# A Spin-Light Polarimeter for Multi-GeV Longitudinally Polarized Electron
Beams

Prajwal Mohanmurthy, and Dipangkar Dutta P. Mohanmurthy and D. Dutta are with
the Department of Physics and Astronomy at Mississippi State University,
Mississippi State, MS 39762.Manuscript received July 11, 2013

###### Abstract

The physics program at the upgraded Jefferson Lab (JLab) and the physics
program envisioned for the proposed electron-ion collider (EIC) include large
efforts to search for interactions beyond the Standard Model (SM) using parity
violation in electroweak interactions. These experiments require precision
electron polarimetry with an uncertainty of $<$ 0.5 %. The spin dependent
Synchrotron radiation (SR), called ”spin-light,” can be used to monitor the
electron beam polarization. In this article we develop a conceptual design for
a “spin-light” polarimeter that can be used at a high intensity, multi-GeV
electron accelerator. We have also built a Geant4 based simulation for a
prototype device and report some of the results from these simulations.

###### Index Terms:

Polarized electrons, synchrotron radiation, spin light, differential
ionization chambers.

##  I Introduction

The determination of the longitudinal polarization of the electron beam is one
of the dominant systematic uncertainties in any parity violating electron
scattering (PVES) experiment. In order to achieve the desired high precision,
the polarization of the electron beam must be monitored continuously with an
uncertainty of $<$0.5%. These ambitious goals can be achieved if multiple
independent and high precision polarimeters are used simultaneously. In
addition to being precise, the polarimeters must be non-invasive and must
achieve the desired statistical precision in the shortest time possible.
Compton and Møller polarimeters are typically the polarimeters of choice for
these experiments and are essential to achieve the desired precision. However,
a complimentary polarimetry technique based on the spin dependence of
synchrotron radiation, referred to as “spin-light,” can be used as a relative
polarimeter. A spin-light polarimeter could provide additional means for
improving the systematic uncertainties and when calibrated against a
Compton/Møller polarimeter it could provide a stable continuous monitoring of
the beam polarization. We develop the conceptual design for a continuous
polarimeter based on “spin-light”. The proposed spin-light polarimeter can
achieve statistical precision of $<$ 1% in measurement cycles of less than 10
minutes for 4 - 20 GeV electron beams with a beam currents of $\sim$ 100
$\mu$A.

TABLE I: A comparison of the Compton, Møller and Spin-light polarimeters. Compton | Spin-Light | Møller
---|---|---
non-invasive, | non-invasive, | invasive
continuous | continuous |
analyzing power | analyzing power | analyzing power
energy dependent | energy dependent | energy independent
high currents | moderately | low currents
| high currents |
target is 100% | no target | target is $<$ 10%
polarized | needed | polarized
(requires stable laser) |  |
electron & photon | beam left & right | no independent
detection are | detectors provide | measurements
two independent | two independent | possible
measurements | measurements |
high precision | high precision | high precision
absolute polarimeter | relative polarimeter | absolute polarimeter
Best reported [3] | expected | Best reported [1]
instrumental | instrumental | instrumental
uncertainty: 0.4% | uncertainty: 0.6% | uncertainty: 0.47%
Best reported [3] | estimated | Best achieved [2]
absolute | absolute | absolute
uncertainty: 0.5% | uncertainty: $\sim$5% | uncertainty: 0.85%

Møller and Compton polarimeters have a proven track record of very high
precision, the JLab Hall-C Møller polarimeter has an instrumental uncertainty
of 0.47% [1] and absolute uncertainty of 0.85% [2], while the Compton
polarimeter used in the SLD experiment achieved an instrumental uncertainty of
0.4% [3] and an absolute uncertainty of 0.5% [3], hence they are essential for
any PVES program. However, a spin-light polarimeter would have a few
operational and instrumental advantages over conventional polarimeters, such
that when used in parallel with Compton/Møller polarimeters they might help
reduce the systematic uncertainties and achieve the very high precision
essential for the future PVES program. For example, Møller polarimeters use a
polarized Fe target, and the polarization of the target is difficult to
determine and may depend on the beam intensity. Moreover, Møller polarimeters
operate at low current, and are invasive to the primary experiment. Compton
polarimeters require a stable laser (the photon target) and are very sensitive
to backgrounds. The proposed spin-light polarimeter is a target free device,
hence it should be easier to operate over long periods, with its stability
governed just by the stability of the electron beam. Moreover, this novel
polarimeter would facilitate cross-checks and systematic studies when used
with other conventional polarimeters. On the other hand, one of the
disadvantages is that the proposed device can achieve comparable instrumental
uncertainties only as a relative polarimeter, whereas the absolute
polarization is what is required in the PVES experiments. Nevertheless a
precise and stable relative polarimeter can be a very useful device. The spin-
light polarimeter could be used in conjunction with a Compton polarimeter,
such that the difficult to operate Compton polarimeter is used for calibration
and the easier to operate and stable spin-light polarimeter is used to
continuous monitor the beam polarization. Moreover, only the Møller and the
spin-light polarimeters allow measurement of the transverse component of a
longitudinally polarized electron beam. The key features of conventional
polarimeters and a spin-light polarimeter are summarized in Table I.

In this article we discuss the theory behind the spin-light polarimeter and
present a complete conceptual design for such a polarimeter. In order to study
some of the systematic uncertainties of the novel spin-light polarimeter we
have built a Geant4 [4] based simulation of the polarimeter. The results from
these simulations are also presented here. This work was inspired by a 1993
proposal by Karabekov and Rossmanith [5].

