For the best separation, the beam spot at FPF4 has to be minimized, which can be performed by reducing the first and second order geometric (primarily hori-zontal) and chromatic aberrations in the focal planes via fitting of the multipole strengths. All multipoles of the preseparator (12 quadrupoles, 10 sextupoles and 4 octupoles) can be tuned to achieve optimal settings for theBρrange from 2 to 20 Tm and maintain the first order ion-optical layout described in [62].
Besides reducing the aberrations and maintaining the first order layout of the separator, some transfer map elements were deliberately fitted with an off-set to improve the overall transmission. The offoff-sets used in this work were determined by the GSI Super-FRS group. The element(a|δδ)at FPF2 was fit-ted to(0.0003/δ2ma x)in order to reduce the maximal current of the sextupole FPF2KS11 (see Fig. 6.1) for a rigidity of 20 Tm. Although the last offset is only required for 20 Tm, it was kept for all rigidities in order to provide always the same fit conditions. All objective transfer map elements, their desired values and the transversal planes of their acquisition are listed in Tab. 6.1. The hor-izontal beam width inside the dipoles had to be kept constant by controlling the corresponding transfer map elements in the planes before the first dipole P0 and after the last dipole P1. This was required for preserving the first order resolving power at FPF2
R1,F P F2= (x|δ)/((x|x)∆xi)≈ −2.6/(1.65∆xi), (6.1) which corresponds top/∆p=1576for∆xi=1mm.
Optimal multipole settings were obtained with the help of the multiparamet-ric fit-procedure in COSY via the Levenberg-Marquardt algorithm [63]. Since the maximal order of the multipole elements is three, the fitting was performed using transfer maps truncated up to 3rdorder.
There is an alternative to truncating the transfer maps to 3rdorder for the fitting. It is generating 3rdorder maps from 3rdorder polynomial representa-tion of the magnetic field, based on a least squares fit (further such maps will be named LS maps). In Fig. 6.3 the integral non-uniformity of the Super-FRS
68 6 Application: Super-FRS preseparator optics
Table 6.1:Transfer map elements used as objectives for the optimization of the settings of the Super-FRS preseparator. The elements without description are optimized for reducing aberrations. The units of a transfer map element (zjf|zki)correspond to a fraction with the units ofzjfin the numerator and the units ofzkiin the denominator. If several quantities are placed after the ”|”
symbol, the denominator is equal to the product of all corresponding units.
(x,a,y,b,l,δ)have the units of (m, rad, m, rad, m, 1).
Plane Element Desired value Description
P0 (x|a) 0.172 Value to preserve 1storder resolution P0 (b|b) 0 Lattice parameter.
FPF1 (x|a) 0 FPF2 (x|a) 0
FPF2 (y|b) 0.05 An offset for better transmission FPF2 (y|δ) 0
FPF2 (x|aa) 0 FPF2 (x|aδ) 0 FPF2 (x|δδ) 0
FPF2 (a|δδ) 0.0003/δ2ma x An offset to reduce max. current in FPF2KS11.
FPF2 (y|bδ) 0
FPF2 (x|δδδ) 0
FPF3 (b|b) 0.03 An offset for better transmission
P1 (x|δ) 0
P1 (a|δ) 0
FPF4 (x|a) 0
FPF4 (y|b) 0.0012 An offset for better transmission FPF4 (x|x) 2 Value to preserve 1storder resolution FPF4 (x|aa) 0
FPF4 (x|aδ) 0 FPF4 (x|δδ) 0 FPF4 (a|δδ) 0 FPF4 (y|bδ) 0 FPF4 (x|aaa) 0 FPF4 (x|δδδ) 20∆pma x
δ2ma x
An offset for better transmission
6.3 Fitting the preseparators optics 69
preseparator dipole together with its 3rdorder least squares fit and its 3rdorder Taylor expansion around the pointx=0is shown. It is obvious that fitting of the preseparators optics will lead to different optima for the truncated transfer maps and the LS maps.
−20 −15 −10 −5 0 5 10 15 20
x
, cm
−2 0 2 4 6 8
Int eg ral no n-u nif orm ity , 1 0
−4Original curve
3
rdorder least squares fit 3
rdorder Taylor e pansion
Figure 6.3:The integral non-uniformity of the magnetic field of the NC Super-FRS preseparator dipole forI=575A together with its 3rdorder Taylor expan-sion and least squares fit.
