LHC Project Note 198 1999-09-14
[email protected]
Proposal of a Gas Ionization BSM for the LHC A. Arauzo1, C. Bovet1, J. Buon2 and P. Puzo2 1
CERN, SL Division
2
Laboratoire de l'Accélérateur Linéaire, Orsay, France
Keywords: Gas-ionization, LHC
Summary A new type of Beam Size Monitor (BSM) for the LHC is proposed here. This monitor is a residual-gas ionization detector and its working principle is based on the measurement of the acceleration given to ions by the space charge field of the beam. Such a BSM is able to measure the transverse beam size over the whole LHC energy range.
1.
Introduction This paper describes in detail the proposal for a Gas-Ionization Beam Size Monitor in the LHC. This type of detector was first developed by the Orsay group for the FFTB at SLAC [1, 2].The conception of the LHC detector differs, as it is go-ing to be used for the first time with a proton beam and for short bunch to bunch separation. The motivation behind this project is the design of a Beam Size Monitor (BSM) useful over all the energy range, from injection (450 GeV) to collision (7 TeV). In the following sections the principle of detection is explained, along with the experimental scheme and the main components. Theoretical calculations are
This is an internal CERN publication and does not necessarily reflect the views of the LHC project management.
made in order to reproduce the experimental conditions and the possible measurements. The steps involved in these calculations are described, along with the more important results obtained. The algorithms used in the simulations are valid for elliptic beam, nevertheless, for simplicity, most of the calculations were performed for the case of a round beam. This is a good approximation for the LHC, where beams have a small ellipticity. Moreover, most results presented are very general and the conclusions can be extrapolated for an elliptic beam. However, the most remarkable features, obtained in the case of a flat beam, have been also explained. The nominal beam parameters used for the LHC have been taken from the Yellow Book [3].
2 Detector principle The detection principle involves the ionization of a gas target by the beam and the subsequent acceleration of the ions by the space charge field of the beam itself. The gas target is the residual gas of the vacuum chamber (the beam pipe), or the gas injected with a leak to increase the local pressure up to a value compatible with the machine high vacuum and high ion production rate. The present study will show that the injection of Helium gas could be an option to increase the resolution at low beam dimensions (high energies). Due to the high velocity of the beam, the electrical field created is transverse, and for the particular case of a round beam, will be radial. Consequently, the created cations will acquire a radial velocity in the transverse plane.
3 Experimental scheme Fig. 1 shows a possible experimental layout for the detection system. The ions accelerated by the space charge field of the beam are collected through a slit cut in the vacuum pipe wall. The width of this slit is a parameter which can be adjusted as a function of the number of ions needed for measurements, and the required resolution. At the slit, we have a given distribution (spectrum) of ion velocities or energies which is a function of the beam dimensions. This distribution in velocities is transformed into a distribution of hitting positions in a detector by the application of a longitudinal magnetic field, i.e. parallel to the beam axis and perpendicular to 2
Beam pipe
slit w
d=2
Ro = 5 cm
.5 cm
O
θ = 63o
OR
CT
TE
DE
D
Figure 1: Experimental layout. The continuous lines represent the trajectories of ions with the highest velocity. The dashed lines correspond to the lowest velocities.
the ion velocity. This field is of the order of a few hundred gauss and it is shielded inside the beam pipe. With a collector in the horizontal position, the detector will be installed at a vertical distance Ro from the slit centre, making an angle θ with the horizontal plane. The slit width can range between 0.1 cm and 0.5 cm, while the slit length is of 2 cm. The parameters Ro and θ are fixed to values which depend on the geometry and possible dimensions of the detector. For the LHC beam parameters [3], these values can range from: for Ro , 3 - 10 cm, and for θ, from −20o to −80o . The position for the detector has been chosen so that it profits from a focalisation effect due to the geometry of the design, as can be seen in Fig. 1. This focalisation is most interesting for the highest ion velocities (continuous lines in 3
Fig. 1) and it is not actually important at the lowest velocities (dashed lines in Fig. 1). In this way the spot width due to the aperture in the slit can be reduced for a certain narrow range of velocities. The reason for choosing the maximal resolution to be at the highest velocities comes from the shape of the velocities distribution, as will be seen in the following section.
