# II.C. Position and Momentum

We now begin our exploration of the non-destructive measurement of
position and momentum, using the deposited ionization energy (Section 6)
and limited in accuracy by the multiple scattering in the medium
(Section 5). The goal is to measure the vector momentum which we
indicated in Figure I.1 as part of the general task of the detector
designer.

To that end we first look at particle motion in magnetic fields in
Section 7. Since the helical trajectory depends on the magnitude of the
momentum, we deploy gaseous detectors in Section 8, which use localized
ionization deposition to locate points along the trajectory. Since the
charge is collected on electrodes, we also must explore motion in
electric fields. Finally, if greater spatial accuracy is needed, silicon
detectors are employed, as explicated in Section 9. The points are fit
to a hypothesis regarding the trajectory. The simplest example of a
straight line is treated in the Appendices. A more complex helical path
hypothesis yields best-fit estimates for the vector momentum and their
errors of the particles leaving the track points.

# 7. Magnetic Fields

"*Of Newton with his prism\-\--voyaging through strange seas of thought,
alone*."\....William Wordsworth

"*The wheel is come full circle*."\....Shakespeare, 'King Lear'

The use of magnetic fields in particle detectors has several aspects. As
we will discuss, magnetic fields are used to prepare the "beams" that
impinge on experiments. In addition, the creation of a large volume with
a magnetic field and associated detectors allows us to measure the
trajectory of charged particles in the field (the jargon is "tracking")
by detecting the ionization caused by the particle passage. Tracking
allows us to measure the position and momentum of charged particles (see
Section 13 for some examples of tracking in large detectors).

## 7.1 Solenoidal Fields

We begin by considering electromagnets in "colliding beam" applications,
i.e. where 2 oppositely directed beams collide head on. The most common
topology for such an experimental electromagnet is the solenoid.
Consider a solenoid coil of N total turns, each carrying a current I,
with full length L and radius a. The magnetic field in CGS units is, at
a point along the axis where the ends subtend angles
![](media/image1.emf) and ![](media/image2.emf), defining n as the
turns/length = N/L.

$B = \frac{2p\text{nI}}{c}\left( \text{cos}q_{1} + \text{cos}q_{2} \right)$
(7.1)

The result at the centerline of a very long solenoid approaches the
elementary result which follows from Amperes law.

$\begin{array}{r}
B(\text{centerline}) = \frac{4p\text{nI}}{c}\left\lbrack \frac{\left( \frac{L}{2} \right)}{\sqrt{\left( \frac{L}{2} \right)^{2} + a^{2}}} \right\rbrack{ºB}_{o} \\
 \\
B_{o} \rightarrow 4p\frac{\text{nI}}{c}\text{as}L \rightarrow ¥,\text{CGS}
\end{array}$ (7.2)

In MKS units the result for the magnet center line is (see Table g3.1,
1/c ![](media/image3.emf));

$B \rightarrow m_{o}\text{nI},\text{MKS}$ (7.3)

In the center of an infinitely long solenoid the field depends only on
the current carried by each turn I and the turns per unit length n. For
reference purposes, 1 Tesla is 10 kG or 1 N/(Am^2^). The electromagnetic
constants that are needed for all our calculations can be found either
in Table 3.1 or in Table 1.1.

For example, to produce a 20 kG field with 1 turn per centimeter, n,
requires 16 kA of current I. Thus a copper bus (water-cooled) of 1
centimeter diameter at 16 kA produces a 2 Tesla solenoidal field. The
energies stored in such magnets may be enormous,
$U_{B}\text{\textasciitilde}B^{2}\underline{V}$. Typical experiments
in high-energy physics or magnetic fusion devices now utilize magnets
with stored energies in the 10-100 MJ range.

## 7.2 Dipole Fields - Fringe Fields

We can also look at a typical "fixed target" topology for the field i.e.
where a beam is directed to a target at rest in the laboratory. A
"dipole" magnet is illustrated schematically in Fig.7.1a. The iron
"return yoke" shapes the field produced by the coils and produces a
roughly constant field in the air gap. Given that the physical magnet
length is \~ L, the field has a finite spatial extent. There is a
\"fringe field" due to the constraints imposed by Maxwell's equations.
For example, in the absence of sources, if the field ramps down (from
magnitude B~0~) in the z direction over a characteristic distance,
$\ell$, then a component B~z~ is induced. Since all fields are finite in
extent, a "fringe field" is guaranteed to exist.

$\int_{}^{}{\overrightarrow{B} \cdot d\overrightarrow{x}}\text{\textasciitilde}B_{o}(L + l),magnetic\text{ ital }length = L + l$
(7.4)

Outside the sources, the curl of the magnetic field is zero. Looking at
one component of that vector equation, we find that the $¶_{y}B_{Z}$ is
equal to $¶_{Z}B_{y}$. Therefore, B~Z~ arises because the change of the
y component of the field is nonzero. An estimate of the size of Bz and
its sign is given in Eq. 7.5 and is shown schematically in Fig. 7.1c.
Note that what started as a simple uniform field has now grown a lens,
i.e. B field proportional to deviation from magnet centerline, at both
it's exit and entrance. We discuss magnetic lenses further in Section
7.6 and 7.7 below.

