Issue 76

Beam Dynamics Newsletter

3.9 Harmonytron: Vertical FFA with HNJ Acceleration

Yoshiharu Mori, Institute for Integrated Radiation and Nuclear Science, Kyoto University, Japan.
Yujiro Yonemura, Hidehiko Arima and Nobuo Ikeda, Dept. of Engineering, Kyushu University, Japan.


The past decade has seen a marked increase in demands for high intensity and high energy hadron accelerators. There is interest in generating intense beams of secondary particles such as pions, muons and neutrons, and several studies have been carried out to design a high intensity machine which is able to accelerate protons or deuterons to an energy of 500 MeV/u - 1 GeV/u and achieve these goals. Both linear and circular accelerators have reached beam powers (beam energy \( \times \) averaged beam current) of about 1 MW. However, conventional ring accelerators such as cyclotrons and synchrotrons struggle to reach higher levels of 10 MW or more. Here, we propose the new idea of a fixed field, continuous wave (cw) accelerator using a technique known as harmonic number jump (HNJ) acceleration. The machine adopts the concept of a vertical scaling FFA to eliminate transition energy  [1].

Fixed magnetic field ring accelerators such as cyclotrons or FFAs  [2] have the capability for high intensity beam acceleration. In the cyclotron, however, it has been recognized that the beam bunch shape is deteriorated by space charge forces, and beam extraction becomes difficult, when the beam current exceeds 5 mA. In horizontal FFA accelerators, either scaling or non-scaling, cw operation with a fixed radio frequency (rf) field has difficulties in accelerating non-relativistic heavy particles such as protons. The revolution period is different on each turn for such particles, and synchronization can be maintained only by varying the rf frequency.

The idea behind harmonic number jump (HNJ) acceleration, first proposed by Ruggiero  [3], is to, in effect, achieve synchronization for non-relativistic particles and yet have a constant rf frequency. It turns out that such a scheme can also be used for accelerating high energy (relativistic) muons in a scaling FFA  [4, 5].

The condition for synchronization in a circular accelerator is

(1) \begin{equation} \label {HM_eq1}f_\mathrm {rf}=\frac {h_i}{T(E_i)}\,, \end{equation}

where \( T(E_i) \) is the revolution period on the \( i \)th turn, when the energy is \( E_i \), and \( f_\mathrm {rf} \) is the frequency of the rf field. \( h_i \) is an integer number called the harmonic number. Ruggiero  [3] showed that it is possible to adhere to this condition in a non-isochronous ring with fixed rf frequency by changing the harmonic number \( h_i \) by an integer on each turn.

In general, however, the HNJ acceleration scheme meets some difficulties when accelerating heavy particles such as protons or deuterons over a wide range of medium (non-relativistic) energies. In ordinary strong (AG) focusing circular accelerators, problems can occur at transition energy, leading to beam loss caused by collective beam instabilities such as the negative mass instability. In HNJ acceleration, the sign of the harmonic number increment changes as transition energy is crossed; the harmonic number decreases below transition energy and increases above. However, if the concept of transition energy ceased to become relevant, HNJ acceleration could become an efficient method for accelerating high intensity proton and ion beams.

Transition energy could be inherently eliminated if the momentum compaction were zero. This is true of a linear accelerator but not in a normal circular machine. A vertical scaling FFA accelerator, despite being a ring accelerator, does however have this unique feature; its momentum compaction is zero because it has a constant orbit radius whatever the beam energy.

The idea of the vertical scaling FFA accelerator (vFFA) was originally proposed by Ohkawa in 1955  [6] and recently analyzed in detail by Brooks  [7]. It has the unusual feature that the beam moves vertically and its orbit radius is always constant during acceleration. Constant frequency rf acceleration then becomes possible for relativistic particles, e.g. electrons which may be why Ohkawa named such a machine an “electron cyclotron”. Since the orbit radius is always constant, momentum compaction is zero and no transition energy exists. This makes the vFFA a suitable candidate for proton and ion beams using fixed-frequency HNJ acceleration. We have given this new type of machine the name “Harmonytron”.

