Chapter 2. Luminosity of heavy ion collisions
Contents
Chapter 2. Luminosity of heavy ion collisions .........................................................................26
2.1. Strategy of the collider operation...................................................................................27
2.2. Luminosity limitations ...................................................................................................28
2.2.1. Bunch number limitations .......................................................................................28
2.2.2. Beta function in the CP ...........................................................................................29
2.2.3. Bunch intensity. Low limit for the bunch length ....................................................30
2.2.4. Optimum bunch length ...........................................................................................32
2.2.5. Beam-beam parameter and Lasslet tune shift .........................................................33
2.3. The beam parameters. Possibility of the future development ........................................35
2.4. References ......................................................................................................................37
26
2.1. Strategy of the collider operation
The goal of the heavy ion program at the NICA facility is to reach the average luminosity of
1027 cm-2s-1 at the ion energy of 3.5 x 3.5 GeV/u. Additional requirement relates to the
luminosity distribution along the interaction region. High resolution of the vertex position in
the MPD detector is suggested to provide using Inner Tracker (IT) on the basis of Silicon
Strip Detector (SSD). The cost of the IT is sufficiently dependent of its length, which is
designed to be of 50 cm. Correspondingly, maximum part of the luminosity has to be
localized inside the IT.
The average luminosity value is given by the following formula
Texp
L 
 Lt dt
0
Texp  T prep
,
(2.1.1)
where Texp is the experiment duration, Tprep is the beam preparation time.
An obvious requirement to the collider characteristics is its permanent luminosity during an
experiment cycle. One of major effects, which lead to the luminosity decrease during a cycle
of a collider operation is intrabeam scattering (IBS) inside each bunch of both beams
circulating in the rings. It leads to increase of the bunch length and cross-section. To suppress
this detrimental effect one can use one of cooling methods applicable for heavy ions –
stochastic or electron one. Possibilities of the cooling methods are compared in Chapter 6.5.
Here we suppose that the beam cooling stabilizes the luminosity during experiment and the
beam preparation time is sufficiently shorter than the experiment duration. At such conditions
the mean luminosity is equal to the peak one practically.
The IBS growth rate in the horizontal plane is larger than in the vertical one. It occurs due to
presence of horizontal dispersion in the ring leading to coupling of the horizontal and
longitudinal ion motion. To have a round beam cross-section in the Collision Point (CP) it is
proposed to operate the collider rings at equal betatron tunes in both transverse planes. In this
case a residual coupling of the ring optics structure (due to an imperfection in the lens
adjustment, longitudinal magnetic field of the detector solenoid, etc.) forms equal beam
emittances and beam dimensions in the horizontal and vertical planes. (Such a mode of the
collider operation is used, for instance, at RHIC).
For the round beams the peak luminosity can be calculated in accordance with the following
formula:
L
 
Fcoll f  s*  ,
4
 
N b2
*
(2.1.2)
where Nb is the ion number per bunch,  is the transverse unnormalized r.m.s. emittance, * is
the beta function in the collision point, s is the rms value of the longitudinal beam size. Fcoll
is the collision repetition frequency, which is equal to the bunch revolution frequency Frev
multiplied by the bunch number Nbunches. The factor f is defined by the formula
27
 
1
f  s*  

 

exp( u 2 )du
.