##  II Synchrotron Radiation

At the typical energies and magnetic fields of present day terrestrial
accelerators, the intensity and angular distribution of SR from an electron
moving along a curved path, under the influence of a magnetic field, is
described with high precision by classical electrodynamics [6]. The total
radiation rate for highly relativistic electrons, $P^{clas}$ is $\propto
E^{4}$ as given by the Larmor formula:
$P^{clas}=\frac{2}{3}\frac{e^{2}\gamma^{4}c}{R^{2}}$, where $e$ is the
electron charge, $v_{e}$ is the velocity of the electron, $c$ is the velocity
of light in vacuum, $\beta=\frac{v_{e}}{c}$,
$\gamma=\frac{1}{\sqrt{1-\beta^{2}}}=\frac{E}{m_{e}c^{2}}$ is the Lorentz
boost, $m_{e}$ is the rest mass of the electron, and, $R$ is the radius of
curvature of the electron orbit. The angular distribution of the radiated
power is given by

$\frac{dP}{d\Omega}=\frac{e^{2}\gamma^{4}c}{4\pi
R^{2}}\frac{(1-\beta\cos\theta)^{2}-(1-\beta^{2})\sin^{2}\theta\cos^{2}\phi}{(1-\beta\cos\theta)^{5}},$
(1)

where the angles $(\theta,\phi)$ are measured with respect to the direction of
the electron’s motion. For highly relativistic electrons ($\gamma>>1$), the
radiation is Lorentz boosted in the forward direction with an opening angle
$\theta\approx 1/\gamma$. The classical theory also tells us that SR is
strongly linearly polarized [7], with $P_{\sigma}=\frac{7}{8}P$ and
$P_{\pi}=\frac{1}{8}P$, where $\sigma$-component is the one where the electric
field lies in the plane of the electron orbit while $\pi$-component is the one
where the electric field lies in the plane perpendicular to the orbital plane.
Experimentally, these properties of SR have been demonstrated with high
precision for a wide range of frequencies [8] \- [11].

Classical theory also tells us that the cone of SR passes over a fixed angle
$\theta$ in retarded 111accounts for the finite delay between the photon
emission and detection time $\Delta
t^{{}^{\prime}}\approx\frac{\Delta\theta}{\omega_{0}}=\frac{1}{\gamma\omega_{0}}$,
where $\omega_{0}=v_{e}/R\approx c/R$ is the Larmor frequency for the
electron. For a distant observer the corresponding time interval is $\Delta
t=\Delta t^{{}^{\prime}}(1-\beta)\approx 1/(2\gamma^{3}\omega_{0})$. Hence the
spectral width of the radiation is $\Delta\omega\approx 1/\Delta
t=2\gamma^{3}\omega_{0}$. This implies that the radiation is strong at high
harmonics of the Larmor frequency $(\omega_{0})$ and can be considered as
continuous. However from quantum mechanics we know that the radiation at
frequency $\omega$ consists of photons of energy $\hbar\omega$. Thus there
must exist a sufficiently strong acceleration such that a single photon will
carry away all of the electron’s energy ($\gamma m_{e}c^{2}=\hbar\omega_{c}$).
The critical magnetic field strong enough to provide this acceleration is
found to be $B_{crit}=\frac{m^{2}c^{3}}{e\hbar}$ = 4.41$\times$109 Tesla. The
frequency of the photon radiated under the influence of $B_{crit}$ is called
the critical frequency ($\omega_{c}$) and is given
by$\omega_{c}=\frac{3}{2}\gamma^{3}\omega_{0}=\frac{3}{2}\gamma^{3}c/R$. From
this we get the critical energy
$E_{crit}=m_{e}c^{2}\sqrt{\frac{m_{e}cR}{\hbar}}\approx$ 106 GeV. The
extremely large values of the critical field and critical energy help explain
why the classical theory is successful at the energies and fields accessible
at present day accelerators. However, it turns out that several quantum
effects appear at considerably lower electron energies.

###  II-A Quantum Theory of Synchrotron Radiation

The exact expression for SR intensity including quantum corrections was
calculated by Sokolov, Ternov and Klepikov, based on the solution to the Dirac
equation in the framework of quantum electrodynamics [12]. They showed that at
energies above a few 100 MeV, there would be fluctuations in the radius of the
electron orbit leading to radial oscillations [13] of the electron trajectory
and quantum widening of the trajectory similar to Brownian motion. These
oscillations and widening of the trajectory are essential in determining the
dynamics of electrons in an accelerator, specially storage rings. Sokolov and
Ternov also developed the mathematics required to describe the spin of
relativistic electrons moving in an external electromagnetic field [14, 15],
which allowed them to calculate the electron spin related properties of SR.

The power radiated by electrons undergoing transitions $n\rightarrow
n^{{}^{\prime}}$ (related to the radius of the electron orbit), $s\rightarrow
s^{{}^{\prime}}$ (quadratic fluctuation of the radius) and $j\rightarrow
j^{{}^{\prime}}$ (spin orientations with respect to the magnetic field),
integrated over angles and summed over polarization states, is given by [14,
15]:

$P=P^{clas}\times\frac{9\sqrt{3}}{16\pi}\sum_{s^{{}^{\prime}}}\int_{0}^{\infty}\frac{ydy}{(1+\xi
y)^{4}}I^{2}_{ss^{{}^{\prime}}}(x)F(y),$ (2)

where,

$\displaystyle F(y)$ $\displaystyle=$
$\displaystyle\frac{1+jj^{{}^{\prime}}}{2}[2(1+\xi
y)\int_{y}^{\infty}K_{5/3}(x)dx$ (3) $\displaystyle+$
$\displaystyle\frac{1}{2}\xi^{2}y^{2}K_{2/3}(y)-j(2+\xi y)\xi yK_{1/3}(y)]$
$\displaystyle+$
$\displaystyle\frac{1-jj^{{}^{\prime}}}{2}\xi^{2}y^{2}\left[K_{2/3}(y)+lK_{1/3}(y)\right]$

where $P^{clas}$ is the classical expression for SR power radiated,
$y=\frac{\omega}{\omega_{c}}$, $x=\frac{3}{4}\frac{\xi\gamma^{3}y^{2}}{(1+\xi
y)^{2}}$, $I_{ss^{{}^{\prime}}}$ are the Laguerre functions, $K_{n}(x)$ are
modified Bessel functions, and $\xi=\frac{3}{2}\frac{B}{B_{crit}}\gamma$ is
the critical parameter. Because, $B_{crit}$ = 4.41$\times$ 109 Tesla we have
$\xi<<1$ for magnetic fields used at all man-made accelerators and the above
expression can be expanded in terms of the critical parameter $\xi$ to get
[16];