![](media/image4.wmf)

Fig. 7.1: a) Layout of a dipole electromagnet, b) By as a function of z,
c) Fringe field Bz as a function of z, and d) Fringe field Bx as a
function of z.

$\begin{array}{r}
\overrightarrow{Ñ}x\overrightarrow{B} = 0 \\
¶_{y}B_{z} = ¶_{z}B_{y}\text{\textasciitilde} \pm \frac{B_{o}}{l} \\
B_{z}\text{\textasciitilde}(¶_{z}B_{y})y\text{\textasciitilde} \pm B_{o}\frac{y}{l}
\end{array}$ (7.5)

The divergence of the magnetic field is also zero. The $¶_{Z}B_{Z}$ must
be compensated by, for example, $¶_{x}B_{x}$. That implies the existence
of an x component of the field proportional to the second z partial
derivative of *B~y~*. (See Fig. 7.1d.)

$\begin{array}{r}
\overrightarrow{Ñ} \cdot \overrightarrow{B} = 0 \\
¶_{x}B_{x}\text{\textasciitilde} - ¶_{z}B_{z}\text{\textasciitilde} - y(¶_{z}^{2}B_{y}) \\
B_{x}@ - \text{xy}(¶_{z}^{2}B_{y})
\end{array}$ (7.6)

The "fringe fields" *B~Z~* and *B~x~* vanish at the centerline,
B~x~(0,0) = B~z~(0,0) = 0 and hence act on off axis particles as lenses.

A photo of a "fixed target" style dipole magnet installed in a
high-energy physics experiment is shown in Fig. 7.2. This photo is a
concrete realization of the schematic diagram already shown in Fig.
7.1a. In that photo the coils of the electromagnet and the iron return
yoke which shapes the field can be readily identified. A fixed target
"bubble chamber", such as that used to make the picture shown in Fig.
6.6, is also immersed in a reasonably uniform dipole field. A prepared
"beam" of particles of roughly the same momentum and all moving with
about the same angle impinges on the "bubble chamber" from below in Fig.
6.6. Assuming that momentum goes inversely with radius of curvature (Eq.
7.8 below), the momentum of the beam particles is clearly larger than
that of the decay products since the radius of curvature of the beam is
seen to be large.

## 7.3 Particle Motion in a Uniform Field

Let us now look at the trajectory of a particle, of charge q, in a
uniform solenoidal magnetic field oriented along the z axis. Since we
are doing numerical estimates, we will use MKS units here. The force
$\overrightarrow{F}$ due to the magnetic field is given by the Lorentz
force equation, Table 3.1. Since this force is perpendicular to the
velocity, the power dissipated is zero. Hence, the momentum vector is of
constant magnitude.

![](media/image5.wmf){width="6.354166666666667in"
height="5.864583333333333in"}

Fig. 7.2: Photograph of an analysis magnet showing the coils of the
electromagnet and the iron yoke. (Photo - Fermilab)

$\begin{array}{r}
\overrightarrow{F} = m\overrightarrow{a} = q(\overrightarrow{v}x\overrightarrow{B}) = d\frac{\overrightarrow{p}}{\text{dt}} \\
\overrightarrow{F} \cdot \overrightarrow{v} = 0 \Rightarrow \left| \overrightarrow{p} \right| = \text{const}
\end{array}$ (7.7)

Since there is no force along the field direction, the path in z, as a
function of arc length, is a straight line with a constant direction
cosine, α~Z~ = dz/ds. In the transverse plane, (x,y), the equations of
motion imply a solution where the momentum vector simply rotates in that
plane by an azimuthal "bend" angle φ~B~.

The centrifugal force in the non-relativistic case can be equated to the
Lorentz force to yield an expression for the radius of curvature of the
momentum vector in the x-y plane.

$\begin{array}{r}
F_{\text{cent}} = \frac{\text{mv}^{2}}{r} = \text{qvB} \\
p = \text{mv},r = a \\
a = \frac{p}{\text{eB}}
\end{array}$ (7.8)

In MKS units, 1/e is 33.3 when the momentum, p, is given in units of GeV
(10^9^ eV), the magnetic field, B, is given in kG, and the radius of
curvature, a, is given in meters. For example, a 100 GeV proton in a 10
kG field has a radius of a curvature of 336 meters.

Since, in the non-relativistic case,
$\overrightarrow{p} = m\overrightarrow{v}$ Eq. 7.7 becomes

$d\frac{\overrightarrow{p}}{\text{dt}} = \frac{q}{m}\left( \overrightarrow{p}´\overrightarrow{B} \right)$
(7.9)

The form of Eq. 7.10 implies circular motion which we could prove by
assuming a circular trajectory and substituting into the equations of
motion. Since the forces is transverse to momentum it always acts so as
to bend the path into a circular arc just as in the case of a satellite
in Earth orbit. The frequency can be read off directly, ωp = qpB/m, and
is given by the "cyclotron frequency", $w$~c~, quoted in Table 1.1.

$w_{c} = \frac{\text{eB}}{m}$ (7.11)

For constant field we have a constant rotation frequency which is
important in the design of cyclotrons because then simple radio
frequency (r.f.) drivers of fixed frequency can be used. For reference
$w_{P}$ = 9.6 x 10^6^ rad/(sec kG) and $w_{e}$ = 1.7 x 10^10^ rad/(sec
kG) for protons and electrons respectively, (see Table 1.1.) For
example, a 10 MeV proton cyclotron has a final T = 10 MeV or p = 141
MeV. If the bending field is 10 kG then the protons at a full energy
have a 0.47 m orbital radius. The r.f. system works at $w$ = 9.6 x 10^7^
rad/sec or f \~ 15 MHz. These values are typical of cyclotron
parameters.

The situation is modified in considering relativistic motion.
Non-relativistic motion is circular with a fixed frequency, whereas in
relativistic motion the inertial mass increases by a factor of γ as is
evident in Eq. 7.8, p = γmv. Thus, *ω* goes from *ω~c~* to eB/γm which
means that the radio frequency power must be applied at variable
frequency. For a practical high-energy circular accelerator, we need to
increase the magnetic field B as the particles gain energy, or γ, in
order to keep the beam within a fixed radius vacuum pipe. For example,
the Fermilab Tevatron magnets are "ramped" up to full field of \~30 kG
during the acceleration cycle which takes \~10 sec to go up to \~ 1 TeV.

It is this effect which defines the ultimate size of a circular
accelerator. The physics of magnetic solids implies that there is a
maximum field which we can get from iron magnets of about 2.0 Tesla due
to domain saturation; all the domains are aligned at that point.
Therefore, the radius of an accelerator must grow proportionally to the
maximum momentum
$a\text{\textasciitilde}\frac{p_{\text{max}}}{e}B_{\text{max}},\text{}B_{\text{max}}\text{\textasciitilde}\text{20}\text{kG}$.
For example, a 1 TeV (10^12^ eV = 1000 GeV) accelerator would have a
1.65 km radius. The use of higher field "superconducting" electromagnets
allows us to somewhat relax this limit but only by a factor of 2 or 3.

A concrete visualization of the circular trajectories is given in Fig.
6.6. This is a "bubble chamber" picture of an event featuring the
production of two neutral particles, which leave no ionization in the
chamber, each decaying into two charged particles. We can tell that the
charged particle pairs are positive and negative since the sense of the
rotation in the magnetic field in which the chamber is immersed is
opposite for the elements of the pair. Looking at Eq. 7.8 the sense of
rotation in a magnetic field depends on the charge. In the lower pair it
is also clear that one of the decay products is much more energetic,
\"stiffer\", than the other one. The higher momentum track has a much
larger radius of curvature. The lower most charged particle in the pair
has a lower momentum.

We can also observe that momentum conservation, (Appendix A), in the
decays means that the neutral parent should point back to the production
vertex which is visible. In the case of the upper pair the plus and
minus daughter tracks roughly share the momentum, so that the charged
pair bisector goes close to the production vertex. In the lower pair the
neutral parent particle's momentum vector must be near to the direction
of the "stiff" track of the pair. Indeed that track by itself
approximately points back to the production vertex. We can look at these
bubble chamber pictures and do the vector momentum conservation
kinematics with rulers and protractors which is a very instructive
exercise highly recommended to the student. You really believe in
energy-momentum conservation after you do the kinematics yourself by
hand.

## 7.4 Momentum Measurement and Error

It\'s also possible to use solid electromagnets without an air gap,
simply as an iron core wound with exciting coils. The deflection angle,
![](media/image6.emf), on traversing a length, L, in a direction
transverse to the magnetic field, B, is given in small angle
approximation as the length divided by the radius of curvature, a. (See
Appendix F and Eq. 7.8.)

$f_{B}\text{\textasciitilde}\frac{L}{a} = \frac{\text{eLB}}{p}$ (7.11)

Using Eq. 7.8 for the radius of curvature, we find that the bend angle
is the transverse momentum impulse divided by the momentum.

$\begin{array}{r}
\text{}\varphi_{B}\ \text{} = \ ({\Delta p}_{T}\frac{)_{B}}{p} \\
({\Delta p}_{T})_{B}\text{} = \ \text{eLB}\  = \ 0\text{.}\text{03}\text{BL}\left( \frac{\text{GeV}}{\text{kGm}} \right)
\end{array}$ (7.12)

This result is similar to the treatment that was given for multiple
scattering where the physics defined the transverse momentum impulse
which was independent of the incident particle save for its charge. The
bend angle goes as the inverse of the incident particle momentum. For
example, 1.5 meters of iron magnetized to near saturation at 18 kG gives
a momentum impulse of roughly 0.82 GeV. Thus a 8 GeV particle bends
\~0.1 rad \~ 6^o^ in traversing the iron.