3.9.1 Vertical Scaling FFA for Harmonytron

In a vertical scaling FFA accelerator, the magnetic field strength changes exponentially in the vertical direction to preserve zero-chromatic beam optics with constant orbit radius. The field at \( x=0 \) is given by

\[ B_y(0)=B_0\exp (my). \]

Here \( y \) is the vertical axis in (\( x \)-\( y \)) coordinates and the characteristic number, \( m \), is expressed as

\[ m=\frac {n}{\rho }, \]

where \( \rho \) is the orbit curvature and \( n \) is the field index, defined by

\[ n=\frac {\rho }{B_y}\,\frac {\mathrm {d}B_y}{\mathrm {d}y}. \]

\( B_x \) and \( B_y \), derived from \( \mathrm {div}\vec {B}=0=\mathrm {curl}\vec {B} \), are

(2) \begin{equation} \label {HM_eq2} B_x=-B_y(0)\sin (mx)\qquad B_y=B_y(0)\cos (mx).

The linearized particle motion in the transverse direction, which from eq. (2) is subject to a skew quadrupole magnetic field, can be described by betatron equations in skew coordinates. Since the beam orbit curvature, \( \rho \), is constant whatever the particle momentum, \( n \) becomes constant and the betatron tunes stay constant, which complies with zero chromaticity.


Figure 1: Stability region in FD doublet lattice presented in Table 1 when the number of cell is 16. Horizontal axis shows \( m=n/|\rho | \) and vertical axis a ratio of magnetic field strength of F and D magnets, respectively.

The characteristic number \( m \) is related to the orbit displacement, \( y_d \), between initial momentum, \( p_i \), and final beam momentum, \( p_e \), through \( m=(1/y_d)\ln (p_e/p_i) \). If \( p_e/p_i \) equals 3 and \( y_d \) is less than 2 m, a value of \( m \) greater than 0.55 m\( ^{-1} \) should be chosen. Table 1 shows typical machine parameters for a vertical scaling FFA accelerator aimed at accelerating protons from 50 MeV to 500 MeV. The stability of beam optics in the vertical scaling FFA is determined by \( m \) and the lattice configuration. Figure 1 shows the stability region in a FD doublet lattice when the number of cells is 16.

Table 1: Beam parameters of vertical scaling FFA for proton Harmonytron

Particle type protons
Number of cells 16
Lattice FD doublet
Initial harmonic number 80
Change of harmonic number/turn 1
Injection energy ( MeV) 50
Extraction energy ( MeV) 500
Circumference(m) 64
\( m=n/\rho \): field index number (m\( ^{-1} \)) 1
rf frequency (MHz) 117.8
Maximum magnetic field(T):F 2.5
Orbit elevation (m) 1.23
Orbit separation at maximum energy(m) 0.11
3.9.2 HNJ acceleration in vertical scaling FFA

From the synchronization condition of HNJ acceleration given by eq. (1), the required energy gain to jump the harmonic number by \( \Delta _ih \) between turns \( i \) and \( i+1 \) can be written as  [4]

(3) \begin{equation} \label {HM_eq3} E_{i+1}-E_i=\frac {\Delta _ih}{f_\mathrm {rf}\left (\dfrac
{\partial T}{\partial E}\right )}. \end{equation}

\( T \) is the revolution time, piecewise linearized around the particle energy. In ordinary circular accelerators, the relative change in revolution period of a particle with momentum can be expressed via the slippage factor defined as,

(4) \begin{equation} \label {HM_eq4}\eta =\alpha _p-\frac 1{\gamma ^2}. \end{equation}

\( \alpha _p \) is the momentum compaction factor and \( \gamma \) is the Lorentz factor. When \( \eta \)=0 the particle is at transition energy, given by

\[ \gamma _t=\frac 1{\sqrt {\alpha _p}}\,. \]

Below transition energy, \( \eta <0 \) and \( \displaystyle \frac {\partial T}{\partial E}<0 \), so the harmonic number needs to decrease during acceleration. Above transition energy, in contrast, the harmonic number should increase. The rf phase synchronizing the particle’s motion is normally less than \( \pi \) below transition energy and greater than \( \pi \) above transition to ensure stable beam acceleration. Therefore, a fast rf phase jump becomes essential when accelerating through \( \gamma _t \). A horizontal scaling FFA, whose momentum compaction is positive, has difficulty avoiding transition in accelerating protons or ions across a wide energy range. On the other hand, in a vertical scaling FFA accelerator, the momentum compaction is always zero so the transition energy is infinite; effectively transition energy does not exist, the beam is always “below transition” and measures otherwise taken to avoid beam loss at transition crossing are unnecessary.

