**Self-Bunching of a Coasting Beam in the Accumulator**

**ProtonPlan2-doc-198**

Dave McGinnis

February 4, 2007

[[Beam Admittance of a Coasting Beam]{.underline}]{.smallcaps}

The first part of this note is almost a direct copy of Sacherer's
formulation[^1]. Assume that the cavity is located azimuthally φ in the
ring. The single particle equations of motion are:

$\frac{\text{dE}}{\text{dt}} = \frac{q}{T_{o}}V(t)2\pi\sum_{n}^{}{\delta(\varphi - 2n\pi)} = \frac{q}{T_{o}}V(t)\sum_{n}^{}e^{- \text{jn}\varphi}$
(1)

$\frac{d\varphi}{\text{dt}} = \omega_{o} + \frac{\text{ηω}_{o}}{\beta^{2}\gamma}\frac{E}{E_{o}} = \omega_{o} + \kappa E$
(2)

For a distribution of particles ψ(φ,E,t) in φ-E phase space, the
continuity equation is:

$\frac{\partial\psi}{\partial t} + \frac{\partial\left( \overset{˙}{\varphi}\psi \right)}{\partial\varphi} + \frac{\partial\left( \overset{˙}{E}\psi \right)}{\partial E} = 0$
(3)

Reduces to the Vlasov equation

$\frac{\partial\psi}{\partial t} + \left( \omega_{o} + \kappa E \right)\frac{\partial\psi}{\partial\varphi} + \frac{q}{T_{o}}V(t)\sum_{n}^{}e^{- \text{jn}\varphi}\frac{\partial\psi}{\partial E} = 0$
(4)

If the voltage is small, we can break the solution into a stationary
solution plus a small perturbation:

$\psi(\varphi,E,t) = \psi_{s}(E) + \psi_{x}(\varphi,E,t)$ (5)

$\psi_{x}(\varphi,E,t) = \sum_{n}^{}{\psi_{n}(E,t)}e^{- \text{jn}\varphi}$
(6)

$\frac{\partial\psi_{n}}{\partial t} - \text{jn}\left( \omega_{o} + \kappa E \right)\psi_{n} = - \frac{q}{T_{o}}V(t)\frac{\partial\psi_{s}}{\partial E}$
(7)

Assume that the voltage is a sine wave. For small perturbations, the
density modulation will also be a sine wave:

$\begin{array}{r}
V(t) = \text{Re}\left\{ \overset{\sim}{V}(\omega)e^{j\omega t} \right\} \\
\psi_{n}(E,t) = \text{Re}\left\{ {\overset{\sim}{\psi}}_{n}(E,\omega)e^{j\omega t} \right\}
\end{array}$ (8)

The amplitude of the density modulation is:

${\overset{\sim}{\psi}}_{n} = j\frac{q}{T_{o}}\overset{\sim}{V}(\omega)\frac{\frac{\partial\psi_{s}}{\partial E}}{\omega - {n\omega}_{o} - n\kappa E}$
(9)

The distribution density is normalized to unity:

$\iint_{}^{}{\psi(\varphi,E,t)}d\varphi\text{dE} = 1$ (10)

The beam current is ω~o~ times the charge per radian

$\begin{array}{r}
I(\varphi,t) = \text{qN}\omega_{o}\int_{}^{}\psi(\varphi,E,t)\text{dE} \\
I_{\text{DC}} + \text{Re}\left\{ \sum_{n}^{}{e^{j(\omega t - n\varphi)}{\overset{\sim}{I}}_{n}(\omega)} \right\}
\end{array}$ (11)

where

${\overset{\sim}{I}}_{n}(\omega) = \text{qN}\omega_{o}\int_{}^{}{{\overset{\sim}{\psi}}_{n}(E,\omega)}\text{dE}$
(12)

At the location φ=0, the admittance of the beam at frequency ω is:

$Y_{b}(\omega) = \sum_{n}^{}Y_{n}(\omega) = \sum_{n}^{}\frac{{\overset{\sim}{I}}_{n}(\omega)}{\overset{\sim}{V}(\omega)}$
(13)

