11institutetext: CERN, Geneva, Switzerland

# Preparing contributions to CERN reports (school, workshop, and conference
proceedings)

Scientific Text Processing Team

###### Abstract

The huge development of nuclear physics facilities for fundamental and applied
Science purposes has required, in the last three decades, ion beams of
multicharged ions of growing current. Multiple ionization implies the
formation of dense and energetic plasmas to take place. This in turn requires
specific plasma trapping configurations which are fulfilled in essentially two
types of ion sources: EBIS (Electron Beam Ion Sources) and ECRIS (Electron
Cyclotron Resonance Ion Sources). Multiple ionization is also obtained in
laser generated plasmas (LIS - Laser Ion Sources), where the high energy
electrons and the extremely high electron density allow step-by-step
ionization. This report then discusses the atomic physics background at the
basis of highly charged ions production and describes the scientific and
technological features of the most advanced ion sources of the aforementioned
types. A particular attention is paid to ECRIS and their last developments,
since at the moment they represent the most effective and reliable machines
for feeding modern accelerators.

###### Abstract

This document explains instructions for authors who want to prepare electronic
copy to be published as a CERN Report.

##  0.1 Introduction

Research on ion sources is basically imposed by the growing request of intense
currents of multicharged ions by modern nuclear physics facilities. The
physical reason of so demanding requests can be found in the rules od ion
acceleration in circular machines like cyclotrons. In the latter case, the
energy scales with the square of the charge over mass ratio:
$E\propto\displaystyle\frac{q}{A}^{2}$. Intense currents are obviously
requested in order to reduce the acquisition time in low cross section nuclear
physics experiments, or to increase the production rate of exotic beams in
modern facilities like SPIRAL2, FAIR, etc. For sake of example, the FRIB
(Facility for Rare Isotope Beams) project at MSU - Michigan (USA) has beam
requirements of $I\geq 280\ muA$ for $U^{34+}$, where the beam will be
delivered to a $400\ MeV/A$ LINAC for $U$ and $900\ MeV/A$ for proton beams.
Similarly, the FAIR facility at GSI will be based on a $1mA$ beam of
$U^{28+}$. Multicharged ion production requires an extreme condition for the
matter, turned under the state of plasma: several techniques to produce
plasmas in laboratories are available, thus fixing the characteristics of the
ion beam: they differ each other for the electron temperature, electron
density and plasma lifetime that can be achieved by means of the different
heating methods. The most simple way to create plasmas is by means of
electrical discharge in under vacuum tubes; other ways are by means of
electron beams passing through neutral gases or, in a more complicated manner,
by means of electromagnetic waves interacting with gases or vapors and in
presence of a well shaped magnetic field. The latter method is preferable for
HCI formation, since the magnetic field makes longer the time spent by the
ions to suffer multiple ionizing collisions with energetic electrons. In the
previous chapters many pages have been devoted to the description of plasma
properties when the plasma heating is provided by means of microwaves and in
presence of the ECR condition, and for this reason such plasmas have been
named ECR plasmas.

The characterization of the different types of ion sources can be done by
tacking into account some fundamental quantities (quality factors). In
particular, as the ionization up to high charge states requires long times, it
is clear that the ion confinement time will be a crucial parameter to
determine the maximum achievable charge state. The number of electrons
available is also important, because it is clearly related to the number of
electrons able to ionize the atoms down to inner shells. Hence a qualitative
criterion for evaluating the ability of a given plasma to provide HCI is given
by the quality parameter $Q$ defined as the product of the electron density
$n_{e}$ with the ion confinement time $\tau_{i}$:

$Q=n_{e}\tau_{i}$ (1)

The following proportionality is valid:

$<q>\propto n_{e}\tau_{i}$ (2)

Motivations leading to such rules will be more clear in the next sections. At
the moment it is worth to mention that each source, according to its peculiar
characteristics, will have a specific capability to increase the electron
density or the ion confinement time, then to maximize the $Q$ parameter.
Although $Q$ can be considered as the main parameter to determine the quality
of ion source performances, the electron temperature (or more generally the
average electron temperature in the plasma) also plays a key role: it fixes
the maximum charge state can be achieved depending on the given ionization
potential. In terms of extracted currents, a too long confinement time would
limit the amount of ions extracted from the system per time unit. Well
confined plasmas can then produce extremely high charge states, paying
something in terms of current intensity. Mathematically we have:

$I\propto\frac{n_{e}}{\tau_{i}}$ (3)

Since 3 displays a inverse proportionality between current and confinement
time and a direct dependence on the electron density (as in case of $Q$), the
electron density $n_{e}$ becomes the key parameter if one wants to maximize
both currents and charge states.

In order to complete the set of quality parameters we have to take into
account the total background pressure in the chamber where the plasma is
created. Fully ionized plasmas are rare, and in many cases the neutral atoms
and the lowly charged ions may decrease the mean charge state because of
charge exchange processes (whose cross section is considerably higher than the
cross section of electron recombination). Hence another quality criterion can
be inferred, and it states that:

$\displaystyle\frac{n_{0}}{n_{e}}\leq f\left(T_{e},A,z\right)$ (4)

with $f$ that is a function of the atomic mass $A$, of the atomic number $z$
and of the electron temperature $T_{e}$;

Finally, in order to perfectly match the ion sources with the accelerators the
beam quality must be also improved, especially in terms of brightness and
emittance.

Other additional characteristics of ion sources are important, as the
reliability, the lifetime of the system, the maintenance and the last, but not
the least, the cost (for building and for ordinary operations).

##  0.2 Atomic Physics background for HCI production

Collisional dynamics among plasma particles are of primary importance for
ionization rates determination and for dealing with a great number of other
phenomena like electrons and ions lifetime evaluation in magnetically confined
plasmas. An atomic physics background is therefore needed to individuate the
most significant plasma parameters influencing the ion sources performances.
Basically, collisions dynamics are mainly determined by the electron
temperature, the electron density and the background pressure.

It is convenient to introduce some physical quantities which are useful for
treatments of collisional dynamics. In particular, once defined the cross
section, we can introduce the mean free path, the mean time between a
collision and the following one, and finally the collision frequency. The
analysis of the various collisions in plasmas will be carried out principally
in terms of the collision frequency.

If $\sigma$ is the cross section of a given collision, then the mean free path
will be given by:

$\lambda_{m}=\frac{1}{n\sigma}$ (5)

where $n$ is the density 111Here we name a general density with $n$. The
electron and ion densities are usually distinguished by using the notation
$n_{e}$ and $n_{i}$.. Once defined the $\lambda_{m}$ parameter, the time
between two successive collisions will be:

$\tau=\frac{\lambda_{m}}{v}$ (6)

where $v$ is the velocity of the colliding particles. The collision frequency
can be easily determined from the taugen, and it is given by:

$\frac{1}{\tau}=\frac{v}{\lambda_{m}}=n\sigma v$ (7)

Generally in plasmas all the particles (electrons, ions and neutrals) follow
some energy distribution function, 222In many cases it is convenient to use a
maxwellian-like distribution, for which a temperature can be determined.
Anyway we will see that for low pressure and high temperature plasmas the
energy distribution function does not follow a pure maxwellian shape, but
nevertheless it can be viewed as a superposition of several plasma populations
with maxwellian distributions at different temperatures. then we cannot
consider a single velocity in freqcollgen. This expression can be still used
to determine $\nu$, but we have to average over all the possible values of
$\sigma v$ 333We must average not only on $v$, but also on $\sigma$ because in
general $\sigma\equiv\sigma(v)$.. Then we obtain:

$\nu=n\overline{\sigma v}$ (8)

Assumptions done so fare are quite general and no specifications have been
given about the several types of plasma collisions.

Figure 1:  A schematic overview of the principal collision occurring in
plasmas. the abbreviations $e$, $n$, $i$ mean electrons, neutrals and ions
respectively.

Figure 1 features a schematic overview of the main plasma collision processes,
and a classification has been done according to the number of colliding
particles: multiple collisions are characteristics of plasmas, because they
occur thanks to the long range coulomb interaction. The binary collisions are
more similar to the collisions in gaseous systems; however in this case non-
elastic and ionizing collisions occur because of the high energy content of
the plasma particles, especially of electrons.

###  0.2.1 Multiple-scattering collisions

Note that in plasmas any electric field can be felt by charged particles only
within a distance that is named Debye Length - $\Lambda_{D}$:

$\Lambda_{D}=\left(\frac{\epsilon_{0}K_{B}T_{e}}{ne^{2}}\right)^{\frac{1}{2}}$
(9)

this means that a proof particle feels the electromagnetic interaction due
only to those plasma particles located inside the Debye sphere 444The Debye
sphere is the sphere with radius $\Lambda_{D}$.. Hence, more than a cross
section linked to the binary interaction between the proof particle and a
particle of the Debye sphere, the possibility to be subjected to
$90\textdegree$ scattering after many collisions (each one giving small
scattering angles) must be investigated. Then we can appositely define a cross
section, $\sigma_{90\textdegree}$, given by the formula555The quantities in
the formula are expressed in the c.g.s. system:

$\sigma_{90\textdegree}=8\pi\left(\frac{z_{1}z_{2}e^{2}}{Mv^{2}}\right)^{2}\ln\left(\frac{\Lambda_{D}}{b_{min}}\right)$
(10)

where $z_{1}$ and $z_{2}$ are the charges of the to particles, $e$ is the
electron charge, $M$ in the mass of the colliding particle (if one considers
the other particles much more massive) and the term
$\displaystyle\frac{\Lambda_{D}}{b_{min}}$ is the so called Coulomb
logarithmic. This equation, except the logarithmic, is practically the
Rutherford formula for charged particles scattering (binary collision). The
Rutherford cross section is almost two order of magnitude lower than the
eq,sigma_novanta. This means that a $90\textdegree$ scattering due to a single
collisions is much less probable than multiple deflections due to many
collisions 666This result was formerly guessed by Chandrasekhar ..

A more precise representation of the multiple collisions can be given in the
center of mass system, and the collision frequencies can be calculated by
taking into account collisions among the two different plasma species:
electrons and ions. Omitting, for sake of brevity, the mathematical
description, we can use the final formulas to understand what is the role that
each type of collision plays in the plasma:

$\displaystyle\nu_{90\textdegree}^{ee}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{sp}}=5\cdot
10^{-6}n\frac{\ln\left(\displaystyle\frac{\Lambda_{D}}{b}\right)}{T_{e}^{\frac{3}{2}}}$
(11) $\displaystyle\nu_{90\textdegree}^{ei}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{sp}}\sim 2\cdot
10^{-6}zn\frac{\ln\left(\displaystyle\frac{\Lambda_{D}}{b}\right)}{T_{e}^{\frac{3}{2}}}$
(12) $\displaystyle\nu_{90\textdegree}^{ii}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{sp}}\sim
z^{4}\left(\frac{m_{e}}{m_{i}}\right)^{\frac{1}{2}}\left(\frac{T_{e}}{T_{i}}\right)^{\frac{3}{2}}\nu^{ee}_{90\textdegree}$
(13)

$n_{e}$ is in $cm^{-3}$ and $T_{e}$ in $eV$.

Each frequency is usually called characteristic Spitzer collision frequency.
Once given the collision frequency777We talk about the collision frequency for
sake of simplicity, but strictly speaking the term ’́collision’́ is uncorrect
as we should speak about $90\textdegree$ scattering. we can easily determine
the time between two successive events:
$\nu_{90\textdegree}^{ee}\simeq\nu_{90\textdegree}^{ei}$, and
$\tau_{90\textdegree}^{ee}\simeq\tau_{90\textdegree}^{ei}$; on the contrary,
if we consider ions colliding against electrons (i-e collisions), we have:

$\tau_{90\textdegree}^{ie}\simeq\left(\frac{m_{i}}{m_{e}}\right)\tau_{90\textdegree}^{ei}$
(14)

Thermalization among different plasma species is governed by collisions. The
time needed for a $90\textdegree$ deflection and that needed for energy
exchange. It comes out that:

$\displaystyle\tau_{m}^{ee}$ $\displaystyle\sim$
$\displaystyle\tau_{90\textdegree}^{ee}\sim\tau_{90\textdegree}^{ei}$ (15)
$\displaystyle\tau_{m}^{ii}$ $\displaystyle\sim$
$\displaystyle\tau_{90\textdegree}^{ii}\sim\left(\frac{m_{i}}{m_{e}}\right)^{\frac{1}{2}}\tau_{90\textdegree}^{ei}$
(16) $\displaystyle\tau_{m}^{ei}$ $\displaystyle\sim$
$\displaystyle\tau_{m}^{ie}\sim\frac{m_{i}}{m_{e}}\tau_{90\textdegree}^{ei}$
(17)

According to the scheme 1, the Spitzer collisions are the dominant processes
in highly ionized and low pressure plasmas, and totally regulate the
thermalization processes. From above formulas it comes out:
$\tau_{m}^{ee}\sim\tau_{90\textdegree}^{ee}$ and
$\tau_{m}^{ei}\gg\tau_{90\textdegree}^{ei}$, implying a much easier electron-
electron thermalization than electron-ion one. For sake of example, at typical
electron energies ranging from few $eV$888This temperature is typical for the
ion sources used to produce lowly charged ion beams at high currents (up to
tens of $mA$): these sources are usually defined Microwave Discharge Ion
Sources (described in section LABEL:). to some $keV$999For plasmas of ECRIS
for higly charged ions. the $\nu_{ee}$ varies from $10^{5}\div 10^{6}$
$sec.^{-1}$ ($T_{e}=10\ eV$) to $10^{2}\div 10^{3}$ $sec.^{-1}$ ($T_{e}=1\
KeV$), so that electron thermalization requires few $\mu s$, in the first
case, or $ms$ in the second case. The ion thermalization proceeds on a
$\frac{m_{i}}{m_{e}}$ (i.e. about $10^{3}\div 10^{4}$) longer timescale. In
ECRIS, where the ions lifetime is shorter than the time required for the ion
collisional heating, the ions remain cold. This result has a immediate
consequence on the emittance, that can be thereby maintained sufficiently
low101010it strongly depends on the ion temperature..

Another consequence deriving from equations set relencoll is that all the
electrostatic collisions frequencies are much smaller than some other
characteristic frequency of the plasma, in particular the Larmor frequency (in
case the plasma would be magnetized), and the microwave frequency (in case the
plasma is sustained by electromagnetic fields).

$\omega_{RF}\gg\nu_{ee}\ \ \ \mbox{and also:}\ \ \ \omega_{RF}\gg\nu_{ei}$
(18)

under this condition the plasma is called collisionless.

