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58 Nuclear Medicine Physics

Accelerating potential

b′

a′

c′

FIGURE 2.17

Phase stability in a synchrocyclotron. The points a and a



correspond to successive passages

through the ion accelerating gap in the same direction in a synchronous orbit. Ions reaching the

acceleration gap slightly later (b) will be accelerated, bringing their phase in the next equivalent

acceleration gap passage (b



) closer to the synchronous orbit. Asimilar effect occurs with the ions

arriving at the synchronous orbit in advance (c and c



). The point d corresponds to instability, as

the delayed or in advance ions in the vicinity of this point tend to increase.

accelerating gap later than the cluster center (corresponding to point c in the

example) will be decelerated, and the corresponding mass decrease will lead

to an increase in revolution frequency, bringing them closer to the cluster

center for the next passage. As a consequence, the ions in the same cluster

oscillate around a synchronous orbit. In each passage through the accelerat-

ing gap, they are not only accelerated but also grouped together by this phase

stability effect.

The accelerating process and phase stability restrict the accelerating ion

phase in relation to the accelerating magnetic ﬁeld in such a way that, unlike

the conventional cyclotron, it is not absolutely necessary to use high acceler-

ating electrical ﬁeld values. Therefore, with the aim of reducing high electric

power production and insulation problems, the individual accelerations (i.e.,

the accelerating electrical ﬁeld amplitude) are typically lower than in a

conventionalcyclotron,resultingin a much higher number of revolutions [74].

2.2.8 The Isochronous Cyclotron

2.2.8.1 Thomas Focusing and Working Principle

An alternative method of overcoming the relativistic mass increase limit

imposed on maximum kinetic energy, which does not involve modulation

of the accelerating electrical ﬁeld frequency (and the consequent decrease

in beam intensity), can be inferred from the cyclotron resonance condi-

tion: Increasing the cyclotron magnetic ﬁeld at the rate determined by the

relativistic mass increase enables the revolution period to be maintained

(isochronism).

This magnetic ﬁeld radial increase violates the stability condition result-

ing from the Kerst–Serber equations or, in other words, results in a loss of

Cyclotron and Radionuclide Production 59

the beam magnetic focusing effect, making the acceleration process impossi-

ble. The proposed solution by J. H. Thomas [75] is magnetic ﬁeld azimuthal

periodic modulation, which creates an axial focusing mechanism even in the

presence of an increasing radial magnetic ﬁeld. In addition, the magnetic ﬁeld

radial increase corresponds to a negative ﬁeld index, thus increasing radial

focusing, as predicted by the relevant Kerst–Serber equation. The result is the

isochronous cyclotron, in which the magnetic ﬁeld is, therefore, highly nonho-

mogeneous, both radially and azimuthally (which is why it is also known as

the azimuthally varying ﬁeld (AVF) cyclotron).

Thomas focusing can be understood with the aid of Figure 2.18, which illus-

trates the azimuthal cut (for a constant radius) perpendicular to the median

plane between the cyclotronmagnetic ﬁeld poles, transferredto a paper plane.

The distance between the magnetic poles has two alternating values, a larger

one (corresponding to the top region) and a smaller one (the valley region)

along the azimuth around the cyclotron center. For each top-valley pair,

the azimuthal periodicity element is termed the sector, thus explaining the

alternative name for this cyclotron: sector focused.

The bend in the magnetic ﬁeld lines in the transition zone between a top

and a valley implies that there is a magnetic ﬁeld azimuthal component.

Figure 2.18 depicts the magnetic ﬁeld lines for this geometry, allowing for

qualitative variation analysis of this component (B

) along the same ﬁeld

line and between different ﬁeld lines. In both cases, its respective direction

is inverted when crossing the medium cyclotron plan and the top and valley

axial geometric center, where it is zero.

Even inside a D, far away from the electrical ﬁeld interaction, the equi-

librium orbit of an ion in this magnetic ﬁeld is not a perfect circle arc. The

magnetic ﬁeld axial component and ion velocity vector component along the

cyclotron medium plan produce a Lorentz force equivalent to a centripetal

force. However, variations in the magnetic ﬁeld axial component mean that

the values are different in tops and valleys, corresponding to variations in

the bending radius, which will be bigger when this component is smaller and

vice versa, as illustrated in Figure 2.19. The oscillating orbit around a circle

FIGURE 2.18

Isochronous cyclotron azimuthal cut transferred to a paper plane showing the magnetic ﬁeld

lines (and an example of the ﬁeld axial and azimuthal component in a space point) resulting

from the tops and valleys.

