Structural characterization of amorphous materials using x-ray scattering

Transcription

1 Structural characterization of amorphous materials using x-ray scattering Todd C. Hufnagel Department of Materials Science and Engineering Johns Hopkins University, Baltimore, Maryland Funding for work on scattering from metallic glasses provided by NSF-DMR, DOE-BES, ARO, ARL Scattering data presented collected at SSRL (7-, 1-), APS (1-ID), NSLS (X-14A)

2 Outline 1. How is scattering from amorphous materials different from diffraction from crystalline materials?. A gentle introduction to the theory of scattering from amorphous materials, and the structural information you get. 3. The path to enlightenment: From raw data to real-space structure 4. Experimental techniques and considerations 5. Resonant scattering 6. Resources

3 X-ray diffraction (crystalline materials) Incident x-rays Os ı Eq Os Diffracted x-rays.hkl/ planes (ŝ ŝ ) Scattering vector: q π λ Bragg s Law: q = 4π sin θ λ = π d hkl

4 Example: Mixed amorphous + crystalline structure Composite consisting of ~4 vol. % Ta particles in Zr-based metallic glass matrix Ta Metallic glass matrix Ta Ta Ta Ta Ta Ta q (Å 1 )

5 Crystalline Scattering is strong (intense peaks) Amorphous Scattering is weak Scattering concentrated into a few sharp diffraction peaks Scattering spread throughout reciprocal space (all values of q) Data easily interpreted using simple equations Detailed analysis required to obtain real-space information

6 Amorphous scattering theory I Scattered amplitude from a single atom: A n D f n exp. i Eq Er n / Scattered intensity from n atoms (arbitrarily arranged):! I eu.eq/ D X n A n! X m A m f n D Atomic scattering factor Er n D Position of nth atom D X n D X m f n exp. i Eq Er n /! X X fm f n exp.i Eq Er nm / n m f m exp.i Eq Er m/! Er nm D Vector joining nth and mth atoms

7 Amorphous scattering theory II Assume that the material is isotropic (so we can average over all orientations of rnm) to obtain the Debye scattering equation: X f m f n sin qr nm I eu.q/ D X m n qr nm This applies to any assemblage of identical scattering units (e.g. molecules) so long as they scatter independently and are randomly oriented.

8 Example: Ehrenfest relation Amorphous case is too hard; instead, consider scattering from diatomic gas: d If the molecules scatter independently, then: I(q) m n ( sinqr mn f m f n = f 1 + sinqd ) qr mn qd If the atoms are point scatters, then f is independent of q and the scattering has a maximum at q max = 1.3(π) d!! P. Ehrenfest Proc. Amsterdam Acad. 17, 1184 (1915) A. Guinier X-Ray Diffraction (Dover, 1994), pp

9 Amorphous scattering theory III Define a new quantity, the structure factor, S(q): S.q/ I eu.q/ N hf.q/i N D number of atoms S(q) is related to the real-space structure, the pair distribution function ρ(r), by a Fourier transform:.r/ ı D Z 1 4q.S.q/ 1/ sin qr qr dq ı D average atomic density Note the limits on the integral!

10 Data in reciprocal space and real space Reciprocal space Structure factor (a) Real space Pair distribution function (b) S(q) ρ(r) (Å 3 ) q (Å 1 ) r (Å) S(q) = ρ(r) ρ = 1 8π 3 I (q) N f (q) 4πq (S(q) 1) RDF = 4πr ρ(r) g(r) = ρ(r) ρ sin qr qr dq RDF, 4πr ρ(r) (Å 1 ) Real space Radial distribution function r (Å)

11 Physical interpretation of real-space functions S(q) = I (q) N f (q) ρ(r) ρ = 1 8π 3 4πq (S(q) 1) sin qr qr dq g(r) = ρ(r) ρ RDF = 4πr ρ(r) Figure from Y. Waseda, Anomalous X-Ray Scattering for Materials Characterization (Springer, )

