en.wikipedia.org website review
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SEO Keyword summary for en.wikipedia.org/wiki/bremsstrahlung
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be displaystyle
Focus keyword
Short and long tail
Short Tail Keywords displaystyle left right |
long Tail Keywords (2 words) boldsymbol beta beta decay hbar comega physical review charged particle |
long Tail Keywords (3 words) due to bremsstrahlung dot boldsymbol beta conservation of energy physical society aps khn c ebert speed of light terrestrial gamma-ray flashes |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/bremsstrahlung page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
Page title
Title length
bremsstrahlung wikipedia
Meta description
Meta description legth
Meta description SEO
No meta relevance in the description detected !
Content SEO
Number of Words
Spam detected?
Headings
Heading distribution
Heading normalisation
Heading SEO impact
Emphasis (bold and italic)
Emphasis SEO impact
Images
Number of images
Images dimensions
Image alt descriptions
Images SEO impact
wikipedia free encyclopedia displaystyle bar qequiv epsilon pfrac gamma cleftdot beta frac leftboldsymbol cdot dot boldsymbol right textstyle mathbf sqrt varepsilon paparallel vfrac aequiv vdot paperp leftaperp vright leftaparallel egamma mpme approx dpdomega cfrac lefthat times leftlefthat righttimes rightright left hat domega dpaparallel vdomega sin theta cos clambda hnu jvnu over vgrm ffvnu jnu grm pepropto nrm mev eta zequiv zbar hbar equiv zll gtextffbornsqrt zgg gtextffclasssqrt leftln zeta rightgamma gtextffapprox lnbtextmaxbtextmin bmax bmin brm maxvnu min hmev max zrightright cpartial tinu nabla inu knu tmathbf bnu tefrac ehnu ktextbte omega exphbar krm bte tez etexth textpte yll ktextmax dpmathrm mec nine pequiv nee neni yfrac lambda textb ktextbttextezi etexthapprox textbhbar mtextektextbttexte ktextmaxpropto ltextc leftfrac ktextbteright textp yapprox lnyegamma yegamma beginalignedpmathrm int textpinfty ninektextmaxgytextp exgypfrac ytextpinfty dyy ytextp yright exytextpyomega textpendaligned gytextp pmathrm ninekrm btefrac gyrm mathrm temathrm textff tfrac neniz gtextb ktextbtemec ilambda dlambda kleftfrac rightfrac hcevapprox vmathrm pmkv beginalignedd sigma alpha textfine leftmathbf frightleftmathbf irightfrac idomega fdphi fleftefcleftmathbf frightcos fright ilefteicleftmathbf irightcos iright rightrightqquad imathbf fefcleftmathbf flefteicleftmathbf irightqquad irightleftmathbf frightsin isin fcos phi leftefcleftmathbf frightlefteicleftmathbf irightc irightleft rightrightendaligned textfineapprox etextkinif eif eifetextkinifmec efeihbar beginalignedtheta isphericalangle fsphericalangle fmathbf textangle between planes text endaligned beginaligneddomega dtheta fsin fendaligned beginalignedmathbf comega frightfrac frightleftcos icos rightendaligned vgg eiomega isum limits beginalignedi asqrt delta iln isqrt ileftdelta rightdelta cdelta pflefteicpicos ilefteicpicos pfdelta clefteicpicos irightleftdelta irightrighti acpflefteicpicos irightln efpfcefpfcrighti leftdelta efdelta pfcright itimes leftleftleftefpfcrightrightrightleftleft ileftefpfcrightleftdelta rightleftleftdelta pfcrightsqrt irightrightrightleftleftefpfcrightleft ileftefpfcrightrightrightleftleftleftdelta irightrightright irightleftef efpf rightpfcleft irightefpfcdelta rightrightleftdelta cleftdelta pfcrightpflefteicpicos pfcrightleftleftdelta ileftei ipfcleftdelta pfm pfcrightlefteicpicos irightleftleftdelta apfcleftdelta pfcrightleftdelta ifrac aleftdelta leftleftdelta irighttimes eicpicos efleft rightrightpfcleft irightdelta ileft pfc iefrighteicpicos rightleftdelta ileftleftdelta rightleftef efpfcrightrightleftdelta ileftfrac pfcdelta efrighteicpicos irighti ialefteicpicos irightendaligned beginalignedafrac idelta iendaligned icon edit wikidata wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/bremsstrahlung
Mobile rendering
Mobile optimizations
Responsive design detected (mobile css)
No flash detected !
Mobile improvement
Marketing / lead generation for en.wikipedia.org/wiki/bremsstrahlung
Social Media
Facebook shares | Facebook likes | ||
Facebook comments | Tweets | ||
Google +1 |
Conversion form
Search form
Analytics
Online presence
SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
bremsstrahlung found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
Favicon icon found?
Robots.txt found?
Sitemap found?
