# Non-destructive Measurements

The particles emitted in a collision, Fig. I.1, have a position,
momentum, charge, and other properties (e.g. lifetime) which we wish to
measure. If the interaction with a detecting medium transfers but little
energy to that medium, the measurement is called non-destructive. Within
this category we distinguish between measurements of time and velocity
(Sections 2-4) and measurements using ionization (Section 6) but limited
by scattering (Section 5) of the trajectory of the particles in the
detecting medium (Sections 7-9).

In the first case physical processes which depend on velocity are used,
while in the latter the trajectory in electric and magnetic fields is
used (Sections 7-9) to infer the position, momentum, and charge of a
particle.

# II.A. Time and Velocity

In Section 2, we examine the photoelectric effect whereby light is
converted to an electrical signal. These devices are commonplace; for
example photocells which are used to control the doors of elevators. By
utilizing photomultiplier tubes and scintillating materials, precise
time measurements can be made. These "clocks" are then used to measure
velocity as the time to go to a fixed distance. In Section 3 we turn to
the Cerenkov radiation which is emitted at a velocity dependent angle
when a charged particle moves with a velocity which exceeds the velocity
of light in a medium. The emitted light can be converted to an
electrical signal using the techniques of Section 2. Section 4 concerns
x-ray radiation emitted by a charged particle in a medium with index of
refraction \<1. This "sub-threshold" Cerenkov photon emission is used to
measure particles moving at velocities very near to the speed of light.

# 2. The Photoelectric Effect, Photomultipliers, Scintillators

"*There is a crack in everything, that\'s how the light gets
in*."\....Leonard Cohen

"*Time is natures way of keeping everything from happening at
once*."\....Anon

Of the physical scattering processes introduced in Section 1, the
photoelectric effect is our starting point. The effect is due to the
absorption of a photon by an atom with subsequent electron emission. We
first derive the photoelectric cross section and examine the regime (low
photon energy) where it dominates. This Section then takes up our first
extended discussion of a detector, the photomultiplier tube.
Applications of the device include time of flight measurements with fast
scintillators, and coincidence logic. A discussion of light collection
in scintillation counters explores both classical "light pipes" and
"wavelength shifting" techniques. Other devices also require a knowledge
of the photoelectric effect. For example see Section 4 where the
transition radiation detector performance depends critically on
photoelectric absorption of the emitted x rays.

## 2.1 Interaction Hamiltonian

As we saw from the data shown in Section 1, the low energy interaction
of a photon is dominated by the photoelectric effect. This effect was
first explained quantitatively by Einstein, using the new quantum theory
and treating the photon as a particle. A schematic of the process in
which there is an incident plane wave with wave number k, wavelength λ,
and frequency ![](media/image1.emf), is shown in Fig. 2.1. Also
indicated is the energy level diagram appropriate to the kinematics. The
atom is initially in a bound state with negative energy
-![](media/image2.emf) and is described by a wave function uo(r). The
final state for the electron has momentum p and exists in the ionization
continuum as a free particle.

![](media/image3.wmf){width="4.680555555555555in"
height="1.7638888888888888in"}

Fig. 2.1: Energy level diagram for the kinematics of the photoelectric
effect.

The free particle Hamiltonian H~o~, is modified in the presence of an
electromagnetic field by an interaction term arising from a replacement
of the momentum,
$\overrightarrow{p} \rightarrow \overrightarrow{p} - \text{ie \{}\overrightarrow{A}$
where $\overrightarrow{A}$ is the vector potential of the
electromagnetic field. This replacement is valid in both classical
mechanics and quantum mechanics. It leads to an interaction term in the
Hamiltonian, H~I~, which is proportional to the vector potential,
$\overrightarrow{A}$, of the photon and the momentum
$\overrightarrow{p}$ of the electron.

$\begin{array}{r}
H_{o} = \frac{p^{2}}{2}m \\
\overrightarrow{p} \rightarrow \overrightarrow{p} - \text{ie \{}\overrightarrow{A}
\end{array}H_{I}\text{\textasciitilde}\left( \frac{\text{ie \{}\overrightarrow{p} \cdot \overrightarrow{A}}{m} \middle| \right)$
(2.1)

The kinematics for classical energy and momentum conservation are

$\begin{array}{r}
 - e + hw = \frac{p^{2}}{2}m \\
{\underline{\overrightarrow{p}}}_{e} + \overset{\rightarrow}{hk} = \overrightarrow{p}
\end{array}$ (2.2)

Ignoring the small energy of the recoil atom, the final state energy is
simply the kinetic energy of the ionized electron. The wave vector
$\overrightarrow{k}$ refers to the initial photon state, while
$\overrightarrow{p}$ refers to the final state free electron. The
initial electron momentum ${\underline{\overrightarrow{p}}}_{e}$ is not
observed. Therefore this free variable is "integrated over" in the sense
that the initial state wave function, u~o~(r), contains the probability
to obtain different initial electron momenta.

## 2.2 Transition Amplitude and Cross Section

In non-relativistic perturbation theory the lowest order transition
amplitude is the matrix element, A, of the interaction potential between
the unperturbed initial and final states.

