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274:). But if there is a space between them (and the length of this has been correctly chosen), the overall effect is focusing in both horizontal and vertical planes. A lattice can then be built up enabling the transport of the beam over long distances—for example round an entire ring. A common lattice is a FODO lattice consisting of a basis of a focusing quadrupole, 'nothing' (often a bending magnet), a defocusing quadrupole and another length of 'nothing'.
267:) which show that it is impossible for a quadrupole to focus in both planes at the same time. The image on the right shows an example of a quadrupole focusing in the vertical direction for a positively charged particle going into the image plane (forces above and below the center point towards the center) while defocusing in the horizontal direction (forces left and right of the center point away from the center).
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The simplest magnetic quadrupole is two identical bar magnets parallel to each other such that the north pole of one is next to the south of the other and vice versa. Such a configuration will have no dipole moment, and its field will decrease at large distances faster than that of a dipole. A
719:
911:{\displaystyle {\begin{aligned}{\vec {B}}_{\text{normal}}&=\left({\begin{matrix}K\cdot y,&K\cdot x,&0\end{matrix}}\right)\\{\vec {B}}_{\text{skew}}&=\left({\begin{matrix}J\cdot x,&-J\cdot y,&0\end{matrix}}\right)\\\end{aligned}}}
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The quadrupoles in the lattice are of two types: 'F quadrupoles' (which are horizontally focusing but vertically defocusing) and 'D quadrupoles' (which are vertically focusing but horizontally defocusing). This situation is due to the laws of
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Magnetic field lines of an idealized quadrupole field in the plane transverse to the nominal beam direction. The red arrows show the direction of the magnetic field while the blue arrows indicate the direction of the
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A charged particle beam in a quadrupole magnetic field will experience a focusing / defocusing force in the transverse direction. This focusing effect is summed up by a focusing strength
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The same equation will be true for the y direction, but with a minus sign in front of the focusing strength to account for the field changing directions.
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Steven M. Lund, Transverse
Particle Dynamics, US Particle Accelerator School (USPAS) Lectures on “Beam Physics with Intense SpaceCharge”
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If an F quadrupole and a D quadrupole are placed immediately next to each other, their fields completely cancel out (in accordance with
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determines whether (for a fixed particle charge and direction) the quadrupole focuses or defocuses particles in the horizontal plane.
714:{\displaystyle x^{\prime \prime }(z)+{\frac {\gamma ^{\prime }(z)}{\gamma ^{2}(z)\beta (z)}}x^{\prime }(z)+\kappa (z)x(z)=0}
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The components of the ideal magnetic field in the plane transverse to the beam are given by the following (see also
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501:{\displaystyle p=\gamma mv=\beta \gamma mc,\qquad \beta ={\frac {v}{c}},\quad \gamma =(1-\beta ^{2})^{-1/2}}
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of the field, the dipole terms cancel and where the lowest significant terms in the field equations are
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and two opposing magnetic south poles. The steel is magnetized by a large
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on a positive particle going into the image plane (away from the reader)
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238:. Therefore a 'lattice' of electromagnets is used to bend, steer and
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https://people.nscl.msu.edu/~lund/uspas/bpisc_2017/lec_set_02/tpd.pdf
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in the coils of tubing wrapped around the poles. Another design is a
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stronger version with very little external field involves using a
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and particles in the magnetic will behave according to the ODE
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is the field gradient of the normal quadrupole component and
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is the field gradient of the skew quadrupole component. The
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Equations of motion and focal length for charged particles
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layout but with the current in one of the coils reversed.
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Four bar magnets configured to produce a quadrupole
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46:. Quadrupole magnets are useful as they create a
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158:At the particle speeds reached in high energy
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990:{\displaystyle \mathrm {T} /\mathrm {m} }
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302:which depends on the quadrupole gradient
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1101:"Conventional Magnets for Accelerators"
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555:{\displaystyle \kappa ={\frac {G}{}}}
81:In some designs of quadrupoles using
515:. The focusing strength is given by
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242:a charged particle beam.
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1076:Quadrupole Magnetic Field
236:electrostatic deflection
89:pole tips: two opposing
295:{\displaystyle \kappa }
234:is more effective than
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16:Group of four magnets
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361:{\displaystyle =p/q}
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62:. This is used in
28:Quadrupole magnets
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1127:Quadrupole magnet
1125:Media related to
1010:{\displaystyle K}
954:{\displaystyle J}
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381:{\displaystyle q}
315:{\displaystyle G}
265:Maxwell equations
85:, there are four
30:, abbreviated as
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50:whose magnitude
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146:, are used to
136:electromagnets
109:Main article:
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99:Helmholtz coil
83:electromagnets
48:magnetic field
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121:storage ring
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134:Quadrupole
1137:Categories
1063:References
66:focusing.
44:quadrupole
883:⋅
877:−
866:⋅
836:→
803:⋅
789:⋅
759:→
682:κ
665:′
645:β
627:γ
610:′
606:γ
585:′
582:′
544:ρ
526:κ
483:−
469:β
465:−
453:γ
433:β
420:γ
417:β
405:γ
339:ρ
290:κ
230:and thus
204:×
54:with the
32:Q-magnets
1021:See also
368:, where
152:electron
963:SI unit
511:is the
142:of the
123:of the
36:magnets
921:where
766:normal
56:radial
1104:(PDF)
263:(the
240:focus
148:focus
140:linac
87:steel
843:skew
154:beam
150:the
60:axis
737:).
74:=3
1139::
166::
78:.
1106:.
1005:K
984:m
979:/
974:T
949:J
929:K
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894:0
889:,
886:y
880:J
872:,
869:x
863:J
856:(
852:=
833:B
821:)
814:0
809:,
806:x
800:K
795:,
792:y
786:K
779:(
775:=
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721:.
709:0
706:=
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700:z
697:(
694:x
691:)
688:z
685:(
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673:z
670:(
661:x
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651:z
648:(
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636:(
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615:(
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590:(
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562:,
547:]
541:B
538:[
534:G
529:=
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473:2
462:1
459:(
456:=
449:,
444:c
441:v
436:=
429:,
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423:m
414:=
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402:=
399:p
376:q
356:q
352:/
348:p
345:=
342:]
336:B
333:[
310:G
215:,
212:)
208:B
200:v
196:+
192:E
188:(
185:q
182:=
178:F
72:k
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