text:
\section{Introduction}

Large longitudinal electrical fields required for the acceleration of charged particles are achieved by resonant systems. RF cavities as oscillators with high quality factors assure this excitation of such fields. As a consequence of the small bandwidth, most RF systems in particle accelerators are operated using sinusoidal RF voltages, which are inherently non-linear.

The approach to longitudinal beam dynamics assuming a linear RF voltage is nonetheless a very relevant approximation in many cases. The contribution of the non-linearity of the RF voltage to longitudinal beam dynamics is nonetheless essential to keep high-intensity beams stable in many circular accelerators.

Additional non-linearity in the form of one or more extra RF systems, again with sinusoidal voltages, is actually often introduced intentionally. This allows to modify the longitudinal beam parameters like bunch length, bunch spacing, as well as the longitudinal stability most efficiently.

A brief, general overview to the longitudinal beam dynamics in synchrotrons describing the interaction between single particles or bunches with the RF systems is introduced for the case an arbitrary normalized amplitude. Attention is then given to the model case of a linear RF waveform. This special case not only allows to gain insight into the longitudinal motion in the centre of basically any bucket, but also outlines the differences with respect to the non-linear case.

Examples for longitudinal manipulations to adjust the bunch parameters show the versatility of double- and multi-harmonic RF systems. The synchrotron frequency distribution of the individual particles in a bunch has a strong impact on its longitudinal stability. Techniques are introduced to improve the stability by making use of non-linear voltage, notably a combination of two sinusoidal RF voltages, the double-harmonic RF system.

\section{Longitudinal beam dynamics}
\label{sec:LongitudinalBeamDynamics}

The momentum, $p$ of a relativistic particle in an accelerator can be written in the form of
\begin{equation}
    p = m v = m_0 \beta \gamma c \, ,
\end{equation}
where $m_0$ is the rest mass of the particle, $\beta = v/c$ its relative velocity ratio with respect to the speed of light in vacuum, $c$ and $\gamma = m/m_0$ the relativistic mass factor.

Synchrotrons profit from the fact that particles pass through the accelerating systems for some thousands to millions of turns. The momentum change per turn is therefore moderate with respect to the total particle momentum. When the particle passes through the RF station once, it can be safely assumed that the momentum gain or loss remains small compared to the total one. The momentum change, $dp/dt$ is hence expressed in its discrete form. Within a single turn the momentum changes by $\Delta p$, and $\Delta t$ becomes the revolution period, $T_\mathrm{rev} = 1/f_\mathrm{rev} = \beta c/(2 \pi R)$. The momentum change with time can be transformed according to
\begin{equation}
    \dot{p} = \frac{dp}{dt} \simeq \frac{\Delta p}{\Delta t} = \frac{\Delta p}{T_\mathrm{rev}} = \frac{m_0 \beta c^2}{2 \pi R} \Delta (\beta \gamma) = \frac{m_0 \beta c^2}{2 \pi R} (\beta \Delta \gamma + \gamma \Delta \beta) = \frac{\Delta E}{2 \pi R} \, .
    \label{eqn:MomentumDerivativeToEnergyGainPerTurn}
\end{equation}
To replace $\Delta \beta$ in Eq.~(\ref{eqn:MomentumDerivativeToEnergyGainPerTurn}) by $\Delta \gamma$ the relativistic relations
\begin{equation*}
    \beta (\gamma)= \sqrt{1-\frac{1}{\gamma^2}}  \hspace{1em} \text{and its derivative} \hspace{1em} \frac{\Delta \beta}{\Delta \gamma} \simeq \frac{d \beta}{d \gamma} = \frac{1}{\beta \gamma^3}
\end{equation*}
have been applied.

For the synchronous particle as a reference, momentum and energy changes can be related to the derivative of the bending field during acceleration. A particle with the mass $m$ and the charge $q$ moving with the velocity $v$ in a perpendicular magnetic field of the strength $B$ experiences the Lorentz force, $F_\mathrm{L} = q v B$. This force must compensate the centripetal force, $F_\mathrm{Z} = m v^2/\rho$, which translates to
\begin{equation}
    F_L = F_Z \hspace{1em} \Longrightarrow \hspace{1em} p = q \rho B \, ,
\end{equation}
where $\rho$ represents the bending radius in the dipole magnets. Using Eq.~(\ref{eqn:MomentumDerivativeToEnergyGainPerTurn}) the energy gain or loss can then be written as
\begin{equation}
    \Delta E_\mathrm{turn} = 2 \pi q \rho R \dot{B} \, .
    \label{eqn:energyChangePerTurnDueToBdot}
\end{equation}

Any energy change, $\Delta E_\mathrm{turn}$ must be compensated by the RF system to keep the particle on a central orbit which defines the voltage, $V g(\phi)$ during its passage though the cavities and hence
\begin{equation}
    \Delta E_\mathrm{turn} = q V g (\phi) \, ,
    \label{eqn:energyChangePerTurnDueToVoltage}
\end{equation}
where $V$ is the absolute peak voltage amplitude and $g(\phi)$ its normalized amplitude.

Combining Eq.~(\ref{eqn:energyChangePerTurnDueToBdot}) with Eq.~(\ref{eqn:energyChangePerTurnDueToVoltage}) and assuming a sinusoidal RF amplitude, $g(\phi) = \sin(\phi)$ results in the conventional relation defining the synchronous phase
\begin{equation}
    \sin \phi_\mathrm{S} = \frac{qV}{2 \pi q \rho R \dot{B}}
\end{equation}
at which the so-called synchronous particle receives exactly the energy change required to stay on the central orbit with the increasing or decreasing bending field.

For any other particle with a phase offset of $\Delta \phi = \phi -
\phi_\mathrm{S}$ with respect to the synchronous particle the time derivative of its energy offset, $\Delta E$ is expressed as
\begin{equation}
    \frac{d}{dt} \Delta E = q V \frac{\omega_\mathrm{rev}}{2 \pi} \left[ g(\phi) - g( \phi_\mathrm{S} ) \right] \, ,
    \label{eqn:hamiltonEquationDeltaE}
\end{equation}
where $\omega_\mathrm{rev} = 2\pi f_\mathrm{rev}$ is the angular revolution frequency.

The relationship which links frequency and momentum offset in a synchrotron is called phase slip factor and is defined as\footnote{The phase slip factor is sometimes defined with the opposite sign, e.g.~\cite{bib:dome1984}.}
\begin{equation}
\eta = - \frac{\Delta \omega/\omega}{\Delta p/p} = \alpha_c - \frac{1}{\gamma^2} = \frac{1}{\gamma_\mathrm{tr}^2} - \frac{1}{\gamma^2} \, ,
\end{equation}
with the momentum compaction factor $\alpha_c = (\Delta R/R)/(\Delta p/p) = 1/\gamma_\mathrm{tr}^2$. The momentum compaction factor is a property of the magnet lattice~\cite{bib:bovet1970}. The relative frequency deviation $\Delta \omega/\omega$ can be either with respect to the revolution, $\Delta \omega_\mathrm{rev}/\omega_\mathrm{rev}$ or the RF frequency $\Delta \omega_\mathrm{RF}/\omega_\mathrm{RF}$, where $\omega_\mathrm{RF} = h \omega_\mathrm{rev}$. The ratio between RF and revolution frequency is called harmonic number, $h = \omega / \omega_\mathrm{rev}$.

The phase change of a particle due to a frequency offset is given by~\cite{bib:pirkl1995, bib:montague1977}
\begin{equation}
    \frac{d}{dt} \phi = - h \Delta \omega_\mathrm{rev} \, ,
    \label{eqn:frequencyOffsetToPhaseDerivative}
\end{equation}
where the minus sign is a consequence of the choice of the RF phase, $\phi$ as a variable. A particle with positive revolution frequency offset arrives too early in the cavity and experiences the RF voltage at a lower phase with respect to the synchronous reference particle.

Replacing $\Delta p/p = 1/\beta^2 \Delta E/E$ and using Eq.~(\ref{eqn:frequencyOffsetToPhaseDerivative}) results in~\cite{bib:dome1984}
\begin{equation}
\frac{d}{dt} \phi =  h \omega_\mathrm{rev} \frac{\eta}{E \beta^2} \Delta E = \frac{h \eta \omega_\mathrm{rev}}{pR} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right) \, .
\label{eqn:hamiltonEquationDeltaPhi}
\end{equation}
Equations~(\ref{eqn:hamiltonEquationDeltaE}) and (\ref{eqn:hamiltonEquationDeltaPhi}) form a set of Hamilton's equations as their structure matches
\begin{equation}
    \frac{dq}{dt} = \frac{ \partial H}{\partial p} \hspace{1em} \text{and} \hspace{1em} \frac{dp}{dt} = - \frac{ \partial H}{\partial q} \, ,
\end{equation}
with $\phi$ and $\Delta E/ \omega_\mathrm{rev}$ as canonically conjugated variables $q$ and $p$:
\begin{eqnarray}
\frac{d}{dt} \phi & = & \frac{h \eta \omega_\mathrm{rev}}{pR} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)  \label{eqn:firstHamiltonEquation} \\
\frac{d}{dt} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right) & = & \frac{q V}{2 \pi} \left[ g(\phi) - g( \phi_\mathrm{S} ) \right] \label{eqn:secondHamiltonEquation} \, .
\end{eqnarray}
The choice of variables $\phi$ and $\Delta E / \omega_\mathrm{rev}$ defines the explicit form of the Hamiltonian, and it is convenient to calculate invariant areas in the longitudinal phase space using these variables.
Combining both equations Eq.~(\ref{eqn:firstHamiltonEquation}) and Eq.~(\ref{eqn:secondHamiltonEquation}) leads to the Hamiltonian of the synchrotron motion
\begin{equation}
    H \left( \phi , \frac{\Delta E}{\omega_\mathrm{rev}} \right) = \frac{1}{2} \frac{h \eta \omega_\mathrm{rev}}{p R} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)^2 - \frac{qV}{2 \pi} \left[ \int g(\phi) d \phi - g(\phi_\mathrm{S}) \phi \right] \, .
\end{equation}
It describes the behaviour of a particle in the coordinates of phase, $\phi$, and energy offset $\Delta E$. For any given particle trajectory in the longitudinal $\phi$-$\Delta E/\omega_\mathrm{rev}$-phase space the value of the Hamiltonian varies very slowly with respect to the dynamics of the system.