$\displaystyle P$ $\displaystyle=$ $\displaystyle
P^{clas}[(1-\frac{55\sqrt{3}}{24}\xi+\frac{64}{3}\xi^{2})$ (4)
$\displaystyle-$
$\displaystyle(\frac{1+jj^{{}^{\prime}}}{2})(j\xi+\frac{5}{9}\xi^{2}\frac{245\sqrt{3}}{48}j\xi^{2})$
$\displaystyle+$
$\displaystyle(\frac{1-jj^{{}^{\prime}}}{2})(\frac{4}{3}\xi^{2}+\frac{315\sqrt{3}}{432}j\xi^{2})+...]$

These quantum corrections to the classical expression for charge radiation
($P^{clas}$) involve contributions from the electron recoil effects of
radiation, interference of the charge radiation and radiation due to the
intrinsic magnetic moment of the electron, magnetic moment radiation due to
Larmor precession, magnetic moment radiation due to Thomas precession,
interference between the Larmor and Thomas radiation and radiation due to the
anomalous magnetic moment of an electron [16]. The lowest order spin-dependent
correction is of order $\xi$ and the lowest order spin-flip correction term is
of order $\xi^{2}$. The difference (of order $\xi$) between the expression for
power radiated by polarized and unpolarized (spin averaged) electron beams has
the form,

$\displaystyle P^{pol}-P^{unpol}$ $\displaystyle=$ $\displaystyle-j\xi
P^{clas}\int_{0}^{\infty}\frac{9\sqrt{3}}{8\pi}y^{2}K_{1/3}(y)dy.$ (5)

The above expression is directly related to the spin polarization of an
electron beam $j$, hence the difference $P^{pol}-P^{unpol}=P^{spin}$ can be
called “spin-light” [14, 15]. This offers a new possibility for visual or
direct observation of the polarization characteristics of an electron beam by
determining the SR power at a fixed range of spectral frequency.

The spin dependence of the SR was verified at the VEPP-4 storage ring in
Novosibirsk [17], using a 3 pole wiggler magnet (called a magnetic snake). The
intensity of the SR produced by the wiggler for transversely polarized
electrons was monitored while the beam was periodically depolarized using a RF
field. The measured variation in SR intensity with polarization matched
exactly with the expectation from the Sokolov-Ternov theory. This spin
dependent part of the SR has been successfully used at the VEPP-4 to monitor
the transverse polarization of the electron beam.

The spin-flip term has a unique effect on circular accelerators. The radiation
probability with spin flip is given by
$w^{\uparrow\downarrow}=\frac{1}{\tau}\left(1+j\frac{8\sqrt{3}}{15}\right)$,
where $j=$ 1 is for spin along the magnetic field and $j=$ -1 is for spin
opposite to the magnetic field [14]. Using this relation, Sokolov and Ternov
had predicted that, over time the beam in a circular accelerator would
eventually become polarized opposite to the direction of the magnetic field.
In other words the beam of a circular accelerator becomes transversely
polarized over time via self polarization. This phenomena known as the
Sokolov-Ternov self polarization was first observed at the French storage ring
[18] at Orsay and is now routinely used to polarize beams at circular electron
accelerators such as DESY.

##  III Spin Light

In the discussion above the spin orientation $j$ is relative to the magnetic
field that produces the SR, with $j=$ 1 along the magnetic field. Polarized
electron beams have longitudinal ($p_{z}$) and transverse ($p_{\perp}$)
components relative to the beam direction and the transverse polarization have
vertical and horizontal components. For a vertically oriented magnetic field,
the total SR power from transversely polarized electrons, ignoring spin flip
terms and other terms of order $\xi^{2}$, is given by [14]:

$\displaystyle P_{\gamma}(tran)$ $\displaystyle=$
$\displaystyle\frac{9n_{e}}{16\pi^{3}}\frac{ce^{2}}{R^{2}}\gamma^{5}\int_{0}^{\infty}\frac{y^{2}dy}{(1+\xi
y)^{4}}\oint d\Omega(1+\alpha^{2})^{2}$ (6) $\displaystyle\times$
$\displaystyle[K^{2}_{2/3}(z)+\frac{\alpha^{2}}{1+\alpha^{2}}K^{2}_{1/3}(z)$
$\displaystyle-$ $\displaystyle p_{\perp}\xi
y\frac{1}{\sqrt{1+\alpha^{2}}}K_{1/3}(z)K_{2/3}(z)],$

where $n_{e}$ is the number of electrons,
$z=\frac{\omega}{2\omega_{c}}(1+\alpha^{2})^{3/2}$, and $\alpha=\gamma\psi$,
where $\psi$ is the vertical angle in the frame of the moving electron. The
rest of the symbols are as defined in previous instances. The polarization
dependent term in the above expression is an even function of the vertical
angle therefore when integrated over all angles it makes the total SR power
spin dependent. Thus by measuring this spin dependence in the total SR power
radiated one can measure the transverse polarization of the electron beam.

On the other hand, the total SR power from longitudinally polarized electrons,
ignoring spin flip terms and other terms of order $\xi^{2}$, is given by [14]:

$\displaystyle P_{\gamma}(long)$ $\displaystyle=$
$\displaystyle\frac{9n_{e}}{16\pi^{3}}\frac{ce^{2}}{R^{2}}\gamma^{5}\int_{0}^{\infty}\frac{y^{2}dy}{(1+\xi
y)^{4}}\oint d\Omega(1+\alpha^{2})^{2}$ (7) $\displaystyle\times$
$\displaystyle[K^{2}_{2/3}(z)+\frac{\alpha^{2}}{1+\alpha^{2}}K^{2}_{1/3}(z)$
$\displaystyle+$ $\displaystyle p_{z}\xi
y\frac{\alpha}{\sqrt{1+\alpha^{2}}}K_{1/3}(z)K_{2/3}(z)],$

