A Summary of the Design, Development, and Testing of Photomultiplier
Tubes and Base Electronics for the BTeV Experiment

A Report for the BTeV Experimental Group and a Tutorial for the
Phototube Initiate

**May 31, 2001**

**Tim Brennan**

**Teacher Fellow, 2000-2001**

**Fermi National Accelerator Laboratory**

**Introduction**

This account of photomultiplier tubes, base electronics, and testing is
intended to be a summary of the work I have done for Fermilab's BTeV
experiment over the past several months. It is written to serve many
purposes.

One purpose is somewhat self-indulgent; this document serves as a
self-assessment of my level of understanding of this project. Though I
have a background in electrical engineering, my knowledge of
photomultiplier tubes was negligible and my experience in particle
physics was none. I struggled mightily with many aspects of this project
and the act of summarizing the results has clarified many, though not
all, of my misunderstandings.

Another reason for creating this document is to account for the state of
the project as I leave it. From the beginning of this Fellowship, it was
unclear which project(s) might demand my attention, how deeply I might
get involved with any given project, and whether I might "dabble" in a
variety of projects. Looking back on the year, it seems clear that the
opportunity to understand photomultiplier tubes, design base
electronics, test the combinations of tubes and bases, and report on the
results has filled out the calendar in a meaningful way. Though I do not
consider myself to be an expert at any one of the tasks mentioned, I do
feel as though I've developed, in a broad-based way, a greater personal
understanding regarding the detection of light in particle collision
experiments. Clearly, though I may be leaving, the development of tubes
and bases will continue. For the benefit of the person who next takes up
the task of advancing this project, I write what I hope to be a valuable
review of this year's efforts.

This document exists for yet another reason. I have learned throughout
this year that there are many people, even some deeply involved with
particle physics, for whom the topic of photomultipliers is something of
a mystery. For some folks, their experience with photomultiplier tubes
is something of a distant memory; for others, there has never been such
an experience. Many rely on the expertise of a few, placing their trust
in an industry and technology that are well developed. Photomultiplier
tubes, for some, exist as "black boxes" - they are used without being
completely understood. Though I make no claim to complete understanding
myself and though this is obviously not the definitive document
regarding photomultiplier tubes, it is my sincere wish that my work and
this summary may act as a "primer" of sorts. For those who truly are
experts in this field, I imagine this document to be of little benefit.
What I hope for, with those readers, is that I have addressed the topic
in such a way as to have committed no egregious errors.

**Acknowledgments**

As this is my final exercise before leaving, it seems like an
appropriate time and place to give thanks. I would like to thank Harry
Cheung for his mentoring throughout the year, providing me with
everything from technical insights to test equipment to tours of
experimental halls. Thanks also to Joel Butler, one of the BTeV
spokespersons, for taking me on as a Fellow and for finding time for the
occasional chat about life within and without the physics community.
Thank you to Bob Jones and his wonderfully capable technical team, for
providing me with first-class technical support; I'm sure Bob will not
miss my frequent requests for "just one more" electronics assembly. I
wish to give special thanks to Jeff Appel for his invaluable support,
guidance, and friendship, particularly through some difficult moments.
Elliott McCrory has also provided guidance and valuable insight into the
workings of Fermilab; though we met late in the year, his impact on my
Fellowship was significant, uplifting, and inspirational.

This has been a very challenging year for me, personally and
professionally. To Tom Jordan, Liz Quigg, Spencer Pasero, Marge Bardeen,
and all of the folks in Fermilab's Education Office, I offer much praise
and endless thanks. Without your vision of how an educator can
participate in the world of high-energy physics, and without your faith
in the ways that that world can positively influence the classroom, my
year would have been far less valuable. Without your friendship, my year
would have been far less enjoyable. I encourage you all -- if I haven't
left you drained -- to shepherd the next Teacher Fellow as you have
shepherded me.

Finally, I can never thank my wife, Beth, sufficiently for her boundless
faith in me, and her endless support of me. To get here, from Vermont,
together, she left a job that she loved, with no assurance that it would
be there upon our return. To go back, she will again leave a job - and
some newly found friends and colleagues - that she loves, with no
assurance that we will return to Illinois. Her sacrifices have been
difficult ones; her insistence in making them, on my behalf, has been
astonishing. I hope that in my lifetime I may, somehow, give back to her
all she has given me. I believe it was Linus Pauling who, when asked
why, after a lifetime filled with scientific successes, he continued to
work on important scientific and social issues with a diligence
unnecessary for a "retired" person, responded, "I want my wife to be
proud of me." With a partner like Beth, I can think of no greater creed
by which to live. I look forward to our continued adventures together.

**Photomultiplier Tube Theory of Operation**

A photomultiplier tube (PMT) is a device that creates an electric charge
in proportion to the amount of light energy that it receives. The charge
is typically integrated throughout some critical time period and is
taken as a measure of the input light energy. The operation of the PMT
is critically dependent upon an important physical phenomenon: the
photoelectric effect.

As the name suggests, the photoelectric effect involves both photons
(particles of light) and electrons (particles of matter). Under the
right conditions, a metallic plate will release electrons when struck by
a single photon. The number of electrons liberated depends, in some
special ways, upon both the intensity and the frequency of the light
wave. Curiously, if the frequency of the light source drops below a
critical, "cut-off" frequency, no electrons will be liberated, no matter
how intense the light source. This frequency is a function of the metal
used - each metal has a different cut-off frequency that depends upon
the binding energy between that atom's nucleus and its orbiting
electrons. Hence the frequency of the light acts as a switch; the flow
of electrons can be interrupted by allowing the frequency of the light
to drop below the cutoff. Once above the cutoff frequency, the number of
liberated electrons increases with the intensity of the light source.
The relationship is essentially a linear one, at a given frequency.