4 Simulations and Results In this section the principle of detection is explained in more detail, from the ionization of the gas by the beam to the cation position at the detector. The results from simulations based on the evolution of one cation as well as for a statistical distribution of cations are presented. Finally, from the results of these simulations, the possibilities for the detector and its resolution as a function of the beam energy are discussed. In all the calculations, the nominal conditions of the LHC beam have been assumed. Thus, the bunches contain 1011 protons and are separated in time by 25 ns. The bunch dimensions are also those expected for the nominal conditions, and will depend on the beam energy.
4.1 Ion generation 4.1.1 Ionization cross section The ionization of the atoms or molecules in a gas target by a particle beam can be studied from different points of view. Within the Born approximation, the ionization of an atom by a relativistic particle can be described as resulting from the exchange of a quasi-real photon. The ionization cross section can then be calculated by relating it to data on the photoionization cross section. The contribution from Rutherford ionization also has to be taken into account. This results from the short distance interaction between the incident particle and the atomic electron. The energy transfer is high enough to take the atomic electron as free. Within this theory, the total ionization cross section, σ, for He [4] has been found to be approximately σ = 0.3 Mb (for γ = 105 ). The ionization can also be described within the Bethe Theory. The ionization cross section is calculated from the total cross section of inelastic scattering. This
4
theory was developed to analyse the ionization stopping power of fast charged particles in matter. Rieke and Prepejchal [5] have described the primary ionization cross section of gaseous atoms and molecules for high energy electrons and positrons employing the Bethe theory [6, 7]. The measurements performed using several gases are successfully described with the following relation, extracted from the Bethe theory [6, 7]:
where
σ = Ax1 + Bx2
(1)
x1 = β −2 ln (β 2 /(1 − β 2 )) − 1
(2)
x2 = β −2
(3)
The energy dependence is accurately described by the equation: σ = (1.874 · 10−20 ) · C [ (M 2 /C) x1 + x2 ] cm2
(4)
where the quantity M 2 is known as the total dipole-matrix element squared for ionization, measured in units of the Bohr radius squared. The enhancement of the effective stopping is caused by the relativistic compression of the electric field. In the direction of flight the field is reduced√by (1 − β 2 ), whereas in the transverse direction it is enhanced by a factor of 1/ 1 − β 2 . Thus, as the velocity approaches the velocity of light, the electric field expands in the transverse plane, so increasing its influence on the electrons of the medium. Generally the total cross section is calculated, which has a contribution from the ionization cross section as well as from the excitation cross section (as was the case for the measurements performed in He). Table 1 shows the values of M 2 and C for both He and H2 [5]. Table 1.
He H2
M2 C 0.774 ± 0.030 7.653 ± 0.037 0.695 ± 0.015 8.115 ± 0.021
With these values, the ionization cross section is calculated as a function of en+ ergy. For instance, σ = 0.30 Mb for H+ 2 and σ = 0.31 Mb for He at low LHC + + energy (γ = 478); σ = 0.37 Mb for H2 and σ = 0.39 Mb for He at high LHC energy (γ = 0.746 · 104 ). 5
Using this theory, we obtain σ = 0.46 Mb for He+ (γ = 105 ), compared with the σ = 0.30 Mb found using the first method described [4] . The difference in the values are an estimate of the theoretical uncertainty, and either of the values can be taken as an approximative value for the ionization cross section. As far as multiple successive ionizations are concerned, the first ionised cations are accelerated very quickly away by the beam, so that, double ionization has little chance to take place. 4.1.2 Counting rates The rate at which the ions arrive at the detector depends on the ionization rate (cations created per unit of time) and the angular acceptance of the collector slit. In other words, the arrival rate is directly related to the pressure of the residual gas, the beam intensity, the ionization cross section and the slit dimensions. The standard conditions are bunches with Np = 1011 protons, and a pressure of the residual gas of 10−10 Torr at room temperature. This pressure, supposing the gas to be ideal, gives a density of ng = 3.29 · 106 molecules/cm3. Assuming a length for the slit of l = 2 cm, a width w = 0.2 cm, and an ionization cross section σ = 0.30 Mb, the number of ions hitting the detector per second is: Np ng l σ (α/2 π)/T ' 100000 ions/s.