The spin dependent term in the above expression is an odd function of the
vertical angle therefore when integrated over all angles it goes to zero and
the total SR power for longitudinally polarized electrons is spin independent.
However, the power radiated into the space above ($0<\psi<\pi/2$) and below
($-\pi/2<\psi<0$) the orbital plane of the electron are different and the
difference between them is spin dependent. Therefore, by measuring this
spatial asymmetry one can monitor the longitudinal polarization of the
electron beam. For $\xi y<<1$, if we divide the above expression by the energy
of the radiated photon, $E_{\gamma}=\frac{3}{2}\frac{\hbar c}{R}\gamma^{3}y$
we get the total number of photons radiated into a finite horizontal angle
$\Delta\theta$ as,

Figure 1: (a) The total SR power radiated, per MeV (red), and the spin
dependent difference in power radiated, per MeV (blue), above and below the
orbital plane, for 11 GeV, longitudinally polarized electrons in a 4 T
magnetic field and 100 $\mu$A current, and 10% detector efficiency. (b) The
total number of SR photons per MeV and the number of “spin-light” photons per
MeV, and (c) the asymmetry $\frac{\Delta N}{N}$ as a function of photon
energy. (d) The time for 1% statistical uncertainty, in seconds.

$\displaystyle N_{\gamma}$ $\displaystyle=$
$\displaystyle\frac{3}{4\pi^{2}}\frac{1}{137}\frac{I_{e}}{e}\gamma\Delta\theta\int_{y_{1}}^{y_{2}}ydy\int_{-\alpha}^{\alpha}(1+\alpha^{2})^{3/2}$
(8) $\displaystyle\times$
$\displaystyle\left[K^{2}_{2/3}(z)+\frac{\alpha^{2}}{1+\alpha^{2}}K^{2}_{1/3}(z)\right]d\alpha,$

where $I_{e}$ is the beam current. The difference in the photon flux radiated
in the the space above and below the electron orbit is given by,

$\displaystyle\Delta N_{\gamma}(p_{z})$ $\displaystyle=$
$\displaystyle\frac{3}{\pi^{2}}\frac{1}{137}\frac{I_{e}}{e}p_{z}\xi\gamma\Delta\theta\int_{y_{1}}^{y_{2}}y^{2}dy\int_{0}^{\alpha}\alpha(1+\alpha^{2})^{3/2}$
(9) $\displaystyle\times$ $\displaystyle K_{1/3}(z)K_{2/3}(z)d\alpha$

To examine the size and characteristics of the spin dependence we have
numerically integrated the above two expressions for longitudinally polarized
electron with 100% polarization, in a 4 Tesla magnetic field, with $I_{e}$ =
100 $\mu$A, and $E_{e}$ = 11 GeV. We have integrated over a horizontal angular
acceptance of $\Delta\theta$ = 10 mrad, and a vertical acceptance of
$\alpha=\pm$1\. The characteristic spectra of SR and spin-light obtained from
these numerical intragrations are shown in Fig. 1. The total power radiated
$P_{\gamma}$(long) and the spin dependent difference of power radiated above
and below the orbital plane of the electron $\Delta P$(long) are shown as a
function of photon energy in Fig 1 (a). The number of SR photons
$N_{\gamma}$(long) and the number of spin-light photons $\Delta
N_{\gamma}$(long), as function of photon energy are shown in Fig 1 (b). The
asymmetry defined as $A=\frac{\Delta N_{\gamma}(long)}{N_{\gamma}(long)}$ as a
function of photon energy is shown in Fig. 1 (c).

Figure 2: (a) The spin dependent asymmetry as a function of ratio of photon
energy to the critical photon energy, for electron beam energy, Ebeam = 4 – 20
GeV. (b) The spin dependent asymmetry for magnetic field, Bwigg = 2 – 5 Tesla.

Figure 3: (top) The total SR power radiated as a function of the electron beam
energy, for a 100 $\mu$A beam current and Bwigg = 4 Tesla. (bottom) The
vertical size of the SR beam spot as a function of the electron beam energy,
at a distance of 10 m from the wiggler magnet.

Fig. 1 (c) indicates that one should measure the hard tail of the SR spectrum
($E_{\gamma}>$ 500 keV) and avoid the soft part of the spectrum where the
asymmetry is low and changes rapidly with energy. Although the asymmetry is
small $\sim$ 10-4 the photon flux is high, even at the hard tail of the
spectrum, allowing a rapid determination of the asymmetry, with 1% statistical
uncertainty within a few tens of seconds ($\frac{\delta
A}{A}=\frac{1}{A\sqrt{2N}}$) as shown in Fig 1 (d). The energy dependence of
the asymmetry for $E_{e}$ = 4 – 20 GeV and the magnetic field dependence of
the asymmetry for $B_{wigg}$ = 2 – 5 T are shown in Fig 2(a) and Fig 2(b)
respectively. These figures demonstrate that a spin-light based polarimetry is
a very promising technique at intermediate energies and can be used to monitor
the polarization of 4 – 20 GeV electrons in very rapid measurement cycles,
with high statistical precision.

Although the size of the asymmetry increases with increasing electron beam
energy it should be noted that the total power of SR increases as the fourth
power of the electron beam energy and increases linearly with beam current.
Moreover, the vertical size of the SR spot decreases with increasing electron
beam energy as shown in Fig. 3 (bottom). These factors impose practical
limitations on the highest electron beam energies and the highest currents at
which a spin-light polarimeter would be feasible. We estimate that it is best
suited for the 4 - 20 GeV energy range for currents less than $\sim$ 10 mA.

##  IV A Conceptual Design

The two basic components of a spin-light based polarimeter are the source of
SR and the X-ray detector which can measure the spatial asymmetry.

###  IV-A The SR Source - Wiggler

A three pole wiggler magnet with a magnetic field that has uniform magnitude
but reversed direction at each pole and a short-long-short pole arrangement is
well suited as a source of SR. The three poles must be symmetric about the
center such that the line integral of the magnetic field in the direction of
the motion of the electron, $z$, must be zero (i.e. $\int B(z)dz$ = 0),
ensuring that it does not affect the electron beam transport and its spin
direction (beyond the wiggler). The field being of opposite polarity at the 3
poles, flips the sign of the spin dependent spatial asymmetry from any two
adjacent poles and hence when measured simultaneously it can help reduce
systematic uncertainties arising from the vertical motion of the beam.