If no electrons are emitted below the cutoff frequency -- no matter how
intense the light wave -- then thinking of light as a wave cannot be
completely correct. How, then, does one explain the mechanism of the
photoelectric effect? One can visualize this effect by thinking of
*quantized particles* of light -- which Einstein dubbed "photons" --
colliding with the surface atoms of the metals and liberating electrons
via collisions. The greater the intensity is, the greater the number of
incident photons and, hence, the greater the number of liberated
electrons.

But that's only part of the story. Each photon has an energy that
depends upon its frequency. If the frequency of an individual photon is
too low, its energy will be insufficient to the task of liberating the
electrons, which are bound to the atom by electromagnetic forces.
Indeed, rather than speaking of "collisions," physicists would say that
an electron "absorbs" the energy of the incident photon and, if the
absorbed energy was great enough to overcome the binding electromagnetic
force, the electron will be freed. While it is possible for *multiple*
low-energy photons to "collide" with an electron virtually
simultaneously, thereby contributing a *total* energy capable of
releasing the electron from the bounds of the nucleus, in practice, this
is rare. Therefore, the number of electrons liberated by light of the
"wrong" frequency is, to a first approximation, zero.

So, assuming the incident photons are energetic enough, where do these
liberated electrons go? If the metal is inside a photomultiplier tube,
then the electrons are released from an initial metal plate (the
photocathode) into an electric field. The field is set up in such a way
as to accelerate the electrons toward a second metal plate where, upon
colliding with the plate, more electrons are liberated. The incoming
electrons have such great velocity (and, therefore, kinetic energy) that
each one can liberate 3-4 more electrons. A third plate, held at a
voltage that is higher still than the second, will replicate the
process. This can continue for as many stages as is practical (typically
6-12) with the same kind of yield at each stage. A tremendous
multiplicative factor can be achieved in this manner; a ten stage PMT
with a gain of 3 electrons per stage will have a gain of 3^10^ or
roughly 60,000. Thus, in this example, for every electron liberated at
the photocathode, 60,000 will arrive at the last stage (the anode).

There are many types of PMT construction techniques. Each technique
optimizes one or more of the following variables that are important to
the PMT user: cost, electron collection efficiency, size, gain, response
time, and linearity (Philips, page 1-14). In general, one can say that a
prerequisite of the geometry of the successive plates (dynodes) is that
electrons liberated from one plate should be physically directed toward
the next plate. In other words, though an electric field will be used to
accelerate the electrons from one dynode to the next, there should be no
physical barriers in the electrons' path. Clearly, a wide variety of
strategies will achieve this goal; some of these are shown below (figure
from Philips, page 1-15):

![](media/image1.png){width="5.983333333333333in"
height="4.320138888888889in"}

The voltages at each dynode need to be both large enough to effectively
collect the liberated electrons and stable enough to create known,
reliable electric fields. In addition, the first stage and the last
stage(s) need to be particularly efficient in order to maximize the
efficiency of the PMT.

The first stage of the PMT is the photocathode to first dynode. The
voltage between these two plates needs to be great enough so that any
electrons emitted by the cathode can be efficiently swept up by the
first dynode. Interestingly, the effect of this voltage is negligible as
long as it remains above a certain threshold, typically on the order of
tens of volts. When the voltage is greater than this threshold, the
cathode acts like a current source; the number of electrons is
proportional to the incident flux of photons (Philips, page 1-11).
Typically, the first dynode-to-cathode voltage is set at 2-3 times the
nominal voltage difference between the dynode stages.

The dynode-to-dynode voltage level has a great impact on the overall
gain of the PMT. Indeed, it is the usual way in which a particular gain
is achieved. There are two physical effects that determine the gain.
Some liberated electrons do not make it, for one reason or another, to
the next stage. The percentage of previously emitted electrons that
strike the next dynode is called the *collection efficiency, l*. On the
other hand, there is a multiplying effect; a single electron can
liberate more than one electron at a given dynode. This is called the
*secondary emission coefficient, s*. Each of these effects is an
increasing function of dynode voltage. In fact, the relationship between
l, s, and V, the interdynode voltage, is:

l \* s = k \* V ^a^ (Philips, page 1-12)

> where k = constant of proportionality that varies with PMT

a = an exponent with a value between .65 and .75

The product of these two effects (l and s) can be called the gain (g~i~)
for a given stage of the PMT. The overall gain of the PMT is simply the
product of the individual stage gains. In the simplest arrangement, all
of the interdynode voltages are equivalent and all of the gains are
equivalent, as well. For an n-stage PMT, the overall gain would be:

(g)^n^ = (k \* V^a^)^n^

g ^n^ = k^n^ \* V^an^

g ^n^ = k' \* V ^(.7n)^ (allowing a = .7, k^n^ to be replaced by k')

Here it is now plain to see that the gain is proportional to an
exponential function of the interdynode voltage. The exact exponent
depends upon the number of stages - in the final equation above, if one
assumes that a is approximately .7, then a ten stage tube will result in
an exponent of, roughly, 7.

Finally, when considering the interdynode voltages, one needs to
carefully consider the last one or two stages. These are the stages that
must provide the greatest number of electrons; any performance
variations that are functions of current will be most apparent here.
There are, at least, two separate issues to consider.

First, if the dynode voltages are created by a resistor divider circuit
(shown below, Figure 2, Philips, page 1-3), it is important to make the
maximum dynode current a small portion of the overall resistor string
current. If the dynode current does become an appreciable percentage of
the string current, then the voltage at the dynodes will drop by an
amount equal to the resistor-dynode current product and the gain of that
stage will drop as well. Typically, the resistors are chosen such that
the string current is 10-100x the maximum dynode current. In addition,
capacitors are placed across the dynodes to supply the dynode charge
while keeping the dynode voltage relatively fixed. One attendant problem
with these solutions is this: the resistors will dissipate heat
according to I^2^R and this heat can cause fluctuations in tube and
scintillator performance, particularly in the case where many PMTs and,
perhaps more importantly, many scintillators are expected to operate in
close proximity to one another. Additionally, the storage capacitors can
be prohibitively large and expensive. For these reasons and others,
Kerns has advocated the use of a transistorized base. More will be said
about this later.