12 Data reduction 1: Raw data Intensity (counts) Zr57Ti5Cu18Ni8Al1 bulk metallic glass Transmission geometry (.8 mm diameter cylinder) 8.7 kev x-rays (APS beamline 1-ID) MAR image plate 3 mm camera length Scattering vector magnitude, q (Å -1 ) 14 16

13 Data reduction 1I: Apply corrections 5 4 Corrected Raw data Intensity (counts) Scattering vector magnitude, q (Å -1 ) 14 16

14 Data reduction 1I: Apply corrections 5 1. Detector dead time Intensity (counts) Measured intensity (counts per second) 14x µs shaping time 1 µs shaping time 1 True intensity (arbitrary units) 3 4x Scattering vector magnitude, q (Å -1 ) 14 16

15 Data reduction 1I: Apply corrections Intensity (counts) Detector dead time. Background (substrate/container) 3. Absorption 4. Polarization 5. Multiple scattering O r O 1 x h 1 y Scattering vector magnitude, q (Å -1 ) 14 16

16 Data reduction 1I: Apply corrections Intensity (counts) Detector dead time. Background (substrate/container) 3. Absorption 4. Polarization 5. Multiple scattering 6. Inelastic processes (Compton, fluorescence) Scattering vector magnitude, q (Å -1 ) 14 16

17 Data reduction III: Normalize to e - units 5 Intensity (electron units) Coherent independent scattering Scaled intensity I coh.q/ D hf.q/f.q/i Scattering vector magnitude, q (Å -1 ) 14 16

18 Data reduction IV: Normalization, S(q) 4 I eu.q/ D I elastic.q/ C I coh.q/ C I inc.q/ D I obs.q/ 3 S.q/ D Normalization constant Measured (arb. units) Calculated/tabulated I eu.q/ hf.q/ihf.q/i D I obs.q/ I coh.q/ I inc.q/ hf.q/ihf.q/i S(q) Calculated/tabulated Scattering vector magnitude, q (Å -1 ) 14 16

19 Data reduction IV: Normalization, S(q) 4 S.q/ D I eu.q/ hf.q/ihf.q/i D I obs.q/ I coh.q/ I inc.q/ hf.q/ihf.q/i 3 Warren s large angle method: Ieu(q) should oscillate about the independent coherent scattering at large q. S(q) Oscillating about S(q)=1 is equivalent to oscillating about hf.q/f.q/i Scattering vector magnitude, q (Å -1 ) B. E. Warren, X-Ray Diffraction (Dover, 199)

20 Data reduction IV: Normalization, S(q) 4 S.q/ D I eu.q/ hf.q/ihf.q/i D I obs.q/ I coh.q/ I inc.q/ hf.q/ihf.q/i 3 Norman/Krogh-Moe ( integral ) method: Makes use of the fact that ρ(r) at low r to obtain: S(q) D R 1 q I coh.q/ dq ı hf.q/ihf.q/i R 1 q I obs.q/ dq : Scattering vector magnitude, q (Å -1 ) N. Norman, Acta Crystallogr. 1, 37 (1957) J. Krogh-Moe, Acta Crystallogr. 9, 951 (1956)

21 Data reduction V: Real-space functions Positions of peaks Coordination shell distances Pair distribution function,!(r) (Å -3 ) ρ(r) ρ = 1 8π 3 Upper limit set by experiment 4πq (S(q) 1) sin qr qr dq. 5 1 Distance from central atom, r (Å) 15

22 Data reduction V: Real-space functions Radial distribution function, 4!r "(r)(atoms Å -1 ) RDF = 4πr ρ(r) Inspect the low r region...should be zero! (Large oscillations are a sign of trouble with normalization, corrections, and/or damping.) 4r ı Area under peak = coordination number 4 6 Distance from central atom, r (Å) 8 1

23 Damping and the high q limit S(q) Damping function RDF (atoms Å -3 ) Scattering vector magnitude, q (Å -1 ) Scattering vector magnitude, q (Å -1 )