In Fig. 6.4 the phase space distributions generated with 13thorder optics and optima obtained using LS as well as truncated maps are shown. Depending on the particular experimental application either 3rdorder transfer maps can be used for the fitting. If it is important to capture a maximal number of particles from the whole phase volume, the best option would be to perform the fitting using LS maps. Although the resulting horizontal phase space is convex for smallxanda(Fig. 6.4a)), the fit conditions lead to the best confinement of the whole beam. Controversially, if only a small fraction of the phase space is required, the best option would be to use truncated high order maps for the fitting. Indeed, in Fig. 6.4b)the area with high density in the middle of the spot is slightly more upright and narrow than in Fig. 6.4a).
70 6 Application: Super-FRS preseparator optics
Figure 6.4:The transverse horizontal phase space distribution in the FPF4, gen-erated using 13th order computation. The optimal multipole strengths were obtained with dipole maps, generated using a 3rdorder least squaresB field representationa), and with dipole transfer maps of 3rd order, obtained via truncation of 13thorder mapsb).
Discussion of the optimal multipole settings
The dependencies of the optimal multipole strengths in preseparator optics on Bρ(Fig. 6.5) have very similar shapes compared to the corresponding integral non-uniformities in the dipole magnet field distribution (Fig. 6.6) although with different sign to compensate the effect of the dipoles. The magnets chosen for comparison are labeled as in Fig. 6.1. The curve for octupole FPF3KO13 in Fig. 6.5 has a deviating shape, which is likely influenced by the vertical octupole component of the dipole and by thevertical fit conditions. The vertical octupole component is not shown here but it has a similar characteristic as the horizontal sextupole component in Fig. 6.6 with a different sign.
In Tab. 6.2 and 6.3 the mean integral multipole components of the dipole and the discussed preseparator multipoles normalized with respect toBρare presented, respectively. From these numbers it can be deduced that the mean quadrupole component of the dipole is about as large as 4% of the quadrupole strengths of FPF1QT13 and FPF4QT11, the mean sextupole component of the dipole is 16% of FPF1KS11 and the mean octupole component of the dipole is 12% of FPF2KO11. These values roughly explain the correspondence between the relative change spreads in Figs. 6.5 and 6.6.
6.3 Fitting the preseparators optics 71
−0.015 0.000 0.015 0.030 0.045
Rel. change, %
a)
FPF1QT13FPF4QT110 4 8 12
Rel. change, %
b)
FPF1KS11FPF4KS112 4 6 8 10 12 14 16 18 20 Bρ, Tm
−1.6
−1.2
−0.8
−0.40.00.4
Rel. change, %
c)
FPF2KO11 FPF3KO13
Figure 6.5:Relative change of the optimal multipole strengths ver-sus the particle rigidity Bρ for two quadrupoles a), two sex-tupolesb)and two octupolesc) labeled in Fig. 6.1.
−0.4 0.0 0.4
Rel. change, %
a)
Quadrupole
80 0 80
Rel. change, %
b)
Sextupole
2 4 6 8 10 12 14 16 18 20 Bρ, Tm
20 0 20
Rel. change, %
c)
OctupoleFigure 6.6: Relative change of the dipoles integral horizon-tal non-uniformities of a) 1st (quadrupole), b) 2nd (sextupole) and c) 3rd (octupole) orders versus the particle rigidity.
72 6 Application: Super-FRS preseparator optics
Table 6.2: Mean integral multipole components of the Super-FRS dipole normalized with respect to the mag-netic rigidity.
Quadrupole 0.015 m−1 Sextupole -0.003 m−2 Octupole 0.027 m−3
Table 6.3: Mean integral multipole strengths for selected Super-FRS quadrupoles, sextupoles and oc-tupoles normalized with respect to the magnetic rigidity.
FPF1QT13 0.356 m−1 FPF4QT11 0.353 m−1 FPF1KS11 -0.0206 m−2 FPF4KS11 -0.0172 m−2 FPF2KO11 -0.146 m−3 FPF3KO13 -0.0923 m−3