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18 1 NUMERICAL CALCULATION OF MOTION AND FIELDSwhere ¢))¡., ) & ¡.,¡., ) & ) ) ) w G¡.,¡., ) ) & ) ) w{R¡-,¡., ) & ) w¡.,¡-, ) ) ) & ) w¡.,¡., ) ) & ¡-,, and denotes differentiation at constant ¡-,yZz x VDXZY ¡-,XX¡., ) &*) ) w U¡.,¡-, ) ) ) ) & ¡.,¡-, ) ) &*) w¡-,¡., ) ) ) & ¡.,These derivatives ¢ VDXZYand ¡ VEXZYof the electric and magnetic fields are calculated from the vector fields ¢ ^g b‰bb‰g‰j ¢b g ^j ¡ ‰g>>b‰g‰j ¡b g ^>^ Rq>Tq- :T'PR ^ GPMR ^ U”q¡ Ž MMM&#uorqWs&*)¢ ¡ w) ) uo qWs& o q ¢Ws¢ )-x yZz|&#¡ ) ).x yZz w @) )¢ ) ) x yZz (1.2.20)¡ /) ) x yZz w U) ) uorq s&*) @ orq ¢s ) x yZz q ¢&#¢ ) x yz U ) ) )) )¢ ) ) x yZz o q¢ ) ) ) x yZzq oWs U o q ¢Wsq&*) ) ) ) /)ts¡ ) ) )Wx yz w*R&#, ¡ ¡-,and VDXZY B(1.2.21)&# VEXY x yZz q ¡.,¡&#VEXZY x yZz q ¡.,¢('§#"kS"e ,¡ §©"$Œ"[ and their derivativesj andj , following eqs. (1.2.8) and (1.2.13).1.2.4 Calculation of the time of flight^ §^ ^ § ^ ^ ^The time of flight eq. (1.2.6) involves the r (' derivatives* ¦'?> ,eq. (1.2.18). In the absence of electric field eq. (1.2.7) however reduces to the simple form>~¦', etc. that are obtained from(1.2.22) ;29732541.3 Calculation of ¡ and its Derivatives§©"$Œ"[ and derivatives are calculated in various ways, depending whether field maps or analytic representations of¡optical elements are used. The basic means are the following.1.3.1 Extrapolation from 1-D axial field map [5]A cylindrically symmetric field (e.g., using BREVOL) can be described by an axial 1-D field map of its longitudinalcomponent†\ > 7/‘ , while the radial component on axis ¡|’ §#"/ C\ is assumed to be zero. ¡ Ž §#"$is obtained at any point along the -axis by a polynomial interpolation from the map mesh (see section§#"$¡„Ž “\ § 1.4.1). §#"/ Then the field components §#"/ , at the position of the particle, §#"/ are obtained from Taylor ’ ¡ ¡}Žexpansions truncated at the fifth order insymmetry(hence, up to the fifth order derivative ^> ¡ ŽP ¡ Ž§#" \ ), assuming cylindrical^ § ¡ Ž §#" \ w§#"/\ O §#"¡ Ž T\ §#"¡ Ž M¡ Ž ¡ ’ ¡„Ž(1.3.1)^ §^ §By differentiation with respect § toFinally a conversion from the§§§^ ^ ând , up to the second order, these expressions provide the §#"/ derivatives of .coordinates to ¡ the Cartesian coordinates of zgoubi is performed, thus §#"kS"e §©"/§#"/ w§#" \ w@ ^§#" \ OG ^§#" \ providing the expressionsj needed in the eq. (1.2.8).1.3.2 Extrapolation from Median Plane Fields^ §^ ^ In the median §#"kS" \ plane, , and its derivatives with respect § to ¡}• or , may be calculated from analytical models(e.g. in Venus magnet - VENUS, and sharp edge multipoles SEXQUAD and QUADISEX) or numerically by polynomialinterpolation from 2-D field maps (e.g. CARTEMES, TOSCA).