2

 
 u 
1   *s  
    
(2.1.3)
This factor is close to unity when the longitudinal rms beam size is much less than the value
of beta function at the CP, and decreases at with an increase of s.
To reach maximum peak luminosity one needs to satisfy the following obvious requirements:
- maximum collision repetition rate (that corresponds to the maximum bunch number in
the rings);
- minimum beta function in the CP;
- maximum bunch intensity;
- minimum beam emittance and
- minimum bunch length.
2.2. Luminosity limitations
In this chapter the luminosity limitations related to the main physical effects and technical
parameters of the collider systems are considered. The collider ring circumference is assumed
to be about 225 m (that is limited by available space in the existing building).
2.2.1. Bunch number limitations
The peak luminosity is proportional to the bunch number in the rings. The maximum bunch
number is limited by the following main factors:
- “electron cloud” effect;
- multibunch instability;
- technical parameters of the injection system.
2.2.1.1. Electron cloud
The ion beam circulating in a ring provides inside a vacuum chamber a radiofrequency
electric field. The electric field accelerates electrons (produced by residual gas ionization and
when stray particles hit the chamber walls) towards the system's chamber wall. If the number
of emitted electrons per impinging electron, given by the Secondary Electron Yield (SEY) of
the wall surface, is greater than unity, the electron density inside the vacuum chamber
increases exponentially creating a so-called electron cloud. This usually ends in catastrophic
drop in system performance by increasing the vacuum pressure, producing beam instabilities,
causing beam loss and/or interference in beam diagnostics [2.2.1]. This effect is often referred
also to as Beam Induced Multipacting (BIM).
The proton storage ring of the Budker Institute of Nuclear Physics (BINP, Novosibirsk) in
1965 [2.2.2] and the Ion Storage Ring at CERN in 1972 [2.2.3] are perhaps the first machines
to suffer from electron clouds. In the 1990's, electron clouds were observed in many
accelerators with positively charged particles (the PSR at LANL, the PF at KEK, the PEP-II
at SLAC, the SPS at CERN, etc), often acting as a fundamental limit to machine performance
resulting from the aforementioned effects.
28
The phenomenon is quite sensitive to a host of accelerator parameters including beam bunch
intensity, beam bunch spacing, beam dimensions, chamber geometry, and properties of the
chamber surface material. If a certain threshold is crossed, an electron cloud quickly develops
or disappears. The least well known property is the SEY dependence on electron energy,
especially for low energy electrons. It plays a crucial role because it determines the number of
surviving electrons when the electric field is absent, a condition that occurs regularly during
the gap between the passage of two consecutive beam bunches.
Treatment of the BIM as a resonance effect leads to very simple limitation to the ion bunch
spacing Lbb (distance between neighbor bunches) inside a round vacuum chamber of the
radius a [2.2.4]:
Lbb 
a2
,
re ZN b
(2.2.1)
where re is the electron classical radius, Z is the ion charge number. For a = 35 mm (see
Fig. 1.8) and the bunch intensity of Nb = 109 it gives for uranium ions Lbb  1.8 m. However
this estimate is far too stringent, so the electron clouds occur even for bunch spacings much
larger than that suggested by Eq. 2.2.4. For instance the current performance of the RHIC
parameters is limited by the electron cloud effect when the bunch spacing is less than about
35 m [2.2.5]. Therefore the electron cloud effect will be considered in more detail at the
technical design stage of the NICA collider.
2.2.1.2. Multibunch instability
The multibunch instability can occur when a bunch train circulates in the ring which vacuum
chamber includes resonance elements [2.2.6]. Electric fields provided by one of the bunches
can distort the motion of other bunches, that can lead to increase of coherent oscillation
amplitude (both – longitudinal and transverse). This effect can limit as the bunch number in
the ring as the bunch intensity. The instability threshold is more sensitive to parameters of the
elements common for both collider rings. The design of the NICA collider RF system
presumes usage of two RF cavities – each of them is installed in its own ring. Preliminary
estimations show that in this case the multibunch instability does not limit the collider
performance practically. However, design of the vacuum chamber of the interaction region
(common for the both rings) has to be performed tacking into account its RF properties.
2.2.1.3 Technical parameters of the injection system
Main technical limitation of the bunch number in the ring is related with an achievable pulse
duration of the injection kicker kick. To avoid distortion of circulating bunches during
injection of a new one the time spacing between bunches has to be longer than about 0.5kick.
Here we assume kick  100 ns (which lies in well established rang of parameters) that allows
to have up to 10 – 15 bunches into each ring. Parameters of the injection system are discussed
in the Chapter 6.4 of this Report.
2.2.2. Beta function in the CP
29
At small value of the beta function in the collision point its maximum value depends on the
distance from the CP to the nearest quadrupole lenses. Neglecting the influence of the
detector magnetic field on the particle motion, the dependence of the beta function on the
distance from the CP s is given by the formula:
 s    * 
s2
*
.
(2.2.2)
The MPD design presumes that the minimum distance from the quadropole lens to the CP Lq
is no less than 5 m. To avoid the ion losses in the low beta quadrupoles the vacuum chamber
aperture a has to be larger than the beam rms radius a = n at n in the range 4  6. This
gives the following approximate limitation for the *:
  n
*
2
L2q
a2
,
(2.2.3)
where  is unnormalized rms emittance.
This limitation is sufficient at small ion energy, when the beam emittance has to be large
enough to avoid tune shift limitation (see Chapter 2.2.5). In the presented project the beta
function in the CP is chosen to be 0.5 m. At the ion energy of 1 GeV/u it requires the aperture
of the quadrupoles of the low beta insertion of about 80 mm. (The estimations tacking into
account the optic design of the interaction region are given in the Chapter 6.3.2.) The
possibility to increase the luminosity in the low energy region is related with a design of large
aperture lenses. At maximum ion energy the * can be decreased with corresponding increase
of the peak luminosity.
2.2.3. Bunch intensity. Low limit for the bunch length
The ion bunch intensity and the single bunch luminosity can be limited by the following
effects:
- Laslett tune shift,
- Beam-beam effect,
- Coherent instabilities.
Tune shift and beam-beam limitations are considered in the Chapter 2.2.5. They can be
avoided by appropriate increase of the beam emittance. In this chapter we discuss the coherent
longitudinal and transverse instabilities, which threshold slightly depends on the beam
emittance.
The proposed NICA injection chain is developed to guarantee the achievement of the bunch
intensity of 109 ions and provide a technical reserve required for future development of the
facility. At given bunch intensity the coherent instabilities limit the peak beam current. The
ratio between peak and mean beam current is determined by the bunch length, therefore the
threshold of the coherent instabilities limits the achievable bunch length from the low side.
The threshold current of the coherent longitudinal (microwave) instability can be estimated by
the following formula [2.2.7]:
30
I
4 Fl Amc 2  2   p2
Z L eZ
,
(2.2.4)
where ZL is the longitudinal coupling impedance including space-charge impedance
Z G
a
Z L,sc  0 2L and impedance of the vacuum chamber. Here Z0 = 377 Ohm, G L  1  2 ln is
r
2
the longitudinal form factor, a is the vacuum chamber aperture, r is the beam radius. e is the
elementary charge, ,  are the relativistic factors, mc2 is the nucleon rest energy, A is the ion
1
1
atomic number.   2  2 is the ring momentum slip factor.