24 More on damping and Fourier transform The Nyquist theorem puts a limit on the real space resolution: r D =q max.' : Å for these data/ q [S(q) 1] (a) Damping function RDF, 4pir ρ(r) (Å 1 ) (b) q (Å 1 ) r (Å) R. Lovell, G. R. Mitchell, and A. H. Windle. Acta Crystallogr. Sec. A, 35, (1979) T. C. Hufnagel, R. T. Ott, and J. Almer. MRS Symp. Proc. 913, Z18-1 (6)

25 Extracting MRO information from the RDF Write the RDF as: ρ(r) 4πr 1 = g(r) 1 = A ( r exp r ) λ Exponential decay envelope in terms of a screening length, λ sin ( ) πr D + φ Oscillations due to short-range order (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 % Ta 4% Ta 1 4% Ta!=5.3±. Å. g(r) r(g(r)-1) % Ta!=3.8±. Å r(å) r (Å) 15 5 Analysis based on A. Bodapati et al. J. Non-Cryst. Sol. (in press) See also C.W. Outhwaite, Stat. Mech., 188 (1975)

26 Experimental geometries and detectors Step scanning Focusing graphite monochromator Position-sensitive x-ray detector Excellent energy resolution possible (with the right detector) Slow scanning (hours) May be count-rate (dead time) limited Beam slits X-ray beam from monochromator Detector slits Area detector Ion chamber (incident beam monitor) Evacuated enclosure Sample Excellent signal-to-noise ratio Very fast scans (seconds) Measure q in multiple orientations simultaneously (anisotropic specimens) Poor energy resolution Limited angular range Eq jj Image plate Specimen Eq?

27 Data collection considerations Need to measure I(q) to the largest q possible: q max D 4 D 4E hc q max.å 1 / E.keV/ The elastic, single-scattering intensity I(q) must be separated from all other sources of intensity. (This usually involves both experimental design and data analysis.) High signal-to-noise is essential at all values of q, especially at high q: ρ(r) ρ = 1 8π 3 4πq (S(q) 1) sin qr qr dq

28 Anomalous dispersion of x-rays Write the atomic scattering factor for x-rays as f (q,e) = f (q) + f (q,e) + i f (q,e) Energy-independent atomic form factor Energy-dependent anomalous scattering factors f', f'' (electrons) f (q,e) is unique for each element: f Ni f Ni f Zr f Zr f' (electrons) Scattering cross-section (1/cm) q (nm -1 ).6!E = -1 ev!e = -5 ev!e = -1 ev Energy (kev) Zr K absorption edge E=17998 ev X-ray energy (ev)

29 Energy resolution is critical Position-Sensitive Detector Output intensity (cts/monitor cts) K RR! k (1/Å) elastic K " RR kev energy Mo RR #E = 394 ev Ge RR #E = 11 ev Complete spectrum allows experimental removal of inelastic contributions from Compton and resonant Raman scattering Compton scattering nominally eliminated down to ~3 Å -1 below Ge edge and ~5-1 Å below Mo edge. H. Ishii and S. Brennan (unpublished)

30 Partial pair distribution function analysis Amorphous NiZr studied by combined resonant x-ray scattering and Reverse Monte Carlo simulations. Total structure factor S(q) Partial structure factors Partial distribution functions More complicated alloys are... more complicated. J. C. de Lima et al., Phys. Rev. B 67, 941 (3)

31 Resonant x-ray scattering on amorphous materials For a sample with m elements, the total scattered intensity is a weighted sum of m(m+1)/ partial structure factors: I(q) = n α β x α f α (q,e) f β (q,e)s αβ(q) Because f (q, E) only varies rapidly with E near an absorption edge, we can make a differential measurement for an element of interest (A here): ( ) I(q,E) E q = x A β E [ ] f A (q,e) fβ (q,e) + f A(q,E) f β (q,e) S Aβ (q) Only pair correlations that involve atoms of type A contribute to the total scattering For instance, we can make scattering measurements at two energies below an absorption edge, and take the difference. The resulting real-space information is a distribution function specific to the element of interest.

