Longitudinal Schottky Pickup for the Recycler

Ed Cullerton, Martin Hu

Beams Division/RF & Instrumentation

**Abstract:** A longitudinal schottky pickup has been built for the
recycler. It is to be used as a very sensitive beam current monitor, and
as a tool to measure synchrotron frequencies. The pickup is based on a
pickup designed for the anti-proton source accumulator, and has a
resonant frequency of 79.203 MHz.

# Logitudinal Schottky Pickup Basics

In a circular particle accelerator/storage ring, a single particle with
charge $e$and revolution frequency $f_{0}$ can be represented as

$i(t) = \text{ef}_{0}\sum_{n = - \infty}^{\infty}{\text{exp}(j2{n\pi f}_{0}t) = \text{ef}_{0} + 2\text{ef}_{0}\sum_{n = 1}^{\infty}{\text{cos}(2{n\pi f}_{0}t)}}$

as observed at a point on the accelerator. In frequency space the above
are lines at$f_{0}$ and its harmonics. If we sum the above equation over
$N$ particles, each with slightly different revolution frequencies, then
the lines become bands, and the r.m.s. current per band is proportional
to the square root of the number of particles:

$i_{\text{rms}} = 2\text{ef}_{0}\sqrt{\frac{N}{2}}$,

The total power in a Schottky band remains constant, independent of $n$,
the harmonic number, but the power spectral density is proportional to
the number of particles in the beam divided by the frequency spread of
the beam:

P.S.D. $$$i^{2} > \frac{}{\Delta f}$ ,

where $\Delta f$ is related to the momentum dispersion function, $\eta$,
of the lattice, and $\frac{\Delta p}{p}$, the momentum spread of the
beam, as follows:

$\Delta f = \text{nf}_{0}\eta\frac{\Delta p}{p}$ ,

As $n$ increases, the power spectral density decreases, resulting in
worse signal-to-noise ratios. Furthermore, the separation between two
adjacent bands is roughly (Fig. 0)

$f_{0} - {n\Delta f}_{0}$,

for large $n$. For sufficiently large $n$, the bands start overlapping
each other. Let $f_{n}$ denote the nth harmonic of the revolution
frequency chosen for the pickup resonance, $f_{n} = \text{nf}_{0}$. In
order to avoid overlapping bands, we need

$f_{0} - {n\Delta f}_{0}$$0$,

or equivalently

$\frac{f_{0}^{2}}{{\Delta f}_{0}} > f_{n}$.

The upper frequency limit below overlapping for the recycler detector is
about 1 GHz.

$(n - 1)f_{0}$ $\text{nf}_{0}$

$f_{0}$

frequency

#  $\frac{(n - 1){\Delta f}_{0}}{2}$ $\frac{{n\Delta f}_{0}}{2}$

# 

# Fig. 0 Schottky bands separation $f_{0} - {n\Delta f}_{0}$for large $n$. 

In a bunched beam, the particles will execute synchrotron oscillations
with respect to the synchronous particles, and the arrival time of
particles at the detector is modulated by an additional term

$\tau\text{sin}(\Omega t + \phi)$,

Where $\tau$,$\Omega$ and $\phi$ are the amplitude, frequency and phase
of the particles with respect to the synchronous particles. Then the
longitudinal signal becomes

$i(t) = \text{ef}_{0} + 2\text{ef}_{0}\sum_{n = 1}^{\infty}{\sum_{k = - \infty}^{\infty}{j_{k}(2\pi\text{nf}_{0}\tau)\text{cos}\left\lbrack \left( 2\pi\text{nf}_{0} + k\Omega \right)t + k\phi \right\rbrack}}$,

where $j_{k}$ is the Bessel function of order $k$. Displayed in
frequency, the revolution lines consist of a central line ($j_{0}$ term)
and satellites spaced at $\Omega$ and amplitudes determined by the
Bessel coefficients.

# 

# Pickup Design

![](media/image1.jpeg){width="5.997222222222222in"
height="2.7618055555555556in"}The longitudinal schottky can be viewed as
a quarter wave open-circuit transmission line stub. The beam pipe is the
center conductor of a coaxial transmission line, and an outer shell on
the outside of the beam pipe acts as the outer conductor of the coaxial
transmission line. To simulate the effect of an open-circuited
transmission line, a ceramic gap is placed in the beam pipe. A schematic
of the pickup is shown in fig.1. The quarter wave structure has a
resonant frequency of 79.203 MHz ($n$= 882). A type N connector is used
to extract the signal when the structure is resonating. A braided wire
is soldered from the center pin of the type N connector and is connected
to the center beam pipe using a metal "worm" clamp. The resonant
frequency of the structure is tuned by adjusting the position of outer
shell. To match the impedance of the resonant structure to 50 Ohms, the
position of the metal worm clamp is adjusted. With these two
adjustments, the impedance, at a given resonant frequency, can be
matched very well to 50 Ohms. The Fermilab assembly drawing for the
schottky pickup is 0231.000 MD-322188.

Fig. 1. Longitudinal schottky pickup

# Pickup measurement

The schottky pickup has a resonant frequency of 79.203 MHz. A 1-port s11
calibration was done to a network analyzer to look at the impedance seen
looking into the schottky pickup. The impedance is to matched to 50 Ohms
at the resonant frequency. A plot of the impedance seen looking into the
schottky pickup is shown in fig.2. The impedance is also shown on a
smith chart plot in fig.3.

![](media/image2.jpeg){width="5.449305555555555in"
height="2.6131944444444444in"}

![](media/image3.jpeg){width="5.147222222222222in"
height="2.907638888888889in"}

Fig.2. Impedance seen looking into the schottky pickup

Fig.3. Impedance plotted on a smith chart

Markers are placed at points were the real impedance is equal to the
imaginary impedance. These points are used to determine the Q of the
pickup. The Q is derived using the following equation.

resonant frequency

Q(unloaded) = \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

F2 -- F1

F1 = first frequency point where the real impedance is equal to the
imaginary impedance

F2 = second frequency point where the real impedance is equal to the
imaginary

impedance

Using the smith chart shown in fig.3, F1 and F2 are determined, and the
unloaded Q and loaded Q is calculated below. The loaded Q for a well
matched circuit is about ½ the unloaded Q.

Q(unloaded) = 416.8

Q(loaded) = 208.4

# Pickup installation

The schottky pickup is in the MI 60 section. A low noise amplifier is
used to amplify the signal at the output of the schottky pickup. The
amplifier used is a M/A Com A75-3 10-500 MHz low noise amplifier. It
runs off a 15 volt power supply, has about 18 dB of gain, and has noise
figure of 1.7 dB. Photographs of the installed pickup are shown in the
figures below.

![](media/image4.jpeg){width="2.7in" height="2.025in"}

Fig.4. Logitudinal Schottky Fig.5. Low noise Amplifier