$\begin{array}{r}
A = \langle f|H_{I}|i\rangle \\
\text{  }\text{\textasciitilde}\frac{e}{m}\int_{}^{}e^{i\frac{\overrightarrow{p} \cdot \overrightarrow{r}}{h}}(\overrightarrow{A} \cdot \overrightarrow{p})u_{o}(r)d\overrightarrow{r}
\end{array}$ (2.3)

Details of the calculation are given in Appendix C. Here we simply
sketch the steps.

We need an expression for the bound state wave function of the inner
shell electrons in the lowest angular momentum state. Recall that the
square of the wave function is the probability density. Since the
characteristic size for the state is the Bohr radius, a, the wave
function is contained in a volume \~${pa}^{3}$ or
$\left| u_{o}(r) \right|^{2}\text{\textasciitilde}\frac{1}{{\pi a}^{3}}$.
Referring to Appendix B and C, the bound state wave function is:

$\begin{array}{r}
u_{o}(r)\text{\textasciitilde}\frac{1}{\sqrt{{pa}^{3}}}e^{\frac{- r}{a}} \\
a_{o} = \frac{D}{\alpha},a\text{\textasciitilde}\frac{a_{o}}{Z}
\end{array}$ (2.4)

The inner electrons in an atom "see" the full charge, Z, of the nucleus
unscreened by the other electrons and thus are bound tightly. Therefore,
their radius is reduced with respect to the hydrogenic Bohr radius by a
factor 1/Z.

The matrix element A is proportional to ![](media/image4.emf) where
![](media/image5.emf) is the outgoing electron deBroglie wavelength
$\lambda_{\text{DB}}\frac{ºh}{p}$. The smallness of the amplitude is due
to the poor overlap integral for A since
$a\text{>>}\lambda_{\text{DB}}$. Basically, it is very unlikely to find
a high momentum in the initial state. Thus the photoelectric cross
section, $s_{\text{PE}}$, scales in a complicated way with photon
energy. The main physics contribution is the
$\left( \frac{\lambda_{\text{DB}}}{a} \right)^{8} = \left( \frac{h}{\text{pa}} \right)^{8}$
behavior coming from the square of the matrix element. As before, $D$ is
the Compton wavelength of the electron.

$\begin{array}{r}
s_{\text{PE}}\text{\textasciitilde}{\alpha D}^{2}\left\lbrack \frac{\text{mc}^{2}}{hw} \right\rbrack\left\lbrack \frac{\lambda_{\text{DB}}}{a} \right\rbrack^{5} \\
\text{\textasciitilde}\frac{1}{w^{\frac{7}{2}}} \\
hw\text{\textasciitilde}\frac{p^{2}}{2}m(\text{Eq}\text{.}2\text{.}2)
\end{array}$ (2.5)

The cross section falls rapidly with incident photon energy; in this
approximation as ![](media/image6.emf). Therefore the photoelectric
effect is only important at low photon energies as is, indeed, clear
looking back at Section 1.

If the photon has an energy which is equal to the energy of one of the
inner electrons, an enhanced absorption will occur. The simple
approximate generalization of the Bohr result for different principle
quantum numbers, n, and for inner electrons bound to atoms with atomic
number Z is given below. The effect for E~n~ is taken care of by the
replacement
$U = \frac{e^{2}}{r} \rightarrow U = (\text{Ze})(e\frac{)}{r}$ or
$\alpha \rightarrow Z\alpha$.

![](media/image7.emf) (2.6)

Some representative data on the intensity of the emitted x-ray for the
inverse process in various heavy elements is shown in Fig. 2.2. In this
inverse process, ![](media/image8.emf), elements under electron
bombardment emit x-ray photons. The photon emission occurs when the
atom, ![](media/image9.emf), de-excites by an inner electron transition
accompanied by the emission of an x-ray with a quantized wave length.
The scale for, ![](media/image10.emf) or keV energies, as expected from
Eq. 2.6.

![](media/image11.png){width="3.013888888888889in"
height="3.888888888888889in"}

Fig. 2.2: Data on the intensity of emitted x-rays as a function of
wavelength in various heavy elements. The "quantized" lines
corresponding to the binding energies of the inner electrons are very
evident. (From Ref. 1.8, with permission.)

Data on the relationship between the square root of the emitted inverse
wavelength and the atomic number is shown in Fig. 2.3. As expected from
Eq. 2.6, the bound state energies, and therefore the emitted x-ray
frequencies are proportional to Z2, which is called Mosley\'s Law. The
mechanism of x-ray emission by electron bombardment is the basis of
medical and dental radiology and is now a commonplace.

![](media/image12.png){width="3.7916666666666665in"
height="2.5965277777777778in"}

Fig. 2.3: Data on the relationship between the x-ray wavelength and the
atomic number. The $\hslash\omega$ = E \~Z^2^ behavior expected from Eq.
2.6 is observed. (From Ref. 2.9, with permission.)

