Luminosity Limitations for Electron-Ion Collider[^1]

V. A. Lebedev

Thomas Jefferson National Accelerator Facility

12000 Jefferson Avenue, Newport News, VA 23606

Email -- lebedev@jlab.org

**Abstract.** The major limitations on reaching the maximum luminosity
for an electron ion collider are discussed in application to the
ring-ring and linac-ring colliders. It is shown that with intensive
electron cooling the luminosity of 10^33^ cm^-2^s^-1^ is feasible for
both schemes for the center-of-mass collider energy above approximately
15 GeV. Each scheme has its own pros and cons. The ring-ring collider is
better supported by the current accelerator technology while the
linac-ring collider suggests unique features for spin manipulations of
the electron beam. The article addresses a general approach to a choice
of collider scheme and parameters leaving details for other conference
publications dedicated to particular aspects of the ring-ring and
linac-ring colliders.

# Introduction

Currently HERA at DESY and CEBAF at Jefferson Lab are two major players
in the study of structure of the proton and light ions. HERA operates
with polarized electrons and unpolarized protons, while CEBAF operates
with polarized electrons and polarized protons or nuclei. HERA^\[1\]^ is
an electron-proton collider with very high center-of-mass energy (320
GeV) and moderate luminosity (1.7⋅10^31^ cm^-2^s^-1^), while
CEBAF^\[2,3\]^ is a fixed target machine with moderate energy and
practically unlimited luminosity. The energy of the electron beam at
CEBAF can presently be varied from 0.6 to 6 GeV and the machine is
expected to be upgraded to 12 GeV^\[4\]^ within about 5 years, boosting
the center-of-mass energy of electron-proton collisions to 4.8 GeV.
CEBAF luminosity for operation into the 4π CLAS detector of Hall B is
mainly limited by the detector to about 10^34^ cm^-2^s^-1^. The
luminosity is a few orders of magnitude higher for two other halls,
which use spectrometers for particle detection.

For further progress in the study of nucleon spin structure a machine in
the intermediate range of energies is required. The energy range of 15
to 50 GeV is currently considered to be interesting, with a request for
the luminosity to be 10^33^ cm^-2^s^-1^ or above. For physics of
interest the effective luminosity of the collider is proportional to the
square of each beam polarization,
$L_{\text{effective}} = L_{o}p_{e^{2}}p_{i^{2}}$, and therefore
achieving polarization above 70% for both electron and ion beams is of
primary importance. Naive model of electron-quark collisions would
require a ratio of the electron to proton energy to be about 1 to 6. We
will not impose such a constraint but, as one will see later, an
optimization of machine parameters yields close energy ratio.

Two general concepts for the collider have been suggested. The first is
the classical ring-ring collider^\[5\]^, where ion and electron beams
are stored in independent storage rings of the same circumference. The
second concept is the linac-ring collider^\[6\]^, colliding protons in a
storage ring against electrons from an energy recovery linac. Below we
will try to analyze advantages and disadvantages of each scheme, as well
as major factors limiting the collider luminosity. Reaching 10^33^
cm^-2^s^-1^ luminosity with minimum cost will be our major optimization
criterion. Because the machine luminosity generally grows with energy,
we will also address the question at which energy each of the considered
schemes can achieve the required luminosity. Although the parameter list
and the luminosity optimization considered below are carried out for the
electron-proton collider all results are also applicable to the
electron-ion collider.

# LUMINOSITY Limitations 

There are three major limitations on the beam brightness. The first is
the space charge effect in the ion beam. It causes nonlinear dependence
of particle betatron tune on amplitude (Laslett tune shift) and,
consequently, the loss of the particle motion stability in the case of
large tune shift. This limits the phase density of the ion beam and is
one of the major limitations at low ion energy. The second one is the
beam-beam effect at the interaction point (IP) which limits the density
of beams for both collider schemes. The third one is intrabeam
scattering (IBS) in the ion beam. Although the intensity of IBS
decreases with increasing energy, it is still an important limitation
even at highest energy considered for the collider. In the estimates
below we presume that the horizontal and vertical beam emittances, as
well as the horizontal and vertical beta-functions at the IP, are equal
for each of the beams (round beams). However, we do not necessarily
consider the beam sizes for each of two beams to be equal. In this case
the luminosity is determined by the following formula,

$L = \frac{N_{e}N_{i}f_{0}}{2\pi\left( \sigma_{i}^{2} + \sigma_{e}^{2} \right)}$
, (1)

where $N_{e}$ and $N_{i}$ are the number of electrons and ions per
bunch, $\sigma_{e}^{}$ and $\sigma_{i}^{}$ are the electron and ion beam
sizes, and $f_{0}$ is the bunch frequency.

## 1.1 Laslett Tune Shift Limit

We will start our consideration from the Laslett tune shift. Its value
is determined by the following formula

${\Delta\nu}_{i} = \frac{Z^{2}e^{2}N_{i}R}{2\sqrt{2\pi}\ m_{i}c^{2}\gamma_{i^{3}}\beta_{i^{2}}\sigma_{\text{si}}\varepsilon_{i}}$
, (2)

where *Ze* is the ion charge, *m~i\ ~*= *Am~p~* is the ion mass, *R* is
the storage ring mean radius, *γ~i~* and *β~i~* are the ion relativistic
factors, *ε~i~* is the ion beam emittance, and $\sigma_{\text{si}}$ is
the rms longitudinal beam size. Substituting the ratio *N~i~*/*ε~i~*
from the above equation into the luminosity formula and choosing the
beta-function at the IP to be equal to the bunch
length,$\beta^{} = \sigma_{\text{si}}$, one obtains the following
remarkably simple formula for the luminosity limit due to the Laslett
tune shift:

$\begin{array}{r}
L_{\Delta\nu} = \sqrt{\frac{2}{\pi}}\frac{\eta}{\ Z}\frac{I_{e}B_{d}}{e^{2}\left( 1 + \sigma_{e}^{2}/\sigma_{i}^{2} \right)}\gamma_{i^{2}}\beta_{i}{\Delta\nu}_{i}\overrightarrow{} \\
\ \left( \frac{\eta}{0\text{.}2} \right)\left( \frac{I_{e}}{1\ A} \right)\left( \frac{B_{d}}{4\ T} \right)\left( \frac{{\Delta\nu}_{i}}{0\text{.}\text{05}} \right)\left( \frac{\frac{E_{i}}{A}}{\text{15}\ \frac{\text{GeV}}{\text{nucleon}}} \right)^{2}\frac{1\text{.}\text{06} \cdot \text{10}^{\text{33}}\ \text{cm}^{- 2}s^{- 1}}{Z\left( 1 + \sigma_{e}^{2}/\sigma_{i}^{2} \right)}\ \text{.}
\end{array}$ (3)

Here *B~d~* is the magnetic field of ring dipoles and $\eta$ is the
fraction of the machine circumference covered by the dipoles, *I~e~* is
the electron beam current, and we expressed the ring radius through the
bending magnetic field, the beam momentum and the dipole occupation
factor $\eta$.

The values of parameters in Eq. (3) with the exception of the electron
beam current and ion beam energy are well determined and cannot be
significantly increased. In particular, the electron cooling sets the
maximum value of Laslett tune shift, *∆ν*, to about 0.05 determined by
the distance to the nearest non-linear resonance. Actually, the cooling
cools the beam to the equilibrium between cooling and heating coming
from non-linear resonances and intrabeam scattering. If machine
parameters are chosen so that the last does not dominate, the beam is
cooled as far as it is not heated by the resonance. In absence of
cooling higher order resonances play more significant role and *∆ν* is
smaller. In this case the emittance is rather determined by the ion
source emittance and the emittance growth in the course of acceleration.
It is currently expected that ${\Delta\nu}_{i} = 0\text{.}\text{01}$ can
be achieved without cooling.

The luminosity limit is proportional to the occupation factor and
magnetic field of the dipoles. Therefore superconducting dipoles are
strongly favored even in the case of comparatively small energy (\~10-20
GeV), when conventional wisdom prefers a normal conducting synchrotron.
The request for two IPs suggests that the ion ring should be a
racetrack. Then, the interaction regions and other systems such as RF
system, beam injection, electron cooling and Siberian snakes for
suppression of depolarizing spin resonances should be located in the
straight sections. The length of the straight sections does not depend
much on the ring energy and expected to be slightly above 100 m. Using a
strong magnetic field in the dipoles decreases the ring circumference
and increases the luminosity limit. But at a proton energy of about 20
GeV the bending radius of SC dipoles is quite small (\~10-20 m) which
complicates the dipole design for very high magnetic field. The choice
of superferric dipoles with about 4 T field is a reasonable compromise.
Further increase of the magnetic field makes dipoles more complicated
but does not bring significant gain in the machine circumference because
it starts to be limited by the length of the straight sections. Taking
all of the above into account we can estimate *η* = 0.2 for 4 T dipoles.
For an energy of 20 GeV/nucleon, this yields a ring circumference of
about 500 m.

The luminosity limit of Eq. (3) does not depend on the ion beam current
and is proportional to the electron beam current. The recent
commissioning of B-factories^\[7,8\]^ suggests that an electron beam
current of 2 A can be achieved in a storage ring with parameters
required for the collider. That determines that to reach 10^33^
cm^-2^s^-1^ luminosity one needs a proton energy above 15 GeV for the
ring-ring scheme. For the linac-ring scheme the major limitation for
electron beam current comes from the electron injector^\[6\]^. It sets
the current maximum to about 0.2-0.3 A. In this case the energy of
proton beam needs to be above about 30 GeV to achieve 10^33^ cm^-2^s^-1^
luminosity, taking into account the factor of almost 2 which can be
obtained with an electron beam size significantly smaller than the ion
beam size, $\sigma_{e}^{2}\text{<<}\sigma_{i}^{2}$. As one will see
below such a choice is not limited by other constraints.

Note that in the absence of ion beam cooling ∆*ν* is about five times
smaller. For both schemes this requires an additional increase of ion
energy by factor of$\sqrt{5}$to achieve 10^33^ cm^-2^s^-1^ luminosity.
One can also see from Eq. (3) that the luminosity per nucleon *LA* is
proportional to *A*/*Z* and therefore practically does not depend on *Z*
for fully stripped ions. As will be seen below intrabeam scattering puts
more severe request for the ion beam energy increase in the absence of
cooling.

## 1.2 Luminosity Limit due to Beam-beam Effects

Beam-beam effects in colliders are well known. They shift the betatron
tunes of the beams and set the following two luminosity limits,

$\begin{array}{r}
L_{i} = \frac{I_{i}Ε_{i}\xi_{i}}{e^{3}{Z\beta}_{i}^{}}\frac{2}{1 + \sigma_{i}^{2}/\sigma_{e}^{2}} = \\
\ \left( \frac{I_{i}}{1\ A} \right)\left( \frac{Ε_{i}}{\text{15}\ \text{GeV}} \right)\left( \frac{\xi_{i}}{0\text{.}\text{02}} \right)\left( \frac{7\ \text{cm}}{\beta_{i}^{}} \right)\frac{2}{1 + \sigma_{i}^{2}/\sigma_{e}^{2}}1\text{.}\text{85} \cdot \text{10}^{\text{33}}\ \text{cm}^{- 2}s^{- 1}\ ,
\end{array}$ (4)