The intensity and the asymmetry both increase with increasing field strength,
while the pole length decreases with increasing field strength. Therefore a
field strength of 4 T is a judicious choice for the wiggler field. A 10 mrad
bend can be achieved with a pole length of 10 cm. Thus the total magnet length
is 40 cm, and the spacing between the poles is optimized for ease of
extraction and detection of the SR beam. A separation of 1 m between the poles
allows for collimators to be placed that can separate the SR beams spots from
the different poles. The small pole length ensures that the effect of spin-
flip inducing SR and the fluctuation of the SR power are negligible ($<$
0.1%).

Wiggler magnets are regularly used at light sources around the world such as
the Advanced Photon Source (APS) at Argonne National Lab and Spring8 in Japan.
Some of these magnets are well suited for a spin-light polarimeter [22].

Figure 4: A position sensitive ionization chamber developed at the APS and
SPring-8 with a resolution of 5$\mu$m when operated at photon flux of
5.0$\times$1012 8 keV photons/sec. [23] Note that the image shown here would
constitute just one half of a differential ionization chamber required for
spin-light polarimetry.

####  IV-A1 Effect of the wiggler on the electron beam

A non-invasive polarimeter is highly desirable and hence we must study the
effect of fluctuations related to the quantum nature of SR produced by the
wiggler. The effect of SR on the electron beam were carefully studied for the
recirculating arcs [25], and the same methods can be used to calculate the
influence of the wiggler on critical beam parameters. As described in Ref.
[24] and [25], the distribution of energies lost by individual electrons in
bending through some angle $\theta$ is given by a convolution of the
distribution of the number of photons emitted and the distribution of energies
of those photons. The number of photons emitted by a particular electron per
radian bend will be distributed according to Poisson statistics about a mean
value given by [24], $n=\frac{5}{2\sqrt{3}}\frac{\gamma}{137}=20.62E,$ where
$n$ is the mean number of photons per radian bend, and $E$ is the beam energy
in GeV. The average energy of the photons emitted is
$E_{c}=\hbar\omega_{c}=\frac{3}{2}\frac{\hbar c\gamma^{3}}{R}$ [24]. Therefore
the mean energy fluctuation is given by $\Delta E=\sqrt{n}E_{c}$. It is
interesting to note that the energy fluctuation depends only on the electron
beam energy and the bend radius of the wiggler. A beam of 11 GeV electrons in
a 4T wiggler with a 10 m bend radius and a bend angle of 10 mrad, gives
$n\sim$ 2 and $E_{c}$ = 199 keV. Therefore $\Delta E/E\sim$ 2.5$\times$10-5,
which is comparable to the fluctuations due to the recirculating arcs of the
JLab accelerator [25].

The SR power spectrum usually peaks at angles of $\pm 1/\gamma$ with respect
to the electron direction. However, if an electron emits on the average two
photons in a magnet, the angular distribution of the momentum kick received by
each electron is peaked in the direction of the electron’s motion. The
magnitude of the transverse kicks generated by the emission of a photon with
energy $E_{c}$ in the direction $\theta_{\gamma}=1/\gamma$ with respect to the
electron direction is given by [24],
$\Delta\theta_{e}=\frac{E_{\gamma}\sin{\theta_{\gamma}}}{E_{e}}=11.3\times
10^{-9}\frac{E_{e}(GeV)}{R(m)}$. The r.m.s. kick from the emission of $n$
photons is given by $\sqrt{n}\Delta\theta_{e}$. Thus for a 11 GeV beam bend by
10 mrad the r.m.s. kick is $\sim$ 1.5 $\times$ 10-8 rad, which is negligible.

Thus the wiggler magnet would have negligible influence on the electron beam
and a spin-light polarimeter can be used for non-invasive monitoring of the
beam polarization.

####  IV-A2 Influence of the wiggler bend direction

The wiggler bend direction was chosen to be beam-left for the conceptual
design (see for example Figs. 8 & 9). Since the bend direction is transverse
to the asymmetry direction one does not expect any systematic influence due to
the choice of the bend direction. However, the design includes two symmetric
pairs of ionization chambers placed on either side of the beam, and since the
bend is small (10 mrad), it should be possible to build symmetric pairs of
collimators and slits on both sides of the beam. With such a setup the
independence of the spin-light asymmetry with respect to the wiggler bend
direction can be directly verified during calibration and commissioning of the
device. For the stability of operation, changes in bend direction during
regular operation is not desirable.

###  IV-B The X-ray Detector - Ionization Chamber

The detector used to measure the spatial asymmetry must be sensitive to X-rays
in the range of about 500 keV to 2.5 MeV and must be able to pick out a small
asymmetry from a large spin independent background, it must be radiation hard,
have low noise and be able to withstand high rates of $\sim$ 1012 photons/sec.
Ionization chambers (IC) are well known for their high rate capability when
operated as an integrating detector (i.e. in current mode), low electronic
noise and radiation hardness. Argon/Xenon is an attractive candidate for use
as an ionization medium, its high atomic number (18/54) and density (when
compressed) gives it a high stopping power for hard X-rays and low energy
gamma [26]. Over the last two decades, room temperature, high pressure ($>$ 50
atm, 0.55 g/cc) xenon (HPXe) ionization chambers have been developed with high
detection efficiency in the 50 keV - 2.0 MeV range [19, 20, 21].

Figure 5: (left) A schematic of the split anode plate. (right) The dual
differential ionization chamber for spin-light polarimetry.