The term \( \partial T/\partial E \) in eq. (3) is related to the slippage factor through

\[ \frac {\partial T}{\partial E}= \frac {\eta \,\gamma ^2\,C}{M_0c(\gamma ^2-1)^\frac {3}{2}},

where \( C \) is the circumference of the ring, \( c \) is the speed of light and \( M_0 \) is the rest energy (938.2 MeV for protons). Since the momentum compaction is zero in the Harmonytron FFA, eq. (3) can be expressed as,

(5) \begin{equation} \label {HM_eq5}E_{i+1}-E_i=-\frac {\Delta _ih\,M_0\,(\gamma ^2_i-1)^{\frac
32}}{f_\mathrm {rf}} \,\frac {c}{C}. \end{equation}

As can be seen from this equation, since \( f_\mathrm {rf} \) is constant, the required energy gain per turn is a proportional to \( \Delta _ih \), which should be negative for acceleration. Figure 2 shows the energy gain per turn given by eq. (5) for the parameters shown in Table 1 and a change of harmonic number per turn \( \Delta _ih=-1 \). In this case, the harmonic number is 80 at the injection energy of 50 MeV and decreases to 47 at the maximum energy of 509 MeV, as shown in Fig. 3.


Figure 2: Energy gain per turn as a function of kinetic energy.



Figure 3: Harmonic number as a function of turn number.

Figure 2 shows that the energy gain required for each turn in HNJ acceleration increases largely according to the beam energy. Figure 3 shows the harmonic number change as a function of turn number. The energy gain per turn is given by the rf voltage \( V_i \) and phase \( \Phi _i \) through

(6) \begin{equation} \label {HM_eq6}E_{i+1}-E_i=QeV_i\sin {\Phi _i}, \end{equation}

where \( Qe \) is the electric charge of the particles. Here, either \( V_i \) or \( \Phi _i \) should be varied to give the required energy gain. The energy gain per turn as a function of turn number is shown in Figure 4.


Figure 4: Energy gain per turn as a function of turn number.

Figure 5 shows the change of required rf voltage, \( V_i \), as a function of the turn number when the stable phase \( \Phi _i \) is constant at 60\( ^\circ \). The change of stable phase, \( \Phi _i \), when the rf voltage is constant at \( V_i=50 \) MV, is shown in Figure 6. The larger increases in energy come towards the end of beam acceleration so that the beam orbit turn separation may become largest on the final turn. This is a good feature for beam ejection, especially in cw operation, since special extraction devices may not be needed. As an example, if \( m=1.0 \) m\( ^{-1} \) as in Table 1, the orbit separation at the final turn is about 11 cm, which should be large enough for beam extraction.

As can be seen from Fig. 5 and Fig. 6, with HNJ acceleration, the rf voltage and/or phase have to be changed to satisfy the energy gain per turn shown in eq. (3). Some schemes to change the rf voltage or phase during acceleration were proposed by Ruggiero in his original paper  [3]; however, there are practical difficulties in realizing them. Moreover, in HNJ acceleration for medium energy, heavy ions, the energy change per turn (Fig. 4) becomes so large that standard adiabatic conditions in longitudinal focusing (synchrotron oscillations) may not be well-enough satisfied to stay within the large longitudinal beam acceptance.


Figure 5: Required rf voltage as a function of turn number when the stable phase is \( 60^\circ \).



Figure 6: Stable rf phase as a function of turn number when the rf voltage is 50 MV.