###  0.2.2 Binary Collisions

The binary collisions in plasmas can be divided, according to figure 1, into
two groups: the electron-neutral collisions (elastic and inelastic), and the
ionizing collisions. The former play an important role especially in case of
high pressure plasmas with a low degree of ionization and a low electron
temperature 111111For high pressure we mean, in this case, $10^{-3}\div
10^{-4}\ mbar$, i.e. the operating pressures of microwave discharge plasmas
(MDP) (see section LABEL:). Low temperature here means $T_{e}\leq 15\ eV$,
that also in this case is typical of MDP.. As featured in figure 1, these
collisions provide the thermalization in this kind of plasmas, playing the
same role of the Spitzer collisions in case of highly ionized and low pressure
plasmas. As for the creation of multi-charged ions in ECRIS these collisions
are not so important, then we focus our attention on the ionizing collisions.

Figure 2:  Ionization frequencies versus the electron energy. For high
electron energies the experimental data have been fitted with the Bethe-Born
theoretical formula.

The figure 2 shows the trend of the ionization frequency versus the electron
energy for different ion species. It can be noticed that $\nu_{i}$ is
parameterized by the background pressure; hence, to determine the real
ionization rate, we have to multiply the frequency shown in figure by the
operating pressure of our ion source. In addition, the trend shown in figure 2
considers all the possible ionizing events, i.e. the production rates for all
the charge states that can be produced by a single collision. However it can
be demonstrated that generally a single ionizing collision expels just one
electron from the atom’s shells, thus requiring, the process to achieve highly
charged ions, many collisions. For higher electron energies, $\nu_{i}$ can be
approximated by the Bethe and Born formula:

$q_{i}\propto\frac{1}{E}\ln E\ \ \ \mbox{or:}\ \ \
\nu_{i}\propto\frac{1}{\sqrt{E}}\ln E$ (19)

It is clear from figure 2 that the ionizing frequency increases after a
threshold value (corresponding to the ionization potential), and then slowly
decreases for higher electron energies. If we take into account the usual
maxwellian distribution function of the electrons, then we have to replace the
electron energy in the $x$ axis of figure 2 with the electron temperature.
This means that a not negligible amount of ionizing collisions occur also
below the energy threshold, they being due to the electrons of the
distribution’s tail. For example, in microwave discharge plasmas the electron
temperature is usually $\sim 10\ eV$, but nevertheless the atoms can be easily
ionized because of the tail electron component, whose energy is high enough to
provide effective ionizing collisions.

In case of ECRIS, where the production of highly charged ions is of primary
importance, a particular relationship among electron density, electron
temperature, and ion confinement time has to be achieved. This discussion
leads to the so called multiply-charged ion (MI) production criterion. As
mentioned above, the ionization down to inner shells of atoms required a Step
by Step process, because the cross section of single ionization per collision
are much higher than the multi-ionization.

Figure 3:  Single-impact ionization cross section $\sigma_{0_{z}}$ compared to
successive-impact ionization cross sections against electron energy (Ar).

This is confirmed by the figure 3, which shows that the cross section for
single ionization due to a single collision is on the order of $10^{-16}\
cm^{2}$, then, for multi-ionizations, it rapidly decreases. It comes out that
the probability to expel two electrons is one order of magnitude lower, to
expel three electrons it is two order of magnitude lower, and so on. The
figure also shows the cross section for single ionizations but in case of
highly charged ions (transitions $z\rightarrow z+1$ ).

According with the considerations done in section
LABEL:sec,scheme_of_collisions, the time needed for the transition from charge
state $z_{1}$ to charge state $z_{2}$ takes a time, on average:

$\tau_{z_{1}z_{2}}=\frac{1}{\left[n_{e}\sigma_{z_{1}z_{2}}\left(v_{e}\right)v_{e}\right]}$
(20)

and averaging on a maxwellian distribution121212We usually average the
collision parameters on a maxwellian distribution. As said in the previous
pages, the energy distributions of ECRIS plasmas differ from maxwellian
trends, however they can be considered as superpositions of three different
distributions, each one having its own temperature. Then, for sake of
simplicity, we assume that the results obtained for a maxwellian distribution
are valid for each plasma population, with the further consideration that the
warm component is the most effective to produce highly charged ions. Since now
we will use the parameters of the warm population to determine the ionizing
frequencies of ECRIS plasmas. We have:

$\displaystyle\tau_{z}$ $\displaystyle=$
$\displaystyle\frac{1}{\left[n_{e}S_{z}\left(T_{e}\right)\right]}$ (21)
$\displaystyle S_{z}\left(T_{e}\right)$ $\displaystyle=$
$\displaystyle\langle\sigma_{z}\left(v_{e}\right)v_{e}\rangle\left(T_{e}\right)$
(22)

only if the ion confinement time is longer than the time required for the
given ionization (i.e. the time eq,time_of_ionization the transition
$z\rightarrow z+1$ takes place. Then the condition to produce highly charged
ions in plasmas can be written as:

$n_{e}\tau_{i}\geq\frac{1}{\left[S_{z}\left(T_{e}\right)\right]}$ (23)

In plasmas with long confinement times the step by step ionization provides
highly charged ions. The equation criterioMI can be rewritten by substituting
the $S_{z}\left(T_{e}\right)$ parameter, obtaining:

$\xi n_{e}\tau_{i}\geq 5\cdot 10^{4}\left(T_{e}^{opt}\right)^{\frac{3}{2}}$
(24)

where $\displaystyle\xi\equiv\sum_{j=1}^{N}q_{j}$ is the number of subshells
in the atom outer shells.

Figure 4:  Golovanivsky’s diagram of the $\left(n_{e}\tau_{i}\right)$ criteria
for the production of highly charged ions. The ions enclosed by circles are
completely stripped; then some combinations of electron temperature, electron
density and ion confinement time allow to produce completely stripped ions;
inside brackets uncompletely stripped ions are shown: they can be produced
with the corresponding plasma parameter of ions enclosed in circles.

The criterion which comes from eq,MIcriterion is of paramount importance in
ECRIS, as it fixes the plasma operational parameters needed for good
performance sources. Some numerical examples can be given about the required
densities, confinement times and electron temperatures. If one would to obtain
completely stripped light ions, he needs $T_{e}\simeq 5\ eV$, with
$n_{e}\tau_{i}\geq 10^{10}\ cm^{-3}$. The required electron temperature is
given by the energy threshold of the given charge state that one would like to
obtain.

$T_{e}^{opt}\simeq 5W_{thr}$ (25)

A more general picture of the criteria linked to the plasma parameters is
given by the figure 4, which features what combinations of temperatures,
densities and confinement times are able to produce different ion species with
different charge states.

###  0.2.3 Recombination effects

The charge state of highly charged ions may decrease because of some electron-
ion recombination process. The recombination mechanisms are of three different
types:

  * •

radiative recombination: free electrons are captured by ions with the
following emission of a photon;

  * •

dielectronic recombination: it is a three steps process, where an electron
firstly excite an ion, then it is captured, and finally a photon is emitted;

  * •

charge exchange recombination: highly charged ions, colliding with lowly
charged species and/or with neutrals, capture some electrons from the
electronic cloud of the other particle, thus decreasing their own charge
state;

By analyzing the cross sections of each mechanism it comes out that only the
charge exchange recombination play a remarkable role in ion sources. Because
of its own nature, the recombination rates must depend on the background
pressure: the higher is the number of the neutral or lowly charged ions, the
higher is the probability of charge exchange collisions. Again, in order to
calculate the correct parameters for ECRIS, the time needed for a single
charge exchange process must be evaluated. The charge exchange collision time
must be longer than the ionization time for a given charge state (see equation
eq,MIcriterion), i.e. $\tau_{exch}>\tau_{i}\geq\tau_{z\rightarrow+1}$.
Bypassing the mathematical passages, we can finally write:

$\frac{n_{0}}{n_{e}}\leq 7\cdot
10^{3}\xi\frac{\sqrt{A}}{z\left[T_{e}^{opt}\right]^{\frac{3}{2}}}$ (26)

where $n_{0}$ is the density of neutrals, and $A$ is the atomic mass number.

Then, in order to obtain Argon fully stripped ions ($Ar^{18+}$) at
$n_{e}\simeq 10^{12}\ cm^{-3}$, we need an operative pressure of $\sim
10^{-7}\ mbar$, corresponding to a number of neutrals per cubic centimeter
$\sim 10^{9}\ cm^{-3}$. These numbers are typical of many modern ECRIS like
SERSE, where the minimum operative pressure is $10^{-8}\ mbar$, but it usually
works at $1\div 4\cdot 10^{-7}\ mbar$.

In summary, the analysis of collision dynamics in plasmas put in evidence the
importance of the low background pressure and of plasma confinement by means
of magnetic fields. For low pressures ($p<10^{-4}\ mbar$) the mean free path
for ionizing collisions becomes bigger than the plasma itself131313Plasma
dimensions are comparable with the metallic chambers usually employed in ion
sources for plasma sustaining, i.e. some tens of centimeters.:
$\lambda_{i}\geq 300\ cm$. Only by means of a magnetic field it is possible to
have some collisions in times shorter than the ion lifetime. Electrons spiral
around the field lines and in addition they are reflected by magnetic mirror,
so that the go forth and back inside the magnetic trap and collide sometime
with neutral atoms or charged ions.

##  0.3 Ion sources for high charge states and high current intensity

In the following the principal ion sources able to fulfill the modern
accelerators requests in terms of high charge states and high current are
itemized.

  1. 1.

PIG, i.e. Penning Ion Sources;

  2. 2.

EBIS: i.e. Electron Beam Ion Sources;

  3. 3.

LIS: i.e. Laser Ion Sources;

  4. 4.

ECRIS: i.e. Electron Cyclotron Ion Sources.

On the basis of the discussion done in the previous section, the comparison of
the data (both for currents and for charge states) plays in favor of ECRIS,
which in fact are the most used in the laboratories spread over the world.

provided that the ions stay for long enough time available to subsequent
ionizing collisions with electrons.

###  0.3.1 PIG sources

The PIG source are composed of two cathodes positioned at the extremity of an
hollow anode. A gas is injected into the device and the ionization is provided
by the electrons emitted from the cathode and traveling inside the anode. By
virtue of their negative potential, the cathodes repel the electrons, which
oscillate many times inside the device, thus ionizing the atoms up to high
charge states. In this way it is possible to obtain high ionization efficiency
and then highly charged ions.

In many cases a strong magnetic field is used to further confine the electrons
inside the anode. In this case the electron beam is focused in a restricted
region inside the device, thus leading to the formation of a negative
potential well which is able to confine the ions. The improvement of the
confinement immediately leads to the increase of the maximum obtainable charge
state, according to the expression eq,qmed_propto_Q, as the ions have more
time to be ionized.

Because of very high electron densities($n_{e}\leq 10^{13}$) and of very low
operative background pressures multiply charged ions can be created with a
relatively high efficiency. The ion confinement time it’s enough to reach
medium and high charge states, and it can be estimated by means of [Wolf]:

$\tau_{i}=\frac{r^{2}B}{T_{e}}$ (27)

where $r$ is the plasma radius, $B$ is the magnetic field in Tesla and $T_{e}$
the electron temperature expressed in eV. Considering the usual values of the
magnetic field, electron temperature and plasma radius $\tau_{i}$ lies in the
range $10-100\ \mu sec$. The achievable charge states are for example $8^{+}$
for Xenon, but for this element charge states up to $16^{+}$ have been
produced.

The figure 5 features the characteristics of a PIG source.

The beam extraction can occur in two different ways: it can be axial, or
alternatively the extraction hole can be located on the anode side. In the
case shown in figure the extraction hole is located in the lateral side of the
anode.

In the following some of the most important parameters characterizing the PIG
sources are reported:

Figure 5: Schema di una PIG ion source.

  * •

The magnetic field lies in the range $0,1T-1T$ and can be produced by means of
room temperature magnets. Usually uniform magnetic field are used, but some
attempts of mirror-like fields have been done.

  * •

The potential difference between the anode and the cathodes lies in the range
$100-1000\ V$.

  * •

the extracted current is on the order of
$\displaystyle\frac{10}{A}\div\displaystyle\frac{100}{A}\displaystyle\frac{mA}{cm^{2}}$,
where $A$ is the ion’s mass; then the current depends on the ion mass and on
the anode’s surface.

In the following some advantages and disadvantages of PIG sources are
summarized:

  * •

ADVANTAGES: intense current of multi-charged ions; quite easy methods to
produce metallic ions; possibility to operate as an internal source for
cyclotrons.

  * •

DISADVANTAGES: beam instability; not so high quality of the extracted beam;
time variation of the beam quality (it varies during the source operations);
short lifetime of highly charged ions, with the consequence that maximum
observed charge states are $12^{+}\div 14^{+}$.

###  0.3.2 Electron Beam Ion Sources

The EBIS sources (Electron Beam Ion Source) are some characteristics similar
to the PIG ones: an electron beam is focused by means of a strong magnetic
field in the central part of the the solenoid generating the field, thus
creating a plasma with ions confined by the potential well due to the focused
electron beam. A simplified picture of this mechanism is shown in figure 6.
The atoms filling the plasma chamber are ionized by the electron beams. Once
became ions, they are confined by the potential well, and their confinement
time increases as their charge state increases.

Cylindrical electrodes surrounding the device can be used to increase the
potential well, thus increasing the ion confinement time, producing higher
charge state. The first of figures 6 schematically describes the EBIS
apparatus. The solenoids for plasma confinement is shown, and also the
electron beam shape used for the ionization. Furthermore the extraction system
is shown.

Figure 6:  Schema di una EBIS.

The EBIS are largely used as injectors for Synchrotron of completely stripped
light ions (Dubna, Saclay, Stoccolma). However they can be also used as
injector of highly charged heavy ions (e.g. at CERN or LNBL).

For very high charge states the maxima currents are not higher than some $nA$,
but nevertheless they are able to produce charge states up to $X_{e}^{53+}$
and $X_{e}^{54+}$ (JINR \- Dubna, Russia) (although for $X_{e}^{54+}$ currents
do not exceed one $nA$). Some experimental tests carried out at the Los Alamos
National Laboratories, USA, have demonstrated that the creation of totally
stripped Uranium ions is possible, even if the extracted current is on the
order of few tens of ions per second. Although these machines are completely
useless for accelerator, they allow to carry out some interesting experiments
about atomic physics.

Obviously to reach such high charge states the second term of the expression
eq,qmed_propto_Q has been exploited. Because of this trapping optimization,
these sources are usually named EBIT - Electron Beam Ion Trapping. The
expression eq,current_vs_ne_taui explains only few ions per second are
produced in the Los Alamos laboratory, as the considerable increase of the
confinement time is not accompanied by a correspondingly increase of the
electron density.

Finally we can summarize the main advantage and disadvantage of the EBIS:

  * •

ADVANTAGES: high charge states can be easily produced; the beam quality is
quite good because of the low temperature ions (the emittance is negatively
affected by high temperature ions); large possibility to obtain various time
of pulsed ($\mu s$ beams) and continuous wave beams.

  * •

DISADVANTAGES: low currents, high realization costs, very complex device.

###  0.3.3 Laser Ion Sources (LIS)

The Laser Ion Sources [Wolf] operates thanks to the interaction of focused
laser beams interacting with solid targets.

Figure 7:  Schematic view of a LIS source.

If the power density of the laser beam is enough high then a rapid
vaporization of the target material occurs, and a plasma plume is generated
expanding at supersonic velocities (higher than $10^{4}\ m/sec$). The figure 8
shows the expanding plume; this photo was done during a recent experiment
carried out at LNS with a Langmuir Probe, able to characterize the plasma
plume in terms of temperature and density for different times from laser
pulse. This figure helps to understand what are plasma dimensions: the probe
was put $1.5\ cm$ from the Tantalum target, and the plume front reach the
metallic tip (as shown in the picture) after one $\mu s$ approximately. The
transverse plasma dimension is comparable with the probe tip length, then it
is about $1\ cm$.

Figure 8:  Plasma plume in a LIS source and the experimental apparatus for
Langmuir probe measurements of laser generated plasma properties.

Electrons generated during the target evaporation are heated by the remaining
part of the laser radiation up to several tens of $eV$ if a Q-switched Nd:YAG
laser, operating at 1064 nm wavelength, with 9 ns pulse duration and 1-900 mJ
energy, is used. Such a laser operates at INFN-LNS, where a LIS is used
instead of an ion source as a powerful experimental apparatus to study the
physics of plasma produced by laser ablation.

Usually the laser beam is focused through a convergent lens on a metal target
(Ta, Ag, Cu, etc.) placed inside a vacuum chamber at $10-6\ mbar$. The laser
beam incidence angle is $<90\textdegree$ respect to the target surface and the
spot diameter is $1\ mm$. The plasma plume is emitted along the normal to the
irradiated target whatever the direction of laser beam with respect to the
normal direction.

In general laser energy varies, in the various laboratories where such devices
are used, from $1\ J$ to $50\ J$, with pulse lengths up to $1\ \mu sec$ and
repetition rate up to $1\ Hz$ for $CO_{2}$ lasers. The electron temperature
$T_{e}$, and the charge states distribution depend on laser properties as:

$T_{e}\approx\left(P\overline{Z}\right)^{\frac{2}{7}}$ (28)

where $P$ is the laser power. The figure 7 shows in particular the scheme of
the LNS’s laser source; the system is composed by different parts:

  * •

the laser beam generator;

  * •

the optical system for laser focalization;

  * •

the vacuum chamber were the target is located;

  * •

several diagnostics tools especially for ion properties analysis (TOF - Time
of Flight measurements, IEA - Ion Energy Analyzers, etc.);

  * •

pumping system for vacuum.

The main limitation to the use of LIS in the accelerator facilities is linked
to the low repetition rate. This is mainly due to the laser pumping, which
requires a lot of time especially in case of high power - high intensity
lasers. Anyway, as mentioned above, LIS are powerful methods to investigate
non equilibrium plasma physics, and they find applications in many field of
fundamental and applied physics (one of the most recent applications regards
the cultural goods).

The most important application of laser sources in the accelerator physics is
connected to the possibility that the expanding plumes can work as
accelerators themselves. It has been demonstrated that strong electric field
can be generated inside the plasma plume, and these field can accelerate the
ions up to $MeV$ energies in case of laser at high intensity. The possibility
to obtain small accelerators thanks to non equilibrium expanding plasmas is
under investigation in many laboratories, and also at LNS some interesting
results have been obtained with low intensity lasers, accelerating ions up to
$10\ keV$ .

If the plasmas produced by laser ablation are used as ion source the
performances summarized in table 1 can be reached:

Charge State | Pow. Dens. | Current. | Pulse time.
---|---|---|---
$1\div 2$ | $10^{9}\ \ \frac{W}{cm^{2}}$ | $10\div 100\ \ mA$ | $20\ \ \mu s$
$z\geq 10^{+}$ | $\geq 10^{12}\ \ \frac{W}{cm^{2}}$ | $0.5\div 10\ \ mA$ | $5\div 10\ \ \mu s$
Table 1: Characteristic performances of Laser Ion Sources.

##  0.4 Fundamentals about magnetic confinement of plasmas

Confinement of charged particles, and then of plasmas, is of primary
importance in many field of physics. In case of ion sources, the ions must
live for a long time inside the plasma, so that they are subjected to a
sufficiently number of collisions. Confiment can be ensured by electric or
magnetic field. While electric field effect on charged particles is quite
trivial to discuss, the role of non-uniform magnetic field requires more
details to be clearly explained. Magnetic confinement can be studied either in
a single particle or fluid approximation. Let us start to recall the main
equations of the fluidodynamics applied to plasmas. The general equation which
describe the momentum transport in a plasma subjected to electric and magnetic
fields is:

$mn\left[\frac{\partial\vec{u}}{\partial
t}+\left(\vec{u}\cdot\vec{\nabla}\right)\vec{u}\right]=qn\left(\vec{E}+\vec{u}\times\vec{B}\right)-\vec{\nabla}p$
(29)

in the one fluid approximation, i.e. when the fluid equations are written for
a mixture of ions and electrons and the average mass density and velocity are
calculated, we can do some approximations which simplify considerably the
model. We suppose that the particles move with velocities less than $c$ (as in
case of the single particle approximation), we take into account the plasma
quasi-neutrality (i.e. $n_{e}\simeq_{i}$), we neglect the ratio $m_{e}/m_{i}$
and the gravitational term, we consider only long-spatial-scale phenomena,
with the further restriction that only low-frequency phenomena occur, and we
finally consider that the system remain isotropic at all times. Hence the set
of the so-called magneto-hydro-dynamics becomes:

$\displaystyle\frac{\partial\rho_{m}}{\partial
t}+\vec{nabla}\cdot\rho_{m}\vec{V}=0$ (30) $\displaystyle\ \ \ \mbox{mass
continuity equation}\rho_{m}\frac{\partial\vec{V}}{\partial
t}=\frac{\vec{J}\times\vec{B}}{c}\nabla p$ (31) $\displaystyle\ \ \
\mbox{momentum transport}\vec{E}+\frac{\vec{V}\times\vec{B}}{c}=\eta\vec{J}$
(32) $\displaystyle\ \ \ \mbox{Ohm's
law}\vec{\nabla}\times\vec{E}=-\frac{1}{c}\frac{\partial\vec{B}}{\partial t}$
(33) $\displaystyle\vec{\nabla}\times\vec{B}=\frac{4\pi\vec{J}}{c}\ \ \
\mbox{Maxwell equations}$ (34)

from the second of the equations eq,MHD1$\div$eq,MHD5 it comes that, for
plasmas in a stationary state:

$\vec{\nabla}p=\vec{J}\times\vec{B}$ (35)

from the equation eq,condizioneEquilibrioMHD it follows that the currents and
the magnetic field lines lie on isobaric surfaces, and the vector $\vec{J}$
and $\vec{B}$ are orthogonal each other.

Furthermore, the equation eq,MHD1, by making use of the equation:

$\vec{\nabla}\times\vec{B}=\mu_{0}\vec{j}$ (36)

becomes:

$\vec{\nabla}\left(p+\frac{B^{2}}{2\mu_{0}}\right)=\frac{1}{\mu_{0}}\left(\vec{B}\cdot\vec{\nabla}\right)\vec{B}$
(37)

Assuming that $\vec{B}$ weakly varies along its direction, we can write:

$p+\frac{B^{2}}{2\mu_{0}}=costant$ (38)

In the equation eq,sommapressMHD $\displaystyle\frac{B^{2}}{2\mu_{0}}$ is the
magnetic pressure. The sum of the kinetic pressure and of the magnetic
pressure is constant, then in a plasma with a density gradient the magnetic
pressure will be high where the particle pressure is low (i.e. in the plasma
periphery), and it will be low where the plasma density is high. This
diamagnetic effect is due to the so-called diamagnetic current, which arises
as a consequence of the pressure gradient. The intensity of the diamagnetic
effect can be evaluated by making the ratio between the two terms of the
equation eq,sommapressMHD. Such a ratio is usually indicated as the $\beta$
parameter:

$\beta\equiv\frac{\sum nkT}{\displaystyle\frac{B^{2}}{2\mu_{0}}}$ (39)

Generally, $\beta$ can be used to determine the goodness of the plasma
confinement:

a)

$\beta\ll 1$ then the magnetic pressure is much higher than the plasma kinetic
pressure; this is the condition for an optimal confinement;

b)

$\beta>1$ the plasma pressure is higher than that due to the magnetic field,
then the plasma cannot be magnetically confined;

c)

$\beta=1$ the diamagnetic effect generate an internal magnetic field exactly
equal to the external one; then two regions exist: one where only the magnetic
field is present, the other one where only the plasma, without magnetic field,
exists.

It can be demonstrated that only in case of $\beta\ll 1$ the plasma is well
confined, otherwise the equilibrium is quite unstable.

The most simple device for plasma confinement was early investigated by Fermi
and it is named Simple Mirror141414Fermi proposed the simple mirror
configuration, with sources of the magnetic fields moving towards the center
of the system, as a possible mechanism involved in the generation of cosmic
rays.. In this magnetic configuration the field is provided by two solenoids
with coinciding axes, located to definite distances each other. Figure
LABEL:fig,SimpleMirror features the shape of the field lines.

The principal constraints for charged particles confinement are here
summarized:

  1. 1.

a magnetic field gradient is needed $\vec{\nabla}_{\parallel}B$, directed
along the direction of the field itself;

  2. 2.

even in the simple cases, as for Simple Mirror configurations, the field maybe
axisymmetric but it must have a radial component;

  3. 3.

the velocity vector of the charged particle must form a particular angle with
the field lines. A boundary exists for this angle, and for angles smaller than
this the particle is no more confined.

Figure 9:  Magnetic field lines produced in a simple mirror configuration. The
density of the field lines increases going to right, and this generates the
gradient which works as a mirror for charged particles.

In figure 9 only one of the two mirrors is displayed, and the Larmor motions
and the frift around the axis are shown. Along the mirror axis $z$ charged
particles experience a force which is able to confine them provided that some
specific conditions are satisfied.

The correct expression of the $F_{z}$ we have to average it on a gyro-period.
For sake of simplicity we consider a particle moving along the mirror axis.
Then the $v_{\varphi}$ is constant and depending on the $q$ sign it will be:
$v_{\varphi}=\mp v_{\perp}$. Furthermore we have that $r=r_{L}$ and then:

$\bar{F}_{z}=\mp\frac{1}{2}qv_{\perp}r_{L}\frac{\partial B}{\partial
z}=\mp\frac{1}{2}q\frac{v_{\perp}^{2}}{\omega_{c}}\frac{\partial
B_{z}}{\partial z}=-\frac{1}{2}\frac{mv_{\perp}^{2}}{B}\frac{\partial
B_{z}}{\partial z}$ (40)

and in case of the eq,campoparabolicoshirkov approximation:

$\bar{F}_{z}=-\frac{mv_{\perp}^{2}}{B}B_{1}z$ (41)

then the force is directed in the opposite direction with respect to the
particle motion. We can also define the so-called il magnetic moment of the
charged particle as:

$\mu\equiv\frac{1}{2}\frac{mv_{\perp}^{2}}{B}$ (42)

The the eq,Fzmedia becomes:

$\bar{F}_{z}=-\mu\left(\frac{\partial B}{\partial z}\right)$ (43)

generalizing for a whatever gradient along the particle motion:

$\vec{F}_{\parallel}=-\mu\frac{\partial B}{\partial
s}=-\mu\vec{\nabla}_{\parallel}B$ (44)

where $d\vec{s}$ is a line element along $\vec{B}$ direction.

The main property of $\mu$ is that it is an adiabatic invariant for the
particle motion151515To read more about the adiabatic invariants in plasmas
see . The confinement of charged particles in mirror-like devices can be
studied in terms of $\mu$ invariance.

The breakdown of the adiabatic invariance of $\mu$ expels the particle from
the confinement. The adiabatic invariance is valid as long as the $B_{z}$
component slowly varies as $z$ varies during a gyro-periodo; this condition
can be written in the following manner:

$\frac{r_{L}}{L}\ll 1$ (45)

i.e. the Larmor radius $r_{L}$ must be much smaller than $L$, the
characteristic length for the $B$ variation161616This properties of the $\mu$
invariance is of primary importance also for ECRIS physics, as it will be
demonstrated in section LABEL:. If such condition is not satisfied non
adiabatic effect begin to play their role, leading to the expulsion of the
particle from the confinement.

The mirror effect in terms of the $\mu$ invariance can be explained as
follows: consider a proof particle moving from the center to the periphery of
the magnetic bottle171717The Simple Mirror systems are often named m̈agnetic
bottles.̈; it sees a growing magnetic field; as $\mu$ must be kept constant
the perpendicular component of the velocity $v_{\perp}$ must increase because
of the $B$ icrease. As the magnetic field does not make any work on the
system, then the kinetic energy remains constant. Hence as $v_{\perp}$
increases the $v_{\parallel}$ decreases. If $B$ is sufficiently high at the
device periphery then there will be a mirror point where $v_{\parallel}=0$ and
the particle is reflected.

##  0.5 Ion confinement in ECR plasmas

The magnetic confinement is effective only for particles which do not
experience so many collisionn between the mirror points. This condition is
often fulfilled for the electrons (for ECR Ion Sources is fulfilled almost for
any exsisting device) but only rarely for ions. In ion sources, in fact, ions
are cold, and their collisionality is a factor $\frac{m_{i}}{m_{e}}$ larger
than for electrons. This makes them poorly magnetized at least in the plasma
core. The conservation of plasma quasi-neutrality thereby makes the ion
confinement a direct consequence of the electrons one. Presently, several
models have been proposed to explain the ion confinement in magnetically
confined plasmas, but only few experimental results are available to finally
evaluate the best one. The most commonly accepted model assume that well
confined electrons into the plasma core provide a sufficiently strong negative
potential dip that incorporates multichrged ions. This theory applies only for
the plasma core, while for the whole volume a more general approach must be
taken into account. The quasi-neutrality must be preserved everywhere, then an
ambipolar diffusion process arises. The slower and the faster species can be
determined by analyzing the collisions times when collisions are responsible
of particles deconfinement. The slower species determine the sign of the
plasma potential, i.e. the potential which arises to accelerate the slower
species and to accelerate the faster. Tonks and Langmuir formerly described
the properties of plasmas located inside absorbing boundaries181818For
absorbing boundaries we usually mean the walls of a metallic plasma chamber.:
the preservation of the quasi-neutrality means that $n_{e}\simeq Zn_{i}$, with
$Z$ mean ion charge state, and $j_{e}\simeq j_{i}$; then the electron and the
ion densities must remain the same, along with the currents of particles
leaving the plasma. Starting from this assumption it can be affirmed that in
ECRIS plasma halo the electrons191919Because of their higher mobility, due to
the low plasma halo temperature and its high density, the electrons are more
mobile than ions in the plasma periphery, then they tend to leave the plasma
with an higher rate than ions., having
$\displaystyle\frac{\tau_{ei}}{\tau_{ii}}<1$, leave the plasma more easily
than ions, thus creating a positive potential at the plasma boundary which
accelerates ions and tends to retain the escaping electrons. On the contrary,
into the plasma core, the electrons are less collisional, because their
temperature is very high. Then they are well confined and generate a hot
electrons cloud which retains the ions. The negative potential dip can be
indicated as $\Delta\phi$:

$\Delta\phi\propto-\frac{T_{i}}{ze}$ (46)

Assuming that the bouncing time of ions inside the plasma trap is $\tau_{b}$
we obtain:

$\tau_{b}\simeq\frac{B_{max}}{B_{min}}\left[\frac{plasma\ length}{\langle ion\
velocity\rangle}\right]$ (47) $\tau_{i}\simeq
e^{\displaystyle\frac{Z\Delta\phi}{KT_{i}}}$ (48)

with $Z=ze$. The final situation can be visualized in figure LABEL:potdip2,
where the complete shape of the plasma potential is shown. Note that the
plasma potential is globally positive, but the ḧollow shape,̈ with the small
dip in the plasma core, confines the ions.

According to this model electrons at high energies ($T_{e}>10\ keV$) provide a
stabilizing effect for the plasma because they practically confine the ions.
In order to increase the ion lifetime, we have to increase the potential
deepness.

This model for the ion confinement has been put in crisis by some preliminary
results obtained by our group in collaboration with the Kurtchatov Institute
of Moscow. In section LABEL: some preliminary results about a possible
alternative scheme of ion confinement will be presented.

###  0.5.1 Onset of MHD instabilities and stabilized B-minimum traps

The simple mirror configuration allows to confine that particles obeying to
the adiabatic adiabatic invariance of the magnetic moment. Gibson, Jordan and
Lauer proved experimentally that single charged particles remained trapped in
a magnetic mirror in manner completely consistent with the prediction of the
adiabatic theory. Then this theory predicted and the experiment confirmed that
single particles are satisfactorily trapped in a magnetic mirror field.
However, the trapping of a single particle in some field configuration does
not mean that a plasma has a stable equilibrium. Other than the losses induced
by the collisions, which can change the velocity vector direction putting it
inside the loss cone, other instabilities may arise increasing the particles
losses. Because of the magnetic confinement, the simple mirror plasmas have a
non isotropic energy distribution function. This leads to some instabilities
in the velocity space, the so-called loss cone instability. However, some
hydrodynamical instabilities may be more dangerous. The most important
instability involving simple mirror plasmas in the so called flute
instability. This instability for magnetically confined plasmas is equivalent
to the well known Rayleigh-Taylor instability, which arises in neutral fluids
and also in plasmas subjected to the gravitational force. In case of
magnetically confided plasmas, the magnetic field lines curvature triggers the
instability. In figure LABEL:fig,flute_instability a schematic representation
of the mechanism at the basis of the flute instability is reported.

Only if the magnetic field lines curve into the plasma everywhere, then the
confinement will be stable. This result gives us some practical information to
how realize optimal plasma traps where plasma are stably magnetically
confined.

n 1962 Ioffe first reported on plasma-confinement experiments in a magnetic
field configuration that had the desiderable feature that the magnetic field
increased in every direction away from the plasma boundary, and that did not
have the undesirable feature of a region where the magnetic field went to zero
inside the plasma (as in case of cusp configurations), producing an open wide
loss cone. Such a configuration can be obtained as a superposition of two
magnetic field structures: the one created by two solenoids (simple mirror),
and the other one created by six conductors surrounding the plasma chamber (an
hexapole). In this way the magnetic field increases in every direction that
can be seen from the plasma center. In fact the field produced by the hexapole
increases along the plasma radius, and it is constant along the plasma axis.
This provides a stabilizing effect also radially, being the simple mirror
already able to confine the plasma axially. This configuration can also be
thought as a multi-mirror device: then the particle is reflected in many
points traveling along the field lines. Ioffe and his co-workers at Kurchatov
Institute of Moskow used a device known as the PR-5 (Probkotron-5) to
demonstrate that the hydromagnetic instability could be suppressed with a
minimum-B configuration. The figure 10 schematically shown the field lines of
such a field.

Figure 10:  Sistema magnetico (a) e struttura del campo ottenuto dalla
sovrapposizione di un campo prodotto da due solenoidi e di un campo prodotto
da un esapolo (b).

More details about the structure of the minimum-B field are discussed in
section LABEL:. Note that the geometrical locus where $B$ is constant are
ellipsoids that are usually defined as (egg-shaped surfaces). As there are no
axial symmetries, the $\mu$ adiabatic invariance cannot be invoked to study
the particles confinement. Another adiabatic invariant can be used for
B-minimum configurations, i.e.:

$J=\oint v_{\parallel}dl$ (49)

$J$ essentially determines the field lines length between two reflection
points. The $J$ invariance implies that even after the reflection the particle
will move along the same field line, or at least along a field line with the
same $J$ value. Field lines with the same $J$ value define a surface over
which particles with fixed values of $\displaystyle\frac{W}{\mu}$ will move.

Then the system is realized so that the $costant\ J$ surfaces do not intercept
the plasma chamber walls. Up to now we have pointed out the abitlity of a
minimum-B configuration in ensuring the MHD stability, but no numbers are
given about the relative values of the confining magnetic field, i.e. about
the mirror ratio at the extraction and injection side, and also in the radial
direction.

S. Gammino and G. Ciavola proposed, in 1990, the so-called High-B Mode concept
, stating that only an appropriate optimization, in terms of mirror ratios, of
the confining magnetic field, leads to the exploitation of the electron
density as expected from the frequency scaling, thus leading to an effective
increase of the sources performances. Such a principle does not conflict with
the Geller laws (that will be discussed in the next section), but limits their
applications to well confined plasmas.

The main aspect of the HBM concerns the possibility to give some numbers about
the actual configuration of the magnetic field. These results have been
obtained by virtue of MHD considerations. First, note that in section LABEL:
we determined that a stable MHD equilibrium can be reached only if the
magnetic pressure is higher than the plasma pressure202020For plasma pressure
we mean the usual kinetic pressure due to the particle temperature and
density. Then we have to estimate the plasma pressure in ECRIS, we have to
compare it with the magnetic pressure (which depends on the maximum field that
can be produced by magnet), and finally we can affirm whether or not a given
magnetic configuration is stable according to the MHD criteria for
stability212121Note, from section LABEL:, that magnetically confined plasmas
can be considered MHD-stable only if the $\beta$ parameter is appreciably
smaller than one.

We report the expression for the $\beta$ parameter:

$\displaystyle\beta=\frac{n_{e}KT_{E}}{\displaystyle\frac{B^{2}}{2\mu_{o}}}$
(50)

well confined plasmas are characterized by the $\beta$ parameter:

$0.005<\beta<0.01$ (51)

the plasma electron density can be roughly estimated by means of the
eq,cutoffSL:

$n_{e}\simeq n_{cut-
off}=\frac{m_{e}}{e^{2}}\epsilon_{0}\omega_{RF}^{2}=\frac{\epsilon_{0}}{m_{e}}B^{2}_{ECR}$
(52)

with:

$\omega_{RF}=\frac{qB_{ECR}}{m}$ (53)

Hence, by substituting the eq,cutoffMHD into the eq,betaconf, and considering
the equation eq,condbeta:

$\frac{\epsilon_{0}}{m_{e}}B^{2}_{ECR}KT_{e}<0.01\frac{B^{2}}{2\mu_{0}}$ (54)
$\left(\frac{B}{B_{ECR}}\right)^{2}>100KT_{e}2\frac{\mu_{0}\epsilon_{0}}{m_{e}}$
(55)

we also have:

$\mu_{0}\epsilon_{0}=\frac{1}{c^{2}}\ \ \ \ ;\ \ \ \ m_{e}c^{2}=511\ KeV$ (56)

and then we obtain:

$\left(\frac{B}{B_{ECR}}\right)^{2}>\frac{200KT_{e}}{511\ KeV}$ (57)

the above equation is of primary importance for the operations of modern
ECRIS, as it gives some useful information on practical preparation of an
experiment.

Figure 11:  Experimental confirmation of the HBM concept. The extracted
current increases as the ratio $B/B_{ECR}$ increases, but it stops when
$B/B_{ECR}=2$, in perfect agreement with the equation eq,HBM in case of
$T_{e}\simeq 10\ keV$.

For instance, if $KT_{e}=10\ KeV$, then:

$\frac{B}{B_{ECR}}>2$ (58)

These results have been confirmed by a great amount of the experimental data.
One of them is shown in figure 11, showing the trend of the extracted current
from the SERSE source of the INFN-LNS for different values of the radial
mirror ratio $\displaystyle\frac{B}{B_{ECR}}$. Note that the extracted current
rapidly increases, then tends to saturate when
$\displaystyle\frac{B}{B_{ECR}}>2$, confirming that the electron density
reaches its maximum value only when the plasma stability condition is
established.

More precise scaling laws can be determined by taking into account all the
typical values of the magnetic field in the plasma trap: the injection, the
extraction and the radial field, all of them compared with the field at ECR
and with that at the bottom of the configuration (usually named $B_{m}in$ or
$B_{0}$). More details about the magnetic field structure of modern ECRIS are
discussed in section LABEL:. All the scaling laws for the magnetic field are
presented in section LABEL:.

Finally, note that the relation eq,HBM strictly links the magnetic
configuration to the used frequency ($B_{ECR}$ is fixed by the microwave
frequency). Hence, to effectively improve the ECRIS performances one has to
increase both the microwave frequency and the magnetic field, in particular
the mirror ratios, of the plasma trap.

##  0.6 Semplified treatment of wave propagation in a magnetized plasma

In strongly magnetzed plasmas the direction of the external magnetic field is
a cause of anisotropy of the propagation of the electromagnetic waves. The
electron motion is in fact not free in any direction, and waves are susteined
in different manners according with their propagation direction. Skipping all
the theoretical details, let’s here introduce the propagation constant also in
case of a generic angle of propagation:

$k^{\prime}_{\displaystyle\theta}={\omega\over
c}\left[1-{X(1+iZ-X)\over(1+iZ)(1+iZ-X)-\displaystyle{{1\over
2}Y^{2}_{T}}+\sqrt{\displaystyle{{1\over
4}Y^{4}_{T}}+Y^{2}_{L}(1+iZ-X)^{2}}}\right]^{\displaystyle{1\over 2}}$ (59)
$k^{\prime\prime}_{\displaystyle\theta}={\omega\over
c}\left[1-{X(1+iZ-X)\over(1+iZ)(1+iZ-X)-\displaystyle{{1\over
2}Y^{2}_{T}}-\sqrt{\displaystyle{{1\over
4}Y^{4}_{T}}+Y^{2}_{L}(1+iZ-X)^{2}}}\right]^{\displaystyle\frac{1}{2}}$ (60)

Hence we have found two waves traveling along an arbitrary direction defined
by $\theta$ with $\displaystyle k^{\prime}_{\theta}$ and $\displaystyle
k^{\prime\prime}_{\theta}$. As a function of $X$, $\displaystyle
k^{\prime}_{\theta}$ is more similar to a wave propagating in an isotropic
plasma, then the associated wave will be called ordinary wave; the wave with
$\displaystyle k^{\prime\prime}_{\theta}$ is named extraordinary wave. Anyway
in plasma physics another classification prevails, according to the scheme
shown in figure 12. Usually, the waves propagating along the magnetic field
direction are named $R$ and $L$ waves, according to their polarization222222As
it can be intuited by the denomination, the $R$ wave is a right hand polarized
wave, the $L$ a left hand polarized wave. The waves propagating along the
perpendicular direction to $\vec{B}_{0}$ are named $O$ and $X$, according to
the orientation of the wave electric field with respect to the magnetostatic
field direction.

Figure 12:  Scheme of the wave propagation in anisotropic plasmas. Figure 13:
Diagram showing the possible orientations of the electric field with respect
to the magnetostatic field, and also the possible polarizations in waves
propagating in magnetized plasmas.

If $\vec{k}\perp\vec{B}_{0}$ and $\vec{E}\parallel\vec{B}_{0}$, then we have
the $O$ mode. Otherwise, if $\vec{k}\perp\vec{B}_{0}$ and
$\vec{E}\perp\vec{B}_{0}$, whatever will be the direction of $\vec{E}$, the
wave will be called $X$ mode.

Figure 14:  Schematic representation of the possible ways to inject
electromagnetic waves into a column of a magnetized plasma. The different
modes can be excited, according to the location of the waveguide providing the
electromagnetic field, and also according to the wave polarization inside the
waveguide.

Many other waves can be generated in plasmas, many of them involving ions,
collective fluid motions, and so on. By applying the kinetic theory it can be
also determined the theory of waves propagating in a warm plasma. See
reference [Travelpiece] for more details.

We next consider the cutoffs and resonances of the $R$, $L$, $O$ and $X$
waves. For the R waves, $k$ becomes infinite at $\omega=\omega_{c}$. In this
case the microwave frequency is equal to the Larmor frequency, hence the wave
is in resonance with the cyclotron motion of the electrons. This resonance is
of primary importance for ECRIS and takes the name Electron Cyclotron
Resonance. More generally the cutoffs and the resonances of the various
propagation modes can be summarized by reporting the different
$\displaystyle\frac{c^{2}k^{2}}{\omega^{2}}$ formulas obtained from the
eq,7.2.57 and eq,7.2.58. A schematic summary of the main electron
waves232323With the statement electron waves are usually indicated the high
frequency waves, i.e. the electromagnetic waves. For these waves the ions can
be considered as an immobile medium, as the wave frequency is to high to have
some ions response to the electromagnetic field. It is usually assumed that
ions ensure the quasi-neutrality, but they do not take part to the
electromagnetic wave propagation. is given in table
LABEL:tab,cutoffs_and_resonances.