60 Nuclear Medicine Physics

Valle

Valley

To p

Valley

To p

FIGURE 2.19

Diagram showing bending radius, assuming a completely closed hypothetical orbit in an

isochronous cyclotron with a four-sector proﬁle, as depicted in Figure 2.18.

and the ion velocity vector present a radial component alternately pointing

toward the cyclotron center or the outside.

A vertical Lorentz force results from the magnetic ﬁeld azimuthal compo-

nent and the velocity vector radial component, which always point toward

the cyclotron median plane, providing axial focusing.

This behavior of the magnetic ﬁeld azimuthal component, ion velocity

vector, and, therefore, the Thomas focusing occurs not only in relation to

azimuthal modulation in this ﬁeld but also for any other geometry with the

same periodicity and symmetry type, such as sinusoidal modulation. In any

example, it is greater, as the relative ﬁeld variation between top and val-

ley increases (directly associated with the bending of ﬁeld lines) and can be

conﬁgured to ensure an overall focusing effect even in an increasing radial

ﬁeld.

2.2.9 Focusing Reinforcement Using the Alternating Gradient Principle

Thomas axial focusing can be reinforced if the tops and valleys do not occur

in the same azimuthal position in different radii but in a spiral-shaped dis-

tribution from the center (Figure 2.20). Thus, in each top-valley transition,

the magnetic ﬁeld will have a radial component whose gradient in succes-

sive transitions is alternately centripetal or centrifugal. Crossing these spatial

regions where the magnetic ﬁeld radial components are felt, the ion, given

Cyclotron and Radionuclide Production 61

Valle

Valley

To p

FIGURE 2.20

Three isochronous cyclotron sectors viewed from above, where the borders between tops and

valleys are spiral shaped according to the alternating gradient principle. A closed hypothetical

path is also represented.

its azimuthal speed component, will be under the axial Lorentz force in a

direction that is alternately in and out of focus. The resulting general effect is

efﬁcient focusing, as established in the alternating gradient principle [76]. This

principle is simply the application to the charged particle beam dynamics

of the geometrical optics principle, which establishes that the focal distance

of a converging and diverging lens group with an equal absolute focal dis-

tance value is always positive (i.e., corresponds to a general focusing effect),

regardless of the order of the lens pair (Figure 2.21). In addition to being pro-

portional to the difference between the ﬁeld values in tops and valleys, the

amplitude of this focusing effect depends on the limit line proﬁle between

them, increasing in proportion to the increase in the angle that the tangent to

the limit line makes with the radial direction, as can be seen from the relevant

geometry.

From this analysis, it can also be concluded that the ion path in the region

where it is under an out-of-focus force is more nearly perpendicular to the

transition boundary than when it crosses a focusing region, so that it remains

for a comparatively longer time in this region under a focusing force. This

fact is reﬂected in a second overall axial focusing effect associated with the

spiral shape of the tops and valleys [77], reinforcing the alternating gradient

principle (which would be felt even if the time taken to cross the in-focus and

out-of-focus regions were the same).

Due to these effects, an isochronous cyclotron with spiral tops and valleys—

also called an FFAG (ﬁxed-ﬁeld

∗

alternating-gradient) cyclotron—can easily

∗

With regard to magnetic ﬁeld time invariance.

62 Nuclear Medicine Physics

conv

div

FIGURE 2.21

Illustration of the alternating gradient principle using an optical analogy: the overall effect of the

path through a converging lens and a diverging lens of the same distance (in modulus) is always

converging, no matter which lens is ﬁrst. The explanation relies on the proportionality between

the deviation angle and the distance from the radius to the optical axis.

generate axial focusing forces that are much greater than those usually

obtained through a magnetic ﬁeld radial decrease in a conventional cyclotron.

These forces are strong enough to create a pronounced axial focusing effect,

even with a radial increase in the magnetic ﬁeld in order to maintain the reso-

nance condition for the relativistic energies of the accelerating ion beams. For

this reason, the axial focusing characteristic of the FFAG cyclotron is known

in jargon terms as strong focusing, as opposed to the weak focusing obtained in

the conventional cyclotron.