$\begin{array}{r}
L_{e} = \frac{I_{e}Ε_{e}\xi_{e}}{e^{3}{Z\beta}_{e}^{}}\frac{2}{1 + \sigma_{e}^{2}/\sigma_{i}^{2}} = \\
\ \left( \frac{I_{e}}{1\ A} \right)\left( \frac{Ε_{e}}{5\ \text{GeV}} \right)\left( \frac{\xi_{e}}{0\text{.}\text{035}} \right)\left( \frac{7\ \text{cm}}{\beta_{e}^{}} \right)\frac{2}{1 + \sigma_{e}^{2}/\sigma_{i}^{2}}1\text{.}\text{08} \cdot \text{10}^{\text{33}}\ \text{cm}^{- 2}s^{- 1}\ ,
\end{array}$ (5)

which corresponds to the tune shift limitations in the ion and electron
beams. Here $E_{i} = \text{Am}_{p}c^{2}\gamma_{i}$ and
$E_{e} = m_{e}c^{2}\gamma_{e}$ are the energies of the ion and electron
beams, and the parameters
$\xi_{i} = e^{2}\text{ZN}_{e}\beta_{i^{}}/\left( 4{\pi E}_{i}\sigma_{e}^{2} \right)$
and
$\xi_{e} = e^{2}\text{ZN}_{i}\beta_{e^{}}/\left( 4{\pi E}_{e}\sigma_{i}^{2} \right)$
correspond to the linear tune shifts in the ion and electron beams.
There is not much freedom in choice of the parameters in the above
equations. The achievable value of betatron tune shift for the ion beam
depends on the beam cooling. We currently believe that with effective
electron cooling (10 to 100 s damping time) *ξ~i~* can be close to 0.02
per IP while without cooling it should be an order of magnitude lower.
The beta-function at the IP is determined by the ion bunch length and is
limited by the beam separation, the chromaticity of the final focus and
its aperture. All these considerations limit *β~i~* to the range of 6 to
10 cm for ion beam energy from 15 to 30 GeV. The luminosity limit
described by Eq. (4) does not really depend on the collider scheme (both
schemes have similar ion rings) and for the ion energy of 15-30 GeV sets
the required value of the proton current to the range of 1-2 A.

In the case of the luminosity limit due to electron beam tune shift
described by Eq. (5) the electron beam parameters depend on the collider
scheme. Considerable experience acquired on the electron-positron
colliders suggests that *ξ*~e~ = 0.035 can be achieved. For the
linac-ring scheme with single IP when one can accept quite significant
electron beam emittance growth after the collision the tune shift can be
as large as one^\[6\]^ and for practical machine parameters the
beam-beam effects do not limit *ξ*~e~. With two interaction points the
tune shift will be limited by the emittance growth after the first
collision and *ξ*~e~≤0.2.

For the linac-ring scheme there is another limitation on the product
*ξ~e~ξ~i~* which usually puts more severe limitations on possible
collider parameters. In this case the interaction of electron beam with
the ion beam transfers an electromagnetic excitation from the ion beam
head to its tail and thus acts similar to the transverse impedance of
the ion ring causing the ion beam kink instability above the
threshold^\[9,10\]^. Taking into account an estimate of Ref. \[6\] and
more accurate simulation results of Ref. \[11\] the threshold can be
parametrized in the following form,

$\xi_{e}\xi_{p} \leq \frac{\nu_{\text{si}}}{5\pi}\frac{\beta_{e}^{}}{\sigma_{\text{si}}}$
, (6)

where *ν~si~* is the dimensionless synchrotron tune. Taking into account
that the synchrotron tune for a high energy proton synchrotron hardly
can be made more than 0.01 and *ξ~p~* is desired to be about 0.005-0.01
that limits *ξ~p~* to be less than about 0.1.

## 1.3 Intrabeam Scattering

Another important limitation on the beam brightness is determined by
intrabeam scattering (IBS). For the above collider parameters the
longitudinal energy spread in the beam frame is significantly smaller
than the transverse one. That allows one to get comparatively simple
formulas to describe IBS. In this case IBS transfers the energy from the
transverse degrees of freedom to the longitudinal one and the growth
rate can be approximated by the following formula^\[12\]^,

$\frac{d}{\text{dt}}\left( \theta_{{||}^{2}} \right) \equiv \frac{d}{\text{dt}}\left( \frac{\overline{p_{{||}^{2}}}}{p} \right) = \frac{\left( \text{Ze} \right)^{4}N_{i}}{4\sqrt{2}A^{2}m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}}\left\langle \frac{\Xi_{||}\left( \theta_{x},\theta_{y} \right)}{\sqrt{\theta_{x^{2}} + \theta_{y^{2}}}}\frac{L_{C}}{\sigma_{x}\sigma_{y}} \right\rangle_{s}$
. (7)

Here

$\Xi_{||}(x,y) \approx 1 + \frac{\sqrt{2}}{\pi}\text{ln}\left( \frac{x^{2} + y^{2}}{2\text{xy}} \right) - 0\text{.}\text{055}\left( \frac{x^{2} - y^{2}}{x^{2} + y^{2}} \right)^{2}$,
(8)

averaging is performed along the beam orbit,
$\sigma_{x} = \sqrt{\varepsilon_{x}\beta_{y} + D^{2}\theta_{{||}^{2}}}$,
$\sigma_{y} = \sqrt{\varepsilon_{y}\beta_{y}}$,
$\theta_{x} = \sqrt{\varepsilon_{x}/\beta_{x}}$and
$\theta_{y} = \sqrt{\varepsilon_{y}/\beta_{y}}$ are the beam sizes and
angular spreads along the ring, and

$L_{C} = \text{ln}\left( \sqrt{\sqrt{\frac{\pi}{2}}\frac{\sigma_{x}\sigma_{y}\sigma_{\text{si}}\gamma_{i}}{N_{i}}\left( \frac{\text{Am}_{p}c^{2}\gamma_{i^{2}}\beta_{i^{2}}\left( \theta_{x^{2}} + \theta_{y^{2}} \right)}{2e^{2}} \right)^{3}} \right)$
(9)

is the Coulomb logarithm.

