Ewald's sphere

The Ewald sphere is a geometric construction used in electron, neutron, and x-ray diffraction which shows the relationship between:

the wavevector of the incident and diffracted beams,
the diffraction angle for a given reflection,
the reciprocal lattice of the crystal.

It was conceived by Paul Peter Ewald, a German physicist and crystallographer.^[1] Ewald himself spoke of the sphere of reflection.^[2] It is often simplified to the two-dimensional "Ewald's circle" model or may be referred to as the Ewald sphere.

Ewald construction

A crystal can be described as a lattice of atoms, which in turn this leads to the reciprocal lattice. With electrons, neutrons or x-rays there is diffraction by the atoms, and if there is an incident plane wave $\exp(2\pi i\mathbf {k_{0}} \cdot \mathbf {r} )$ ^[a] with a wavevector $\mathbf {k_{0}}$ , there will be outgoing wavevectors $\mathbf {k_{1}}$ and $\mathbf {k_{2}}$ as shown in the diagram^[3] after the wave has been diffracted by the atoms.

The energy of the waves (electron, neutron or x-ray) depends upon the magnitude of the wavevector, so if there is no change in energy (elastic scattering) these have the same magnitude, that is they must all lie on the Ewald sphere. In the Figure the red dot is the origin for the wavevectors, the black spots are reciprocal lattice points (vectors) and shown in blue are three wavevectors. For the wavevector $\mathbf {k_{1}}$ the corresponding reciprocal lattice point $\mathbf {g_{1}}$ lies on the Ewald sphere, which is the condition for Bragg diffraction. For $\mathbf {k_{2}}$ the corresponding reciprocal lattice point $\mathbf {g_{2}}$ is off the Ewald sphere, so $\mathbf {k_{2}} =\mathbf {k_{0}} +\mathbf {g_{2}} +\mathbf {s}$ where $\mathbf {s}$ is called the excitation error. The amplitude and also intensity of diffraction into the wavevector $\mathbf {k_{2}}$ depends upon the Fourier transform of the shape of the sample,^[3]^[4] the excitation error $\mathbf {s}$ , the structure factor for the relevant reciprocal lattice vector, and also whether the scattering is weak or strong. For neutrons and x-rays the scattering is generally weak so there is mainly Bragg diffraction, but it is much stronger for electron diffraction.^[3]^[5]

References

^ Ewald, P. P. (1921). "Die Berechnung optischer und elektrostatischer Gitterpotentiale". Annalen der Physik (in German). 369 (3): 253–287. Bibcode:1921AnP...369..253E. doi:10.1002/andp.19213690304.
^ Ewald, P. P. (1969). "Introduction to the dynamical theory of X-ray diffraction". Acta Crystallographica Section A. 25 (1): 103–108. Bibcode:1969AcCrA..25..103E. doi:10.1107/S0567739469000155.
^ ^a ^b ^c John M., Cowley (1995). Diffraction physics. Elsevier. ISBN 0-444-82218-6. OCLC 247191522.
^ Rees, A. L. G.; Spink, J. A. (1950). "The shape transform in electron diffraction by small crystals". Acta Crystallographica. 3 (4): 316–317. Bibcode:1950AcCry...3..316R. doi:10.1107/s0365110x50000823. ISSN 0365-110X.
^ Peng, L.-M.; Dudarev, S. L.; Whelan, M. J. (2011). High energy electron diffraction and microscopy. Oxford: Oxford University Press. ISBN 978-0-19-960224-7. OCLC 656767858.

Notes

^ In some physics texts the $2\pi$ is omitted

External links

Crystallography

Key concepts

Structure

Systems	Cubic Hexagonal Monoclinic Orthorhombic Tetragonal Triclinic

Phase
transition

Defects

Grain boundary	Disclination CSL Growth Abnormal growth

Algorithms

Associations	Europe France Germany UK US Japan

Category
Commons

Read other articles:

Agrilus oblatus

Agrilus oblatus Klasifikasi ilmiah Kerajaan: Animalia Filum: Arthropoda Kelas: Insecta Ordo: Coleoptera Famili: Buprestidae Genus: Agrilus Spesies: Agrilus oblatus Nama binomial Agrilus oblatusKerremans, 1900 Agrilus oblatus adalah spesies kumbang yang tergolong ke dalam famili Buprestidae. Spesies ini juga merupakan bagian dari ordo Coleoptera. Spesies Agrilus oblatus sendiri merupakan bagian dari genus Agrilus yang mencakup sekitar 3.000 spesies.[1] Nama ilmiah dari spesies ini per...