32 Resonant x-ray scattering from (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 Cu K edge E=8979 ev Ta L III edge E=9881 ev Zr K edge E=17998 ev Average atom Average atom Average atom RDF (atoms/å) (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 % Ta 4% Ta RDF (atoms/å) (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 % Ta 4% Ta RDF (atoms/å) (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 %Ta 4%Ta r (Å) Cu environment r (Å) Ta environment r (Å) Zr environment 7 8 Cu DDF (atoms/å) (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 % Ta 4% Ta Ta DDF (atoms/å) (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 4% Ta Zr DDF (atoms/å) (Zr 7 Cu Ni 1 ) 9-x Ta x Al 1 % Ta 4% Ta r (Å) % Ta: CN=13.7 ±.5 4% Ta: CN=13.1 ± r (Å) 4% Ta: CN=13.7 ± r (Å) Zr-edge data published in T. C. Hufnagel and S. Brennan, Phys. Rev. B 67, 143 (3)

33 Lessons I ve learned the hard way 1. Characterize the dead time of your detector at the x-ray energies you use, and over a large range of count rates.. Either exclude the Compton scattering experimentally, or count all of it (so you can calculate and subtract it later). 3.If doing a transmission experiment, measure the actual absorption of your specimen (μt product). 4. If possible, measure the fluorescence from the specimen separately from the elastic scattering. (This tells you a lot about potential alignment problems.) 5. Rather than doing a few lengthy scans, do many scans of short duration (to avoid systematic errors). 6. Do the data analysis in near-real time on the first few scans, to identify problems early.

34 What s important 1. Signal-to-noise ratio (particularly at high q). Energy resolution on the scattered beam side (critical for resonant scattering) 3. Energy resolution on the incident beam side (resonant scattering only) 4. Minimizing background (anything that is not elastic scattering from your specimen) What s not (within reason) Resolution in q space; brightness (divergence)

35 X-ray amorphous Intensity (arb. units) Laboratory source Cu Kα radiation (λ=1.54 Å) Intensity (arb. units) Synchrotron source E=17898 ev (λ=.693 Å) Scattering vector magnitude (Å -1 ) Scattering vector magnitude (Å -1 ) Is it amorphous? First peak symmetric Second peak at ~ qmax (for a metallic glass) No other scattering features present Combine with other techniques (TEM, DSC) Fundamental problem: All you get (from scattering) is the RDF, which is only sensitive to pair correlations.

36 Beyond the RDF: Fluctuation microscopy T. C. Hufnagel, Nature Mat. 3, 666 (4) (figure adapted from P. M. Voyles) The variance contains information about higher order (3- and 4-body) correlations Normalized variance Zr 59 Ta 5 Cu 18 Ni 8 Al 1 Zr 57 Ti 5 Cu Ni 8 Al Scattering vector magnitude, k (nm -1 ) J. Li, X. Gu, and T. C. Hufnagel. Microscopy and Microanalysis 9, 59 (3)

37 Fluctuation x-ray microscopy Complement to FEM great potential for looking at polymers, biologics, self-assembled materials Future: Higher spatial resolution Harder x-rays Resonant scattering 7 µm thick layer of 77 nm latex spheres E = 1.83 kev (6.77 Å) L. Fan et al., MRS Symp. Proc. 84, QC6.7 (5)

38 References and resources Books - B. E. Warren, X-Ray Diffraction (Dover, 199) - T. Egami and S. Billinge, Underneath the Bragg Peaks: Structural Analysis of Complex Materials (Pergamon, 3) Dissertations/SSRL reports from the Bienenstock group (Fuoss, Kortright, Ludwig,Wilson,Ishii...) Software - Matlab routines ( ~bren/files/amorphous/) - Billinge group software ( ftp/pub/billinge/)