4Ž ^ ˜ ¡§ ^"– ^ ˜ ¡ ^¡wTGThe scalar potential used for the calculation of ^¡@w>"To ^ G> ¡T ¡¡ T>• ^ ˜ ¡ ^‰g‰j ¡ bb¡ >s>X4b^T ¡T ¡T ^s>Psw^ww>˜P ¡4^¡ P>^>˜4¡ PsPq1.3 Calculation of ¡ and its Derivatives 19Median plane antisymmetry is assumed, which results in¡ Ž ¡ Ž wr¡ Ž (1.3.2)§#"kS" \ †\¡ – §#"kS" \ †\§#"kS"e §#"kS" §#"kS"e wr¡ – §#"kS" ¡|– §#"kS"e ¡ • §#"kS"¡|• Accomodated with Maxwell’s equations, this results in Taylor expansions below, for the three components of (here, ¡stands for ¡|• ¡ \ ) §©"$Œ" T ¡ Ž (1.3.3)§#"kS"e ô ^^ §^ §^ § ^ ¡|– §#"kS"e ^^ ^ §^ ^ ¡|• §#"kS"e ¡—w@ Ro*^> ô*^P @> ^which are then differentiated one by one with respect to § , , or , up to second or fourth order (depending on opticalelement or IORDRE option, see section 1.4.2) so as to get the expressions involved in eq. (1.2.8).^ §^ ^ §^ §^ ^ 1.3.3 Extrapolation from arbitrary 2-D Field Maps2-D field maps that §#"kS"e give the §#"$Œ"[ three components §#"kS"e , and at §©"$ each node of a-elevation map may be used. and its derivatives at any pointŽ ¡|– are calculated by polynomial interpolation¡¡ ¡|• §#"$Œ"[ followed by Taylor expansions in , without any hypothesis of symmetries (see section 1.4.3 and keywords MAP2D,MAP2D-E).1.3.4 Interpolation in 3-D Field Maps [6]In 3-D field maps ¡and its derivatives up to the second order with respect § to , , or are calculated by means of asecond order polynomial interpolation, from 3-D U 3-point grid (see section 1.4.4).U1.3.5 2-D Analytical Field Models and ExtrapolationSeveral optical elements such as BEND, WIENFILT (that uses the BEND procedures), QUADISEX, VENUS, etc., aredefined from the expression of the field and derivatives in the median plane. 3-D extrapolation of these off the median isdrawn from Taylor expansions.1.3.6 3-D Analytical Models of FieldsIn many optical elements such as QUADRUPO, SEXTUPOL, MULTIPOL, EBMULT, etc., the three components of ¡and their derivatives with respect § to , or are obtained at any step along trajectories from analytical expressionfollowing§#"kS"edrawn from the scalar potential ˜¡ Ž¡ Ž" etc. (1.3.4)> " ^" ^^ §Multipoles^ § ^ ^ ^ §electro-magnetic multipoles with poles (namely, QUADRUPO ( ž @to 10), EBMULT ( ž @Ÿž to 10)) is [7] q) to DODECAPO ( ž G), MULTIPOL ( ž j ( 8š›œ \to R ) in the case of magnetic and§#"kS"e ^ §^ ^
Page 1 and 2: CEA DSM DAPNIA-02-395 (2002)ZGOUBI
Page 3 and 4: Table of contentsPART A Description
Page 5: PART ADescription of software conte
Page 8 and 9: 8¤¥ PARTICUL Particle characteris
Page 10 and 11: 10ELECTROMAGNETIC ELEMENTSDecapoleD
Page 13 and 14: !!%" && 131 NUMERICAL CALCULATION
Page 15 and 16: 1.2 Integration of the Lorentz Equa
Page 17: 1.2 Integration of the Lorentz Equa
Page 21 and 22: in fourth order. The coefficients
Page 23 and 24: 4 ´ ´""4§> ´4 >4>1.4 Calculati
Page 25 and 26: A cylindrically symmetric field can
Page 27: Introducing also ¡The derivatives
Page 30 and 31: 30 3 SYNCHROTRON RADIATION¾ÍôVT
Page 32 and 33: 32 3 SYNCHROTRON RADIATIONÉ Éwher
Page 35 and 36: 354 DESCRIPTION OF THE AVAILABLE PR
Page 37 and 38: 4.2 Definition of an Object 37where
Page 39 and 40: 4.2 Definition of an Object 39\Pqî
Page 41 and 42: 4.2 Definition of an Object 41¹q
Page 43 and 44: 4.3 Declaration of options 43\P4.3
Page 45 and 46: 4.3 Declaration of options 45END or
Page 47 and 48: 4.3 Declaration of options 47· £
Page 49 and 50: 4.3 Declaration of options 49{ww{{q
Page 51 and 52: 4.3 Declaration of options 51GASCAT
Page 53 and 54: 4.3 Declaration of options 53·{N$
Page 55 and 56: 4.3 Declaration of options 55PARTIC
Page 57 and 58: 4.3 Declaration of options 57RESET:
Page 59 and 60: 4.3 Declaration of options 59if qî
Page 61 and 62: 4.3 Declaration of options 61P4©@4
Page 63 and 64: 4.3 Declaration of options 63:£¨
Page 65 and 66: u2 >04.4 Optical Elements and relat
Page 67 and 68: 4.4 Optical Elements and related nu
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4.5 Output Procedures 123¨ ¢ < ¤
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4.5 Output Procedures 1254·444"4HI
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4.5 Output Procedures 127PICKUPS: B
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4.5 Output Procedures 129wSPNPRNL,
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4.5 Output Procedures 131TWISS: Cal
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4.6 Complementary Features 133U@©@
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4.6 Complementary Features 135¢ww@
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Keywords and input data formatting
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146 Keywords and input data formatt
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Examples 231oÑwü¦INTRODUCTIONSev
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2331 MONTE CARLO IMAGES IN SPES 2Fi
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235********************************
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237zgoubi data file.800 MeV/c KAON
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2393 IN-FLIGHT DECAY IN SPES 3Figur
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241Excerpt of zgoubi output : histo
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243Excerpt of zgoubi output : first
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245zgoubi data file (begining and e
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247zgoubi data file.MICROBEAM LINE,
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2499 HISTO HISTOGRAHISTOGRAMME DE L
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253INTRODUCTIONThe basic zgoubi FOR
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REFERENCES 255References[1] F. Méo
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Indexacceleration, 56, 58, 74, 155,
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