RAY TRACING TIIROUGH MAGNET SYSTEMS. E. A. Taylor. The Spectrometer Facilities Group (SFG) has written a simple Ray Tracing

Transcription

1 RAY TRACING TIIROUGH MAGNET SYSTEMS SLAC-TN December 1971 INTRODUCTION E. A. Taylor The Spectrometer Facilities Group (SFG) has written a simple Ray Tracing program which is intended as a supplement to TRANSPORT. The program allows the designer to trace the paths of relativistic charged particles through systemscomposed of any of the commonly used magnetic elements. The elements may be shifted or rolled with respect to the central axis in order to calculate the required positional accuracy. The algorithms used, in general, operate by doing a piecewise integration of the differential equation of motion, and provide a solution which is approximately true to all orders. The program MAGRAY is written in FORTRAN IV, with a few WATFIV "format- free' statements which allow a very convenient input. The program can be accessed and used on the WYIBUR terminal with the command "Use WYL.SX.EAT.MAGRAY on WYIQO~. Since the program is only about 150 statements in length, it is relatively easy to understand and to modify for the introduction of special effects, or the modification of the output format. A companion program ACCEPT is described in this same note. ACCEPT uses MAGRAY to calculate the solid angle acceptance of a magnet system. It requires additional input information on the apertures, or vacuum chamber sizes, and generates a map of the 8 acceptance, together with a record of which ele- ment limited the beam. DESCRIPTION OF MAGRAY INPUT Three types of input statements are expected. In order of their occurrence they are: 1. The "SYSTM" statement must have two numbers: N, the number of ele- ments in the system being examined, and PS, the momentum of the system, GeV/c.

2 -2-2. N "ELEMJWI" statements with three numbers each. The description of the elements is covered below. 39 Finally, an unlimited number of "RAY" statements. The initial parameters of each ray are described by 5 numbers: a. X in centimeters b. 8 or dx/dz in milliradians c. y in centimeters or dy/dz in milliradians e. p or difference in momentum from the system momentum, in percent. GEOMETRY The program uses a right hand coordinate system. Fig. 1 indicates the coordinates as seen by the particle, i.e. particle going into the paper, in the direction Z. Y RAY.. -._ FIG. 1 Coordinate System

3 -3- ELENEXL'S OF THE SYSTEM The elements will be described by Type, which is the first of three numbers for Type 1 each element. - Drift Drift has only one dimension, length,in meters. Thus a drift of 150 centimeters would have a statement The spaces between the numbers are required by the format-free input of WATFIV. The third number, 0, is a dummy. Type 2 - Quadrupole A quadrupole is described by length in meters, and a gradient, kilogauss/cm. In order to provide the maximum versatility to the program, a quadrupole may be rotated 45' from its usual orientation by giving it a negative length. Fig. 2 shows the sign convention for quadrupoles and also bending magnets and sextupoles. The polarity of the magnet assumes negative particles going into the paper. Examples: A quadrupole 50 cm long, required to be defocussing in the x-plane, with a field of I2 Kilogauss at 10 cm radius would have the statement

4 -4- I POSITIVE LENGTH (4 m NEGATIVE IENGTH (b) QUADRUPOIZ WITH POSITIVE GRADIENT POSITIVE IXNGTH NEGATIVE LENGTH (d),bendingg MAGNET WITH POSITJYE FIEID ,.^ POSITIVE LENGTH 5% I f?? (f) NEGATIVE LE!NGTH SMTUPOI8 WITH POSITIVE BETA FIG. 2 Sign Convention for Electrons going into the paper.

