194:. The beta function is typically adjusted to have a local minimum at such points (in order to minimize the beam size and thus maximise the interaction rate). Assuming that this point is in a drift space, one can show that the evolution of the beta function around the minimum point is given by:
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This implies that the smaller the beam size at the interaction point, the faster the rise of the beta function (and thus the beam size) when going away from the interaction point. In practice, the
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Typically, separate beta functions are used for two perpendicular directions in the plane transverse to the beam direction (e.g. horizontal and vertical directions).
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is a function related to the transverse size of the particle beam at the location s along the nominal beam trajectory.
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of the beam line elements (e.g. focusing magnets) around the interaction point limit how small beta star can be made.
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where z is the distance along the nominal beam direction from the minimum point.
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The value of the beta function at an interaction point is referred to as
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the beam is assumed to have a
Gaussian shape in the transverse direction
261:{\displaystyle \beta (z)=\beta ^{*}+{\dfrac {z^{2}}{\beta ^{*}}}}
76:{\displaystyle \sigma (s)={\sqrt {\epsilon \cdot \beta (s)}}}
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It is related to the transverse beam size as follows:
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108:is the location along the nominal beam trajectory
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297:"Introduction to Transverse Beam Optics"
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182:(also called Twiss parameters).
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135:{\displaystyle \sigma (s)}
180:Courant–Snyder parameters
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327:Accelerator physics
22:accelerator physics
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101:{\displaystyle s}
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18:beta function
303:. Retrieved
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282:References
251:∗
247:β
225:∗
221:β
205:β
192:beta star
186:Beta star
155:ϵ
121:σ
60:β
57:⋅
54:ϵ
37:σ
321:Category
276:aperture
144:Gaussian
305:24 June
85:where
300:(PDF)
307:2024
16:The
20:in
323::
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241:2
237:z
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217:=
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127:s
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49:=
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43:s
40:(
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