce on the object re mde, with the help of the lens, to convert into point on the imge. In tht plne (t given distnce from the lens), the rys from ll other points on the object must similrly be mde to convert into the corresponding points in the imge. There is one-to-one correspondence between the object nd its imge. In virtul imge the rys from given point is diverging fter pssing through the lens. They re, however, converging into point if extrpolted bckwrds. To observe virtul imge second lens is necessry, mking the diverging rys converge into rel imge. If this is the lens of the eye, we will get the impression tht we see n (enlrged) imge plced on the sme side of the lens s the object. b) Mke sketch of n experimentl setup where you obtin rel imge of n object. If you wnt, you cn use the illustrtion in figure 1 s strting point (see ppendix). 1
See figure. The mgnifiction of your imge will depend on your choices of position nd strength of the lens. Observe tht the imge is inverted, tht the rys going through the center re stright lines nd rys tht re prllel before the lens cross t the bck focl plne. c) Wht is the distnce between the lens nd the bck focl plne in your construction? In the illustrtion, the distnce to the bck focl plne is shown s 18 mm nd represents the distnce from the lens to the point where the two rys tht ws prllel to the opticl xis before the lens cross ech other. The point of this question is to let the student show tht he or she knows bout the concept, so mesuring with ruler is OK. d) Wht kind of informtion bout the object cn we get from the bck focl plne? In the bck focl plne, rys diffrcted (scttered) in given direction will meet t point. We find the Fourier trnsform, the diffrction pttern, of the object in the bck focl plne. If our object is diffrction grting tht is illuminted with prllel bem of light, we will find regulr diffrction pttern in the bck focl plne 1. e) Mke sketch showing how the imge from b) with the use of n dditionl lens cn ct s n object for nother rel imge. You my do the construction in the figure from question b) or you my mke new one. 1 cf. Frunhofer diffrction. 2
As pointed out erlier, the figure cn look very different depending on your choice of the strength of the lenses (position of the bck focl plne) nd distnce between the objects (Object nd Imge 1) nd lenses (Lens 1 nd Lens 2). The rel imge does not hve to be mgnified s it is in this illustrtion. Of cuse, the illustrtion bove is not correct, even if it would be ccepted s correct nswer t the exm. If the light rys re to chnge direction t the rrowhed of Imge 1, we would hve to put semitrnsprent screen in tht position. Without such screen, we would not be ble to see the tip of the rrow in Imge 2. With the size of Lens 2 tht is used bove, the red, dotted line shows where the rrow would be truncted in Imge 2. 3
Exercise 2 Crystllogrphy Lttice A c Lttice B Figure 2. Illustrtions of Brvis lttice A nd B Figure 2 illustrtes two different point lttices (nmed A nd B) both hving orthogonl xil systems (90 o between the unit vectors of the cell). ) Wht does lttice point represent in Brvis lttice? A unitcell cn be described by Brvis point lttice where ech lttice point represents bsis or motif. The bsis/motif cn be single tom or collection of mny toms. (The rrngement of the toms must be so tht the symmetry of the Brvis lttice is preserved.) b) Wht type of Brvis lttices is consistent with lttice A nd B in figure 2? Lttice A is consistent with tetrgonl primitive lttice nd lttice B is consistent with cubic body centered lttice. c) Use the copy of the figures in the ppendix nd illustrte the position of set of (111) nd (102) plnes for lttice A nd B so tht the corresponding size of the plnr distnces re shown. 4
