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One of the options open for improving linear ion accelerators is to use focusing by a high-frequency (hf) quadrupole field. The theory of linear accelerators with hf quadrupole focusing has been put forth in a number ofeontributionsto the literature; the reader will find a list of publications and reference list of formulas in [1]. A model of such an accelerator is now under study at IFVE (Institute of High-Energy Physics).

FROM UDC 621.384.643 In carrying out certain methods of multiturn proton injection into a circular accelerator or storage ring, it is necessary to change the average energy of the particles at the exit of the linear acceleratorinjector in an almost linear fashion during a current pulse. In principle, particle energy can be changed simultaneously with beam debunehing. For this purpose, the frequency of the buncher voltage must be shifted with respect to the frequency of the accelerating field in the injector by an amount( 1) where r is the length of the proton current pulse; Ago is the phase spread of the used portion of the voltage variation in the debuncher. The value of/~go is determined by the tolerance in the linearity of the modulation law. For example, for a permissible relative error in the rate of change of the average momentum,equal to 0.1 and a beam pulse length r = 15 psec, we obtain Ago = 0.90 rad and Af = 9.54 kHz.In the variables r = go -gos, g = (P -Ps)/Ps (go and p are the phase and momentum of a particle), let the phase volume of the beam at the exit of the linear accelerator be bounded by the ellipse ~2 g~Then at the exit of the debuncher the phase volume is enclosed by the curvewherel is the drift length to the debuncher; U D and T D are the voltage amplitude and the flight-time factor in the debuncher gap; E 0 is the rest energy of a particle; /3 and T are the relative velocity and energy of the partides; ~ is the wavelength of the hf field in the linear accelerator; go0 is the instantaneous value of the phase of the hf field in the debuncher for which the synchronous particle passes through the center of the debuncher gap, %(t)=2aAl(t----~).(5)The parameter k is positive if the particle bunch arrives in the debuneher on the rising portion of the sinusoidal voltage and is negative for the descending portion.For the following evaluations, we limit ourselves to a linear rise of the debuncher voltage. Then the condition for optimal debunching has the form k = k 0. In this case, the phase volume of the beam at the debuncher exit is enclosed by the ellipse

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