Contents

## The Lecture Series 1st Lecture Motivation Transport of a ray Equivalence of Transport of ONE Ray Û Ellipse

Danza, Maria, Host has reference to this Academic Journal, PHwiki organized this Journal An Introduction to Ion-Optics Series of Five Lectures JINA, University of Notre Dame Sept. 30 Dec. 9, 2005 Georg P. Berg The Lecture Series 1st Lecture: 9/30/05, 2:00 pm: Definitions, Formalism, Examples 2nd Lecture: 10/7/05, 2:00 pm: Ion-optical elements, properties & design 3rd Lecture: 10/14/05, 2:00 pm: Real World Ion-optical Systems 4th Lecture: 12/2/05, 2:00 pm: Separator Systems 5th Lecture: 12/9/05, 2:00 pm: Demonstration of Codes (TRANSPORT, COSY, MagNet) 1st Lecture Motivation, references, remarks (4 – 8) The driving as long as ces (9) Definitions & first order as long as malism (10 – 16) Phase space ellipse, emittance, examples (17 – 25) Taylor expansion, higher orders (26 – 27) The power of diagnostics (28 – 30) Q & A 1st Lecture: 9/30/05, 2:00 3:30 pm: Definitions, Formalism, Examples

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Motivation Manipulate charged particles (b+/-, ions, like p,d,a, ) Beam lines systems Magnetic & electric analysis/ separation (e.g. St. George) Acceleration of ions Who needs ion-optics anyway Over 6109 people have – I hope so – happy lives without! A group of accelerator physicists are using it to build machines that enables physicists to explore the unkown! Many physicists using accelerators, beam lines in addition to magnet system (or their data) needs some knowledge of ion-optics. This lecture series is an introduction to the last group in addition to I will do my best to let you in on the basics first in addition to than we will discuss some of the applications of ion-optics in addition to related topics. Introductory remarks Introduction as long as physicists Focus on ion-optical definitions, in addition to tools that are useful as long as physicist at the NSL & future users of St. George recoil separator. Light optics can hardly be discussed without lenses & optical instruments ion-optics requires knowledge of ion-optical elements. Analogy between Light Optics in addition to Ion-Optics is useful but limited. Ion-optical & magnet design tools needed to underst in addition to electro-magnet systems. Ion-optics is not even 100 years old in addition to (still) less intuitive than optics developed since several hundred years Galileo Telescope 1609 Optics in Siderus Nuncius 1610 Historical remarks: Ruther as long as d, 1911, discovery of atomic nucleus

Basic tools of the trade Geometry, drawing tools, CAD drafting program (e.g. AutoCad) Linear Algebra (Matrix calculations), first order ion-optics (e.g. TRANSPORT) Higher order ion-optics code to solve equation of motion, (e.g. COSY Infinity, GIOS, RAYTRACE (historic) Electro-magnetic field program (solution of Maxwells Equations), (e.g. finite element (FE) codes, 2d & 3d: POISSON, TOSCA, MagNet) Properties of incoming charged particles in addition to design function of electro-magnetic facility, beam, reaction products (e.g. kinematic codes, charge distributions of heavy ions, energy losses in targets) Many other specialized programs, e.g as long as accelerator design (e.g. synchrotrons, cyclotrons) not covered in this lecure series. Literature Optics of Charged Particles, Hermann Wollnik, Academic Press, Orl in addition to o, 1987 The Optics of Charged Particle Beams, David.C Carey, Harwood Academic Publishers, New York 1987 Accelerator Physics, S.Y. Lee, World Scientific Publishing, Singapore, 1999 TRANSPORT, A Computer Program as long as Designing Charged Particle Beam Transport Systems, K.L. Brown, D.C. Carey, Ch. Iselin, F. Rotacker, Report CERN 80-04, Geneva, 1980 Computer-Aided Design in Magnetics, D.A. Lowther, P. Silvester, Springer 1985 Ions in static or quasi-static electro-magnetic fields Lorentz Force q = electric charge B = magn. induction E = electric field v = velocity For momentum analysis the magnetic as long as ce is preferred because the as long as ce is always perpendicular to B. There as long as e v, p in addition to E are constant. Force in magnetic dipole B = const: p = q B r p = mv = momentum = bending radius Br = magn. rigidity For ion acceleration electric as long as ces are used. Dipole field B perpendicular to paper plane Radius r Object (size x0) General rule: Scaling of magnetic system in the linear region results in the same ion-optics Note: Dispersion dx/dp used in magnetic analysis, e.g. Spectrometers, magn. Separators, x p p+dp (1)

