Homework for Diffraction-MSE 603: Solutions May 2002

Transcription

1 Hmewrk fr Diffratin-MSE 603: Slutins May An x-ray beam f 1.5 Å impinges n a Ge single rystal sample with an inient angle θ lse t the ritial angle θ f the Ge surfae. Taking int aunt the absrptin, the nrmal mpnent f the internal wavevetr K 0n is given by: n k0 θ θ iµ K0 / k0, where k 0 π/λ, λ is the x-ray wavelength, an µ 40 m -1 is the linear absrptin effiient fr Ge. (a) Using θ (δ) 1/, alulate the ritial angle fr Ge at 1.5 Å. Density ρ 5.3 g/m 3 fr Ge. Frm lass ntes, θ (δ) 1/ λ [ A] ρ[ g / m ] 5.67 mra (b) The penetratin epth τ f the x-ray beam int the sample is given by the imaginary part f K 0n via τ 1/[Im(K 0n )]. Calulate τ at θ 0, θ, an 3θ. Sine µ/k 0 40x10-8 x1.5/(π) 9.6x10-7 is muh less than θ 3.1x10-5, we an ignre µ/k 0 at θ 0. Therefre, 1.5/(4πx ) 1 Å. At θ θ, K 0 n k0 iµ / k0 k0 µ / K 0 n ik 0 θ, an τ λ/(4πθ) 0 iπ / 4 k e. Thus τ 1/[Im(K 0n )] 1.5/[4πx(9.6x10-7 ) 1/ xsin(3π/4)] 17 Å. At θ 3θ, we use the full frmula: K0 n k0 θ θ iµ / k0, an the imaginary ( θ θ ) + ( µ / k ) ) 1/ 4 part: 1 µ / k0 Im( K 0n ) k0 0 sin artan. Sine µ/k 0 θ θ << θ 1/ µ / k0 µ, we have: Im( K0n ) k0 ( θ θ ). Thus 1/ ( θ θ ) ( θ θ ) 1/ τ ( θ θ ) µ / Å. () Shw that at θ >>θ, τ θ/µ, whih is ue t the simple effet f absrptin. 1/ ( Frm (b) we have τ θ θ ) µ θ µ, if θ >>θ. () Lg nt an alulate n-line the refletivity urve fr 3-30keV x-rays, assuming an inient angle f 0.5, an a RMS rughness f 0Å an 5Å respetively. This shul be easy an fun. The nly thing is that we nee t be areful abut units n x-ray energies. This web site uses ev instea f kev. (e) In (), what is the pssible ause fr the glith at arun 10 kev? The glith at arun 10 kev is ue t the K-ege f germanium, 11.1 kev. 8

2 . A 1 µm-thik film f single rystal Zn 0.07 Mn 0.93 Te is grwn epitaxially n a substrate ZnTe (001). It has the same fae-entere ubi (f) lattie struture at rm temperature as the substrate, exept that the lattie nstant (ab) is 6.3 Å fr Zn 0.07 Mn 0.93 Te, an 6.10 Å fr ZnTe. The surfae nrmal is alng a ubi axis whih is hsen t be the -axis. An x-ray iffratin experiment is perfrme with an inient x-ray energy f 8.3 kev. (a) Assuming that the thin film is mpletely relaxe, alulate the Bragg angles f the (00) refletins fr bth Zn 0.07 Mn 0.93 Te an ZnTe. X-ray wavelength: λ 1.4/ Å. -spaing: 6.3 Å / 3.16 Å fr Zn 0.07 Mn 0.93 Te, an 3.05 Å fr ZnTe. Bragg angle: θ B sin -1 (λ/) fr Zn 0.07 Mn 0.93 Te, an θ B fr ZnTe. (b) The measure angular with f the Zn 0.07 Mn 0.93 Te (00) refletin in a raial san is θ Assuming that strain an be ignre, estimate the average main size in the thin film. Average main size: L λ/( θ sθ B ) 1.494/(0.05xπ/180xs13.67 ) 3.5x10 3 Å. () In reipral lattie units f Zn 0.07 Mn 0.93 Te, the (00) refletin f Zn 0.07 Mn 0.93 Te urs at l. What is the l value fr the ZnTe (00)? l ZnTe x6.3/ () Make a sketh f the reipral pints (000), (00), (00) an (0) fr bth Zn 0.07 Mn 0.93 Te an ZnTe. (00) (0) (000) (00) 3. Same as Prblem. At lw temperatures (<60 K) the Zn 0.07 Mn 0.93 Te bemes antiferrmagneti, an the rystal lattie transfrms int rthrhmbi, resulting in 3 types f mains that -exist in the sample. The rthrhmbi istrtin in eah type f mains an be expresse by: (ε7x10 4 ) a >a(1 ε), b >b(1 ε), >(1+ε), a >a(1 ε), b >b(1+.75ε), >(1.75ε), a >a(1+.75ε), b >b(1 ε), >(1.75ε), (a) Repeat () in Prblem. (Ignre the ZnTe). fr -type fr b-type fr a-type.