5 -5- Type 3 - Bending Magnets A Bending magnet requires a length in meters, and a field in kilogauss. As shown in Fig. 2, a positive length bends particles in the horizontal plane, and a negative length bends particles in the vertical plane. Example: A bending magnet 3 meters long with a field of 18 Kilogauss which bends negative particles upwards would have the statement: Type 4 - Sextupole A sextupole requires a length in meters and a Beta in Ki.logauss/cm2. The choice of positive or negative length is made by referring to Fig. 2. Example: A sextupole of type 2(f) but of opposite polarity is 0.5 meter long, and has a field of 3 Kilogauss at 10 cm rad. The Beta is 3/100 = 0.03 b,/cm2, and the statement reads: Type 5 - Rotated Pole Face If a bending magnet has its entrance face or its exit face other than perpendicular to the central ray, there is a focussing or defocussing effect nitude on all The only of the rays except the central ray. information required on the statement is the sign and mag- Pole Face angle. The convention for this is the same as in TRANSPORT. Fig. 3 Example of Positive Bl and Positive E!2

6 -6- In Fig. 3, Bl and B2 are both positive, and will have the effect of defocussing in the plane of the paper, and focussing in the trans- verse plane. A Rotated Pole Face statement must either precede or follow a Bending magnet statement. The third number is usually zero, as noted on page 7a. Example: Let the magnet in the example of Type 3 be a common rectangular magnet which bends 4.64, but is tipped so that its front face is lo inclined and its rear face The three statements are then: Type 6 - Rotation of axes In order to examine the effect of rolling a quadrupole or sextupole, the coordinate system is rolled in the opposite direction before the component and then back an equal amount afterwards. Roll of a Bending magnet is treated on page 7a. Positive angle of roll is defined in Fig. 1. Example: Suppose the quadrupole in the example of Type 2 is to be rolled lo ccw, i.e. top moved left as the beam sees it, These statements would then be o Type 7 - Translation of Axes The effect of a lateral error in placement of a quadrupole or

7 -7- sextupole is examined by moving the axes in the opposite direction and then back again after the component. The x movement is stated first then the y movement. Example: Suppose the quadrupole in the previous example be 0.1 cm high and 0.25 cm to the right. The statements are then: i i Type 8 - Delta Quadrupole The effect of a small quadrupole can be introduced at the ends of the components by this Type. It differs from a regular quadrupole only in that it is treated as an impulse function, changing angles but not the x or y, and its stated length is not added into the system. The magnitude of the impulse can be considered a product of Length X gradient. Example: Suppose a quadrupole impulse equal to 1% of the example of Ty-pe 2 be required. The statement would be: X2, or alternately, Type 9 - Delta Sextupole Sextupole impulses are introduced in the same manner as above. Thus a correction which is to be 1% of the example of Type 4 would require a statement:

8 -7a- ERRORS IN POSITION OF THE ELEMENTS QUADRUPOLES AND SEXTUPOLES Most of the errors in position of these elements can be described by change of DRIFT, Element Type 1, or ROTATION OF AXES, Element Type 6, or TRANSLATION OF AXES, Element Type 7. YAW and PITCH errors are not treated directly in the program, but can be approximated by translation of the center of the element, or if necessary, by rewriting that portion of the program which traces the ray through quadrupoles or sextupoles. BENDINGMAGNITS Since a Bending Magnet deflects the central ray, it must be treated differently. The most serious error that can occur in Bending Magnet position is ROLL. Small ROLL errors can be introduced by use of the third number of Element Type 5, Pole Face Rotation, which is normally a zero. The ROLL of a Bending Magnet following this element is given in milli- radians, a positive value indicating a clockwise rotation of the Bending Magnet, as seen by the beam. Example: Suppose the Bending Magnet in the example on page 6 is to have a 10 mr, counter-clockwise ROLL error. The three statements become: o If the Bending Magnet would otherwise not require Pole Face information, the statements are:

9 -8-20 GeV/c Spectrometer A good example of the usage of this program is the study of the 20 GeV Spectrometer. This system has never behaved according to the predictions of TRANSPORT, because the first bending magnet is asymmetric in order to allow a very close approach to the primary beam. It has a large notch in the iron, and its coils have all of the return loops on one side of the aperture. The magnet was measured for harmonic content, and this has been included by the DELTA QUAD, and DELTA SEXT elements. In addition, the first quad of the system was rolled counter clockwise 0.7 degrees early in the optics test in an effort to correct some of the aberrations. A portion of the output is reproduced. Fig. 4 is a listing of the first 13 elements, and Fig. 5 is the trace of ray No. 4 through the first 13 elements. The input for this portion of the system is: SYSTEM STATEMENT OOl ELEMENT STATEMENTS l 7 0 i