![](media/image2.png){width="5.983333333333333in"
height="4.320138888888889in"}

The other issue to consider is the space charge effect. When electrons
are liberated from a dynode, they will not all leave at once, nor will
they all have the same kinetic energy. A group of electrons, therefore,
will spread out in space and arrive at the next dynode at different
times. The electrons, of course, are negatively charged and will create
an electric field in opposition to the electric field that is set up by
the (positive) voltage between the two dynodes in question. In small
numbers the electric field created by the earliest arriving electrons
will create only a modest oppositional field. The late arriving
electrons will have enough kinetic energy to easily overcome such
opposition. But, in larger numbers, the earliest arriving electrons can
create an opposing field large enough to inhibit the late arriving
electrons. This "space charge effect" can prevent the PMT from operating
as a linear instrument; the gain of the PMT at the uppermost levels of
input light energy may be diminished when compared to the gain at the
lower input levels. The likeliest locations for this space charge effect
are in the final few stages of the PMT where the number of electrons is
greatest. One simple way to address this effect is to increase the
voltages at the last few stages; the resulting increase in the
electrons' kinetic energy is often adequate for overcoming the
oppositional field. This additional field strength, however, also
diminishes the useful lifetime of the tube; the final few plates will
wear down more quickly due to the additional interactions of photons and
electrons.

**Electronics Design**

My primary responsibility throughout this design and test project has
been to design a ten-stage (and later a six-stage) base electronics
package that was to provide the proper voltages to the dynodes, couple
the resulting signal out to an integrator, and create as linear a
response as possible to a wide range of input light energy. In addition
to these universal requirements of a base, there were several other key
design goals, driven by specific BTeV needs:

-   low heat dissipation

-   small volume

-   AC coupling of anode signal

Each goal was met with a different design choice. In order to dissipate
as little heat as possible, I chose to use a transistorized base, a la
Kerns. To keep the package small, we created a long, narrow printed
circuit board, populated with (mostly) surface mount electronics. AC
coupling was achieved, naturally, through the use of a capacitor, though
the value of that capacitor had to be chosen carefully within the
tradeoff of circuit performance versus size. Each of these choices is
considered below. In addition, the topic of linearity is of such primary
importance that it will be considered in greater depth, in the testing
section.

Low Heat Dissipation

The use of transistors in a base allows one to accomplish two
simultaneous, and somewhat competing, goals that would not be possible
to attain with the traditional, resistive, base. The two goals are to
maintain dynode voltages that do not vary with appreciable dynode
current and to provide that additional current while dissipating as
little heat as possible within the base. It will be instructive to look
at the resistive base strategy to see why these goals are difficult
within that model.

As stated earlier, in order to keep the dynode voltages from varying
during the time when current is supplied to those dynodes, a large
standing current is required in the resistive string. If that current is
large, then the meager, additional current to be supplied to the dynodes
will create a small, hopefully tolerable, additional drop in voltage
across the resistors. If the additional voltage drop is objectionable,
one would increase the standing current by dropping the resistive values
accordingly. One proceeds in this manner until the dynode current is a
small enough fraction of the resistor current. The large standing
current through the resistors, of course, creates additional heat
according to I^2^R. It was our desire to keep that heat to a minimum
since our tubes and bases would ultimately be placed in assemblies that
number in the thousands, and the scintillator performance (to say
nothing of the tube and base performance) is well known to vary with
temperature.

A transistorized base, however, can provide the fixed voltages necessary
for the dynodes while simultaneously providing both large dynode
currents and low heat dissipation. To see why, one must consider the
operation of the transistor: current that flows into the base of the
transistor is amplified -- by a gain factor that has a guaranteed
minimum value for any given transistor model -- into current that flows
from the collector to the emitter (see Figure 3a below). While that
current amplification is occurring, the base-emitter voltage is fixed at
approximately one diode drop, or roughly .5 volts. In practice, the base
of the transistor is connected to the junction of two resistors, the
values for which have been chosen so that they create an appropriate
dynode voltage (see Figure 3b below). The emitter is connected to the
dynode. Its voltage is just .5 volts lower than that junction, a
negligible drop considering the usual levels of voltage associated with
dynodes. The transistors are cascaded together with the emitter of one
transistor being connected to the collector of the next one. The dynode
currents flow through the collector-emitter junctions and only a small
portion of that current is necessary to flow into the bases. Since it is
only the small base currents, and not the dynode currents, that flow
through the resistor string, the resistor values themselves can be much
larger, the standing current can be much lower, and the overall heat
dissipation drops according to I^2^R.

![](media/image3.png){width="5.121527777777778in"
height="3.7736111111111112in"}

Figure 3 (a) Typical transistor characteristics and (b) Transistors used
to supply dynode current and voltages

Of course, during the supply time for the dynodes, the dynode current
does heat up the transistors by the same amount that that current would
have heated the resistors. But the duty cycle of that supply is
typically very low and the overall heat dissipation of the circuit can
be approximated, to a first degree, by looking at the "no load"
condition. Indeed, it is under this assumption that the dissipative woes
of the resistive base are perhaps most clearly seen. For a resistive
base, the resistor values are chosen to be low so that, *in the rare
case that dynode current needs to be supplied*, the additional voltage
drop is negligible. At all other times, the resistors simply heat up
their environment.

A short series of calculations can demonstrate the advantage of the
transistorized base. Let's assume a 1000-volt supply and a need to
provide a maximum of 1 mA of dynode current. In a resistive base, we
would choose our resistors so that the 1 mA dynode current was, at most,
1/10^th^ of the standing current through the resistors. Such a choice
would force a standing current of, at least, 10 mA. The power dissipated
in the resistive string can be calculated according to VI, where V is
the overall high voltage supply. This circuit would dissipate, at all
times, (1000 V \* 10 mA) or 10 watts.