The second term of the Hamiltonian is identified as the normalized potential
\begin{equation}
    W (\phi) = \frac{1}{\cos \phi_\mathrm{S}} \left[ \int g(\phi) d \phi - g(\phi_\mathrm{S}) \phi \right] \, ,
    \label{eqn:normalizedPotential}
\end{equation}
which is basically the integral of the RF voltage. The Hamiltonian can hence be written as
\begin{equation}
    H \left( \phi , \frac{\Delta E}{\omega_\mathrm{rev}} \right) = \frac{1}{2} \frac{h \eta \omega_\mathrm{rev}}{p R} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)^2 - \frac{qV\cos\phi_\mathrm{S}}{2 \pi} \, W (\phi) \, .
    \label{eqn:GeneralHamiltonian}
\end{equation}

The general equation of motion is obtained by differentiating Eq.~(\ref{eqn:firstHamiltonEquation}) in time and substituting the result in Eq.~(\ref{eqn:secondHamiltonEquation}). This general equation can be written as
\begin{equation}
    \frac{d^2}{dt^2} \phi + \frac{ \omega_\mathrm{S}^2 }{ \cos \phi_\mathrm{S} } \left[ g(\phi) - g(\phi_\mathrm{S}) \right] = 0 \hspace{1em} \text{with} \hspace{1em} \omega_\mathrm{S}^2 = - \frac{ h \eta \omega_\mathrm{rev} q V \cos \phi_\mathrm{S} }{ 2 \pi p R} \, ,
    \label{eqn:GeneralEquationOfMotionAndSynchrotronFrequency}
\end{equation}
where $\omega_\mathrm{S}$ is the angular frequency of the longitudinal oscillation in phase and energy of the particle.

With the help of Eq.~(\ref{eqn:firstHamiltonEquation}) the energy offset variable can be changed from $\Delta E/\omega_\mathrm{rev}$ to $\dot{\phi}/\omega_\mathrm{S}$, and the Hamiltonian Eq.~(\ref{eqn:GeneralHamiltonian}) for an arbitrary RF voltage function $g(\phi)$ can be rewritten in the elegant form of
\begin{equation}
    H \left( \phi, \frac{\dot{\phi}}{\omega_\mathrm{S}} \right) = \frac{1}{2} \left( \frac{ \dot{\phi} }{ \omega_\mathrm{S}} \right)^2 + W(\phi)
    \label{eqn:GeneralHamiltonianPhiPhiDot}
\end{equation}
again with the normalized potential according to Eq.~(\ref{eqn:normalizedPotential}). Compared to Eq.~(\ref{eqn:GeneralHamiltonian}) the Hamiltonian has been divided by the factor of $k = - q V \cos \phi_\mathrm{S} / (2 \pi )$. The stable phase angle $\phi_\mathrm{S}$ is not necessarily unambiguously defined as an angle but indicates the energy gain or loss per turn of the synchronous particle.

\section{Linear longitudinal dynamics}

The longitudinal dynamics of a particle under the influence of a linear RF field is mainly a simplified mathematical model to describe the motion in the linear regions of a non-linear RF amplitude. Figure~\ref{fig:SinusoidalLinearVoltage} compares sinusoidal, $g(\phi) = \sin \phi$ and linear, $g(\phi) = \phi$ normalised RF amplitudes.
\begin{figure}[htb]
	\centering\includegraphics[width=.4\linewidth]{images/SinusoidalLinearVoltage}
	\caption{Sinusoidal (blue) and linear (red) RF amplitude.}
	\label{fig:SinusoidalLinearVoltage}
\end{figure}
For the case of no net acceleration, $\phi_\mathrm{S} = 0$ and $g(\phi_\mathrm{S})=0$, the synchronous particle is located at the zero crossing of the RF voltage, where linear and sinusoidal RF amplitudes are almost identical. It is worth noting that the saw-tooth waveform has a large frequency spectrum which technically makes the generation of large voltages unfavourable, and it is therefore of only limited practical interest for implementation.

For a linear RF amplitude the Hamilton equations Eq.~(\ref{eqn:firstHamiltonEquation}) and Eq.~(\ref{eqn:secondHamiltonEquation}) simplify to
\begin{eqnarray}
\frac{d}{dt} \phi & = & \frac{h \eta \omega_\mathrm{rev}}{pR} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)  \label{eqn:firstHamiltonEquationLinearVoltage} \\
\frac{d}{dt} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right) & = & \frac{q V}{2 \pi} \phi \label{eqn:secondHamiltonEquationLinearVoltage} \, .
\end{eqnarray}
The corresponding Hamiltonian for this linear longitudinal particle motion can be written as
\begin{equation}
     H \left( \phi , \frac{\Delta E}{\omega_\mathrm{rev}} \right) = \frac{1}{2} \frac{h \eta \omega_\mathrm{rev}}{p R} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)^2 - \frac{1}{2} \frac{qV}{2 \pi} \phi^2 \, .
     \label{eqn:HamiltonianLinearVoltage}
\end{equation}

The explicit form of this Hamiltonian can again be significantly simplified by changing the variable describing the energy offset of the particle. Equation~(\ref{eqn:secondHamiltonEquationLinearVoltage}) actually translates this energy offset, $\Delta E/\omega_\mathrm{rev}$ into a drift velocity, $d\phi/dt = \dot{\phi}$ with respect to the synchronous particle. In the new set of variables $\phi$ and $\dot{\phi}/\omega_\mathrm{S}$ the explicit form of the Hamiltonian becomes
\begin{equation}
    H \left( \phi , \frac{\dot{\phi}}{\omega_\mathrm{S}} \right) = \frac{1}{2} \left( \frac{\dot{\phi}}{\omega_\mathrm{S}} \right)^2 + \frac{1}{2} \phi^2 \, ,
    \label{eqn:HamiltonianLinearVoltageNormalized}
\end{equation}
applying the same normalization factor, $k$ as for Eq.~(\ref{eqn:GeneralHamiltonianPhiPhiDot}).

The relation describes the equation of a circle, $r^2 = x^2 + y^2$. Therefore, the trajectories of all particles under the influence of a linear RF amplitude can be represented as circles in the longitudinal $\phi$-$\dot{\phi}/\omega_\mathrm{S}$-phase space~(Fig.~\ref{fig:VoltagePotentialPhaseSpaceLinearVoltage}):
\begin{equation*}
    \dot{\phi} (\phi) = \omega_\mathrm{S} \sqrt{2 H_\mathrm{trajectory} -  \phi^2} \, .
\end{equation*}
The RF potential is then a parabolic function, $W \propto \phi^2$.
\begin{figure}[htb]
	\centering
	\includegraphics[width=0.329\linewidth]{images/SingleHarmonicLinearPhaseSpaceConstructionAllframe0001}
	\includegraphics[width=0.329\linewidth]{images/SingleHarmonicLinearPhaseSpaceConstructionAllframe0024}
	\includegraphics[width=0.329\linewidth]{images/SingleHarmonicLinearPhaseSpaceConstructionAllframe0040}
	\caption{Amplitude (top), resulting potential (middle) and normalized longitudinal phase space (bottom) in coordinates of $\phi$ and $\dot{\phi}/\omega_\mathrm{S}$. The phase space trajectories are just circles and the oscillation frequency is the same for all particles.}
	\label{fig:VoltagePotentialPhaseSpaceLinearVoltage}
\end{figure}
As indicated by the trajectories (bottom) all particles moreover have the same synchrotron frequency, independent of their energy or phase deviation with respect to the bucket center.

\section{Sinusoidal RF voltage}

For the most common case of a single sinusoidal RF voltage the term $V\phi$ in Eq.~(\ref{eqn:secondHamiltonEquationLinearVoltage}) is replaced by $V \sin \phi$ which introduces a simple non-linearity. As explained above, sinusoidal RF voltages are most common in accelerators with conventional RF systems since large voltages are easily achieved in the resonant RF cavities. The Hamilton equations for this case become
\begin{eqnarray}
\frac{d}{dt} \phi & = & \frac{h \eta \omega_\mathrm{rev}}{pR} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right) \\
\frac{d}{dt} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right) & = & \frac{q V}{2 \pi} \left( \sin \phi - \sin \phi_\mathrm{S} \right) \label{eqn:secondHamiltonEquationSinusoidalVoltage}
\end{eqnarray}
and can be combined to the Hamiltonian
\begin{equation}
    H \left( \phi , \frac{\Delta E}{\omega_\mathrm{rev}} \right) = \frac{1}{2} \frac{h \eta \omega_\mathrm{rev}}{p R} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)^2 + \frac{qV}{2 \pi} \left[ \cos\phi - \cos \phi_\mathrm{S} + (\phi - \phi_\mathrm{S}) \sin \phi_\mathrm{S} \right] \, .
\end{equation}
This standard expression, and its detailed analysis of stable parameter regions in the phase space of $\Delta \phi$ and $\Delta E/\omega_\mathrm{rev}$, can be found in a large number of text books and review articles~\cite{bib:montague1977,bib:pirkl1995,bib:tecker2006,bib:tecker2019}.

