Lifetime Studies at Metrology Light Source and ANKA

The Metrology Light Source MLS , situated in Berlin Germany is an electron storage ring operating from 105 MeV to 630 MeV and is serving as the national primary radiation source standard from the near infrared to the extrem ultraviolet spectral region [1]. In its standard user mode, the lifetime is dominated by the Touschek effect. Measurements and analysis of the Touschek lifetime as a function of beam current and RF Voltage will be presented and compared to measurements done at the A amp; 730; ngstro mquelle Karlsruhe ANKA electron storage ring Karlsruhe, Germany which operates at 0.5 to 2.5 GeV [2

Applying a Sociopolitical Equity Framework to a Developmental Algebra Curriculum

The Thinking With Algebra (TWA) curriculum, a National Science Foundation Project (DUE 2021414), applies a sociopolitical equity framework (Gutiérrez, 2012) to strengthen student participants’ mathematics achievement outcomes and mathematics identity development. Instructor and student data indicate improvements in these two dimensions. This project also expands knowledge about creating equitable math learning environments that support underrepresented students’ mathematical success

The Multiple Usages of Thinking With Algebra

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Students\u27 Perspectives of Thinking With Algebra

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Characteristics of two outstanding elementary teachers of mathematics: Implications for teacher education

This study identifies characteristics of two “outstanding” elementary teachers of mathematics who were different in style and who taught in different settings. The intent is to determine what characteristics make these different teachers outstanding in hopes of helping preservice teachers improve their teaching of mathematics. Preliminary findings indicate that both teachers were (1) focused on children’s learning of mathematics; (2) focused on the mathematical solution methods used by students; (3) believed that all of their students could learn mathematics; (4) were enthusiastic and dedicated to the profession of teaching; and (5) cared deeply about their students and emphasized the necessity of building relationships with them

Analytic calculation of the electric field of a coherent THz pulse

Accelerators can serve as sources for intense THz radiation by producing sub-ps long electron bunches, which generate synchrotron radiation coherently in the THz regime. In this paper, we present an analytic method to calculate the electric field pulse emitted by an electron bunch with arbitrary bunch shape. Our method is applied to a Gaussian bunch as well as a complex bunch profile resulting from a nonlinear beam dynamics simulation

Lifetime Studies at Metrology Light Source and ANKA

Abstract The Metrology Light Source (MLS), situated in Berlin (Germany) is an electron storage ring operating from 105 MeV to 630 MeV and is serving as the national primary radiation source standard from the near infrared to the extrem ultraviolet spectral region INTRODUCTION To provide users of synchrotron radiation with temporally stable experimental conditions, the lifetime τ of the stored beam with current I is a parameter of concern. This is valid for machines with a decaying beam such as ANKA and MLS, but as well for machines operated in top up mode such as BESSY II in Berlin. In 2012, the standard user operation at MLS yielded a lifetime of 3.5 hours at 150 mA beam current. Although reasonable due to the energy, this is a low value compared to 16 hours at ANKA and it would benefit the users of synchrotron radiation if it could be improved. THEORY There are two major loss mechanisms determining the lifetime of the electrons in an accelerator: The scattering of the electrons with residual gas atoms and the scattering of the electrons with other electrons within the bunch. The latter is known as the "Touschek effect", named after Bruno Touschek who first observed the effect at the small AdA electron-positron collider. The two contributions are called gas lifetime and Touschek lifetime respectively. The gas lifetime depends on the pressure P and a scattering cross section σ gas for particle losses, for which the interested reader is referred to The cross section itself is a function of the acceptance of the accelerator δ acc = Δp max /p 0 , while the pressure P depends in some respect on the beam current. The electrons in a bunch perform transverse betatron oscillations. Being an incoherent motion, this leads to * [email protected] Coulomb scattering. During the scattering process, transverse momentum gets transferred to longitudinal momentum. If the particles momentum deviation exceeds the momentum acceptance it will be lost. The resulting Touschek lifetime depends on the rate of scattering processes and therefore on the density within the bunch, i.e. on the bunch volume and the bunch current. Furthermore, it depends on the momentum acceptance with the power of three with σ x,y,s being the rms-bunch sizes and length and D(ξ) being a slowly varying function with respect to the acceptance δ acc . D also depends on the optical functions around the ring through ξ. The loss rates from Touschek effect and gas scattering add to the total loss rate 1/τ which can be measured. Multiplying the number of particles N (or the stored current I) to the Touschek lifetime τ T results in a constant: N · τ T = const. Therefore, when plotting I · τ for a Touschek dominated lifetime a constant can be expected with respect to current. Acceptance Touschek lifetime and gas lifetime depend on the acceptance of the accelerator. Two acceptances are important here, and whichever is the smallest is the limiting one: • RF-acceptance • Geometrical acceptance. The RF-acceptance δ acc,RF approximately depends on the applied cavity voltage V as [4] The geometrical acceptance depends on the minimal aperture of the vacuum chamber a(s) and the dimensions of the beam. For MLS a first order approximation considering only the horizontal plane is: with D x being the horizontal dispersion function. In the upper plot of Predictions In the lower part of EXPERIMENT In LIFETIME IMPROVEMENT The lifetime at MLS can be improved if the acceptance is only RF-limited even to larger voltages than 300 kV. To do so, the geometrical acceptance has to be improved. In order to find optics with an increased Touschek lifetime, brute force optics scans using a Fortran code were performed In Eq. 4, the horizontal dispersion function D x is in the denominator. By decreasing the dispersion function at the place with minimum aperture, the geometrical acceptance δ acc,geom can be improved. At MLS each quadrupole is powered independently. By tuning some quadrupoles of one family against the remaining ones of that family, the dispersion function was tuned to be zero at the septum. In The peak lifetime with respect to cavity voltage is now located at 500 kV, being the maximum applicable cavity voltage at the moment. In The total lifetime increase is as high as 80 %. By solely increasing the acceptance, and the change in optical functions, the lifetime would have been expected to increase by about 30 %. An explanation for the additional increase could be a strong halo around the beam due to intra-beam scattering. With this, the effect of leading the beam through the centre of the vacuum chamber at the septum could be explained as well. CONCLUSION AND OUTLOOK The theory of the Touschek effect describes the dependencies on energy and acceptance well. By understanding the different loss mechanisms and the methods to manipulate the different acceptances, it was possible to generate a new user optics with an by 80 % improved lifetime. To completely explain the total lifetime increase, further measurements are needed. To further increase the lifetime, alternate optics determined by optics scans will be tested ACKNOWLEDGMENT The authors like to thank Andreas Jankowiak (HZB) and Gerhard Ulm (PTB) for supporting this work. REFERENCES [1] R. Klein et al., Phys. Rev. ST-AB 11, 110701, 2008. [2] A.-S. Müller et al., "Energy Calibration Of The ANKA Storage Ring", Proceedings of EPAC 2004. [3] T. Goetsch, "Lifetime Studies at Metrology Light Source and Angströmquelle Karlsruhe" -Diploma Thesis, Karlsruhe Institut für Technologie, May 2013. [4] M. Sands, "The Physics Of Electron Storage Rings -An Introduction", National Technical Information Service, Springfield, Virginia, 1970. [5] J. le Duff, "Current And Current Density Limitations In Existing Electron Storage Rings", In: Nucl. Instr. and Meth. in Physics Research, 1985

Thinking with Algebra: A Project and Perspective

Thinking with Algebra (TWA) is a project to develop algebra curriculum for college classes. The overall goal is to develop curricular materials that will help prepare students conceptually for college algebra. The project is described and the importance of algebraic structure as a theoretical consideration is explained