In a similar, transistorized base, with the same assumptions, we could
choose resistors that were much larger. To choose correctly, we need to
understand the additional current flowing through the resistors during
dynode supply time. If we assume a very modest gain of 40 in our
transistors, then the additional resistor current would equal the base
current which would be, at most, 1 mA/40, or .025 mA. The resistors
would then be chosen to create 10x this amount of standing current. The
power dissipation would then be (1000 V \* .25 mA), or .25 watts. The
decrease in power dissipation is, of course, a factor of 40. Choosing
transistors with greater gains will result in greater power savings.

Small Volume

Previous negotiations with the Hamamatsu organization led to the
selection of a PMT with a one-inch diameter. Though we wished for the
PMT and base to take up as little space as possible, there was a degree
of freedom in the length of the assembly. So, though the cross-section
of the assembly was restricted to a one-inch diameter, the base
electronics could be as long as necessary. The electronics fit
comfortably on a 3.00-inch board. This length, when combined with a
"seating" height above a socket and the thickness of an end cap, created
an overall length of 3.41 inches.

I followed the lead of a previous experiment in choosing a housing
appropriate for the base electronics. The KTeV group had created a tube
that covered the electronics and was secured by screws that passed
through the tube and terminated into a metallic end. I modified the KTeV
drawings and selected materials that were appropriate for the BTeV base.
One important difference between the KTeV and BTeV housing tubes was
that the BTeV housing needed to be made of a non-conductive material.
PVC piping of the right diameter turned out to be an inexpensive and
easily worked source; ten feet of one-inch (outside) diameter PVC pipe
costs about \$3.00 and would yield approximately 30 housings! The end
cap needed to be metallic and easily worked. I chose a standard grade of
aluminum (6061-T6); this is the same choice made by the KTeV group.
Mechanical drawings for each of these pieces are shown below.

One additional note about heating: if necessary, holes could be drilled
in the housing to allow for heat to naturally dissipate away from the
electronics. We considered doing this as a standard; to date, though,
there is little evidence showing a need for such ventilation.

![](media/image4.png){width="6.513888888888889in"
height="4.691666666666666in"}

Figure 4. End Cap for Base Electronics Assembly

![](media/image5.png){width="6.248611111111111in"
height="4.511805555555555in"}

Figure 5. Housing for Base Electronics Assembly

AC Coupling

The output of a PMT is available at the anode. For the BTeV project, it
had been previously decided that a positive power supply would be used.
(For a summary of the tradeoffs between positive and negative supplies,
see Appendix C.) This meant the anode would be held at some high
voltage, typically 1000 volts and such a large voltage cannot be passed
onto the integrating electronics; capacitors are used to block the DC
voltage while passing along the AC portion of the signal.

Though any capacitor will block the DC voltage, one must choose the
right capacitor to pass along the AC portion with minimal distortion.
The output section of the base is a simple high-pass filter, or a
differentiator. The voltage across the, standard, 50-ohm output resistor
is approximated by:

Vo(t) \~ 50\*C \* dVc/dt (Horowitz and Hill, page 22)

I ran spreadsheet simulations of this output based on an estimated input
current that is represented below in Figure 6. This particular signal
would create an integrated output of approximately 200 pC, which seemed,
at the time, to be a reasonable design maximum. Examination of the
various output signals (Figures 7a-7c) reveals important information. As
the size of the capacitor grows, the maximum output approaches the IR
product of, roughly, 4 mA and 50 ohms, or 200 millivolts. In addition,
the shape of the output signal approaches the shape of the input signal.
Finally, increasing capacitance generates smaller and smaller overshoot
voltages on the back end of the signal. If there were no other
considerations, the choice for the largest available capacitor would be
clear. Capacitor size, however, grows with the product of capacitance
and voltage. The coupling capacitor, in this application, would be
expected to withstand a minimum of 1500 volts and general engineering
safeguards would push that choice to 2000 or even 3000 volts. As the
capacitance value increases toward the microfarad range, the size of the
capacitor becomes prohibitively large, given the volume constraints
considered earlier. Indeed, locating *standard inventory* capacitors
with a 3000-volt rating and a .022 uF value became problematic; I
eventually settled for two .01 uf capacitors in parallel. The
performance of these parallel capacitors is closely modeled by Figure
7b.

Clearly, from a signal point of view, it would be desirable to get
larger valued capacitors; the limited volume prevents that choice for
now. It may be true that surface mount capacitors with 3000-volt ratings
and large capacitor values will become available for future use.
Alternately, this coupling capacitor would be unnecessary in a scenario
where the anode was grounded but still positive with respect to the
cathode, that is, with a negative power supply.

![](media/image6.wmf)

Figure 6. The Estimated Input Current into the AC coupling capacitor

![](media/image7.wmf)

Figure 7a. Voltage Output with .0022 uF capacitor

![](media/image8.wmf)

Figure 7b. Voltage Output with a .022 uF capacitor

![](media/image9.wmf)

Figure 7c. Voltage Output with .22 uF capacitor

Having selected the capacitors and analyzed the stages that would
benefit from having transistors, I designed the circuit in Figure 8.
Later, I will refer to two versions of this design: the uniform voltage
and the tapered voltage design. In the uniform voltage design, resistors
R9, R10, and R11 all have values of 560 Kohms, resulting in a uniform
voltage distribution from dynode 1 through dynode 10 and the anode. In
the tapered version, R9 is a 1.1 Mohm and R10 and R11 are both 1.8 Mohm
resistors. The resulting voltage distribution is "tapered" in that the
voltage difference between dynodes 9 and 10 is roughly twice that of the
previous stages whereas the final two stages have voltage differences
that are three times the earlier stages. The tapered base helps to
eliminate the space charge effect mentioned earlier.

![](media/image10.png){width="5.331944444444445in"
height="6.875694444444444in"}

Figure 8. Ten Stage Base Schematic

**Test Setup**

A critical function of the PMT/base combination is that it should
deliver a linear output in response to changes in input light energy. In
order to test these bases and tubes for linearity, I used a testing
station originally developed for the KTeV program. A block diagram is
show below; the station currently resides in the front room of Lab 5.