In the smooth focusing approximation, for equal horizontal and vertical
emittances, $\varepsilon_{x} = \varepsilon_{y}$, equal betatron tunes,
$\nu_{x} = \nu_{y}$, and small contribution of energy spread into the
beam size,$D(\Delta p/p)\text{<<}\sigma_{x}$, the momentum spread growth
rate can be written in the following form

$\Lambda_{||} \equiv \frac{1}{\theta_{{||}^{2}}}\frac{d}{\text{dt}}\left( \theta_{{||}^{2}} \right) = \frac{Z^{4}}{8A^{2}}\frac{e^{4}N_{i}L_{C}}{m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}\varepsilon_{x^{\frac{3}{2}}}\theta_{{||}^{2}}}\sqrt{\frac{\nu_{x}}{R}}$
. (10)

The heating of the longitudinal degree of freedom, consequently, causes
cooling for both transverse degrees of freedom; but there is another
mechanism, which additionally heats the horizontal degree of freedom. At
regions with non-zero dispersion, changes in longitudinal momentum
change the particles reference orbits, which additionally excites the
horizontal betatron motion,

$\frac{{d\varepsilon}_{x}}{\text{dt}} = \left\langle A_{x}\frac{{d\theta}_{{||}^{2}}}{\text{dt}} \right\rangle_{s}$
, (11)

where

$A_{x} = \frac{D^{2} + \left( D^{'}\beta_{x} + \alpha_{x}D \right)^{2}}{\beta_{x}}$
. (12)

Finally, one can write for the emittance growth rates

$\begin{array}{r}
\begin{bmatrix}
\Lambda_{x} \\
\Lambda_{y}
\end{bmatrix} \equiv \frac{1}{\varepsilon_{x,y}}\frac{{d\varepsilon}_{x,y}}{\text{dt}} = \frac{\left( \text{Ze} \right)^{4}N_{i}}{8\sqrt{2}A^{2}m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}\varepsilon_{x,y}} \\
\ \left\langle \frac{1}{\sqrt{\theta_{x^{2}} + \theta_{y^{2}}}}\frac{L_{C}}{\sigma_{x}\sigma_{y}}\begin{bmatrix}
2A_{x}\Xi_{||}\left( \theta_{x},\theta_{y} \right) - \frac{\beta_{x}}{\gamma_{i^{2}}}\Xi_{}\left( \theta_{x},\theta_{y} \right) \\
 - \frac{\beta_{y}}{\gamma_{i^{2}}}\Xi_{}\left( \theta_{y},\theta_{x} \right)
\end{bmatrix} \right\rangle_{s}\ \text{.}
\end{array}$ (13)

where

$\ \Xi_{}(x,y) \approx 1 + \frac{2\sqrt{2}}{\pi}\text{ln}\left( \frac{\sqrt{3x^{2} + y^{2}}}{2y^{2}}x \right) + \frac{0\text{.}\text{5429}\text{ln}(\frac{y}{x})}{\sqrt{1 + \text{ln}^{2}\left( \frac{y}{x} \right)}}$
. (14)

The energy conservation requires
$\Xi_{}(x,y) + \Xi_{}(y,x) = 2\Xi_{||}(x,y)$ which for considered
approximation is fulfilled with accuracy better than 1%.

In the smooth focusing approximation for equal emittances and betatron
tunes we obtain for the horizontal emittance growth rate,

$\Lambda_{x} = \frac{Z^{4}}{\text{16}A^{2}}\frac{e^{4}N_{i}L_{C}}{m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}\varepsilon_{x^{\frac{5}{2}}}}\sqrt{\frac{R}{\nu_{x}}}\left( \frac{2}{\nu_{x^{2}}} - \frac{1}{\gamma_{i^{2}}} \right)$
. (15)

Note that for considered collider the length of the straight sections is
close to the length of arcs and therefore Eq. (15) underestimates this
growth rate by about factor of two.

## 1.3 Choice of Basic Parameters

The limitations considered above allow one to choose the basic machine
parameters for both collider schemes. As a major goal we will consider
achieving the luminosity of 10^33^ cm^-2^s^-1^ for the electron-proton
collider with the minimum machine energy and, consequently, the minimal
cost. In this case, the Laslett tune shift (see Eq. (3)) and the
electron beam current limitations set the minimum energy of the proton
ring. They are 15 GeV for the ring-ring collider and 30 GeV for the
linac-ring collider.

The distance between bunches is determined by the beam separation after
collision and was chosen to be 3 m for both schemes. The limitation
coming from the beam-beam effects determines the proton beam current and
the energy of the electron beam. Main parameters of the considered
colliders are shown in Table 1.