2.2.10 Aspects of Quantitative Characterization

An analogous but symmetrical magnitude to the ﬁeld index is usually used

in the quantitative characterization of the magnetic ﬁeld in an isochronous

cyclotron, known as the average ﬁeld index,

n, given by

n =

, (2.76)

where r is the radial distance to the isochronous cyclotron center and

repre-

sents the average value of the magnetic ﬁeld axial coordinate (B

) for a course

along the path in the median plane:



ds, (2.77)

where s is the coordinate corresponding to a course along the path with total

length S.

Cyclotron and Radionuclide Production 63

The resonance condition implies that the revolution angular frequency ω

is equal to the applied electrical ﬁeld constant angular frequency ω. For an

ion with charge q, mass m, and rest mass m

, the latter is given by

ω =

, (2.78)

where B

is the axial magnetic ﬁeld in the cyclotron center in which the radial

coordinate is zero. The bend radius variation constant that is present in the

accelerating ion path in this kind of cyclotron leads to the deﬁnition of ω

average terms, with the implicit concept of equivalent radius equal to the arc

described, divided by the covered azimuthal angle, which can be approached

by the radial coordinate relating to the cyclotron center, except in the ﬁrst

revolutions. Thus, if m is the relativistic mass:

. (2.79)

The resonance condition is maintained for each radial coordinate value

by the average axial magnetic ﬁeld radial modulation in such a way that the

revolutionfrequency is kept equal to theaccelerating electricalﬁeld frequency,

implying that

= B

γ, (2.80)

where γ characterizes the relativistic mass increase:

γ =



1 −



−1/2

. (2.81)

From the differentiation of the expression that relates the magnetic ﬁeld

axial coordinate average value to the value it assumes in the cyclotron cen-

ter, the average ﬁeld index needed to maintain the isochronous resonance

condition can be written in terms of γ:

n =

dγ

. (2.82)

In FFAG cyclotrons, magnetic ﬁeld characterization is produced with the

aid of the ﬂutter function, φ. This is a function of the radial coordinate (only)

that describes the variation in magnetic ﬁeld intensity between the tops

and valleys. It is deﬁned as an average quadratic deviation along a cir-

cle (corresponding to a radial coordinate value) of a function Φ, chosen

64 Nuclear Medicine Physics

so that its average value along a revolution is the unit and conforms to

the equation:

Φ. (2.83)

Similar to B

, Φ is a function of the radial and azimuthal θ coordinates. The

ﬂutter function is then given by

ϕ =

2π



(Φ −

Φ)

dθ, (2.84)

Φ =

2π



Φ dθ = 1. (2.85)

The complete description of betatron oscillations in a generic isochronous

cyclotron is a complex problem demanding the use of approximations in its

resolution and often presenting the results as series expansion. A large part

of this complexity is related to the fact that the average ﬁeld index, the ﬂutter

function, and the angle ζ that the tangent to the limit line between the tops and

the valleys makes with the radial direction (crucial, not only to axial focus-

ing but also to the corresponding free oscillations) are functions of the radial

coordinate, which does not correspond to the (variable) bending radius of the

equilibrium orbit. Moreover, as a function of these two variables, the mag-

netic ﬁeld gradients have clearly different values and orientations. As a ﬁrst

approach, the result for free radial oscillations is the same as the one described

by the corresponding conventional cyclotron Kerst–Serber equation, that

is (using the average ﬁeld index), the proportionality coefﬁcient between

the free radial oscillation frequency and the resonance frequency Ω

given by

√

1 +

n. (2.86)

As far as axial free oscillations are concerned, as a ﬁrst approximation

the proportionality coefﬁcient between the oscillating frequency and the

resonance frequency Ω

will be given by

=−

n +ϕ(1 +2 tan

ζ) +···. (2.87)

The ﬁrst term in this expression is analogous to the one expressed by the

Kerst–Serber equation for free axial oscillations in a conventional cyclotron,

whereas the term that follows is the diverging effect compensation of a pos-

itive average ﬁeld index, given by the magnetic ﬁeld azimuthal variation

(quantiﬁed by the ﬂutter function) and the shape of the transition line between

the tops and valleys.

Cyclotron and Radionuclide Production 65

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