$\vec{B}_{0}=0$ | $\omega^{2}=\omega_{p}^{2}+k^{2}c^{2}$ | (light waves)
---|---|---
$\vec{k}\perp\vec{B}_{0},\ \vec{E}\parallel\vec{B}_{0}$ | $\frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}}$ | ($O$ wave)
$\vec{k}\perp\vec{B}_{0},\ \vec{E}\perp\vec{B}_{0}$ | $\frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}}{\omega^{2}}\frac{\omega^{2}-\omega_{p}^{2}}{\omega^{2}-\omega_{h}^{2}}$ | ($X$ wave)
$\vec{k}\parallel\vec{B}_{0}$ | $\frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}/\omega^{2}}{1-\left(\omega_{c}/\omega\right)}$ | (R wave)
$\vec{k}\parallel\vec{B}_{0}$ | $\frac{c^{2}k^{2}}{\omega^{2}}=1-\frac{\omega_{p}^{2}/\omega^{2}}{1+\left(\omega_{c}/\omega\right)}$ | (L wave)

In the table $\omega_{h}=\sqrt{\omega_{c}^{2}+\omega_{p}^{2}}$ is the so
called upper hybrid frequency. This frequency practically corresponds to the
frequency of the Langmuir oscillations in non magnetized plasmas.
Electrostatic electron waves across $\vec{B}$ have this frequency, while those
along $\vec{B}$ have the usual plasma oscillations with $\omega=\omega_{p}$.
Furthermore, from table LABEL:tab,cutoffs_and_resonances it comes out that the
$R$ wave suffers the resonance reported before (i.e. the ECR resonance), as
its index of refraction goes to infinity as $\omega\rightarrow\omega_{c}$.
Also the $X$ mode has a resonance, but at the upper hybrid frequency. Then
inside in a magnetized plasma, where an $X$ mode is traveling, the so called
Upper Hybrid Resonance (UHR) occurs if:

$\omega^{2}=\omega_{p}^{2}+\omega_{c}^{2}=\omega_{h}^{2}$ (61)

The $L$ wave does not suffer any resonance, as well as the $O$ mode. Instead
they suffer a cutoff that can be determined by the equations reported in table
LABEL:tab,cutoffs_and_resonances when the index of refraction, i.e.
$\displaystyle\frac{c^{2}k^{2}}{\omega^{2}}$, goes to zero. A powerful method
to visualize the cutoffs and the resonances of the various modes is to plot
them on the so called CMA diagram242424The CMA diagram takes the name from
Clemmow, Mullaly and Allis who proposed the graphical representation of the
many cutoffs and resonances of electromagnetic waves propagating in a cold
plasma. This diagram is reported in figure 15.

Figure 15:  The Clemmow-Mullaly-Allis diagram for the classification of waves
in a cold plasma.

When the propagation occurs at a given angle with respect to the magnetic
field, the phase velocities change with the angle. Some of the modes listed
above with $\vec{k}\parallel\vec{B}_{0}$ or $\vec{k}\perp\vec{B}_{0}$ change
continuously into each other. Such a complicated picture is greatly simplified
by the CMA diagram, which however works only in case of cold plasma
approximation ($T_{e}=T_{i}=0$). Any finite temperature modification implies a
so great number of complications that the diagram becomes completely harmful.

The diagram can be viewed as a plot of the $\omega_{c}/\omega$ versus
$\omega_{p}^{2}/\omega^{2}$, or equivalently a plot of the magnetic field
versus the plasma density. For a given frequency $\omega$, any experimental
situation characterized by $\omega_{c}$ (i.e. the magnetic field) and by
$\omega_{p}$ (i.e. the plasma density) is denoted by a point on the graph. The
total plane is divided into several zones, and the boundaries of each zone are
the cutoffs and the resonances mentioned above. For example, the upper hybrid
resonance can be easily found in the graph: considering an $X$ wave
propagating from a region of high magnetic field, inside a plasma with a fixed
value of density (for example $\omega_{p}^{2}/\omega^{2}=0.7$), the value of
the magnetic field where the UHR occurs can be easily determined on the graph.

Figure 16:  A detailed view of the Clemmow-Mullaly-Allis diagram showing the
region of the so-called overdense plasma, i.e. that region where the plasma
density exceeds the cutoff of the ordinary ($O$) mode.

The vertical line at $\omega_{p}^{2}/\omega^{2}=1$) is the so called $O$
cutoff, and it corresponds to the cutoff density in an unmagnetized plasma.
Above this value the plasma will be called overdense, as shown in figure 16.
The small diagrams in each region delimitated by the cutoffs and the
resonances (i.e. the eight-shaped or ellipsoidal small diagrams) indicate not
only which wave propagates, but also the qualitative variation of the phase
velocity with the angle (considering that the magnetic field is directed along
the vertical direction).

As mentioned above, the kinetic theory provides more information about the
wave propagation then the macroscopic fluid approach and the single particle
investigation. The main assumption of both fluid and single particle approach
is that no velocity distributions of plasma particles are taken into account.
This limits the applicability of such approaches only to low temperature
plasmas, or to that plasmas for which the particle velocities are near to the
fluid element velocity. The Landau damping, the ion waves and all that waves
at harmonics of the cyclotron frequency are out of the fluid or single
particle scheme as they all depend on plasma temperature. For example, in case
of the $O$ mode, whose dispersion relation is reported above and it was based
on a single particle approach, if we applied the kinetic theory we find that
other possible propagation modes there are in proximity of the cyclotron
harmonics $n\omega_{c}$.

##  0.7 Plasma heating by resonant-stochastic wave-electrons interaction

In figure LABEL:fig,accelerated_plasma a simple picture of the plasma-wave
interaction is shown. The figure represents the one dimensional plasma in the
phase space: along the vertical axis the plasma velocity is reported, and the
colors of the plasma slab represent the different plasma electrons energies;
note also that the plasma, in this one dimensional model, extends along the
$z$ direction. The further an electron is from the $z$ axis, the higher is its
kinetic energy. Then the electron temperature is strictly connected with the
vertical thickness of the slab. A wave carrying an electric field of a given
amplitude interacts with the plasma. For sake of simplicity we suppose that
the wavelength is comparable with the plasma dimension. The wave field
accelerates the plasma particles according to the sign of the electric field.
In a frame traveling with the wave, no time variations there are, and the
plasma organizes itself as shown in figure: where the electric field is
stronger the electrons gain more energy, otherwise they remain in proximity of
the $z$ axis. At the beginning of the process no variations of the slab area
occur: the interaction is completely adiabatic, then non conservative effects
are not taken into account at first approximation. The plasma retains its
temperature, because the slab thickness remains unchanged, but nevertheless a
great amount of electrons are strongly accelerated. Such a process may be
enough to provide a great number of energetic electrons abe to ionize atoms.
This explains why before we said that no electron heating is needed for good
performances ion sources. However, collisions (in case of collisional
plasmas), or stochastic effects lead to the breakdown of the adiabatic
invariance and to the particle spread in the phase space.

Figure 17:  Spread in the phase space of the electron velocity due to the
activation of the stochastic effects.

The figure 17 shows that after some time the stochastic effects begin to play
a role in the plasma-wave interaction dynamics. This leads to the enlargement
of the plasma area in the phase space, being this a sign of an effective
heating.

In ECRIS an effective heating is experimentally observed; on the other side we
know that the electron lifetimes are on the order of $msec$ for well confined
plasmas. Then the non-adiabatic effects, which provide the stochasticity,
arise in times shorter than $msec$. Reasonable estimations and numerical
simulations give a time on the order of few $nsec$ to observe non adiabatic
effects in such plasmas (see section LABEL: for more details). Then the
collisionless plasma heating is completely determined by the stochasticity, as
demonstrated also by Liebermann and Lichtemberg .

According to their theory, the stochasticity in ECR plasmas is provided by the
multi-passages through the resonance, as long as the forgetting mechanism
introduced before works well. In order to better understand how the heating
develops in multi-passage interactions, it is better to introduce the
separatrix concept. Particles which resonantly interact with waves can be
considered as trapped by the wave potential. The resonant condition is
achieved if the well known equation: $\omega=kv_{e}$ is satisfied. Again,
consider a frame moving with the wave velocity: particles moving with a
similar velocity can oscillate into the wave potential252525As before, we
consider a one dimensional model to explain the particle trapping in the wave
field at the resonance. $\varphi(z)$. Then the electron moves with a velocity
$v_{e}$, while the wave potential, that can be indicated with $\varphi(z)$:

$\varphi=\varphi_{0}\sin\left(\omega t-kz\right)$ (62)

is stationary and sinusoidal, as the frame moves with $v_{\varphi}$. Because
of the energy conservation we can write:

$\frac{1}{2}mv_{e}^{2}+e\varphi(z)=\epsilon=const.$ (63)

then the velocity $v$ of the electron is:

$v_{e}=\pm\sqrt{\frac{2\epsilon-2e\varphi(z)}{m}}$ (64)

this velocity can be graph in the phase space $z-v_{e}$.

Figure 18:  Spread in the phase space of the electron velocity due to the
activation of the stochastic effects.

Particles at the bottom of the wave potential (i.e. perfectly resonant with
the wave) stay exactly at the center of each separatrix, and their trajectory
in the phase space is just a point (their velocity with respect to the wave is
zero, then they must lie on the $z$ axis). If an electron is moved from the
bottom of the potential, it will begin to oscillate around the equilibrium
point, similarly to mechanical oscillators with small amplitude oscillations
around the equilibrium. As long as the perturbation is small enough, the
motion around the minimum will be harmonic. In the phase space, these motions
are represented as ellipsoidal trajectories, as shown in figure 18. The
electron velocity may be higher and higher, then the ellipsoidal trajectories
become bigger and bigger. Actually, if the electron energy begins appreciable
with respect to $\varphi(z)$, the motion is no longer harmonic and non linear
effects arise. If the electron stays on the top of the $\varphi(z)$ potential,
then the electron velocity will be zero according to the equation 64. In this
case the electron has the same velocity of the wave, but as it stays on top of
the wave potential, its equilibrium is not stable, and chaotic motions may
arise. When this condition is satisfied the phase space separates in different
zones. Inside each zone the particle is trapped by the wave. The figure 18
features the separatrix of a sinusoidal wave interacting with a single
particle. This theory, named pendulum model, explains easily how the non
linearity and the stochastic effects can play a fundamental role in the
collisionless wave-particle interaction. That electrons whose velocity differ
considerably from the wave phase velocity are not trapped and their velocity
is only slightly perturbed by the wave. If the separatrix areas present some
non uniformities, for example because of localized wave patterns as for laser
beams, for electron beam plasmas and so on, the heating becomes more
effective, as such systems have other sources of stochasticity.

Next we discuss about the heating in ECRIS through electron multi passages
across the ECR zones. An easier treatment is possible if we consider a Simple
Mirror configuration, because in that case a one dimensional model can be
applied and the bouncing of the electrons inside the trap are subjected only
to the potential $\mu B$. Then the equation for the $z$ variables becomes:

$z=z_{0}\cos\left(\omega_{b}t+\psi_{0}\right)$ (65)

where $\omega_{b}$ is the bouncing frequency. The perpendicular component of
the wave electric field can be simply written as:

$E_{\perp}\left(t,z,r,\theta\right)=E_{\perp_{0}}\cos\left(\omega
t-kz+\theta_{\perp}\right)$ (66)

where $\theta_{\perp}$ is the initial perpendicular phase. By substituting the
equation eq,zmotion into the eq,perpelectricfield we obtain:

$\displaystyle
E_{\perp}\left(t,z,r,\theta\right)=E_{\perp_{0}}\cos\left(\omega
t+\theta_{\perp}-kz_{0}\cos\left(\omega_{b}t+\psi_{0}\right)\right)=$
$\displaystyle=\sum_{n}A\exp{\left[\left(\omega-\omega_{c}\right)t+\theta_{\perp}+\theta_{b}\right]}$
(67)

where:

$\theta_{b}=\omega_{b}t+\psi_{0}$ (68)

Then the equation eq,composedmotion takes into account the cyclotron
frequency, the frequency of the wave and the bouncing frequency. The Fourier
expansion puts in evidence that the composition of the bounce and cyclotron
frequencies leads to an effective multi-waves interaction: i.e. we may imagine
the single particle interacting with a couple of waves at the resonance. Then
each wave-particle interaction has a proper separatrix and many zones like
that shown in figure 18 are present in the phase space262626Here we talk about
the perpendicular phase space, i.e. that one formed by
$p_{\perp}-\theta{\perp}$.. The figure LABEL:eq,ECRseparatrix shows the
structure of the several separatrix in the phase space in proximity of the ECR
zone.

Figure 19:  The many separatrix occurring in proximity of the ECR zones: they
are overlapped.

It is clear that all the separatrix are overlapped, being the distance between
them proportional to $\omega_{b}$. The lowest separatrix intercepts the plasma
slab; this means that electrons of the plasma can be trapped into the lowest
separatrix. According to its own kinetic energy, to the phase with respect to
the wave, etc., the motion will evolve into the several sepratrix in a
complicated manner, taking into account also non linearity in the wave
particle interactions. As the separatrix are overlapped, the electron
perpendicular momentum (then its energy) will increase more and more passing
from a separatrix to the subsequent one, as schematically shown in figure. In
the real situation the orizontal separatrix extension at the resonance is
limited, then the electron will escape after a fixed time from the interaction
zone. But because of its bouncing motion, it will pass again through the
resonance. As the former interaction was regulated by a stochastic motion in
the phase space, then it will arrive again to the ECR with a completely random
phase, thus being able to gain more energy. This process goes on for many
passages through the resonance. However, the higher is the electron energy,
the higher is $\omega_{b}$. When the bouncing frequency becomes comparable
with the Larmor frequency, then the separatrix split, and the electron cannot
pass through them. This effect corresponds to the activation of some adiabatic
invariants: the electron heating stops because the phase randomization
process, strictly connected to the separatrix overlapping, does not work
anymore.

Once discussed the physical mechanism we next report a more quantitative
treatment of the heating process.