(5)
where α is the opening angle of the slit, α = arctan(w/d), d = 2.5 cm is the vacuum chamber radius, and T is the bunch to bunch separation, T = 25 ns. For the case of a predominant dissociated state for the H2 when the beam is 1 on , we obtain the same rate for the cation H + , assuming σ(H) = 0.5 σ(H2 ) and the same gas density. Saturation can be reached if the rate of ionization is so high that the gas has no time to redistribute. If this is the case, a hole in the gas distribution is made, and the ionization regime saturates. For a gas with a Maxwell-Boltzmann velocity distribution at room temperature, the mean velocity of a Hydrogen molecule is 1.3 · 106 mm/s. So in less than 1 µs, the gas can rearrange itself within the beam dimensions. For the standard conditions (P = 10−10 Torr, σion = 0.30 Mb, Np = 1011 protons/bunch), approximately 4 cations are created per cm in 1 µs. The number of molecules per cm in the transverse area occupied by the beam (roughly 3 mm2 ) is of the order of 100.000, hence, saturation is not reached. 1
This was found to be the case in the FFTB [4], thus we have considered it also to be the case for the LHC
6
The bunch filling in the LHC ring also has to be taken into account. The beam in fact fills only 2835 of the 3564 avalaible buckets, which is about 80%. The bunch distribution in the PS is what ultimately determines the fill pattern in the LHC. In the PS there are 81 bunches separated by 8 empty buckets. For the detector measurements, a triggering scheme could be used to avoid the influence of the empty bunches. The measuring time depends on the number of bunches which act on the cation before it arrives at the slit. This restriction in the measuring time introduces a reduction factor in the counting rate, which can oscillate between 0.7 and 0.5 for H+ . Taking a factor 0.6 and an efficiency for the detector of 50%, one expects a counting rate of 30000 ions per second.
4.2 The Transverse Electric Field [8] As mentioned before, the electric field is mainly in the transverse plane due to Lorentz contraction (480 < γ < 7500 in the LHC). Its longitudinal variation follows the longitudinal distribution of the protons, which is assumed to be Gaussian (with a rms value σz ). In the transverse plane the beam will also be Gaussian, with a rms value which can be slightly different in the horizontal and vertical planes. A transverse 2D Gaussian distribution of charge will give a characteristic electrical field which is null at the axis and goes through a maximum in both directions. The position of this maximum depends on the beam dimensions. For an elliptic beam (σx > σy ) the space charge field is numerically described by the error functions [9, 10], and it is calculated as a function of the transverse dimensions for a given longitudinal position within the bunch. For a round bunch, the calculation is simplified by the axial symmetry, giving a radial electrical field which is a maximum for a distance from the centre of the beam given by r = 1.58 σ. For the general case of an elliptic beam, a code “Tefi” has been developed, which includes the round beam as a particular option. The calculations are made using double precision. The field gives an outward acceleration to the cations created by the beam itself. In the particular case of a round beam, which will be mostly used in this study, this acceleration is also radial, which therefore introduces a radial velocity. The largest velocities are reached by the cations created near the longitudinal axis of the beam. This is due to the slow drift of the ions during the 25 ns interval between bunches, which means that several bunches will pass around the ion before it leaves the transverse area of the bunch. Since the ions created close to the beam axis take longer to leave than those created at the bunch edge, they receive more 7
kicks and reach a larger velocity. The cations will continue to receive kicks when they drift from the bunch area up to the vacuum chamber wall. In the following chapters most of the discussion involves round beams, except in some cases, where flat beams are considered (see chapter 6).
4.3 One ion dynamics 4.3.1 Single kick: only one bunch The evolution of one ion as the bunch passes is obtained from a given threedimensional initial position for the generated cation. The velocity of the beam is taken as c (the velocity of light), and the magnetic field due to the beam is neglected (see section 5). The code (“cation”) performs the calculation of the position and velocity of the cation as a function of time (or s, the longitudinal coordinate of the ion with respect to the bunch rest system). The bunch length is considered as 8σz (from −4σz to +4σz ). The space charge field acts on the ion from its initial position s0 , where it has been created, up to the final position, at the end of the bunch passage, s = −4σz . The cation evolution is calculated by means of a numeric integration procedure in time. The sampling time interval, δt, is given by δt c = σz /10; lower values for δt give the same results. At every step, the space charge field is calculated at the initial position of the cation. With this value, the new position and velocity values for the cation are evaluated. Once the evolution due to one bunch is calculated, the next step is to consider the kicks given by successive bunches until the ion reaches the vacuum pipe wall. 4.3.2 The whole evolution A successive train of bunches every 25 ns is considered. A distance d = 2.5 cm is assumed between the beam axis and the vacuum chamber. By considering the kick given by each bunch and the drift during that kick as well as the free drift between bunch passages, the whole evolution is obtained. An example of this evolution is shown in Fig. 2. Due to the discontinuity of the beam (the bunches themselves), the cation velocity increases in steps. Typically an ion receives about 10-20 kicks during its flight to the chamber wall. A kick is strongest when the radial position of the ion
8
is close to the maximum of the electric field. At further radial distances the kicks are weaker and the acceleration decreases.