Closer to home, we all have an electron accelerator in our home -- the
T.V. set -- with which we can do experiments. Inside there is an
electron gun which accelerates through about 10 kV, an energy needed to
excite the phosphers on the screen. We can use permanent magnets to make
a \~ 1 kG field extending over about 1 cm. Thus, a refrigerator magnet
put in close proximity to the TV screen will cause a noticeable
deformation of the TV image. Note that a 10 keV energy electron has a \~
100 keV momentum, to compare to the 300 keV momentum impulse due to a 1
kG field extending over 1 cm. The student is encouraged to do this
experiment.

We recall that the transverse momentum impulse due to multiple
scattering has a characteristic two-dimensional scattering energy of
![](media/image7.emf) or 14 MeV. Traversing the 1.5 meters of iron,
given that the radiation length of iron is 1.76 cm (Table 1.2), a
particle crosses 85 radiation lengths and therefore suffers a multiple
scattering momentum impulse of 0.13 GeV.

$({Dp}_{T})_{\text{MS}} = \frac{E_{S}}{\sqrt{2}}\sqrt{\frac{L}{X_{o}}}(\text{Section}3)$
(7.13)

The ratio of the momentum impulses is about 16%. It defines the
achievable momentum resolution in such a system, since which is limited
by the multiple scattering. Note that the best possible resolution,
(∆p~T~)~MS~/(∆p~T~)~B~ scales as ![](media/image8.emf) Therefore, it is
extremely costly to improve performance by increasing the length, and
there is a natural limitation to the magnetic field when you reach
saturation in the iron, B \~ 2 T. Thus, the example given above is
fairly typical and difficult to improve on much.

A photo of a typical solid iron magnet is shown in Fig.7.3. The iron
toroid in this device is used both for the detection and identification
of muons. We can see that the steel is excited by the copper coils which
are visible in the photograph. The absorber mass is useful in removing
particles which interact in the steel and do not escape into the muon
tracking system which is located outside the steel and thus not visible.
The momentum measurement is limited, as discussed above, to about 16%.
We will return to the topic of hadronic cascades, "punch through" and
muon detection in Section 12 and Section 13.

![](media/image9.png){width="6.03125in" height="4.759722222222222in"}

Fig. 7.3: Photograph of iron toroids for the detection and
identification of muons. (Photo - Fermilab)

If we are operating in vacuum rather than in steel, the angle of
deflection in the field goes as ![](media/image10.emf), and thus the
fractional error in the momentum measurement goes as
![](media/image11.emf). The achievable resolution in this case is
ultimately limited by alignment and the systematic errors of the
position measuring detectors limit the final accuracy of the angular
measurement. The fractional momentum resolution, in the case of vacuum
scales as p, whereas in the case where the multiple scattering limits it
is a constant.

Assume that the bend angle is measured using detectors which make
measurements having an error dx~T~ along an arc length L. For bend angle
![](media/image12.emf) and momentum impulse $({Dp}_{T})_{B}$;

$\begin{array}{r}
\frac{1}{p} = \frac{f_{B}}{(}{Dp}_{T})_{B} \\
d(\frac{1}{p}) = \frac{{df}_{B}}{({Dp}_{T})_{B}}\text{\textasciitilde}\frac{(\frac{\text{dx}_{T}}{L})}{({Dp}_{T})_{B}} \\
\frac{\text{dp}}{p^{2}}
\end{array}$ (7.14)

We have not specified the number of detectors, nor their optimal
deployment along the path length of the track. Thus Eq. 7.14 is only
indicative of the order of magnitude of the momentum error.

For example, with detectors having ![](media/image13.emf) (e.g.
diffusion limited gas detectors, see Section 8) deployed over L = 1 m in
a field with momentum impulse ![](media/image14.emf) = 1 GeV, a 100 GeV
particle is measured with a momentum accuracy of \~ 1%.

The multiple scattering fractional error is a constant,
![](media/image15.emf). Thus, it is conventional to parameterize the
momentum resolution of a tracking device as the statistical sum ("fold
in quadrature", see Appendix J) of measurement and multiple scattering
errors characterized by the constants a and b respectively.

![](media/image16.emf) (7.15)

## 7.5 Exact Solutions - Cartesian and Cylindrical Coordinates

Consider now the exact solution for the orbit in a perfectly uniform
magnetic field, B, oriented along the z axis. We use the arc length, s,
as a track parameter since the velocity, ![](media/image17.emf), is
constant. Eq. 7.7 using $\overrightarrow{p}\text{and}s$ becomes

$\begin{array}{r}
\text{ds} = \text{vdt} \\
pd\frac{\overrightarrow{p}}{\text{ds}} = q(\overrightarrow{p}x\overrightarrow{B})
\end{array}$ (7.16)

This equation implies that the momentum rotates with frequency,
$\omega_{s} = \text{qB}$ or (see Eq. 7.11) by angle $f_{B}$ = s/a. For
details see Appendix F. The solutions given in Appendix F for z are
straight lines.

