Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics
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48 Nuclear Medicine Physics
2.2.4.3 Dynamic Focusing
In addition to energy variations, the particles passing through the gap
between the Ds experience a variable electrical field during the transit time.
The accelerator field time variation is responsible for dynamic focusing. In
this portion of the radio frequency cycle, when the field decreases its ampli-
tude the converging force at the beginning of the gap has a greater amplitude
than the diverging force at the end, resulting in an overall converging effect.
During the other acceleration half-cycle quadrant, the effect is inverted and
the final result is diverging.
The amplitude of the dynamic focusing effect decreases as the particle
energy (and therefore the orbital radius) increases, as the time the particle
needs to go through the gap decreases and, as a consequence, the electrical
fieldvariation amplitudedecreases.The analyticalexpressionfor theresulting
deflection [69] using the same notation is
(
Δχ
)
dynamic
≈−
qV
0
sen ϕ
E
c
r
z. (2.54)
It is, therefore desirable for the ions to go through the acceleration gap
when the potential difference between electrodes reaches maximum ampli-
tude and starts to decrease. However, this advantage cannot be realized in
revolutions corresponding to the inner cyclotron zone, where the need to
ensure the resonance condition with the best possible benefit in terms of mag-
netic focusing leads to a magnetic field value that is slightly larger than the
resonance value. In the first stages of acceleration, this dynamic unfocusing
is, therefore, responsible for the loss (by collision with the D surface) of many
ions with considerable axial components.
2.2.4.4 Combining the Focusing Effects
Having considered the two electric focusing aspects separately, it is now
important to make a combined assessment considering a number of accel-
eration gaps. This study can be achieved by analogy with an electrostatic lens
sequence combination [64,69,70].
Examining the expressions, it can be seen that the static focusing effect
only prevails over the dynamic—resulting in efficient focusing—in the ini-
tial passages through the acceleration gap and over a restricted phase range
corresponding to very small phases. The static focusing effect resulting from
a change in speed over the acceleration gap quickly decreases with energy.
Therefore, the electric field variation effect—dynamic focusing—tends to be
stronger in most revolutions. For that reason, the equivalent lens focal dis-
tance will be relatively short for low-energy ions in the appropriate phase but
will become very long for high-energy ions. Taking into account the expres-
sion that relates kinetic energy to the cyclotron orbit square of the radius and
considering that the ion is experiencing the average potential V
m
effect, for
Cyclotron and Radionuclide Production 49
each one of the k times it passes through the acceleration gap to gain this
energy:
E
c
= kqV
m
=
1
2
q
2
m
B
2
r
2
⇔ r =
1
B
2 k
m
q
V
m
. (2.55)
When observing the spiral path transferred to a plane, this informative
expression helps us understand that the gap between successive lenses (with
an increasing focal distance) is proportional to the square root of an integer
series. We can formally express axial motion under the electrical field influ-
ence between the electrodes mathematically as a differential equation that is
a function of the independent variable k, the number of times the ion goes
through the acceleration gap [69]:
d
2
z
dk
2
+
πqV
0
sen ϕ
E
c
z = 0. (2.56)
Bearing in mind that the axial coordinates and their variation rate (corre-
sponding to the ion deflection) slowly change, a possible solution is [68,69],
for positive ϕ
i
:
z = A
E
c
sen ϕ
1/4
sen
⎛
⎝
√
π
k
0
qV
0
sen ϕ
E
c
dk +δ
⎞
⎠
. (2.57)
This expression, in which A and δ are integral constants, shows that even if
there is electric focusing, the axial oscillation amplitude increases in propor-
tion to the 4th root of the ratio between the kinetic energy and the phase sine
(therefore, very slowly). It also grows with k
1/4
when the phase is constant.
In the initial phase of acceleration, it is important to emphasize this electric
focusing resulting effect, otherwise, the axial coordinate would change a lot
faster, in line with the kinetic energy or with k.
In negative phases, the motion is no longer oscillatory but exponential,
either increasing or decreasing. Regardless of the initial conditions, exponen-
tial growth will prevail, resulting in an increase in the absolute value of the
ion axial coordinate and, therefore, a defocus situation [69].
