– by Alex, Nadeesha and Louis –

Choosen example reaction: ⁴⁵V(p,γ)⁴⁶Cr
⇒important for ⁴⁴Ti production rate in core collapse Supernova (Horoi et al. Nucl.Phys.A 718, (2003))
Beam: ⁴⁵V¹³⁺ of 135 MeV (1% enegry spread)

Command:

OV 1 1 0 ; {order 1, phase space dim 1 INCLUDING ENERGY, # of parameters 0}

Output:

Explanation:

R₁₁ = 1.0 is “Magnification”, i.e. no magnification provided with the dipole.
R₁₂ = 0.0 is “Focusing”, i.e. no focusing is provided by the dipole.
R₂₁ = 2.2 is approximate “Path length”,which is 0.5+0.78+1.0 meters
R₂₂ = 0.0 mean that output angles does not depend on input angles.
R₁₅ = -0.41 is ToF difference due to different initial position.
R₂₅ = -0.35 is ToF difference due to different initial angle.

Reaction summary for ⁴⁵V+p → γ+⁴⁶Cr, E_kin(⁴⁵V)=135 MeV
The maximum γ energy is 8.481 MeV. The minimum γ energy is 7.249 MeV.
The maximum ⁴⁶Cr energy is 132.634 MeV. The minimum ⁴⁶Cr energy is 131.402 MeV. The maximum ⁴⁶Cr angle is 0.13 degrees.

If we assume a normalized emittance of 0.6 π mm mrad provided from ReA3 at 3.0 MeV/u the geometric emittance is 0.6/0.08 = 7.5 π mm mrad.
With the spot size on the target, designed to be ±0.5 mm (1σ), the anglular spread then is ±15 mrad.
Adding the calculated spread from the reaction with the target of 0.13 deg = ±2.26 mrad we get ±17.26 mrad of angular spread.
⇒COSY beam input:

SB 0.0005 **0.01726** 0 0.0005 **0.01726** 0 0 0.0047 0 0 0 ;

Output for this beam passing the dipole:

The distance between the center of Q7 and the focal plane FP2 is 6.085 m as extracted from the script file.

*Bρ the magnetic rigidity was extracted from COSY by comand CONS(CHIM)

Fit result for parabola X_max² = σ₁₁ = a·K² - 2ab·K + ab² + c is:

a = 0.000362572706
b = 0.0834414129893
c = 3.506973286e-07

The consistency check for beam size calculation in Q7 shows:
X_calculated = 3.13mm, X_cosy_read_out = 3.16mm at the Q7 exit.
Emittance calculated from fit parameters ε= √(ac) / L² = 3.05E-7 π m rad = 0.305 π mm mrad.
Start beam emittance= XX · AX= 0.2 π mm mrad, so the emittance grows from the start to the focal point.
This emittance grow is possibly due to the optics aberrations. That is because the input COSY file does a 4th-order-calculation. Also the measured points don't fit the parabola ideally, i.e. the optics is non-linear.

The effective length of Q2 was reduced by 3% to 0.291m with the drift lenghts before and after increased by 0.0045m.
This led to a resolving power change from 445 to 102 for the standard particle of the script. As an educated guess we changed the gradient of Q2 by 3% to compensate the length reduction. The resolving power recoverd to ~400.

This Q2 gradient was then used as the starting value for the fit routine, that minimized the inverse of the resolving power. Additionally the gradients of Q3 and Q4 were set as free parameters of the fit, too. The best fit value after 38 steps was 463, wich is even 5% better! then the original resolving power.

Off course then one has to check if the max angle and energy acceptance is still met. The fit values for Q2,Q3 and Q4 stayed close to the starting values so one has to keep in mind that the fit routine might not have discovered a global minimum but only a local one.

Beam position. Longitudinal offset.

We populated “X-θ” phase-space with rays distributed on the border of ellipse by the loop:

LOOP PHI 0.0 360 5;
SRXX:= XX*COS(PHI*3.14159/180);
SRAX:= AX*SIN(PHI*3.14159/180);
ENDLOOP;

Rays in another planes are on-axis:

SRYY:= 0.0;
SRAY:= 0.0;
SRDE:= 0.0;

The rays now look like this (not all the rays are displayed):

Created loop for target position. Observed piece-wise curve for resolution vs target offset (meters) (see figure below) in case of 4th order calculation. If order reduced to the 1st, than curve is smooth, but maximum resolution is shifted. The resolution drops by 5% at the offsets (-5 mm, +7 mm).

Beam position. Transverse offset.
The resolution drops by 5% at the beam offsets (-0.8 mm, +0.5 mm). The offset is of the beam half-size order.

Beam size.
Change beam size with SCALING_FACTOR = (0.5..1.5):

SRXX:= XX*COS(PHI*3.14159/180)*SCALING_FACTOR

Resolution reasonably drops with larger beam size.

Dipole 4 position. Transvers offset

Dipole 4 position. Longitudinal offset

Dipole 4 position. Pitch

⁴⁵V+p → γ+⁴⁶Cr

Beam is set to 132MeV ⁴⁶Cr¹⁴⁺

RP 132.0 46*PARA(1) 14*PARA(2);
{Design acceptances, depends on kinematics of a specific reaction!}
XX:=0.0005;
AX:=0.01726;
YY:=0.0005;
AY:=0.01726;
DE:=0.0047;

Recoil beam fits the SECAR very well.

±1 charge state

Added extra rays:

{Define special rays for different charge states}
SR 0 -AX 0 0 0 0.0 0 1/14 2;
SR 0 0 0 0 0 0.0 0  1/14 2;
SR 0 AX 0 0 0 0.0 0 1/14 2;

SR 0 -AX 0 0 0 0.0 0 -1/14 3;
SR 0 0 0 0 0 0.0 0 -1/14 3;
SR 0 AX 0 0 0 0.0 0 -1/14 3;

{Beam 45V14+}
SR 0 -AX 0 0 0 3/132 -1/46 0/14 5;
SR 0 0 0 0 0 3/132 -1/46 0/14 5;
SR 0 AX 0 0 0 3/132 -1/46 0/14 5;

One can see how Wien filter cleans the recoils from the projectile beam at FP2.