Although the energies of electron beams for the linac-ring and the
ring-ring colliders are equal they are set by different limitations. For
the ring-ring collider the energy is determined by the beam-beam tune
shift of electrons in the field of protons. For the linac-ring collider
the energy is determined by the kink instability threshold; and
parameters are chosen so that the beam is at the instability threshold
of Eq. (6). The threshold of the kink instability is proportional to the
synchrotron tune and therefore achieving highest possible tune is
desirable. Expressing the tune through the bunch length and the energy
spread we obtain the following formula:

$\nu_{s} = \frac{\gamma_{i}}{1 - \gamma_{i^{2}}}\frac{{q\sigma}_{\text{si}}}{2\pi R}\frac{\text{eV}_{0}}{\text{Am}_{p}c^{2}\theta_{||}}$
, (16)

  ---------------------------------------- --------------- ---------- ---------------- ----------
  **TABLE 1. Parameters of Ring-ring and                                               
  Linac-ring Colliders with Luminosity of                                              
  10^33^ cm^-2^s^-1^**                                                                 

                                           **Ring-ring**              **Linac-ring**   

                                           Electrons       Protons    Electrons        Protons

  Center of mass energy, GeV               13.8                       19.3             

  Kinetic beam energy, GeV                 3               15         3                30

  Circumference, m                         420             420        \-               650

  Betatron tunes                           ≈11             ≈12        \-               ≈16

  Critical kinetic energy, GeV             \-              7.2        \-               9.7

  Beam current, A                          1.7             1.8        0.27             2.7

  Beta-functions at IP, cm                 7               7          7                7

  Rms normalized beam emittance, mm⋅mrad   780             2.26       130              1.4

  Rms beam size at IP, μm                  96              96         40               55

  Laslett tune shift                       \-              0.05       \-               0.05

  Beam-beam tune shift per IP              0.032           0.006      0.15             0.0029

  Synchrotron tune                         \-              0.0075     \-               0.007

  Rms momentum spread                      6⋅10^-4^        8⋅10^-4^   \<5⋅10^-3^       8⋅10^-4^

  Rms bunch length, cm                     ≤ 3             7          ≤ 0.3            7

  Longitudinal intrabeam scattering        \-              110        \-               80
  lifetime, $\Lambda_{{||}^{- 1}}$, s                                                  

  Transverse intrabeam scattering          \-              370        \-               150
  lifetime, $\Lambda_{x^{- 1}}$, s                                                     
  ---------------------------------------- --------------- ---------- ---------------- ----------

where *V*~0~ is the total voltage of the ion ring RF system, and *q* is
the ring's harmonic number. Further decreasing of the momentum spread,
$\theta_{||}$, is limited by the intrabeam scattering and the single
bunch longitudinal instability. Therefore the only free parameter we
have is the RF voltage. Its increase requires increasing the momentum
compaction factor and, consequently, decreasing the betatron tune, which
leads to a fast growth of the transverse IBS growth rate. As result a
compromised value *ν~x~*≈16 has been chosen.

Note that the threshold of the kink instability was computed in the
linear theory, which overestimates the instability threshold. We
currently work on a more detailed non-linear simulation of the
instability and we believe that this more accurate theory should
increase the instability threshold by a factor of two. Consequently, the
electron beam energy could be decreased to about 1.5 GeV, but in the
case of two IPs this additional decrease of the electron beam energy is
limited by deterioration of the electron beam emittance after the first
collision.

If fully striped ions are used in the machine optimized for protons the
Laslett tune shift is going to be the major limitation for luminosity.
Actually, in this case A/Z≈2, and the energy per nucleon is about two
times lower. Substituting this into Eq. (3) yields that the luminosity
per nucleon, *LA*, is also two times smaller than for the proton case.
Usually one would like to keep the same bunch length and the beam size
at the IP and, consequently, the same beam emittance. Then, Eq. (2)
yields that the number of ions should be 4*Z* times less than the number
of protons and the ion beam current should be 4 times less than the
proton beam current. That determines that the tune shift in the electron
beam is one forth of the tune shift for the electron-proton case, and
the tune shift of the ion beam is the same as for the proton beam. The
IBS increments grow by approximately Z/2 times and, as one will see in
the next section, they can be compensated by increased strength of the
electron cooling.

# ELECTRON COOLING 

The intrabeam scattering in the ion beam is so strong that it is
impossible to reach the required luminosity without strong cooling of
the ions. Electron cooling is the only cooling method, which works for
the ion density discussed above. For the collider parameters the
velocity spread in the ion beam is sufficiently large and cooling can be
considered non-magnetized. Then the cooling force in the beam frame
is^\[13\]^:

$F^{'}(v^{'}) = \frac{4\pi\ n_{e}^{'}Z^{2}e^{4}}{m_{e}}\int_{}^{}{\frac{v^{'} - v_{e}^{'}}{\left| v^{'} - v_{e}^{'} \right|^{3}}f_{e}\left( v_{e}^{'} \right)dv_{e^{3}}^{'}}$
, (17)

where $f_{e}\left( v_{e}^{'} \right)$ is the electron distribution
function. The longitudinal velocity spread for the electron beam
accelerated in an electrostatic accelerator is usually much smaller than
its transverse energy spread. In our case of very high energy electron
cooling, it is expected that the electron beam will be accelerated by a
low frequency linear accelerator with energy recovery, and therefore the
longitudinal velocity spread is expected to be significantly higher. For
the estimate we assume that the longitudinal and transverse energy
spreads of electrons are equal. Then, the friction force can be
approximated by the following formula:

$F^{'}(v^{'}) \approx \frac{4\pi\ n_{e}^{'}Z^{2}e^{4}L_{\text{Ce}}}{m_{e}}\frac{v^{'}}{\left( v^{} + 2\text{.}4\overline{v_{e^{2}}^{'}} \right)^{\frac{3}{2}}}$
, (18)

where
$\overline{v_{e^{2}}^{'}} = \overline{v_{\text{ex}^{2}}^{'}} = \overline{v_{\text{ey}^{2}}^{'}} = \overline{v_{\text{ez}^{2}}^{'}}$,
and *L~Ce~* is the Coulomb logarithm.