$Y_{n}(\omega) = - j\frac{{q\omega}_{o}}{n\kappa}I_{\text{dc}}\int_{}^{}{\frac{\frac{\partial\psi_{s}}{\partial E}}{E - \frac{\omega - {n\omega}_{o}}{n\kappa}}\text{dE}}$
(14)

$Y_{n}(\omega) = \frac{{q\omega}_{o}}{n\kappa}I_{\text{dc}}\left( \pi\frac{\partial\psi_{s}}{\partial E}|_{E = \frac{\omega - {n\omega}_{o}}{n\kappa}} - \text{jP}\text{.}V\text{.}\left( \int_{}^{}{\frac{\frac{\partial\psi_{s}}{\partial E}}{E - \frac{\omega - {n\omega}_{o}}{n\kappa}}\text{dE}} \right) \right)$
(15)

[[Beam Stability]{.underline}]{.smallcaps}

If the energy spread is not large then Y~n~ is the beam impedance around
the frequency ω\~nω~o~. If there is a narrowband impedance in the ring,
Z~cav~, the a current modulation of the beam will produce a voltage:

$\overset{\sim}{V}(\omega) = \left( {\overset{\sim}{I}}_{\text{ext}}(\omega) + {\overset{\sim}{I}}_{\text{beam}}(\omega) \right){\overset{\sim}{Z}}_{\text{cav}}(\omega)$
(16)

In turn, the voltage will modulate the current:

${\overset{\sim}{I}}_{\text{beam}}(\omega) = {\overset{\sim}{Y}}_{b}(\omega)\overset{\sim}{V}(\omega)$
(17)

The modulation voltage becomes

$\overset{\sim}{V}(\omega) = \frac{{\overset{\sim}{Z}}_{\text{cav}}(\omega)}{1 - {\overset{\sim}{Z}}_{\text{cav}}(\omega){\overset{\sim}{Y}}_{b}(\omega)}{\overset{\sim}{I}}_{\text{ext}}(\omega)$
(18)

which is stable for

$\text{Re}\left\{ {\overset{\sim}{Z}}_{\text{cav}}(\omega){\overset{\sim}{Y}}_{b}(\omega) \right\} < 1$
(19)

The impedance of a cavity is given as:

${\overset{\sim}{Z}}_{\text{cav}}(\omega) = \frac{j\text{ωω}_{c}\frac{R_{\text{cav}}}{Q}}{\left( \omega_{c}^{2} - \omega^{2} \right) + j\frac{\text{ωω}_{c}}{Q}}$
(20)

If the cavity resonant frequency is centered on harmonic **n**
(ω~c~=nω~o~) and the cavity bandwidth is much larger than the frequency
spread of the beam then:

$1 > R_{\text{cav}}\frac{{q\omega}_{o}}{n\kappa}I_{\text{dc}}\pi\frac{\partial\psi_{s}}{\partial E}|_{E = \frac{\omega - {n\omega}_{o}}{n\kappa}}$
(21)

Assume that the distribution function is Gaussian:

$\psi_{s} = \frac{1}{(2\pi)^{1\text{.}5}}\frac{1}{\sigma_{E}}e^{- \frac{1}{2}\left( \frac{E}{\sigma_{E}} \right)^{2}}$
(22)

Where σ~E~ can be determined from the 95% momentum spread:

$\sigma_{E} = \frac{1}{4}{\Delta E}_{\text{95}\text{\%}}$ (23)

The gradient of the distribution is:

$\frac{\partial\psi_{s}}{\partial E} = - \frac{E}{\sigma_{E^{2}}}\psi_{s}$
(24)

Which has a maximum at E=-σ~E~ of:

$\frac{\partial\psi_{s}}{\partial E}|_{\text{max}} = \frac{1}{\pi}\sqrt{\frac{\text{32}}{\pi e}}\left( \frac{1}{{\Delta E}_{\text{95}\text{\%}}} \right)^{2}$
(25)

The limit on cavity impedance becomes

$R_{\text{cav}} < \sqrt{\frac{\pi e}{\text{32}}}n\eta\frac{E_{T}/q}{I_{\text{dc}}}\left( \frac{{\Delta E}_{\text{95}\text{\%}}}{{\beta E}_{T}} \right)^{2}$
(26)