Figure 2.13 shows the profile of some ion paths along a D taking electric
focusing only into account.
As the radius increases, focusing becomes more and more important, due to
the nonuniformity of the magnetic field. This is why it cannot be disregarded
and, in fact, becomes predominant. The final combined result of magnetic
and electric focusing is efficient ion beam axial focusing for the ions that are
not lost in the first revolutions where the electric effect prevails. A combined
consideration of the various expressions for axial deflections resulting from
the aforementioned electric and magnetic focusing effects leads to the predic-
tion (experimentally confirmed [64]) of a beam axial profile (transferred to a
50 Nuclear Medicine Physics
z
r
0
FIGURE 2.13
Bidimensional diagram of a D section of some ion paths under the electric (de)focusing effect.
plane) resembling an envelope, whose cross thickness increases until it is lim-
ited by the internal D opening and afterward decreases in amplitude with the
increase in ion energy (and increasing orbital radius), due to the decreasing
magnetic field (Figure 2.14).
2.2.5 Phase Relations and Maximum Energy
2.2.5.1 Path and Phase in the First Revolutions
Eventhough ionscan beemitted fromthesourceduring theentireelectric field
half cycle, corresponding to an accelerating force between the ion source and
the D, the resulting wide spatial distribution of ions quickly narrows, mak-
ing all the ions cross the acceleration gap in approximate phases distributed
around the electric field peak.
Exact ion path analysis is complex, because it depends to a great extent on
detailed knowledge of the electric and magnetic fields, especially the electric
field between the ion source and the Ds. However, a good approximation can
be achieved, illustrating the conclusion of the previous paragraph, by study-
ingthe initialion path—the first revolutions—onthe basisof somesimplifying
assumptions.
z
r
0
FIGURE 2.14
Bidimensional diagram of a D section of cyclotron ion vertical motion. The lines are the ion paths,
considering only the electric field effects. The dotted lines represent the combined effects of the
magnetic and electric fields after the magnetic field has become predominant.
Cyclotron and Radionuclide Production 51
First, a point-like ion source and a uniform electric field between the Ds can
be assumed. A gap between the Ds big enough to prevent the ions from enter-
ing during the first revolutions can also be considered (which is completely
true as the gap is usually several centimeters long to allow for ion source posi-
tioning and the insertion of testing probes and beam diagnostics). Finally, it
can be assumed that the ions do not suffer any radial or axial deflection in
relation to the ideal orbit, thus preventing the radial and axial focusing effects
from masking the phase relations to be explained.
Using a coordinate system with its origin in the ion source and with the x
and z-axes lying along the electrical field and magnetic field, respectively (the
open D surfaces are parallel to the y-axes), the motion in the xy plane can be
described by the equations:
m
d
2
x
dt
2
= qE +qB
dy
dt
, (2.58)
m
d
2
y
dt
2
=−qB
dx
dt
, (2.59)
in which q and m are the accelerating ion and mass charge, respectively, and
B is the magnetic field modulus (assumed to be uniform for the purposes
of simplification and because it is the cyclotron central zone, which is not
usually modulated to secure axial focusing); whereas the electrical field ε
with the amplitude ε
0
and angular frequency ω is given by
ε = ε
0
cos
(
ω(t +t
0
)
)
. (2.60)
The possibility of the ions beginning their path in different accelerator cycle
phases of the electrical field is considered by the introduction of t
0
.
By introducing a (unique) magnetic field axial component value corre-
sponding to the fundamental condition of the cyclotron resonance, the motion
equations can be written as
d
2
x
dt
2
=
q
m
ε
0
cos
(
ω(t +t
0
)
)
+ω
dy
dt
, (2.61)
d
2
y
dt
2
=−ω
dx
dt
. (2.62)
Considering that the ions are resting at source in the initial moment, the
motion equation solutions are
x =
qε
0
2mω
2
[
sen(ωt
0
) sen(ωt) − t sen
(
ω(t +t
0
)
)
]
, (2.63)
y =
qε
0
2mω
2
[
cos(ωt
0
) sen(ωt) −t cos
(
ω(t +t
0
)
)
−2 sen(ωt
0
)
(
1 −cos(ωt)
)
]
.
(2.64)
52 Nuclear Medicine Physics
From these expressions, the initial ion paths can be illustrated in dia-
grams that show the different phase values at the moment of emission, as
in Figure 2.15. The position of the various ions (emitted in different phases)
for the same ωt value is represented by the series of white points in the center
connected by dotted lines.
The ion depicted by the corresponding curve ϕ = 0 leaves the source when
the electric field reaches maximum value. It is in perfect resonance with the
electric field, crossing the y-axes to ω · t = π,2π,3π,4π, etc. An initial phase
equal to ϕ =−π/2 corresponds to an ion emitted from the source at the exact
moment that the field accelerates in the positive x-axis direction, so that it is
under maximum acceleration in this direction, describing a larger orbit than
the one emitted in phase zero. Figure 2.15 also illustrates several intermediate
phases, and it is significant that the increasingly positive phases correspond
to ions emitted when the field is progressively lower on emission and nearer
to the point of inverting its direction, thus initially describing only a small arc.
The limit phase for acceleration in this x-axis direction is ϕ =+π/2. For
a hypothetic point-like source that allows for emission in all directions
ωt = 3π
ωt = 2π
ωt = π
ωt = 3π/2
ωt = 5π
/2
–10
10 (cm)
5
5
–5 –10 10
–π/2
–π/3
–π/6
π/6
π/3
φ = 0
–5
FIGURE 2.15
Initial ion paths in a cyclotron, numerically calculated, for a sinusoidal electrical field that is uni-
form along the xx axis (maximum amplitude 25 kV/cm, 20 MHz frequency) and a 13.05 kGauss
axial magnetic field. Revolutions start from the center, in different phases of the accelerating
electrical field.
Cyclotron and Radionuclide Production 53
from this phase value, the ions will be accelerated in the opposite direc-
tion and the respective path will be symmetrical to the one corresponding
to ϕ =−π/2.
After three-fourths of the cycle (assuming it began the moment the ions
corresponding to ϕ =−π/2 were emitted), all the ions are near the median
line between the Ds (corresponding to x = 0), almost independently of the
field value at the moment of emission. This “gathering” is repeated at each
half cycle with an asymptotic approximation to the median line between the
Ds, corresponding to the position of the emitted ions in phase zero. As accel-
eration increases, the motion of the ions approaches what it would be if they
had all been emitted at the moment when the accelerating electric field had
reached its maximum value, in what can be termed the phase focusing around
the zero phase.
Resonant acceleration would take place even if the electric field did not
reach zero inside the Ds. However, in practice, the D surfaces are very close to
each other, and the electric fields cannot penetrate very deeply inside the Ds.
Therefore, the larger orbit shapes are controlled only by the magnetic field
(which is closer to the arc of a circle than a spiral) (Figure 2.16).
ωt = 3π
ωt = 2π
ωt = π
ωt = 3π/2
ωt = 5π/2
–10
10 (cm)
5
5
–5 –10 10
–π/2
–π/3
–π/6
π/6
π/3
φ = 0
x
D
–x
D
–5
FIGURE 2.16
First revolutions cyclotron ion path evaluated under the same conditions as Figure 2.15 but
assuming the electric field is zeroin the planes x =±x
D
. The spiral orbits turn into circlesegments
outside x
D
. Phase focusing and spatial concentration are not as complete as in the uniform field
situation but are still observable.
54 Nuclear Medicine Physics
2.2.6 Maximum Kinetic Energy: The Relativistic Limit
The fundamental equation for cyclotron resonance, as previously expressed,
and the resulting basic working principle are based on equal applied electric
fieldfrequency and ion revolutionfrequency, resultingin a directproportional
relationship between this frequency value and the magnetic field (both con-
sidered as constants). In practice, the electrical field frequency is kept constant
but this is not the case with ion revolution frequency. When the accelerated
ion kinetic energy surpasses its rest mass by some percentage units, the rel-
ativistic effects can no longer be disregarded. Among these, the increasing
mass directly interferes with the validity of the resonance equation.
A transit time increase inside the Ds corresponding to this increase implies
a phase change for the ions entering the acceleration gap, resulting in
progressive delay in relation to the existing electrical field, until they are
eventually slowed down by this field.