To find the damping decrement one needs to perform averaging of the
force over betatron motion. For the sake of this estimate we consider
that the ion longitudinal velocity spread is much smaller than the
transverse one,
$\overline{v_{x^{2}}^{'}},\overline{v_{y^{2}}^{'}}\text{>>}\overline{v_{z^{2}}^{'}}$,
and we put that $v_{z}^{'} = \beta_{i}{c\theta}_{||}$,
$v_{x}^{'} = \sqrt{2}\beta_{i}\gamma_{i}{c\theta}_{x}\text{cos}\phi_{x}$
and
$v_{y}^{'} = \sqrt{2}\beta_{i}\gamma_{i}{c\theta}_{y}\text{cos}\phi_{y}$,
where a factor of $\sqrt{2}$ takes into account that the amplitude is
$\sqrt{2}$ times larger than the rms value. After performing averaging
and returning back to the lab frame we obtain the following estimate for
the damping decrements:

$\begin{array}{r}
\tau_{{||}^{- 1}} \approx \frac{\text{16}\pi\ n_{e}Z^{2}e^{4}L_{\text{Ce}}}{3m_{e}m_{i}\gamma_{i^{5}}\beta_{i^{3}}c^{3}}\frac{\eta_{c}}{\left( 2\theta_{x^{2}} + 0\text{.}\text{66}\left( \theta_{{||}^{2}}/{\gamma_{i}}^{2} \right) + 1\text{.}6\theta_{e^{2}} \right)\sqrt{\left( \theta_{{||}^{2}}/{\gamma_{i}}^{2} \right) + 2\text{.}4\theta_{e^{2}}}}\ , \\
\tau_{{}^{- 1}} \approx \frac{\text{16}\pi\ n_{e}Z^{2}e^{4}L_{\text{Ce}}}{3m_{e}m_{i}\gamma_{i^{5}}\beta_{i^{3}}c^{3}}\frac{\eta_{c}}{\left( 2\theta_{x^{2}} + 1\text{.}2\left( \theta_{{||}^{2}}/{\gamma_{i}}^{2} \right) + 2\text{.}8\theta_{e^{2}} \right)^{\frac{3}{2}}}\ \text{.}
\end{array}$ (19)

Here the decrements are defined as
$\tau_{{}^{- 1}} \equiv 1/\varepsilon d\varepsilon/\text{dt}$ and
$\tau_{{||}^{- 1}} \equiv 1/\overline{{\Delta p}^{2}}d\overline{{\Delta p}^{2}}/\text{dt}$,*θ~e~*
is the angular spread in the electron beam, *η~c~* is a fraction of the
ring orbit used for cooling, and the following two approximate equations
have been used to perform averaging:

$\begin{array}{r}
\int_{0}^{\pi}{\int_{0}^{\pi}\frac{\text{dxdy}}{\pi^{2}}}\frac{1}{\left( \text{cos}^{2}x + \text{cos}^{2}y + \Delta^{2} \right)^{\frac{3}{2}}} \approx \frac{2}{3}\frac{1}{\Delta\left( 1 + \frac{2}{3}\Delta^{2} \right)}\ , \\
\int_{0}^{\pi}{\int_{0}^{\pi}\frac{\text{dxdy}}{\pi^{2}}}\frac{\text{cos}^{2}x}{\left( \text{cos}^{2}x + \text{cos}^{2}y + \Delta^{2} \right)^{\frac{3}{2}}} \approx \frac{1}{1\text{.}\text{55}\Delta\left( 1 + \left( \frac{2}{1\text{.}\text{55}} \right)^{\frac{2}{3}}\Delta^{2} \right)^{\frac{3}{2}}}\ ,
\end{array}$ (20)

Assuming that electron beam size is three times larger than the rms ion
beam size in the cooling section,
$r_{\text{eb}} = 3\sqrt{\varepsilon_{i}\beta_{c}}$ we can finally
rewrite Eq. (19) in the following form

$\begin{array}{r}
\tau_{{||}^{- 1}} \approx \frac{8\ Z^{2}e^{3}I_{0}L_{\text{Ce}}}{\text{27}m_{e}m_{i}\gamma_{i^{5}}\beta_{i^{4}}c^{4}\varepsilon_{i^{2}}}\frac{\eta_{c}}{\left( 1 + \left( 0\text{.}\text{33}\frac{\theta_{{||}^{2}}}{\gamma_{i^{2}}} + 0\text{.}8\theta_{e^{2}} \right)\frac{\beta_{c}}{\varepsilon_{i}} \right)\sqrt{\left( \theta_{{||}^{2}}/{\gamma_{i}}^{2} \right) + 2\text{.}4\theta_{e^{2}}}}\ , \\
\tau_{{}^{- 1}} \approx \frac{4\sqrt{2}\ Z^{2}e^{3}I_{0}L_{\text{Ce}}}{\text{27}m_{e}m_{i}\gamma_{i^{5}}\beta_{i^{4}}c^{4}\varepsilon_{i^{\frac{5}{2}}}}\frac{\eta_{c}\sqrt{\beta_{c}}}{\left( 1 + \left( 0\text{.}6\frac{\theta_{{||}^{2}}}{\gamma_{i^{2}}} + 1\text{.}4\theta_{e^{2}} \right)\frac{\beta_{c}}{\varepsilon_{i}} \right)^{\frac{3}{2}}}\ \text{.}
\end{array}$ (21)

which will be used for the cooling time estimates considered below. Here
$I_{0} = {\pi r}_{e^{2}}n_{e}\beta_{i}c$is the electron beam current.