Definition of BEAM as long as mathematical as long as mulation of ion-optics What is a beam, what shapes it, how do we know its properties Beam parameters, the long list Beam rays in addition to distributions Beam line elements, paraxial lin. approx. higher orders in spectrometers System of diagnostic instruments Not to as long as get: Atomic charge Q Number of particles n Defining a RAY Ion-optical element Code TRANSPORT: (x, Q, y, F, 1, dp/p) (1, 2, 3, 4, 5, 6 ) Convenient easy to use program as long as beam lines with paraxial beams Code: COSY Infinity: (x, a, y, b, l, dK, dm, dz) Needed as long as complex ion-optical systems including several charge states different masses velocities (e.g. Wien Filter) higher order corrections Not defined in the figure are: dK = dK/K = rel. energy dm = dm/m = rel. energy dz = dq/q = rel. charge change a = px/p0 b = py/p0 All parameters are relative to central ray properties Not defined in the figure are: dp/p = rel. momentum l = beam pulse length All parameters are relative to central ray central ray Note: Notations in the Literature is not consistent! Sorry, neither will I be. TRANSPORT Coordinate System Ray at initial Location 1

Transport of a ray Ray at initial Location 0 Ray after element at Location t 6×6 Matrix representing optic element (first order) Note: We are not building r in addition to om optical elements. Many matrix elements = 0 because of symmetries, e.g. mid-plane symmetry (2) TRANSPORT matrices of a Drift in addition to a Quadrupole For reference of TRANSPORT code in addition to as long as malism: K.L. Brown, F. Rothacker, D.C. Carey, in addition to Ch. Iselin, TRANSPORT: A computer program as long as designing charged particle beam transport systems, SLAC-91, Rev. 2, UC-28 (I/A), also: CERN 80-04 Super Proton Synchrotron Division, 18 March 1980, Geneva, Manual plus Appendices available on Webpage: ftp://ftp.psi.ch/psi/transport.beam/CERN-80-04/ David. C. Carey, The optics of Charged Particle Beams, 1987, Hardwood Academic Publ. GmbH, Chur Switzerl in addition to Transport of a ray though a system of beam line elements Ray at initial Location 0 (e.g. a target) Ray at final Location n 6×6 Matrix representing first optic element (usually a Drift) xn = Rn Rn-1 R0 x0 Complete system is represented by one Matrix Rsystem = Rn Rn-1 R0 (3) (4)

Geometrical interpretation of some TRANSPORT matrix elements Wollnik, p. 16 Focusing Function (xa) Wollnik = dx/dQ physical meaning = (xQ) RAYTRACE = R12 TRANSPORT Achromatic system: R16 = R26 = 0 Defining a BEAM (Ellipse Area = p(det s)1/2 Emittance e is constant as long as fixed energy & conservative as long as ces (Liouvilles Theorem) Note: e shrinks (increases) with acceleration (deceleration); Dissipative as long as ces: e increases in gases; electron, stochastic, laser cooling Attention: Space charge effects occur when the particle density is high, so that particles repel each other Warning: This is a mathematical abstraction of a beam: It is your responsibility to verify it applies to your beam 2 dimensional cut x-Q is shown e = Ö s11s22 – (s12) 2 “““““` Emittance (5) Equivalence of Transport of ONE Ray Û Ellipse Defining the s Matrix representing a Beam

The 2-dimensional case ( x, Q ) Ellipse Area = p(det s)1/2 Emittance e = det s is constant as long as fixed energy & conservative as long as ces (Liouvilles Theorem) Note: e shrinks (increases) with acceleration (deceleration); Dissipative as long as ces: e increases in gases; electron, stochastic, laser cooling 2 dimensional cut x-Q is shown s = æs11 s21 ü ès21 s22 þ Real, pos. definite symmetric s Matrix s-1 = 1/e2 æs22 -s21 ü è-s21 s11 þ Inverse Matrix ss-1 = æ1 0 ü è 0 1þ Exercise 1: Show that: = I (Unity Matrix) 2-dim. Coord.vectors (point in phase space) X = X T = (x Q) æx ü èQþ Ellipse in Matrix notation: X T s-1 X = 1 Exercise 2: Show that Matrix notation is equivalent to known Ellipse equation: s22 x2 – 2s21 x Q + s11Q2 = e2 (6) Courant-Snyder Notation s = æs11 s21 ü ès21 s22 þ In their famous Theory of the Alternating Synchrotron Courant in addition to Snyder used a Different notation of the s Matrix Elements, that are used in the Accelerator Literature. For you r future venture into accelerator physics here is the relationship between the s matrix in addition to the betatron amplitue functions a, b, g or Courant Snyder parameters æ b -a ü è-a g þ = e Transport of 6-dim s Matrix Consider the 6-dim. ray vector in TRANSPORT: X = (x, Q, y, F, l, dp/p) Ray X0 from location 0 is transported by a 6 x 6 Matrix R to location 1 by: X1 = RX0 Note: R maybe a matrix representing a complex system (3) is : R = Rn Rn-1 R0 Ellipsoid in Matrix notation (6), generized to e.g. 6-dim. using s Matrix: X0 T s0-1 X0 = 1 Inserting Unity Matrix I = RR-1 in equ. (6) it follows X0 T (RTRT-1) s0-1 (R-1 R) X0 = 1 from which we derive (RX0)T (Rs0 RT)-1 (RX0) = 1 (7) (9) (8) The equation of the new ellipsoid after trans as long as mation becomes X1 T s1-1 X1 = 1 where s1 = Rs0 RT (10) (6) Conclusion: Knowing the TRANSPORT matrix R that transports one ray through an ion-optical system using (7) we can now also transport the phase space ellipse describing the initial beam using (10)