To study the behaviour of particles close to the centre of the bunch, it is convenient to introduce $\Delta \phi = \phi - \phi_\mathrm{S}$ which brings the Hamiltonian in the form
\begin{equation}
    H \left( \phi , \frac{\Delta E}{\omega_\mathrm{rev}} \right) = \frac{1}{2} \frac{h \eta \omega_\mathrm{rev}}{p R} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)^2 + \frac{qV}{2 \pi} \left[ \cos(\phi_\mathrm{S} + \Delta \phi) - \cos \phi_\mathrm{S} + \Delta \phi \sin \phi_\mathrm{S} \right] \, . \label{eqn:hamiltonianSinusoidalVoltage}
\end{equation}
In the limit of $\Delta \phi \ll 1$, valid for particles close to the synchronous one, the first part of the potential term can be rearranged using
\begin{eqnarray}
    \cos(\phi_\mathrm{S} + \Delta \phi) & = & \cos(\phi_\mathrm{S}) \cos \Delta \phi - \sin \phi_\mathrm{S} \sin \Delta \phi \\
    & \simeq & \cos \phi_\mathrm{S} \left( 1- \frac{1}{2} \Delta \phi^2 \right) - \sin \phi_\mathrm{S} \Delta \phi
\end{eqnarray}
such that Eq.~(\ref{eqn:hamiltonianSinusoidalVoltage}) is approximated to
\begin{equation}
    H \left( \Delta \phi , \frac{\Delta E}{\omega_\mathrm{rev}} \right) \simeq \frac{1}{2} \frac{h \eta \omega_\mathrm{rev}}{p R} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)^2 - \frac{1}{2} \frac{qV}{2 \pi} \cos \phi_\mathrm{S} \Delta \, \phi^2 \, . \label{eqn:hamiltonianSinusoidalVoltageSmallAmplitudes}
\end{equation}
As in Eq.~(\ref{eqn:HamiltonianLinearVoltage}) the explicit dependency of $\Delta \phi$ and $\Delta E / \omega_\mathrm{rev}$ is again quadratic, indicating that trajectories with constant $H ( \Delta \phi , \Delta E / \omega_\mathrm{rev} )$, as well as small phase and energy deviations with respect to the synchronous particle at the stable phase are elliptic (or circular, depending on the normalization of coordinates). Each particle performs synchrotron oscillations in the longitudinal phase space, constantly exchanging phase and energy offset, with the synchrotron frequency according to Eq.~(\ref{eqn:GeneralEquationOfMotionAndSynchrotronFrequency}).

Figure~\ref{fig:AcceleratingDecelratingBucketCropped} illustrates the longitudinal phase space for the case of an accelerating bucket.
\begin{figure}[htb]
	\centering\includegraphics[width=.4\linewidth]{images/AcceleratingDecelratingBucketCropped}
	\caption{The gray shaded area occupied by particles indicates the longitudinal emittance. The separatrix (red) indicates the limit of closed trajectories.}
	\label{fig:AcceleratingDecelratingBucketCropped}
\end{figure}
For any given trajectory the value of the Hamiltonian is constant. The extremes (Fig.~\ref{fig:AcceleratingDecelratingBucketCropped}) of two particles on the same trajectory are the case zero phase (1) or energy deviation (2). For these particles the Hamiltonian is expressed as
\begin{eqnarray}
    H \left( \Delta \phi = 0, \frac{\Delta E}{\omega_\mathrm{rev}} \right) = H_\mathrm{trajectory}  & = & \phantom{-} \frac{1}{2} \frac{h \eta \omega_\mathrm{rev}}{p R} \left( \frac{\Delta E}{\omega_\mathrm{rev}} \right)^2 \hspace{1em} \mathrm{and} \\
    H \left( \Delta \phi, \frac{\Delta E}{\omega_\mathrm{rev}} = 0 \right) = H_\mathrm{trajectory} & = & - \frac{1}{2} \frac{qV}{2 \pi} \cos \phi_\mathrm{S} \Delta \, \phi^2 \, .
\end{eqnarray}

The longitudinal emittance, $\varepsilon_\mathrm{l}$ is the surface in the longitudinal phase space occupied by the ensemble of all particles particles on stable trajectories around the synchronous particle, the so called bunch. In the example shown in Fig.~\ref{fig:AcceleratingDecelratingBucketCropped} it is represented by the gray shaded area. Choosing a canonically conjugated set of variables its surface is preserved, just like an incompressible fluid, during acceleration or RF manipulations. For the common choice of time and energy offset, the unit of the longitudinal emittance becomes $[\varepsilon_\mathrm{l}] = [\pi \Delta \tau \Delta E] = \mathrm{eVs}$. The duration from centre of the bunch to the maximum phase excursion of the encircling trajectory is $\Delta \tau$ (half the total bunch length), and the height of the trajectory is $\Delta E$ (half the total energy width).

For any given trajectory, independent of the non-linearity of the RF waveform, the longitudinal emittance can be calculated according to
\begin{equation}
    \varepsilon_\mathrm{l} = \frac{2}{h \omega_\mathrm{rev}} \int_{\Delta \phi_\mathrm{i}}^{\Delta \phi_\mathrm{f}} \Delta E (\Delta \phi) \, d(\Delta \phi) \, ,
\end{equation}
which is the surface within the surrounding trajectory $\pm \Delta E (\Delta \phi)$ of the bunch.

\section{Controlling bunch parameters with multi-harmonic RF systems}

Due the the resonant nature of conventional RF systems, this section concentrates on the combination of an accelerating voltage delivered by multiple RF stations, each of which generates a sinusoidal voltage. In the most simple extension of two harmonic RF systems, the voltage $V \sin \phi$ is replaced by $V_1 \sin \phi + V_2 \sin( h_2/h_1 \phi + \phi_1) ]$, where $h_2/h_1$ is the harmonic number ratio and $V_2$ is the voltage of the second harmonic RF system. Already for an RF voltage composed of two sinusoidal amplitudes, analytical estimates of the bucket parameters are only possible for few special cases. Generally, the trajectories can be calculated numerically using Eq.~(\ref{eqn:GeneralHamiltonianPhiPhiDot}).

\subsection{Bunch length manipulations}

Two examples of a double harmonic RF voltage, with $h_2/h_1=2$ and $V_2/V_1=0.5$ are illustrated in Fig.~\ref{fig:doubleHarmonicBunchShorteningLengtheningModeExample}.
\begin{figure}[htb]
	\centering
	\includegraphics[width=0.45\linewidth]{images/DoubleHarmonicRFVoltageBunchShorteningMode}
	\includegraphics[width=0.45\linewidth]{images/DoubleHarmonicRFVoltageBunchLengtheningMode}
	\caption{Sum of two RF voltages (red) with a harmonic ratio of $h_2/h_1=2$ (black and blue). The amplitude of the higher harmonic system is half the amplitude of the fundamental one. The effect on the beam depends strongly on whether the voltages are at the same phase (left) or in opposite phase (right) at the centre of the bunch (dashed line).}
	\label{fig:doubleHarmonicBunchShorteningLengtheningModeExample}
\end{figure}
When both the sinusoidal amplitudes are in phase at the position of the bunch, the local voltage gradient is increased, decreasing the bunch length. This configuration is called bunch shortening mode and results in a larger peak current. Nonetheless this case is has a strong impact on longitudinal stability as shown in Sec.~\ref{sec:bunchShorteningMode}. In the opposite case, when both RF systems are in counter-phase in the bunch lengthening mode, the local voltage gradient is decreased, which stretches the bunch and reduces the peak longitudinal line density.

The most important application of the bunch lengthening mode is the reduction of the instantaneous current is the reduction of the space charge effect~\cite{bib:ferrario2014}. The lower the instantaneous beam current, the fewer particles are squeezed into a given part of the circumference, hence reducing the electric forces between particles. Figure~\ref{fig:doubleHarmonicBunchLengtheningModeExamplePSB} shows the RF voltage used in the Proton Synchrotron Booster (PSB) at CERN during the low energy part of the acceleration ramp (left) and the resulting longitudinal distribution of the bunch.
\begin{figure}[htb]
	\centering
	\includegraphics[width=0.42\linewidth]{images/DoubleHarmonicRFVoltageBunchLengtheningModePSB}
	\includegraphics[width=0.48\linewidth]{images/DoubleHarmonicTomographicReconstructionBunchLengtheningModePSB}
	\caption{The RF voltage during the low energy part of the acceleration cycle in the CERN Proton Synchrotron Booster (PSB) used two harmonics a $h_1 = 1$ and $h_2 = 2$. The measured bunch distribution under these conditions (right) is stretched such that the time projection, the bunch profile has two peaks around the depleted bunch center.}
	\label{fig:doubleHarmonicBunchLengtheningModeExamplePSB}
\end{figure}
The important bunch flattening and depletion of the distribution in the centre due to the inverted voltage gradient reduces the maximum longitudinal line density and significantly reduces the space charge effect.

\subsection{Adjustment of the bunch distance}

Not only the bunch length, but also the distance of the bunches, normally an integer multiple of the RF wavelength, can be controlled with double-harmonic RF systems. The circumference of the Proton Synchrotron (PS) at CERN is exactly four times larger than the four sandwiched rings of its injector, the aforementioned PSB. An RF harmonic of $h_\mathrm{PS} = 8$ in the the PS thus corresponds to two bunches with a spacing of half a turn in the corresponding single-harmonic RF voltage at $h_\mathrm{PSB} = 2$.

Adding some small RF voltage at the revolution frequency, $h_\mathrm{PSB} = 1$, allows to control the bunch spacing and adapt it to a different harmonic number in the PS. With this technique the circumference ratio is virtually moved to $h_\mathrm{PS}/h_\mathrm{PSB} = 7/2$ for the transfer of beams for the Large Hadron Collider (LHC) between PSB and PS~\cite{bib:hancock2010}.