![](media/image11.png){width="5.379166666666666in"
height="4.156944444444444in"}

Figure 9. The PMT Linearity Test Station

There were two critical software utilities that provided me with control
over the test station. One, called "calibf," allowed me to choose start
(opaque) and end (transparent) positions of the graded wheel. Before
running any linearity test, I would invoke calibf in order to set the
graded wheel to its start position. In addition, after major changes
occurred in the test station -- or after power outages or interruptions
that forced a resetting of the electronics -- I would run the calibf
routine in order to recalibrate the start and end positions.

The other critical routine is called "burrfera" after two of the types
of analog to digital converters used in the collection of data. Though
burrfera grants the user an abundance of test strategies, I used only
three.

Choosing "1" from the burrfera menu would begin a linearity test. During
the test, the graded wheel would slowly advance from opaque to
transparent allowing greater input light energy to get to the
photocathode of the PMT. The program would record digitized versions of
two measurements of the input light energy (from two pin-diodes placed
near the fluorescing dye solution in the laser/wheel box) and a
digitized, integrated measurement of the output signal. These data were
stored in a file entitled "pmt.out" and were subsequently saved with
encoded names at the end of any worthwhile test.

Choosing "4" from the burrfera menu would allow the operator the
opportunity to select different filter wheels. These wheels effectively
set the maximum amount of light that was to be passed onto the
photocathode. An opportune time to select these filters was at the end
of each linearity test. At that time, the graded wheel was positioned at
the transparent position; selecting the new filter wheels and checking
the response of the PMT/base to this amount of light allowed the
operator the chance to see if, at the maximum input, the output was
still in the measurement range of the analog-to-digital converter (ADC).
The filter wheels could be chosen in such a way as to maximize the
output signal from the PMT/base without saturating the ADC associated
with that signal. The ADC associated with the output, by the way, was a
LeCroy 2249W converter with a maximum count of roughly 2000. Each count
was the equivalent of .25 pC. allowing a linearity test of up to \~ 500
pC.

To implement the selection of the filter wheels, an operator would need
the ability to fire the laser a small number of times without altering
the graded wheel position. Such an ability could be attained by choosing
"12" from the burrfera menu. Generally, I opted for 10 pedestal readings
followed by 20 output readings, choices I could make in response to
program prompts. Over the course of the 20 readings, I checked the
output counts to see if they were near the top of the range of the ADC.
In general, I accepted any settings that created maximum outputs of
1600-1700 counts, though I had no absolute standard by which I made my
selection. Once I was satisfied with the filter wheel selections, I
would leave the burrfera routine, reset the graded wheel to the start
position (through calibf as described above), and return to burrfera to
begin a new linearity test.

**General Testing Comments**

The goal for any PMT and base is to produce a linear output in response
to light energy input. At this test station, the input light energy to
the PMT could be manipulated in any number of ways. The graded wheel, of
course, provided a mechanism for attenuating the fixed output of the
laser from essentially zero to some, pre-determined, maximum level. The
filter wheels helped to establish what that maximum level would be, for
any given test. The amount of light into the PMT itself could be altered
by adjusting the position of the fiber optic cable with respect to the
PMT photocathode window. Similarly, the fiber optic cable could be moved
to different distances at other points in the station; variations in
those distances would enhance or diminish the ability of the cable to
pick up light. All of these techniques were required, at one time or
another, to force the output of the PMT/base into a range that
corresponded to the full-scale range of the 2249W analog-to-digital
converter. Whenever possible, I tried to create a test whereby the
output of the PMT/base corresponded to at least 1700 counts at maximum
light input to the PMT.

The measurement of the input light was actually done well "upstream"
from the PMT window. Indeed, the "input" to the PMT was measured by
pin-diodes stationed near the fluorescing dye through which the laser
light was directed. The dye fluoresced at 440 nm, a wavelength typical
for testing PMTs. The pin-diodes, in turn, were chosen for their
sensitivity to light at this wavelength and for their linear response to
light. Previous work, completed by the KTeV experimental group, had
verified that these pin-diodes could be used as reliable, linear,
detectors of light energy.

Though the pin-diodes were quite linear in their response to light, it
was not unusual for the output of the pin-diodes to drift significantly
throughout a day of testing. This "drift" was likely the result of laser
light variations (with temperature?) since an increase in the measured
input light (at the pin-diodes) was accompanied by an actual increase in
the PMT/base output. In addition to this laser drift, variations in the
level of the dye, variations in the positions of the pin-diodes with
respect to the dye, and the occasional use of filters over the inputs of
the pin-diodes all, at one time or another, altered the amount of light
entering the pin-diodes. All in all, the magnitude of the signal at the
pin-diodes varied, from test to test, in some dramatic ways. Therefore,
as you will notice, the x-axis scales in the test results section will
vary by large factors. The effect of all this is that, when comparing
one test with another, there is no *absolute* meaning to the x-axis
numbers. Fortunately, the scale of the x-axis has no bearing on the
judgment of linearity.

What is crucial to the judgment of linearity, however, is the y-axis
scale. In any given test, I wanted to know two things, both of which
depended absolutely on the y-axis values. First, how linear was the
output response? Second, at what y-axis value did the output become
unacceptably non-linear?

**Evaluation Methods**

In order to address the two questions above, I needed some evaluation
tools. I used four Fortran programs, written by Harry Cheung, to
accomplish several intermediate goals:

\- read in the data from the data files,

\- subtract the pedestals (the non-zero number of counts, for each
analog-to-digital converter, in response to zero input) from the data,

\- bin the data into histograms,

\- generate error bars and means for each data bin, and

\- generate an output file to be used by a statistical package.

The statistical package -- called the Physics Analysis Workstation, or
PAW -- allowed me to find best-fit lines (or functions) for all or any
portion of my data set. In addition, PAW would generate a measure of the
"goodness of fit," something that is worth elaborating upon.