 tr
Fl is the factor depending on distribution function shape: Fl = 1 for the standard Keill-Schnell
criterion and it can reach a value of about 10 for distribution function with long tails. The rms
relative momentum spread p is limited from the upper side by the collider acceptance. In the
estimations below we assume that maximum acceptable value corresponds p = 10-3.
In the case of a bunched beam this criterion limits a peak current value, which is related with
the ion number in the bunch as follows:
I peak 
ZeN b
Trev
C
,
2  s
(2.2.5)
where Trev is the revolution period, s is the rms bunch length.
The threshold current of the transverse coherent instability can be estimated using SchnellZotter criterion [2.2.7]:
I  4 Ft
A mc 2  Q
Q ,
Z 2 e Zt R
(2.2.6)
where R is the mean ring radius, Q is the betatron tune. Zt is the transverse coupling
impedance including space-charge impedance
Z t  i
Z0 R  1
1
 2
2 2  2
  a
r 
and impedance of vacuum chamber. Ft is the factor depending on distribution function shape:
Ft = 1 for the standard Schnell-Zotter criterion and it can reach a value of about 10 for
distribution function with long tails.
The betatron tune spread Q suppresses the bunch coherent oscillations (due to Landau
damping) and an increase of the tune spread leads to increase of the threshold current of
transverse microwave instability. An effective tune spread for n-th mode of coherent dipole
oscillations one can estimate as [2.2.8]:
Qn 
(Q  n)   2  p2  Qsc2
31
,
where the ring chromaticity  is determined as   Q /p / p , here Q is the tune shift
corresponding to the momentum shift p/p. Qsc is the incoherent tune shift (see
Chapter 2.2.5 below).
At the uranium bunch intensity of 109 ions the rms bunch length corresponding to the
threshold peak current of the longitudinal instability (2.2.4) is between 20 and 30 cm. At the
same conditions the bunch intensity is below than the threshold of transverse instability more
than one order of magnitude (the estimations are done for the collider parameters presented in
the Chapter 6).
Electron cooling application for the luminosity preservation leads to formation of a dense core
in the ion distribution function (see Chapter 6.5.2). The expected momentum spread in the
care is about 3 times less than its initial value (310-4). In this case a stabilization of the
longitudinal coherent motion one can expect due to long tails in the distribution (factor Fl in
the formula 2.2.4).
The requirements to the vacuum chamber impedance and stability of the longitudinal motion
under the electron cooling are the issues addressed to the collider technical design stage.
2.2.4. Optimum bunch length
The bunch length is chosen as a compromise between two contradicting requirements. On the
one hand the bunch length has to be as small as possible to avoid a “hour glass” effect and to
provide the luminosity concentration in the central part of the detector (Inner Tracker or
Silicon Strip Detector SSD). On the other hand a small bunch length leads to increase of the
bunch peak current that can provoke coherent instability (as it is described in the previous
chapter) and requires high RF voltage for matching of the bunch with the RF. The last
limitation is economical one – the required RF voltage amplitude determines the cost of the
RF system. For reduction of the RF voltage amplitude to reasonable level (of 100 – 200 kV)
the RF system will be operated at high harmonics of the revolution frequency.
An increase of the bunch length leads to decrease of the factor (2.1.2) (and corresponding
decrease of the peak luminosity) that is referred to as the “hour glass” effect. At the beta
function in the CP of 0.5 m, to have this factor above 90% the bunch length has to be no
longer than 30 cm (see Fig. 2.2.1). This value is chosen as the design value of the bunch
length in the presented Project.
At finite value (non-zero) of the bunch length the luminosity is distributed along the
interaction region. The efficiency of the SSD detector can be estimated as:
LS sD / 2
FSVD 
dL
ds
ds
 LS sD / 2

