Abstract
This paper reviews nonresonant, meV-resolution inelastic x-ray scattering (IXS), as applied to the measurement of atomic dynamics of crystalline materials. In conjunction with a companion paper on spectrometers and sample science (Part I, also in this handbook), it is designed to be an introductory, though in-depth, look at the field for those who may be interested in performing IXS experiments or those desiring a practical introduction to harmonic phonons in crystals at finite momentum transfers. The treatment of most topics emphasizes practical issues, as they have occurred to the author the last two decades spent introducing meV-resolved IXS in Japan, including designing and building two IXS beamlines, spectrometers, and associated instrumentation, performing experiments, and helping other scientists. This paper, Part II, focuses on scattering theory and on calculations, emphasizing relations and ideas that have been of practical use in experiments. This includes an introduction to many issues related to scattering from harmonic phonons, including basic nomenclature, intensities, anti-crossings, sum rules, and issues related to calculations and modeling. It is designed for a reader who has a basic knowledge of reciprocal space as used in x-ray scattering and wishes to extend this to atomic dynamics.
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Notes
- 1.
Note that for the purposes of considering atomic dynamics, formulas for neutron scattering can generally be converted to x-ray scattering by replacing the coherent neutron scattering length, b, by ref(Q), where re is the classical radius of the electron, re = e2∕mc2 ∼ 2.81 fm, and f(Q) is the x-ray form factor for the relevant atom or ion at momentum transfer Q = |Q|.
- 2.
We take f to be normalized to Z, the number of electrons in the atom at Q = 0.
- 3.
We emphasize the distinction between a primitive cell as (one of) the smallest repeatable unit whose translation can generate the entire structure, as different from a unit cell that is often larger and is chosen for convenience. For example, the cubic unit cell used to describe diamond contains eight atoms, while the primitive cell contains only two atoms. The phonon dispersion is then described by 6 = 3 × 2 phonon branches, not 3 × 8 = 24. Using the larger cell in a model would cause a lot of unnecessary computation and lead to many phonon branches with zero intensity.
- 4.
The term “pseudo-harmonic” is often used to indicate a harmonic model where the atomic interaction terms are scaled with temperature to account for the effect of thermal expansion. This leads to harmonic phonons at any fixed temperature, but since strictly speaking, a harmonic solid does not undergo thermal expansion, the model termed pseudo-harmonic.
- 5.
In one case a sample with an interesting magnetic structure appeared to show such a violation, but the lack of symmetry in the spectra was eventually traced to a software bug in a counter card driver. The card was actually in use in many places, but there were very few that used it at the low rates of the IXS experiment, and, precisely in the low-rate region, the software error became noticeable.
- 6.
We generally use boldface quantities to indicate three vectors (in either real space or reciprocal space) and dual arrow superscripts to indicate matrices such as \(\overrightarrow {\overrightarrow \Phi }_{\ell d\ell 'd'}\). In some cases, such as the eigenvector matrix for all atoms, \(\overrightarrow {\mathbf {e}}_{\mathbf {q}}\), a single vector over a boldface quantity indicates an extended one-dimensional matrix of many vectors.
- 7.
For example, in the case of CaAlSi, the presence of a soft mode introduces large differences in the Debye-Waller factor for different atoms, with factors of two easily possible at room temperature (Kuroiwa et al. 2008).
- 8.
Most work, including that here, emphasizes the effect of non-harmonicity on line shapes and frequency, neglecting the effect on polarization. However, such effects must be present, if only in that a non-harmonic shift in the frequency may move a mode closer or further from an anti-crossing, leading to a change in polarization. One would also expect such effects to appear more generally, but the usual treatment separating the intensity and the line shape does not facilitate its discussion. Lovesey (1984) also suggests that direct non-harmonic effects on polarization tend to vanish at high-symmetry points of the Brillouin zone.
- 9.
As an exercise, one can replace the Ewald sum by calculating many force constant matrices out, say, to some range R – one finds, as might be expected, the phonon behavior is generally well predicted until you approach small momentum transfers (∼1/R).
- 10.
This usage of the term “shell” in the shell model should not be confused with the shells (first neighbors, second neighbors, etc.) sometimes referred to in BvK models.
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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, Hiroshi Fukui, and Kazuyoshi Yamada. I think Mark Sutton for kindly pointing me to his review paper. 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 work 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, and 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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Baron, A.Q.R. (2019). High-Resolution Inelastic X-Ray Scattering Part II: Scattering Theory, Harmonic Phonons, and Calculations. 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_52-2
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