The inverse mean free path, in (gm/cm2)^-1^, as a function of the photon
energy incident on lead is shown in Fig. 2.4. We calculate the inner
electron binding energy for lead for the first two principle quantum
numbers. The computed energies from Eq. 2.6 are 91 and 23keV
respectively. As seen in Fig. 2.4, there is pronounced structure at an
incident photon energy corresponding to these two energies. We compare
the photoelectric cross section to the Thompson cross section,
$\sigma_{T}$, for photon elastic scattering off free electrons given in
Table 1.1 and derived later in Section 10. Assuming that Thompson
scattering is incoherent, the sum over Z electrons is simply Z times the
cross section off a single electron.

![](media/image13.png)

Fig. 2.4: The inverse mean free path,
$\frac{1}{\left\langle L \right\rangle}\rho$, for photons incident on
Pb. The distinct physical mechanisms of photoelectric effect, Compton
scattering and pair production are evident, along with the "resonant"
behavior near the bound state energies indicated by Eq. 2.6. (From Ref.
B.1, with permission.)

$\begin{array}{r}
s_{\text{PE}}\text{\textasciitilde}\frac{\text{32}p}{3}\sqrt{2}(Z\alpha)^{4}Z\left( \frac{m}{hw} \right)^{\frac{7}{2}}(\alpha D)^{2} \\
s_{T}\text{\textasciitilde}\frac{8p}{3}Z(\alpha D)^{2} \\
\frac{s_{\text{PE}}}{s_{T}}\text{\textasciitilde}4\sqrt{2}(Z\alpha)^{4}\left( m_{e}\frac{c^{2}}{hw} \right)^{\frac{7}{2}}\text{.}
\end{array}$ (2.7)

Therefore, for heavy atoms where ![](media/image14.emf), we expect the
photoeffect to dominate over Thompson scattering for energies, $hw$ \<
mc^2^ \~ 0.51 MeV. Indeed, this is the observed behavior and the
approximate energy region where the 2 cross sections are comparable,
(See Fig. 1.10). For example, in Pb, a photon with 10 keV energy has
![](media/image15.emf) \~ 6.8 x 10^5^ or ![](media/image16.emf) Mb. This
estimate from Eq. 2.7 is reasonably close to the value which can be read
off from Fig. 1.10.

At photon energies below about 1 MeV in lead the photoelectric effect
dominates. Above 1 MeV the Compton effect (relativistic elastic photon
scattering), see Section 10, then dominates briefly before pair
production rises rapidly to be the major physical effect at high energy.
At low energies a power law behavior is observed, as expected from Eq.
2.5 which gives a straight line on the log-log plot of Fig. 2.5. The
mean free path, \<L\>, for photon attenuation is shown, as a function of
photon energy, in different materials in Fig. 2.5 for energies from 1
keV to 100 GeV. At high energies the constant value of the pair
production cross section is evident as is strong dependence on the
atomic number,
$\left\langle L \right\rangle\text{\textasciitilde}X_{o}\text{\textasciitilde}\frac{1}{Z}$,
where X~o~ is the radiation length, to be defined later.

![](media/image17.png){width="6.496527777777778in"
height="3.3180555555555555in"}

Fig. 2.5: Mean free path, ![](media/image18.emf), as a function of
photon energy for different materials. At low energies the photoelectric
Z^5^ behavior is displayed, while at high energies the energy
independent Xo \~ 1/Z pair production behavior is observed. (From Ref.
1.1.)

At intermediate energies the incoherent nature of the elastic Compton
scattering implies that, \<L\> \~ independent of Z, as is seen for
photons of 1 to 10 MeV energy

At still lower energies where the photoelectric effect is important we
see a complicated Z dependence. Looking at Eq. 2.7 we see that the
photoelectric cross section goes as Z5. Clearly the overlap integrals
also vary strongly with energy. As an example, picking a photon energy
of, $hw$ = 10 keV, we find that the photoelectric cross section in lead
is \~2.0 x 105 barns. The corresponding mean free path is 0.0015 gm/cm2,
in reasonable (rough) agreement with the data given in Fig. 2.5.

Note that there are also quantum mechanical effects associated with the
emission due to the energy levels shown in Fig. 2.1. As indicated in
Appendix B, there is a reflection which can be large. For example if the
potential well of the metal is deep, U~o~= 10 eV, E = 0.10 eV means a
reflection coefficient of R \~ 0.67 which significantly retards the
emission process.

## 2.3 The Angular Distribution

The angular distribution of the emitted photoelectron is shown for
several photon energies in Fig. 2.6. In the low energy case,
$hw < \text{mc}^{2}$, there is a classical dipole pattern (see Section
10), which can be seen to arise from the
$\overrightarrow{p} \cdot \overrightarrow{A}$ factor in Eq. 2.3 for
![](media/image19.emf). The electron is emitted preferentially in the
direction of the accelerating transverse electric field\
of the photon. For high energies, a "searchlight" pattern (see Section
10) develops, ![](media/image20.emf) leading to the electron being
thrown forward along the incident photon direction.

![](media/image21.wmf){width="4.127777777777778in"
height="3.4555555555555557in"}

Fig. 2.6: Angular distribution of the photoelectron with respect to the
incident photon direction for several recoil e velocities.