The "goodness of fit" can be evaluated via a statistical method known as
a "chi squared" test. In general, this test allows one to evaluate the
statistical significance of the connection between any two columns of
data. It is frequently -- though not exclusively, as we shall see --
used to evaluate how likely a data set is to have been generated by
chance. For example, if one rolled a six-sided die 3000 times and
collected 515 "ones," 498 "twos", etc., the chi-squared test would
generate a number that would allow one to evaluate the likelihood that
the die was a fair one. A low chi-square number would indicate that the
actual results matched the expected results. The larger the chi-square
number, the greater the deviation is from an expected result. In
practice, one would calculate the chi-square number and determine, via a
table or computer program, the probability that the particular results,
say, with the die, could occur by chance.

Though Adler (page 227) and Moore (page 606) use an alternate, and less
widely applicable definition, I used the mathematical definition of the
chi-squared test as given by Bevington (page 67):

Χ^2^ = Σ  (observed data -- expected data)^2^

variance of data

= Σ  (observed data -- expected data)^2^

(standard deviation)^2^

Before evaluating this expression, the data need to be processed. The
data are first adjusted for pedestal offsets, and then binned. In any
given x-bin there will be some number of y-values. From those y-values,
one computes a mean and a variance. The means from each of the bins are
used to perform a least-squares regression analysis, which yields a
best-fit line. It is this best-fit line that provides the "expected"
value in the chi-squared computation; for any given x-value there is a
corresponding "fitted" y-value. The "observed data" are mean values of y
for each x-bin. Each difference is squared and divided by the variance
for that bin. The sum is over the number of x-bins.

A fitted function that matches the data set well will produce
differences between the bin means and the fitted values that are
approximately equal to the standard deviation of the bin (see Appendix A
for justification). Each term should contribute, on average, a value
that is close to 1. The sum, therefore, should be -- again, for
functions that fit the data set well -- roughly equal to the number of
terms, or bins. The *best* best fits will have ratios of Χ^2^ to the
number of terms that are approximately equal to 1. Larger ratios are
indications of poorer fits; one may always fit a "best" line but that
doesn't mean the data is truly linear.

In each of the graphs that follow, one may notice a "chi2/ndf" number in
the upper right hand corner. This is essentially the ratio -- the
"reduced chi-squared" number (Bevington, pg. 195) -- that is discussed
above. The "ndf" is the "number of degrees or freedom," a number that is
smaller than the number of bins by two since there are two constraints
placed upon a linear fit function (a slope and an intercept). This
number is valid for the region of the graph for which a linear fit has
been established. That region can be seen in each graph; it is the
region for which the function is drawn.

Using this "chi2/ndf" as a metric, I attempted to establish, for any
given base and tube under test, the answers to the two questions that
motivated this section: how linear is the output response? and, at what
y-axis value does the output become unacceptably non-linear?

The graphs shown in Figures 10 a, b, and c show the results of a
ten-stage base and tube under a variety of conditions. In each case, I
used a single tube: a Hamamatsu ten-stage tube, model number 5800. A
"uniform" base is one in which the interdynode voltages are all
equivalent, save for the cathode-to-first stage which is typically three
times larger than the subsequent stages. A "tapered" base follows a
different pattern. It is the same as the uniform base until the final
three stages which are, respectively, two times, three times, and three
times greater than the earlier stages. It is apparent that the tapered
base is more effective at creating a linear performance than the uniform
base. Indeed, one may suppose that the larger interdynode voltages of
the final three stages provide greater kinetic energies to the incoming
electrons, allowing those electrons to overcome the oppositional field
created by the space charge effect. A direct comparison of the tapered
and uniform bases at 900 volts (Figures 10a and 10b) shows that the
tapered base is respectably linear through a range of about 1500 counts
(roughly 375 pC.); this is an increase of about 50% greater range than
the uniform base. This range is extended even further when the high
voltage is reduced to 700 volts; Figure 10c shows a linear response up
to nearly 1800 counts or 450 pC. In order to get approximately 400 pC of
output charge at both 700 and 900 volts, I manipulated the light -- by
adjusting the final gap between the light fiber and the PMT window -
such that the input was much greater at 700 volts than at 900. I don't
have a satisfying explanation for the difference in performance though
it seems reasonable to question the response of the cathode as a
function of reception area; are its responses to inputs over narrow and
wide areas equivalent?

![](media/image12.png){width="3.1284722222222223in"
height="3.6194444444444445in"}

Figure 10a. Uniform ten-stage base at 900 volts

![](media/image13.png){width="3.1368055555555556in"
height="4.059027777777778in"}

Figure 10b. Tapered ten-stage base at 900 volts

![](media/image14.png){width="3.0006944444444446in"
height="3.8722222222222222in"}

Figure 10c. Tapered ten-stage base at 700 volts

Ultimately, the BTeV experimenters would like to use a six-stage base
for their photo-detectors. Several different manufacturers have
responded with prototype, six-stage, bases. The results of some of these
tests will be discussed here. Results of all of my tests are available
at a Fermilab website; see Appendix B for information regarding those
files. The first set of figures (Figure 11a and Figure 11b) feature
Hamamatsu six-stage bases, model number 5380. All bases for the
six-stage Hamamatsu tubes are tapered. Figure 11a shows the performance
of tube #CA0119 at 900 volts. It is very linear within the first 1000
counts (250 pC.) with a tendency to "roll off" somewhere above 1200
counts. Figure 11b is similar; it is tube #ZL0093 under the same
conditions. The tendency to become non-linear is somewhat more
pronounced with this tube. Figure 11c is the CA0119 tube again, this
time at 800 volts. Finally, each of the tubes was subjected to 1000-volt
(or greater) conditions, as shown in Figures 11d and 11e. It appears, as
with the ten-stage tubes and bases, that performance is enhanced at
lower voltages.