Figure 2: One cation evolution under the action of a train of bunches. The cation velocity is shown as a function of its radial distance from the beam axis.
4.4 Velocity distribution To simulate the experimental conditions, a statistical study has to be done. A number of ions are generated according to the beam distribution. That implies the generation of normally distributed random numbers, giving out to a tridimensional Gaussian distribution of initial positions similar to those of the charges in the beam itself. As the beam is made of trains of bunches, with a 25 ns separation between successive bunches, the time at which a cation is created, and its time of flight, can not be determined. Here we will not consider any variation of the velocity distribution with time. 9
Figure 3: Histogram of the velocities at the collector for an ensemble of 20000 cations H+ . Calculated for σr = 0.1 cm and σz = 12 cm.
For a round beam, the spectrum or distribution in velocities was obtained for + 20000 cations (H+ , H+ 2 or He ) as a function of the radial position r. The axial symmetry allows us to reduce the transverse dimensions to one radial dimension. The cations reaching the slit are contained within the angle made by the slit edges and the beam axis (taken to be at d = 2.5 cm). The velocity distribution is the same for any angle, however the direction of the velocity vector (i.e. radial) depends on the position where the cation crosses the slit, as in the case of light rays coming from a point source (see Fig. 1). As a consequence, a distribution in position at the slit also implies a distribution in the velocity vector. The code (“cutdis”) calculates the final velocity and position for a distribution of ionized atoms, restricted to an aperture given by the slit width. To minimise the computation time, this restriction is made, when possible, at the ionization step before commencing the calculation. As is shown in Fig. 3, there is a sharp upper cutoff to the velocity. This max10
imum velocity is attained by the cations created near the centre of the beam. The shape of the distribution is mainly explained by the cation radial density function. For low r values the cation density is proportional to r, it then passes through a maximum before slowly decaying to zero at high r values. The longitudinal distribution of the beam gives rise to a decrease of the slope. Due to the beam bunching, for every radial position there is a range of final velocities. It can be seen that for higher longitudinal distances to the beam centre the edge sharps. This behaviour also depends on the beam dimensions, since the smaller the beam for the same intensity, the stronger the kick. For very small beams, as we will see later, the effect on the velocity distribution is peculiar.
Figure 4: The squared maximum velocity of H + cations at the collector as a function of the rms radial beam dimension, σr , on a logarithmic scale.
By applying a linear fit to the cutoff velocity edge, a prediction of the maximum velocity, vf m , can be obtained. This value is a function of the beam dimensions. Fig. 4 shows for H+ , the dependence of the predicted values, vf m , with the rms radial dimension of the beam, σr . Here vf2m in units of c2 is plotted as a 11
function of σr (in cm) on a logarithmic scale. The dependence is approximately linear and it is fitted to: vf2m = (0.0722 ± 0.0012) − (0.0838 ± 0.0006) · ln σr
(6)
This relation is, to first order, divided by 2 for the case of H+ 2 , and divided by 4 for the case of He+ . Fig. 5 shows the same linear dependence for the He+ cations, down to a lower limit in the transverse beam dimension. It can be seen that the linear fit is good for values of σr > 0.02 cm. This limit occurs for σr > 0.05 cm in the case of the H+ , and for σr > 0.03 cm for H+ 2 . An explanation of this phenomenon will be given in section 4.5.2.
Figure 5: The squared maximum velocity of He+ cations at the collector as a function of the rms radial beam dimension, σr , on a logarithmic scale.
Once the cations have reached the slit, they exit the vacuum chamber and have to be detected. To know the energy of every ion, an external magnetic field is applied in order to obtain the velocity from the bending radius. 12
After the cations have been bent, they reach the detector with a spatial distribution related to their velocities. In the next chapters the detection system is described from both the scientific and technical point of view.