Starting from the Canobbio parameters, Liebermann and Lichtenberg formulated
the theory to calculate the maximum value of the electron energy for particles
bouncing between the mirror, then passing many times through the resonance. Be
$t_{e}$ the time effectively spent by the electron in the ECR region:

$t_{e}=\frac{0.71}{\omega}\left(\frac{2\omega}{\delta
v_{\perp_{i}}}\right)^{\frac{2}{3}}$ (69)

where the perpendicular velocity $v_{\perp}$ is given by:

$v_{\perp}\propto\frac{e}{m}Et_{e}$ (70)

i.e., in terms of $\delta$:

$v_{\perp}\propto\frac{E}{\delta^{\frac{2}{3}}}$ (71)

and also:

$v_{\perp}\propto\frac{\sqrt{P}}{\delta^{\frac{2}{3}}}$ (72)

by virtue of the eq,tempoeffettivoECR, we have that $t_{e}$ does not depend on
$v_{\parallel_{R}}$, and it is inversely proportional to $v_{\perp_{R}}$, i.e.
the higher is the transversal velocity, the less effective will be the
particle acceleration.

For multiple passage through the ECR region, the $v_{\perp}$ evolves according
to:

$v_{\perp}^{2}=v_{\perp_{0}}^{2}+v^{2}+2\left(vv_{\perp_{0}}^{2}\cos\theta\right)$
(73)

where $v$ is the maximum velocity gained for each passage, and $\theta$ a
function of the phase. $vv_{\perp_{0}}\cos\theta$ is the stochastic term.
After some passages, the mechanism explained above begins to play a role, the
separatrix split and the heating stops; this mechanism can be explained also
in a different manner. As the velocity increases, some adiabatic invariant
arise. Such invariants depend also on the wave electric field ($\vec{E}$).
Defining with $W_{S}$ the maximum value of the energy that can obtained
through the stochastic heating, we have:

$W_{S}=0.34eEL\sqrt{\left(\frac{\overline{t}}{\tau_{S}}\right)}$ (74)

with:

$\tau_{S}\sim 4\pi\frac{m}{e}\frac{1}{B}\frac{L^{2}}{l^{2}}$ (75)

that is the time necessary to have a phase shift of $2\pi$; $L$ is the
characteristic length of the mirror field, as it can be viewed by the
$\vec{B}$ equation:

$B_{z}=B_{0}\left(1+\frac{x^{2}}{L^{2}}\right)$ (76)

and:

$\overline{t}=\sqrt{\left(\frac{mL}{eE}\right)}\left(1+\frac{l^{2}}{L^{2}}\right)^{\frac{5}{4}}$
(77)

in correspondence of the resonance position $x=l$. In this conditions the
electron is mirrored just inside the ECR zone
([LichtembergI],[LichtembergII]).

The maximum energy achievable by the electrons will be:

$W_{B}\sim 5W_{S}$ (78)

with $W_{B}$ that is usually named absolute stochastic barrier.

To give some estimation of such a barrier, we can substitute some realistic
values into the equations eq,Ws$\div$eq,absstochbarrier, thus obtaining:

$W_{S}\sim 2E^{\frac{3}{4}}\left[\frac{V}{m}\right]\propto
P_{RF}^{\frac{3}{8}}$ (79)

with $E$ and $P_{RF}$ wave’s electric field and power. As for ECRIS we have:

$0.1\leq E\leq 10\ \frac{kV}{cm}$ (80)

the absolute stochastic barrier will lie in the range:

$10\ keV\leq W_{B}\leq 270\ keV$ (81)

The estimation of the maximum energy achievable by ECRIS plasma electrons is
realistic if compared with experiments. This is a remarkable result, as the
single particle theory may be considered somehow oversimplified. However such
calculations fall for modern ECRIS, like VENUS . In section LABEL: we will
largely discuss about the VENUS experimental results which feature the
presence into the plasma of electrons with energies well above the stochastic
barrier imposed by the Liebermann and Lichtenberg theory. We will discuss
other possible heating mechanisms in section LABEL:, but also the above theory
will be investigated thanks to numerical simulations (see section LABEL:), in
order to understand if the adiabatic invariants, activated at high electron
energy, can be broken by additional effects not considered in the Liebermann
and Lichtembeg theory.

Strictly speaking, it can be noticed, by the separatrix treatment, that the
heating may continue if the separatrix splitting does not occur, i.e. if
additional perturbations to the particle motion change the separatrix
amplitudes or provide additional separatrix, in order to have again a
separatrix overlapping. In this way the phase randomization should continue
even above the stochastic barrier. The possible sources of further
randomization of the electron-wave phase relationship will be largely
discussed in section LABEL:.

Then the so-called ËCRIS Standard Model,̈ which has traced the road for the
development of electron cyclotron resonance ion sources in the last twenty
years, is based on experimental evidences and adds the Geller’s scaling laws
and the High-B mode concept. This model have permitted an average increase of
about one order of magnitude per decade in the performance of ECR ion sources
since the time of pioneering experiment of R. Geller at CEA Grenoble [ref].
However nowadays such a trend is limited not only by the technological limits
of the magnets, but also by the experimental evidence of hot electron
population formation that becomes more important for high plasma density (i.e.
for operational frequency of $28\ GHz$ and higher) and for microwave power
above $3\ kW$.

It has been shown that in an homogeneous and unmagnetized plasma the cut-off
frequency assumes the simplest mathematical form, and it is the same of the
$O-mode$ in case of magnetized plasmas. Although in the cylindrical cavities
if ECRIS more than a unique mode propagation is present, so that all the
possible cut-offs should be taken into account, the simplest picture to
determine a relationship between the microwave frequency and the maximum
achievable plasma density considers the $O-mode$ cut-off frequency, so that,
writing it in terms of maximum density as a function of the field frequency,
we obtain:

$n_{cut-off}=\epsilon_{0}m_{e}\frac{\omega^{2}}{e^{2}}$ (82)

More generally, all the possible electromagnetic modes propagating in plasmas
(R, L, O, X) suffer a cut-off. Then the maximum density of the plasma where
these modes propagate cannot overcome some fixed values. Only electrostatic
modes are able to propagate in plasmas of whatever density (see section LABEL:
on Bernstein modes).

The main consequence which come from the equation 82 is that by increasing the
electromagnetic field heating the plasma the electron density can be
increased. In magnetized plasmas the relationships between density and
frequency may change, but the dependence of the extracted current on the
square of microwave frequency has been demonstrated in many experiments. Such
a dependence comes from the eq,cutoffSL by virtue of the 3.

On the basis of these considerations and of the experimental results, twenty
years ago R. Geller proposed the scaling laws [13] which have been for long
time a guideline for the ECRIS community:

$q_{opt}\propto\log B^{\frac{3}{2}}\\\ I^{q+}\propto f^{2}M_{i}^{-1}$ (83)

where $q_{opt}$ is the optimum charge state, $B$ is the peak field of the
magnetic trap, $f$ is the microwave frequency, Iq+ is the intensity of the
charge state $q$ and $M_{i}$ is the mass of ions.

These formulas were verified in several ECRIS spread over the word, including
SERSE at INFN-LNS . However, although the microwave frequency was continuously
increased, in order to fulfill the eq,gellerlaws, the increasing trend of the
sources performances became to saturate especially when it was clear that the
MINIMAFIOS source produced at $16.6\ GHz$ a mean charge state lower than the
SC-ECRIS operating at MSU272727The MINIMAFIOS source was able to extract CSD
with $\langle q\rangle=9$, whereas the SC-ECRIS produced $\langle
q\rangle=12$. It was clear that the confinement, that for the SC-ECRIS was
much better than for MINIMAFIOS, was the other key parameter to ensure that
the benefits provided by the increase of the microwave frequency were
exploited.

##  0.8 Advanced research for ECRIS development

The best compromise in terms of intense currents and charge states are
obtainable by ECR Ion Sources, which coniugate a performing magnetic system
providing long ion confinement times, with an efficient mechanism of plasma
heating capable to boost the density up to values close, or even slightly
above, the cut-off limit. The aforementioned standar model has been followed
as ensured for several decades a constant increase of the sources
performances. Up to now the "ECRIS Standard Model" has traced the road for the
development of electron cyclotron resonance ion sources. Since its formulation
(completed in 1990, when the HBM concept was proposed), this model has been
proven by many experiments carried out with different sources, operating at
different ECR frequencies. The main rules were confirmed by experiments
performed at MSU-NSCL in 1993-94 and in 1995. At power, magnetic fields and
microwave frequencies used up to now the ECRIS obey to the Standard Model: the
extracted current strongly increases as the microwave frequency increases, but
only the increase of mirror ratio can exploit the performances, making
effective the increase of electron density with frequency. Reassuming the
conclusions of the high-b mode concept, and linking them with the experimental
results it is known that the magnetic field distribution in the plasma chamber
must obey to the rules:

a)

the radial magnetic field value at the plasma chamber wall must be
$B_{rad}\geq 2B_{ECR}$;

b)

the axial magnetic field value at injection must be $B_{inj}\simeq 3B_{ECR}$
or more;

c)

the axial magnetic field value at extraction must be about $B_{ext}\simeq
B_{rad}$ ;

d)

the axial magnetic field value at minimum must be in the range
$0.30<\frac{B_{min}}{/}{B_{rad}}<0.45$.

Then according to the Standard Model the ECRIS development is strictly linked
to the improvements in superconducting magnets and of the microwave generators
technology. No scaling laws were put in evidence about the microwave power and
the ECR heating process (ECRH), because the relation between the power and the
magnetic field is not so simple. Some authors studied the RF coupling to the
plasma in terms of the maximum power rate per unit volume and of its
relationship with the beam intensity produced by different ECR ion sources
[ref]; only recently it was observed that the efficacy of the RF to plasma
energy transfer depends also on the amount of the wave energy coupled to the
plasma chamber. Then the cavity design, and in particular the microwaves
injection geometry, is of primary importance for an high RF energy
transmission coefficient into the plasma chamber. Note that the problem of the
wave energy transmission into the plasma must be divided into two parts: the
first is connected to the microwave generator-waveguide-plasma chamber
coupling, while the second one regards the wave-plasma interaction. The
standard model does not allow to argue any prediction about the mechanism at
the basis of electron and ion dynamics for different profiles of magnetic
field and for different microwave frequency. Some preliminary experimental
results obtained with III generation ECRIS (VENUS) put in evidence the limits
of Standard Model, in particular as regards the magnetic field scalings.
Although the standard model has permitted to strongly improve the ECRIS
performances up to the $3_{rd}$ generation ones (ECRIS which operate at $28\
GHz$ and with magnetic field up to $4\ T$), we expect that the technological
limits can be overcame by means of a better understanding of plasma physics,
in particular of the ECR-heating mechanism.

Since 1994 (first evidences of Two Frequency Heating effect) several possible
ways to improve ECRIS performances overcoming the "brute" scaling of the main
operative parameters have been investigated worldwide. In this section the
most important ones will be discussed in detail.

###  0.8.1 Two Frequency Heating

It can be hystorically considered the first non-conventional method of plasma
heating outside the scheme traced by the standard model. Since 1994, the so
called TFH has been used [32],[35] to improve the HCI production by feeding
the plasma with two electromagnetic waves at different frequencies instead of
one. In some cases, even three or more close frequencies or a white noise
generator (WNG) [36] have been used. The TFH demonstrated to be a powerful
method, e.g., in the case in [32] for ${}^{238}U$, it increased the production
of higher charge states (from $35+$ to $39+$) by a factor from two to four and
shifted the peak charge state from $33+$ to $36+$. Unfortunately, neither the
relationship between the two frequencies nor the respective power was
univocally determined. In fact, any source features a different set of
parameters and the optimization is done empirically, just by looking to the
maximization of beam current. Several qualitative explanations have been given
about this phenomenon, all related to the increase of the average electron
temperature $T_{e}$ and the ionization rate by assuming that the crossing of
two resonance surfaces helps the electron to gain more energy. This simple
picture does not explain the reason of the relevant changes in the charge
state distribution (CSD) for different couples of frequencies (even for the
case of minor changes, let us say few megahertz over $14$ or $18\ GHz$), that
can be explained nowadays by the frequency tuning effect. It is important to
underline that, even in the case of TFH applied to many existing sources, a
TWT is often used, the other being a klystron-based generator. The choice of
TWT allows the experimentalists to slightly vary the second frequency: as an
example, a strong variation of ECRIS performances has been observed in TFH
operations in [35]. It was observed that the production of $O^{7+}$ with $60\
W$ emitted by the TWT at the optimum frequency gives the same effect as $300\
W$ from the fixed-frequency klystron. The maximum current is obtained by means
of klystron at $427\ W$ and TWT at $62\ W$ ($I=66\ e\mu A$); in order to
obtain the same current, they needed $P_{RF}=800\ W$ by the klystron in SFH.
Furthermore, in the case of TFH, the current increases almost $20\%$ (from
$57$ to $66\ e\mu A$) when the TWT emitted frequency shifts from $11.06$ to
$10.85\ GHz$, being constant the klystron and TWT emitted power. Then, the TFH
is an effective method to increase the extracted current from ECRIS, but it
can be fully exploited only by means of the frequency tuning. Several
measurements have been carried out with the SERSE ion source. The TFH has been
used either for operation at $14$ or $18\ GHz$, with a clear advantage with
respect to SFH. It was not successful when a $28-GHz$ gyrotron and an $18-GHz$
klystron were used at the same time [39], and that is an evidence that fixed-
frequency generators are not adequate for TFH. The performances obtained by
two-fixed-frequency klystron were also not so good as the ones obtained by a
fixed-frequency generator plus a TWTA.

As an example, for tin production, a current on the order of $2$ to $3\ e\mu
A$ was measured for the charge state $29+$ when a power of $1.4\ kW$ from the
$18-GHz$ klystron and a power of $1.0\ kW$ from the $14.5\ GHz$ klystron were
used. By keeping unchanged all the parameters and the $18\ GHz$ klystron
power, a TWTA operating in the range of $8$ to $18\ GHz$ replaced the $14.5\
GHz$ klystron. Even if the maximum power from the TWT was only $200\ W$, it
permitted one to obtain more than $3\ e\mu A$ of the charge state $29+$ with a
frequency of $17.600\ GHz$. Another interesting operating point was observed
at the frequency around $17.000\ GHz$, which was less critical than any other
setup and permitted one to obtain the state $30+$ with the same beam current,
only by means of minor changes of the gas pressure. It must be underlined that
the vacuum inside the plasma chamber improved by decreasing the total power
from $2.4$ to $1.6\ kW$, which permitted one to move the CSD to higher charge
states and to have an excellent reproducibility, even after weeks. In order to
have a comparison of these results with the performances in SFH mode, the
$Sn^{30+}$ current obtained with the same power from the $18\ GHz$ klystron
was only $0.7\ e\mu A$. All these data are explained by the description of the
microwave-plasma interaction given here and they demonstrate the advantage of
TFH operations, when the chosen frequency has the right electromagnetic field
distribution in the plasma chamber. Moreover the closer frequencies are the
best choice. Not only the couples $18$ and $17.6\ GHz$ or $18$ and $17.0\ GHz$
but also for operation around $14\ GHz$; it was observed that the optimum was
obtained for $14.5\ GHz$ klystron and TWTA at $14.3\ GHz$. This observation is
also coherent with the unsuccessful test with a $28\ GHz$ gyrotron and an $18\
GHz$ klystron.