## 2.4 The Photomultiplier Tube

A major application of the photo-effect appears in the photomultiplier
(PMT) tube. In a PMT use is made of the photoelectric effect to convert
a photon signal to an electrical signal where the freed electrons are
collected as a current.

As a second application, in Section 4 we examine transition radiation
detectors which emit x-ray photons. The photoelectric effect is an
important source of absorption since this type of detector is not
transparent to its own emissions. Thin foils of low Z material are used
as radiators which significantly self absorb the emitted x-rays

Let us now turn to photomultiplier tube kinematics. The PMT have special
photocathodes which efficiently convert light into electrons. In a
solid, the atomic energy level diagram of Fig. 2.1 which is relevant to
single atoms becomes more complex. The uppermost populated atomic level
is smeared into a continuum (the "valence band" = VB) by the mutual
interaction of the electrons in the solid. The electrons remain bound to
the metal. The energy diagram relevant in a metal is shown in Fig. 2.7.
Electrons in the metal are bound by energies eV~WF~ (the "work
function"). Absorption of a photon of energy $\hslash\omega$ leads to
ionized electrons with kinetic energy T, T ≤ ($\hslash\omega$ - eV~WF~).
As the reaction time for this process is very short, PMT are very useful
devices for the accurate measurement of time.

![](media/image22.wmf)

Fig. 2.7: Energy level diagram for a metal.

## 2.5 Time of Flight

A first application of the PMT is the direct measurement of the time of
particle passage between two fixed locations. If a system of particles
of different masses has been prepared to possess a unique momentum, for
example by collimation in a magnetic field (see Section 8), then time of
flight gives us velocity and hence "particle identification" or the mass
of a given particle. Assume a fixed distance L and a time of passage, t.
In Eq. 2.8 the relativistic expressions for energy ε and momentum *p*
are used. (See Appendix A.)

$\begin{array}{r}
\text{vt} = (\beta c)t = L \\
g = \frac{1}{\sqrt{1 - \beta^{2}}} = \frac{e}{m}\text{\textasciitilde}\frac{p}{m} \\
\beta = \sqrt{1 - \frac{1}{g^{2}}}\text{\textasciitilde}1 - \frac{1}{2g^{2}},\beta \rightarrow 1 \\
t\text{\textasciitilde}\frac{L}{c}\left( 1 + \frac{1}{2g^{2}} \right) \\
Dt\text{\textasciitilde}\frac{L}{c}\left( \frac{m_{1}^{2} - m_{2}^{2}}{2p^{2}} \right),\beta \rightarrow 1
\end{array}$ (2.8)

For a given flight path, L, the difference in time of flight for
particles of different mass, ![](media/image23.emf), is proportional to
the difference of the squares of the masses of particles. The time
separation becomes worse as the inverse of the square of the particle
momentum. Therefore time of flight is typically a useful technique only
for slow, almost non-relativistic, particles.

The time of flight method collapses at high energies because, by virtue
of the special theory of relativity, ![](media/image24.emf) independent
of the mass. The required flight path at fixed time resolution goes like
p^2^. For example, to achieve a four standard deviation $\frac{\pi}{K}$
separation in ![](media/image25.emf), m~π~ \~ 0.14 GeV, m~K~ \~ 0.49
GeV, with a detector with a resolution of 300 psec, requires a 3 meter
flight path, L, at 1 GeV and a 12 meter flight path at 2 GeV. As this
example shows, we are normally limited by geometric and fiscal
constraints to rather low momenta.

## 2.6 Scintillators and Light Collection

So far we have not specified how particle passage is detected and made
into a light signal which impinges on the PMT. We now consider the use
of plastic "scintillators" for this purpose. These plastics consist of a
"base" structural plastic which has small additives of "fluors"
dissolved in the base. The real physics of scintillation light
production is rather complicated. An incident charged particle first
interacts with the atoms of the scintillator exciting a "primary"
fluorescent material which de-excites and emits photons in the
ultraviolet. This light is absorbed within a mean free path of \~ 1 mm
by a "secondary" fluorescent material also dissolved in the "base"
plastic which emits in the visible part of the spectrum where the "base"
plastic is fairly transparent.

For example, the primary fluor might be p-terphenyl (PTP) which emits at
4400 $\overset{o}{A}$ with a 5 nsec decay time, while the secondary
fluor might be "POPOP" which emits at 4200 $\overset{o}{A}$ in 1.5 nsec.
A typical base plastic is polystyrene. Suffice it to say that
operationally a plastic scintillator emits blue light with a time
constant for defluorescence $\underset{\text{\textasciitilde}}{}$ 8
nsec.

Plastic scintillator is a basic workhorse detector, used in many
applications requiring charged particle detection. The plastic is easy
to machine and can be cut and polished into shapes specific to a given
application.

A scintillator plus photomultiplier tube system can be used to measure
time of flight since all of the components of this system are very fast.
Let us work through a typical application. A typical plastic
scintillator such as NE102 has a density of about 1 gm/cm^3^, and an
index of refraction *n* of 1.58. It emits blue light with a
characteristic decay time constant of only a few nsec.

![](media/image26.emf) (2.9)

As a "rule of thumb" we get roughly one scintillation photon for every
100 eV of deposited ionization (see Section 6) energy (2% conversion
efficiency). For the case of plastic scintillator, looking in Table 1.2
(see also Section 6), this means a yield of about 500 photons per
centimeter.

$\begin{array}{r}
1\frac{g}{\text{100}}\text{eV} \\
\text{500}\frac{g}{\text{cm}}
\end{array}$ (2.10)

The emitted light needs to be captured and collected. As it is emitted
isotropically inside the plastic, there is a critical angle for total
internal reflection, ![](media/image27.emf). In plastic the critical
angle is about 39o so that of order 1/3 of the emitted photons are
captured within the plastic and move towards one end. A schematic
representation of the capture of the light is shown in Fig. 2.8.

![](media/image28.wmf){width="5.138888888888889in"
height="2.0416666666666665in"}

Fig. 2.8: Schematic of a scintillator with phototube readout showing the
critical angle for light collection.

$\begin{array}{r}
\text{sin}q_{c} = \frac{1}{n} \\
q_{c}\text{\textasciitilde}\text{39}^{o} \\
\frac{D\Omega}{4p} = \left( \frac{1 - \text{cos}q_{c}}{2} \right)\text{\textasciitilde}0\text{.}3
\end{array}$ (2.11)

As a concrete example, a 1/8" (0.32 cm) thick scintillation counter
would capture 48 photons within the critical angle. If the optical
collection efficiency is only 1/2 due to poor surface quality of the
plastic, we still have 24 photons arriving at the phototube and
impinging on the photocathode. The photocathode has some probability to
convert the photon into a photoelectron. That probability, the "quantum
efficiency", at the wavelengths emitted by the scintillator may be of
order 25%, see Fig. 2.9. We now have 6 photoelectrons coming off the
photocathode. Since this is a stochastic or random process, the number
of independently emitted photoelectrons is Poisson distributed and the
inefficiency, or probability to observe no photoelectrons, at a mean,
$\left\langle N \right\rangle$, of 6 is 0.2%, i.e. e^-6^ ≈ 0.002. (See
Appendix J.)

$1 - \varepsilon = e^{- \left\langle N \right\rangle},\varepsilon = \text{efficiency}$
(2.12)

![](media/image29.png){width="4.021527777777778in"
height="3.4902777777777776in"}

Fig. 2.9: Photocathode "Quantum Efficiency" as a function of
![](media/image30.emf) of the light impinging on a PMT. (From Ref.
2.10.)

A plot of the window transmission coefficient for different PMT as a
function of ![](media/image31.emf) is shown in Fig. 2.10. Clearly we can
extend the range of photon energies passed by choosing a window made of
MgF~2~. This ability to pass UV light will be mentioned again in our
discussion of Cerenkov counters in Section 3. Fiscal constraints may
limit us to the use of glass windows with only pass
$\lambda\  > \text{3000}\text{}\overset{o}{A}$ photons. Since
MgF~2~ windows are expensive, one other option is to put a layer of
"wavelength shifter" on a cheap glass window. This material absorbs high
energy photons and re-emits them isotropically at a
![](media/image32.emf) matched to the peak photocathode efficiency. We
will see examples of the wavelength shifting (WLS) of light later.
Clearly better transmission means more photoelectrons and thus higher
detection efficiency.

The photocathode layer is thin, because the photoelectric cross section
is very large at low photon energies. For typical PMT cathode materials
the mean free path for a 1 to 3 eV photon is 10^-5^ cm or 1000
$\overset{o}{A}$. Thus the cathode material may only be \~100 atoms
thick and can be vacuum deposited on the glass window. (See Fig. 2.5 and
extrapolate in energy as 1/ω^7/2^).

![](media/image33.png){width="3.998611111111111in"
height="3.1458333333333335in"}

Fig. 2.10: Transmission of different PMT window materials as a function
of ![](media/image34.emf). (From Ref. 2.10.)

## 2.7 Gain and Time Structure

The rest of the photomultiplier consists of electrodes called "dynodes".
At each dynode the electrons from the previous stage strike the material
after falling through an accelerating voltage. A schematic of a typical
PMT electrode structure is shown in Fig. 2.11. Due to secondary electron
emission there is electron multiplication by a factor
![](media/image35.emf), typically of order 3, at each dynode. Therefore,
the gain, G, for a tube with 14 dynode "stages", *N~d~*, is of order
5x106. This being a geometric multiplication, the total gain is given by
the gain per stage, ![](media/image36.emf), raised to the power, N~d~.

$\begin{array}{r}
d\text{\textasciitilde}3 = \frac{\text{gain}}{\text{dynode}} \\
G\text{\textasciitilde}(d)^{N_{d}} = \text{PMT}\text{gain}
\end{array}$ (2.13)

Note that the variation in gain with applied voltage, which sets δ, is
rapid, which means we need to be careful to regulate the voltage well.
Note also that small manufacturing variations in the gain per stage lead
to large changes in the overall gain. Thus we expect PMT to have
variable gains from tube to tube, which we can equalize by supplying a
slightly different voltage to each tube.

![](media/image37.png){width="3.8409722222222222in"
height="3.8868055555555556in"}

Fig. 2.11: Schematic of the electrode structure of a typical
photomultiplier. (From Ref. 2.10.)

For example, the approximate cross section for e multiplication is given
in Eq. 2.14 where ![](media/image38.emf) = ionization energy and
![](media/image39.emf) = incident electron energy. For
![](media/image40.emf) = 10 eV and ![](media/image41.emf) = 100 eV,
![](media/image42.emf) \~ 6.7 x 10^-17^ cm^2^ or \<L\> = 1.7 x 10^-7^ cm
in a metal such as copper. Thus the dynodes can be very thin.

$s(e^{-} \rightarrow e^{-} + e^{-})\text{\textasciitilde}\frac{\text{pα}^{2}}{{eI}_{o}}$
(2.14)

A contribution to the rise time of the current pulse comes from the
spread in kinetic energy of the ejected photoelectrons. The spread is
partially caused by the spread of energies in the valence band, as is
obvious from looking at Fig. 2.7. Hence there is a spread in transit
time to the first dynode, *dt*. For an initial kinetic energy spread
from zero to T, the time spread, *dt*, for non-relativistic motion in an
electric field *E* is,
$\text{dt} = \frac{\sqrt{2\text{mT}}}{(}\text{eE})$, which for T = 1.2
eV in a field of E = 150 V/cm is *dt* \~ 0.2 nsec.

Yet another contribution to the rise time is the path length difference
from the first dynode to different locations on the cathode, as is
visible in Fig. 2.11. The resultant spread in arrival times scales with
the size of the cathode. Typical values of intrinsic phototube rise time
(for 5 cm diameter tubes) are \~ 2 nsec. These times are well matched to
the scintillator decay lifetimes. (Eq. 2.9.)

Thus our 6 photoelectrons, having dropped through a 14 stage tube,
become 4.6x10-12 C = 4.6 pC of anode signal charge. They\'re delivered
in a rather short time because the photocathode response is effectively
instantaneous and the PMT rise time and the fluorescence time of the
scintillator are very rapid, ![](media/image43.emf). If we assume that
the total signal charge is delivered in 10 nsec, it corresponds to a
current of about 0.46 mA. The photomultiplier output is effectively a
current source. The PMT current delivered into a standard 50Ω coaxial
cable (see Appendix D) is 23 mV. That level of voltage pulse corresponds
to the typical threshold voltages, V~T~, which are covered by the fast
electronics which are commercially available (see Section 2.9).

A photograph of a photomultiplier tube and the associated voltage
distribution for the multiple dynodes is shown in Fig. 2.12 where we get
an idea of the layout of a photomultiplier tube.

![](media/image44.wmf){width="3.55in" height="4.106944444444444in"}

Fig. 2.12: PMT and "base" assembly showing the dynode structure in the
PMT and the system used to supply voltage drop to the successive
dynodes. (Photo - Fermilab.)

An oscilloscope signal trace of the output of a photomultiplier tube
driven by a scintillator is shown in Fig. 2.13. Notice that the peak
output current is 50 mA and the rise time of the signal is a few nsec.
This plot serves to illustrate again the order of magnitude of the
currents and times which were quoted in the numerical example given
above. Note that there is a stochastic width to the peak height of the
pulse. It's fractional spread is determined by the number of independent
primary events or the number of emitted photoelectrons (see Appendix J).
For low signal levels we can alternatively look at the fraction of time
no photoelectrons are emitted and hence, see Eq. 2.12, independently
estimate ![](media/image45.emf). In Fig. 2.13 the fractional spread is
\~ 10% indicating that ![](media/image46.emf) is \~ 100, which is 16
times larger than our numerical example.

![](media/image47.png){width="3.69375in" height="3.5555555555555554in"}

Fig. 2.13: Oscilloscope trace of a PMT current pulse output. Note the
rise time of \~ 2 nsec and the pulse width of \~ 10 nsec full width at
half maximum, FWHM. The peak current is \~ 50 mA and the signal pulse
height spread is \~ 10%. (From Ref. 2.10.)

## 2.8 Wavelength Shifting

There is an interesting alternative solution to the problem of the
collection of scintillation light called "wavelength shifting" (WLS).
Typically the Liouville theorem tells us that when we are bringing light
out and routing it using total internal reflection to a photomultiplier
tube, the "light pipe" must have an area which does not diminish or we
will lose photons. However, these light pipes are unwieldy and ungainly
and mean a loss of active detector area because they take up a finite
amount of space. The required matching photocathode surface area is also
expensive.

There is a way to evade this difficulty by using a fiber which takes the
scintillation light and shifts it to a longer wavelength, captures it by
total internal reflection and finally pipes it out to a phototransducer
of some sort. Now, at first blush this violates the Liouville theorem.
For example, in a typical application one may have a ratio of the
primary scintillator area, A, to the secondary wavelength shifting fiber
area, ![](media/image48.emf), which is a factor of order 104. (See Fig.
2.14b.)