L
,
(2.2.7)
where LssD is its length, L is the luminosity determined by the formula (2.1.1). The luminosity
distribution along the interaction region dL/ds can be evaluated numerically for the bunch at
Gaussian distribution. At the chosen bunch length of s = 30 см the SSD efficiency is about
80% that is acceptable.
32
Fig. 2.2.1. Dependence of f- factor on the rms bunch length in cm, * = 50 сm.
2.2.5. Beam-beam parameter and Lasslet tune shift
In this chapter the luminosity limitations related to space charge effects are considered.
To obtain maximum peak luminosity the beam emittance has to be as small as possible. Main
physical limitations of the beam emittance are the incoherent (“Lasslet”) tune shift and beambeam tune shift (“beam-beam parameter”), which can not exceed some limiting values. In the
estimations presented in this chapter the limiting value for the tune shift was chosen to be
0.05, for the beam-beam parameter – 0.005. Both values correspond to relatively weak space
charge of the beam, and the luminosity level is rather underestimated (“pessimistic”).
Linear part of the tune shift due to interaction with a colliding bunch is called “beam-beam”
parameter. If the beam-beam exceeds some limit value the powerful diffusion process is
began, that leads to fast increase of the beam emittance. For the round beam the beam-beam
parameter is equal to

1  2
,
A 4 2 2
Z 2 rp
Nb
(2.2.8)
where rp is the proton classical radius.
The incoherent (Lasslet) tune shift can be estimated as
Q  
Z 2 rp
Nb
A 4 2 3
Fsc Fb ,
(2.2.9)
where Fsc – image force correction factor (usually Fsc ~ 1). The bunching factor
Fb 
C
2  s
33
(2.2.10)
is equal to ratio of peak to mean bunch current.
Both the values (2.2.8) and (2.2.9) do not depend on the bunch number in the ring and the
describing effects limit so called “single bunch” luminosity.
If the single bunch luminosity is limited by the beam-beam effect it can be expressed via  as
L
A N b c 2 3   s 
 ,
f
Z 2 rp  *C 1   2   * 
(2.2.11)
If the luminosity is limited by the tune shift value Q it is equal to:
L
A N b c 2  s 3 3   s 
  f  * Q .
Z 2 rp  * C 2
 
(2.2.12)
In the general case the luminosity is limited by minimum value from (2.2.11) and (2.2.12).
For NICA parameters the value of the single bunch luminosity is plotted in the Fig. 2.3.2 as
the function of the beam energy at Q = 0.05 and  = 0.005. One can see, that the luminosity
is limited by the tune shift value up to the energy of about 4.5 GeV/u. This is larger than the
maximum ion energy at chosen magnetic rigidity value.
Fig. 2.2.2. Single-bunch luminosity in cm-2s-1 as the function of the beam energy in GeV/u for
U ions. Q = 0.05 and  = 0.005. The calculations performed taking into account the hour
glass effect.
At the bunch number of between 10 and 20 in each ring the total luminosity for U-U
collisions at 3.5 GeV/u lies in the range from 7.31026 and 1.51027 cm-2s-1. These values are
closed to the required level for the MPD operation.
2.2.6. Beam emittance
34
To keep the tune shift below chosen value the rms unnormalized bunch emittance has to be
stabilized at the level of:

Z 2 rp
Nb
C
.
A 4  Q 2  s
2
3
(2.2.13)
The dependence of the emittance on the experiment energy is shown in the Fig. 2.4.1.
Estimations of the beam emittance from at the exit of the injection chain shows that the peak
luminosity of the required value can be achieved even without a beam cooling at the
experiment energy. However, the beam cooling is necessary for a suppression of the
emittance growth during experiment.
2.2.3. Rms unnormalized emittance in mmmrad corresponding to Q = 0.05 as the function
of the beam energy in GeV/u.
2.3. The beam parameters. Possibility of the future development
The luminosity of the order of 1027 cm-2s-1 will be reached owing to the following
peculiarities of the NICA design.
1. Collider operation with low beta function in the interaction point. The short (of
about 10 m) interaction region allows to have maximum beta functions in the triplets
of about 90 m when the beta functions in the collision point are of 0.5 m. At such
conditions the maximum beam radius in the lenses of the low beta insertion section is
about 4 cm that requires reasonable aperture of the lenses.
2. Short bunch length. The rms bunch length of about 30 cm makes possible to reduce
significantly “the hour glass effect” and to obtain required luminosity distribution
along the interaction region. The length of the inner tracker of the MPD is chosen to
be 50 cm that allows to accept by the tracker, at the given bunch length of about 80%
of the luminosity.
35
3. Collider operation at the beam emittance corresponding to the space charge
limit. In the NICA energy range the luminosity is limited by the incoherent tune shift
value. If the ion number per bunch and the tune shift are fixed the luminosity is scaled
with the energy as 33. The formation and preservation of low emittance value
corresponding to the tune shift limit is produced by electron or stochastic cooling
application at the experiment energy.
4. High collision repetition rate. The collider is operated at the bunch number of 10 
20 in each ring. This is achieved at well established injection kicker parameters (the
kicker pulse duration is about 100 ns) by means of injection into the collider of
bunches of a short length. The bunch of the required length is formed in the
Nuclotron after the acceleration. Small longitudinal emittance value required for the
bunch compression in the Nuclotron (see Chapter 5, section 5.3.2) is provided by the
electron cooling of the ion beam in the Booster (Chapter 4, section 4.9).
5. Long luminosity life-time. For the luminosity preservation the beam cooling is
foreseen. In equilibrium between intrabeam scattering (IBS) and the cooling the
luminosity life-time is limited mainly by the ion interaction with the residual gas
atoms. The vacuum conditions in the collider rings will provide the beam life time of
a few hours. The beam preparation time is designed to be between 2 and 3 minutes.
Therefore, the mean luminosity value is closed to the peak one.
The luminosity value at the 15 bunches in each collider ring is presented in the Table 2.1.
Table 2.1.
NICA parameters for U-U collisions.
Circumference
Number of collision points
Beta function in the collision point
Rms momentum spread
Rms bunch length
Number of ions in the bunch
Number of bunches
Incoherent tune shift
Rms beam emittance (geometrical)
at 1 GeV/u
at 3.5 GeV/u
Luminosity per one collision point
at 1 GeV/u
at 3.5 GeV/u
m
m
m
225
2
0.5
0.001
0.3
109
15
0.05
 mm mrad
3.8
0.26
cm-2s-1
6.61025
1.11027
During a technical design of the facility the bunch parameters will be optimized depending on
the experiment energy and ion specie. So, an increase of the bunch intensity allows increasing
the luminosity at the same value of the tune shift. To keep the constant tune shift the beam
emittance has to be increased proportionally to the bunch intensity and the luminosity is
scaled linearly with the ion number.
At the maximum experiment energy the peak luminosity can be increase either – by increase
of the bunch intensity or by decrease of the beta function in the collision point. After
optimization of the collider regime and achievement of a maximum bunch intensity one can
expect increase of the peak luminosity to the level of (2 3)1027 cm-2s-1.
36
At minimum experiment energy the peak luminosity is limited by geometry of the collider
optic structure (chapter 2.2.2). The possibility to increase the luminosity in this energy range
is related with a design of large aperture lenses and an increase of the bunch intensity.
2.4. References
2.2.1. F. Zimmermann and G. Rumolo. Electron cloud build up in machines with short
bunches. ICFA Beam Dynamics Newsletters, 33, April 2004.
2.2.2. G.I. Dimov G.I. Budker and V.G. Dudnikov. In Proceedings of the International
Symposium on Electron and Positron Storage Rings. Universitaires de France, Orsay, 1966.
2.2.3. O. Grobner. Bunch induced multipacting. In Proceedings of the 10th International
Conference on High Energy Accelerators. USSR Academy of Science, Protvino, 1977.
2.2.4. O. Grobner. Beam induced multipacting, Proceedings of PAC'97, Vancouver, 1997.
2.2.5. U. I. Ariz, Electron Clouds in the Relativistic Heavy Ion Collider, C-A/AP/#228,
February 2006.
2.2.6. M.M.Karliner, N.V. Mityanina, V.P.Yakovlev, Multibunch resistive wall instabilities of
an intense beam in storage rings. Proc. of XV International Conference on High Energy
Accelerators HEACC-92 (Hamburg, 1992), Int. J. Mod. Phys. (Proc. Suppl.) 2B (1993).
2.2.7. P. Zenkevich et. al., Problems of Ion beam stability for MUSES storage rings, RIKENAF-AC-18, February 2000, Ion bunch stability in the double storage ring, RIKEN-AF-AC-19,
February 2000.
2.2.8. J. Bosser et. al., Stability of cooled beams, NIM A 441 (2000) 1-8.
37