Navigation and internal links
Navigation
Url seperator
Human readable urls
Number of links
Link SEO Impact
statistics
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w view history
httpsenwikipediaorgwindexphptitlebremsstrahlungoldid1222599797
page information
printable version
radiative combination
radiative losses
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wiki read
particle physics
german
electromagnetic radiation
deceleration
charged particle
electron
atomic nucleus
kinetic energy
photon
law of conservation of energy
continuous spectrum
synchrotron radiation
relativistic particle
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sr
physical constants
evanescent
exponential integral
hartree energy
thermal de broglie wavelength
gaunt factor
cgs
optically thin
radiative cooling
nuclear fusion bremsstrahlung losses
xray tube
rhodium
kv
characteristic k lines
pm
duanehunt law
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characteristic xrays
kramers law
wavelength
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kev
conservation of energy
plancks constant
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elementary charge
picometres
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gamma decay
32p
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plastic
wood
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xmmnewton
rosat
exosat
suzaku
h ii regions
terrestrial gammaray flashes
pair production
finestructure constant
momenta
mass of an electron
absolute value
virtual photon
monte carlo codes
ebert u
beamstrahlung
wiggler synchrotron
freeelectron laser
history of xrays
landaupomeranchukmigdal effect
radiation length
synchrotron light source
merriamwebstercom dictionary
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issn
arxiv
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quantum electrodynamics
eulerheisenberg lagrangian
feynman diagram
guptableuler formalism
path integral formulation
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kleinnishina formula
landau pole
qed vacuum
selfenergy
schwinger limit
uehling potential
vacuum polarization
vertex function
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bhabha scattering
breitwheeler process
compton scattering
delbrck scattering
lamb shift
mller scattering
schwinger effect
photonphoton scattering
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SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
Item | Factor | Pointers | |
---|---|---|---|
PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 65% | A title should reflect the contents of a site. This site has a 50 % match | |
Title Length | 30% | Limit your title to anywhere between 40 and 70 characters. Your title was 27 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
Meta description relevance | 0% | Meta Description should reflect the contents of a site. This site has a 0 % match | |
Number of internal links | 30% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 261 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 4 level 1 folders and 7 folders above or in the first level of navigation. | |
Headings | 30% | Headers should reflect the contents of a site. This site has a 13 % match | |
Links | 10% | Link anchors should to some degree reflect the contents of a site. This site has a 5 % match | |
Image alt tags | 20% | Image alt tags should to some degree reflect the contents of a site. This site has a 7 % match | |
Bold and italic | 21% | Bold and italic tags should reflect the contents of a site to some degree. This site has a 7 % match | |
Html ratio | 30% | Try to keep the html / text ratio as low as possible. More html means longer loading times. Layout should be handled in a serpate css file | |
Image descriptions | 94% | 94.38202247191 % of all images have been described via the "alt" attribute. Describing images with relevant text may lead to better results in the search engines. | |
Page errors | 100% | Pages with no errors display significantly faster on most browsers. We detected 0 errors and warnings | |
WordCount | 20% | An ideal page contains between 400 and 600 words.This page contains 7770 words | |
Server response time | 100% | A fast server speeds up a website. This server responds 82.45% faster then average | |
Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 358 inline style declarations ( <a style="color:green">) with a size of 12507 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
Flash | 100% | Perfect, we detected no flash objects on your page | |
Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
How would you like to have SEO advice for all your pages ?? Start your SEO Dashboard and optimize your website!
en.wikipedia.org images and descriptions
152 images found at en.wikipedia.org Images can improve the user experience for a website by making a pag visually appealing Images can also add extra keyword relevance to a webpage by using alt tags. Images can also slow down a website. If the width and height for a picture is not specified for a browser know in advance how large the image is. A browser must first load the picture and see before it knows how much space should be on the page. Upon reservation In the meantime, the browser can do little but wait. When the height and width for the plate are given in the HTML code, a browser just continues to build for a page while the images load in the background.