![](media/image49.emf) (2.15)

The theorem is evaded by red shifting the light. When the light is
shifted to longer wavelengths, the energy per photon is reduced and
therefore the light is "cooled". Since the information content must be
the same, we have packed the same information into a smaller area since
the information is now "stored" at lower energies. Thermal noise sets
the scale for the smallest allowed storage of information energy. The
random thermal energy is defined by the absolute temperature, T, and
Boltzmann\'s constant, k, \~kT. (See Section 1.)

Boltzmann\'s constant, k, is displayed in Table 1.1. As a mnemonic,
Boltzmann\'s constant times the temperature at room temperature, T \~
300^o^K, is roughly 1/40 eV. Therefore, for example, consider taking
light emitted in the blue at about 4500 ![](media/image50.emf) and
cooling it to 5000 ![](media/image51.emf) in the green, wavelength
shifting by about 500 ![](media/image52.emf). That "cooling" corresponds
to an energy shift of \~ 0.28 eV.

$\begin{array}{r}
\lambda\text{\textasciitilde}\text{4500}\overset{o}{A} \rightarrow \text{5000}\overset{o}{A} \\
De\text{\textasciitilde}0\text{.}\text{28}\text{eV}
\end{array}$ (2.16)

The increased "packing factor" by which we can compress the information
is given by the energy shift, ![](media/image53.emf), divided by kT,
which defines the theoretical upper limit of information content
compression. That ratio is of order 7.3 x 104 in this example, showing
that we can achieve very large compression factors. That in turn allows
us to make scintillation detectors which are effectively almost fully
active since the wavelength shifter fiber uses up only a minuscule
fraction of the area of the detector for optical readout. We will return
to "hermetic" detectors in Section 12.

$\begin{array}{r}
e^{\frac{De}{\text{kT}}} = 7\text{.}3´\text{10}^{4} \\
(\text{kT})_{\text{300}^{o}K}\text{\textasciitilde}\frac{1}{\text{40}}\text{eV}
\end{array}$ (2.17)

A picture of the two different kinds of light pipes is given in Fig.
2.14. Figure 2.14a is a photograph of a traditional scintillation
assembly with an equal area light pipe. Note how cumbersome the light
piping is, even though the equal area law is not even strictly
maintained. In contrast, Fig. 2.14b shows a scintillation assembly with
wavelength shifter (WLS) readout. It should be clear from the photos
that the ability to cool the information allows for a much more compact
and hence "hermetic" readout.

![](media/image54.wmf){width="4.048611111111111in"
height="5.022222222222222in"}

Fig. 2.14a: "Light pipe" assembly without wave length shifting. (Photo -
Fermilab.)

![](media/image55.png){width="5.60625in" height="5.311111111111111in"}

Fig. 2.14b: Array using wavelength shifting fiber. The area ratio of
scintillator plate to WLS fiber is \~ 10^4^. (Photo - Fermilab.)

## 2.9 Coincidence Logic and Deadtime

The signals from the anode of a PMT are oftne used to make logical
decisions, i.e. decisions based on Boolean logic. The basis of these
decisions is first the initial decision between logical "*φ* " and
logical "1" and subsequent AND, OR, and NOT electronic operations on the
signal. A typical 2 fold AND coincidence setup is shown in Fig. 2.15.
Signals on inputs #1 and #2 are first "discriminated" by passing through
circuits that give logical "1" for a fixed time ![](media/image56.emf)
if and only if *V* \> *V*~T~ ("threshold" voltage see Appendix I). These
signals, C~1~ and C~2~, are input to an AND "gate" which gives a logical
"1" output, C~12~, if and only if both of its inputs are "1", or true,
at some point in time (an AND "coincidence" gate). The method commonly
used for most high speed signal connections is to use coaxial cables
loaded with a dielectric medium. Although an extremely useful tool, we
relegate the discussion of these cables to Appendix D.

![](media/image57.wmf){width="3.341666666666667in"
height="2.338888888888889in"}

Fig. 2.15: Schematic of a typical discriminator and coincidence logic
for a 2 input AND operation.

Clearly if a particle beam passes through both scintillation counters 1
and 2, the counting rate of the AND gate, *C*~12~ represents the true
particle beam rate. It is also possible that there are false counts due
to signals accidentally coinciding in time at the inputs to the AND
gate. The timing diagram shown in Fig. 2.16 makes it clear that the time
interval appropriate to an accidental 2 fold AND coincidence,
![](media/image58.emf), is ![](media/image59.emf).

![](media/image60.emf) (2.18)

In Eq. 2.18 ![](media/image61.emf) is the accidental AND rate, *C*~1~
and *C*~2~ are the "singles rates" and ![](media/image62.emf) is the
"live time".

![](media/image63.wmf){width="3.56875in" height="1.7798611111111111in"}

Fig. 2.16: Timing diagram showing a pulse from source #1 of width *τ*~1~
and the range of times for which source #2 of width *τ*~2~ is in
accidental coincidence with source #1.

