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Atomic Electron Motion for Moller Polarimetry in a Double-Arm Mode


CEBAF–PR–96–003

ATOMIC ELECTRON MOTION ¨ FOR MOLLER POLARIMETRY IN A DOUBLE–ARM MODE
arXiv:hep-ex/9602002v1 2 Feb 1996
Andrei Afanasev?
CEBAF, 12000 Je?erson Ave., Newport News, VA 23606, USA and NuHEP Center, Department of Physics, Hampton University, Hampton, VA 23668, USA

Alexander Glamazdin?
CEBAF, 12000 Je?erson Ave., Newport News, VA 23606, USA

January 31, 1996

Abstract We analyse an e?ect of electron Fermi motion at atomic shells on the accuracy of electron beam polarization measurements with a M¨ller polarimeter operating in a double–arm o mode. It is demonstrated that the e?ect can result in either increase or decrease of the measured polarization depending on the detector positions. The e?ect is simulated for the M¨ller o polarimeter to be installed at CEBAF Hall A.

To be submitted to Nuclear Instruments and Methods

? ?

On leave from Kharkov Institute of Physics and Technology, Kharkov 310108, Ukraine Permanent address: Kharkov Institute of Physics and Technology, Kharkov 310108, Ukraine

1

1

Introduction

M¨ller polarimeters are widely used for electron beam polarization measurements in a GeV energy o range. High quality of polarization experiments anticipated at new–generation CW multi–GeV electron accelerators such as CEBAF requires precise measurements of electron beam parameters. One of these parameters is the electron beam polarization. It can be measured by a M¨ller polarimeter. o There is a number of systematic corrections which should be accounted for when obtaining electron beam polarization from the asymmetry measured with a M¨ller polarimeter. They may o be related to the knowledge of foil magnetization, accidentals, backgrounds, etc. An important systematic correction is due to electron Fermi–motion at atomic shells [2]. This correction is in principle di?erent from the others listed above because it enters on the level of elementery ee-cross section. By analogy with radiative corrections, this correction may be called internal as opposed to external corrections. Therefore, the e?ect is present in any design of the M¨ller polarimeter, o operating in either single–arm or coincidence modes. Before the paper [2] was published, the e?ect of Fermi–motion at atomic shells was belived to negligible. However, the discrepancy between polarization values measured with M¨ller and Compo ton polarimeters at SLC/SLD demonstrated that this correction may be large, it was estimated to be 14% in this particular case [3]. In this paper, we report on some results obtained while developing a M¨ller polarimeter for Hall o A at CEBAF. The paper is organized as follows. Section 2 describes the formalism for polarized ee-scattering and the e?ect of a target electron motion on the spin asymmetry. Section 3 lists basic results, and the conclusions are presented in Section 4.

2

Formalism

M¨ller polarimetry is based on scattering of polarized electron beam on a polarized electron target. o The polarization dependent cross-section for the electron-electron scattering is given by [1]
¨ ¨ dσ M oll dσ M oll = 0 ? (1 + d?? d?

Pib Aij Pjt ),
i,j

(1)

(i, j = x, y, z), where Pib (Pjt ) are components of the beam (target) polarization, Aij are the asymmetry parameters, ¨ and dσ M oll /d?? is the cross–section for the unpolarized particles. Here we use the coordinate system 0 with z–axis the electron beam direction and x–(y-) axis coplanar (normal) to the reaction plane. Using one–photon exchange and the ultrarelativistic limit for the unpolarized cross–section and the nine asymmetry parameters one has
M¨ α2 ?2 (4 ? sin2 Θ? )2 dσ0 oll = γ , d?? 4m2 sin4 Θ?

(2) (3)

Azz = ?

(7 + cos2 Θ? ) sin2 Θ? , (3 + cos2 Θ? )2 2

?Axx = Ayy = Azx = Axz = ?

sin4 Θ? , (3 + cos2 Θ? )2 2 sin3 Θ? cos Θ? , γ(3 + cos2 Θ? )2

(4) (5) (6)