$z = z_{o} + \alpha_{z}s,s = 0\text{at}z = z_{o},\alpha_{Z} = \text{constant}$
(7.17)

The direction cosines of the momentum rotate by φ~B~ in the (x,y) plane.

$\begin{pmatrix}
\alpha_{x} \\
\alpha_{y}
\end{pmatrix} = \begin{pmatrix}
\text{cos}f_{B} & \text{sin}f_{B} \\
 - \text{sin}f_{B} & \text{cos}f_{B}
\end{pmatrix}\begin{pmatrix}
\alpha_{\text{xo}} \\
\alpha_{\text{yo}}
\end{pmatrix}$ (7.18)

The "bend angle", φ~B~ is the angle of azimuthal rotation of the
momentum vector. The orbit is a helix.

We integrate to get the coordinates. Initial position and momentum
components are indicated by the *0* subscript (see Appendix F). The
orbit is a circle in the (x,y) plane.

$\begin{array}{r}
(x - x_{o}) = - (\frac{a}{p})(p_{y} - p_{\text{yo}}) \\
(y - y_{o}) = (\frac{a}{p})(p_{x} - p_{\text{xo}}) \\
(x - x_{o} - {a\alpha}_{\text{yo}})^{2} + (y - y_{o} + {a\alpha}_{\text{xo}})^{2} = \\
a(\alpha_{x^{2}} + \alpha_{y^{2}}){ºa}_{T}^{2}
\end{array}$ (7.19)

The solutions given in Eqs. 7.18 and 7.19 are exact. Although a
completely uniform field is not possible (see Fig. 7.1), these solutions
are still useful. For example, we can differentially "track" a particle
through an inhomogeneous field because we can make a series of small
"steps", treating the field as locally uniform during each step, and
applying the exact solution. This is a useful strategy, for example in
"Monte Carlo" programs which are used to construct a computer model of a
device. (See Appendix K.)

The forms given above are in Cartesian coordinates which are perhaps
more appropriate to fixed target geometries. In collider experiments the
experimenter is more likely to have built detectors around a basic
solenoid in an axial geometry for which cylindrical coordinates are more
appropriate. For example see Fig. 9.6 and Fig. 9.7 for two Si detector
geometries.

The angular definitions are given in Fig. 7.4. A particle emitted from
the origin with a transverse momentum vector labeled by the azimuthal
angle ![](media/image18.emf) describes a circle in the x,y plane and the
trajectory is evaluated at radius r. The azimuthal angle
![](media/image19.emf) labels the position of the particle as it crosses
radius r, whereas the azimuthal angle ![](media/image20.emf) labels the
angle of the momentum vector at that point. (See Appendix F.)

![](media/image21.wmf){width="3.3777777777777778in"
height="3.011111111111111in"}

Fig.7.4: Definitions used in constructing the path of a particle in a
purely axial solenoidal field emitted from the origin at angle φo. The
particle at radius r is located at angle φ\' and its momentum vector at
that point has an azimuthal angle φ\'\'.

We define a momentum transverse to the magnetic field, p~T~, along with
the corresponding transverse radius of curvature,
![](media/image22.emf). (See also Eq. 7.19.)

$\begin{array}{r}
p_{T} = p\text{sin}q \\
a_{T} = \frac{p_{T}}{{qB}_{o}}
\end{array}$ (7.20)

The initial angle of emission at the origin is specified by the initial
momentum components. The exit position, $\varphi^{'}$, has no solution,
if sin \> 1, or if ![](media/image23.emf) is less than half the radius.
That is because the circular path has a small enough radius that it
never reaches r. These are called "loopers" and they continue to rotate
in the magnetic field until they drift out along the z direction as
determined by Eq. 7.17, ![](media/image24.emf). They are visible as
circles in Fig. 6.6 where the drift direction is parallel to the line of
sight and therefore, not apparent.

$\begin{array}{r}
\text{tan}f_{o} = \frac{p_{\text{yo}}}{p_{\text{xo}}} \\
\text{sin}(f^{'} - f_{o}) = - (\frac{r}{2}a_{T}) \\
\text{tan \{}f
\end{array}^{''} = \left\lbrack \frac{{\text{sin}f_{o} - (\frac{r}{a_{T}})\text{cos \{}f}^{'}}{{\text{cos}f_{o} + \left( \frac{r}{a_{T}} \right)\text{sin \{}f}^{'}} \middle| \middle| \right\rbrack,f^{''} - f_{o} = f_{B}$
(7.21)

The small bend angle approximation to the exact expressions is given
below, in the special case ![](media/image25.emf)

![](media/image26.emf) (7.22)

The exit angle ![](media/image27.emf) is basically the azimuthal "bend
angle" which is roughly the distance traveled, \~r, divided by the
transverse radius ![](media/image28.emf). This is in accord with the
Cartesian case. The circular trajectory can be approximated by two
straight lines bent by the angle ![](media/image29.emf) at the center of
the path. Therefore, the label of the exit point is simply the bend
angle divided by a factor of 2. Clearly, these small angle
approximations have a nice transparent geometric interpretation as
illustrated in Fig. 7.5.