[[Numerical Examples]{.underline}]{.smallcaps}

The Accumulator 53 MHz RF system (h=84) that is used to decelerate the
injected beam for momentum stacking consists of two cavities with an
average Q of 176 and an R/Q of 164Ω.[^2] For the present Accumulator
stacking lattice used for Run II, η=0.010 (at the core orbit), γ=9.38,
E~T~=8850MeV, f~o~=628930Hz. At an intensity of 100x10^10^ particles, a
guassian distribution for the momentum spread will be stable if the 95%
momentum spread is greater than 11.1MeV for a cavity impedance of 57 kΩ
(2 x 164Ω x 176). A 95% momentum spread of 11.1MeV corresponds to a
fundamental frequency width (∆f~o~) of 7.9Hz. The beam admittance at
h=84 is shown in Figure 1. The Nyquist plot is shown in Figure 2.

For an intensity of 3 x 4.7x10^12^ which is the projected intensity for
three Booster batches injected into the Accumulator for SNUMI, the beam
current, I~dc,~ is 1.42A. For SNUMI, the Accumulator lattice would have
an η of 0.023.[^3] If the Booster 53 MHz bucket longitudinal emittance
is 0.08 e-sec and the momentum stacking dilution in the Accumulator has
a dilution of 25% , then the Accumulator longitudinal emittance would be
84 x 0.300 eV-sec which corresponds to a 95% momentum spread of 15.9MeV.
The beam admittance plot is shown in Figure 3. The Nyquist plot for a
cavity impedance of 57 kΩ (2 x 164Ω x 176) is shown in Figure 4.

[[RF Feedback]{.underline}]{.smallcaps}

The Nyquist plot in Figure 4 is clearly unstable by a factor of three.
The beam can be made stable by reducing the impedance of the cavities.
Using RF feedback will reduce the cavity impedance as seen by the beam
without increasing the power required by the cavity power amplifier.
Assume that the cavity is a current to voltage transformer:

${\overset{\sim}{V}}_{\text{cav}}(\omega) = {\overset{\sim}{Z}}_{\text{cav}}(\omega){\overset{\sim}{I}}_{\text{gen}}(\omega)$
(27)

Assume that a portion of the cavity voltage is tapped from the cavity
and fed back into the input of the cavity:

$\Delta{\overset{\sim}{I}}_{\text{gen}}(\omega) = {\overset{\sim}{Y}}_{\text{fb}}(\omega){\overset{\sim}{V}}_{\text{cav}}(\omega)$
(28)

where the gain of the feedback loop is scaled to the cavity impedance:

${\overset{\sim}{Y}}_{\text{fb}}(\omega) = \frac{- g}{R_{\text{cav}}}e^{- j\left( \omega - \omega_{c} \right)\tau_{\text{fb}}}$
(29)

where g is a scaleable gain and τ~fb~ is the delay of the feedback loop.
The effective cavity impedance seen by the beam is:

${\overset{\sim}{Z}}_{\text{eff}}(\omega) = \frac{{\overset{\sim}{Z}}_{\text{cav}}(\omega)}{1 - {\overset{\sim}{Z}}_{\text{cav}}(\omega){\overset{\sim}{Y}}_{\text{fb}}(\omega)}$
(30)

It can be shown empirically that the RF feedback loop will be stable if:

$g - 1 < \frac{2Q}{\omega_{c}\tau_{\text{fb}}}$ (31)

Figure 5 shows the Nyquist plot if an RF feedback gain of 5 (14dB) is
used. The maximum delay for a feedback gain of 5 is about 280nS. In
Figure 5, a delay of 150nS was used which decreases the stability at
other revolution lines. However there is about a factor of two stability
margin for all revolution lines.