Morethan loss by residualmolecule dispersion inside the cyclotronvacuum
chamber, the advent of relativistic effects is an important factor supporting
the option of a high accelerating electrical field amplitude. Obviously, if the
resulting phase jump from the revolution and resonance frequencies were
different, there would be no consequences if the applied potential to the Ds
correspondedtothe maximum energyreachedafterone acceleration only. The
smaller the maximum accelerating electrical field value, the quicker the effect
of the difference will become evident. In a quantitative analysis, considering a
uniformand constant magneticfield with modulusB, this becomesclear when
we relate the phase change to the kinetic energy increase for each revolution.
Let us consider the electrical field frequency f applied to the Ds (corre-
spondingto the cyclotronresonancecondition expressed)and the accelerating
(charge q) ion revolution frequency f
r
given by
f =
qB
m
0
, (2.65)
f
r
=
qB
m
, (2.66)
wherem
0
and m are the ion restmass and the ion relativisticmass, respectively.
The phase variation ϕ for each revolution N will be given by [70]:
dϕ
dN
= 2π
f − f
r
f
r
= 2π
m
m
0
−1
. (2.67)
From the relativistic equivalence between mass and energy:
m
m
0
=
E
E
0
=
E
0
+E
c
E
0
= 1 +
E
c
E
0
, (2.68)
Cyclotron and Radionuclide Production 55
whereE
c
and E
0
arethe ion kinetic energy and the ion rest energy, respectively,
defined as relativistic. The previous expressions lead to
dϕ
dN
= 2π
E
c
E
0
. (2.69)
On the other hand, the kinetic energy increase in each revolution can be
related to the maximum applied potential to the Ds by
dE
c
dN
= 2qV
M
sen ϕ. (2.70)
The factor 2 resultsfrom the fact that in each revolutionthe ion goes through
the accelerating gap between the Ds twice. From this expression:
dϕ
dN
= 2qV
M
sen ϕ
dϕ
dE
c
=−2qV
M
d(cos ϕ)
dE
c
. (2.71)
Comparison of this expression of the phase variation in each revolution
with the previous one leads to
d(cos ϕ)
dE
c
=−
π
qV
M
E
c
E
0
. (2.72)
where ϕ
0
is the ion initial phase. Taking into account the fact that the kinetic
energy at the moment of emission can be considered zero, integrating this
expression results in
cos ϕ
0
−cos ϕ =
π
qV
M
E
0
(E
c
)
2
2
. (2.73)
Even though this equation emerges after some simplifying assumptions, it
enables us to conclude that for a certain phase change in which the ions find
the electrical field when they arrive at the acceleration gap, the greater the
potential amplitude applied to the Ds, the greater the kinetic energy obtained
will be. As previously noted, in the first revolutions (even before the rel-
ativistic effects start to become relevant), independently of the phase at the
moment of emission, all the ions go through the acceleration gap with a phase
value very similar to the corresponding potential peak (ϕ
0
= 0). The greatest
acceptable total phase deflection must be one that permits the ions to arrive
at the gap between the Ds and still find the accelerating half cycle, that is, the
phase limit ϕ = π/2. Substitution of these values would enable us to discover
the maximum kinetic energy in a cyclotron, with the specifications that are
the basis of the simplifying assumptions leading to this expression, such as
uniform and constant magnetic field considerations.
56 Nuclear Medicine Physics
Technical problems relating to the production of high electric potential val-
ues and the related insulation impose limitations on the use of this method
of obtaining the desired kinetic energy in a cyclotron, based on an increase in
the maximum electrical field applied to the Ds in such a way that the number
of revolutions N is small enough to verify the condition:
N
dϕ
dN
≤
π
2
. (2.74)
A different (and complementary) solution is to increase the angular range
of the phase variation, which can be achieved if the magnetic field value is
such that in the first revolutions the ion revolution period is less than that
of the accelerating electrical field frequency. The phase will move toward a
maximum limit of −π/2. Meanwhile, the relativistic effect of mass increase
willgradually be felt,making the phase jump smaller each timein the negative
direction, until finally (when the electrical field and revolution frequency
values are the same, in a transient resonance condition only reached when
the ions are at a considerable distance from the cyclotron center) it becomes
positive, that is, the ions arrive at the acceleration gap with a time delay.
