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Electron Radiated Power in Cyclotron Radiation Emission Spectroscopy Experiments

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Electron Radiated Power in Cyclotron Radiation Emission Spectroscopy Experiments Electron Radiated Power in Cyclotron Radiation Emission Spectroscopy Experiments A. Ashtari Esfahani,1, ∗ V. Bansal,2 S. B¨oser,3 N. Buzinsky,4 R. Cervantes,1 C. Claessens,3 L. de Viveiros,5 P. J. Doe,1 M. Fertl,1 J. A. Formaggio,4 L. Gladstone,6 M. Guigue,2, y K. M. Heeger,7 J. Johnston,4 A. M. Jones,2 K. Kazkaz,8 B. H. LaRoque,2 M. Leber,9 A. Lindman,3 E. Machado,1 B. Monreal,6 E. C. Morrison,2 J. A. Nikkel,7 E. Novitski,1 N. S. Oblath,2 W. Pettus,1 R. G. H. Robertson,1 G. Rybka,1, z L. Salda~na,7 V. Sibille,4 M. Schram,2 P. L. Slocum,7 Y-H. Sun,6 J. R. Tedeschi,2 T. Th¨ummler,10 B. A. VanDevender,2 M. Wachtendonk,1 M. Walter,10 T. E. Weiss,4 T. Wendler,5 and E. Zayas4 (Project 8 Collaboration) 1Center for Experimental Nuclear Physics and Astrophysics and Department of Physics, University of Washington, Seattle, WA 98195, USA 2Pacific Northwest National Laboratory, Richland, WA 99354, USA 3Institut f¨urPhysik, Johannes-Gutenberg Universit¨atMainz, 55128 Mainz, Germany 4Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 5Department of Physics, Pennsylvania State University, State College, PA 16801, USA 6Department of Physics, Case Western Reserve University, Cleveland, OH 44106, USA 7Wright Laboratory, Department of Physics, Yale University, New Haven, CT 06520, USA 8Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 9Department of Physics, University of California Santa Barbara, CA 93106, USA 10Institut f¨urKernphysik, Karlsruher Institut f¨urTechnologie, 76021 Karlsruhe, Germany (Dated: January 10, 2019) The recently developed technique of Cyclotron Radiation Emission Spectroscopy (CRES) uses frequency information from the cyclotron motion of an electron in a magnetic bottle to infer its kinetic energy. Here we derive the expected radio frequency signal from an electron in a waveguide CRES apparatus from first principles. We demonstrate that the frequency-domain signal is rich in information about the electron's kinematic parameters, and extract a set of measurables that in a suitably designed system are sufficient for disentangling the electron's kinetic energy from the rest of its kinematic features. This lays the groundwork for high-resolution energy measurements in future CRES experiments, such as the Project 8 neutrino mass measurement. PACS numbers: 29.40.-n, 23.40.-s, 52.50.Qt I. INTRODUCTION TO CYCLOTRON purely magnetic trap, eliminating the possibility of a full RADIATION EMISSION SPECTROSCOPY Penning trap configuration. Following the invention of the Penning trap [1], low- Current implementations of CRES [6,7] consist of elec- energy electrons bound with electric and magnetic fields trons produced and trapped inside a waveguide. The have been used to make some of the most precise mea- waveguide propagates the cyclotron radiation emitted by surements of fundamental physics values (e.g., the g−2 of the electrons to a receiver with minimal losses. The back- the electron [2,3]). The success of these measurements ground magnetic field within the waveguide consists of was contingent on a well-developed theory relating the two contributions: a strong, uniform, background field, signal from the axial motion of the electron in the trap which is parallel to the axis of the waveguide, and a mag- to the electron's kinematic parameters [4]. netic distortion which forms the magnetic bottle. Recently it has been proposed [5] that the tech- nique called Cyclotron Radiation Emission Spectroscopy (CRES) be used to make precise measurements of the Here we develop a mathematical description that re- energies of electrons trapped in a magnetic bottle. The lates the characteristics of the apparatus, the motion of arXiv:1901.02844v1 [physics.ins-det] 9 Jan 2019 cyclotron radiation from these particles gives direct in- the electron, and the measured signal. In section II, we formation about their total energy. Since an electric field investigate the variation of the cyclotron frequency due would introduce a position-dependent component to the to the electron's motion. In section III, we derive the particle energy, studies of radioactive decay require a electron's radiation spectrum into the waveguide. In sec- tionIV, we study the effects of signal reflections on the measured radiation spectrum. In sectionV, we apply the formulae from sectionsII, III, andIV to two examples of ∗ [email protected] y Now at Sorbonne Universit´e and Laboratoire de Physique magnetic bottle configurations. Finally, in sectionsVI Nucl´eaireet des Hautes Energies,´ CNRS/IN2P3, 75005 Paris, and VII, we demonstrate that there is sufficient informa- France; [email protected] tion in the signal to reconstruct the kinematic parameters z [email protected] of the electron. 