[[Tracking]{.underline}]{.smallcaps}

The previous formulas were tested with a tracking program. Each particle
is assigned a phase with respect to the RF phase, φ~RFp~. The applied RF
voltage changes the energy of each particle **p** by the amount

${\Delta E}_{\text{RF}_{p}} = \text{qV}_{\text{rf}}\text{sin}\left( \varphi_{\text{RF}} - \varphi_{\text{RF}_{p}} \right)$
(32)

where q is the charge of the macro-particle. The energy change for each
particle **p** due to beam loading

${\Delta E}_{\text{BL}_{p}} = q\sum_{n}^{}{V_{c_{n}}\text{cos}\left( \frac{n}{h_{\text{RF}}}\varphi_{\text{RF}_{p}} \right)} + q\sum_{n}^{}{V_{s_{n}}\text{sin}\left( \frac{n}{h_{\text{RF}}}\varphi_{\text{RF}_{p}} \right)}$
(33)

$V_{c_{n}} = \frac{R}{Q}Q\frac{I_{c_{n}} + I_{s_{n}}\text{tan}\left( \varphi_{D_{n}} \right)}{1 + \text{tan}^{2}\left( \varphi_{D_{n}} \right)}$
(34)

$V_{s_{n}} = \frac{R}{Q}Q\frac{I_{s_{n}} - I_{c_{n}}\text{tan}\left( \varphi_{D_{n}} \right)}{1 + \text{tan}^{2}\left( \varphi_{D_{n}} \right)}$
(35)

where:

$\text{tan}\left( \varphi_{D_{n}} \right) = Q\frac{\omega_{c^{2}} - \left( {n\omega}_{o} \right)^{2}}{\omega_{c}{n\omega}_{o}}$
(36)

where ω~c~ is the resonant frequency of the cavity. The Fourier
components of the beam current are determined from:

$I_{b}\left( \varphi_{\text{RF}_{k}} \right) = \sum_{n}^{}{I_{c_{n}}\text{cos}\left( \frac{n}{h_{\text{RF}}}\varphi_{\text{RF}_{k}} \right)} + \sum_{n}^{}{I_{s_{n}}\text{sin}\left( \frac{n}{h_{\text{RF}}}\varphi_{\text{RF}_{k}} \right)}$
(37)

where I~b~ is the instantaneous beam current at the particle **k** with
a phase coordinate φ~RFk~

$I_{c_{n}} = 2\text{qh}\text{.}_{\text{rf}}f_{o}\sum_{p}^{}{\text{cos}\left( \frac{n}{h_{\text{RF}}}\varphi_{\text{RF}_{p}} \right)}$
(38)

$I_{s_{n}} = 2\text{qh}\text{.}_{\text{rf}}f_{o}\sum_{p}^{}{\text{sin}\left( \frac{n}{h_{\text{RF}}}\varphi_{\text{RF}_{p}} \right)}$
(39)

because phase coordinate in this tracking algorithm is referenced to the
RF bucket, the coefficients I~cn~, I~sn~ are non-zero only if the ratio
of n/h~rf~ is an integer.

During one turn, the change in energy coordinate of the each particle
advances by:

$E_{p_{m + 1}} = E_{p_{m}} + {\Delta E}_{\text{RF}_{p}} + {\Delta E}_{\text{BL}_{p}}$
(40)

The phase coordinate of particle p advances by:

$\varphi_{\text{RF}_{p_{m + 1}}} = \varphi_{\text{RF}_{p_{m}}} - 2\pi h\eta\frac{E_{p_{n + 1}}}{{\beta E}_{c}}$
(41)

The phase coordinate is then referenced back to the center of the RF
bucket.

$\varphi_{\text{RF}_{p_{m + 1}}} \leftarrow \varphi_{\text{RF}_{p_{m + 1}}} - \varphi_{\text{RF}}$
(42)

The phase of the RF bucket then advances by:

$\varphi_{\text{RF}_{m + 1}} = 2\pi\frac{f_{\text{RF}_{n + 1}} + f_{\text{RF}_{n}}}{2}T_{o}$
(43)

The results of a tracking simulation for a guassian distribution with a
95% momentum width of 15.89 MeV, a beam current of 1.41A,  and η= 0.023,
γ=9.38, E~T~=8850MeV, f~o~=628930Hz is shown in Figure 6. The impedance
used in the tracking simulation is 57 kΩ (2 x 164Ω x 176) which
corresponds to the Nyquist plot of Figure 4. The simulation shown in
Figure 6 clearly shows self-bunching of the distribution.

