Longitudinal Space Charge Impedance
Wakefields
A wake potential $V_W$ is defined by:
\[\mathcal{E} _{ind} = -\nabla V_W \rightarrow \Delta V_W = -\int_L\mathcal{E} _{ind}\cdot dl\]The wakefunction is defined as the wake field gain per unit charge:
\[W(\tau) = \frac{\Delta V_w}{q} = -\frac{1}{q}\int_0^{l_{ind}} \vec{\mathcal{E}}_{ind}(s)\cdot \vec{ds}\]The particle’s energy gain due to the experienced wake potential is given by:
\[U = \Delta E = q \Delta V_W = q^2 W\]where:
\[t = \frac{s}{\beta c}-\tau\]and our normalized charge density function is defined as:
\[Q = \int_{-\infty}^{\infty}\lambda(\tau)d\tau = qN_b\]In a bunch distribution, we must convolute our charge density to yield the total induced wake potential for a particle from all other particles:
\[V_{ind}(\tau) = -\int_{-\infty}^{\infty}\lambda(\tau')W(\tau-\tau')d\tau' = (\lambda * W)(\tau)\]Our charge density $\lambda(\tau)$ and wake function $\mathcal{W}(\tau)$ can be re-written in frequency domain as bunch spectrum $S(f)$ and impedance $Z(f)$:
\[S(f) = \mathcal{F}\{\lambda(\tau)\} = \int_{-\infty}^{\infty}\lambda(\tau)e^{-i\omega\tau}d\tau \\ Z(f) = \mathcal{F}\{W(\tau)\} = \int_{-\infty}^{\infty}W(\tau)e^{-i\omega\tau}d\tau\]Using fourier properties, our induced voltage in the frequency domain is:
\[V_{ind}(\tau) = \mathcal{F}^{-1}\{S(f)Z(f)\} = -\int_{-\infty}^{\infty} S(f) Z(f) e^{i2\pi f \tau} df\]For harmonics $n = f/f_{rev}$:
\[V_{ind}(\tau) = -f_{rev}\sum_{n=-\infty}^\infty S(nf_{rev})\mathcal{Z}(nf_{rev})e^{i2\pi f_{rev}\tau}\]Small Amplitude Oscillations
As a reminder, in normal synchrotron oscillations, we have that the particles within a bunch make quasi-elliptical orbits at the synchrotron frequency $\Omega_s = \omega_s$ in ($\Delta E- \Delta t$) phase space according to the following:
\[\begin{aligned} \Delta t &= \Delta t_0\cos(\omega_s t)\cr\cr \Delta E &= \Delta E_0 \sin(\omega_s t) \end{aligned}\]To model this motion as an induced voltage, we require the spectrum and impedance:
\[V_{ind} = -qN_b\int_{-\infty}^\infty S(f)Z(f)e^{i\omega \tau}df\]Using the Jacobi-Anger expansion:
\[e^{i z\cos\theta}=J_0(z)+2\sum_{n=1}^\infty i^nJ_n(z)\cos(n\theta)\]and that the Bessel functions can be approximated for $0<z«\sqrt{a+1}$:
\[J_a(z) \approx \frac{1}{\Gamma(a+1)}\left(\frac{z}{2}\right)^a\]Where:
\[\Gamma(n) = (n-1)!\]Therefore for small z:
\[e^{i z \cos \theta} \approx J_0(z)+2iJ_1(z)\cos(\theta) \approx1 + i z \cos \theta\]Replacing for:
\[z = 2\pi f \tau_0 \quad\theta = \omega_s t \quad \omega = 2\pi f \quad\tau = \tau_0 \cos(\omega_s t)\]We then get:
\[e^{i 2\pi f\tau_0\cos(\omega_s t)} = 1 + i 2\pi f\tau_0\cos(\omega_st)\]Or:
\[\boxed{e^{i\omega\tau} = 1 + i\omega \tau}\]Re-substituting into our equation for induced voltage yields:
\[V_{ind} = -qN_b\int_{-\infty}^\infty S(f)Z(f)(1+i\omega\tau )df\]Defining:
\[\begin{aligned} Z_0 &= \int_{-\infty}^\infty S(f) Z(f) df = \int_{-\infty}^\infty S(f) R(f) df\cr Z_1 &= \int_{-\infty}^\infty S(f) Z(f) i \omega df = 2\pi\int_{-\infty}^{\infty}S(f)X(f)fdf \end{aligned}\]Our expression simplifies to:
\[\boxed{V_{ind} = -qN_b(Z_0 + \tau Z_1 + ...)}\]For an impedance defined by:
\[Z(f) = R(f) + j X(f)\]As it is defined by the fourier transform of the real wake potential, it is hermitian (as is the spectrum) if it is hermitian, $R(f)$ is even and $X(f)$ is odd:
Equations of Motion
If we now recite our equation of motion:
\[\ddot{\phi}+\frac{h\omega_s^2\eta}{2\pi\beta_s^2E_s}\left(qV_g g(\phi)+qV_{ind}\right)=0\]Or equivalently:
\[\ddot{\tau} + \frac{\eta}{\beta_s^2E_sT_{rev}}q\left(V_g g(h\omega_s\tau)+V_{ind}(\tau)\right) = 0\]Which substitutes to:
\[\ddot{\phi}+\frac{\Omega_s^2}{\cos\varphi_s}g(\phi)=\frac{\Omega_s^2}{\cos\varphi_s}\frac{q N_b}{V_g}\left(Z_0+Z_1(\frac{\phi}{h\omega_s})\right)\]or more consicely:
\[\boxed{\ddot{\phi}+\frac{\Omega_s^2}{\cos\varphi_s}\left(g(\phi)+\frac{q N_b}{V_g}\left( Z_0+Z_1 \frac{\phi}{h\omega_s}\right) \right) = 0}\]