3 + + -type b-type a-type (00) (000) (0) (00) (b) Hw many iffratin peaks yu expet t see arun the (00)? Arun the (00)? An arun the (0)? Frm abve piture, we expet t see tw peaks arun (00) an (00), but three peaks arun (0). () Assuming that the ppulatin rati f the ifferent mains is N a :N b :N 1:1:3 in the sample, what are the relative integrate intensities f the iffratin peaks arun the (00) an arun the (00)? Arun the (00): nly -type mains ntribute t the lw-q peak but bth a- an b-types ntribute t the high-q peak. S the relative intensities shul be: I lw-q :I high-q N :(N a +N b ) 3:. Arun the (00): nly a-type mains ntribute t the lw-q peak but bth - an b-types ntribute t the high-q peak. S the relative intensities shul be: I lw-q :I high-q N a :(N b +N ) 1:4. 4. A pwer iffratin pattern is btaine n silin, using Cu Kα raiatin (λ1.54å). The rystal struture f silin is ubi with a lattie nstant f 5.43 Å. Its unit ell ntains 8 silin atms late at: (0, 0, 0), (1/, 1/, 0), (1/, 0, 1/), (0, 1/, 1/), (1/4, 1/4, 1/4), (3/4, 3/4, 1/4), (3/4, 1/4, 3/4), an (1/4, 3/4, 3/4). What are the Miller inies f the first tw lines in the pwer pattern, an what is the rati f the integrate intensity f the first t that f the sen? First tw lines fr a silin pwer pattern shul be (111) an (0). This an be erive by alulating the struture fatrs fr lwest (hkl) integers an lk fr the nnvanishing nes: F ( hkl) f i π ( h / + k / ) i π ( h / + l / ) i π ( k / + l / ) i π ( h / 4+ k / 4+ l / 4) [ 1 + e + e + e ][ 1 + e ] F (100) 0; F (00) 0; F (110) 0; F (0) 8 f ; F (111) 4 f (1 i); F () spaings fr (111) an (0): ( 111) a Å ( 0) a 1.90 Å 8 8

4 Bragg angles fr (111) an (0): Atmi sattering fatrs f Appenix 10]: sinθ λ λ θ ( 111) sin 14.1 λ θ ( 0 ) sin 3.6 f 1 f (111) f (0) fr Si (111) an (0) [frm Cullity, f (0.159Å f (0.60Å Lrentz-Plarizatin fatrs LP fr (111) an (0): -1-1 ) 8.68 ) s 14.1 LP(111) sin 14.1 s s 3.6 LP(0) sin 3.6 s Multipliity fatrs m hkl fr (111) an (0): m 111 8, m 0 1. Putting all these fatrs tgether, the rati f the integrate intensities fr the (111) an (0) pwer iffratin line is J J w w (111) (0) F (111) F (0) LP(111) m LP(0) m ( ) ( ) The benzene mleule C 6 H 6 has six arbns at the rners f a hexagn with a C-C bn f 1.39 Å. Suppse that a sattering pattern frm a sample f benzene vapr is rere with λ1.54 Å x-rays. Negleting the sattering frm H, at apprximately what sattering angle θ will the first peak ur? Benzene vapr is amrphus an thus the x-ray sattering nsiere here is a wie angle sattering (WAXS) pattern. The first peak rrespns t the largest spaing between the atms, whih is a.78 Å, an urs at π Q 1.3 4π sinθ π r: 1.3. λ λ 1.54 Thus: θ sin 1.3 sin r: θ a

5 6. A iffratmeter using Cu Ka raiatin n a metal isk llets the (111) iffratin peaks f Cu, Ni, an Au an the integrate peak intensities are in the ratis 0.5:6.:1, respetively fr the three f rystals. (a) Assuming that the sample is a hmgenus mixture f the three metals, (e.g. a presse pwer mixture), estimate the mpsitin in At %; (b) Assuming that the sample has a pure gl film ver a hmgenus mixture f Cu an Ni, estimate the thikness f the gl film; () The Ni refletin is signifiantly wier than the thers, with a FWHM f 0.3. A referene sample f silin has a FWHM f 0.1. What es this say abut the grain size f the nikel in the sample? [ Dave: Cul yu please pst yur slutin here? Thanks! -- Qun ]