In the lst cse, the coordinte system hs been rotted to mke the visuliztion esier (note tht the right hnd rule still is obeyed). It is explicitly sked for sets of plnes, so more thn one plne should be drwn. This cn be done s shown bove, nd/or the plne through the origin cn be drwn. More thn one plne is lso necessry to mrk the distnce (d-vlue) between the plnes in the figure. Exercise 3 Interction with mtter Cu Kα 1 X-rys re commonly used when performing X-ry diffrction experiments. ) Describe nd illustrte with drwing the principles of how n X-ry tube is constructed nd how X-rys re generted. Electrons re ccelerted over voltge from the filment (cthode) to the Cu trget (node) in vcuum tube. As the electrons (high energy) interct with the trget, energy from the electrons is trnsferred to the Cu toms in the trget (round 98 % s het). X-rys generted by two different processes will be produced. These re generlly described s chrcteristic X-rys nd bremsstrhlung ( breking rdition ) (bremsestråling in Norwegin). The Bremsstrhlung occurs when electrons re slowed down inside the trget. It is described s white electromgnetic rdition mening tht it is rdition with continuous specter of energies/wve lengths. (The energy of the Bremsstrhlung X- rys is rnging from E = 0 nd up to the initil energy of the ccelerting electrons hitting the Cu-trget in the X-ry tube.) Upon energy trnsfer from the ccelerted electrons to Cu toms, toms re excited (ex. core electron will go to higher energy sttes, leving core hole behind). As the toms return to their ground stte, the core electron holes cn be filled by electrons from higher electron levels s illustrted. In this sitution, the energy difference between the two electronic sttes is relesed s electromgnetic rdition in the X- ry energy regime. Since the number of protons is different for ech element, the 5
energy of the X-ry photons tht is relesed is different for ech element. Therefore this rdition is clled chrcteristic X-rys (for the given element). (Cf. the processes we utilize in EDS-nlysis in SEM nd similr.) b) Would you describe Cu Kα X-rys s chrcteristic X-rys or bremsstrhlung X-rys (breking rdition)? Argue for your nswer. The Cu Kα X-rys re chrcteristic X-rys since they hve specific energies/wve lengths chrcteristic for the energy differences between the electron shells in the Cu tom (K nd the L subshells of Cu). In the X-ry tube both Cu Kα 1, Cu Kα 2 nd Cu Kβ X-rys re produced. c) Wht does ech of the symbols K, α 1, α 2 nd the β describe? The symbols re used in the Siegbhn nottion where the Ltin cpitl letter describe the shell with the core hole (here the K-shell) tht is filled upon the de-excittion process. α 1, α 2 nd β describes different X-ry energies due to the different electron trnsitions occurring when the electron hole is filled. In the cse of Cu Kα 1 nd α 2, the nottion describes trnsitions of electrons from the L subshells to the K shell nd β trnsitions from the M shell to the K shell. The electron trnsitions with the highest probbility (resulting in the highest X-ry intensity) re described with the first letter in the Greek lphbet, nd less likely trnsition re described with descending letters in the lphbet. Likewise describes the numbers in the Siegbhn nottion different trnsition probbility where 1 indicte higher electron trnsition probbility thn 2 (i.e. Cu Kα 1 X-rys hve higher intensity thn the Cu Kα 2 X-rys). A scttering process is ctegorized either s elstic or inelstic. d) Give one exmple of n elstic nd one exmple of n inelstic scttering process tking plce in smple. You cn choose if the smple is irrdited with electrons, X-rys or neutrons. There re mny possible nswers to this exercise. Any scttering processes where energy hs been trnsferred from the probe to the specimen during the scttering process, resulting in ny kind of excittion (nd secondry processes) re exmples of inelstic scttering processes. For elstic scttering processes, energy is not trnsferred to the specimen upon the scttering process. Exmples re Brgg scttering of electrons, X-rys or neutrons used in diffrction studies. 6
Exercise 4 Diffrction Brggs lw is commonly written s 2dsinθ = nλ ) Mke sketch tht indicte wht the symbols represents nd show how to deduce the expression. The generl expression for the structure fctor is: F g = F hkl = Σf j (θ)exp(2πi(hu j +kv j +lw j )) b) Wht does the symbols represent nd wht re the vlues of f(θ) for X-rys t θ = 0? h, k nd l re the Miller indices of the Brgg reflection g. The coordinte of tom j within the crystl unit cell is given r j = u j +v j b+w j c. The sum goes over the number of toms within the crystl unit cell, nd f j (θ) is the tomic scttering fctor of tom j. The vlue of the tomic scttering fctor depends on the type of probe (electron, neutron or X-ry). For X-rys t θ = 0 the vlue of f(θ) is equl to Z (the tomic number of tom j). Two smples re investigted with powder X-ry diffrction. One smple consist of phse with Brvis lttice s illustrted with A in figure 2, the other smple consist of phse with lttice s illustrted in B. The lttice prmeter is 4 Å (both in lttice A nd B) nd c = 7 Å (in lttice A). In your XRD experiments, imging tht you hve rndom orienttion of the grins in the smples. c) At which 2θ ngle would you see the first reflection in the powder diffrctogrm from the smple with lttice A? 7
