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High-Resolution Inelastic X-Ray Scattering I: Context, Spectrometers, Samples, and Superconductors

Synchrotron Light Sources and Free-Electron Lasers

Abstract

This paper reviews nonresonant, meV-resolution inelastic x-ray scattering (IXS), as applied to the measurement of atomic dynamics of crystalline materials. It, in conjunction with a companion paper on theory and calculations (Part II), is designed to be an introductory, though in-depth, look at the field for those who may be interested in performing IXS experiments or in understanding the operation of IXS spectrometers or even those desiring a practical introduction to harmonic phonons in crystals at finite momentum transfers. The treatment of most topics begins from ground level, with an emphasis on practical issues, as they have occurred to the author in 15 years spent introducing meV-resolved IXS to Japan, including designing and building two IXS beamlines, spectrometers and associated instrumentation, performing experiments, and helping users. After a short introduction to the broader field of IXS, this paper, Part I, discusses the relative merits of IXS as compared to other methods of investigating atomic dynamics, especially inelastic neutron scattering (INS). A very brief overview of spectrometer operation and the types of spectra observed is also given. The paper then focuses on the main issues relevant for spectrometer design, including a brief survey of relevant formulas from dynamical diffraction and an in-depth discussion of how various design issues have been addressed in the different types of operating spectrometers, including spectrometers using spherically figured analyzers and the relatively new “post-sample-collimation” (PSC) systems. Finally, there is discussion of the types of experiments that have been carried out with a brief discussion of the many types of crystalline samples that have been investigated and detailed discussion on measurements of superconductors.

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Notes

  1. 1.

    A comprehensive treatment of many aspects of IXS, but not the meV-resolved work discussed here, can be found in Schülke (2007)

  2. 2.

    The magnitude of the momentum transfer, \(\vert \mathbf{Q}\vert = (4\pi /\lambda )\sin (2\Theta /2)\), is determined by the scattering angle, \(2\Theta \), and the incident x-ray wavelength, \(\uplambda\), and there is no coupling between energy transfer and momentum transfer so, different than a neutron triple-axis spectrometer, fixed Q scans are done without moving the sample or the two-theta arm.

  3. 3.

    Momentum transfer and energy transfer are completely decoupled with x-rays so, for IXS, scanning energy transfer at constant momentum transfer requires no motion of the sample or analyzers.

  4. 4.

    Scanning Q at a fixed energy transfer is possible, as are more sophisticated correlated scans, but can be dangerous, except in well-defined simple cases. This danger is because (1) IXS energy resolution is typically Lorentzian, with long tails, so that the intensity at a fixed energy can respond to changes in intensity of spectral features that are far away – thus, it can be necessary to know the entire energy spectrum at a particular momentum transfer to properly interpret the intensity at one energy – and (2) dynamical spectra are generally complex, with many modes for materials with larger unit cells, so even in the absence of long tails, it can be important to know the local shape of the spectrum to evaluate intensity changes at a particular energy.

  5. 5.

    Note that this reference takes \(k = 1/\lambda\), dropping the factor of 2π that is commonly used today.

  6. 6.

    Sometimes the reflectivity is defined in terms of intensity per unit area, and this will lead to an additional factor of | b | in Eq. (3).

  7. 7.

    For a symmetric reflection, b = −1, for any plane wave, the angle of incidence will be the same as the angle of reflection, with the Bragg reflection only reducing the intensity.

  8. 8.

    Note that sometimes the extinction length is just taken as the value at \(\upeta = 0\) which will lead to replacing the factor of 2 by \(\uppi\) in Eq. (9).

  9. 9.

    Beam from an undulator insertion device usually has a complex frequency spectrum with several peaks and is called “pink” to distinguish it from the broader “white” beam (without narrow peaks, but falling off at high and low energy) that is provided by wigglers or bending magnets.

  10. 10.

    Reducing temperature to reduce W and keep the Debye-Waller factor near unity both increases reflectivity and the bandwidth. However, these is a net win in throughput for a fixed final bandwidth.

  11. 11.

    This thickness is needed to preserve good resolution. See Said et al. (2011).

  12. 12.

    For thinner (\(<\sim 0.5\) mm thick) analyzers, one can do the “dicing” using reactive ion etching, but this is not possible for thicker where severe aspect ratios (\(\sim \) 50:1) are needed to keep the grove width small (Finkelstein 2005).

  13. 13.

    The same restriction exists also for INS, but, often, exemplary plots of the Q-E plane are shown for low-dimensional samples – so the limit is not always obvious. Meanwhile, the new chopper spectrometers can also rotate the sample and do post-selection of event to choose only those in high symmetry directions.

  14. 14.

    Here we will use “BCS” to refer to the original discussion (Bardeen et al. 1957) where phonons were introduced in a weak-coupling limit without an explicit relation between the coupling and the detailed phonon spectra, while Eliashberg theory (Eliashberg 1960) allowed a direct relation based on application of Migdal’s approximation (Migdal 1958) ignoring vertex corrections – effectively assuming interesting electronic states have energies much larger than phonon energies.

  15. 15.

    We use tetragonal notation where the [100] direction in real space is along the Cu-O bond and the [001] direction is normal to the Cu-O plane. For the moment, and as is often done, we neglect differences between (100) and (010) though, for most materials, these are not completely equivalent.

  16. 16.

