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Tracks through a TM110-mode (deflecting) rf cavity with all magnetic and electric field components. NOT RECOMMENDED--See below.
Parallel capable? : yes
Parameter Name Units Type Default Description
PHASE $DEG$ double 0.0 phase
TILT $RAD$ double 0.0 rotation about longitudinal axis
FREQUENCY $HZ$ double 2856000000 frequency
VOLTAGE $V$ double 0.0 peak deflecting voltage
PHASE_REFERENCE   long 0 phase reference number (to link with other time-dependent elements)
VOLTAGE_WAVEFORM   STRING NULL $<$filename$>$=$<$x$>$+$<$y$>$ form specification of input file giving voltage waveform factor vs time
VOLTAGE_PERIODIC   long 0 If non-zero, voltage waveform is periodic with period given by time span.
ALIGN_WAVEFORMS   long 0 If non-zero, waveforms' t=0 is aligned with first bunch arrival time.
VOLTAGE_NOISE   double 0.0 Rms fractional noise level for voltage.
PHASE_NOISE $DEG$ double 0.0 Rms noise level for phase.
GROUP_VOLTAGE_NOISE   double 0.0 Rms fractional noise level for voltage linked to group.
GROUP_PHASE_NOISE $DEG$ double 0.0 Rms noise level for phase linked to group.
VOLTAGE_NOISE_GROUP   long 0 Group number for voltage noise.
PHASE_NOISE_GROUP   long 0 Group number for phase noise.
START_PASS   long -1 If non-negative, pass on which to start modeling cavity.
END_PASS   long -1 If non-negative, pass on which to end modeling cavity.
GROUP   string NULL Optionally used to assign an element to a group, with a user-defined name. Group names will appear in the parameter output file in the column ElementGroup

NB: Although this element is correct insofar as it uses the fields for a pure TM110 mode, it is recommended that the RFDF element be used instead. In a real deflecting cavity with entrance and exit tubes, the deflecting mode is a hybrid TE/TM mode, in which the deflection has no dependence on the radial coordinate.

To derive the field expansion, we start with some results from Jackson[17], section 8.7. The longitudinal electric field for a TM mode is just

E_z = - 2 i E_0 \Psi(\rho, \phi) \cos \left(\frac{p \pi z}{d}\right) e^{-i\omega t},
\end{displaymath} (77)

where $p$ is an integer, $d$ is the length of the cavity, and we use cylindrical coordinates $(\rho, \phi, z)$. The factor of $-2i$ represents a choice of sign and phase convention. We are interested in the TM110 mode, so we set $p=0$. In this case, we have
E_x = E_y = 0
\end{displaymath} (78)

and (using CGS units)
\vec{H} = - 2 i E_0 \frac{i \epsilon \omega}{c k^2} \hat{z} \times \nabla \Psi e^{-i \omega t}.
\end{displaymath} (79)

For a cylindrical cavity, the function $\Psi$ for the $m=1$ aximuthal mode is
\Psi(\rho, \phi) = J_1 (k \rho) \cos \phi,
\end{displaymath} (80)

where $k = x_{11}/R$, $x_{11}$ is the first zero of $J_1(x)$, and $R$ is the cavity radius. We don't need to know the cavity radius, since $k = \omega/c$, where $\omega$ is the resonant frequency. By choosing $\cos\phi$ for the aximuthal dependence, we'll get a magnetic field primarily in the vertical direction.

In MKS units, the magnetic field is

\vec{B} = \frac{2 E_0}{k c} e^{-i \omega t} \left( \hat{\rho...
...hi} \cos\phi \frac{\partial J_1(k\rho)}{\partial \rho}\right).
\end{displaymath} (81)

Using mathematica, we expanded these expressions to sixth order in $k*\rho$. Here, we present only the expressions to second order. Taking the real parts only, we now have

$\displaystyle E_z$ $\textstyle \approx$ $\displaystyle E_0 k \rho \cos \phi \sin \omega t$ (82)
$\displaystyle c B_\rho$ $\textstyle \approx$ $\displaystyle E_0 \left(1 - \frac{k^2 \rho^2}{8}\right)\sin\phi \cos\omega t$ (83)
$\displaystyle c B_\phi$ $\textstyle \approx$ $\displaystyle E_0 \left(1 - \frac{3 k^2 \rho^2}{8}\right)\cos\phi \cos\omega t$ (84)

The Cartesian components of $\vec{B}$ can be computed easily
$\displaystyle c B_x$ $\textstyle =$ $\displaystyle c B_\rho\cos\phi - c B_\phi\sin\phi$ (85)
  $\textstyle =$ $\displaystyle \frac{E_0}{4} \rho^2 k^2 \cos\phi \sin\phi \cos\omega t$ (86)
$\displaystyle c B_y$ $\textstyle =$ $\displaystyle c B_\rho\sin\phi + c B_\phi\cos\phi$ (87)
  $\textstyle =$ $\displaystyle E_0 \left(1 - \frac{k^2\rho^2 (2 \cos^2\phi + 1)}{8}\right) \cos\omega t$ (88)

The Lorentz force on an electron is $F = -e E_z \hat{z} - e c \vec{\beta} \times \vec{B}$, giving

$\displaystyle F_x/e$ $\textstyle =$ $\displaystyle \beta_z c B_y$ (89)
$\displaystyle F_y/e$ $\textstyle =$ $\displaystyle -\beta_z c B_x$ (90)
$\displaystyle F_z/e$ $\textstyle =$ $\displaystyle -E_z - \beta_x c B_y + \beta_y c B_x$ (91)

We see that for $\rho \rightarrow 0$, we have $E_z = 0$, $B_x = 0$, and
c B_y = E_0 \cos \omega t.
\end{displaymath} (92)

Hence, for $\omega t=0$ and $E_0>0$ we have $F_x>0$. This explains our choice of sign and phase convention above. Indeed, owing to the factor of $2$, we have a peak deflection of $e E_0 L/E$, where $L$ is the cavity length and $E$ the beam energy. Thus, if $V = E_0 L$ is specified in volts, and the beam energy expressed in electron volts, the deflection is simply the ratio of the two. As a result, we've chosen to parametrize the deflection strength simply by referring to the ``deflecting voltage,'' $V$.

Explanation of $<$filename$>$=$<$x$>$+$<$y$>$ format: Several elements in elegant make use of data from external files to provide input waveforms. The external files are SDDS files, which may have many columns. In order to provide a convenient way to specify both the filename and the columns to use, we frequently employ $<$filename$>$=$<$x$>$+$<$y$>$ format for the parameter value. For example, if the parameter value is waveform.sdds=t+A, then it means that columns t and A will be taken from file waveform.sdds. The first column is always the independent variable (e.g., time, position, or frequency), while the second column is the dependent quantity.

next up previous
Next: RFTMEZ0 Up: Element Dictionary Previous: RFMODE
Robert Soliday 2014-06-26