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http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en.svg height: 13 width: 117 description: the free encyclopedia |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c7aa84ae51eb3a7f96fc2792ea71af84fbb014ee height: height attribute not set width: width attribute not set description: {\textstyle \gamma ={1}/{\sqrt {1-\beta ^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/acb0a8377db20e42274444cb181d51b5532b5844 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon _{0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/877675569ff6e6b7e81234da66a97f599aa0e92f height: height attribute not set width: width attribute not set description: {\displaystyle {\dot {\boldsymbol {\beta }}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/702cafc420cc00c54896f6d125112820956aaf6b height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\beta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e337624fba61e83c63d5cb122cf154e1199a3cd0 height: height attribute not set width: width attribute not set description: {\displaystyle p_{a\parallel v}={\frac {2{\bar {q}}^{2}a^{2}\gamma ^{6}}{3c^{3}}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4e88bdd726940af197a0d55eec9489c6f659b8e2 height: height attribute not set width: width attribute not set description: {\displaystyle a\equiv {\dot {v}}={\dot {\beta }}c} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3de061cf75dc837d2cc9f7b1946849fd481e72df height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\beta }}\cdot {\dot {\boldsymbol {\beta }}}=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/016e8059c1f7c3a915576968fb2691560ddff1ea height: height attribute not set width: width attribute not set description: {\displaystyle p_{a\perp v}={\frac {2{\bar {q}}^{2}a^{2}\gamma ^{4}}{3c^{3}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/29facd992f1457962cb0e25caba7fbcd0eebc4a9 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{4}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/978b0d02de464144a9cf9a2d5c36c5e66dae1719 height: height attribute not set width: width attribute not set description: {\displaystyle \left(a\perp v\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fca4eff5d399e7db87f43c7fb7322146fcd76541 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{6}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb64ed8dddf9eae577354af6e05482b77867b6a height: height attribute not set width: width attribute not set description: {\displaystyle \left(a\parallel v\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8da89dffbbabc8a3da04c57de204b0f8590ab99f height: height attribute not set width: width attribute not set description: {\displaystyle e=\gamma mc^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b height: height attribute not set width: width attribute not set description: {\displaystyle e} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7712cd0b9fa594676194f2fe5a03b413ffa5994a height: height attribute not set width: width attribute not set description: {\displaystyle m^{-4}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1c768b7b17fd067146251309fc07449ff84f96 height: height attribute not set width: width attribute not set description: {\displaystyle m^{-6}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8879ed5b3d9df86a3adaa6147f1393d146676b3f height: height attribute not set width: width attribute not set description: {\displaystyle (m_{p}/m_{e})^{4}\approx 10^{13}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9a77777df83cbdbf96f0eaa6e74ba90eba9ce43b height: height attribute not set width: width attribute not set description: {\displaystyle d\omega } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6827cf9b3ff176243f832d8978d17f617ccab7d5 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {dp_{a\parallel v}}{d\omega }}={\frac {{\bar {q}}^{2}a^{2}}{4\pi c^{3}}}{\frac {\sin ^{2}\theta }{(1-\beta \cos \theta )^{5}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/07ba5105b33e9ac88f605cc50663a9e2c16fcc63 height: height attribute not set width: width attribute not set description: {\displaystyle -e} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597 height: height attribute not set width: width attribute not set description: {\displaystyle v} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7620c533057ff764ac2077a003e4a7c14a0450e4 height: height attribute not set width: width attribute not set description: {\displaystyle ze} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/00a9fbcb730da9d55a40e864e1fe71313ad1a6fa height: height attribute not set width: width attribute not set description: {\displaystyle \nu =c/\lambda } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc23768f43b26085f80e1882a94b31d24abd653 height: height attribute not set width: width attribute not set description: {\displaystyle h\nu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e7d311d128ce6679e564de93f758612a8a645a height: height attribute not set width: width attribute not set description: {\displaystyle j(v,\nu )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/39ec522da9516d3ad6fe1628adbbad092ebb4f21 height: height attribute not set width: width attribute not set description: {\displaystyle j(v,\nu )={8\pi \over 3{\sqrt {3}}}{z^{2}{\bar {e}}^{6}n_{i} \over c^{3}m_{e}^{2}v}g_{\rm {ff}}(v,\nu )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/af3f02144c696c16ad34b539c2fae269c85f914e height: height attribute not set width: width attribute not set description: {\displaystyle h\nu >mv^{2}/2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/71f116eb997fffd16b00be0a403669909a6a0d33 