Figure 7 shows the distribution of each turn around the ring in longitudinal phase space simulated by multi-particle beam tracking for different initial beam distributions. The beam parameters used in the tracking are also those shown in Table 1, with constant \( \Phi _i=60^\circ \) and a single rf cavity located locally in the ring. The particle distribution in longitudinal phase space at injection is Gaussian in phase with zero energy spread. Examples of initial beam distributions were simulated with phase spreads of 0.02 rad and 0.1 rad. As can be clearly seen from this figure, although the HNJ acceleration works in principle, the beam acceptance longitudinally is very small. All of the particles are accelerated up to the final energy of 500 MeV when \( \sigma =0.02 \) rad but a large fraction of particles are lost when \( \sigma =0.1 \) rad. Unless these difficulties can be overcome, HNJ acceleration may not be practical.


Figure 7: Particle distribution on each turn of the ring in longitudinal phase space simulated by multi-particle beam tracking for different initial beam distributions.

3.9.3 Adiabatic Condition in HNJ acceleration

It was mentioned above that the energy gain per turn in HNJ acceleration changes largely as a function of turn number as shown in Fig. 4. Thus, preserving the adiabatic condition of synchrotron oscillation during acceleration is important to keep a large phase space acceptance. The criterion for adiabaticity during rf acceleration can be expressed as  [8, 9],

(7) \begin{equation} \label {HM_eq7} \left |\frac 1{\Omega _s^2}\,\frac {\mathrm {d}\Omega
_s}{\mathrm {d}t}\right |\ll 1, \qquad \text {or, equivalently}\qquad \Omega _s\gg \frac 1A\left |\frac
{\mathrm {d}A}{\mathrm {d}t}\right |, \end{equation}

where \( \Omega _s/2\pi \) is the synchrotron frequency and \( A \) is the bucket area. When this condition is satisfied, the particles are well trapped by the rf bucket and properly accelerated. The condition (7) can be evaluated with the adiabatic parameter \( \eta _\mathrm {ad} \) given by the following equation when the rf phase is constant (\( \Phi _i=\Phi _s \))  [9, 10].

(8) \begin{equation} \label {HM_eq8} \eta _\mathrm {ad}=\frac {\Omega _sT_\mathrm {r}}{1-\sqrt
{V_i/V_i+\Delta V)}}. \end{equation}

Here, \( V_i \) is the total rf voltage on the \( i \)th turn and \( \Delta V \) is the increment of rf voltage on the rf cavity after that turn, \( T_\mathrm {r} \) is a transit time at the rf cavity gap. The synchrotron frequency is given by,

\[ \Omega _s =\omega \sqrt {\frac {h\eta \cos \Phi _s}{2\pi \beta ^2\gamma }\, \frac
{eV}{m_0c^2}\,\frac {Q}{A}}, \]

where \( \omega /2\pi \) is the revolution frequency, \( \Phi _s \) is the synchronous phase, \( m_0 \) is the rest mass of a particle, \( V \) is the rf voltage of a cavity and \( Q/A \) is the charge to mass ratio. The parameter \( n_\mathrm {ad} \) is a measure of the adiabaticity of the system, showing how slow is the change in bucket height with respect to the synchrotron frequency. When \( n_{ad}\gg 1 \), the system can be considered to be adiabatic. In the present case given by the parameters of Table 1, the adiabatic parameter becomes \( n_\mathrm {ad}\approx \)2, so that adiabaticity may not be perfectly fulfilled. Thus, the longitudinal acceptance becomes relatively small, as described above.

This problem could be overcome by distributing several rf cavities around the ring and tuning the frequency of each rf cavity  [4, 5]. If the rf system consists of \( N \) rf-cavities, the adiabatic parameter shown in eq. (8) becomes approximately \( N \) times larger than for a single rf cavity. The rf frequency of each rf cavity can be derived from the following equation:

\[ f_{i,j}=f_\mathrm {ref}\cdot {\biggl [ 1+\frac {2j+N+1}{2N}\frac {\Delta _i h}{h_i} \biggr
]}^{-1}. \]

Here \( i \) is the turn number, \( j \) is the cavity number and \( f_\mathrm {ref} \) is a reference rf frequency. As long as \( h_i \) is much larger than its variation \( \Delta _i h \), \( h_0 \) is also much larger than \( \Delta _i h \). The frequency of each cavity is independent of the turn number, and is approximately given by

\[ f_{i,j}\approx f_\mathrm {ref} \biggl [ 1-\frac {\Delta _ih}{h_0}\bigg (\frac {2j+1}{2N}+\frac
12\biggr ) \biggr ]. \]


Figure 8: Result of longitudinal beam tracking when 16 rf cavities (instead of a single rf cavity) are distributed homogeneously around the ring, parameters as in Table 1.