Another recent development, is the split collector ionization chamber that
have turned the IC into a position sensitive device. Position sensitive
ionization chambers are designed to have the collector plate split into two
sections in a zig-zag/backgammon pattern such that each half operates as an
independent ionization chamber. A prototype of such a chamber has been shown
in Fig. 4. These chambers were developed at the APS at ANL and at the SPring-8
light source in Japan. They are used to measure the vertical position of X-ray
beams and have been shown to have a resolution of 5 $\mu$m [23]. These
chambers also have very low dark currents in the $\sim$pA range and have been
operated at photon flux of 5.0$\times$1012 photons/sec. They work by measuring
the difference in counts between the two halves of the chamber, i.e. they are
differential ionization chambers (DIC). A position sensitive DIC operated in
current mode can be used to measure the spatial asymmetry of the SR generated
by longitudinally polarized electrons.

A dual, 1 atm. Ar/Xe differential ionization chamber would be ideal for a
relative polarimeter. A schematic for such an IC is shown in Fig. 5. The
chamber would consist of Ti or stainless steel windows thick enough to cut
down the low energy X-rays ($<$ 50 keV). A pair of split central anode plates
(separated by a thin insulator) would be placed between the cathodes. The
anode plates would be split in a backgammon pattern. The current measured on
each half of the anode plates is amplified with a differential current
amplifier.

Figure 6: A schematic of the collimated beams of synchrotron radiation from 2
adjacent poles of the wiggler magnet incident on the differential ionization
chambers. The figure on the left is a beam’s view of the electrodes of the
DIC. The collimated radiation is shown as oval blobs, while the figure on the
right is an isometric view of the electrodes, without showing the incident
radiation.

A magnified view of the synchrotron radiation from two adjacent poles of the
wiggler magnet incident on a dual DIC is shown in Fig. 6. The left panel of
the figure shows the beam’s view of the dual DIC. The collimated radiation is
shown as oval blobs with the up-down asymmetry represented by the gray shading
of the blob (the collimation scheme needed to achieve this is discussed in the
next section). The right panel shows an isometric view of the electrodes in
the DIC, without showing the incident radiation. The collimated SR beam from
two adjacent poles will be incident on opposite sides of the anodes in the
dual DIC. The spin-light spatial asymmetry (above and below the orbital plane)
will have opposite sign in each half of the DIC because the magnetic field
direction of the adjacent poles of the wiggler are opposite. On the other hand
any spatial asymmetry due to vertical motion of the beam will have the same
sign in the two halves of the dual DIC and hence should cancel to first order.
Thus the dual DIC is essential to ensure that the spin-light polarimeter is
insensitive to vertical beam motion.

####  IV-B1 The Signal from the DICs

If we denote $N^{L(R)}_{SR}$ as the number of SR photons on the left(right) of
the anode plates, $N^{L(R)}_{spin}$ as the number of spin-light photons, and
$\Delta N^{L(R)}_{z}$ as the difference in number of photons introduce by the
vertical beam motion, then the contribution to the measured current from the
top left part of the dual DIC will be (see Fig. 6);

$I_{1}^{L}\propto N^{L}_{SR}+N^{L}_{spin}+\Delta N^{L}_{z},$

similarly the current contribution from the top right of the dual DIC is;

$I_{1}^{R}\propto N^{R}_{SR}-N^{R}_{spin}+\Delta N^{R}_{z}.$

From the simulation studies we estimate that after collimation the size of
these currents will be on the order of $\sim$ 10 nA.

Note the change in the sign of the contribution from spin light photons
because they are generated from adjacent poles of the wiggler while the
contribution from vertical beam motion has the same sign. For the bottom left
and right parts of the dual DIC we get;

$I_{2}^{L}\propto N^{L}_{SR}-N^{L}_{spin}-\Delta N^{L}_{z},$

and

$I_{2}^{R}\propto N^{R}_{SR}+N^{R}_{spin}-\Delta N^{R}_{z}.$

Thus, the signal from the top and bottom halves of the DIC, $S1$, and $S2$ as
shown in Fig. 6, can be written as,

$\displaystyle S1$ $\displaystyle\propto$
$\displaystyle(N^{L}_{SR}+N^{L}_{spin}+\Delta
N^{L}_{z})-(N^{R}_{SR}-N^{R}_{spin}+\Delta N^{R}_{z})$ (10) $\displaystyle=$
$\displaystyle 2N_{spin},$

and

$\displaystyle S2$ $\displaystyle\propto$
$\displaystyle(N^{L}_{SR}-N^{L}_{spin}-\Delta
N^{L}_{z})-(N^{R}_{SR}+N^{R}_{spin}-\Delta N^{R}_{z})$ (11) $\displaystyle=$
$\displaystyle-2N_{spin}.$

Hence $S1-S2\propto 4N_{spin}\propto 4P_{e}$, and the vertical motion related
asymmetry cancels to first order as does the corrections due to transverse
polarization. However, it should be noted that the signals $S1+S2$ is
proportional to the transverse polarization of the electron beam. This
possibility of measuring both the longitudinal and transverse asymmetries in
the same setup, provides further capability for reducing systematic
uncertainties and makes the spin-light polarimeter an extremely versatile
tool.

Figure 7: (left) The number of photons absorbed in a 1-atm, 50 cm long Xe
chamber. (right) The detector response weighted asymmetry.

The number of photons absorbed in the ionization chamber can be calculated by
multiplying the SR spectrum with the absorption function
$A(\lambda)=1-e^{-\mu(\lambda)\cdot t}$, where $t$ is the length of the
chamber, $\lambda$ is the photon wavelength and $\mu$ is the absorption
coefficient which is obtained from NIST database [27]. The number of photons
absorbed in a 50 cm long chamber with 1 atm Xe, is shown in Fig. 7(left). Also
shown in the absorption weighted (or detector response weighted) asymmetry
(right).

###  IV-C Collimation

The spacing between the wiggler poles was chosen to be 1 m to allow adequate
room for the placement of collimators that would separate the SR beams from
each pole. The flight path from the wiggler to the detector is selected to be
10 m, which implies that the SR spot size due to each of the wiggler poles
will fan out over a horizontal length of $\sim$ 10 cm. The vertical width of
the SR spot is only $\sim$ 1 mm. With appropriate placement of collimators on
the wiggler pole entrance and exit faces, it is possible to separate the SR
beam spot from the four different wiggler poles. The magnet system will wiggle
the beam by 10 mrad in the horizontal plane such that each pole of the wiggler
magnet produces a fan of synchrotron radiation in the horizontal plane as
shown in the top view of the magnets (Fig 8).

Figure 8: A schematic of the fan of synchrotron radiation produced as the
electron beam traverses through each of the 4 poles of the wiggler magnet. The
top view of the magnets has been shown. The two colors are used to indicate
that the poles 1 and 4 have opposite polarity compared to poles 2 and 3 and
therefore the sign of the asymmetry for the SR fans of the two colors are
opposite.

For longitudinally polarized electrons this fan of synchrotron radiation will
have an up-down asymmetry in the vertical direction (due to spin-light). A
series of collimators placed at the front and back faces of each pole of the
wiggler magnet and at the center of the central pole of the wiggler will be
used to select small angular ranges from the entire fan of synchrotron
radiation as shown in the top view of the magnets in Fig 9. This collimation
scheme enables separation of the synchrotron radiation from each pole of the
wiggler magnet. Such a separation is necessary because the up-down asymmetry
of the synchrotron radiation has opposite sign for each pole of the wiggler.
Each of the collimated beams of synchrotron radiation will be separated by a
few cm when they are projected into two symmetric dual DICs located at a
distance of 10 m from the wiggler magnet.

Figure 9: A schematic for the slits and collimators used to select small
angular range of the fan of synchrotron radiation. The synchrotron beams from
poles 1 and 2 will be incident on the dual DIC placed on the beam left and the
beams from poles 3 and 4 are incident on the dual DIC placed on the beam
right.

###  IV-D The Complete Polarimeter

Figure 10: A schematic for a differential spin-light polarimeter (not to
scale).

A 3D view of the complete spin-light polarimeter is shown in Fig 10. Each pole
of the wiggler magnet is separated by a distance of 1 m and the two dual
differential ionization chambers are placed 10 m from the last pole of wiggler
magnet. The vertical backgammon split on the central anodes makes the DIC
position sensitive in the vertical direction and hence the signal from the DIC
is sensitive to the up-down asymmetry for each of the collimated beams of
synchrotron radiation. The second dual DIC is necessary to provide an
independent measurement of the up-down asymmetry and help reduced systematic
uncertainties. The main parameters of this conceptual design are tabulated in
Table II.

TABLE II: Parameters of the SR polarimeter at 11 GeV | Beam & Magnet
---|---
Ee, current | (GeV), ($\mu$A), | 11.0, 100
Bwiggler | (T) | 4.0
Pole (total) length | (m) | 0.1 (0.4)
Separation | (m) | 1.0
between poles |  |
bend angle | (mrad) | 10
vert. opening | (mrad) | 0.05
angle |  |
Flight path | (m) | 10
to det. |  |
| SR and detector
Nγ/s | (Hz) | 5.8$\times$ 1013
$\Delta$NSpin/s | (Hz) | 1.8$\times$ 109
Detector | (1 atm Xe, cm) | 50
medium |  |
Nabs/s | (Hz) | 3.1$\times$ 1012
$\Delta$Nfluctuation/s | (Hz) | 7.6$\times$ 106
vert. beam | (mm) | 1.0
spot after 10m |  |
flight |  |

###  IV-E Systematic Instrumental Uncertainties

Some of the major sources of systematic uncertainties for a spin-light
polarimeter include the background asymmetries from processes such as
Bremsstrahlung and false asymmetry due to vertical beam motion, differences in
chamber efficiency and magnetic field non-uniformity between adjacent poles of
the wiggler. The measured experimental asymmetry from a spin light polarimeter
can be written as $A_{expt}=A_{raw}(1+B/S)-A_{B}B/S+A_{F}$, where $S$ and $B$
are the signal and background, $A_{B}$ is the background asymmetry and $A_{F}$
is the false asymmetry due to factors such as vertical beam motion,
differences in the chamber efficiency and differences in the field strength
between adjacent poles. The main advantage of operating the ionization
chambers as differential detectors is that the false asymmetries will cancel
to first order. In addition the visible portion of the synchrotron light can
be used to align the detectors and help control systematic uncertainties. The
3-pole design ensures that the vertical beam motion related false asymmetry
also cancels to first order. However, the size of the background must be small
compared to the signal. In order to address this issue, a full Geant4 [4]
simulation of a proto-type spin-light polarimeter was built. The simulation
was also used to study the effect of the asymmetry associated with the
background. In addition to the built-in synchrotron radiation physics
available in Geant4, we have implemented a spin-light generator using a
parametrized model of spin-light at the nominal running conditions (11 GeV
beam, 4T magnetic field and 10 mrad bend angle). Using this generator we were
able to reproduce the expected photon energy spectrum and the expected
asymmetry as shown in Fig. 11.

Figure 11: The simulated photon spectrum and asymmetry compared to the
calculated spectra for the nominal running conditions.

The exact position and width of the slits/collimators on the front and back
faces of the wiggler magnets were optimized iteratively to obtain the best
separation between the SR photons originating from the different poles of the
magnets. Currently, we are using slits only to illuminate the beam right split
ionization chamber. A 3 mm sheet of lead shielding is also applied in front of
the ion chamber.

The ionization generated from the photons incident on the ion-chamber is
integrated to obtain the signal from the chamber and this signal was used to
study several of the systematic uncertainties, such as the effect of
background radiation, the position/alignment of the collimators and the
alignment of the ion chambers, on the asymmetry signal. The results of these
studies have been tabulated in Table III.

The simulated data show that the background from non-SR radiation is $\sim$
1.6% and most of the background is at energies below 0.5 MeV. If the
background related dilution can be determined to $\sim$ 1% the systematic
uncertainty due of background radiation will be $\sim$ 0.5%. Moreover, in the
proposed setup the background can be determined by measuring the difference in
the signal from the chambers with the wiggler magnets turned on and off during
calibration and commissioning of the device and at some relatively long
interval during regular operations (these background measurements would be
invasive in nature).