![](media/image15.png){width="3.3833333333333333in"
height="4.378472222222222in"}

Figure 11a. Six-stage base at 900 volts (Tube s/n CA0119)

![](media/image16.png){width="2.9409722222222223in"
height="3.795138888888889in"}

Figure 11b. Six-stage base at 900 volts (Tube s/n ZL0093)

![](media/image17.png){width="2.754166666666667in"
height="3.563888888888889in"}

Figure 11c. Six-stage base at 800 volts (Tube s/n CA0119)

![](media/image18.png){width="3.0006944444444446in"
height="3.8722222222222222in"}

Figure 11d. Six-stage base at 1000 volts (Tube s/n ZL0093)

![](media/image19.png){width="3.0006944444444446in"
height="3.8722222222222222in"}

Figure 11e. Six-stage base at 1100 volts (Tube s/n CA0119)

ETI and Photonis also submitted prototype six-stage tubes. The ETI tubes
were tested with uniform and tapered bases; the Photonis tube was tested
only with a tapered base that followed their suggested voltage schedule.
Some of the results are shown here; all of the results are, again,
available at a Fermilab website.

Some results of the ETI tests are shown below. Figures 12a and 12b show
the performance of ETI tube \# 103, with a uniform base. Counter to the
Hamamatsu results, it appears that the six-stage ETI tubes perform
better at higher voltages; the linear region at 1000 volts is roughly
twice as large as the linear region at 800 volts.

![](media/image20.png){width="2.547222222222222in"
height="3.404861111111111in"}

Figure 12a. ETI six-stage tube (s/n 103), with uniform base, at 800
volts

![](media/image21.png){width="2.097916666666667in"
height="2.8159722222222223in"}

Figure 12b. ETI six-stage tube (s/n 103), with uniform base, at 1000
volts

In Figures 13a and 13b, one can see the results of using a tapered base
with an ETI tube. While there is some improvement with an increased high
voltage, the performance with the tapered base is poor when compared to
the uniform base results in Figure 12. It is also possible that tube \#
106 is inferior to tube #103; these are prototypes from ETI, not
standardized products with a mature history.

![](media/image22.png){width="2.8270833333333334in"
height="3.7944444444444443in"}

Figure 13a. ETI six-stage tube (s/n 106), with a tapered base, at 900
volts

![](media/image23.png){width="2.7006944444444443in" height="3.625in"}

Figure 13b. ETI six-stage tube (s/n 106), with tapered base, at 1200
volts

The results from the tests of the Photonis six-stage tube are shown in
Figures 14a and 14b. In each of these tests, one may notice a pronounced
"s" shape to the data. There is no single, satisfactory, linear region,
though the effect seems less pronounced at the lower voltage.

![](media/image24.png){width="2.9256944444444444in"
height="3.9194444444444443in"}

Figure 14a. Photonis, six-stage tube, at 600 volts

![](media/image25.png){width="2.5034722222222223in"
height="3.3680555555555554in"}

Figure 14b. Photonis, six-stage tube, at 1100 volts

**APPENDIX A**

The variance is defined as the square of the standard deviation. Since
the standard deviation is a measure of the average difference between
the mean and the data points, then the non-squared values of the
numerator and denominator, in Bevington's chi-square formula, should, on
average, be approximately equivalent. That is,

\< (observed data -- expected data) \> \~ (variance) ^.5^

or

\< (observed data -- expected data) \> \~ standard deviation

where \< . . . \> indicates the expected value of a term.

Since the numerator and denominator terms are roughly equivalent, then
the contribution from each data point, to the chi-square sum, should be
about 1. Of course, squaring each term will accentuate differences from
unity. Nevertheless, for "well-fitted" functions, the chi-square sum
should be roughly equal to the number of terms in the sum.

**APPENDIX B**

Summary of Directories

The following is a summary of the various directories found at
http://home.fnal.gov/\~tbren/. I have been testing various PMT base
electronics under a variety of conditions and the directories have been
created to sort the tests in some meaningful way. Since the titles of
the directories contain limited information, this document is meant to
describe the contents of each directory.

NEGBASES -- These are files for two types of negative supply base
electronics. As a check on performance I looked at an older, purely
resistive base that had been used at Fermilab. As you can see in the
files labeled "neg\*.\*" the performance of these bases is outstanding.
They are very linear throughout the entire output range (0 to \~ 1600
counts or, equivalently, 0 to 400 pC.). I also tested a transistorized
base that used a negative supply (anode grounded, cathode at -- HV).
Though it's clearly not as linear as the purely resistive base, the
advantage of a transistorized base is lower heat dissipation.

PROTOSIX -- Though I do provide more information below, these tests
should be considered faulty; I had interpreted the manufacturer's data
sheet incorrectly and had the incorrect voltage schedule built into the
base. Please see the "NEWSIX" files below for data with correctly built,
six-stage bases.

These are the files for some six-stage prototype tubes and bases. There
are three sub-categories: six\*.\*, sixc\*.\*, and sixp\*.\* The
six\*.\* files are the files for the six-stage base as originally
designed. The sixc\*.\* files are the same base but with larger output
capacitors across the final two stages (up to .022 uf from the original
.0033 uf). The sixp\*.\* files are for that same base (with the larger
caps) but with the distance between the PMT and the light cable
increased. Since the gain of the six-stage base was reduced from that of
the 10-stage base, I had originally placed the PMT very close to the
light cable, in an effort to create a large output signal (up to 1600
counts). Upon reflection, I thought perhaps that the light might be
striking a very limited area of the photocathode and that there might be
some non-linearity associated with that situation. The distance was
increased by 2-3 cm in each trial, as voltage levels were increased.

In all three of these sub-categories, the voltage level is encoded into
the name of the file. For example, sixc1000.\* would indicate that that
file was data taken with the enhanced capacitor unit, run at 1000 volts.

NEWSIX -- These are the files that contain the data for the correctly
built, six-stage base. There are three main sub-directories: "c19"
contains the data taken with Hamamatsu tube #CA0119, "z93" contains the
data taken with Hamamatsu tube #ZL0093, and "scattertests" contain the
data for six consecutive tests taken with the CA0119 tube, to see if
there were any short-term, drift-like anomalies with the tube/base.