4.5 Distribution at the detector As the cations leave the vacuum chamber they are acted on by the applied longitudinal magnetic field which bends their trajectories changing only the direction of the velocity vector (see Fig. 1). The bending radius, Rb , of the trajectory is given by: Mv (7) eB For the cation H+ , for example, Rb [m] = 3.1316 v [in units of c]/B[T]. With the nominal beam parameters for the LHC at injection, the maximum velocity reaches 0.52 10−3 c. For a magnetic field of 300 gauss, the corresponding radius becomes Rb = 5.4 cm. The initial velocity spectrum is translated into a distribution in radii which causes a distribution of impact positions at the detector. The code (“magnon”) calculates the trajectory for every cation as a function of its velocity and position at the slit. The impact position is where this trajectory crosses the detector, being described by the variable D (see Fig. 1). As was explained in section 3, because of the finite slit width, the focalisation effect has to be taken into account. For any given speed, there is a range of circular trajectories with the same radius but with a different origin and angle. The geometry for the detector is chosen in such a way that all trajectories for the maximum speed are focused at the detector. For lower velocities the focusing is not so good, although, as mentioned before, this is of less importance. This effect can be observed in Fig. 6. It can be seen that the defocalisation is not very strong even for the lower velocities. This is because the right focal conditions for these velocities are not far from the given parameters. Nevertheless, when the detector angle, θ, is quite far away from the optimal one, the dispersion in position in the detector is very high. For example, for Ro = 5.4 cm and B = 300 gauss, with a slit 5 mm wide, the spot size at the detector for the highest velocities is 1 cm when θ = 00 compared with 0.05 cm for the optimal angle of θ = −650 (see Fig. 1). Rb =
13
Figure 6: Scatter plot of the position in the detector as a function of the velocity. 10000 H+ , Ro = 5 cm, θ = −63.4o . Collision energy parameters: σr = 0.03 cm, σz = 7.7 cm, B = 424.4 gauss. Slit width, w = 0.2 cm.
The θ values corresponding to a series of Ro values, for a slit width w = 0.5 mm, are the following: Table 2.
Ro θ (in degrees) 3 cm -50.2 4 cm -58.0 5 cm -63.4 6 cm -67.4 7 cm -70.3 8 cm -72.6 9 cm -74.5 10 cm -76.0 14
For the case of a wider slit of w = 5 mm, the calculated values are only slightly lower, differing by less than one per cent. The distribution of the impact position D has a maximum towards high velocities (see Figs. 9, 10). If focalisation is optimised for the top velocities, the distribution in positions reproduce the distribution in velocities for the maximum values. Fig. 6 shows the dispersion of the impacts for all the trajectories. The effect of the dispersion on the measured velocity distribution is negligible as long as the slit width is much smaller than the spatial resolution of the detector. A small slit will provide a sharper image on the detector, but will reduce the number of counted cations. It is worth saying that the focalisation is only achieved at one point, and that the rms dispersion spot size, δD, in the detector is calculated in the zone of the maximum peak. So the position of impact for the maximum velocity is given with the uncertainty Dm ± δD.
Figure 7: Detector rms spot size, δD, calculated for the maximal velocity as a function of the slit width.
Fig. 7 shows the rms spot size in the detector as a function of the slit width for the maximum velocity. It can be fitted to a parabolic curve. The data are simulated taking the H+ as the cation. The beam dimensions are, σz = 12 cm 15
and σr = 0.1 cm. The injection energy conditions were used to obtain the rms dispersion in the detector for the maximal velocity, δD, as a function of the slit width. The error in the vf m determination results from the error in obtaining the position of impact for the maximum velocity, Dm , or maximum D value. Here we will not try to minimise the uncertainty on the D values. We will just take the uncertainty to be of the order of the spot width of the ions on the detector. When focalisation occurs, the error in Dm is the same as that found in the determination of vf m values from the distribution in velocities. This is accomplished if the spot width is small enough and does not increase the uncertainty. For example for a slit 2 mm wide, the spot size is approximately 100 µm for a typical Dm = 5 cm. These values allow the determination of the beam size, according to Eq. (6), to a theoretical accuracy of 0.5 %. 4.5.1 Very small beams At high energies both the transverse and longitudinal dimensions of the beam are reduced. If the velocity distribution is obtained as a function of the transverse rms σr , then it is clear that a change in the distribution is produced. This change is different for the different species, being a function of the cation mass. For example, for the H+ , the velocity distribution which normally has a sharp edge at the highest velocities, starts to change for σr ≤ 0.5 mm. The smaller the σr , the less sharp the edge, while at the same time some peaks or discontinuities appear. This behaviour can be observed in Fig. 8, where the distribution in velocities is obtained for H+ when σr = 0.01 cm (a value much lower than the nominal lowest dimension at 7 TeV, where σr = 0.03 cm). This alteration in the shape of the velocity distribution comes from the discontinuity in the beam, i.e. the bunches. As σr gets smaller, the electric field becomes stronger and its spatial extension decreases. Thus the width of the maximum peak of the radial dependence of the electric field becomes very thin. The cation can go through the region where the field is a maximum, either within a bunch or between two bunches. In the first case, the kick given to the ion is very high, but in the latter case, no kick is given in that region, leading to a softer ion evolution. This causes a discontinuity in the velocity distribution. As for the peaks observed in Fig. 8, their origin can be understood by looking at the distribution of the number of kicks given to each cation before arriving at the slit (top figure). There is some accumulation when passing from one number 16
Figure 8: Velocity distribution for the H+ cation for σr = 0.01 cm. In the top figure the kick number is shown as a function of the acquired velocity.