The ECRIS group of INFN-LNS, Catania, has tried to explain TFH on the basis of
fundamental processes occurring during the electron-wave interaction. On this
purpose, numerical simulations based on a pure Monte Carlo approach were
developed. The main assumptions are reported in […]. To be opportunely
modeled, the TFH effects can be depicted in the following way:

1)

We double the resonance zone width, so that electrons that do not gain energy
during the first crossing may have another possibility to be heated by the
second one. Figure 20 shows the locations of these two resonance regions
inside the plasma chamber calculated for the initial setup of our simulations.

2)

Preliminary simulations have shown a singificant role played by the ECR
interaction in recovering electrons otherwise contained into the magnetic loss
cones. This effect (a deterministic plug-in of cold electrons) is currently
deemed to play the main role in confining the most of the plasma inside the
volume embedded by the resonance surface. Adding a second resonance, electrons
that have not been plugged by the first wave can be recovered by means of the
second one, with a further increase of the plasma confinement.

Figure 20:  Section of the two resonance surfaces showing the position of the plasma electrons. | $(14+18)GHz$ | $18\ GHz,$ | $18\ GHz,$ | $14\ GHz,$
---|---|---|---|---
| $(1000+300)W$ | $2000\ W$ | $1000\ W$ | $300\ W$
Recovered electrons $[\%]$ | 38.2 | 35.2 | 19.2 | 12.9
Energy (keV) | 1.9589 | 1.2612 | 1.0141 | 1.5509
Table 2:  Percentage of recovered electrons (i.e., electrons that are
recovered by the electromagnetic field) and the electron energy after $50\ ns$
are shown for different heating configurations (SFH and TFH) and at different
powers. Figure 21:  Cumulative probability plot of the electron energy
distribution function for the different simulated modes and power. Once fixed
an energy on the horizontal axis, the correspondent value on the vertical one
gives the relative number of particles below that energy, or the $1-CP$
relative number of electrons above that energy value, where $CP$ is the
cumulative probability.

A series of simulations has been carried out in order to verify the effect of
the TFH. In the case in figure 20, the two resonance zones are far away. This
allows the electrons to randomize their gyro-phase with respect to the
electromagnetic wave. The numerical study has dealt with the case of two
electromagnetic waves at different frequencies supplying the mode $TE_{4\ 6\
22}$ ($1000\ W$) and the $TE_{4\ 4\ 23}$ one ($300W$), respectively. It is
important to notice in table 2 that, by means of the TFH, the final energy is
almost two times higher than in the case of SFH at $18\ GHz$, whereas the
amount of recovered electrons (i.e., equal to $38.2\%$) is two times the one
obtained in single frequency operation ($18\ GHz$; $1000\ W$).

It is the case to remember, according to the saturation curve shown elsewhere
(figure LABEL:), that in order to double the final energy in SFH, we had to
triplicate the power, whereas in TFH, we increase the total power to about
$30\%$. The two frequencies heat the plasma much better than the single one at
$2000\ W$: The final energy is almost $50\%$ higher, and even the amount of
recovered electrons is slightly increased. Then, the effect of the second
frequency is to increase the electron temperature and improve the electron
confinement. This means that also the ion confinement time increases, and,
finally, the extracted ion beam has higher charge states and currents. A more
detailed analysis can be carried out by looking to figure 21. The cumulative
probability is plotted on the vertical axis, with respect to the electron
energy. A point on one of the plotted curves gives the number of
electrons282828Note that the vertical axis has been normalized to $1$. whose
energy is below the corresponding energy value indicated by the abscissa. Let
us define the cumulative probability for a given energy as $CP$, then the
number of particles above a given $E$ value will be, obviously, $1-CP$. It
gives information about the different number of electrons which populates
different energy ranges. It is clear, from the figure, that the $TE_{4\ 4\
23}$ mode is able to remove the greatest amount of electrons from the cold,
initial low energy population. For this mode in SFH the number of the
electrons above $2\ keV$ is higher than the $T_{4\ 6\ 22}$ at any power, being
the $\sim 20\%$ for the former and the $\sim 15\%$ for the latter at $2000\ W$
and even more for $T_{4\ 6\ 22}$ at $1000\ W$. For higher energies, above $6\
keV$, the relative difference increases: for $TE_{4\ 4\ 23}$ $(1-CP)\sim
13\%$, whereas for $TE_{4\ 6\ 22}$ at $2000\ W$ $(1-CP)\sim 8\%$. The gap
between the two modes drops approaching the maximum energy, probably because
of the limited duration of the simulations. This is another evidence of the
potentialities of the frequency tuning. The heating rapidity does not depend
on the power directly, but on the electric field amplitude over the resonance
surface. After $50\ ns$, by means of the TFH all the electrons have the time
to acquire more energy from the wave. This capability to remove great amounts
of electrons from the cold population is one of the main advantages of the
TFH, as it permits to quickly populate the warm plasma component.

The TFH allows the increase of both the electron density and the ion
confinement time, thus increasing $<q>$; it optimizes the electromagnetic wave
energy transfer to the electrons so that a higher quantity of heated electrons
can be created before they leave the confinement region. The experimental
results of TFH obtained with SERSE show an increase of the electron
temperature and of the ion confinement (the CSD peak shifts toward higher
charge states).

###  0.8.2 Frequency Tuning Effect

Many experiments in the last years have shown that significant improvements of
ECRIS performances (both in terms of total extracted current and highly
charged ions production) are obtainable by slightly varying the microwave
frequency in the case of SFH (Single Frequency Heating) [28], that is defined
as a "frequency tuning" effect. It was known that the variation of the
frequency increases the electron density (see eq. eq,frequencyscaling), then
it improve the ECRIS performances. However this occurs only for variations on
the order of $GHz$, and it is strictly connected with the increase of the cut-
off density.

On the contrary several experiments have demonstrated that even slight
variations of the pumping wave frequency may lead to strong variation of the
extracted current. Such variations are on the order of $MHz$ over $GHz$, but
they are sufficient to strongly change the wave propagation inside the plasma
chamber and the mode pattern on the resonance surface.

The first evidence that significant improvements can be obtained by varying
the frequency of the microwaves was given by the different performance
observed for the SERSE and CAESAR ion source when fed by a klystron based or a
travelling-wave-tube (TWT) based generator [4,5] either at 14 and 18 GHz.Other
interesting results came from experiments performed at ORNL and at JYFL [25].

Figure 22:  Comparison between trends of O8+ at 18 GHz for klystron (up to 800
W) and TWT1.

Figure 27 features that a remarkable increase in the production of $O^{8+}$
can be obtained by using the TWT instead of Klystrons; in particular the
current of $O^{8+}$ obtained with the TWT at $240\ W$ is obtained by the
Klystron at $800\ W$, i.e. a power three times higher. It is important to
underline that the two generators operated at two different but close
frequencies; in particular the Klystron was operating at $18.0\ GHz$, while
the TWT at $17.9\ GHz$ (both the amplifiers were fed by DRO-type oscillators).
Such differences of performance was initially explained as a greater frequency
dispersion of the TWT, but a series of measurements carried out with the two
microwave generators, by means of a Spectrum Analyzer, pointed out that the
spectrum of the emitted radiation of a TWT is similar to the one of the
Klystron [37]. Then the only difference between the two generators is the
output frequency, with the further possibility for the TWT to vary the emitted
frequency, thus optimizing the source performances.

In section LABEL:sec,SUPERNANOGAN the experimental set-up used for the
determination of the frequency tuning effect on the SUPERNANOGAN ion source
has been described, together with the set-up of the experiment carried out on
the CAPRICE source at GSI, Darmstadt, which demonstrated as the frequency
tuning strongly affect also the ion beam formation. Fig. 23 shows the current
for $C^{4+}$ obtained with the SUPERNANOGAN ion source in 2005 versus the
microwave frequency, keeping unchanged the power and all the other source
parameters. The figure features strong fluctuations of the extracted current
in a frequency span of about $90\ MHz$.

Figure 23:  Trend of the analyzed $C^{4+}$ current for the SUPERNANOGAN ECR
Ion Source versus the RF frequency.

The frequency was changed in the range $14.44$ to $14.53\ GHz$ with a step of
$1\ MHz$ and it was observed that changes of a few MHz changed the $C^{4+}$
current even of $70\%$. Further experiments including the final validation of
the frequency tuning effect have been carried out with the SUPERNANOGAN source
of CNAO [ref]. The effect was remarkable for this source, as it permitted to
increase by $30\%$ its performance for $C^{4+}$ and by $50\%$ for $H^{3+}$
(these ions are particularly requested for medical applications) and an
additional increase of reliability and availability figures was registered.

Interesting results come from the GSI experiment carried out in 2007. In that
occasion, as stated before, the frequency tuning was demonstrated to strongly
affect also the beam shape, other than the extracted current for each charge
state.

For sake of example fig. 24 reports the shape evolution of a Helium beam
recorded on the first viewer related to an acquisition lasting $150\ sec$
($500\ W$ of microwave power with standard magnetic field profile and
injection pressure of $4.3\cdot 10^{-6}\ mbar$).

Figure 24:  Variation of the extracted beam shape for different microwaves
frequency.

The two helium charge states can be observed as well as the aberrations
introduced by the hexapole. Furthermore, fig. 24 evidently shows that the beam
intensity distribution is inhomogeneous and that this distribution changes
with the microwave frequency, while keeping all the other parameters
constants.

By keeping the source in these conditions and looking to the analyzed beams
after the dipole magnet, it can be clearly observed a change of the $He^{2+}$
beam current during the sweep together with a change of the total current
extracted by the ion source (fig. 25).

Figure 25:  Trend of the $He^{2+}$ extracted current (full line) for different
microwave frequencies. The total extracted current is also shown (dashed
line).

The current increase for some values of the microwave frequency is about the
$50\%$. Such figure can be compared with the fig. 23: as in that case the
current strongly fluctuates with few $MHz$ of frequency variation with a
comparable percentage.

In addition from fig. 23 comes that the total extracted current also varies
with the frequency, but with a lower percentage with respect to the $He^{2+}$
current.

The experiment was repeated by positioning the viewers in different position
of the beam line, but in any case beam transport settings were kept constant:
the effect of the frequency change was clearly visible also in these cases.

Similar observations have been done for Argon beams; for sake of example the
fig. 26 shows the ion current evolution of the $Ar^{8+}$ charge state during
some frequency sweeps (each one $150\ sec$ long): the source conditions were
kept the same during the measurements ($235\ W$ of microwave power with fixed
value of gas pressure and with the same magnetic field profile). The periodic
behavior of these experimental results with respect to the imposed periodic
variation of the microwave supply frequency around the $14.5\ GHz$ value shows
that the described measurements are repeatable and reliable.

Figure 26:  Ar8+ evolution with the frequency : the time is reported on the X
axis (the sweep is $150\ sec$ long), while in the Y axis the beam current is
reported ($100\mu A$ full scale).

Measurements of the $S_{11}$ scattering parameter were carried out in
parallel. The $S_{11}$ parameter is connected with the reflection coefficient,
and the measurements have shown that for some frequencies the amount of the
reflected power increases, thus demonstrating that some resonances are present
inside the cavity with or without plasma. In addition, fluctuation in the
extracted power occurs also in case of quasi-constant reflection coefficient.

This "frequency tuning" effect has been verified experimentally even for
different ion sources [29, 30], and for its comprehension we started the
investigation of plasma properties by means of numerical simulations.

Other experimental results obtained in the last years are connected somehow to
the frequency tuning effect. In particular the results obtained with SERSE in
2001-2002 by using the TWT or the Klystron generators while keeping unchanged
the other source parameters and by considering the net microwave power (the
waveguide losses were about the same for the two cases.

Figure 27:  Comparison between trends of $O^{8+}$ at $18\ GHz$ for klystron
(up to $800\ W$) and TWT for different values of RF power.

Figure 27 features that a remarkable increase in the production of $O^{8+}$
can be obtained by using the TWT instead of Klystrons; in particular the
current of $O^{8+}$ obtained with the TWT at $240\ W$ is obtained by the
Klystron at $800\ W$, i.e. a power three times higher. It is important to
underline that the two generators operated at two different but close
frequencies; in particular the Klystron was operating at $18.0\ GHz$, while
the TWT at $17.9\ GHz$ (both the amplifiers were fed by DRO-type oscillators).
Such differences of performance was initially explained as a greater frequency
dispersion of the TWT, but a series of measurements carried out with the two
microwave generators, by means of a Spectrum Analyzer, pointed out that the
spectrum of the emitted radiation of a TWT is similar to the one of the
Klystron [37].

A series of simulations were carried out in order to verify the effect of
different microwave frequencies on the electron dynamics. It has been
demonstrated in section LABEL: that slight variation of the injection
frequency change the electromagnetic field distribution in the plasma chamber,
as long as the standing wave formation can be guessed also in presence of
plasma. More than the global distribution of the field inside the chamber, the
mode pattern over the resonance surface must be investigated. The resonance
surface can be calculated numerically once known the magnetic field structure,
as it corresponds to that surface with constant field equals to
$\frac{m\omega_{R}F}{q}$. The figure 28 features the resonance surface of the
SERSE source calculated numerically by means of MATLAB. Note that by varying
the frequency of about $50\ MHz$ the field distribution over the surface
changes remarkably. In particular if one consider the zones of maxima the
electric field is more than $10^{2}$ higher in case of figure 28 (b) than in
case of figure 28.

Figure 28:  Simulated resonance surface of the SERSE source with the
distribution of the electromagnetic field (false color representation).

Then the excitation of near modes, even with distances of few MHz, strongly
changes not only the maximum field over the surface, but also the distribution
of minima and maxima zones.

Simulations carried out considering different modes have demonstrated that
some of them are able to heat the electrons up to energies much higher than o

##  0.9 Introduction

Authors must observe the following instructions to ensure consistency and
uniformity in the style and layout of CERN Reports. Wherever possible, each
contribution will go directly to be printed, without any editing, and should
therefore be in its final form.