height: height attribute not set width: width attribute not set description: {\displaystyle g_{\rm {ff}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5e21303faca6e2167f4ecbf2a75aed982e817035 height: height attribute not set width: width attribute not set description: {\displaystyle n_{e}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d50f2c47583fc8729acf6329ed927dfb649e102b height: height attribute not set width: width attribute not set description: {\displaystyle \nu <\nu _{\rm {pe}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee472341c24388a61f4800ad731dffdea61fdfb height: height attribute not set width: width attribute not set description: {\displaystyle h\nu \ll m_{e}v^{2}/2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d8bd5b123e875836b60ce7674a5fe216d21714f2 height: height attribute not set width: width attribute not set description: {\displaystyle \eta _{z}\equiv z{\bar {e}}^{2}/\hbar v} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3893302ce82aeed8e0a4f6c51a3a2b7834e0b7ef height: height attribute not set width: width attribute not set description: {\displaystyle \eta _{\nu }\equiv h\nu /2m_{e}v^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f958b8d9590165395ebce95c4f6d8e2957eaa2 height: height attribute not set width: width attribute not set description: {\displaystyle \eta _{z}\ll 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/641df1ebb03ca4bcb95166eb3908a1a80d2005ba height: height attribute not set width: width attribute not set description: {\displaystyle g_{\text{ff,born}}={{\sqrt {3}} \over \pi }\ln {1 \over \eta _{\nu }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c54fb0758db2065c8e67fe8c7c8578326b8febbe height: height attribute not set width: width attribute not set description: {\displaystyle \eta _{z}\gg 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3c67f130c76cc8dfecceab3f87a2fb4ec7b7538e height: height attribute not set width: width attribute not set description: {\displaystyle g_{\text{ff,class}}={{\sqrt {3}} \over \pi }\left[\ln \left({1 \over \eta _{z}\eta _{\nu }}\right)-\gamma \right]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6d98d2d06a6fc0b8dcabb6608e42b48c943b4967 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma \approx 0.577} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/209de21935d669fa198d17d5de2746742f893de8 height: height attribute not set width: width attribute not set description: {\displaystyle 1/\eta _{z}\eta _{\nu }=m_{e}v^{3}/\pi z{\bar {e}}^{2}\nu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a height: height attribute not set width: width attribute not set description: {\displaystyle h} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a358f8a6fd6669b5623223c8d54123fd6ff66d33 height: height attribute not set width: width attribute not set description: {\displaystyle g_{\text{ff}}\approx \ln(b_{\text{max}}/b_{\text{min}})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/780d4316592868b8ac4a8172428a9517f82d68cd height: height attribute not set width: width attribute not set description: {\displaystyle b_{\max }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/240193543e442fc05c07db68f361859222a9ab85 height: height attribute not set width: width attribute not set description: {\displaystyle b_{\min }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/caad71c2e8b47ad0935e1e8f9a7b9cee936dfb69 height: height attribute not set width: width attribute not set description: {\displaystyle b_{\rm {max}}=v/\nu } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3164cb09e2894535364b4429eef5f399e6910cfc height: height attribute not set width: width attribute not set description: {\displaystyle b_{\rm {min}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8e51520e9b958afdb12ae5a22f307c1999c593c6 height: height attribute not set width: width attribute not set description: {\displaystyle \approx h/m_{e}v} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eb402ad4d6b9f99c4e0f8ff6176fa51643997a55 height: height attribute not set width: width attribute not set description: {\displaystyle \approx {\bar {e}}^{2}/m_{e}v^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/06ce030f6dd8faeeea66451c37a456612edf1b3e height: height attribute not set width: width attribute not set description: {\displaystyle g_{\text{ff}}\approx \max \left[1,{{\sqrt {3}} \over \pi }\ln \left[{1 \over \eta _{\nu }\max(1,e^{\gamma }\eta _{z})}\right]\right]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/24a28ef09f8bf34c86afb43699cad45cb886b4df height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{c}}\partial _{t}i_{\nu }+{\hat {\mathbf {n} }}\cdot \nabla i_{\nu }=j_{\nu }-k_{\nu }i_{\nu }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aee60e45df36ffe5f79eec4e8f7a0ed6d13fcccb height: height attribute not set width: width attribute not set description: {\displaystyle i_{\nu }(t,\mathbf {x} )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7118a0b30c7fbe680079f309ec8d84eebc88a2 height: height attribute not set width: width attribute not set description: {\displaystyle j_{\nu }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/00e8613c72ed7a01a3cceb7be082bf13a3a0fffe height: height attribute not set width: width attribute not set description: {\displaystyle k_{\nu }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/172f1a90292ff1160dabfbc32fd0dcb802fc9f98 height: height attribute not set width: width attribute not set description: {\displaystyle i_{\nu }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e0479695ec724d0a9b579cfe4bbb7ea7329ca3 height: height attribute not set width: width attribute not set description: {\displaystyle i_{\nu }={j_{\nu } \over k_{\nu }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/abeef5ac6573610b8562756124da12de32a424f9 height: height attribute not set width: width attribute not set description: {\displaystyle b_{\nu }(\nu ,t_{e})={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /k_{\text{b}}t_{e}}-1}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4bdd97b9d4696d5539d959bbeee47ed5c8e9a9b3 height: height attribute not set width: width attribute not set description: {\displaystyle j_{\nu }/k_{\nu }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5adf70167568c9a3308031f02f687fa2881c56 height: height attribute not set width: width attribute not set description: {\displaystyle \hbar \omega \ll k_{\text{b}}t_{e}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/63a32fd1246a76e773daea983062368f82076c05 height: height attribute not set width: width attribute not set description: {\displaystyle \exp(-\hbar \omega /k_{\rm {b}}t_{e})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f87552ff818f5dc07fa849381462a1e525c7a3e6 height: height attribute not set width: width attribute not set description: {\displaystyle \hbar \omega /k_{\text{b}}t_{e}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d height: height attribute not set width: width attribute not set description: {\displaystyle y} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/bremsstrahlung_power2.svg/220px-bremsstrahlung_power2.svg.png height: 166 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8 height: height attribute not set width: width attribute not set description: {\displaystyle \omega } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5a3f7abaf27a4ac7748bb5bf151a8e68195eb4 height: height attribute not set width: width attribute not set description: {\displaystyle \omega =\omega _{\rm {p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6c2ee71532c63684ff81bf8440a847c38f4ff3 height: height attribute not set width: width attribute not set description: {\displaystyle t_{e}>z^{2}e_{\text{h}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/539dc2a2b8a3fc24953af0735148a01d49e47589 height: height attribute not set width: width attribute not set description: {\displaystyle \hbar \omega _{\text{p}}/t_{e}=0.1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a01066df6224b3021216d6b70f1b36230d3ff2c6 height: height attribute not set width: width attribute not set description: {\displaystyle e_{1}(y)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9931b1c7a8ec8a93b96cb306d353503acb2f3766 height: height attribute not set width: width attribute not set description: {\displaystyle y\ll 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2da948a47335f1df10cd68df06a2ce21624b73 height: height attribute not set width: width attribute not set description: {\displaystyle k_{\text{max}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2db970f307b2a8900a1e3644dcec78d60e3a8a9b height: height attribute not set width: width attribute not set description: {\displaystyle t_{e}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/057444bf35a0c22b19bcae1ef06e06ecdf8abe56 height: height attribute not set width: width attribute not set description: {\displaystyle 4\pi } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/af9e01555da98e6ea2d40728b7af84b128e92189 height: height attribute not set width: width attribute not set description: {\displaystyle {dp_{\mathrm {br} } \over d\omega }={\frac {8{\sqrt {2}}}{3{\sqrt {\pi }}}}{{\bar {e}}^{6} \over (m_{e}c^{2})^{3/2}}\left[1-{\omega _{\rm {p}}^{2} \over \omega ^{2}}\right]^{1/2}{z_{i}^{2}n_{i}n_{e} \over (k_{\rm {b}}t_{e})^{1/2}}e_{1}(y),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/447c04c9f7ce11f7d3838e3bcd60b8d7adbabdf9 height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{p}\equiv (n_{e}e^{2}/\varepsilon _{0}m_{e})^{1/2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6e049b31d24bd13c5bb2accdf21fd78b327297 height: height attribute not set width: width attribute not set description: {\displaystyle n_{e},n_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b886a57ba05271a9c30079fb70700e2403019175 height: height attribute not set width: width attribute not set description: {\displaystyle \omega <\omega _{\rm {p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fd956644c353edcc41a96c8e8fc169a72695ed61 height: height attribute not set width: width attribute not set description: {\displaystyle \omega >\omega _{\rm {p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26 height: height attribute not set width: width attribute not set description: {\displaystyle e_{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c717fbda8ddc84fdc819f8d6bd9091a53800c6a0 height: height attribute not set width: width attribute not set description: {\displaystyle y={\frac {1}{2}}{\omega ^{2}m_{e} \over k_{\text{max}}^{2}k_{\text{b}}t_{e}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5086600f48abfc2eb74e1c33510f5248c029f4aa height: height attribute not set width: width attribute not set description: {\displaystyle k_{\text{max}}=1/\lambda _{\text{b}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/468d4c15cf690de3ea6649f8a29b7a3360b0c1a6 height: height attribute not set width: width attribute not set description: {\displaystyle k_{\text{b}}t_{\text{e}}>z_{i}^{2}e_{\text{h}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/510a9690dc81b9025ff56554fa41013c531c292b height: height attribute not set width: width attribute not set description: {\displaystyle e_{\text{h}}\approx 27.2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0333402983c85e173c967f64e119d2390693eb94 height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{\text{b}}=\hbar /(m_{\text{e}}k_{\text{b}}t_{\text{e}})^{1/2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd16b57446ec6e77c088714ae3476ef0f856d2b height: height attribute not set width: width attribute not set description: {\displaystyle k_{\text{max}}\propto 1/l_{\text{c}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3ef6c147a5cbe5bbebf0fc197c627b3b817d97 height: height attribute not set width: width attribute not set description: {\displaystyle l_{\text{c}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/80fbddc74794a53a6d8b918fd09ebed5a37858b9 height: height attribute not set width: width attribute not set description: {\displaystyle k_{m}=1/\lambda _{b}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed17e2869e4929478a4803f6626cdeafb77c4d2 height: height attribute not set width: width attribute not set description: {\displaystyle y={\frac {1}{2}}\left[{\frac {\hbar \omega }{k_{\text{b}}t_{e}}}\right]^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/47ef5e40b20f2a545201645ec10877b037ffde88 height: height attribute not set width: width attribute not set description: {\displaystyle dp_{\mathrm {br} }/d\omega } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/37f3b7b1f3a0e56cdc507deffe29cc3f694ba4e9 height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{\text{p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f63a1d62fc262690e6ae46bf9b72413d56b0268d height: height attribute not set width: width attribute not set description: {\displaystyle e_{1}(y)\approx -\ln[ye^{\gamma }]+o(y)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb07d9b66bddd596c52abe343c6d5580d699b99 height: height attribute not set width: width attribute not set description: {\displaystyle y>e^{-\gamma }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a79d8a8091948c22638f2472f242550b955e4a38 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p_{\mathrm {br} }&=\int _{\omega _{\text{p}}}^{\infty }d\omega {\frac {dp_{\mathrm {br} }}{d\omega }}={\frac {16}{3}}{\frac {{\bar {e}}^{6}}{m_{e}^{2}c^{3}}}z_{i}^{2}n_{i}n_{e}k_{\text{max}}g(y_{\text{p}})\\[1ex]g(y_{p})&={\frac {1}{2{\sqrt {\pi }}}}\int _{y_{\text{p}}}^{\infty }dy\,y^{-{1}/{2}}\left[1-{y_{\text{p}} \over y}\right]^{1/2}e_{1}(y)\\[1ex]y_{\text{p}}&=y({\omega \!=\!\omega _{\text{p}}})\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea7647ce6ebd21ab7e88ba8f3932a5af892871c height: height attribute not set width: width attribute not set description: {\displaystyle g(y_{\text{p}}=0)=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7edaa31a75018f1956d3cc584de4ee4174f1a17e height: height attribute not set width: width attribute not set description: {\displaystyle y_{\text{p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a69b1668a011d291027ea9d8a102db5e28461522 height: height attribute not set width: width attribute not set description: {\displaystyle p_{\mathrm {br} }={16 \over 3}{{\bar {e}}^{6} \over (m_{e}c^{2})^{\frac {3}{2}}\hbar }z_{i}^{2}n_{i}n_{e}(k_{\rm {b}}t_{e})^{\frac {1}{2}}g(y_{\rm {p}})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41 height: height attribute not set width: width attribute not set description: {\displaystyle \hbar } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef235ee7b92bebd8a7d9aca160a48be26af3bc0 height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{\rm {b}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b5b4ca2325d82a5c09a94f7ae297b2a09f1e5e height: height attribute not set width: width attribute not set description: {\displaystyle g=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7d19ffec2c37abd0a4eb2ba79dc0413d0cb49798 height: height attribute not set width: width attribute not set description: {\displaystyle p_{\mathrm {br} }[\mathrm {w/m^{3}} ]={z_{i}^{2}n_{i}n_{e} \over \left[7.69\times 10^{18}\mathrm {m^{-3}} \right]^{2}}t_{e}[\mathrm {ev} ]^{\frac {1}{2}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7da7a78740ccf246ffce86fb3aa6c216ecc135 height: height attribute not set width: width attribute not set description: {\displaystyle g_{\rm {b}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f6dfaa938b9d9b63a566268348d6d590a153bb8f height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon _{\text{ff}}=1.4\times 10^{-27}t^{\frac {1}{2}}n_{e}n_{i}z^{2}g_{\text{b}},\,} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/5/54/brem_cross_section-en.svg/220px-brem_cross_section-en.svg.png height: 143 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a7aa99e80e4a971f8341cd0095517e6dc3562e37 height: height attribute not set width: width attribute not set description: {\displaystyle k_{\text{b}}t_{e}/m_{e}c^{2}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/tubespectrum.jpg/220px-tubespectrum.jpg height: 150 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f height: height attribute not set width: width attribute not set description: {\displaystyle i} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a height: height attribute not set width: width attribute not set description: {\displaystyle \lambda } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d21f3e49b0cb106094a850e5be7eff8394e778ee height: height attribute not set width: width attribute not set description: {\displaystyle i(\lambda )\,d\lambda =k\left({\frac {\lambda }{\lambda _{\min }}}-1\right){\frac {d\lambda }{\lambda ^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/82c24522483ceaf1d54224b69af4244b60c3ac08 height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{\min }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/97fe1a9666bc3561a3f170aa551846352d00e918 height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{\min }={\frac {hc}{ev}}\approx {\frac {1239.8}{v}}\,\mathrm {pm/kv} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc8315012b95031411647ab791f8805b79e5fe0 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}d^{4}\sigma ={}&{\frac {z^{2}\alpha _{\text{fine}}^{3}\hbar ^{2}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {d\omega }{\omega }}{\frac {d\omega _{i}\,d\omega _{f}\,d\phi }{\left|\mathbf {q} \right|^{4}}}\\&{}\times \left[{\frac {\mathbf {p} _{f}^{2}\sin ^{2}\theta _{f}}{\left(e_{f}-c\left|\mathbf {p} _{f}\right|\cos \theta _{f}\right)^{2}}}\left(4e_{i}^{2}-c^{2}\mathbf {q} ^{2}\right)+{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\theta _{i}}{\left(e_{i}-c\left|\mathbf {p} _{i}\right|\cos \theta _{i}\right)^{2}}}\left(4e_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right.\\&{}\qquad +2\hbar ^{2}\omega ^{2}{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\theta _{i}+\mathbf {p} _{f}^{2}\sin ^{2}\theta _{f}}{(e_{f}-c\left|\mathbf {p} _{f}\right|\cos \theta _{f})\left(e_{i}-c\left|\mathbf {p} _{i}\right|\cos \theta _{i}\right)}}\\&{}\qquad -2\left.{\frac {\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\sin \theta _{i}\sin \theta _{f}\cos \phi }{\left(e_{f}-c\left|\mathbf {p} _{f}\right|\cos \theta _{f}\right)\left(e_{i}-c\left|\mathbf {p} _{i}\right|c1\cos \theta _{i}\right)}}\left(2e_{i}^{2}+2e_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right],\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd height: height attribute not set width: width attribute not set description: {\displaystyle z} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/160e8cc12a4fe316fcfd6ba8f0f2021062177dcc height: height attribute not set width: width attribute not set description: {\displaystyle \alpha _{\text{fine}}\approx 1/137} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455 height: height attribute not set width: width attribute not set description: {\displaystyle c} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f862f79de9371ec3aee2f6f305021de275e8681f height: height attribute not set width: width attribute not set description: {\displaystyle e_{{\text{kin}},i/f}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e2b3c228801d7bb862ab6dd7f0d795358d030e height: height attribute not set width: width attribute not set description: {\displaystyle e_{i,f}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f85d8dd4b800c85a51c74f0da12052ddc75dae6e height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {p} _{i,f}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d89ab83462dd6747dfd9d063fcbc88a3018b2f05 height: height attribute not set width: width attribute not set description: {\displaystyle e_{i,f}=e_{{\text{kin}},i/f}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+\mathbf {p} _{i,f}^{2}c^{2}}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/98cd4e222a0d3abdd059719e5cb7af7741f07377 height: height attribute not set width: width attribute not set description: {\displaystyle e_{f}=e_{i}-\hbar \omega ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/257e7f4e184cd5ca0743d3e3cc9b0f0f025dce11 height: height attribute not set width: width attribute not set description: {\displaystyle \hbar \omega } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dacc57c01e566a67071ebe2ef8b39bc2f47056fa height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\theta _{i}&=\sphericalangle (\mathbf {p} _{i},\mathbf {k} ),\\\theta _{f}&=\sphericalangle (\mathbf {p} _{f},\mathbf {k} ),\\\phi &={\text{angle between the planes }}(\mathbf {p} _{i},\mathbf {k} ){\text{ and }}(\mathbf {p} _{f},\mathbf {k} ),\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {k} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9f65ce264fbd20b24afd9bddb477056eb149a5 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}d\omega _{i}&=\sin \theta _{i}\ d\theta _{i},\\d\omega _{f}&=\sin \theta _{f}\ d\theta _{f}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4402e07cdb4ba5f630c98650a857a16283210e9c height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}-\mathbf {q} ^{2}={}&-\left|\mathbf {p} _{i}\right|^{2}-\left|\mathbf {p} _{f}\right|^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2\left|\mathbf {p} _{i}\right|{\frac {\hbar }{c}}\omega \cos \theta _{i}-2\left|\mathbf {p} _{f}\right|{\frac {\hbar }{c}}\omega \cos \theta _{f}\\&{}+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\left(\cos \theta _{f}\cos \theta _{i}+\sin \theta _{f}\sin \theta _{i}\cos \phi \right).\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bef287571211f9becbf9a01640f89268b42684 height: height attribute not set width: width attribute not set description: {\displaystyle v\gg {\frac {zc}{137}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24 height: height attribute not set width: width attribute not set description: {\displaystyle \phi } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2e703f1d307e506ce0fc81eddc18244daaba2516 height: height attribute not set width: width attribute not set description: {\displaystyle \theta _{f}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/37ee8c9e12a06e9b3da60d063cd972c652d2d104 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {d^{2}\sigma (e_{i},\omega ,\theta _{i})}{d\omega \,d\omega _{i}}}=\sum \limits _{j=1}^{6}i_{j}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f3020405a454cd99ff1a872e9d1503a4e831ec20 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}i_{1}={}&{\frac {2\pi a}{\sqrt {\delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}}\ln \left({\frac {\delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}-{\sqrt {\delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}\left(\delta _{1}+\delta _{2}\right)+\delta _{1}\delta _{2}}{-\delta _{2}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}-{\sqrt {\delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}\left(\delta _{1}-\delta _{2}\right)+\delta _{1}\delta _{2}}}\right)\\&{}\times \left[1+{\frac {c\delta _{2}}{p_{f}\left(e_{i}-cp_{i}\cos \theta _{i}\right)}}-{\frac {p_{i}^{2}c^{2}\sin ^{2}\theta _{i}}{\left(e_{i}-cp_{i}\cos \theta _{i}\right)^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{f}\delta _{2}}{c\left(e_{i}-cp_{i}\cos \theta _{i}\right)\left(\delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)}}\right],\\i_{2}={}&-{\frac {2\pi ac}{p_{f}\left(e_{i}-cp_{i}\cos \theta _{i}\right)}}\ln \left({\frac {e_{f}+p_{f}c}{e_{f}-p_{f}c}}\right),\\i_{3}={}&{\frac {2\pi a}{\sqrt {\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)^{4}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}}\times \ln \left[\left(\left[e_{f}+p_{f}c\right]\right.\right.\\&\left.\left[4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\left(e_{f}-p_{f}c\right)+\left(\delta _{1}+\delta _{2}\right)\left(\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]-{\sqrt {\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}\right)\right]\right)\\&\left[\left(e_{f}-p_{f}c\right)\left(4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\left[-e_{f}-p_{f}c\right]\right.\right.\\&{}+\left.\left.\left(\delta _{1}-\delta _{2}\right)\left(\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]-{\sqrt {\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}\right]\right)\right]^{-1}\\&{}\times \left[-{\frac {\left(\delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)\left(e_{f}^{3}+e_{f}p_{f}^{2}c^{2}\right)+p_{f}c\left(2\left[\delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right]e_{f}p_{f}c+\delta _{1}\delta _{2}\left[3e_{f}^{2}+p_{f}^{2}c^{2}\right]\right)}{\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}\right.\\&{}-{\frac {c\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)}{p_{f}\left(e_{i}-cp_{i}\cos \theta _{i}\right)}}-{\frac {4e_{i}^{2}p_{f}^{2}\left(2\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]^{2}-4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)\left(\delta _{1}e_{f}+\delta _{2}p_{f}c\right)}{\left(\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)^{2}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}m^{2}c^{4}\sin ^{2}\theta _{i}\left(e_{i}^{2}+e_{f}^{2}\right)-2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\theta _{i}p_{f}c\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)+2\hbar ^{2}\omega ^{2}p_{f}m^{2}c^{3}\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)}{\left(e_{i}-cp_{i}\cos \theta _{i}\right)\left(\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)}}\right],\\i_{4}={}&{}-{\frac {4\pi ap_{f}c\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)}{\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}-{\frac {16\pi e_{i}^{2}p_{f}^{2}a\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)^{2}}{\left(\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)^{2}}},\\i_{5}={}&{\frac {4\pi a}{\left(-\delta _{2}^{2}+\delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)\left(\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)}}\\&{}\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{f}^{2}}{e_{i}-cp_{i}\cos \theta _{i}}}\right.\\&{}\times {\frac {e_{f}\left(2\delta _{2}^{2}\left[\delta _{2}^{2}-\delta _{1}^{2}\right]+8p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\left[\delta _{2}^{2}+\delta _{1}^{2}\right]\right)+p_{f}c\left(2\delta _{1}\delta _{2}\left[\delta _{2}^{2}-\delta _{1}^{2}\right]+16\delta _{1}\delta _{2}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)}{\delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}\\&{}+{\frac {2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\theta _{i}\left(2\delta _{1}\delta _{2}p_{f}c+2\delta _{2}^{2}e_{f}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}e_{f}\right)}{e_{i}-cp_{i}\cos \theta _{i}}}\\&{}+{\frac {2e_{i}^{2}p_{f}^{2}\left(2\left[\delta _{2}^{2}-\delta _{1}^{2}\right]\left[\delta _{2}e_{f}+\delta _{1}p_{f}c\right]^{2}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\left[\left(\delta _{1}^{2}+\delta _{2}^{2}\right)\left(e_{f}^{2}+p_{f}^{2}c^{2}\right)+4\delta _{1}\delta _{2}e_{f}p_{f}c\right]\right)}{\left(\delta _{2}e_{f}+\delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\left(e_{i}^{2}+e_{f}^{2}\right)\left(\delta _{2}p_{f}c+\delta _{1}e_{f}\right)}{e_{i}-cp_{i}\cos \theta _{i}}}\right],\\i_{6}={}&{\frac {16\pi e_{f}^{2}p_{i}^{2}\sin ^{2}\theta _{i}a}{\left(e_{i}-cp_{i}\cos \theta _{i}\right)^{2}\left(-\delta _{2}^{2}+\delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\theta _{i}\right)}},\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9a74c7390371e3b25f147a4127a59f7d2de79468 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}a&={\frac {z^{2}\alpha _{\text{fine}}^{3}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {\hbar ^{2}}{\omega }}\\\delta _{1}&=-\mathbf {p} _{i}^{2}-\mathbf {p} _{f}^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{i}\right|\cos \theta _{i},\\\delta _{2}&=-2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{f}\right|+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\cos \theta _{i}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cfdedbecbb7f9adaad0a019a1466a836696d551e height: height attribute not set width: width attribute not set description: {\displaystyle z^{2}} |
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