of kicks to the next, which gives rise to a peak at the velocity where this occurs. When the evolution with σr is looked at for the H+ 2 , the change is seen to take place at lower σr , and even lower for the He+ . The transition between the two types of velocity distribution therefore takes place at smaller beam dimension for higher ion mass. This can be understood as follows: the transition roughly occurs when an ion receives a kick such that on average the subsequent drift δr, after the bunch passage and during 25 ns, is about the same as the beam radial dimension σr . At a given radial position, the energy of a cation is almost independent of the mass, but the velocity is inversely proportional to the square root of the mass. Therefore the heavier the cation, the slower it passes through the maximum in the electric field. The discontinuity is softer, as if the σr were higher. For He, the distribution changes for σr ≤ 0.2 mm. The two extreme situations of the LHC performance are, at the injection energy, the low energy (LE) regime, and the high energy (HE) regime with colliding 17
beams. The expected parameters related to the beam dimensions for these two extremes, LE/HE, are σr = 1.20/0.303 mm and σz = 13/7.7 cm [3]. From now on, we refer to these extreme parameters as the LE and HE conditions, meaning that we are talking about the expected largest and smallest transverse beam dimensions. Simulations for the “D” distribution in the detector have been performed for + H+ , H + 2 and He for LE and HE conditions. The magnetic field is calculated to have a focalisation condition for the vf m value corresponding to one chosen cation. The calculations were carried out for different values of the detector position (Ro and θ).
+ Figure 9: Histograms of the position distributions for the cations He+ , H+ 2 and H . The
focalisation is optimised for H+ . The upper histograms correspond to LE parameters, and the lower ones to HE parameters.
Once the parameters Ro and θ are fixed, the position in the detector where the 18
focalisation is produced is completely determined, and therefore also the bending radius, since the focalisation condition has only one solution, and the magnetic field must be adjusted accordingly. For any pair of Ro and θ values, there is a fixed position in the detector for the focalisation, and the corresponding magnetic field will be determined by the cation mass and the vf m value of the velocity distribution. If we want this position to be equal to the Ro value, the value of θ which fulfils this condition has to be found. For instance for Ro = 5 cm, the focalisation is accomplished at a position in the detector D = 5 cm for an angle θ = −63.4o (see Table 2). In this case, for the LE conditions (σr = 0.1 cm, σz = 12 cm) and for the H+ cation, a magnetic field of B = 312.5 gauss is needed.
+ Figure 10: Histograms of the position distributions for the cations He+ , H+ 2 and H . The
focalisation is optimised for He+ . The upper histograms correspond to LE parameters, and the lower ones to HE parameters.