Baharestan County

County in Tehran province, Iran Not to be confused with Bahar County.For other places with a similar name, see Baharestan. County in Tehran, IranBaharestan County Persian: شهرستان بهارستانCountyLocation of Baharestan County in Tehran province (center left, yellow)Location of Tehran province in IranCoordinates: 35°32′N 51°12′E / 35.533°N 51.200°E / 35.533; 51.200[1]Country IranProvinceTehranCapitalsGolestan an...

مذهب تصوري

المذهب المعنوي[1] أو التصورية[1] هو نظرية فلسفية تفسر الطابع الكلي للكيانات باعتبارها أُطرًا تصورية تقع داخل العقل الفكري.[2] ونظرًا لوقوعها وسط الاسمية والواقعية، تنتهج الرؤية التصورية المفهوم الميتافيزيقي للمسلمات من المنظور الذي ينفي وجودها في كيانات خارج �...

Данио-рерио

Данио-рерио Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеГруппа:Костные рыбыКласс:Лучепёрые рыбыПодкласс:Новопёрые рыбыИн�...

Himself (Bill Cosby album)

This article is about the album. For the film, see Bill Cosby: Himself. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Himself Bill Cosby album – news · newspapers · books · scholar · JSTOR (April 2020) (Learn how and when to remove this message) 1982 soundtrack album by Bill CosbyHimselfSoundtrack...

Wicked, Wicked

1973 film by Richard L. Bare Wicked, WickedDirected byRichard L. BareWritten byRichard L. BareProduced byRichard L. BareStarring David Bailey Tiffany Bolling Randolph Roberts Scott Brady Edd Byrnes Madeleine Sherwood Arthur O'Connell Indira Danks CinematographyFrederick GatelyEdited byJohn F. SchreyerMusic byPhilip SpringerDistributed byMGMRelease date June 13, 1973 (1973-06-13) Running time95 minutesCountryUnited StatesLanguageEnglish Wicked, Wicked is a 1973 horror-thriller f...

美国众议院

此條目需要补充更多来源。 (2021年7月4日)请协助補充多方面可靠来源以改善这篇条目，无法查证的内容可能會因為异议提出而被移除。致使用者：请搜索一下条目的标题（来源搜索：美国众议院 — 网页、新闻、书籍、学术、图像），以检查网络上是否存在该主题的更多可靠来源（判定指引）。美國眾議院 United States House of Representatives第118届美国国会众议院徽章众议院旗...

世界政區索引

此條目可能包含不适用或被曲解的引用资料，部分内容的准确性无法被证實。 (2023年1月5日)请协助校核其中的错误以改善这篇条目。详情请参见条目的讨论页。各国相关主題列表索引国内生产总值石油储量国防预算武装部队（军事）官方语言人口統計人口密度生育率出生率死亡率自杀率谋杀率失业率储蓄率识字率出口额进口额煤产量发电量监禁率死刑国债 ...

Maude (serie televisiva)

Questa voce o sezione sull'argomento sitcom non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. MaudeTitolo originaleMaude PaeseStati Uniti d'America Anno1972-1978 Formatoserie TV Generesitcom Stagioni6 Episodi141 Durata30 min Lingua originaleinglese Rapporto4:3 CreditiIdeatoreNorman Lear Interpreti e personaggi Beatrice Arthur: Maude Findlay Bill Macy: Walt...

Parasite

ParasiteLogo film dalam bahasa InggrisNama lainHangul기생충 Hanja寄生蟲 Alih Aksara yang DisempurnakanGisaengchung SutradaraBong Joon-hoProduserBong Joon-hoKwak Sin-aeJang Young-hwanDitulis olehBong Joon-hoHan Jin-wonPemeranSong Kang-hoLee Sun-kyunCho Yeo-jeongChoi Woo-shikPark So-damPenata musikJung Jae-ilSinematograferHong Kyung-pyo[1]PenyuntingYang Jin-moPerusahaanproduksiBarunson E&A CorpDistributorCJ EntertainmentTanggal rilis 21 Mei 2019 (2019-05-21)&#...

Large false serotine

Species of bat Large false serotine Conservation status Vulnerable (IUCN 3.1)[1] Scientific classification Domain: Eukaryota Kingdom: Animalia Phylum: Chordata Class: Mammalia Order: Chiroptera Family: Vespertilionidae Genus: Hesperoptenus Species: H. tomesi Binomial name Hesperoptenus tomesiThomas, 1905 The large false serotine (Hesperoptenus tomesi) is a species of vesper bat. It is found only in Malaysia. References ^ Senawi, J.; Hutson, A.M.; Kingston, T.; Francis, C.; ...