Figure~\ref{fig:singleBatchTransferLongitudinalPhaseSpace} shows the measured longitudinal phase space, as well as time and energy projections with $h_\mathrm{PSB} = 2$ complemented by additional voltage at $h_\mathrm{PSB} = 1$.
\begin{figure}[htb]
	\centering\includegraphics[width=.4\linewidth]{images/SingleBatchTransferLongitudinalPhaseSpace}
	\caption{Reconstructed longitudinal distribution of two bunches in the main RF harmonic of $h_\mathrm{PSB} = 2$ under the influence of additional voltage at $h_\mathrm{PSB} = 1$. Instead of a spacing of $286\ \mathrm{ns}$ for the single-harmonic case, the distance between the bunches is reduced to about $245\ \mathrm{ns}$.}
	\label{fig:singleBatchTransferLongitudinalPhaseSpace}
\end{figure}
The revolution frequency of the beam in the PSB at a kinetic energy of $E_\mathrm{kin} = 1.4\ \mathrm{GeV}$ is $1.75\ \mathrm{MHz}$ and the natural bunch spacing would be $\tau_{h=2} = T_\mathrm{rev,PSB}/2 =  286\ \mathrm{ns}$. Applying RF voltage at the revolution frequency reduces the short bunch spacing in the PSB to $\tau_\mathrm{short} = 245\ \mathrm{ns}$. Effectively the bunches are slightly displaced in azimuth with respect to their diametrically opposite position, and the long distance between to bunches becomes $\tau_\mathrm{short} = 327\ \mathrm{ns}$, exactly the period of the RF voltage for the capture into $h_\mathrm{PS} = 7$-buckets in the PS. Since the manipulation does not change the revolution frequencies, the bunch spacings must follow the relation to $T_\mathrm{rev,PSB} = 2 \tau_\mathrm{h=2} = \tau_\mathrm{short} + \tau_\mathrm{long}$.

As illustrated in Fig.~\ref{fig:SingleBatchTransferCarliSketch} it is then sufficient to trigger the ejection and injection kickers such that both bunches from each ring are extracted according to the indicated order.
\begin{figure}[htb]
	\centering\includegraphics[width=.6\linewidth]{images/SingleBatchTransferCarliSketch}
	\caption{Transfer scheme of pairs of bunches from each PSB ring. In the double-harmonic RF configuration, the bunches are slightly displaced in azimuth to achieve the desired spacing suitable for injection into the PS. Illustration courtesy of C. Carli, CERN.}
	\label{fig:SingleBatchTransferCarliSketch}
\end{figure}
The ejection kicker of the PSB is triggered during the short spacing to obtain two bunches spaced by $327\ \mathrm{ns}$. This special scheme, based on the special, non-linear RF voltage shape, has been used operationally to save time during fillings of the Large Hadron Collider (LHC) during a few years.

\subsection{Long and short bunches simultaneously}

Conventional electron storage rings can either be operated with long or short bunches. This mode of operation defines the main types of experiments which can be performed with the synchrotron radiation from these rings. An innovative scheme, based on non-linear RF voltages has been proposed for the storage ring BESSY to produce long and short bunches simultaneously~\cite{bib:hzb2015}. Adding a higher harmonic RF voltage at 3 ($1.5\ \mathrm{GHz}$) and 3.5 times ($1.75\ \mathrm{GHz}$) the frequency of the fundamental RF system at $500\ \mathrm{MHz}$ yield an effective RF voltage as illustrated in Fig.~\ref{fig:LongShortBunchVoltagePotentialLongitudinalPhaseSpace} (top).
\begin{figure}[htb]
	\centering
	\includegraphics[width=.29\linewidth]{images/BESSYLongShortBunchIllustration}
	\includegraphics[width=.7\linewidth]{images/LongShortBunchVoltagePotentialLongitudinalPhaseSpace}
	\caption{RF voltage (top), potential (middle) and longitudinal phase space (bottom) for the simultaneous long (blue) and and short (red) mode, as proposed for the electron storage ring BESSY. Super-conducting RF cavities are the only technique to provide sufficiently large RF voltages. Illustration (left) courtesy of BESSY.}
	\label{fig:LongShortBunchVoltagePotentialLongitudinalPhaseSpace}
\end{figure}
Depending on the azimuth of the bunch the RF voltage either sum up or compensate each other at the position of the bunch. The resulting RF potential~(Fig.~\ref{fig:LongShortBunchVoltagePotentialLongitudinalPhaseSpace}, middle) has either short and deep potential wells for short bunches, or when the voltage of the different RF harmonics cancels at the bunch position, a rather long bathtub like potential well forms, which stretches the bunches in these buckets.

The shape of the buckets~(Fig.~\ref{fig:LongShortBunchVoltagePotentialLongitudinalPhaseSpace}, bottom) illustrates the longitudinal phase space featuring short buckets with large momentum acceptance and comparably long buckets with small bucket height. This particular mode of operations allows to virtually combine two storage rings with different beam parameters into a single one able to simultaneously serve two classes of experiments thanks to this clever application of non-linear RF voltages.

\section{Longitudinal stability with multi-harmonic RF systems}

As shown in Sec.~\ref{sec:LongitudinalBeamDynamics} each particle in a bunch executes synchrotron oscillations around the synchronous particle at the centre of the bucket. This causes a continues exchange of phase and energy offset. As long as the surrounding trajectory of a bunch matches with the longitudinal distribution, the macroscopic bunch distribution and profile remains unchanged.

The distribution of the synchrotron frequencies depends on the non-linearity of the RF voltage. Only in the special case of a linear acceleration voltage, as illustrated in Fig.~\ref{fig:VoltagePotentialPhaseSpaceLinearVoltage}, the frequency of the synchrotron oscillation is identical for all particles.

A bunch can generally be described as an ensemble of harmonic oscillators, each of which with its own resonant frequency. The entire bunch becomes longitudinally unstable when coherent oscillations are excited, which means driving a large fraction of the individual particles simultaneously at their resonance frequency. This idea is illustrated in Fig.~\ref{fig:PedulumEquivalent}.
\begin{figure}[htb]
	\centering\includegraphics[width=.8\linewidth]{images/PedulumEquivalent}
	\caption{Ensemble of oscillators represented by pendulums attached to a horizontal bar. The pendulums are excited by a horizontal movement of the bar, as indicated by the red arrows. When all pendulums have the same length (top), all of them can be excited coherently at their resonance frequency. Introducing a spread of individual resonance frequencies makes the coherent excitation impossible.}
	\label{fig:PedulumEquivalent}
\end{figure}
A number of pendulums are hanging at a horizontal bar which is excited by a horizontal movement. When all oscillators have the same resonance frequency~(Fig.~\ref{fig:PedulumEquivalent}, top), they can be easily excited coherently by a movement at that specific frequency. The entire system becomes very unstable. This behavior changes significantly when the pendulums all have slightly different resonance frequencies. Whichever excitation frequency is chosen to drive them, only a small number of pendulums will start to oscillate and a coherent motion of the entire cannot develop. Macroscopically the set of oscillators remains perfectly stable ~(Fig.~\ref{fig:PedulumEquivalent}, top). This damping due to a spread of resonance frequencies of individual oscillators is also referred to as Landau damping~\cite{bib:hofmann2006}. The wider the spread, the more stable the ensemble of oscillators will be macroscopically.

The same mechanism applies to the stability of a bunch in a bucket. The larger the spread of the resonance frequencies, the synchrotron frequencies of the individual particles, the more stable the bunch will generally be against coherent excitation. To judge the longitudinal stability of a bunch, the distribution of synchrotron frequencies of the particles in the bunch is hence most relevant.

\subsection{Synchrotron frequency distribution}

The synchrotron frequency is the inverse of the time it takes a particle to perform one complete oscillation on a closed trajectory in the longitudinal phase space. The average phase velocity for a given trajectory in the longitudinal phase space can be calculated introducing the action angle variable, $J(H)$. For a particle on a given trajectory, $\dot{\phi} (\phi)$ defined by the constant value of its Hamiltonian, the corresponding action angle becomes
\begin{equation}
    J(H) = \frac{1}{2 \pi \omega_\mathrm{S}} \oint \dot{\phi} (\phi) \, d\phi \, .
\end{equation}
The angular frequency, the synchrotron frequency of the particle on the given trajectory is then calculated according to
\begin{equation}
    \omega (H) = \frac{d}{dJ} H \, ,
\end{equation}
and the general expression for the synchrotron frequency can be expressed as
\begin{equation}
    \frac{ \omega (H) }{ \omega_\mathrm{S} } = \cfrac{ \sqrt{2} \pi }{ \displaystyle \int_{\phi_\mathrm{l}}^{\phi_\mathrm{u}} \cfrac{1}{\sqrt{H - W(\phi) } } \, d\phi } \,  .
    \label{eqn:GeneralSynchrotronFrequencyDistribution}
\end{equation}
The integration in the denominator is computed from the lower boundary, $\phi_\mathrm{l}$ to the upper boundary, $\phi_\mathrm{u}$ of the trajectory with the value $H$ of the Hamiltonian, $\dot{\phi} (\phi_\mathrm{l})$ = $\dot{\phi} (\phi_\mathrm{u}) = 0$.

\subsubsection{Linear RF voltage}

As a simple crosscheck, the synchrotron frequency distribution can be derived starting from the Hamiltonian Eq.~(\ref{eqn:HamiltonianLinearVoltageNormalized}). The integral in the denominator of Eq.~(\ref{eqn:GeneralSynchrotronFrequencyDistribution}) becomes
\begin{equation*}
    \int_{\phi_\mathrm{l}}^{\phi_\mathrm{u}} \cfrac{1}{\sqrt{H - W(\phi) } } \, d\phi = \sqrt{2} \int_{\phi_\mathrm{l}}^{\phi_\mathrm{u}} \frac{1}{\sqrt{\phi^2_\mathrm{u} - \phi^2}} \, d\phi = 2 \sqrt{2} \left[ \arctan \frac{\phi}{\sqrt{\phi^2_\mathrm{u} - \phi^2}} \right]_0^{\phi_\mathrm{u}} = \sqrt{2} \pi \, .
\end{equation*}
The trajectories in the linear bucket are of course symmetric with respect to $\phi = 0$ and $\Delta E$, as illustrated in Fig.~\ref{fig:VoltagePotentialPhaseSpaceLinearVoltage} and hence $H_\mathrm{trajectory} = H(\phi = \phi_\mathrm{l}, \dot{\phi}/\omega_\mathrm{S}^2 = 0) = H(\phi = \phi_\mathrm{u}, \dot{\phi}/\omega_\mathrm{S}^2 = 0) = 1/2 \cdot \phi_\mathrm{u}^2$ is applied to obtain the integral. As expected from the linear Eqs.~(\ref{eqn:firstHamiltonEquationLinearVoltage}) and (\ref{eqn:secondHamiltonEquationLinearVoltage}), all particles have the same synchrotron frequency $\omega(H) = \omega_\mathrm{S}$ when the RF voltage is linear. This also applies to the synchrotron frequency in the centre of the conventional bucket generated by a sinusoidal RF voltage.

