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and
Ze + 2AeYe + [Q2(z) + A ]Ye Q2(z)Y> , (4)
where we have replaced the time derivative in Eq. (2) by
D 00t- v(/&z) for the moving proton beam in the
laboratory frame. In deriving Eqs. (3) and (4), we have
also assumed that the incoherent betatron frequency shift
due to the self-fields of the proton beam is negligible, and
that the maximum value of A6 is much smaller than that of
AP, so that wPo = = undepressed betatron frequency.
Note that the damping factors in Eqs. (3) and (4) depend
on the choice of the frequency distribution function.
3 STABILITY ANALYSIS
Since the analysis here is based on the linearized equa-
tions (3) and (4), the subsequent analysis is necessarily
limited to the linear regime. The axial coordinates in the
laboratory and beam frames are denoted by z and z', re-
spectively. We assume that the origins of these two co-
ordinate systems coincide when t 0, and that the head
of the proton bunch is located at z' 0. For simplicity,
relativistic effects are neglected and the Galilean transfor-
mation z' z + vt is used between coordinate systems.
Substituting Y (z', t) e-APtYP(z' t) and Y(z, t)
e-AeYen (z, t) into Eqs. (3) and (4), and tranforming the
proton equation to the beam frame, yieldand
ZPf + W YP> G(z')e(AP Ae)tYen , (5)
Yen + Q2(t - t) [Yt - - eNY,] 0 , (6)where a sharp-edged line density AP is assumed so that
Q(x) 0 for x < 0, and te -z/ is the time when
a slice of electrons located at the position z in the labora-
tory frame enters the proton bunch. Using the variational
method and applying a Fourier transformation, we derive
from Eqs. (5) and (6) the integral equation
Y(z' k) jz'/V Q2(x)G(vx) [4)(s')4'(x)
o (L3 2 k2)W (X)
-'(s')4)(x)] Y(vx, k) dx , (7)
where Y(z', k) = e(k-AP+Ae)s'Y(z', k)/G(z'), s' z'/v
< L/v, and Y(z', k) is the Fourier transform of YP (z' t).
Here, 4)(t) 4) (z, t) and I'(t) = '(z, t) are the linearly
independent solutions of the homogeneous part of Eq. (6),
and W(x) is the Wronskian of 4)(x) and I'(x). In obtain-
ing Eq. (7), we have assumed that Yn= dYn/dt 0 for
t te. Differentiating Eq. (7) twice leads to the following
equation for Y(z', k), i.e.,
v2d2Y/dz'2 _Q2(s'){1+ [G(z')/(k2 -w)]}g . (8)
Thus, the stability analysis has now resulted in solving
Eqs. (7) or (8) and inverting a Fourier transformation. For
non-uniform line densities, Eqs. (7) or (8) have exact so-
lutions only for very limited cases. A possible approxi-
mation in Eq. (7) can be seen by substituting Y(z', k)
((z', k)4)(s') into Eq. (7), which gives((z', k) =/
z/v ((vi, k)Q2(x)G(vx)4)(x)AF(x)
(wj - k2)W(x)x{1 -[1F(s')/4)Qs')][4)(x)/4'(x)]} dx.
(9)
We assume that (i) 4) and I' are such that 4)A' is a smooth
function and 4)/' is an oscillatory function, and (ii) that
the main motion described in ((z', k) are betatron oscil-
lations at much lower frequency than the electron bounce
motion in the beam, so that the contribution from the fast
oscillatory term containing 4)/' in the integration in Eq.
(9) is negligibly small. Then, neglecting the term propor-
tional to 4)/' in Eq. (9) leads to the approximate solution
Y(z', t) ~ {J(z')/[wp(t - s')]3}4G(z') (s')
xexp -(ip + AP)(t - s') - Aes'-(i/4)J(z') + 2wp J(z')(t - s') ]
where
f//v Q2(x)G(vx)
J~z) =i 2W(X) 4)(x) (x) dx .(10)
(11)Similarly, the solution for Y(z', t) in terms of T(s')
can be derived by making the substitution Y(z', k)
((z', k)T(s') in Eq. (7). Equation (10) indicates that the
instability grows in both time and space consistent with
computer simulation results[11]. In the absence of damp-
ing, the asymptotic temporal growth of the instability in the
beam frame is proportional to eat/2 (where the constant a
is independent of t), and the spatial growth is determined
primarily by the quantity J(z'). Further, we see from Eq.
(10) that the e-p mode "wiggles" in space proportional to
4e-2J/4. Note that Eq. (10) also indicates that the pertur-
bation is eventually damped as long as AP is nonzero. This
is due to the combined effects of finite proton bunch length
and the "one-pass" electron-proton interaction.
The following two examples illustrate the applications of
the theory developed here.
Example A: Uniform Line Densities A1. and A0
When both AP and Ae are uniform, Q2(X) x2, and
G(x) &c3, where w is the electron bounce frequency,
and is a constant. Equations (7) and (8) have the exact
solutions 4)(z') exp(iws'y) and '(z') exp(-iLs'y),
where y {1 + [&o$/(k2 w>]}1/2. We concentrate on
the part of the solution containing 4) here. The part con-
taining I' can be treated in the same manner. The solution
for Y1(z',t) is
Y P (z 't) ~ ,oj en ,(s'- t)- oes k(t- s')+ 2 ws' d k (12)
where z' < L. The inverse Fourier transformation in Eq.
(12) can be carried out analytically only in a few parame-
ter ranges by using the steepest descent method. We limit
discussion here to two cases.
The first case corresponds to Q2 < (s'/[wp (t- s')] <
1. In this range, the unstable mode has the approximate
solution
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Synchrotron-based high-pressure research in materials science, article, Date Unknown; [Los Alamos, New Mexico]. (https://digital.library.unt.edu/ark:/67531/metadc934993/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.