Table 2 presents parameters of the required cooling devices for the
ring-ring and the linac-ring colliders considered in the previous
section. The following issues have determined the choice of the
parameters. The cooling decrements are proportional to the cooling
length and one would like to choose this length as long as possible. To
keep the ion beam inside the electron beam along the entire cooling
length, the ion beam beta-functions should be larger, or equal to the
cooling length. However an increase of beta-functions increases electron
beam size and it decreases the electron density, which compensates the
growth of the decrements due to the decrease of the angular spread. As
one can see from Eq. (21) the effect of beta-function increase on
damping decrements depends on the actual values of parameters. For the
considered here parameters, an increase of beta-functions decreases the
decrements. Also note that an increase of beta-functions raises the
requirements on the angular spread of electrons due to smaller angular
spread in the ion beam. Therefore, we choose the beta-functions to be
equal to the cooling length for both cases. The cathode temperature and
size set the minimum for the angular spread in the electron beam. We
believe that the electron beam acceleration and transport needs to be
done (and can be done) with a sufficiently small additional emittance
growth. Then, the transverse temperature of electrons is equal to the
cathode temperature and the longitudinal temperature was chosen to be
equal to the transverse one. A smaller longitudinal temperature does not
bring any significant gain, while the larger one affects both
decrements.

  ----------------------------------------------------- --------------- ----------------
  **TABLE 2. Electron Cooling Parameters for Ring-ring                  
  and Linac-ring Colliders**                                            

                                                        **Ring-ring**   **Linac-ring**

  Kinetic energy of electrons, MeV                      8.2             16.3

  Peak electron beam current, A                         3               10

  Mean electron beam current in energy recovery linac   0.4             1.3
  and 40 cm bunch, A                                                    

  Electron beam radius, cm                              0.6             0.34

  Electron density in the beam frame, cm^-3^            3.2⋅10^7^       1.7⋅10^8^

  Beta-function of proton beam at the center of cooling 30              30
  section, m                                                            

  Rms proton beam size at the center of cooling         0.2             0.11
  section, cm                                                           

  Cooling section length, m                             30              30

  Rms angular spread in the proton beam,                6.7⋅10^-5^      3.7⋅10^-5^

  Rms transverse velocity spread of ions in the beam    3.4⋅10^7^       3.7⋅10^7^
  frame, cm/s                                                           

  Rms longitudinal velocity spread of ions in the beam  2.4⋅10^7^       2.1⋅10^7^
  frame, cm/s                                                           

  Effective electron temperature in the beam frame, eV  0.2             0.2

  Rms velocity spread of electrons in the beam frame,   1.9⋅10^7^       1.9⋅10^7^
  cm/s                                                                  

  Relative rms energy spread of electrons               6.2⋅10^-4^      6.2⋅10^-4^

  Relative rms angular spread of electrons              3.7⋅10^-5^      1.9⋅10^-5^

  Longitudinal emittance damping time, s                110             70

  Transverse emittance damping time, s                  240             150
  ----------------------------------------------------- --------------- ----------------

As one can see from Table 2, the electron velocity spread is already
comparable to the ion velocity spread and its further increase would
cause fast decrease of the damping decrements. There are two major
issues limiting the angular spread for electrons: uncontrolled
transverse dipole fields and the space charge effects. The first issue
is important only inside the cooling section where achieving the beam
angle variation below 10-20 mrad is not going to be a challenging
problem. This issue is worse for the linac-ring case where the angular
perturbations have to be 2 times lower. The space charge effects are a
major concern at low energy and solution of this problem will be another
challenge. If an energy recovery linac is chosen for the acceleration of
electrons, there is another effect, which can significantly affect the
transverse velocities. This is a time dependent focusing of the
accelerating cavities, which focuses differently particle in the head
and in the tail of the bunch.

If for the ring-ring case the electrostatic acceleration to 8 MeV is
still feasible, this is not a viable option for the linac-ring scheme
with 16 MeV, where the energy recovery linac is the only choice. In that
case to maintain effective cooling for all particles in the ion bunch
the electron bunch should have a bunch length of at least 5σ,
corresponding to about 40 cm. In this case achieving 6⋅10^-4^ energy
spread will pose a challenge. One of possible solutions can be an
addition of higher harmonics to correct the accelerating profile. In
particular adding the third and the fifth harmonics allows one to reach
the accelerating gradient uniformity within ±10^-4^ band for 45 cm bunch
and 3 m bunch spacing.

The parameters of the electron coolers required for both projects are
well beyond current state-of-the-art technology for the beam current,
and the angular and momentum spread in the electron beam. The required
electron beam current for the linac-ring scheme is three times higher
and the angular spread is two times lower making this choice
significantly more difficult, if possible at all. Also note that the
electron beam parameters were calculated for the most optimistic case.
In reality, we may have difficulties to achieve the desired velocity
spread in the electron beam. That will require higher electron beam
current making the project more complicated.

# NO COOLING scenario

Taking into account that the electron cooling is still a significant
pending problem we also consider a collider with no cooling at the top
energy. However some cooling can be used at low energy to shape the
beam. In this case the IBS is the major obstacle in achieving high
brightness in the ion beam; usually the longitudinal IBS is more
limiting. Combining Eqs. (1) and (10) and taking into account that the
bunch length and the beta-function at the IP are equal one obtains the
following luminosity limitation:

$L_{\text{IBS}} = \frac{4A^{2}}{{\pi Z}^{4}}\frac{m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\theta_{{||}^{2}}I_{e}\Lambda_{||}}{e^{5}L_{C}\left( 1 + \frac{\sigma_{e}^{2}}{\sigma_{p}^{2}} \right)}\sqrt{\frac{\varepsilon_{x}\nu_{x}}{R}}$
. (22)

For high energy collider without cooling the emittance is determined by
the injector, and the normalized emittance, *ε~xn~*=*ε~x~γ~i~*, is
conserved. The ratio *R*/*γ~i~* is determined by the magnetic field of
dipoles. Thus, we can write the following scaling for the luminosity
limit:
$L_{\text{IBS}} \propto \gamma_{i^{2}}\theta_{{||}^{2}}I_{e}\Lambda_{||}\sqrt{\varepsilon_{\text{xn}}\nu_{x}}$.
As one can see, the only effective free parameter to compensate a longer
IBS time is the ion beam energy. Table 3 depicts the tentative
parameters of the ring-ring collider with 10^33^ cm^-2^s^-1^ luminosity
and minimum center of mass energy.