10 THIS PAGE INTEIWIONALLY LEFT BLANK.

11 EjLEiMEiNTSTATEMENTS o RAY STATEMEWZS 3 1 o 8-2

12 -lo- ELEMENTS OF THE SYSTEM. MOMENTUM = GEV/C TYPE B C go OUTPUT LISTING OF THE ELEMENTS FIG. 4

13 TRACEOFRAYNUMBER 4 X CM THETAMR Y CM PHIMR LGTHMETERS RAY P= -2.00% DRIFT DELTA QUAD DELTA SEXT vi POLE FACE.90 DEG DEG POLE FACE 4.30 DEG DRIFT g SEXT 3* DRIFT lo ROTATE 0.70 DEG lo QUAD lo I2 ROTATE DEG 4. g6o lo g DRIFT g OUTPUT TRACE OF ONE RAY FIG. 5

14 12- Description of ACCEPT In order to determine the solid angle acceptance of a magnet system, the size of the defining apertures must be added to the ELEMENTS of the system. For this purpose, each of the ELEMENT input, as described in MAGBAY, must have two additional numbers: These will be referred to as No. 4 and No. 5, the aperture at the downstream end of the element. No. 4. If this be zero, there is no check of the aperture at this point. Otherwise, the program checks that absolute (x) does not exceed this dimension in centimeters. No. 5. The program checks that absolute (y) does not exceed this dimension. If No. 5 be zero, No. 4 is treated as the radius of a circular aperture, and the ray coordinates are checked for interference If the aperture be asymmetric or irregular in shape, use Element No. 10. Example: The program, starting at statement 85, checks that x be between 3 and -10 and that y be between 4 and -5, all dimensions in centimeters. The RAY cards and ACCEPT calculates give a only the initial xo, yo, and p of a group of rays sort of polar map of 8 and fi acceptance of the system. aeration of ACCEPT Being given the xo, yo, and p of a bundle of rays, ACCEPT steps through combinations of 8 and 9, using MAGRAY to calculate the beam position after each element. It increases the angle in lmr steps until the ray fails to pass, and then backs up in 0.1 mr steps until it gets through each aperture. At that time it prints out the values of the passing 8 and $6, and the number of the element which stopped any larger beam. Sometimes x or y or p may be so large that even a beam parallel with the center line of the system will not get through. In this case a message is

15 -13- given and the program moves to the next group. If the user of this program requires a mapping of ray groups whose zero angle ray does not pass, the source program may be altered so that it can find and examine the special groups of passing rays. INPUT Input and output are best described by using the 20 GeV Spectrometer as an example. The number of elements is 36. The first several ELEMEXC statements are: , and so on The RAY cards now consist of "group" cards, specifying the initial x, y, and p: o 2 Fig. 6 is a portion of the output which lists the elements, and Fig. 7 gives the required solid angle data. The solid angle is presented in polar form: If you picture a vector rotating in 10' steps, then IUD is the maximum angle of divergence of the ray. It is decomposed into Q and fl by the relations

16 -14-8 = RAD * cos(deg) ld = RAD * sin(deg) Fig. 8 is a hand plot of the information contained in Fig. 7. ELEMENTS OF THE SYSTEM. MO- = GEV/c TYPE B C XAPERTURE YAPERTURE go FIG. 6

17 -15- LARGEST BEAMS PASSED f&d NUMBEFi OF LIMITING EIEMENT xo = 6.0 cm YO = 0.0 cm PO = 2.@ DEG RADMR THETAMR PHIMR E 2: i: I ;c ;; 33 ;z ; 5 ; : $ Z 6:40 f2: 6:80 78'2. 1 lo.go * :g; -6: :. ;: l.ll , *39 :;*zt: $7-7:oo FIG. 7

18 0 IbEdr~T t ~-.-.-_ bi= _ f?e5tf?1 ^- - Cl-ING.-_._--.--_... ELFNEWT -.--_.-.*. )- 4.k + -a - t -l - FIG, 8_