\subsubsection{Sinusoidal RF voltage}

For the more general case of a sinusoidal RF voltage but no acceleration, $\phi_\mathrm{S} = 0$, the exact potential is proportional to $1-\cos \phi = 2 \sin^2 (\phi/2)$. Again, profiting from the symmetry of the trajectories around the bucket centre for the stationary bucket, the denominator integral of Eq.~(\ref{eqn:GeneralSynchrotronFrequencyDistribution}) can be expressed as
\begin{equation}
    \int_{\phi_\mathrm{l}}^{\phi_\mathrm{u}} \cfrac{1}{\sqrt{H - W(\phi) } } \, d\phi = \sqrt{2} \int_0^{\phi_\mathrm{u}} \frac{1}{\sqrt{ \sin^2 (\phi_\mathrm{u}/2) - \sin^2 (\phi/2) }} \, d\phi \, ,
\end{equation}
where $\phi$, $\phi_\mathrm{u}$ denote phase offsets with respect to the bucket center. The Hamiltonian has been replaced by the maximum phase excursion $\phi_\mathrm{u}$ of the trajectory under consideration. With the substitution
\begin{equation}
    \xi(\phi) = \arcsin \left[ \frac{ \sin (\phi/2) }{ \sin (\phi_\mathrm{u}/2)} \right]
\end{equation}
this integral reduces to the conventional form of the elliptic integral of the first kind
\begin{equation*}
    K[\sin (\phi_\mathrm{u}/2)] = \int_0^{\pi/2} \frac{1}{\sqrt{1- \sin^2(\phi_\mathrm{u}/2) \sin^2 \xi}} \, d \xi \, .
\end{equation*}
The synchrotron frequency of a particle on a trajectory with a maximum phase excursion of $\pm \phi_\mathrm{u}$ is expressed in the form of
\begin{equation}
    \frac{ \omega(\phi_\mathrm{u}) }{ \omega_\mathrm{S} } = \frac{ \pi }{ 2 K [ \sin (\phi_\mathrm{u}/2)] } \simeq 1 - \frac{\phi_\mathrm{u}^2}{16} \, .
    \label{eqn:synchrotronFrequencyDistributionStationaryBucket}
\end{equation}
While the synchrotron frequency remains rather constant at the centre of the bucket (Fig.~\ref{fig:StationaryBucketSynchrotronFrequencyDistribution}),
\begin{figure}[htb]
	\centering\includegraphics[width=0.8\linewidth]{images/StationaryBucketSynchrotronFrequencyDistribution}
	\caption{Synchrotron frequency distribution in a single-harmonic stationary bucket versus maximum phase excursion of the trajectory (left) and versus the normalized area, $A/A_\mathrm{B}$ encircled by the trajectory. The total bucket area is $A_\mathrm{B}$. The dashed line shows the approximation according to Eq.~(\ref{eqn:synchrotronFrequencyDistributionStationaryBucket}).}
	\label{fig:StationaryBucketSynchrotronFrequencyDistribution}
\end{figure}
it rapidly decreases towards larger synchrotron oscillation amplitudes. At the separatrix the period of the synchrotron oscillations diverges to infinity. A particle on that trajectory would move very slowly towards the unstable fixed point in-between buckets, but never reach it.

As an extension of Eq.~(\ref{eqn:synchrotronFrequencyDistributionStationaryBucket}) an analytic approximation for the synchrotron frequency distribution versus $\Delta \phi_\mathrm{u} = \phi_\mathrm{u} - \phi_\mathrm{S}$ in an accelerating bucket with the stable phase $\phi_\mathrm{S}$ can be obtained as~\cite{bib:zotter1981, bib:zotter2012}
\begin{equation}
    \frac{ \omega(\Delta \phi_\mathrm{u}) }{ \omega_\mathrm{S} } \simeq 1 - \frac{1 + 2/3 \sin^2 \phi_\mathrm{S}}{16 \, (1-\sin^2 \phi_\mathrm{S})} \Delta \phi_\mathrm{u}^2\, .
\end{equation}
For more complicated RF voltages, like double-harmonic RF systems, analytical approximates can only be calculated for few limited cases~\cite{bib:bramham1977}. However, semi-analytical evaluation Eq.~(\ref{eqn:GeneralSynchrotronFrequencyDistribution}) is most useful.

\subsection{Longitudinal stability and synchrotron frequency spread}

As illustrated in Fig.~\ref{fig:PedulumEquivalent}, the larger the spread of the resonance frequencies of an ensemble of oscillators, the more stable the entire system will be against any single-harmonic excitation. For a bunch in a stationary bucket this is sketched in Fig.~\ref{fig:StationaryBucketSynchrotronFrequencyDistributionBunch}.
\begin{figure}[htb]
	\centering\includegraphics[width=0.8\linewidth]{images/StationaryBucketSynchrotronFrequencyDistributionBunch}
	\caption{Comparison of longitudinal phase space and synchrotron frequency spread for different bucket filling factors. When the bucket area is large compared to the longitudinal emittance (gray shaded area), the synchrotron frequency spread is small (left). Increasing the bucket filling factor also increases the synchrotron frequency spread and improved the longitudinal stability.}
	\label{fig:StationaryBucketSynchrotronFrequencyDistributionBunch}
\end{figure}
The area, $A$, encircled by a trajectory in the longitudinal phase space is equivalent to the total longitudinal emittance, $\varepsilon_\mathrm{l}$ of a matched bunch. Choosing a large RF voltage and a large bucket area for a small emittance bunch~(Fig.~\ref{fig:PedulumEquivalent}, left) may suggest important margins against the loss of particles from the bucket. The gray shaded area indicates the fraction of the bucket occupied by particles. In reality, due to the small synchrotron frequency spread, such a small bunch is prone to longitudinal instabilities. When decreasing RF voltage and bucket area or increasing the longitudinal emittance the spread of synchrotron frequencies covered by the bunch becomes wider (right). As a consequence, such a bunch is more resistant against any excitation. Simply reducing the RF voltage during the acceleration cycle, to avoid that the synchrotron frequency spread becomes too small, is hence an efficient means to improve longitudinal stability in hadron synchrotrons. This technique is also referred to as constant bucket area acceleration.

\subsection{Stability in double-harmonic RF systems}

To improve the longitudinal stability beyond the range achievable by voltage reduction, the non-linearity of the RF voltage can be increased by adding a second RF system. In most cases it operates at an integer multiple of the fundamental RF system, and the normalized RF voltage can be written as
\begin{equation}
    g(\phi) = \frac{1}{V} \left[ \, V_1 \sin \phi + V_2 \sin (h_2/h_1 \phi + \phi_1) \, \right] \, ,
\end{equation}
where $V_{1,2}$ and $h_{1,2}$ are again the voltage amplitudes and harmonic numbers of the two RF systems. The relative phase between both of them is $\phi_1$. Figure~\ref{fig:doubleHarmonicBunchShorteningLengtheningModeExample} introduces the two main modes of operation. When both RF voltages are in phase at the centre of the bunch (bunch shortening mode), the effective gradient increases and the bunch length decreases. In the opposite case (bunch lengthening) the voltage gradient is reduced or even inverted and hence the bunch is stretched.

\subsubsection{Bunch shortening mode}
\label{sec:bunchShorteningMode}

The RF amplitude, potential and resulting longitudinal phase space are shown in Fig.~\ref{fig:DoubleHarmonicBunchShorteningPhaseSpaceConstructionAllRatio2} for a set of test particles launched with phase but no energy error. Both RF systems are in phase at the centre of the stationary bucket.
\begin{figure}[htb]
	\centering

	\includegraphics[width=0.35\linewidth]{images/DoubleHarmonicBunchShorteningPhaseSpaceConstructionAllframe0024}
	\hspace*{3em}
	\includegraphics[width=0.35\linewidth]{images/DoubleHarmonicBunchShorteningPhaseSpaceConstructionAllframe0040}
	\caption{Amplitude (top), resulting potential (middle) and normalized longitudinal phase space (bottom) in coordinates of $\phi$ and $\dot{\phi}/\omega_\mathrm{S}$ for the case of two RF systems in phase with a harmonic and voltage ratio of $V_2/V_1 = 1/2$  and $h_2/h_1 = 2.$}
	\label{fig:DoubleHarmonicBunchShorteningPhaseSpaceConstructionAllRatio2}
\end{figure}
After half a period of the synchrotron frequency for the particles in the centre of the bucket (left), the particles towards the separatrix massively lag behind, indicating a large spread of synchrotron frequencies. This effect is even stronger after a full synchrotron frequency period (right). A comparison of the synchrotron frequency distribution between single- and double-harmonic RF voltage versus the bucket filling factor is plotted in Fig.~\ref{fig:DoubleHarmonicBunchShorteningPhaseSpaceConstructionAllSynchrotronFrequencySpread}.
\begin{figure}[htb]
	\centering\includegraphics[width=0.4\linewidth]{images/DoubleHarmonicBunchShorteningPhaseSpaceConstructionAllSynchrotronFrequencySpread_regenerated.png}
	\caption{Synchrotron frequency distribution versus the bucket filling factor for a single- (blue) and double-harmonic RF voltage (red, $V_2/V_1 = 1/2$  and $h_2/h_1 = 2$).}
	\label{fig:DoubleHarmonicBunchShorteningPhaseSpaceConstructionAllSynchrotronFrequencySpread}
\end{figure}
It confirms that the synchrotron frequency spread for a typical bucket filling ratio, $\varepsilon_\mathrm{l}/A_\mathrm{B}$ of about $40\ \mathrm{\%}$ is more than doubled.