![](media/image30.wmf){width="2.5694444444444446in"
height="3.935416666666667in"}

Fig. 7.5: Small bend angle approximation for motion in a uniform field.
The exit angle is φ\'\' \~ r/a~T~ and the exit position is y \~ φ\'\'
r/2 \~ r^2^/2a~T~, for φ~ο~ = 0.

The circular path made by charged particles in a solenoidal field is
illustrated in Fig. 7.6 which is derived from data taken at a Fermilab
collider detector. Note for the various tracks that the sense of the
rotation can be both positive and negative. Note also that the different
radii of curvature reflect particles of different momenta emanating from
the "event vertex" (recall Fig. I.1).

In contrast to the view in the x,y plane, we show the trajectories in a
similar detector located at CERN, the European particle physics
facility, immersed in a dipole field pointing along the y (vertical)
axis, in the (y,z) view, in Fig. 7.7. In this case the particle
trajectories are much straighter because there is no force in the y
direction. We would have precisely straight line trajectories if y were
plotted vs. s. Since the (y,z) view is not exactly congruent to (y,s)
coordinates, the trajectories are only approximately straight.

![](media/image31.png){width="4.378472222222222in"
height="4.316666666666666in"}

Fig. 7.6: Path of charged particles in a solenoidal field, "(x,y) view".
Note that the sense of the radius of curvature is + or - depending on
charge and that the radius of curvature is proportional to the momentum.
(Photo - Fermilab)

## 7.6 Particle Beam and Quadrupole Magnets

As a final topic we consider the simplest aspects of the preparation of
beams of charged particles. Of necessity, we will need to understand the
next most complicated magnetic multipole; the quadrupole magnet. The
simplest elements of a particle beam transport line consist of dipole
magnets, quadrupole magnets and collimators. In the case of
accelerators, we require an accelerating system, such as a radio
frequency, r.f., cavity and, since many traversals of a magnet are made
in a circular machine, correction elements such as sextupoles. The study
of accelerators is well outside the scope of this volume.

![](media/image32.png){width="6.4944444444444445in"
height="3.160416666666667in"}

Fig. 7.7: Path of charged particles in a dipole field, (y,z) view. Note
that the paths are nearly straight. (Photo - CERN Courier)

In traversing a dipole magnet different momenta have different bending
angles. This momentum "dispersion" effect is like the spread of colors
out of a prism. Clearly, using the thick material in collimators to
absorb the unwanted high and low momenta, we can "momentum select" a
beam to pass only particles with momenta within some limits.

A field free (or "drift") space does not act on a particle. In what
follows, we will represent passage through an element of the beam
transport as a matrix. Multiplying the matrices of the individual
elements can then represent complex systems of elements. The matrix
representation, M~D~, of a straight line trajectory over a distance L
is;

![](media/image33.emf) (7.23)

The arc length is s which is \~ the longitudinal distance L. The beam
axis is conventionally taken to be z, and the motion is mostly along z,
ds \~ dz. Motion in the x and y directions transverse to the beam axis
is assumed to be small angle. It is traditional to use x and a "slope"
dx/ds as the 2 variables used to label particle position and direction.
However, that leads to "ugly" dimensionfull matrices. An alternative
formulation using ![](media/image34.emf) and dx/ds is also included in
what follows in the approximation that ![](media/image35.emf) (the
primed matrices). These objects, e.g. ![](media/image36.emf), have
dimensionless matrix elements.

For a dipole magnet of length L, in the approximation that it imparts a
very small bend angle, the transformation is:

![](media/image37.emf) (7.24)

Note that the gradients in the field, Fig. 7.1, would act as thin lenses
at the entrance and exit of the uniform field dipole. We ignore them
here, but the quadrupole formalism we derive below can be directly
applied if desired. Note that the gradient of ![](media/image38.emf) is,
$B^{'}\text{} = \text{}\partial\frac{B_{Z}}{\partial}y\text{}\text{\textasciitilde}\text{}\frac{B_{o}}{\ell}$.
(Eq. 7.5.)

Consider next the case of a quadrupole magnet. The fields are shown in
Fig. 7.8. The field $\overrightarrow{B}$ is zero at the magnet
centerline by design. By construction there is a linear gradient,
![](media/image39.emf).

$\begin{array}{r}
\overrightarrow{Ñ}´\overrightarrow{B} = 0\text{=>}\frac{¶B_{x}}{¶_{y}} = \frac{¶B_{y}}{¶_{x}} = B^{'} \\
B_{y} = B^{'}x \\
B_{x} = B^{'}y
\end{array}$ (7.25)

![](media/image40.wmf){width="3.098611111111111in"
height="3.011111111111111in"}

Fig. 7.8: Field orientation for a quadrupole.