+ If we were able to detect all three species of cation (He+ , H+ 2 and H ), then
19
by changing the magnetic field we would see the three distributions pass through the focalisation point. The “D” distribution when focalised is about 2 cm wide. As an example, Fig. 9 and Fig. 10 show the results of the expected measurements at the detector for Ro = 5 cm and θ = −630 for the three simulated cations at LE and HE. Fig. 9 shows results with a magnetic field which is optimised for the H+ vf m values, with B = 313 gauss at LE and B = 424 gauss at HE. Fig. 10 is optimised for He+ , with B = 620 gauss at LE and B = 756 gauss at HE. It can be seen that for LE, the H+ (Fig. 9 top left) is a good candidate for measuring the beam dimensions. However, at HE the D distribution gives a poor resolution for vf m determination (Fig. 9 bottom left), which in fact prevents an accurate measurement of the beam dimensions for σr < 500 µm (expected for 2 TeV). An injection of He gas, with a partial pressure of the order of 10−10 T orr, could be a solution, since the He+ cation allows a much more accurate determination of the beam dimensions at HE. Moreover, the He+ cation is also a good candidate for detection at LE, as can be seen in Fig. 10 (top right). 4.5.2 Technical details The experimental set-up for the detector depends to a certain extent on the choice of the preferred cation, with Ro ranging from 3 to 10 cm and the angle θ from -20 to -80 degrees. For instance, for the H+ , Ro = 5 cm and θ = −63o is a good choice, with a magnetic field sweeping from 300 to 450 gauss over the whole energy range. The detector could consist of a pair of micro-channel plates (MCP) which have a spatial resolution of 30 µm rms. As for the MCP dimensions, we have seen (see Figs. 9, 10) that the “D” distribution spread is about 2 cm when focalisation is optimised. This range however also depends on the geometric configuration. For a given distance from the slit to the vacuum chamber centre (d = 2.5 cm), and the focalisation conditions for LHC, the distribution width is proportional to the Ro value. For the conditions considered, microchanel plates with a width of 4-5 cm would be suitable to allow a complete distribution measurement. As we have considered a slit which is 2 cm long, the MCP should have at least this length. In the MCP an electron shower is produced for every cation that enters a micro-channel. The efficiency increases with the energy of the incident particle, reaching saturation at 60 % for energies higher than 2.0 keV. We have calculated for the LHC beam parameters, that the cations to be detected have an energy of around 100 200 eV. The MCP efficiency for energies of the order of 100 eV is less than 1 %. 20
As a consequence, an acceleration of the cations to an energy close to 2.0 keV is needed before they reach the MCP. The gain obtained with the MCP pair is of the order of 107 , i.e. one cation can give a charge of 1 pC. The charge can be collected by a phosphor screen, and the generated photons by a CCD camera. The CCD photodetectors are silicon devices made of an array of typically 400x600 pixels 20x20 µm2 . This sensor can cover the whole range in D, for instance ranging from 1 cm to 6 cm, with a spatial resolution better than 100 µm.
5 Effect of the beam magnetic field on the electrons from ionization Gas ionization creates electrons as well as cations. These electrons are sensitive not only to the electric field of the beam, but also to its magnetic field as they can acquire large velocities. Having a negative charge, the electric force acting on them is attractive. As the electron gets free, it starts to oscillate within the potential well of the bunch during its passage. The maximum velocity acquired by the electron in the oscillation occurs at the centre of the beam, and can range from zero, for the electrons created at the beam axis, to 0.1 c, for the electrons created at the border of the bunch. The dynamics of the electrons is rather complicated since the velocity they acquire can be close to the velocity of light and under these conditions the effect of the magnetic field of the beam is no longer negligible. The magnetic force is longitudinal with respect to the beam and has a magnitude with a ratio v/c compared with the electric force. The angular kick due to the magnetic force is given by the expression [1]: δα =
pz v⊥ /c ' q 2 p⊥ 2 1 − v⊥ /c2
(8)
For v⊥ ' 0.1 c, δ α = 0.05, which is not at all negligible. The value for the cations is at most 0.25 · 10−3. The magnetic force acting on the electrons oscillates with the transverse motion of the electron. Most electrons reach the beam pipe before the passage of the following bunch. Even if they reach the slit, as they leave the vacuum chamber, the external magnetic field bends them in the opposite direction to the cations. The cation detector will therefore not receive impacts from the fast electrons due to ionization. As a consequence, we do not have to consider the trajectories of these particles. 21
6 Measurement of beam ellipticity 6.1 General remarks Now that the results obtained for a round beam have been analysed, we will look at the differences obtained when the beam is no longer round. Taking the nominal parameters of the LHC beam as a reference, the ratio between the horizontal and vertical transversal dimensions is increased to obtain a beam which is larger horizontally than vertically. Our aim is to determine a way of measuring both transverse dimensions of the beam. Thus, we look for the features depending on them. It is seen that as the ellipticity increases, there are some effects in the transverse distributions of final velocity which are noticeable: • The velocities are no longer radial. The vertical component for the elliptic beam is stronger than the corresponding one for a round beam. This causes a deviation of the trajectory from the radial one. • In spite of an initial ellipsoidal distribution of cations, the number of ions arriving at the vacuum chamber wall is almost isotropic. • The distribution width, however, presents azimuthal differences. • The maximum velocity at which the cations arrive is not very sensitive to the ellipticity. The mean velocity measured corresponds to the relation obtained for a round beam when the radial rms equals the mean of the x and y rms, that is σr = σm ≡ (σx + σy )/2. The program (“catdis2”) generates two million cations with a probability given by the charge density, which is Gaussian. From all the initial positions generated, only the radial positions within a certain limited angular aperture2 are chosen. This limit takes into account the studied evolution of the cations. In this way, for a detector in the horizontal position, X-Detector or XD, with a physical aperture of one degree, an initial aperture of one degree is enough, since the cations have a tendency to go upwards. However, for a vertical detector, Y-Detector or Y D, a higher initial aperture is needed for a final angle of aperture of one degree. This last limit increases with the ratio ξ = σx /σy . A limiting initial angle of ten degrees is enough for R up to 6. 2
This artificial aperture is taken to avoid tracking the ions with no chance of going through the
slit.