Although the typesetting rules that are described are of a general nature, the
present document emphasizes their use with LaTeX. Templates and more detailed
information for _Word_ users are available on the DTP Web site (_Word_ Section
at the page http://cern.ch/DTP/dtpcernreport.htm).

Note that it is the author’s responsibility to obtain permission from the
copyright holder if material taken from other sources is included in the
source submitted as a CERN Report.

##  0.10 Generalities on typing

When submitting your electronic source in LaTeX, you must use the cernrep
class file, which is available from the DTP Web site (http://cern.ch/DTP).
Moreover, specific instructions for preparing your LaTeX contribution are
given in the appendices.

###  0.10.1 Format for the text

Take care that your material stays inside the text frame which is 16 cm
$\times$ 24 cm (a A4 sized paper has 3.0 cm margins at the top and at the
bottom, and 2.5 cm left and right). This is expecially true for figures and
tabular content.

##  0.11 Parts of the text

###  0.11.1 Abstract

Your contribution should be preceded by an abstract of not more than 150
words, written as a single paragraph.

###  0.11.2 Sectioning commands and paragraphs

The standard LaTeX commands section (level 1), subsection (level 2),
subsubsection (level 3), subsubsubsection (level 4), should be used for
headings. Do not start a new section on a new page, but continue on the same
page.

In the text, leave a blank line after each paragraph.

###  0.11.3 Equations

Equations should be treated as part of the text, and therefore punctuated
(with a space between the end of the equation and the punctuation mark).
Equations are numbered consecutively throughout the report.

An unnumbered single formula is delimited by [ and ] (alternatively a
displaymath environment can be used), while a numbered equation is generated
with the equation environment. You should never use the $$ construct. The CERN
LaTeX classes automatically load the amsmath class, which offers a large
choice of constructs for typesetting mathematics [1]. See also Appendix .14.2
for some hints about using the amsmath extensions in an optimal way.

For cross-referencing equations use the LaTeX commands label and ref, as
explained in sec:crossref. A simple example, with a cross-reference to a
formula, follows.

    \section{Section title}
    \label{sec:mysection}
    Einstein has expressed the relation between energy $E$
    and mass $m$ in his famous equation~(\ref{eq:einstein}):
    \begin{equation}
    E=mc^2 \label{eq:einstein}
    \end{equation}

###  0.11.4 Figures

#### How to supply figures

Where possible, figures should be prepared electronically. Make sure that the
image is of high quality when printed (in black and white) and is of high
enough resolution (min. 300 dots per inch). We accept PICT, TIFF, PNG,and JPEG
files. Encapsulated PostScript (EPS) files are preferred, but in any case,
send us your illustrations in their _original_ format (PNG, JPEG, ), there is
no need for you to reprocess them yourself.

If you cannot provide electronic copies, or if we consider that your files do
not print out well, you should provide a good-quality paper original. In this
case we will arrange for it to be scanned at CERN and included in the report.

Do not use free-hand lettering for the labelling of figures. All lines should
be drawn in red or black ink and be heavy enough (min. 0.75 point) and all
figures, decimal points, symbols, , large enough and sufficiently spaced to
ensure clarity when printed _at the final size_. If using colour, please
ensure that the figure prints clearly in greyscale and adapt your text knowing
that the difference between coloured items when reproduced in grey is not
obvious.

As already mentioned, before using material such as illustrations taken from
other sources, do not forget to obtain permission from the copyright holder.

#### Positioning and layout

All figures must remain within the page area (16cm $\times$ 24cm), where, if
necessary, the page may be turned 90∘ to accommodate the figure. When this is
done, the caption must be oriented in the same way as the figure, and no other
text may appear on that page. The bottom of the turned illustrations should be
at the right-hand side of the page.

#### Including your figure

Figures are included with the figure environment. In the example that follows
we include an EPS graphics file (myfig.eps, where the extension .eps does not
need to be specified since it is the default) with the includegraphics
command, which is defined in the graphicx package that is loaded by default by
the CERN LaTeX classes.

The figure caption (specified as argument of the caption command), must
_follow_ the figure body, and should be brief. No full stop is necessary
unless the caption is more that one sentence long, in which case full
punctuation should be used.

Example

An example with a cross-reference to a figure follows. The reference is
defined by the label command _following_ the caption command. A cross-
reference is generated with the ref command on the second line. Note the
cross-reference key (fig:myfig) which clearly indicates that it refers to a
figure (see sec:crossref for a discussion of the importance of using good
keys).

    \section{Section title}
    In Fig.~\ref{fig:myfig} we see that ...
    \begin{figure}
    \centering\includegraphics[width=.9\linewidth]{myfig}
    \caption{Description of my figure}
    \label{fig:myfig}
    \end{figure}

#### References to figures

LaTeX’s cross-reference mechanism can be used to refer to figures. A figure is
defined with the label command, which must _follow_ the caption command.

Figures must be referenced in the text in consecutive numerical order with the
help of the ref command. Examples of references to figures and how to produce
them follow (see also tab:predef).

  * •

‘Fig. 3’ produced by, `\Fref{fig:myfig}`,

  * •

‘Figs. 3–5’ produced by, `\Figures~\ref{fig:myfiga}--\ref{fig:myfigb}`,

  * •

‘Figure 3’, produced by, `\Figure[b]~\ref{fig:myfig}`. Note the use of the
optional argument [b], which indicates that the word ‘Figure’ should be
typeset in full, in particular at the beginning of a sentence.

Figures with several parts are cited as follows: ‘Fig. 2(a) and (b), Figs.
3(a)–(c)’.

Figures and illustrations _should follow_ the paragraph in which they are
first discussed. If this is not feasible, they may be placed on the following
page (LaTeX’s float mechanism takes care of this automatically, in principle).
If it is not possible to place _all_ numbered figures in the text, then they
should _all_ be placed at the end of the paper.

###  0.11.5 Tables

Tables are defined the the table environment. Each table should be centred on
the page width, with a brief caption (specified as argument of the caption
command) _preceding_ the table body.

In general, tables should be open, drawn with a double thin horizontal line
(0.4 pt) at the top and bottom, and single horizontal line (0.4 pt) separating
column headings from data.

Like figures, tables must be referenced in the text in consecutive numerical
order with LaTeX’s ref command. Examples of cross-references to tables follow
(commands that can be used to generate the given text strings are shown
between parentheses): ‘Table 5’ (`\Tref{tab:mytab}` or
`\Table~\ref{tab:mytab}`), ‘Tables 2–3’
(`\Tables~\ref{tab:mytaba}--\ref{tab:mytabb}`). The word ‘Table’ should never
be abbreviated.

#### Formatting and layout within the table

Write the headings in sentence case but do not use full stops. Units should be
entered in parentheses on a separate line below the column heading. (If the
same unit is used throughout the table, it should be written in parentheses on
a separate line below the caption.)

Unsimilar items should be aligned on the left, whereas similar items should be
aligned on the operator or decimal point. All decimal points must be preceded
by a digit.

Table captions

Table captions should be brief and placed centrally _above_ the table. No full
stop is necessary unless the caption is more that one sentence long, in which
case full punctuation should be used.

Notes in the table

Notes in tables should be designated by superscript lower-case letters, and
begun anew for each table. The superscript letter should be placed in
alphabetical order moving from left to right across the first row and down to
the last. The notes should then be listed directly under the table.

References cited in tables should appear in consecutive numerical order,
following the order explained above for notes (left to right, top to bottom).
They should appear exactly as references cited in the main body of the
document (and be treated as part of the text in terms of numbering, so be
careful when moving tables containing references).

Example

An instance of a simple table, following the proposed rules follows. Notice
the hline commands at the beginning and end of the tabular environment to
generate the double lines at the top and the bottom of the table, as well as
the single hline command to separate the heading from the data.

    \section{Section title}
    Table~\ref{tab:famous} shows three famous mathematical constants.
    \begin{table}
    \caption{Famous constants}
    \label{tab:famous}
    \centering
    \begin{tabular}{lll}\hline\hline
    symbol & description               & approximate value\\\hline
    $e$    & base of natural logarithm & $2.7182818285$   \\
    $\pi$  & ratio circle circumference to diameter &
                                         $3.1415926536$   \\
    $\phi$ & golden ratio              & $1.6180339887$   \\\hline\hline
    \end{tabular}
    \end{table}

###  0.11.6 Bibliographical references

References should be cited in the text using numbers within square brackets:
‘example [1], example [1, 2], example [1–5]’. This is achieved with the cite
command. One can also write ‘see Ref. [1]’, ‘Refs. [1]–[5]’, and, at the
beginning of a sentence, ‘Reference [2]’. These strings can be produced as
follows (see also tab:predef).

  * •

`\Bref{bib:mybiba}` or `\Ref~\cite{bib:mybiba}` typeset, ‘Ref. [1]’.

  * •

`\Refs~\cite{bib:mybiba}--\cite{tab:mybibb}`, typesets, ‘Refs [1–5].

  * •

`\Bref[b]~\ref{bib:mybibc}` typesets, ‘Reference [3]’. The optional argument
[b] indicates that the word ‘Reference’ should be typeset in full, in
particular at the beginning of a sentence.

Bibliographical references should appear in consecutive numerical order and
should be listed in numerical order at the end of the text. Punctuation can be
used either within or outside the brackets, but please ensure one method is
used consistently throughout the contribution.

#### List of references

The list of references (the bibitem entries) must all be grouped inside a
thebibliography environment, as follows.

    \section{Section title}
    Work by Einstein~\cite{bib:einstein} as well as \Bref{bib:gravitation}
    explains the theory of relativity.
    ...
    \begin{thebibliography}
    \bibitem{bib:einstein}} ...
    \bibitem{bib:gravitation} ...
      ...
    \end{thebibliography}

As shown, each bibitem is identified with a reference key (bib:einstein,
bib:gravitation, ), which allows one to refer to the relevant bibliography
entry with a `\cite{bib:einstein}` command in the text. The information about
each entry can be specified in a separate file, mybib.bbl, which can be read
by the program for generating the list of references (see a LaTeX manual or
Chapter 12 of Ref. [2] for details).

Unless you are near the bottom of the last page of text, do _not_ start a new
page for the list of references, but continue on the same page. Note that in
the list of references it is unnecessary to state the title of an article or
chapter in proceedings or in a collection of papers unless a page number
cannot be quoted, for forthcoming publications.

For abbreviations of names of journals quoted in the references, see the
_Journal abbreviations_ entry available from the Web page at the URL
http://cern.ch/DTP/dtpgrammar.htm292929of Modern Physics site:
http://rmp.aps.org/info/manprep.html also has a list.. The entry _Citation of
references_ on the same Web page shows more details on how to present
references.

If you need to provide a bibliography, this should come after the list of
references.

###  0.11.7 Footnotes

Footnotes are to be avoided. If absolutely necessary, they should be brief,
and placed at the bottom of the page on which they are referred to. Take care
when citing references in the footnotes to ensure that these are correctly
numbered.

###  0.11.8 Referencing structural elements

In this section we give a general overview of LaTeX’s reference mechanism
which makes it easy to reference structural elements. First a label command,
with a unique _key_ as its argument to identify the structural element in
question, must be placed in the source, as follows.

  * •

For sectioning commands, such as section, subsection, subsubsection, the label
command must _follow_ it.

  * •

Inside figure and table environments, the label command must be placed _after_
the caption command.

  * •

Inside an equation environment the label command can be placed anywhere.

  * •

Inside an eqnarray environment, the label command can be used to identify each
line, so that it must be placed _before_ each end-of-line . If for a given
line no line-number has to be produced, a nonumber command should be used.

  * •

Inside an enumerate environment a label command can be associated with each
item command.

  * •

Inside a footnote a label command can be placed anywhere.

As seen in all the examples in this document, for reasons of clarity it is
best to place the label command immediately _following_ the element if refers
to (rather than inserting it inside its contents).

From any place in the document one can refer to a structural element
identified with a label command with the help of a ref command. An example
follows.

    \section{My first section}\label{sec:first}
    Figure~\ref{fig:fdesc} in Section~\ref{sec:second} shows \dots
    \begin{table}
    \caption{table caption text}\label{tab:tdesc}
    ...
    \end{table}
    \begin{equation}
    \exp{i\pi}+1=0\label{eq:euler}
    \end{equation}
    \section{My second section}\label{sec:second}
    In Section~\ref{sec:first} contains Table~\ref{tab:tdesc} and
    Eq.~\ref{eq:euler}
    \begin{figure}
    \centering\includegraphics[...]{...}
    caption{Text of figure caption}
    \label{fig:fdesc}
    \end{figure}

To easily differentiate between references to the various structural elements
it is good practice to start the key with a few characters identifying it
(sec: for a sectioning command, such as section, subsection, , fig: for
figures, tab: for tables, and eq: for equations, including eqnarray
environments (which should have a label command placed before the if you want
to identify the line in question). The second part of the key should identify
the particular element clearly, use of a mnemonic component, such as eq:euler
in the example of the equation reference above. Avoid using keys with only
digits, such as f1, f2, , since, if for any reason structural elements are
eliminated or change position in the source, confusion can result.

###  0.11.9 Appendices

Each appendix should be laid out as the sections in the text. Appendices
should be labelled alphabetically and be referred to as Appendix A, Appendices
A–C, Equations, figures and tables should be quoted as Eq. (A.1) and Fig. A.1,

###  0.11.10 Acknowledgements

If required, acknowledgements should appear as an unnumbered subsection
immediately before the references section.

##  0.12 Spelling and grammar

For more information on English grammar rules and commonly misused words and
expressions (including a guide to avoiding ‘franglais’), please see the files
available from the DTP Web pages.

###  0.12.1 Spelling

CERN uses British English spelling, and ‘-ize’ rather that ‘-ise’. Here we
provide a few examples for guidance:

@lX -il: & fulfil (not fulfill)

-re: centre (not center)

-our: colour (not color)

-gue: catalogue (not catalog, but analog is used in electronics)

-mme: programme (not program, unless referring to a computer program)

-ell-: labelled (not labeled)

-ce/-se: licence (noun), license (verb), practice (noun), practise (verb)

-ize: organization, authorize.

Exceptions to this rule include advise, comprise, compromise, concise, demise,
devise, enterprise, exercise, improvise, incise, precise, revise, supervise,
surmise, surprise, televise.

###  0.12.2 Punctuation

#### Hyphen

Hyphens are used to avoid ambiguity, in attributive compound adjectives
(compare ‘a little used car’ and ‘a little-used car’), to distinguish between
words such as ‘reform’ (change for the better) and ‘re-form’ (form again), and