Figure 9: The beam tracking simulation results for longitudinal beam motion with an initial phase spread of \( \sigma =1 \) rad.

From this, the rf frequency of each cavity also increases monotonically as a function of the cavity number when \( \Delta _ih \) is negative. Moreover, if \( h_0\gg \Delta _i h \), then \( f_i\sim f_\mathrm {ref} \).

Figure 8 shows the result of longitudinal beam tracking when \( N=16 \) rf cavities are distributed homogeneously around the ring using parameters from Table 1. As can be seen clearly from this figure, all of the particles are captured and accelerated up to the maximum energy even if the rms phase spread of the initial particle distribution is 0.4 rad, which is 20 times larger than is possible when a single rf cavity is placed locally in the ring. Thus, the adiabatic condition on the longitudinal beam motion provides useful guidance on the rf cavity requirements to obtain a large longitudinal acceptance in HNJ acceleration.

Preserving longitudinal adiabaticity introduces another important and valuable feature in HNJ acceleration. The rf voltage and/or phase should vary for each turn in accordance with eq. (3), but practically, it is difficult to change the rf voltage or phase of the rf cavity during beam acceleration


Figure 10: . The beam tracking simulation results for the longitudinal beam motions from 0 to 3 turns. (red: 0th turn, green: 1st turn, blue: 2nd turn, magenta: 3rd turn)

Normally, the response time for varying voltage or phase in a high-quality factor rf cavity is relatively slow compared with the acceleration ramping time. Even if it is realized with low-quality factor rf cavites, only pulsed, but not cw, beam operation is possible, which sacrifices operational beam duty factor and reduces the average beam current. However, if the longitudinal adiabatic condition of eq. (7) can be satisfied with multiple rf cavities distributed around the ring, this problem could be overcome. When the rf voltage is constant, the rf phase in HNJ acceleration can be varied during beam acceleration. If the longitudinal adiabatic condition is satisfied during acceleration, the particles will be well captured by the rf bucket and accelerated around the stable phase.

Beam tracking simulations showing results for longitudinal beam motion for different initial beam phase spreads are presented in Fig. 9. In this case, the number of rf cavities homogeneously distributed around the ring is 32, each with rf voltage 1.41 MV, held constant during the beam acceleration.

As Fig. 9 shows, the particles are well captured and accelerated up to the maximum energy following the rf stable phase, and the phase acceptance at beam injection is quite large, at more than 70% of \( 2\pi \). This means that an adiabatic beam capture process can be achieved in an HNJ acceleration scheme using many rf cavities with a small rf voltage, distributed uniformly around the ring. Figure 10 shows beam tracking simulation results for the longitudinal motion ffor the first three turns from an initial phase spread of \( \sigma =2 \) rad. The particles are captured adiabatically and well accelerated in a bucket using harmonic number jumps.


In this paper, the vertical scaling FFA accelerator with a harmonic number jump acceleration (HNJ) scheme, named “Harmonytron”, is proposed for medium energy, heavy particle acceleration in cw operation mode.

Harmonytron has several unique features: a wide range of beam energies becomes possible with a monotonic change of harmonic number in HNJ acceleration since no transition energy exists in the ring, and a large longitudinal acceptance with adiabatic beam capture and acceleration can be realized by distributing many rf cavities homogeneously around the ring.

By keeping enough adiabaticity in the longitudinal motion for beam capture and acceleration by distributing many rf cavities around the ring, HNJ acceleration with a constant rf voltage becomes possible, so that cw operation with large longitudinal acceptance can be realized. This is a breakthrough for HNJ acceleration since it had been thought that either rf voltage or phase should be changed to match the energy gain criterion of integer harmonic number jump during acceleration. Moreover, the beam orbit displacement (turn separation) at the maximum energy is fairly large which could make beam extraction easy for cw beam operation.

Harmonytron is a unique accelerator to realize cw operation and potentially high intensity beam acceleration for non-relativistic heavy particles.

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