The same Geant4 simulation was also used to determine the contribution to the
systematic uncertainty due to the uncertainty in the exact position and width
of the slits and collimators that are placed on the front and back faces of
the poles of the wiggler magnet. These collimators are used to separate the SR
photons originating from the various poles. Thus, any uncertainty in the
position and width of the collimators can give rise to a false asymmetry due
to mixing of spin-light and SR photons from different poles. The simulation
demonstrates that a position/width uncertainty of 100 $\mu$m would lead to a
uncertainty of $<$ 0.2%.

We have also simulated the effects of the finite beam size on the up-down
asymmetry. The simulations were performed for a beam with a Gaussian
distribution of $\sigma$= 100 $\mu$m. The effects of the fringe fields at the
tapered edges of the wiggler poles has also been studied using a Poisson Fish
[28] model of the magnetic field. Although the absolute photon flux was
reduced due the fringe fields the effect on the asymmetry was found to be
minimal.

We have studied the effects of beam halos on the spin-light asymmetry. The
restrictive collimation scheme discussed in section IV-C ensures that the
contribution from beam halo is very limited. For a ratio of peak to halo of
$\sim~{}10^{8}$ the dilution to the spin-light asymmetry is several orders of
magnitude smaller than the spin-light asymmetry. Thus, for a tightly
controlled beam, halos should not pose a serious challenge.

Most of the systematic uncertainties listed in Table III are independent of
the electron beam energy in the 4 - 20 GeV range, however, because of the
decrease in the SR spot size with increasing electron beam energy the
uncertainty due the slit width scales inversely with beam energy.

The proposed device is best used as a relative polarimeter, however, it can be
used as an absolute device if the lower and upper bounds of the energy
sensitivity of the DIC is determined accurately. The absolute value of the
spin light asymmetry depends on the absolute value of the energy window over
which the DIC signals are integrated. It is especially sensitive to the lower
bound because of the rapid change in the SR intensity and the spin-light
asymmetry as a function of decreasing photon energy. For an absolute
measurement the lower bound of the integration window and the sensitivity to
the vertical motion of the beam would be the two dominant sources of
systematic uncertainty. Excellent energy resolutions have been demonstrated
for HPXe ionization chambers [19, 20, 21]. With such high resolution
ionization chambers one should be able to determine the response function and
the lower bound of energy sensitivity of the chamber to better than 2%. Using
a lower bound of 0.5 MeV, a variation of $\pm$ 10 keV in the lower bound
results in a 2.5% change in the calculated asymmetry. An absolute measurement
would be sensitive to vertical beam motion, a $\pm$ 0.1 mm variation in the
vertical beam position can result in a 4% change in the calculated asymmetry.
Thus, a spin-light polarimeter would only be capable of $\sim$ 5% absolute
polarization measurement.

A table of estimated systematic uncertainties is shown in Table III. We
estimate the systematic instrumental uncertainties of a relative polarimeter
to be $<$ 1%.

TABLE III: Systematic instrumental uncertainties for a relative polarimeter at $E_{beam}=11$ GeV. Source | Uncertainty | $\frac{\delta A}{A}$
---|---|---
Dark current |  $\sim$ pA |  $<$ 0.01%
Intensity fluctuations |  $\Delta N\times$ 10-3 |  $<$0.1 %
Beam energy | 1.0$\times$10-3 |  $<$ 0.05 %
Density of chamber gas, | relative difference |  $<$0.01%
Slit width | 100 $\mu$m |  $<$0.2 %
Background related | known to 0.5% | 0.5 %
dilutions | for B/S $\sim$ 0.02 |
Other dilutions | cancel to first order |  $<$ 0.1%
Halo contributions | $10^{-8}$ |  $<$ 0.1 %
Total |  | 0.6 %

##  V Conclusion

Spin light based polarimetry was demonstrated over 30 years ago, but has been
ignored since then. A spin-light polarimeter has several advantages over
conventional polarimeters and when used in conjunction with a Compton
polarimeter it could help provide a new benchmark for precision polarimetry.
The 11 GeV beam at JLab or the electron beam at a future EIC would be well
suited for spin light polarimetry and such a polarimeter would help achieve
the $<$ 0.5 % polarimetry desired by experiments approved for the 12 GeV era
and proposed for the EIC. A 3 pole wiggler with a field strength of 4 T and a
pole length of 10 cm would be adequate for such a polarimeter. A dual position
sensitive ionization chambers with split anode plates is ideally suited as the
X-ray detector for such a polarimeter. The differential detector design would
help reduce systematic uncertainties.

## Acknowledgment

This work was supported in part by the U.S. Department of Energy under
contract # DE-FG02-07ER41528, by the EIC Detector R&D grant from Brookhaven
National Lab. One of us (P.M.) would also like the thank the Jefferson Science
Associates for a JSA Fellowship.

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Prajwal Mohanmurthy Prajwal Mohanmurthy obtained his Bachelor of Science
degree from Mississippi State University in 2012. He was a graduate research
fellow at the High Performance Computing Collaboratory at Mississippi State
University and currently is a graduate research fellow in the Laboratory for
Nuclear Sciences at Massachusetts Institute of Technology. His research
interests are centered around test of standard model and fundamental
symmetries in search of physics beyond the standard model. His recent research
involvements have been geared towards a search for Axionic Dark Matter and the
precision measurement of the mass of neutrinos. He also actively collaborates
to develop beam instrumentation for up and coming facilities and future
accelerators.
---
Dipangkar Dutta Dr. Dipangkar Dutta is an Associate Professor of Physics at
the Mississippi State University Department of Physics and Astronomy. He
obtained his Bachelor of Technology degree from Indian Institute of
Technology, Bombay in 1992 and his doctoral degree in Physics from
Northwestern University in 1999. He was a post-doctoral and senior post-
doctoral fellow in the Laboratory for Nuclear Sciences at Massachusetts
Institute of Technology. His research is focused primarily on precision
measurement of fundamental properties of nucleons. He is also interested in
precision tests of fundamental symmetries and the Standard Model.
---