The quadrupole field, outside of the currents, is derivable from a
magnetostatic potential V~B~, Table 3.1, ![](media/image41.emf) =
![](media/image42.emf), using
$\overrightarrow{B} = - {\overset{\rightarrow}{ÑV}}_{B}$. The force
equation for the motion transverse to the direction of motion, assuming
![](media/image43.emf), is;

$\begin{array}{r}
\overrightarrow{F} = d\frac{\overrightarrow{p}}{\text{dt}} = g\text{mv}d\frac{\overrightarrow{v}}{\text{ds}} = q\left( \overrightarrow{v}´\overrightarrow{B} \right) \\
 = {gmv}^{2}d^{2}\frac{\overrightarrow{x}}{\text{ds}^{2}} = \text{pv}d^{2}\frac{\overrightarrow{x}}{\text{ds}^{2}}
\end{array}$ (7.27)

For example, in the x direction, pvd^2^x/ds^2^ \~ -qvB~y~ = -qvB'x. The
equations of transverse motion are those of a simple harmonic oscillator
with "spring constant" ![](media/image44.emf).

$\begin{array}{r}
\frac{d^{2}x}{\text{ds}^{2}} = + \text{kx} \\
\frac{d^{2}y}{\text{ds}^{2}} = - \text{ky} \\
k = q\frac{B^{'}}{p}
\end{array}$ (7.28)

The dimension of ![](media/image45.emf). The equation of motion is one
with solutions having harmonic oscillation in one transverse coordinate
(called focussing or F type) and diverging hyperbolic trajectories in
the other (call defocusing or D type). For the F type we define initial
conditions ![](media/image46.emf)and look at the solution of the
equations of motion. The first integral of Eq. 7.27 in a "thin lens",
where x is \~ constant, is ![](media/image47.emf). Thus a thin lens
quadrupole imparts an angular impulse which depends on the deviation
from the magnet centerline, ![](media/image48.emf). The "focal length",
f, is then defined to be the longitudinal distance over which an
incident parallel, ![](media/image49.emf), ray, ![](media/image50.emf)
is brought to a focus, ![](media/image51.emf) or

![](media/image52.emf) (7.28)

The geometric proof of this relationship is shown in Fig. 7.9a. All
incident parallel rays are brought to the same point on the z axis, the
focus. Hence the quadrupole acts as a charged particle lens. For example
a 3 m long quadrupole magnet with a 5 kG gradient over a 5 cm aperture
has a 11.2 m focal length for a 100 GeV beam.

![](media/image53.wmf)

Fig. 7.9: a) Illustration of the meaning of focal length. b) A doublet
lens system capturing a divergent beam and making it parallel, "point to
parallel".

The exact solution to the equations of motion comes from integrating Eq.
7.27 twice subject to initial conditions ![](media/image54.emf). For the
focusing case

$\begin{array}{r}
{\begin{pmatrix}
x \\
x^{'}
\end{pmatrix}\text{} = \text{}\left( \begin{matrix}
\text{cos}\varphi_{Q} \\
 - \sqrt{k}\text{sin}\varphi_{Q}\text{}
\end{matrix}\text{}\begin{matrix}
\text{sin}\frac{\varphi_{Q}}{\sqrt{k}} \\
\text{cos}\varphi_{Q}
\end{matrix} \right)\begin{pmatrix}
x_{o} \\
x_{o}^{'}
\end{pmatrix},\text{ \{}M}_{Q}^{'}\text{} = \text{}\begin{pmatrix}
\text{cos}\varphi_{Q} & \text{sin}\varphi_{Q} \\
 - \text{sin}\varphi_{Q} & \text{cos}\varphi_{Q}
\end{pmatrix}
\end{array}\text{} = \ M_{Q}\ \begin{pmatrix}
x_{o} \\
x_{o}^{'}
\end{pmatrix}\text{}\text{}\varphi_{Q} = \ \sqrt{k}\text{}L$
(7.29)

in the y or "defocusing" coordinate, ![](media/image55.emf) in both
![](media/image56.emf) and ![](media/image57.emf). Note that, in our
previous numerical example, φ~Q~ \~ 30^o^ which is not a very thin lens.

In the thin lens case, ![](media/image58.emf) \<\< 1, ignoring the
change in x in drifting through the short quadrupole length, the matrix
![](media/image59.emf) is

![](media/image60.emf) (7.30)

In Eq. 7.30 the ± sign refers to F and D type. The physical content of
Eq. 7.30 is that the position does not change in traversing a thin lens
and the angle ![](media/image61.emf) receives an impulse
![](media/image62.emf)\
(- focussing, + defocusing) proportional to the offset
![](media/image63.emf) from the beam axis.