In each sub-directory, the voltage is encoded into the filename as
before. Occasionally, the final number in such a scenario may be a "1"
or a "2" indicating that the test was the first or second test,
respectively, done at that voltage that day. File "c19100.gif" should
have been saved as "c191100.gif." I will try to rename the file soon.

The C19 tube seems to be much more linear than the Z93 tube. Each tube
was tested with the same base.

TENSTAGE -- These are files for the 10-stage PMT/base combination. Again
the high voltage is implied by the file name, as described above. These
files were generated using unit #18 and show how the linearity is
affected by changes in HV.

There are two broad categories of tests in this directory: "tap\*.\*"
and "ten\*.\*" The "tap" files are data taken with a tapered base, where
the voltage in the final three stages was set to 2x, 3x, and 3x the
nominal interdynode voltage. The "ten" files are earlier data; the
interdynode voltages were uniform. Unit #18 was used in each case.

The tapered base tests show a much-extended linear region, compared with
the uniform base tests.

UNIT18 -- These files show the performance of unit #18 over time.
Encoded in the file names are the dates of tests. On March 1, the unit
demonstrated adequate linearity. By April 4^th^, however, there appeared
a remarkable bending in the "line." I noticed a difference in the x-axis
measurement and adjusted my test set-up to reset the x-axis full-scale
reading to be less than 50,000 counts. The result was the picture seen
in the April 5^th^ picture, one that reflects the performance of March
1^st^.

The meaning of all this is still not clear, though there is a suspicion
that adjustments to the dye level (see the test station figure in body
of the paper) throughout the month of March were misguided; at times the
laser light was not directed through the dye at all, at other times it
was directed through the meniscus. When I committed myself to directing
the laser light directly through the body of the dye, I was not able to
recreate the "remarkable bending" mentioned above. This is true whether
the x-axis readings were limited to \< 20,000 counts or allowed to be \>
300,000 counts, variations that were created by moving the pin-diode's
location or filtering the input to the pin-diode with a thin foil. I
have not tried to recreate the failure mode, though it may be worth
exploring for greater insights.

UNIT22 -- This is a set of files that was generated to share with the
our collaborators in Protvino, Russia while they were running their beam
tests with the bases developed here at Fermilab. They represent the
typical performance of the bases at the beginning of March.

UNIT24 -- This set of files is the same as "UNIT22" above, though with a
different unit (#24).

ETITUBES -- These are files taken with two different six-stage tubes
sent to Fermilab by the British company, ETI. Again, the voltage levels
are encoded into the file name. There are two other codes as well: a 3
or a 6 in the name will indicate tube #103 or #106, respectively, while
a "u" or "t" in the fail name will indicate if the base follows a
"uniform" or a "tapered" schedule, respectively. The pinout for the ETI
tubes is different from the Hamamatsu tubes; the bases, therefore, are
not interchangeable. Therefore, two distinct bases were built to test
the ETI tubes.

PHOTONISTUBE -- A French company, Photonis, also provide Fermilab with a
prototype six-stage base. Again, its pinout is different from both the
ETI and Hamamatsu tubes. And, again, the voltage level is encoded into
the file name.

**APPENDIX C**

Choosing between positive and negative high voltage supplies requires a
consideration of the tradeoffs involved.

A positive supply allows the cathode to be grounded. The potential
difference between the cathode and the chassis, including the window,
then is essentially zero. Phototubes operated with positive supplies,
therefore, do not have to be insulated from one another or from other,
grounded, instruments that may be nearby. Furthermore, the long-term
advantages of a grounded cathode are significant; a large potential
difference between the cathode and the glass window can result in
long-term degradation of, and possible damage to, the window itself.
Positively supplied tubes also provide enhanced "dark current"
performance. That is, the number of electrons that would flow under "no
light" conditions is smaller for positively supplied tubes than for
negatively supplied ones.

The cost for these advantages is borne by the coupling capacitor, and
borne in a number of ways. The capacitor, which essentially averages the
signal that passes through it, may provide a small, positive, overshoot
voltage on the back end of the signal pulse. At high data rates, this
overshoot may introduce errors into the integration scheme; the
"counting" of the electrons will not begin at zero in these cases. To
minimize this possibility, the value of the capacitor should be as large
as possible, as one can see from the diagrams in Figures 7a, b, and c.
When one combines this need with the need to withstand at least 1000
volts (and quite possibly twice this voltage), the size of the capacitor
grows quickly. Its size may outgrow the volume constraints placed upon
the base by other, geometric, considerations. The cost of the capacitor
also rises with capacitance value and voltage rating. Indeed, capacitor
manufacturers may not stock the ideal capacitor for these unusual needs;
the cost of a specially ordered capacitor may be quite a burden.

And so it is that one might be tempted to eliminate this capacitor by
adopting a negative high voltage supply. In this case, the *anode* would
be grounded and the cathode would be held at some large, negative
voltage. Of course, the precautions mentioned in the previous paragraph
would then be necessary. One would have to characterize the dark current
performance more rigorously. In addition, the shielding of the phototube
becomes a more difficult chore; not only does one need to shield the
tube magnetically, but electrically, as well. In addition, one would
need to track the performance of the tubes over time; the degradation to
the window caused by potential differences may require replacing tubes.

The polarity choice, then, must be made with these issues in mind.

**REFERENCES**

Henry L. Adler and Edward B. Roessler, "Introduction to Probability and
Statistics," W. H. Freeman and Company, 1972.

Philip R. Bevington, "Data Reduction and Error Analysis for the Physical
Sciences," McGraw-Hill, Inc., Second Edition, 1992.

Paul Horowitz and Winfield Hill, "The Art of Electronics," Cambridge
University Press, 1982.

C.R. Kerns, "A High-Rate Phototube Base," Fermilab Technical Note,
TM-640, January 22, 1976.

David S. Moore and George P. McCabe, "Introduction to the Practice of
Statistics," W. H. Freeman and Company, 1993.

"Photomultiplier Tubes, Principles and Applications," Philips Technical
Manual.