Axy = Ayx = Azy = Ayz = 0,

(E0 + m) , 2m where α is the ?ne–structure constant, Θ? is the c.m.s. scattering angle, m is the electron mass, and E0 is the energy of the incident electron in the laboratory system. It is seen that at Θ? = 900 the asymmetry parameters Axx , Ayy and Azz are maximal γ= 7 1 1 Azz = ? , Axx = ? , Ayy = , (7) 9 9 9 and the asymmetries Axz and Azx are small within the whole angular acceptance and vanish at 900 . In experiment, the polarized electron beam is incident on a magnetized ferromagnetic foil. The observed symmetry of M¨ller scattering from atomic electrons, o A= N↑↑ ? N↑↓ , N↑↑ + N↑↓ (8)

gives the desirable polarization of the electron beam provided the target polarization is known. Let us consider M¨ller scattering on a moving target electron from a particular atomic shell n. o In the laboratory frame, cosine of the scattering angle is given by cos Θ = cos Θ0 + (? cos Θ)F , (9)

where Θ (Θ0 ) is a lab. scattering with (without) target electron motion, and the (? cos Θ)F is a correction due to Fermi–motion of the target (denoted by PF ), and neglecting higher–order terms ?1 in expansion over E0 , we obtain
n PF (1 ? cos Θ0 ), (10) m where Θ12 is an angle between the momenta of the beam and the target electron. At Θ? = 900 , Eq.(10) becomes

(? cos Θ)F = cos Θ12

(? cos Θ)F = cos Θ12

n PF , E0

(11)

this result was reported earlier [4]. Eq.(11) can also be rewritten in the form Θ = Θ0 1 ?
n PF cos Θ12 , m

(12)

3

In the leading order of PF /m expansion, it reproduces the original L. Levchuk’s result (Eqs. 9-10 of Ref.[2]). However, for the case of large acceptances it is more consistent to use a general result Eqs. (9)-(10) herein, because angles far from Θ? = 900 are involved. Let us summarize the obtained results. Atomic electron motion does not a?ect the values of cross section and scattered electrons energies but it changes the angle of M¨ller scattering. A dominant e?ect comes from target electron o motion parallel (antiparallel) to the direction of the incident beam; an e?ect from transverse target motion is supressed by an extra factor of 2m/E0 . Using the changed M¨ller kinematics described o by Eqs.(9)–(10), one should take a sum over all atomic shells and integrate over Θ12 and PF using √ n a proper atomic wave function. To estimate the e?ect, we assume what PF = 2mεn , εn being an B B electron binding energy at the atomic shell n. Electrons in the atom of iron have the binding energies [5] listed in Table 1. Table 1. Electron binding energies for Shell εn , eV B Number of electrons K(1s) 7112 2 LI (2s) LII (2p1/2 ) 846.1 2 721.1 3
26

F e (atomic moment=2.22?B ) MI (3s) MII,III (3p) MIV,V (3d) + NI (4s) 92.9 2 54.0 6 3.6±0.9 6(3d) + 2(4s) 708.1 3

LIII (2p3/2 )

Thus, scattering from K- and L-shells smears the M¨ller scattering angle by ?Θ/Θ ? ±10% and o ±3% respectively, around Θ? = 900 . Only the electrons on the incomplete M-shell are polarized. (Iron needs 4 more electrons on 3d-shell to complete it. Overlapping 3d and 4s levels in metal yield the observable hyromagnetic ratio of 2.22?B ). The M¨ller asymmetry for electron-atom scattering o may be presented as A= Σ(σn (↑↑) ? σn (↑↓)) , Σ(σn (↑↑) + σn (↑↓)) (13)

where the sum is taken over all n atomic shells, and σn (↑↑) or σn (↑↓) correspond to the M¨ller cross o section with spins parallel or antiparallel, respectively. Only polarized (loosely bound) electrons from the M–shell contribute to the numerator in Eq.(13) r.h.s., whereas all electrons including strongly bound K–, L–shell ones, contribute to the denominator of Eq.(13) r.h.s. Therefore, kinematic smearing due to Fermi–motion a?ects the denominator, rather than the numerator for the asymmetry expression Eq.(13). For a standard geometry of M¨ller polarimeters,when the detector(s) is(are) centred around o Θcm = 900 , it may result in missing electrons scattered from K–, L–shells yielding a higher theoretical estimate for the asymmetry A. It was the basic conclusion of [2] con?rmed later in SLAC/SLC measurements [3]. However, in contrast to original predictions [2], this e?ect does not exceed 2% for single–arm M¨ller measurements at MIT [6]. For the double–arm M¨ller polarimeter at MIT o o [7] a preliminary estimate done by one of us [4] predicts the correction to the M¨ller asymmetry to o be around 12% , but this estimate may change if the magnetic ?eld of the polarimeter quadrupole and boundary conditions are carefully included into the calculations. We found the e?ect to be strongly dependent on positioning the detectors in a double–arm M¨ller polarimeter. o