It still remains a phase range below the π/2 limit in which acceleration is
possible.
This procedure permits the phase range to be increased by a factor of 3
and is mathematically reflected by the appearance of a negative term in
the expression that quantifies the phase variation in terms of number of
revolutions.
On the other hand, in discussing magnetic focusing, the decrease in revo-
lution frequency as a consequence of the small magnetic field radial decrease
necessary for axial focusing has already been described (and the implications
of attempting a practical solution discussed). This (decreasing) variation in
revolution frequency in relation to the cyclotron orbit radius, which empha-
sizes the relativistic effect of increasing mass, should also be considered in a
quantitative analysis of phase variation.
The combined consideration of these factors results in the expression [71]:
dϕ
dN
= 2π
B
c
B
B
0
B
c
−
B
B
c
+
B
0
E
c
B
c
E
0
, (2.75)
where the kinetic energy E
c
and the magnetic field B are functions of the
cyclotron path radius. The magnetic field value for radiuses equal to zero, that
is, the cyclotron center, is represented by B
c
. It is a constant value similar to B
0
,
which is the magnetic field value corresponding to the fundamental cyclotron
resonance condition initially defined, that is, for the oscillation frequency of
the electrical field applied to the Ds and the ion rest mass (corresponding to
energy E
0
). Despite the implied complexity of the integration of this expres-
sion, it is possible to reach a compromise between a magnetic field radial
profile (needed for axial focusing), an electric potential applied to the Ds that
Cyclotron and Radionuclide Production 57
is technically feasible, and maximum accelerating ion kinetic energy, with
regard to the kinetic energy variation in each revolution and along the radius
[70,71]. This limit explains why the cyclotron is not used to accelerating light
particles such as electrons. For protons and deuterons it amounts, in practice,
to around tens of MeVs. The limitation of the fundamental cyclotron princi-
ple to nonrelativistic energies is due to the assumption of a constant magnetic
field and a constant applied electrical field oscillating frequency. Technologi-
cally, it is possible to implement the variation in both frequency and magnetic
field adequately, enabling acceleration to take place with relativistic energies.
This implementation is the basis of the particle accelerators that appeared
after the first cyclotrons, the synchrocyclotron and the isochronous cyclotron.
2.2.7 The Synchrocyclotron
2.2.7.1 Working Principle
The relativistic mass increase limit imposed on maximum cyclotron kinetic
energycan be overcomewhen the electrical field oscillating frequency applied
to the electrodes decreases during the acceleration process, after a decrease
in revolution frequency as a result of the relativistic mass increase. This is the
basis of the synchrocyclotron working mode.
In this modulated frequency cyclotron, the yield in terms of the number
of accelerated particles per time unit is much lower—by about two orders
of magnitude—than the conventional cyclotron, because the particles are not
distributed continuously and almost uniformly along the path. Instead, they
form groups or clusters, which are synchronously accelerated from the center
to the outside with the frequency variation. The process can only restart for a
new particle cluster once a cluster has been accelerated to a high energy level.
2.2.7.2 Phase Stability
Phase stability [72,73] is an important feature of the synchrocyclotron and is
crucialto the success of the acceleration process.It prevents the loss of ions in a
group being accelerated that arrive at the acceleration gap at slightly different
times; and it instead enables them to group themselves in the center of the
cluster, in a kind of longitudinal focusing with space and phase features.
Figure 2.17 illustrates phase stability in a synchrocyclotron. An ion in the
center of the accelerating cluster that reaches the accelerating gap at the
moment the accelerator electric field moves to zero value (point a in the fig-
ure) will always circle in this equilibrium orbit, which is termed synchronous.
Another ion from the same cluster reaching the accelerating gap slightly
before this (illustrated as point b in the figure) will find an accelerator field
that will increase its orbit radius and energy, consequently decreasing its rev-
olution frequency. Thus, in the next passage through the accelerating gap, it
will be closer to the cluster center. In a similar manner, the ions reaching the