The A lttice is primitive. Therefor there will be no systemtic extinction (utslukning in Norwegin) of the kind we find for centered lttices. (The symmetry elements screw xis nd glide mirror plnes cn still led to extinction of some reflexes. In ddition, the tomic rrngement cn be such tht the intensity of given reflex my be so wek tht we don t observe it.) Plnes with lrge plnr distnces (d-vlues) will give rice to reflexes t lower ngles compred to plnes with smll d-vlues, s seen from Brgg s lw. The lrgest d- vlue of lttice A is d 001 = 7 Å nd with XRD using Cu Kα 1 rdition (λ = 0.15406 nm), Brgg s lw predicts tht the Brgg ngle is: θ = sin -1 θ = sin -1 (1.5406 Å/2 7 Å) = 6.3. Hence, you would expect to see the first reflection t 2θ = 12.6. d) At which 2θ ngles would you see the first reflection in the powder diffrctogrm from the smple with lttice B? The lrgest lttice spcing of lttice B is 4 Å (d 100 ). However, the Brvis lttice is body centered nd the intensity of the 100 type of reflections will be zero. This cn be understood by evluting the Brvis lttice where you see tht the {002} type of plnes must hve the sme scttering power (sme electron density) s the {001} plnes. (As cn be seen from the nswers to exercise 2 c), this is the sme cse s we hve for the (222) nd (204) plnes reltive to the (111) nd (102) plnes). Hence, t the Brgg ngle of (001) (nd (111) nd (102)) we will hve completely destructive interference nd zero intensity. Alterntively, one could use the expression for the structure fctor nd come to the sme conclusion (F 100 = 0) or use your previous knowledge from the lectures tht body centered rel lttice will result in fce centered reciprocl lttice. We need to find the lrgest d-vlue tht gives constructive interference. We know tht d-vlues for cubic unit cells re given by: d = /(h 2 +k 2 +l 2 ) 1/2. After the {100} type of plnes, the {110} type of plnes hve the lrgest d-vlues (= /2 1/2 = 4 Å/2 1/2 = 2.8 Å). By doing the sme evlution s for the {100} plnes, we find tht the reflection of the 110 type plnes will not be extinct due to the centring of the lttice. It follows tht the Brgg scttering ngle for the {110} plnes is: θ = sin -1 θ = sin -1 (1.5406 Å/2 2.8 Å) = 16.0. Hence, you would expect to see the first reflection t 2θ = 32.0. e) Wht is multiplicity, nd how does it ffect the observed intensities in powder diffrctogrm? 8
Multiplicity is number tht tells how mny different crystllogrphiclly equivlent plnes we hve (i.e. for cubic crystl {100} represents 6 different plnes: (100), (010), (001), (-100), (0-10) nd (00-1), ll hving the sme d-vlue. {111} represents 8 different plnes: (111), (-111), (1-11), (11-1).etc., ll with the sme d-vlue). For powder diffrction, with rndom orienttion of grins, Brgg scttering from given crystllogrphic plne will occur sttisticlly ccording to the multiplicity of the plne. The intensity mesured in diffrctogrm is therefore relted to the intensity of one reflection I hkl times the multiplicity of the plne. f) If you hve grins with strin s illustrted below, why nd how would tht ffect your XRD dt? Inside the crystl, there will be vrition in the d-vlue for given plne within given rnge (d 1 -d 2 ). This will result in vrition in the corresponding Brgg ngle nd the size of the rnge (d 1 -d 2 ) will ffect how wide the diffrction peks will pper. Hence, strined crystls will show brod diffrction peks in diffrctogrm. Figure 3. Illustrtion of the lttice of strined crystl 9
Appendix Object Opticl xis Illustrtion (Figure 1) tht cn be used s strting point for nswering 1 b). c c Illustrtions (Figure 2) tht cn be used when nswering question 2 c) 10