This effect can even be increased by choosing a higher harmonic ratio, as illustrated in Fig.~\ref{fig:DoubleHarmonicPhaseSpaceConstructionComparisons}
\begin{figure}[htb]
	\centering
	\includegraphics[width=0.35\linewidth]{images/DoubleHarmonicBunchShortening3PhaseSpaceConstructionAllframe0040}
	\hspace*{3em}
	\includegraphics[width=0.35\linewidth]{images/DoubleHarmonicBunchShortening4PhaseSpaceConstructionAllframe0040}
	\caption{Amplitude (top), resulting potential (middle) and normalized longitudinal phase space (bottom) in coordinates of $\phi$ and $\dot{\phi}/\omega_\mathrm{S}$ for the case of two RF systems in phase with a harmonic and voltage ratios of $V_2/V_1 = 1/3$ and $h_2/h_1 = 3$ (left), $V_2/V_1 = 1/4$ and $h_2/h_1 = 4$ (right). The plots show the situation after one period of the synchrotron frequency in the center of the bucket.}
	\label{fig:DoubleHarmonicPhaseSpaceConstructionComparisons}
\end{figure}
for the harmonic number ratios of $1/3$ and $1/4$ in the centre of the bucket. Clearly the lag of particles with phase of momentum offset with respect to the synchronous particle increases quickly. Approaching however, the separatrix the motion in longitudinal phase space get faster again. The effect also manifests in the synchrotron frequency distribution~(Fig.~\ref{fig:DoubleHarmonicPhaseSpaceConstructionComparisonsSynchrotronFrequencySpread}, red),
\begin{figure}[htb]
	\centering
	\includegraphics[width=0.329\linewidth]{images/DoubleHarmonicBunchShortening3PhaseSpaceConstructionAllSynchrotronFrequencySpread_regenerated}
	\hspace*{4em}
	\includegraphics[width=0.329\linewidth]{images/DoubleHarmonicBunchShortening4PhaseSpaceConstructionAllSynchrotronFrequencySpread_regenerated}
	\caption{Synchrotron frequency distribution versus the bucket filling factor for the two cases shown in Fig.~\ref{fig:DoubleHarmonicPhaseSpaceConstructionComparisons}. The harmonic and voltage ratios in these examples are $V_2/V_1 = 1/3$ and $h_2/h_1 = 3$ (left), $V_2/V_1 = 1/4$ and $h_2/h_1 = 4$ (right).}
	\label{fig:DoubleHarmonicPhaseSpaceConstructionComparisonsSynchrotronFrequencySpread}
\end{figure}
again compared to the single-harmonic case (blue). A higher-harmonic RF system a $h_2/h_1 = 4$ is operated very successfully in Super Proton Synchrotron (SPS) at CERN to stabilize the high-intensity beam for the LHC~\cite{bib:shaposhnikova1998,bib:shaposhnikova2005}. The thresholds of longitudinal instability are increased by a large factor.

Although the spread in the bunch centre grows further with the harmonic number ratio, the synchrotron frequency function, $\omega/\omega_\mathrm{S}$ may not be monotonic develop saddle points or even invert its slope towards the outer part of the bucket. In such local regions with no gradient or inversion, a large number of particles will again have the same synchrotron frequency, making it very susceptible to any excitation. The longitudinal stability in this case becomes even worse than the single-harmonic case which, even in the bunch centre, guarantees a small frequency spread. Populating these regions of the bucket with particles must be avoided by either keeping the longitudinal emittance small enough or reducing the voltage of the second harmonic RF system.

To longitudinally stabilize a beam with a double-harmonic RF voltage, the harmonic number and voltage ratios must therefore be carefully chosen. The larger the harmonic ratio, the bigger the synchrotron frequency spread for the same voltage, but at the expense of potentially reducing the usable fraction of the bucket area.

\subsubsection{Bunch lengthening mode}

The configuration where the two RF system are in counter-phase at the bunch position is very beneficial to reduce the peak current, and hence the longitudinal space charge effect in the stretched bunch as shown in the example of Fig~\ref{fig:doubleHarmonicBunchLengtheningModeExamplePSB}.

At first sight also the synchrotron frequency distribution as plotted in Fig.~\ref{fig:DoubleHarmonicBunchLengtheningPhaseSpaceConstructionAllSynchrotronFrequencySpread} (left) seems very attractive to stabilize a bunch.
\begin{figure}[htb]
	\centering
	\includegraphics[width=0.35\linewidth]{images/DoubleHarmonicPhaseSpaceConstructionAllSynchrotronFrequencySpread_regenerated}
	\hspace*{4em}
	\includegraphics[width=0.35\linewidth]{images/DoubleHarmonicPhaseSpaceConstructionAllSynchrotronFrequencySpread5DegreesPhaseError_regenerated}
	\caption{Synchrotron frequency distribution versus the bucket filling factor in a double-harmonic RF system with both voltages in counter phase ($V_2/V_1 = 1/2$ and $h_2/h_1 = 2$. In case of a perfect relative phase (left), the synchrotron frequency vanishes at the bunch centre. A small, unavoidable relative phase error of only $\Delta \phi = 5^0$ is however sufficient to change the synchrotron frequency distribution significantly (right).}
	\label{fig:DoubleHarmonicBunchLengtheningPhaseSpaceConstructionAllSynchrotronFrequencySpread}
\end{figure}
Close to the bunch centre particles move very slowly with respect to the synchronous particle. The synchrotron frequency then grows for larger bucket filling ratios and decreases again when approaching the separatrix. Although the spread is extremely large in this ideal case, a small phase error between both RF systems has a detrimental impact. Already with a small residual phase error~(Fig.~\ref{fig:DoubleHarmonicBunchLengtheningPhaseSpaceConstructionAllSynchrotronFrequencySpread}, right), unavoidable in a real accelerator, the slope flips and regions with the same synchrotron frequency are found in different part of the central part of the bunch. Such a synchrotron frequency distribution is not beneficial for longitudinal stability. In bunch shortening the synchrotron frequency distribution is insensitive to such small phase errors due to both voltages adding up in phase at the bucket centre.

The longitudinal stability at the flat-top in the CERN PS has been checked with beam for the three different case of single-harmonic RF at $h_1=21$, adding RF voltage at $h_2=84$ ($h_2/h_1 = 4$) in counter phase and in phase.
As can be seen in the mountain range plot of four exemplary bunches in Fig.~\ref{fig:SingleDoubleHarmonicStabilityPSExample},
\begin{figure}[htb]
	\centering\includegraphics[width=\linewidth]{images/SingleDoubleHarmonicStabilityPSExample}
	\caption{Measured evolution of the bunch profile in the PS during $15\ \mathrm{ms}$ at the flat-top in the PS at CERN. Only four bunches of a train of $18$ are shown. In the single-harmonic case (left) strong quadrupole oscillations are observed, which partially persist when adding voltage at the fourth harmonic in counter phase to stretch the bunches (middle). With the voltage at the fourth harmonic in phase, the oscillations are entirely damped (right).}
	\label{fig:SingleDoubleHarmonicStabilityPSExample}
\end{figure}
the bunches exhibit strong bunch length oscillations (left). A constant bunch-to-bunch phase advance is observed, indicating a coupled-bunch oscillation. With the higher-harmonic RF system in counter-phase longitudinal stability is improved marginally. When flipping the phase of the higher harmonic RF voltage, the oscillations entirely disappear and the bunches are stable.

Although the longitudinal peak density is highest when both RF systems are in phase, the bunches are most stable, with no sign of quadruplolar coupled-bunch oscillations. As has been shown above, this stability is achieved thanks to the enlarged synchrotron frequency spread introduced by an intentional non-linearity caused by the second RF system.

\section{Conclusion}

Multi-harmonic RF systems allow to manipulate the longitudinal beam parameter like bunch length and spacing. The reduction of space charge effects by reducing the peak line density in proton synchrotrons or the simultaneous operation of short and long bunches in an electron storage ring are examples of the wide range of applications.

The non-linearity of the RF voltage applied to a charged particle beam in accelerators is essential to keep high-intensity beams longitudinally stable. In many cases additional non-linearity in the form of multi-harmonic RF systems is intentionally implemented, to manipulate the longitudinal beam parameters like bunch length or distance between bunches. Of equal importance is the improvement of the longitudinal beam stability which is achieved with multi-harmonic RF systems.

The most common case, double-harmonic RF, is installed in most high-intensity electron and proton synchrotrons. With appropriately chosen parameters, the non-linearity due to a high-harmonic RF system induces a wider spread of synchrotron frequencies of the particles in a bunch. The associated gain in longitudinal beam stability is significant and allows to extend the performance reach in terms of bunch intensity and longitudinal beam density well beyond the possibilities with a conventional, single-harmonic RF system.

\section{Acknowledgement}

The author would like to thank Wolfgang H\"ofle, Andreas Jankowiak, Erk Jensen, Danilo Quartullo, Markus Ries, Elena Shaposhnikova and Frank Tecker for discussions and material provided in preparation of this course. Thomas Bohl is acknowledged for his valuable corrections of the manuscript and numerous suggestions for improvements.