    An attentive reader may note that the \(\sim \) 35 meV mode appears to be split near gamma also in (a). However, in (a) what is seen is an anti-crossing between two modes (e.g., Part II, section 2.11.3) as can be seen in (e) where it is still present and (f) where the frequencies have been shifted and it is removed. In contrast, in (c) and (d) the mode is really split by the magnetic ordering.

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Acknowledgements

I am grateful to several scientists who kindly read and offered comments on preliminary versions of this paper including Sunil Sinha, Rolf Heid, Aleksandr Chumakov, Hiroshi Fukui, Kazuyoshi Yamada, and Yuri Shvyd’ko. I thank the scientists who kindly shared some of the details of the beamlines with me, including Ahmet Alatas, Ayman Said, Michael Krisch, and Claudio Masciovecchio. The paper is based on work carried out at SPring-8. I would like to express my deep appreciation to the many scientists in all parts of SPring-8 that I have had the pleasure of working with over the last, nearly, two decades, as well as many collaborators outside SPring-8. This review paper is based on experience gained during many proposals including 2001B 0203 0481 0482 0508 0575 3607, 2002A 0182 0279 0280 0520 0537 0559 0560 0561 0562 0627, 2002B 0151 0178 0179 0180 0243 0248 0249 0287 0382 0383 0529 0539 0565 0593 0594 0632 0668 0709, 2003A 0022 0081 0153 0175 0235 0284 0357 0555 0637 0638 0683 0716, 2003B 0019 0132 0206 0248 0359 0397 0574 0693 0743 0744 0745 0755 0766, 2004A 0322 0439 0510 0519 0577 0582 0590 0634, 2004B 0003 0070 0204 0343 0491 0597 0632 0635 0722 0730 0736 0752, 2005A 0039 0061 0146 0147 0148 0157 0330 0369 0428 0475 0567 0596 0616 0712 0751, 2005B 0082 0093 0124 0253 0266 0295 0346 0441 0443 0484 0603 0623 0650 0731 0736, 2006A 1023 1039 1057 1081 1181 1226 1242 1272 1273 1291 1345 1376 1379 1417 1430 1453 1467 1502, 2006B 1053 1082 1089 1146 1186 1204 1235 1259 1299 1311 1337 1352 1356 1405 1417, 2007A 1109 1118 1125 1125 1222 1234 1279 1281 1301 1374 1436 1441 1473 1505 1507 1523 1539 1561 1612 1647 1671, 2007B 1053 1062 1099 1114 1118 1197 1198 1215 1322 1328 1336 1343 1375 1444 1538 1614 1640 1662, 2008A 1058 1064 1125 1140 1204 1205 1394 1456 1491 1522 1568 1582 1584 1587 1588 1626, 2008B 1381 1403 1473 1178 1108 1326 1584 1240 1144 1169 1491 1634, 2009A 1054 1093 1146 1189 1203 1224 1274 1290 1299 1358 1379 1436 1451 1492 1506 1548, 2009B 1074 1114 1126 1150 1165 1286 1323 1423 1439 1548 1555 1584 1609 1619, 2010B 1108 1112 1177 1185 1206 1353 1354 1392 1410 1453 1497 15271538 1575 1579 1593, 2011A 1051 1075 1104 1117 1136 1154 1180 1256 1271 1300 1304 1366 1373 1452 1502, 2011B 1122 1213 1215 1314 1332 1336 1353 1388 1406 1408 1423 1425 1536 1590, 2012A 1102 1115 1122 1156 1219 1237 1243 1250 1255 1354 1362 1390 1406 1417 1452 1506 1583, 2012B 1080 1125 1159 1196 1226 1236 1277 1283 1343 1356 1358 1364 1439 1577 1596 1658, 2014A 1026 1059 1076 1086 1089 1100 1106 1122 1131 1154 1207 1231 1235 1240 1346 1368 1378 1385 1434 1678 1687, 2014B 1052 1066 1068 1130 1143 1159 1182 1222 1269 1271 1290 1365 1381 1465 1381 1465 1536 1545 1739 1760 1761 1175 1192.

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  1. Materials Dynamics Laboratory, RIKEN SPring-8 Center, RIKEN, 1-1-1 Kouto, 679-5148, Sayo, Hyogo, Japan

    Alfred Q. R. Baron

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  1. Helmholtz-Zentrum Berlin, Berlin, Germany

    Eberhard Jaeschke

  2. Zentrum für Synchrotronstrahlung, TU Dortmund, Dortmund, Germany

    Shaukat Khan

  3. FS-CFEL-1, DESY, Hamburg, Germany

    Jochen R. Schneider

  4. SLAC National Accelerator Laboratory Linac Coherent Light Source, Stanford University, Menlo Park, California, USA

    Jerome B. Hastings

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Baron, A.Q.R. (2015). High-Resolution Inelastic X-Ray Scattering I: Context, Spectrometers, Samples, and Superconductors. In: Jaeschke, E., Khan, S., Schneider, J., Hastings, J. (eds) Synchrotron Light Sources and Free-Electron Lasers. Springer, Cham. https://doi.org/10.1007/978-3-319-04507-8_41-1

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    High-Resolution Inelastic X-Ray Scattering I: Context, Spectrometers, Samples, and Superconductors
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    DOI: https://doi.org/10.1007/978-3-319-04507-8_41-2

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    High-Resolution Inelastic X-Ray Scattering I: Context, Spectrometers, Samples, and Superconductors
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    DOI: https://doi.org/10.1007/978-3-319-04507-8_41-1

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