2 II. MOTION AND CYCLOTRON FREQUENCY OF AN ELECTRON IN A MAGNETIC BOTTLE A. The need for a magnetic trap The angular cyclotron frequency, Ωc, of an electron with kinetic energy Ke and mass me, in a magnetic field B, is given by eB eB Ωc = = 2 ; (1) γme me + Ke=c FIG. 1. Axial motion of an electron in a magnetic bottle. where e is the elementary charge, c is the speed of light, The magnetic trap has depth ∆B and maximum value Bmax. and γ the electron's Lorentz factor. For a known mag- The electron's pitch angle is defined as the angle between the netic field, a measurement of the frequency of an elec- electron's momentum vector and the direction of the local tron's cyclotron radiation is also a determination of its magnetic field. If the electron's pitch angle at the bottom of kinetic energy [6]. The frequency resolution of the mea- the trap satisfies Eq. (4), the electron undergoes an oscilla- surement, and therefore the energy resolution, improves tory axial motion inside the trap. The turning point for the with increasing observation time. Therefore a no-work electron corresponds to the position when the pitch angle is trap is necessary for an electron to be observed for a suf- 90 degrees. ficiently long time. trap. For every electron, we define the pitch angle at the bottom of the trap to be θbot. Due to conservation of en- B. Magnetic bottle and pitch angle definition ergy, the condition on pitch angle for a trapped electron is A magnetic bottle consists of a local minimum in the r ! magnitude of background magnetic field. The behavior −1 ∆B θbot ≥ sin 1 − ; (4) of a charged particle in a magnetic bottle has been well- Bmax described [4,8], so here we highlight only the elements important for our results. If we define an electron's in- where Bmax is the maximum value of the magnetic field stantaneous pitch angle, θ(t), as the angle between the and ∆B is the trap depth. local magnetic field and the electron's momentum, then Existing CRES experiments operate at a background the kinetic energy for an electron undergoing cyclotron field of 1 T, which we use for all examples throughout. motion can be decomposed to its parallel and perpendic- A 4 mT trap depth on this background field can trap ular components as electrons with pitch angles greater than 86 degrees. As a consequence, electrons trapped in the magnetic bottle Ke = Kek + Ke? with a pitch angle other than 90 degrees at the bottom of the trap will undergo periodic axial motion as depicted 1 p2 (2) = 0 cos2 θ(t) + µ(t)B(t); in Fig.1. 2 me where p is the magnitude of the electron's initial mo- 0 C. Time-varying cyclotron frequency of an electron mentum and µ is the equivalent magnetic moment of the electron, given by The magnetic field experienced by an electron varies 2 2 1 p0 sin θ(t) with time due to its axial motion, resulting in a time- µ(t) = : (3) varying cyclotron frequency, given by 2 me B(t) eB(t) In the adiabatic regime, where the change in the mag- Ωc(t) = 2 : (5) netic field direction is slow compared with the cyclotron me + Ke=c frequency, an electron's equivalent magnetic moment is Additionally, the electron's cyclotron motion causes it a constant of motion. For the remainder of this deriva- to radiate, reducing its kinetic energy and therefore in- tion µ is treated as time-independent and the term µB(t) creasing its cyclotron frequency. This energy loss can be behaves as a magnetic potential energy. Electrons with expressed as pitch angles of 90 + δθ or 90 − δθ degrees will have the dK (t) same motion; therefore we will only consider electrons e = −P (t); (6) with pitch angles between 0 and 90 degrees. The pitch dt angle approaches 90 degrees for an electron exploring re- where P , the power radiated by the electron, can be as- gions of increasing magnetic field, whereas the pitch angle sumed to be constant over short times. The energy ra- decreases for an electron approaching the bottom of the diated is much smaller than the electrons initial total 3 energy. Therefore, the instantaneous frequency of radia- tion emitted by the electron can be derived from Eq. (1) as 1.2 eB(t) P t 1 Power [a.u.] Ωc(t) ' 2 1 + 2 ; (7) me + K0=c mec + K0 0.8 where K0 is the initial kinetic energy of the electron. 0.6 Because the cyclotron radiation is observed for a finite amount of time, the frequency is shifting by the electron 0.4 P t power loss 2 . mec +K0 Existing CRES experiments operate with midly rela- 0.2 tivistic electrons, we take a 30 keV electron for examples 00 10 20 30 40 50 60 70 throughout.
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