4

3

Dependence on the detector positions

The ’smearing’ of the kinematics due to target electron motion changes asymmetry of the M¨ller o scattering. This e?ect should be carefully calculated for any speci?c experimental set–up, since the magnitude of the e?ect depends on target thickness (via multiple scattering), polarimeter acceptances, magnetic optics, electron beam parameters, etc. In our study, we have found a new e?ect due to Fermi–motion of atomic electrons. The e?ect is a dependence of the measured M¨ller asymmetry on the relative position of detectors in a double–arm o mode. This e?ect can be understood from Figs. 1 and 2. Fig. 1 demonstrates the ratio of M¨ller o asymmetry neglecting the e?ect of electron Fermi–motion to the same quantity but with the Fermi– motion included in the calculation, as a function of the detectors displacement from the symmetric around Θ? = 900 positions. Note that for the chosen electron energies and target thickness, the angular smearing due to Fermi–motion for K–electrons is an order of magnitude larger than due to multiple scattering in the target. Di?erent displacements of the detectors in a double–arm mode result in a di?erent e?ect due to Fermi motion on the M¨ller asymmetry. The correction is positive o and reaches its maximum value for the case of symmetric position of the detectors, centered at the angle corresponding to 900 –scattering angle in c.m.s.(denoted A in Fig. 2.) If the detectors are moved simultaneously toward larger (position B) or smaller (position C) angles, the Fermi–motion correction to the M¨ller asymmetry becomes negative. In absence of multiple scattering and the o displacements lager than B or smaller than C (e.g., position D), the Fermi–motion correction to the asymmetry would be exactly –100%. It means that we can observe a zero M¨ller asymmetry o scattering polarized electron beam on atomic electrons with nonzero net polarization! The reason is that for this geometry of the experiment, we detect only the electrons scattered on unpolarized atomic shells. This polarization asymmetry of the detected electrons if |?Θd /?Θacc | ≥ 0.5 is completly due to multiple scattering of electrons in the target. It may provide a direct measure of multiple scattering e?ect (i.e., target thickness). Further displacing the detectors, the multiple scattering e?ect dies o? exponentially, and the asymmetry of the electrons detected in coincidence approaches zero, but we may still observe a considerable amount of M¨ller electrons in coincidence o coming from the tails of momentum distributions for atomic electrons. The calculations in Fig. 1 were done for illustration, treating the polarimeter schematically as a target + a pair of detectors, with no magnets involved in the system. The acceptance (normalized to unity), the Fermi–motion smearing and multiple scattering angles were the same as for the realistic case described below. Simulation of the Fermi–motion e?ect was done for the M¨ller polarimeter of CEBAF Hall A. o A detailed description of this polarimeter is given in Ref. [8]. The polarimeter is designed for coincidence mode operation. The magnetic system includes two quadrupoles and a dipole. The simulation was done by RAYTRACE combined with a Monte–Carlo code for simulating M¨ller o and multiple scattering. The results of the simulation are presented in Figs. 3, 4 for the electron beam energy E0 = 0.8 GeV and target thickness = 17.6?m. The distribution of M¨ller electrons o in the detector plane is demonstrated in Fig.3 for one of the detectors. For the other detector, the distribution is symmetric. The axis X is perpendicular to the reaction plane, and Y –axis indicates displacement with respect to the beam axis. The square area dashed with lines having a positive slope demonstrates the detector centered at Y0 corresponding to Θ? = 900 (ΘA = 35.7 mrad). It provides the angular acceptance ?Θ/Θ=10.5%. The square dashed with lines having a negative slope demonstrates the detector displaced along the Y –axis by distance d. The second–arm 5

detector is displaced symmetrically with respect to the beam axis. The e?ect of this displacement is shown in Fig.4 and appears to be in qualitative agreement with the results of Fig.1 obtained for a simpli?ed model of the polarimeter. As can be seen from Fig.4, the Fermi–motion e?ect is positive and maximal, reaching ≈10%, for small displacement d, wereas for large values of d, the e?ect is negative and may reach the magnitude of ?100% completely eliminating the M¨ller asymmetry. o The plot in Fig.4 is asymmetric with respect to d=0 due to dipole dispersion (it would become symmetric if plotted vs. angular shifts). The plateau (instead of a maximum like in Fig.1) is caused by additional boundary conditions set by the polarimeter magnetic system acting like an e?ective collimator. It should be noted that the larger is the angular acceptance, the less Fermi motion a?ects the measured M¨ller asymmetry. We choose the beam energy E0 = 0.8 GeV for illustration because for o higher energies, the angular acceptance of the polarimeter becomes large (25–42% for E0 =1.6–6.0 GeV) and the maximum positive Fermi–motion e?ect is small, not exceeding 3%.