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Moreira dos Anjos-Santos, Lucas, Michele Salles El Kadri, Raquel Gamero and Telma Gimenez. "Developing English Language Teachers' Professional Capacities through Digital and Media Literacies: A Brazilian Perspective." Leadership and Personnel Management: Concepts, Methodologies, Tools, and Applications. IGI Global, 2016. 407-430. Web. 21 Jan. 2020. doi:10.4018/978-1-4666-9624-2.ch019
Moreira dos Anjos-Santos, L., El Kadri, M. S., Gamero, R., & Gimenez, T. (2016). Developing English Language Teachers' Professional Capacities through Digital and Media Literacies: A Brazilian Perspective. In I. Management Association (Ed.), Leadership and Personnel Management: Concepts, Methodologies, Tools, and Applications (pp. 407-430). Hershey, PA: IGI Global. doi:10.4018/978-1-4666-9624-2.ch019
Moreira dos Anjos-Santos, Lucas, Michele Salles El Kadri, Raquel Gamero and Telma Gimenez. "Developing English Language Teachers' Professional Capacities through Digital and Media Literacies: A Brazilian Perspective." In Leadership and Personnel Management: Concepts, Methodologies, Tools, and Applications, ed. Information Resources Management Association, 407-430 (2016), accessed January 21, 2020. doi:10.4018/978-1-4666-9624-2.ch019
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Developing English Language Teachers' Professional Capacities through Digital and Media Literacies: A Brazilian Perspective
Lucas Moreira dos Anjos-Santos (Monash University, Australia & CAPES Foundation, Brazil), Michele Salles El Kadri (State University of Londrina, Brazil), Raquel Gamero (State University of the North of Parana, Brazil) and Telma Gimenez (State University of Londrina, Brazil)
Source Title: Leadership and Personnel Management: Concepts, Methodologies, Tools, and Applications
Copyright: © 2016 |Pages: 24
DOI: 10.4018/978-1-4666-9624-2.ch019
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This chapter aims to demonstrate how a group of educators from a southern Brazilian state university designed and implemented formative workshops to sustain English language teachers' professional development through digital and media literacies. The chapter maps important changes that have happened in language teacher education in Brazil and the convergences these changes share with digital and media literacies coming from a sociocultural paradigm. It also presents and discusses the extent to which the instructional material the group of educators produced for the continuing education of English language teachers integrated 21st century skills and the standards from the TESOL technology framework. As a way to evaluate the instructional material, the chapter analyzes the representations and identities schoolteachers constructed when engaging with digital and media literacies through the instructional material. The chapter concludes by advocating more social, political and collaborative future research in language teacher education and digital and media literacies.
Chapter Preview
The foundational discourses that shape and actively transform educational practices are often multiple and contradictory. Similarly, the roles language teachers perform in promoting adequate and challenging conditions for children and young people to learn in the so-called networked society are varied. On the one hand, teachers themselves are expected to use, to critically appreciate and to engage with diverse social practices. On the other hand, it is part of teachers' professional responsibility to plan and design situated and meaningful language pedagogies. These pedagogies, in turn, should aim not only to prepare young people for the future but also to help them critically engage with and transform a world in constant change and fluidity.
Such challenges become even more pressing if educators shift their focus to digitally mediated language practices. Language teacher educators are confronted by a myriad of discourses and face challenges on to how to develop, design and implement teacher education programs aligned with the contemporary digitally mediated landscape. Similarly, teacher educators aim to create opportunities for schoolteachers to critically interrogate the assumptions deriving from the imperatives surrounding digital technology use in the network society.
Based on such scenario, a number of teacher educators from a southern Brazilian state university developed and implemented formative workshops for English language teachers to foster their digital and media literacies. These workshops were funded by the Brazilian Federal Agency for Higher Education Professional Development (CAPES) program called 'New Talents'. The program, which took place from 2011 to 2012, aimed to support proposals that could offer extracurricular activities for schoolteachers and students from secondary education. Such extracurricular activities were meant to promote scientific knowledge dissemination and contribute to the professional development of teachers. The proposal under which the reported experience occurred was entitled "Education of teachers and students from the secondary education: opportunities, practices, exchange of knowledge/experiences, leverage of the creativity and alignment to the challenges of the globalized society".
In order to offer formative workshops for teachers, we developed instructional materials that explored digital and media literacies and that created conditions for teachers themselves to produce activities for secondary school students. The instructional materials developed and the formative workshops designed are the objects of analysis of this chapter. Therefore, the objectives of this chapter are:
To demonstrate how a group of educators from a southern Brazilian state university designed and implemented formative workshops to sustain English language teachers' professional development through digital and media literacies;
To analyze to what extent the 21st century skills and the standards from TESOL technology framework were integrated in the formative workshops designed and implemented;
To analyze the representations and identities language schoolteachers constructed when engaging with digital and media literacies.
To consider sustainable ways of providing English language schoolteachers with professional development opportunities through digital and media literacies.
The remainder of the chapter is organized into five broad sections. In the first two sections, we present and discuss important changes that have occurred in language teacher education in Brazil and in digital and media literacies. We then move on to present and analyze the instructional material designed to develop English teachers' language professional capacities through digital and media literacies. The analysis focuses on to what extent the instructional material integrates and develops the 21st century skills and the parameters from TESOL technology framework. In the fourth section, we analyze the representations and identities language schoolteachers constructed when engaging with digital and media literacies and the implications they pose for future research. Finally, we conclude the chapter stating the need for finding sustainable ways of developing language teachers' professional capacities through digital and media literacies.
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Since the earliest beginnings of the Polish Chamber of Milk, i.e. since 2008, we have closely cooperated on drawing up and executing the marketing concept as well as internal and external communication strategies of the Chamber. We comprehensively manage current operations of the organisation and conduct public relations activities. Currently, the Polish Chamber of Milk is the largest and fastest developing organisation associating entities of the dairy industry in Poland.
Our tasks includes, among other things, drawing up promotion and information materials as well as running the press office of the Polish Chamber of Milk (www.izbamleka.pl).We cooperate in creating and executing educational and social campaigns whose aim is to promote Polish milk, including "Milk kids" (Dzieciaki mleczaki), "I am crazy about milk" (Mam kota na punkcie mleka) or "Excellent milk" (Doskonałe mleko).
For years, we have been actively involved in carrying out information and educational activities related to the Breast Cancer Early Detection Programme. We cooperate with the Central Coordination Centre and Provincial Coordination Centres on prophylactic campaigns of health programmes. We have conducted numerous operations in cooperation with the Independent Public Complex of Health Care Facilities (SPZOZ), research institutes, foundations and associations, including: organisation of meetings, conferences and events devoted to breast cancer early detection preventive care, drawing up promotion and information materials, running a press office.
We are also the originators and authors of a website gathering information (authorised and approved by scientists) on preventive healthcare and breast examination. On www.badajpiersi.pl every woman will find information and answers to the following questions: how to take care of breasts and how to carry out a self-examination, what are the high risk factor, what examination a woman should undergo to make sure that she is healthy and where diagnostics tests can be run.
Our involvement in promoting healthy lifestyle led to establishing the Prevention and Health Institute (instytutprofilaktyki.pl) in 2014. That is why we treat this project like our own "child". The Institute provides complex services related to nutrition consultation and education, diet-related prevention of diseases of affluence and dietary therapy of chronic disorders, in the case of which good nutrition plays a vital role.
Our close cooperation with the Institute consists in drawing up and executing a marketing strategy, comprehensive project management, public relations activities, preparing information materials and running a press office.
Since 2012, we have run the www.coolturalnybialystok.pl website – an up-to-date guide to the world of culture and entertainment in Podlasie and an unofficial cultural information centre.
Every day, we provide up-to-date information on cinema premieres, concerts, meetings with authors, conferences, workshops, exhibitions, etc. To cut a long story short, we post information on simply everything that happens in Białystok and its vicinity. Our missions is to promote culture and entertainment events. We provide patronage for the most interesting events, support niche culture and various activities in our region. We are open to all forms of artistic manifestations of our readers who want to share their work with us.
For years, we have successfully cooperated with the Oncology Centre in Bialystok by carrying out education and information activities related to breast cancer prevention. We are involved in executing preventive care programmes in schools, organising meetings, conferences and special events (World Cancer Day, No Tobacco Day, Mother's Day) and running a press office.
Moreover, we conduct comprehensive public relations on information and education related to cervical cancer early detection on the request of the University Teaching Hospital in Bialystok. Our tasks consisted in: organising meetings and conferences promoting healthy lifestyle and preventive care, promoting a free cytological examination programme for women and running a press office.
We are the originators and the sole organiser of the largest international dairy industry conference in East-Central Europe. It is the only such big and prestigious forum which annually gathers guests from the whole world: dairy industry experts, representatives of embassies, ministries of agriculture, agricultural advisors, members of government administration and international business organisations, such as EDA, Copa-Cogeca, FIL/IDF, DGAgri.
The Forum is a meeting place where crucial and current issues related to the industry are discussed. It is a unique opportunity to exchange ideas and opinions between the representatives of state institutions, companies and dairy industry experts.
We developed and executed a project aiming at increasing a competitive position of medium-sized dairy companies by implementing ecological solutions. Our tasks consisted in analysing ecological needs of a company, providing environmental consultancy, drawing up a ecological solution implementation plan and conducting an employee training.
Under the Green milk project, we have held over 80 training sessions with about 1,200 participants.


