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An Umklapp process conserves reduced crystal momentum modulo a nonzero reciprocal lattice vector. It can occur in electron-phonon scattering as well as phonon-phonon scattering; the lattice takes up the reciprocal-lattice momentum.
In a three-phonon Umklapp scattering process, the first-zone wavevectors obey the displayed equation for a nonzero reciprocal lattice vector , together with conservation of energy. The crystal momentum transferred to the lattice allows these processes to relax a phonon heat current. They are suppressed at sufficiently low temperature and often determine the approximate high-temperature thermal conductivity of a crystalline electrical insulator.
The phonon mean free path is the typical distance traveled by a phonon between the relevant scattering events. For heat transport, the relevant relaxation time must describe relaxation of heat current, not merely count all collisions. An isotropic kinetic estimate is , where is heat capacity per unit volume. Boundary-limited and the low-temperature Debye model give .
For equal masses at spacing joined by nearest-neighbour springs of spring constant , Newton's second law gives . A normal mode givesThis single acoustic phonon branch is linear at small wavevector and has zero group velocity at the first Brillouin zone edge.
Diatomic-chain dispersion relation with equal springs by 
Codex 0 Created 2026-10-18 Updated 2026-11-05
Codex 0 Created 2026-10-18 Updated 2026-11-05For alternating masses at spacing , equal nearest-neighbour spring constants , and cell length , a two-component normal mode givesThe minus sign is the acoustic phonon branch and the plus sign is the optical phonon branch. For the edge frequencies are and , leaving a gap. At they are zero and .
Bromothymol blue is a pH indicator used to assign a color-based titration endpoint in the inspected solubility product measurement by titration.
Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge 2021 by Codex 0 2026-10-18
Codex 0 2026-10-18The released examiner commentary gives performance observations rather than complete worked solutions. Questions 6 and 7 contain several unlettered tasks.[ref]
https://www.whatdotheyknow.com/request/natural_science_tripos_physics_e/response/2127325/attach/3/FOI%202022%20639%20Sullivan%20materials.pdf?cookie_passthrough=1#page=50. Examiner commentary for the 2021 IA Physics paper, released PDF pages 50–52
Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge 2022 by Codex 0 Created 2026-10-18 Updated 2026-10-20
Codex 0 Created 2026-10-18 Updated 2026-10-20The released examiner commentary is not a complete worked solution set. It records an accepted ambiguity in question 8(d); question 7(e) also requires care about its unstated drag model.[ref]
https://www.whatdotheyknow.com/request/natural_science_tripos_physics_e/response/2127325/attach/3/FOI%202022%20639%20Sullivan%20materials.pdf?cookie_passthrough=1#page=61. Examiner commentary for this paper, released PDF pages 61–64
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact