Salix alba: tidy up some formula and fix % bug in $use \% to avoid parse errors$

2014-10-28T17:47:58Z

tidy up some formula and fix % bug in <math> use \% to avoid parse errors

en>John of Reading: /* Industrial Agriculture Energy Flows */Typo fixing, replaced: is has → has using AWB

2013-04-18T08:33:24Z

Industrial Agriculture Energy Flows: Typo fixing, replaced: is has → has using AWB

← Older revision		Revision as of 10:33, 18 April 2013
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	My name is ~~Michale~~ and ~~I am studying English Literature~~ and ~~Physics at Munith~~ / ~~United States~~.<br><br>~~Here~~ is ~~my weblog kitchen home improvement~~ ([~~http:~~//~~www~~.~~homeimprovementdaily~~.~~com Full Review~~])		In [[accelerator physics]], the term '''acceleration voltage''' means the effective [[voltage]] surpassed by a charged [[particle]] along a defined straight line. If not specified further, the term is likely to refer to the ''longitudinal effective acceleration voltage'' <math>V_\parallel</math>.

			The acceleration voltage is an important quantity for the design of [[microwave cavity\|microwave cavities]] for [[particle accelerator]]s. See also [[shunt impedance]].

			For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. The following considerations are generalized for time-dependent fields.

			{{ambox
			\|image = [[File:Info.svg\|30px]]
			\|text = There are several '''variant definitions''' for the terms [[shunt impedance]] and [[acceleration voltage]] relating to transit time dependence.<ref name="lee">{{cite book \| url=http://books.google.com/?id=VTc8Sdld5S8C&printsec=frontcover&dq=Accelerator+physics#v=onepage&q=Accelerator%20physics&f=false \| title=Accelerator physics \| publisher=[[World Scientific]] \| date=2004 \| edition = 2nd \| last1 = Lee \| first1 = Shyh-Yuan \| isbn = 978-981-256-200-5 }}
			</ref><ref name="wangler">{{cite book
			\|title= RF Linear Accelerators
			\|last1= Wangler \|first1= Thomas \|authorlink1=
			\|coauthors=
			\|editor1-last= \|editor1-first= \| editor1-link=
			\|edition= 2nd
			\|date= 2008
			\|publisher= [[Wiley-VCH]]
			\|location=
			\|isbn= 978-3-527-62343-3
			\|pages=
			\|url= http://books.google.de/books?id=OJdgVI-UrikC&printsec=frontcover
			}} (slightly different notation)</ref> To clear this point, this page differentiates between ''effective'' (including transit time factor) and ''time-independent'' quantities.
			}}

			== Longitudinal voltage ==

			The longitudinal '''effective acceleration voltage''' is given by the kinetic energy gain experienced by a particle with velocity <math>\beta c</math> along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge,<ref name="wangler" />

			<math> V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s </math>.

			For resonant structures, e.g. [[Superconducting Radio Frequency\|SRF cavities]], this may be expressed as a [[Fourier transform\|Fourier integral]], because the fields <math>\vec E,\vec B</math>, and the resulting [[Lorentz force]] <math>\vec F_L</math>, are proportional to <math>\exp(i \omega t)</math> ([[eigenmode]]s)

			<math> V_\parallel(\beta) = \frac{1}{q} \vec e_s \cdot \int \vec F_L(s) \exp\left(i \frac{\omega}{\beta c} s\right)\,\mathrm d s = \frac{1}{q} \vec e_s \cdot \int \vec F_L(s) \exp\left(i k_\beta s\right)\,\mathrm d s </math> with <math>k_\beta = \frac{\omega}{\beta c} </math>

			Since the particles [[kinetic energy]] can only be changed by electric fields, this reduces to

			<math> V_\parallel(\beta) = \int E_s(s) \exp\left(i k_\beta s\right)\,\mathrm d s </math>

			=== Particle Phase considerations ===

			Note that by the given definition, <math>V_\parallel</math> is a [[complex number\|complex quantity]]. This is advantageous, since the relative phase between particle and the experienced field was fixed in the previous considerations (the particle travelling through <math>s=0</math> experienced maximum electric force).

			To account for this [[Degrees of freedom (physics and chemistry)\|degree of freedom]], an additional phase factor <math>\phi</math> is included in the [[eigenmode]] field definition

			<math> E_s(s,t) = E_s(s) \; \exp\left(i \omega t + i \phi \right) </math>

			which leads to a modified expression

			<math> V_\parallel(\beta) = e^{i \phi} \int E_s(s) \exp\left(i k_\beta s\right)\,\mathrm d s </math>

			for the voltage. In comparison to the former expression, only a phase factor with unit length occurs. Thus, the ''absolute value'' of the complex quantity <math> \| V_\parallel(\beta) \| </math> is independent of the particle-to-eigenmode phase <math>\phi</math>. It represents the maximum achievable voltage that is experienced by a particle with optimal phase to the applied field, and is the relevant physical quantity.

			=== Transit time factor ===

			A quantity named ''transit time factor''<ref name="wangler" />

			<math>T(\beta) = \frac{\|V_\parallel\|}{V_0} </math>

			is often defined which relates the effective acceleration voltage <math>V_\parallel(\beta)</math> to the '''time-independent acceleration voltage'''

			<math>V_0 = \int E(s)\,\mathrm d s</math>.

			In this notation, the effective acceleration voltage <math>\|V_\parallel\|</math> is often expressed as <math>V_0 T</math>.

			== Transverse voltage ==

			In symbolic analogy to the longitudinal voltage, one can define effective voltages in two orthogonal directions <math>x,y</math> that are transversal to the particle trajectory

			<math> V_{x,y} = \frac{1}{q} \vec e_{x,y} \cdot \int \vec F_L(s) \exp\left(i k_\beta s\right)\,\mathrm d s </math>

			which describe the integrated forces that deflect the particle from its design path. Since the modes that deflect particles may have arbitrary polarizations, the ''transverse effective voltage'' may be defined using polar notation by

			<math> V_\perp^2(\beta) = V_x^2 + V_y^2, \quad \alpha = \arctan \frac{\tilde V_y}{\tilde V_x} </math>

			with the ''polarization angle'' <math>\alpha</math>
			The tilde-marked variables are not absolute values, as one might expect, but can have positive or negative sign, to enable a range <math>[-\pi/2,+\pi/2]</math> for <math>\alpha</math>. For example, if <math>\tilde V_x = \| V_x \|</math> is defined, then <math>\tilde V_y = V_y \cdot \exp(-i \arg V_x) \in \mathbb R</math> must hold.

			Note that this transverse voltage does '''not''' necessarily relate to a real change in the particles energy, since magnetic fields are also able to deflect particles. Also, this is an approximation for small-angle deflection of the particle, where the particles trajectory through the field can still be approximated by a straight line.

			== References ==
			{{Reflist}}

			[[Category:Accelerator physics]]

en>FrescoBot: Bot: link syntax/spacing

2012-07-20T13:05:07Z

Bot: link syntax/spacing

New page

My name is Michale and I am studying English Literature and Physics at Munith / United States.<br><br>Here is my weblog kitchen home improvement ([http://www.homeimprovementdaily.com Full Review])

Ecometrics - Revision history

Salix alba: tidy up some formula and fix % bug in use \% to avoid parse errors

en>John of Reading: /* Industrial Agriculture Energy Flows */Typo fixing, replaced: is has → has using AWB

en>FrescoBot: Bot: link syntax/spacing

Salix alba: tidy up some formula and fix % bug in $use \% to avoid parse errors$