Chapter 2. Luminosity of heavy ion collisions
Contents
Chapter 2. Luminosity of heavy ion collisions
2.1. Strategy of the collider operation
2.2. Luminosity limitations
2.2.1. Bunch number limitations
2.2.2. Beta function in the CP
2.2.3. Bunch intensity. Low limit for the bunch length
2.2.4. Optimum bunch length
2.2.5. Beam-beam parameter and Lasslet tune shift
2.3. The beam parameters. Possibility of the future development
2.4. References
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-2s-1at the ion energy of 3.5x3.5GeV/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
,(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 onepractically.
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:
,(2.1.2)
whereNbistheionnumberper 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
.(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 tothe 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 ionizationand when stray particles hitthe chamber walls) towards the system's chamber wall.If the number of emitted electrons per impinging electron, given by theSecondary Electron Yield (SEY) of the wall surface, is greater than unity, the electron densityinside the vacuum chamber increases exponentially creating a so-called electron cloud. Thisusually ends in catastrophic drop in system performanceby increasingthe vacuum pressure, producing beam instabilities, causing beam loss and/or interference inbeam 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 inmany accelerators with positively charged particles (the PSR at LANL, the PF at KEK, thePEP-II at SLAC, the SPS at CERN, etc), often acting as a fundamental limit to machineperformance resulting from the aforementioned effects.
The phenomenon is quite sensitive to ahost of accelerator parameters including beam bunch intensity, beam bunch spacing, beamdimensions, chamber geometry, and properties of the chamber surface material. If a certainthreshold is crossed, an electron cloud quickly develops or disappears. The least well knownproperty is the SEY dependence on electron energy, especially for low energy electrons. Itplays a crucial role because it determines the number of surviving electrons when the electricfield is absent, a condition that occurs regularly during the gap between the passage of twoconsecutive 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]:
,(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 Lbb1.8 m.However this estimate is far too stringent, so the electron clouds occur even for bunch spacings much larger than thatsuggested 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 about35 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 0.5kick.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
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:
.(2.2.2)
The MPD design presumes that the minimum distance fromthe quadropole lens to the CP Lqis no lessthan 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 *:
,(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 currentis 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]:
,(2.2.4)
where ZL is the longitudinal coupling impedance including space-charge impedance and impedance of the vacuum chamber. Here Z0 = 377 Ohm, is 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 atomic number. is the ring momentum slip factor.
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:
,(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 Schnell-Zotter criterion [2.2.7]:
,(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
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]:
,
where the ring chromaticity is determined as , here Q is the tune shift corresponding to the momentum shift p/p.Qsc is the incoherent tune shift (see Chapter2.2.5 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 (310-4). 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 referredto 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:
,(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.
Fig. 2.2.1. Dependence of f- factoronthe 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 beam-beam 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
,(2.2.8)
where rp is the proton classical radius.
The incoherent (Lasslet) tune shift can be estimated as
,(2.2.9)
whereFsc – image force correction factor (usually Fsc ~ 1). The bunching factor
(2.2.10)
isequaltoratioofpeaktomeanbunchcurrent.
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.
Ifthesingle bunch luminosityislimitedbythebeam-beameffectit can be expressed via as
,(2.2.11)
IftheluminosityislimitedbythetuneshiftvalueQit is equal to:
.(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 7.31026 and 1.51027 cm-2s-1. These values are closed to the required level for the MPD operation.
2.2.6. Beam emittance
To keep the tune shift below chosen value the rms unnormalized bunch emittance has to be stabilized at the level of: