Split-Pole Spectrograph

Annotated picture of the SE-SPS, An plain picture is here : File:SPS Magnet.png

SE-SPS COSY simulation. An plain picture is here File:SPS Sketch With Cosy.png

The Super Enge Split-Pole Spectrograph is a magnet spectrometer to measure the spectrum of nuclear reactions. The concept and design were developed by Harald A. Enge at 1967, aimed to have a broad-momentum range spectrograph with $p_{max}/p_{min}\approx 2.8$ or $E_{max}/E_{min}\approx 8$. The spectrometer was originally located at the Wright Nuclear Structure Laboratory (closed at 2013), at Yale University. It was moved to FSU in the fall of 2013. It consists of a reaction chamber, a split-pole magnetic spectrograph, a position-sensitive ionization drift chamber, and a plastic scintillator. It has an angular acceptance of 128 msr (vertical ±40 mrad, horizontal ±80 mrad). The maximum B-field is 1.63 T with a radius of curvature from 511 mm to 920 mm. The mean radius is 600 mm. The advantage of the split-pole instead of a single-pole magnet is the aberration (x|θ²) and (x|φ²) are almost zero .

The Super Enge Split-Pole Spectrograph is an upgrade of the Yale Enge SPS. The major change is the redesign of the backward silicon detector array to the SABRE.

SPS Experiment Guide

Media:SPS_Experiment_Guide.pdf

SPS Operating Procedures

I created this section as a place to store procedures for the chamber swaps, however, I expect there are other things we might want to document here. -p

• Target Chamber Swaps

Online Simulator

https://fsunuc.physics.fsu.edu/SPS/

There was another simulator called websps developed by Gordon. But I am unable to revive it after the server upgraded from Ubuntu 20.04 to Ubuntu 22.04, so I write a new one using javascript. The missing part is the energy loss calculation.

Magnet

Design of a Split-pole spectrograph. Take from Ref.

The primary goal of a spectrograph is resolving momentum. A general discussion of magnetic spectrographs can be found at Ref. and Ref. . There are many designs from a simple single dipole to a combination of multiple dipoles and quadrupoles.

The SPS magnet was designed for a large solid angle, large resolving power, and correction of kinematic broadening. Using two-directional focusing and second-order focusing spectrograph can achieve a large sold angle and resolving power. Second-order focusing means the second-order terms in the acceptance angles vanish, i.e. no aberration.

The SPS contains 2 separate poles enveloped by a single coil. The split provides second-order double focusing over a broad range of momenta. The magnet can be rotated from 0 to 55 degrees in the lab. The magnetic field has an upper limit of 1.63 T (or 16.3 kG).

This table is taken from B.P. Kay Ph.D thesis (2007)

Property	Symbol	Value
Orbital radius	$\rho$	511 to 920 mm
Resolving power	$p/\Delta p$	1st order of $(x|\theta )$4290 (at $\theta =\pm 80$ mrad)
Acceptance	Horizontal	160 mrad
	Vertical	80 mrad
Dispersion	$D=(x|\delta )$	1.96
Magnification	$M_{x}=(x|x)$	0.39
	$M_{y}=(y|y)$	2.9
Maximum field	$B$	1.63 T

Transfer matrix and COSY INFINITY simulation

An illustration of the coordinate of an optical element. This is taken from H.A. Enge NIM 162, 161 (1979).

Notice :

need to fill up. Any 1st few orders transfer matrixes?

The entrance coordinates of the beam are $x_{1},y_{1},\theta _{1},\phi _{1}$ wiht momentum $\delta =p/p_{0}$, and coordinate at exit are $x_{2},y_{2},\theta _{2},\phi _{2}$. The entrance and exit coordinates are related by

$x_{2}=f_{x}(x_{1},y_{1},\theta _{1},\phi _{1},\delta )$

using Taylor expansion:

${\frac {x_{2}}{\rho }}=(x|x){\frac {x_{1}}{\rho }}+(x|\theta )\theta _{1}+(x|\delta )\delta +(x|\theta ^{2})\theta _{1}^{2}+...$

In the above expansion, the term $(x|x)$ is the magnification in the x-direction. $(x|\delta )$ is the dispersion, and $(x|\theta ^{2})$ is aberration. The focal plane is the z-position that $(x|\theta )=0$, i.e. the exit $x_{2}$ does not depend on the entrance angle.

Kinematic broadening

Kinematic correction of spectrometer. Taken from H. A. Enge NIM 162, 161 (1979)

Kinematic broadening is the broadening of focus for the same reaction state. After a reaction, the angle and momentum of the recoil particle are related that the entrance angle $\theta _{1}=f(\delta )$ is a function of momentum. For each energy state, the relation between the angle and momentum is unique. For example, in a 2-body transfer reaction, the momentum vector is

$(p_{x},p_{z})=(k\sin(\theta ),\gamma \beta {\sqrt {m^{2}-k^{2}}}+\gamma k\cos(\theta ))$

where $k,\theta$ are the momentum and the scattering angle at the CM frame, $\gamma ,\beta$ are the Lorentz factor from Lab frame to CM frame, and $m$ is the mass of the particle. All 5 coefficients are constant for a fixed energy state. And the different state is characterized by $k$. Defined the kinematic factor K:

$K={\frac {1}{p}}{\frac {dp}{d\theta _{1}}}={\frac {\beta E\sin(\theta )}{k+\beta {\sqrt {m^{2}-k^{2}}}\cos(\theta )}}$

The kinematic broadening can be corrected by shifting the focal plane by

$\Delta z=-DM\rho K,D=(x|\delta ),M=(x|x)$

Focal plane detector

Front view of the opened camerabox. The SPS focal plane detector with the front window removed is at the bottom.

Side Cross section view of the SPS focal plane detector. Taken from B.P. Kay Ph.D. thesis (2007).

Notice :

The drift ion chamber was repaired in summer 2018

The focal plane detector consists of an ion drift chamber with a set of delay lines to detect the position of a particle along the focal plane and a plastic scintillator to detect the energy of the incoming particle. Using the energy loss of the particle through the ion chamber with the energy deposited in the scintillator, particles of different charges and masses can be identified.

The typical pressure of the drift chamber is 70 to 300 Torr of isobutane gas [HC(CH3)3]. The pressure controls the density of the gas and affects the bias voltage, it further affects the drift velocity.

Table of pressure and bias voltages. Data was taken from the Ph.D. thesis of Erin Good (2020)

Gas pressure (Torr)	Anode bias (V)	Cathode plate bias (V)
70	+1050 to +1035	-550 to -500
80	+1150	-550
100	+1250	-600
110	+1200 to 1320	-620 to -600
125	+1425	-650
130	+1360	-725
150	+1500	-700

From bottom to top, the cathode plate, drift region (contains four biased field-shaping wire grids), Frisch grid (grounded), three anode wires, and pickup pads (which are with the delay lines). Electrons induced by any radiation will drift upward, pass the Frisch grid, are accelerated by the anodes, and hit the pickup pads. The pickup pads are strips with 45° against the anode wires, almost parallel to the particle trajectories. Each pickup strip is 0.09" (2.286 mm) wide and 1.4" (35.56 mm) long, and spaced 0.01" (0.245 mm). A total of 440 lead-coated copper strips with a 5 ns delay per strip results in a nominal total delay of 2.2 μs. Every 10 strips share a delay chip. The position of the hit position can then be determined by the time difference at the end of the delay line.

There are two position-sensitive delay lines (separated by 42.8625 mm) in the focal plane detector. By reconstructing the particle trajectory using the position information of both delay lines, the resolution can be enhanced by correcting for the kinematic shift of the reaction.

After passing the drift chamber, the particles will be stopped and detected in a plastic scintillator with a photomultiplier tube (PMT) at each end. Together with the energy loss, obtained by the cathode in the drift chamber, a ΔE-E particle identification can be done.

Outline of the algorithm

There are 9 readouts channels from the focal plane detector:

readout	type of signal
cathode	energy loss
Front delay line Left	timing
Front delay line Right	timing
Front anode	energy loss
Rear delay line Left	timing
Rear delay line Right	timing
Rear anode	energy loss
PMT Left	energy loss
PMT Right	energy loss

The PID is usually using one of the PMT energy and either the cathode or anode energy.

The coordinate at the Focal Plane is the conventional one, where z-axis is perpendicular to the focal plane detector, y-axis is the vertical, and x-axis is the z-axis cross y-axis. The positions of the front and Rear planes are constructed by the timestamp. Suppose the timestamp is in ns.

$x_{1}={\frac {t_{FL}-t_{FR}}{2}}{\frac {1}{2.1}};~~~x_{2}={\frac {t_{BL}-t_{BR}}{2}}{\frac {1}{1.98}}$

The position at the center of the focal plane is

$x_{avg}=x_{1}+x_{2}$

However, for different reactions, there is a z-offset, so that the focal plane is shifted

$x_{avg}=\left({\frac {1}{2}}-{\frac {z_{o}}{D}}\right)x_{1}+\left({\frac {1}{2}}+{\frac {z_{o}}{D}}\right)x_{2}$

where $D=42.8625~[{\textrm {mm}}]$ is the distance between the front and rear delay lines.

Calculation of the z-offset

The z-offset depends on the reaction, the angle $\theta$, and the magnetic field of the spilt-pole. Suppose we know the KE $T_{b}$ and momentum $P$ of the ejectile or the interested particle that goes into the split-pole, The reaction is denoted as a(A,B)b, where a is the beam (projectile), A is the target, B is the heavy recoil (residual),and b is the recoil (ejectile).

$P^{2}=(m_{b}+T_{b})^{2}-m_{b}^{2}=T_{b}(T_{b}+2m_{b})~[{\textrm {MeV/c}}]$

$\rho ={\frac {P}{cZB}}~[{\textrm {m}}]$, where $c=299.792458$, $Z$ is the charge number, and $B$ is the magnetic field in Tesla.

$k={\frac {v\sin(\theta )}{m_{b}+m_{B}-v\cos(\theta )}},~~~v={\sqrt {\frac {m_{a}m_{B}T_{a}}{T_{b}}}}$

$z_{o}=-\rho \delta _{x}M_{x}k[{\textrm {m}}]$

where $\delta _{x}=1.96$ is the x-dispersion and $M_{x}=0.39$ is the x-magnification.

Kinematic correction of the focal plane

As pointed out before, the kinematic broadening can be corrected. In SPS, the dispersion D is 1.96, magnification is 0.39.

Simulated rays near the focal plane.	An animation on the shift of the focal panel. An optimum is reached at FP = -42 mm.

construction of Xavg (X-average) on the virtual focal plan (a liner plane in this case). Need to redraw the picture, the Y-axis should be Z-axis, and it should be rotated 180 degree, so the particle is from bottom to top.

A parallel shift of the focal plane maybe not be enough. Suppose the best focal plan is given by a function $z=f(x)$. The 2 positions extracted from the front and rear delay lines are $x_{1},x_{2}$, and the distance between the front and rear delay lines is $d$. The X-avg is the solution of the equation:

$f(x)={\frac {x_{2}-x_{1}}{d}}\left(x-{\frac {x_{2}+x_{1}}{2}}\right)$

For a linear tilted plane $f(x)=mx+z_{0}$, the X-avg is

$X_{avg}={\frac {x_{1}^{2}-x_{2}^{2}-2dz_{0}}{2(dm+x_{1}-x_{2})}}$

SABRE

Installing particle shield on SABRE (photo taken on May 5, 2022)

SABRE is a Silicon Array for Branching Ratio Experiments with the SPS. Its predecessor is the Yale Lamp Shade Array (YLSA). SABRE sits at backward angles from the target and covers roughly 30% of 4π. SABRE has both thick and thin dead-layer detectors, with the thin dead-layer detectors capable of reaching ~200 keV thresholds for protons and deuterons.

CeBrA

Solid Works drawing of the fully planned array, which will consist of 13 CeBr₃ detectors.

The Cerium Bromide Array (CeBrA) is a γ-ray detector array designed to be used in conjunction with the SE-SPS. Comprised of low-background CeBr₃ scintillators with Hamamatsu photomultipliers. There are currently 7 commissioned detectors in the array with varying size crystals (2-1x1 inch, 4-2x2 inch, and 1-3x4 inch crystal detectors; a schematic for the full array on is shown on the right). The goal of CeBrA is to establish coincident events with the light-ions detected in the focal plane detector of the SE-SPS and the corresponding γ-rays from the excited recoiling nucleus, which are called particle-gamma coincidences. The scattering chamber used for CeBrA differs from the usual sliding-seal chamber that is used with the SE-SPS. With a hemisphere shape, it sits at a fixed 35^o angle relative to the SE-SPS and allows for a more detailed study of the electromagnetic transitions from nuclei excited in reactions using the SE-SPS.

Current array setup for CeBrA as it was used in the Summer 2023 REU experiments studying the 52Cr(d,pγ)53Cr and 34S(d,pγ)35S reactions, which was a follow up from the previous REU from 2022.

Repositories

https://github.com/sesps

Contact

• Jeff Blackmon mailto:blackmon@lsu.edu
• Mark-Christoph Spieker mailto:mspieker@fsu.edu
• Ingo Wiedenhoever mailto:iwiedenhoever@fsu.edu
• who should be contacted?