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Performing a change of variables the ~ and ¢ integrals can be factored. If, further, the betatron phase is approximated as varying linearly near the interaction point, so that y(t)~ct /By—@9Q,t (modulo nT), then the Hamiltonian can be written as

Figure 1 Performing a change of variables the ~ and ¢ integrals can be factored. If, further, the betatron phase is approximated as varying linearly near the interaction point, so that y(t)~ct /By—@9Q,t (modulo nT), then the Hamiltonian can be written as

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TABLE I. Operating tunes for the different resonances. ),=0.11.
TABLE I. Operating tunes for the different resonances. ),=0.11.
FIG. 1. Emittance against the number of kicks used in the distributed force treatment. Cases 1 and 2 are for the simula- tion of Sec. ITI, with no longitudinal radiation, while cases 3 and 4 are for Sec. IV, with longitudinal radiation and a longitudinal- ly distributed weak beam. The cases correspond to (1) 40, —Q,=3 line, c?#/By>=3, o,/By=0.33, (2) 6Q,=2 line, c7/By=1, o,/By=1, (3) 4Q,=3 line, o)/By)=0.33, (4) 62, —Q, =2 line, 7; /Bo= 1.
FIG. 1. Emittance against the number of kicks used in the distributed force treatment. Cases 1 and 2 are for the simula- tion of Sec. ITI, with no longitudinal radiation, while cases 3 and 4 are for Sec. IV, with longitudinal radiation and a longitudinal- ly distributed weak beam. The cases correspond to (1) 40, —Q,=3 line, c?#/By>=3, o,/By=0.33, (2) 6Q,=2 line, c7/By=1, o,/By=1, (3) 4Q,=3 line, o)/By)=0.33, (4) 62, —Q, =2 line, 7; /Bo= 1.
TABLE II. Typical parameters used in the simulations.
TABLE II. Typical parameters used in the simulations.
FIG. 2. Emittance as a function of synchrotron amplitude in the impulse model for (a) the fourth-order and (b) the sixth- order betatron resonance, and their first two synchrotron side- bands.
FIG. 2. Emittance as a function of synchrotron amplitude in the impulse model for (a) the fourth-order and (b) the sixth- order betatron resonance, and their first two synchrotron side- bands.
FIG. 3. Emittance as a function of synchrotron amplitude for he 4Q,—@Q,=3 resonance, for three different strong-beam yunch lengths, in the distributed model. The impulse-model ‘urve is also shown for comparison.
FIG. 3. Emittance as a function of synchrotron amplitude for he 4Q,—@Q,=3 resonance, for three different strong-beam yunch lengths, in the distributed model. The impulse-model ‘urve is also shown for comparison.
FIG. 4. Emittance as a function of strong beam bunch length in the distributed model for (a) the fourth-order and (b) the sixth-order betatron resonance, and their first two synchrotron sidebands. c?/B,~=1.
FIG. 4. Emittance as a function of strong beam bunch length in the distributed model for (a) the fourth-order and (b) the sixth-order betatron resonance, and their first two synchrotron sidebands. c?/B,~=1.
FIG. 5. Emittance as a function of bunch length for the 4Q,..—Q,=3 and 6Q, =2 resonances, in the distributed and im- pulse models. Here the two beams have equal bunch lengths (o,=0o, =0,), and radiation has been incorporated longitudi- nally.
FIG. 5. Emittance as a function of bunch length for the 4Q,..—Q,=3 and 6Q, =2 resonances, in the distributed and im- pulse models. Here the two beams have equal bunch lengths (o,=0o, =0,), and radiation has been incorporated longitudi- nally.
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