  ------------------------------------------------ ----------- -----------
  **TABLE 3. Tentative Parameters of Ring-ring                 
  Collider with Luminosity of 10^33^ cm^-2^s^-1^               
  and without Cooling**                                        

                                                   Electrons   Protons

  Center of mass energy, GeV                       42.6        

  Kinetic beam energy, GeV                         3           150

  Circumference, m                                 1600        1600

  Beam current, A                                  1.7         0.8

  Beta-functions at IP, cm                         5           10

  Rms normalized beam emittance, mm⋅mrad           110         6.6

  Rms beam size at IP, μm                          30          64

  Laslett tune shift                               \-          2⋅10^-4^

  Beam-beam tune shift per IP                      0.023       0.002

  Rms momentum spread                              6⋅10^-4^    8⋅10^-4^

  Rms bunch length, cm                             ≤ 3         10

  Longitudinal intrabeam scattering lifetime,      \-          17
  $\Lambda_{{||}^{- 1}}$, hour                                 

  Transverse intrabeam scattering lifetime,        \-          23
  $\Lambda_{x^{- 1}}$, hour                                    
  ------------------------------------------------ ----------- -----------

In the case of the linac-ring collider the ion ring energy should be
increased by another factor two or three. As one can see, although
achieving the desired luminosity is still feasible with high-energy ion
ring, its energy becomes disproportionally high.

# Discussion 

Achieving the luminosity of 10^33^ cm^-2^s^-1^ for the electron-proton
collider is a challenging problem. Two possible collider schemes have
been proposed. They are the ring-ring and the linac-ring colliders. Both
of them require electron cooling of the proton beam at the collision
energy. The electron beam parameters for the cooler are far beyond of
the current electron cooling technology, and the development of cooling
is one of the highest priorities of the project R&D. In the comparison
of two collider schemes considered above, the electron beam current for
the linac-ring collider has been limited to 200-300 mA. Even with this
optimistic choice supported by estimates, but staying far away from the
current technology level, the luminosity of 10^33^ cm^-2^s^-1^ requires
two times higher energy for the ion ring and significantly more
complicated electron cooling. Although, at the present time the
linac-ring collider does not look competitive to the ring-ring if a new
machine with ultimate luminosity is built, the linac-ring collider can
be a good choice for already existing high energy synchrotrons. In the
case of RHIC collider it does not look reasonable to build a full
circumference electron ring, but in a ring of smaller circumference the
beam-beam effects are much worse. A linac with energy recovery may be a
much better choice.

If a collider optimized for the electron-proton collisions is used for
the electron-ion collisions with fully stripped ions the ultimate
luminosity for the electron-ion mode is expected to be about half of
luminosity for the proton-electron case with approximately the same
requirements for the electron cooling.

# Acknowledgments

The author is grateful to I. Ben-Zvi, Ya. Derbenev, A. Hutton, L.
Merminga, R. Li, G. Krafft, S. Nagaitsev and Yu. Shatunov for useful and
stimulating discussions. I also would like to thank A. Bogacz and M.
Tiefenback for help in the editing of the article.

# References

1\. Barber, D, P., "Electron and Positron Polarization at HERA; Past and
Future," *These proceedings*.

2\. Douglas, D. R., York R. C., and Kewisch J., *Proc. of the 1989
Particle Accelerator Conference*, 1989, pp.557-559.

3\. Bowling, B., et al., *Proc. of the 1991 Particle Accelerator
Conference*, 1991, pp. 446-448

4\. "The Science Driving the 12 GeV Upgrade of CEBAF" edited by L.
Cardman, Jefferson Lab, 2000.

5\. Koop, I, A., et al., "Conceptual Design Study of the Electron-Proton
Storage Ring Collider with Polarized Beams," *These proceedings*.

6\. Merminga L., et al., "An Energy Recovery Electron Linac-On Ring
Collider," *These proceedings*.

7\. Funakoshi Y. , et al., "KEKB Performance," *Proceedings of the 2000
European Part. Accel. Conf.*, 2000. pp. 38-44.

8\. Seeman J.T., et al., "Status Report of PEP-II performance,"
*Proceedings of the 2000 European Part. Accel. Conf.*, 2000. pp. 28-32.

9\. Li R. and Bisognano J., "A Strong-Strong Simulation on the Beam-Beam
Effect in a Linac/Ring B-Factory," *Proceedings of the 1993 Part. Accel.
Conf.*, 1993. pp. 3473-3475.

10\. Perevedentsev E.A., Valishev A.A., "Characteristics and possible
cures of the Head-Tail Instability of Colliding Bunches," *Proceedings
of the 2000 European Part. Accel. Conf.*, 2000. pp. 1223-1225.

11\. Li R., Lebedev V.A., Bisognano J. J., "Analysis and Simulation of
Beam-Beam Kink Instability in a Linac-Ring Electron-Ion Collider," *To
be submitted to* *Proceedings of the 2001 Part. Accel. Conf*.

12.Lebedev V.A., et al., *NIM-A* **391**, 176-187 (1997).

13.Budker G. I. and Skrinsky A. N., *Usp.Fiz.Nauk* **124**, 4,
p.561(1978).

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[^1]: ^^ Work supported by the US DOE under contract #DE-AC05-84ER40150