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Confessions of a social media editor: 'I'm not confident this will be a real field five years from now'
Despite how fundamental it is to the success of brands and publications, social media continues to get a bad rap — be it for enabling the worst aspects of humanity, privileging the superficial or whittling down our collective attention span. Those who work in the space are subject to the same criticisms and also the same concerns.
Fashion media has become especially reliant on the medium to drive traffic to their respective sites, regularly tailoring their content (down to the title formats) based on what receives the most clicks. Facilitating this across multiple platforms are social media editors, the elusive humans behind many of the fashion and beauty feeds consumers know and love. Along with handling all outbound social content, these editors tend to have their hands in video creation, content strategy, analytics and more.
For our latest confessions, in which we grant anonymity for honesty, we spoke to a social media editor with ten years of experience, who currently works for a popular fashion and beauty site. Of her vague title, she said: "At this point, it's used to describe everyone from a 22-year-old in their first professional role to someone 10 years older with a great deal of media experience." Read on for her thoughts on the stigma attached to her job, the frustrating concept of "clickbait" and the uncertain future of her career.
Do you feel like there's a stigma attached to working in social media, and if so, what is it?
I think there's a general perception in the world right now that we're responsible for all the clickbait that everyone hates on social media. While there certainly are some Facebook pages that do nothing but churn out that kind of thing, most of us have very legitimate backgrounds in media, fashion or journalism, and we're always trying to balance the fact that we have to make you click with the fact that we hate clickbait, too.
[That term] is a pet peeve of mine, because "clickbait" actually means something: It refers to a super dramatic headline that is misleading or inflammatory, that clicks through to a different story than you were promised. If we say, "See the first photo of Taylor Swift and her new love," and it's a photo of Taylor and her new kitten, that's clickbait. If it's a photo of Taylor and the new guy she's dating, though, then it's not. Just because you have to click to get the full experience of the story doesn't mean that it's clickbait. Think of those clicks as a form of payment for enjoyable content.
What is the hardest or worst part of your job?
Social as an industry is ever-changing, which makes it fun and exciting on a day-to-day basis, but a little scary when contemplating the long term. We're all really at the mercy of the platforms and what they decide to do as they face the pressure to become profitable businesses, and if they decide brands and publishers aren't a part of the experience their users want, there isn't really anything we can do about it. I don't think that's going to happen — at least not entirely — but no one is seeing the kind of growth on social they used to, and it's just a lot harder to add value in my role than it used to be, through no real fault of my own.
It's frustrating to feel like you're failing when what is actually happening is the tools you're using just aren't as effective. My job was just beginning to be a real field five years ago, and I'm not confident it will be a real field five years from now — or at least not a field where there's room for growth. I know I'm in for another career pivot within the next 5-10 years, and I have some ongoing existential dread about what that will look like.
What are some social media "secrets" used by brands and publications that the average person isn't aware of?
The way businesses do social [isn't much like] the way consumers use social. Yes, our content appears on Facebook, but we use scheduling dashboards, link shorteners, image templates and a lot of tools that the average person doesn't use when they find a funny link and post it on their page. We're often scheduling content well in advance, at a much higher volume than most people realize (often 40-50 posts per day), as well as being responsive to news and creating new content on the fly — and that's just organic content. W also have access to suites of advertising tools, paid promotions and post formats that simply aren't available to individuals.
Consumers have very little idea how we do our jobs, and they're always super paranoid about the wrong thing. We get tons of comments if we feature a brand or a celebrity a lot, with people saying things like, "Did so-and-so pay you to post this?" and the answer is just, "No, they didn't" — because if they did, we would clearly disclose that. It's an FTC violation if we don't. But then most people don't react at all to the ads we do post and often interact with them, not realizing that that just confirms our ability to target consumers by interests, location, age, income, marital status and many other characteristics that would freak people out if [they knew about it]. Yes, it's all anonymized, but the Illuminati isn't putting all that content about the Kardashians in your feed. Big Data is.
Do you think social media editors get enough credit in the fashion industry for all the work they do?
They definitely end up being the unsung heroes of the fashion and beauty game. We get a lot of flack if the content doesn't succeed, but editors get the credit when it does, and we don't always get a say in the content we're given to promote. Since we don't have a byline, we're not perked the same way editors are; we get way fewer gifts and press trip offers and, when we do get anything, it's rarely the really good stuff.
Honestly, though, it's fine. I make anywhere from 20-50 percent more than most of our editors, and I consciously chose to leave editorial for a higher-paying track. I miss the press trips a lot, but I can afford to take my own vacations now, and no one is ever going to gift me a down payment on an apartment.
How does working in social media affect your personal presence online?
I don't have time for it, honestly. Some social editors live and breathe it and love maintaining their social presence, but if I have to choose in any given moment between doing what I do for money and doing it for free, why would I do it for free? I think many of us tend to be behind-the-scenes types anyway, and it's not really possible to get the amount of social content you need to for work if you're maintaining the kind of public social life that editors with a big following have. Most of the social editors I know who have a big personal presence had it already and parlayed it into a full-time job. I've rarely seen it go the other way around.
Plus, now that I've done this for a while, it's clear it's really not the quality of your photos or your strategy that makes an individual get huge on social — it's a lot of things that, quite frankly, aren't priorities for me, like being super thin or taking lots of photos of yourself in really fashion-y outfits.
I'm not worried about building my personal brand to get my next job. I can just pull up a Powerpoint deck of analytics showing that I'm capable of building one. I actually think it's harder for editors and people who do work that isn't as quantifiable; as a result, they feel more pressure to develop a real personal brand. I have data; I don't need window dressing.
Do you ever get sick of social media? How do you avoid getting caught up in it?
I definitely have days where I'm sick of my job like anyone else, but in general, I just kind of love the internet and love talking to people, and I would never wish it all away. I've learned that unplugging fully doesn't really work for me, and I actually think putting pressure on young people in digital to unplug — or telling them they're going to be unhealthy or unhappy if they don't — is alarmist and unfair.
What's the weirdest or most aggravating experience you've had while dealing with the online community?
People say the most ridiculous things when they think no one is reading. When I was newer in this field and cared more about other peoples' opinions, we'd get messages calling out something like a small typo that were like, "Fire your social media editor." I would respond and just say, "Hi! I'm the social media editor, my name is X, and I'll pass along your comment if you really think I should be fired, but I really don't want to have to move back in with my parents." Basically, I was doing the same thing you're supposed to do if you're kidnapped: Use your name, try to humanize yourself, make yourself relatable. Most people fell all over themselves to apologize. They'd tell me they didn't realize anyone read their messages, that they were having a bad day or (my favorite) that they thought they were talking to a robot — which actually could be the case today, but was not common at the time.
There's much more dialogue now about internet bullying and digital communities, so I kind of have the opposite problem today. We have followers who care way too much, messaging us constantly with feedback about content, community and even strategy. Like, I appreciate your investment in our brands, but maybe you should go grab drinks with a friend IRL tonight instead, you know?


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Schrieffer family looking for closure
The dignity of funeral rites -- religious celebrations of life that give resolution to death -- starts the healing process for many. But for Mary Forstrom of Cloquet, the violent way in which it is believed her son died -- run down by a truck aft...
The dignity of funeral rites -- religious celebrations of life that give resolution to death -- starts the healing process for many.
But for Mary Forstrom of Cloquet, the violent way in which it is believed her son died -- run down by a truck after being beaten in an alley in West Duluth -- has provided the family with no closure. The body of Erik Wayne Schrieffer, 27, has not been found.
"I've known for one week and two days," said Forstrom last week. "But he's been missing for nearly three weeks."
Testimony to, and evidence of, the beating and the injuries Schrieffer sustained lead police to believe he died around 1 a.m. on Jan. 24. He wasn't reported missing until Jan. 29.
Joseph Arden Wehmanen, 32, Hermantown, was arrested on Jan. 31 and formally charged with second-degree murder on Monday, according to Duluth police Sgt. Tim Jazdzewski. Wehmanen is jailed on $325,000 bail.
Late Friday afternoon, Forstrom found comfort at Concordia Lutheran Church in Superior.
Seated on a pew inside the empty church where she attended Sunday school as a child and confirmation classes as a teen-ager, Forstrom looked toward the sanctuary where her son was made a child of God through Holy Baptism.
Forstrom has been told that based on evidence at the crime scene, her son could not have survived the injuries without immediate medical treatment. "I know he's dead," Forstrom said, admitting her strength comes from her faith.
"Well, of course it's from God. We all know that. There's no two ways about it. And at the same time I'm angry," Forstrom said.
But she said over the past few horrific days, friends and family have done a lot of reminiscing. There is a theme that permeates the reaction of his friends who admit their love for Schrieffer:
"He had an underlying strength and conviction to do the right thing," writes Forstrom, having gathered some thoughts and memories on paper, prior to her interview at the church. "He was a natural wanderer and lived a life some of us can only imagine."
Forstrom said her son was gentle and soft-spoken, with the ability for "profound thinking and writing" -- he had an "enigmatic flip side" that kept him aloof, and sometimes his family didn't hear from him for stretches of time. As a member of the Boilermakers Local No. 647, he traveled on his job for weeks at a time.
As a child, she writes, he loved his dog, Spirit, a stuffed Santa doll, books, cartoons, skiing and snowboarding and hunting with his father and younger brother, Kyle. He grew up in Superior and graduated from Superior Senior High School.
In addition to her written thoughts, Forstrom on Friday shared some photographs:
A slightly faded color photo shows the sweet, puffy face of a newborn, cradled in his mother's arms. Although the mother's face is not shown, the hospital name band is visible on her wrist, the nursery of the former St. Joseph's Holy Family Hospital in Billings Park is slightly out of focus, in the background.
"Moms ... we all work so hard to keep our kids safe. It's the moms who are hit the hardest," said Forstrom.
And then she picked up a more contemporary photo, showing her son at 25, cradling his own son, Zakary. Schrieffer's lips are brushed against the infant's forehead -- a moment of tenderness captured forever on film.
"He was not untouched by heartache," wrote Forstrom, "but he remained hopeful. He had plans for the future, especially with Zakary."
There are other photos: a chubby-cheeked 4-year-old holding a puppy, a slightly nervous looking adolescent as a greeter at St. Paul's Lutheran Church in Billings Park, a smiling, white-robed teen-ager at confirmation, a reflective man looking out over the Arizona landscape at dusk.
But the most recent shows her son last fall, the afternoon sun in his face, his jacket thrown over his shoulder. "He's saying good-bye," said Forstrom.
There is a message that arises from the grief: "Don't forget to hug your kids, call your mom. You never know when it's going to be your last conversation.
"I can't even think about the future right now," said Forstrom. "I have a loving family and a wonderful husband, a deep spiritual faith. But it's everything I can do to go to work."
Forstrom is a physical therapist assistant at Beverly Health and Rehabilitation Center in Superior. She lives in Cloquet with her husband, Leon. Her youngest son, Kyle, 23, lives in Superior and celebrated his college graduation just two days before his brother is believed to have died.
Forstrom's former husband, Gene Schrieffer, father of Erik and Kyle, lives in Anoka with his wife Susan.
The family is planning a memorial.
"Time will tell," said Forstrom. "We'll know someday. Sometimes you just have to stand back and let the professionals do their job."
She said her faith gives her peaceful images even after visiting the crime scene.
"For some crazy reason, I feel Erik's lying outside, just like he's lying in a casket ... just a part of the earth ... lying there with no jacket.
"God knows what to do," said Forstrom. "I have to have faith, otherwise I'd just fall into a big, old black hole."
She said she feels an empathy for the mother of the suspect. "It's not in her control. She's a mom just like the rest of us."
Susan Anderson is a reporter for the Superior Daily Telegram, a Murphy McGinnis Newspaper.


text:
You were looking for organic. Total number of found items is 8.
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