4

Summary

We studied an e?ect of atomic electron Fermi–motion for a double–arm M¨ller polarimetry. We o demonstrate that this e?ect may be either positive or negative depending on positioning of the detectors. If the detectors are centered at 900 – scattering angle (in c.m.s.), the correction has a maximum positive value. For detectors shifted simultaneously toward larger/smaller angles, the e?ect becomes negative and may reach –100% completely eliminating the observed spin asymmetry. On the other hand, for a single–arm measurement the Fermi–motion correction remains positive despite the shift in the detector position. Magnetic ?elds may essentially change electron kinematics, therefore, it is necessary to do detailed simulation of the polarimeter optical system in order to consistently calculate the electron Fermi–motion e?ect.

Acknowledgements
This work was supported by the US Department of Energy under contract DE–AC05–84ER40150. A.A. would like to acknowledge useful discussions with his colleagues at CEBAF, Hampton University and KhPTI. A.G. would like to thank V.G. Gorbenko for collaboration in writing the Monte–Carlo simulation code for CEBAF Hall A M¨ller polarimeter. o

6

References
[1] A.A. Kresnin and L.N. Rosentsveig, Journ. Exp. Theor. Phys. (USSR) 32, 1957, p.9353 (Soviet Physics JETP (1957) 288). [2] L.Levchuk, The intra–atomic motion of bound electrons as a possible source of a systematic error in electron beam polarization measurements by means of a M¨ller polarimeter, Preprint o KhPTI 92–32, Kharkov, 1992; Nucl. Instr. and Meth. A311 (1994) 496. [3] M. Swartz, H.R. Band, F.J. Decker et al., Nucl. Instr. and Meth. A363 (1995) 526. [4] A. Afanasev, Parallel talk at Spin ’94, Bloomington, IN, September 1994, unpublished. [5] AIP Handbook, (1993). [6] J. Arrington, E.J. Beise, B.W. Fillipone et al., Nucl. Instr. and Meth. A311 (1992) 39, and B. Fillipone, private communication. [7] K.B.Beard, R. Madey, W.M. Zhang et al., Nucl. Instr. and Meth. A361 (1995) 46. [8] M¨ller polarimeter for Hall A, A.Glamazdin, V.Gorbenko et al., CEBAF, January 1996. o

Figure Captions
Figure 1. Fermi–motion correction to the M¨ller asymmetry for 26 F e target as a function of o detectors displacement, ?Θd being the displacement angle with respect to Θ? = 900 , ?Θacc being the angular acceptance, and A(A0 ) being the asymmetry with (without) the Fermi–motion correction. Figure 2. Schematic positions of the detectors. The angle ΘA corresponds to Θ? = 900 , and the positions A, B, C, and D correspond to ?Θd /?Θacc = 0, 0.5,–0.5, >0.5, respectively, in Fig.1. Figure 3. Simulated distribution of M¨ller events in the detector plane of CEBAF Hall A o M¨ller polarimeter. Dashed square areas show di?erent positions of the detectors. The notations o are explained in the text. Figure 4. Simulated Fermi–motion e?ect for CEBAF Hall A M¨ller polarimeter as a function o of the detectors displacement d, as shown in Fig.3.

7

Position A
p p p p p p p p p p p p p

p p p

Detector 1

Position C
p p p p p p p p p p p p p

p p p

Detector 1

Target
p p p t p p p p

p

A
p p p p p p p p p p p p

A

Beam -

Target
p p t p p p

p

p

A
p p p p p p p p p p p p

p

A

Beam -

p p p p p

p p

Detector 2
p p p

Detector 2

Detector 1

Detector 1

Position B
p p p p p p p p p p p p p

p p p

Position D
p p p p p p p p p p p p p

p p p

Target
p p p t p p p p

p

A
p p p p p p p p p p p p

A

Beam -

Target
p p t p p p

p

p

A
p p p p p p p p p p p p

p

A

Beam -

p p p p p

p p p p p

Detector 2

Detector 2



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