Figure 7 shows the result of a tracking simulation with the cavity
impedance of 23.1kΩ which corresponds to an RF feedback gain of 8dB. An
impedance of 23.1kΩ is just on the edge of stability. The tracking
simulation shows a small amount of self-bunching. Figure 8 shows the
result of a tracking simulation with the cavity impedance of 11.6kΩ
which corresponds to an RF feedback gain of 14dB. An impedance of 11.6kΩ
is has a factor of two stability margin and Figure 8 shows no sign of
self bunching.

Figure 9 shows the tracking simulation results for an impedance of
11.6kΩ but with a rectangular momentum distribution. As Equation 15
suggests, even though the peak density is less for the rectangular
momentum distribution, the simulation shows bunching at the edge of the
distribution where the slope of the distribution is the greatest.

[[Conclusion]{.underline}]{.smallcaps}

For the projected SNUMI intensity of 14.1x10^12^ particles in the
Accumulator at a 95% momentum spread of 15.9MeV, and an RF feedback gain
of 14dB, the beam should be stable with a factor of two margin. This
result can be tested by cooling 100mA of antiproton beam on the stacking
lattice (η=0.10) to a frequency width of 7.9 Hz.

![](media/image1.wmf){width="4.752083333333333in"
height="4.065277777777778in"}

Figure 1. Accumulator beam admittance at h=84 for η=0.010,
∆E~95%~=11.1MeV, I~dc~=100mA

![](media/image2.wmf){width="4.747916666666667in"
height="3.747916666666667in"}

Figure 2. Nyquist Plot at near h=84 for η=0.010, ∆E~95%~=11.1MeV,
I~dc~=100mA , 2 cavities with an R/Q of 164Ω and a Q of 176.

![](media/image3.wmf){width="4.752083333333333in"
height="4.066666666666666in"}

Figure 3. Accumulator beam admittance at h=84 for η=0.023,
∆E~95%~=15.9MeV, I~dc~=1410mA

![](media/image4.wmf){width="4.747916666666667in"
height="3.7916666666666665in"}

Figure 4. Nyquist Plot at near h=84 for η=0.023, ∆E~95%~=15.9MeV,
I~dc~=1410mA , 2 cavities with an R/Q of 164Ω and a Q of 176.

![](media/image5.wmf){width="4.747916666666667in" height="3.8in"}

Figure 5. Nyquist Plot at near h=84 for η=0.023, ∆E~95%~=15.9MeV,
I~dc~=1410mA , 2 cavities with an R/Q of 164Ω and a Q of 176 and a
feedback gain of 5 with a delay of 150nS.

![](media/image6.wmf){width="4.759027777777778in"
height="3.1319444444444446in"}

Figure 6. Tracking simulation for a guassian distribution with a 95 %
momentum width of 15.89 MeV, a beam current of 1.41A,  η= 0.023, γ=9.38,
E~T~=8850MeV, f~o~=628930Hz, and R=57 kΩ (2 x 164Ω x 176). The vertical
axis is the momentum distribution, the horizontal access is time in the
simulation. The duration of the simulation is 33mS.

![](media/image7.wmf){width="4.759027777777778in"
height="3.1319444444444446in"}

Figure 7. Tracking simulation using the same parameters as shown in
Figure 6 except for cavity impedance of 23.1kΩ. The vertical axis is the
momentum distribution, the horizontal access is time in the simulation.
The duration of the simulation is 33mS.

![](media/image8.wmf){width="4.751388888888889in"
height="3.2131944444444445in"}

Figure 8. Tracking simulation using the same parameters as shown in
Figure 6 except for cavity impedance of 11.6kΩ. The vertical axis is the
momentum distribution, the horizontal access is time in the simulation.
The duration of the simulation is 33mS.

![](media/image9.wmf){width="4.759027777777778in"
height="3.198611111111111in"}

Figure 9. Tracking simulation using the same parameters as Figure 8
except the momentum distribution is rectangular. The cavity impedance of
11.6kΩ. The vertical axis is the momentum distribution, the horizontal
access is time in the simulation. The duration of the simulation is
33mS.

[^1]: "Stochastic Cooling Theory", F. Sacherer, CERN-ISR-TH/78-11

[^2]: "ARF1 Final Amp", R. Pasquinelli, Pbar Note 478, May 23, 1988

[^3]: "A 2 MegaWatt Multi-Stage Proton Accumulator", D. McGinnis, Beams
    Document 1782, Nov. 2005