## The Quadrupole Doublet

Note that in geometric optics we can have simultaneous focusing in both
transverse coordinates in a single lens. In particle beams, the
fundamental constraint $\overrightarrow{Ñ}´\overrightarrow{B} = 0$
implies focusing (F) in one coordinate, and defocusing (D) in the other.
Therefore, the simplest "element" exhibiting overall beam focusing in
both transverse dimensions is not the single lens, but the quadrupole
doublet which we treat here in the simplest possible (thin lens)
approximation. Note that this approximation is often useful as a first
iteration to a complete solution, for example as "starting values" to a
complicated nonlinear computer code.

As the simplest application consider the quadrupole "doublet" of equal
strength magnets arranged as in Fig. 7.9b. We can find the conditions
needed to capture a beam diverging from a point target and make it
parallel. The full "transport matrix" is the product of the matrices
appropriate to traversing each beam element, ![](media/image64.emf)
![](media/image65.emf) The parallel condition (for vanishing target
size, x~0~ = 0, y~0~ = 0) is, ![](media/image66.emf)
![](media/image67.emf) or ![](media/image68.emf) where M refers to
![](media/image69.emf). The final beam size is ![](media/image70.emf)
The algebra in the simplifying thin lens case, which we omit as an
exercise for the reader, has a solution where the quadrupole focal
length must be;

![](media/image71.emf) (7.31)

and the final beam size is, as can be visualized in Fig. 7.9b;

![](media/image72.wmf) (7.32)

This solution is found by requiring both an x and y parallel beam. The
matrix element M~22~ must vanish simultaneously for both x and y motion.
Those constraints lead to a quadratic equation for f. Note that, as Fig.
7.9b indicates, a cylindrical limiting magnet aperture, see Fig. 7.11,
means an asymmetric angular acceptance for the doublet. That fact is
also clear from Eq. 7.32 where, if ![](media/image73.emf), then
![](media/image74.emf). If the first element is y focussing, then the
angular acceptance is the largest for y.

As a second example, the optical envelope for a thick lens doublet is
shown in "point to point" focus conditions, M~12~ = 0, in Fig. 7.10.
Clearly, since the first quadrupole is y defocusing (x focusing) the
capture angles for a circular quadrupole aperture is asymmetric
![](media/image75.emf). Complete computer optimization codes exist for
both beam lines and accelerators. However, they require "starting
values" being complex nonlinear systems without analytic solutions.
Thus, a good physical intuition is called for even in designing such
systems.

![](media/image76.png){width="3.2916666666666665in"
height="2.013888888888889in"}

Fig. 7.10: Plot of beam envelope for a thick lens quadrupole doublet in
a "point to point" optical configuration.

The design of secondary beams is an art unto itself, as is the design of
accelerators. We have shown here only the simplest cases in order to
give a flavor of the physics. Magnets should be thought of as detectors
in the concrete sense that we use them to prepare beams to do
experiments and to exert forces on particles so that the use of position
measuring devices (Section 8 and 9) allow us to determine the particle
charge and momentum. A photo of a beam line dipole and a quadrupole used
in the Fermilab accelerator complex is shown in Fig. 7.11. These magnets
are appropriate to prepared beams, with vacuum beam pipes of \~ 10 cm
diameter.

![](media/image77.png){width="5.071527777777778in" height="6.45625in"}

Fig. 7.11: Magnets appropriate for high-energy particle beams. a) A
dipole. Note the 2 coils surrounded by the steel "yoke". b) A
quadrupole. Note the 4 coils surrounding the cylindrical vacuum pipe.
Outside the coil is a steel yoke and water cooling plumbing. Photos -
Fermilab)

## References

Section 7: Magnetic Fields

\[1\] Superconducting Magnet Technology for Accelerators, R. Palmer,
A.F. Tollestrup, Annual Review of Nuclear and Particle Science,
[34]{.underline} 247 (1984).

\[2\] R.L. Gluckstern, Nucl. Inst. Meth., [24]{.underline} 381 (1963).

\[3\] R. Yamada, et al., Nucl. Inst. Meth., [138]{.underline} 567
(1976).

\[4\] An Introduction to the Physics of Particle Accelerators, D.A.
Edwards, M.J. Syphers, FNAL FN (1988).

\[5\] [The Physics of Charged-Particle Beams]{.underline}, J.D. Lawson,
Clarendon Press (1988).

Table of Contents

II.C. Position and Momentum [1](#ii.c.-position-and-momentum)

7\. Magnetic Fields [2](#magnetic-fields)

7.1 Solenoidal Fields [2](#solenoidal-fields)

7.2 Dipole Fields - Fringe Fields [3](#dipole-fields---fringe-fields)

7.3 Particle Motion in a Uniform Field
[5](#particle-motion-in-a-uniform-field)

7.4 Momentum Measurement and Error [8](#momentum-measurement-and-error)

7.5 Exact Solutions - Cartesian and Cylindrical Coordinates
[11](#exact-solutions---cartesian-and-cylindrical-coordinates)

7.6 Particle Beam and Quadrupole Magnets
[15](#particle-beam-and-quadrupole-magnets)

7.7 The Quadrupole Doublet [20](#the-quadrupole-doublet)

References [23](#references)