Mathilda Roos

Swedish writer (1852–1908) This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (May 2021) (Learn how and when to remove this message) Mathilda RoosMathilda RoosBorn(1852-08-02)2 August 1852Stockholm, SwedenDied17 July 1908(1908-07-17) (aged 55)Danderyd, SwedenNationalitySwedishOccupationwriter Lovisa Mathilda Roos (pen name, M. Rs.; 2 August 1852 – 17 Jul...

Sintashta

Archeological site in Chelyabinsk Oblast, Russia SintashtaСинташта́Location of the Sintashta culture (violet)Shown within Continental AsiaShow map of Continental AsiaSintashta (Russia)Show map of RussiaLocationChelyabinsk Oblast, RussiaCoordinates52°29′10.4″N 60°11′17.8″E / 52.486222°N 60.188278°E / 52.486222; 60.188278TypeSettlement Sintashta[a] is an archaeological site in Chelyabinsk Oblast, Russia. It is the remains of a fortified settl...

Niños Héroes

6 Mexican teenage military cadets who died in the Battle of Chapultepec (1847) For the Mexico City Metro station, see Niños Héroes / Poder Judicial CDMX metro station. For the Monterrey Metro station, see Niños Héroes metro station (Monterrey). Image based on the medal given to the cadets Monument to the Niños Héroes in Chapultepec Park, Mexico City. The Niños Héroes (Boy Heroes, or Heroic Cadets) were six Mexican military cadets who were killed in the defence of Mexico City during th...

Porto di Genova

Questa voce o sezione sull'argomento porti non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Porto di GenovaStato Italia Regione Liguria Provincia Genova Comune Genova MareMar Ligure Tipocontainer, officine navali, traghetti, crociere Passeggericirca 2 094 701 (2021) Traffico mercicirca 48831000 t (2021) Coordin...

العلاقات المجرية الإيطالية

العلاقات المجرية الإيطالية المجر إيطاليا المجر إيطاليا تعديل مصدري - تعديل العلاقات المجرية الإيطالية هي العلاقات الثنائية التي تجمع بين المجر وإيطاليا.[1][2][3][4][5] مقارنة بين البلدين هذه مقارنة عامة ومرجعية للدولتين: وجه المقارنة الم�...

COVID-19 pandemic in Saint Vincent and the Grenadines

COVID-19 pandemic in Saint Vincent and the GrenadinesDiseaseCOVID-19Virus strainSARS-CoV-2LocationSaint Vincent and the GrenadinesFirst outbreakWuhan, Hubei, ChinaArrival date11 March 2020(4 years, 4 months, 1 week and 4 days)Confirmed cases1,818 (as of 15 April 2021)[1]Active cases131 (as of 15 April 2021)[2]Recovered1,677 (as of 15 April 2021)Deaths10 (as of 15 April 2021)Fatality rate0.55% (as of 15 April 2021)Government websitehttp://www.gov.vc/ Part o...

Kabupaten Sarolangun Bangko

Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: Kabupaten Sarolangun Bangko – berita · surat kabar · buku · cendekiawan · JSTOR Kabupaten Sarolangun Bangko merupakan sebuah bekas kabupaten di Provinsi Jambi, Indonesia. Sejak tanggal 4 Oktober 1999, Ka...

Bertrand Barère

French politician, freemason and journalist (1755–1841) Barere redirects here. For the Russian pianist, see Simon Barere. The CitizenBertrand BarèrePortrait of Barère by Jean-Louis Laneuville (1794)Member of the Chamber of Representativesfrom Hautes-PyrénéesIn office3 June 1815 – 13 July 1815Preceded byJean LacrampeSucceeded byJean-Baptiste DarrieuxConstituencyTarbesMember of the Council of Five Hundredfrom Hautes-PyrénéesIn office22 October 1795 – 26 December 17...

Pieter Timmers

Pour les articles homonymes, voir Timmers. Pieter Timmers Informations Nages Nage libre Nationalité Belge Naissance 21 janvier 1988 (36 ans) Neerpelt, Belgique Taille 2,00 m (6′ 7″)[1] Club BRABO Entraîneur Ronald Gaastra Records Grand bassin 100 m nl : 47 s 80 (RN)200 m nl : 1 min 47 s 01 Petit bassin 50 m nl : 21 s 22 (RN)100 m nl : 46 s 54 (RN)200 m nl : 1 min ...

PROFILPELAJAR.COM