The transport of rays in addition to phase ellipses in a Drift in addition to focusing Quadrupole, Lens 2. 3. Matching of emittance in addition to acceptance Lens 2 Lens 3 Focus Focus Increase of Emittance e due to degrader Focus A degrader / target increases the emittance e due to multiple scattering. The emittance growth is minimal when the degrader in positioned in a focus As can be seen from the schematic drawing of the horizontal x-Theta Phase space. as long as back-of-the-envelop discussions! Emittance e measurement by tuning a quadrupole Lee, p. 55 The emittance e is an important parameter of a beam. It can be measured as shown below. s11 (1 + s12 L/ s11 – L g) + (eL)2/s22 xmax = Exercise 3: In the accelerator reference book s22 is printed as s11 Verify which is correct ¶Bz/¶x l Br g = (Quadr. field strength l = eff. field length) L = Distance between quadrupole in addition to beam profile monitor Take minimum 3 measurements of xmax(g) in addition to determine Emittance e (11) (12)

Emittance e measurement by moving viewer method The emittance e can also be measured in a drift space as shown below. s11 + 2 L1 s12 + L1 2 s22 (xmax(V2))2 = L = Distances between viewers ( beam profile monitors) ¾½¾¾¾½¾¾¾¾½¾® L1 L2 Viewer V1 V2 V3 Beam (xmax(V3))2 = s11 + 2 (L1 + L2)s12 + (L1 + L2 )2 s22 where s11 = (xmax(V1))2 e = Ö “““““` s11s22 – (s12) 2 Emittance: Discuss practical aspects No ellipse no e Phase space! (13) (14) Taylor expansion Linear (1st order)TRANSPORT Matrix Rnm ,l Note: Several notations are in use as long as 6 dim. ray vector & matrix elements. Remarks: Midplane symmetry of magnets reason as long as many matrix element = 0 Linear approx. as long as well designed magnets in addition to paraxial beams TRANSPORT code calculates 2nd order by including Tmno elements explicitly TRANSPORT as long as malism is not suitable to calculate higher order ( >2 ). TRANSPORT RAYTRACE Notation Rnm = (nm) (15) Solving the equations of Motion Methods of solving the equation of motion: 1) Determine the TRANSPORT matrix. 2) Code RAYTRACE slices the system in small sections along the z-axis in addition to integrates numerically the particle ray through the system. 3) Code COSY Infinity uses Differential Algebraic techniques to arbitrary orders using matrix representation as long as fast calculations (16)

Discussion of Diagnostic Elements Some problems: Range < 1 to > 1012 particles/s Interference with beam, notably at low energies Cost can be very high Signal may not represent beam properties (e.g. blind viewer spot) Some solutions: Viewers, scintillators, quartz with CCD readout Slits (movable) Faraday cups (current readout) Harps, electronic readout, semi- transparent Film (permanent record, dosimetry, e.g. in Proton Therapy) Wire chambers (Spectrometer) Faint beam 1012 ® 103 (Cyclotrons: MSU, RCNP, iThemba) Diagnostics in focal plane of spectrometer Typical in focal plane of Modern Spectrometers: Two position sensitive Detectors: Horizontal: X1, X2 Vertical: Y1, Y2 Fast plastic scintillators: Particle identification Time-of-Flight Measurement with IUCF K600 Spectrometer illustrates from top to bottom: focus near, down- stream in addition to upstream of X1 detector, respectively IUCF, K600 Spectrometer Higher order beam aberrations Detector X1 X2 3 rays in focal plane 1. 2. 1. 2. 3. Example Octupole (S-shape in x-Q plane Other Example: Sextupole T122 C-shape in x-Q plot 3. T1222 T126

Q & A Question now ASK! Any topic you want to hear in addition to I havent talked about Let me know! End Lecture 1

## Danza, Maria Host

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