Summary
Spin motion equations in a rotating rest frame in which the particle’s velocity and acceleration vectors are instantaneously at rest are derived. These equations are used to calculate the corrections due to betatron oscillations to theg - 2 frequency in weak focusing storage devices. The corrections are found up to second order in the (nonlinear) focusing field. Nonresonant and resonant solutions are derived and both magnetic and electric focusing fields are treated. Quantitative estimates are also given of depolarization effects near resonance.
Riassunto
Si deducono le equazioni di moto per lo spin in un sistema di riferimento in cui i vettori velocità ed accelerazione sono costanti del moto. Si usano queste equazioni per calcolare le correzioni alla frequenza dig - 2 in un sistema di accumulazione a fuocheggiamento debole. Si ricavano le correzioni al secondo ordine nel campo fuocheggiante (non lineare). Si discutono soluzioni non risonanti e risonanti, trattando sia il fuocheggiamento elettrico che magnetico. Inoltre si dà una valutazione quantitativa degli effetti di depolarizzazione in prossimità della risonanza.
Реэюме
Выводятся уравнения движения спина во врашаюшейся системе координат, в которой векторы скорости и ускорения частицы в данный момент равны нулю. Эти уравнения испольэуются для вычисления поправок, обусловленных бетатронными колебаниями, к частотеg - 2 в слабо фокусируюших накопительных устройствах. Определяются поправки вплоть до второго порядка по (нелинейному) фокусируюшему полю. Выводятся нереэонансные и реэонансные рещения. Рассматриваются магнитные и злектрические фокусируюшие поля. Также приводятся деполяриэапионные зффекты вблиэи реэонанса.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Fierz andV. L. Telegdi:Quanta, edited byP. G. O. Freund (Chicago, Ill., 1970), p. 206, eq. (18).
S. Granger andG. W. Ford:Phys. Rev. Lett.,28, 1479 (1972).
J. Bailey, F. J. M. Farley, H. Jöstlein, G. Petrucci, E. Picasso andF. W. Wickens: CERN proposal PH I/COM-69/20 (1969).
F. J. M. Farley:Phys. Lett.,42 B, 66 (1972).
V. Bargmann, L. Michel andV. L. Telegdi:Phys. Rev. Lett.,2, 435 (1959).
As a concrete example of this, consider the effects of pitch motion in the most recent Michigan electrong - 2 experiment (6). Hereω/2π=ω p /2π=1.5 MHz,ω 0 /2π=3.8 MHz. In this caseω min corresponds ton′=1,n=5 and givesω min/2π=0.1 MHz. So even if the field is sufficiently nonlinear for terms withm±m′=5 to be important onlyT ≳ 10 µs is required, compared with ∼ 2000 µs storage time of a typical data run. If terms withm±m′=5 give a negligibly small contribution to\(\dot \varphi \), then an even smaller averaging period is sufficient.
J. C. Wesley andA. Rich:Phys. Rev. A,4, 1341 (1971).
Note added in proofs. An exhaustive discussion of depolarization effects near spin resonances for storage rings and accelerators has been given byYa. S. Derbenev, A. M. Kondratenko andA. N. Skrinskij: Novosibirsk preprint I Ya. F 44–70 (1970). We thank these authors for pointing out this paper to us.
D. T. Wilkinson andH. R. Crane:Phys. Rev.,130, 852 (1963).
E. Klempt: Ph. D. Thesis, University of Bonn (1969) (unpublished).
J. Bailey andE. Picasso:Progr. Nucl. Phys.,12, 43 (1970).
J. Bailey, W. Bartl, G. von Bochmann, R. C. A. Brown, F. J. M. Farley, M. Giesch, H. Jöstlein, S. Van der Meer, E. Picasso andR. W. Williams:Nuovo Cimento,9 A, 369 (1972).
H. Goldstein:Classical Mechanics (Reading, Mass., 1959), p. 107.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Field, J.H., Fiorentini, G. Corrections to theg - 2 Frequency in weak focusing storage devices due to betatron oscillations. Nuov Cim A 21, 297–328 (1974). https://doi.org/10.1007/BF02724809
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02724809