Clearly, a detector with a fast response time is desirable since it
makes for a reduced rate of false coincidences. Typically we use logic
times, $t_{i}$, which are just sufficient to allow for the intrinsic
time "jitter" of the detector if we are in a high background rate
environment. The reason is that this is the shortest live time we can
use consistent with full efficiency for real coincidences. Therefore, it
is the minimum accidental rate. An example of one cause of PMT time
"jitter" was shown in Fig. 2.11. Scintillator and PMT logical decisions
can be made with live times ![](media/image64.emf) 5 nsec, for spatially
small scintillators. For example, if *C*~1~ = *C*~2~ = 1 MHz, then with
![](media/image65.emf) = 5 nsec the accidental counting rate is
![](media/image66.emf) = 10 kHz. Digging low rate true signals out of
high rate backgrounds often requires a higher level of coincidence, with
corresponding lower accidental rate.

Why were photons not seen as quanta prior to the twentieth century? An
example of the numerology should help to see why. Suppose you have a 1 W
light bulb and you view it with an "eye" of area 1 mm x 1 mm. Assuming a
photon of 1 eV, the number of photons entering your eye at a distance of
10 m from this rather dim bulb is still 10^10^ γ/sec or 10 photons/nsec.
Since humans function as millisecond detectors, we perceive the light as
a continuous "fluid".

In the language of coincidence, with τ\~10^-3^ sec, Cτ is \~ 10^7^, and
the eye integrates over millions of photons. As another example the
fluorescent lights in your home are flickering on and off at f = 60 Hz.
Since your eyes integrate due to persistence of vision you are unaware
of this behavior and believe you are reading this book under a steady
light. Clearly, fast devices are needed to detect the real nature of our
environment, and most modern detectors react on the scale of nsec.

We close this Section with a short discussion of the concept of detector
"deadtime". Suppose we have a detector which is dead to subsequent input
signals for a time ![](media/image67.emf) after producing a signal. A
typical example would be a Geiger counter which must recharge after a
pulse discharges it. Define ![](media/image68.emf) to be the observed
output counting rate and *C* to be the true incident rate. The fraction
of time the detector is dead is simply
![](media/image69.emf)![](media/image70.emf). Therefore the rate which
is lost is;

![](media/image71.emf) (2.19)

Clearly, both detectors and electronics with fast pulse recovery, or
small dead time ![](media/image72.emf), are of value in that the
observed rate is close to the true rate as long as
C![](media/image73.emf)\<\<1. For example, if C = 1MHz and
![](media/image74.emf) = 5 nsec, then 1 + C![](media/image75.emf) =
1.005 or \~0.5% loss of signal. The desire to measure the true reaction
rate efficiently is why particle detectors have a premium placed on
short deadtime.

## References 

Section 2: The Photoelectric Effect, Photomultipliers, Scintillators

\[1\] [Quantum Physics]{.underline}, S. Gasiorowicz, John Wiley & Sons,
Inc. (1996).

\[2\] [Theory and Practice of Scintillation Counting]{.underline}, J.B.
Birks, Pergamon Press (1964).

\[3\] Photomultiplier Tubes, Principles and Applications, Philips
Photonics (1995).

\[4\] [The Art of Electronics]{.underline}, P. Horowitz, W. Hill,
Cambridge University Press (1980).

\[5\] [Pulse Electronics]{.underline}, R. Littauer, McGraw-Hill (1965).

\[6\] R.L. Garwin, Rev. Sci. Inst. [31]{.underline} 1010 (1960).

\[7\] [Techniques of High Energy Physics]{.underline}, D.M. Ritson,
Wiley Interscience (1961).

\[8\] [X-Rays in Atomic and Nuclear Physics]{.underline}, N.A. Dyson,
Longmans (1973).

\[9\] [SCIFI93, Workshop on Scintillating Fiber Detectors]{.underline},
A.D. Bross, R.C. Ruchti, M.R. Wayne, World Scientific (1995).

\[10\] "Photomultipliers and Accessories" Thorn EMI Electron Tubes
(1994).

Table of Contents

II\. Non-destructive Measurements [1](#non-destructive-measurements)

II.A. Time and Velocity [1](#ii.a.-time-and-velocity)

2\. The Photoelectric Effect, Photomultipliers, Scintillators
[2](#the-photoelectric-effect-photomultipliers-scintillators)

2.1 Interaction Hamiltonian [2](#interaction-hamiltonian)

2.2 Transition Amplitude and Cross Section
[3](#transition-amplitude-and-cross-section)

2.3 The Angular Distribution [9](#the-angular-distribution)

2.4 The Photomultiplier Tube [10](#the-photomultiplier-tube)

2.5 Time of Flight [11](#time-of-flight)

2.6 Scintillators and Light Collection
[11](#scintillators-and-light-collection)

2.7 Gain and Time Structure [15](#gain-and-time-structure)

2.8 Wavelength Shifting [18](#wavelength-shifting)

2.9 Coincidence Logic and Deadtime [21](#coincidence-logic-and-deadtime)

References [25](#references)