22
The differences in the velocity distribution at the XD and the YD has to be analysed to allow the determination of the transverse beam dimensions. When compared to the distribution for a round beam, which is azimuth independent, it can be seen that the XD distribution is broader and the YD distribution is narrower. The ratio of their widths can be a measure of the ratio of the transverse dimensions of the beam. As the distribution broadens, the maximum also decreases, giving a measure of the beam shape. The change in the velocity distribution comes from the higher vertical component of the velocity. The cations, mainly those created near the beam axis, tends to go in the vertical direction avoiding the XD. In this way, the proportion of cations arriving at the XD at lower velocities is increased, with less cations arriving at the highest velocities. In the Y D the opposite occurs, with lot of ions created near the centre reaching the detector, even if their initial position is not within the physical angle corresponding to a round beam (radial velocities). This asymmetry is further enhanced by the initial flat distribution of created cations. The focalisation effect is not altered very much by the flatness of the beam. It is true that the velocities are no longer radial, but as the cations reach the slit, the deviations from a radial velocity are very small. As a matter of fact, the dependence of the velocity direction with the final position direction (vy /vx vs. y/x) can be fitted to a straight line with a slope very near to 1, giving deviations of less than 5 % for ξ ≤ 3. When we look for the focalisation point for a given position for the detector, Ro , the required tilt, θ needed to obtain Dmax = Ro , changes with the flatness of the beam. Hence the optimal value is different in both detectors. But it is worth saying, that the focalisation is not very sensitive to changes in this angle of up to around five degrees. To overcome this problem, the detector parameters are chosen with the optimal tilt for a round beam, which is a good compromise for both detectors for whatever the R ratio. + Looking at the distributions for the three cations, H+ , H+ 2 and He , in the detector, a very good resolution for the three distributions is seen, with no overlapping.
23
6.2 Detector physical location We have seen that for the case of a flat beam, two detecting devices will be needed, one in the horizontal plane and the other in the vertical plane. We have analysed the possible experimental conditions concerning a suitable place for the detector along the LHC ring. The gas-ionization beam size detector is foreseen to be installed at the insertion point 6, thus the simulations have been performed with the beam parameters predicted at this point. IR6 is an insertion for the beam dump system. There are some room for the detection system near quadrupole 4, before the kicker. The transverse beam parameters predicted at this position are the following; βx = 205 m and βy = 575 m, with the corresponding rms values, σx = 330 µm and σy = 538 µm at the highest energy of 7 TeV. The low energy values, at 450 GeV, are calculated by multiplying by the inverse square root of the energy ratio, i.e. approximately 3.9. This gives σx = 1308 µm and σy = 2120 µm at 450 GeV. For simplicity, we exchange x for y to have the flatness horizontally. Taking the parameters for the highest energy, the distribution in velocities, or in position at the detector, is fine for He+ and for H+ 2 , but is already deformed for H+ . For the larger beams at low energy, calculations were performed with various ratios of ξ = σx /σy for the same mean value, σm = (σx +σy )/2 = 0.17 cm. The maximum velocity for the cations is azimuthally uniform. Thus we obtain the same vf m value for the horizontal and the vertical distribution. This common value is not dependent on ξ. Moreover, the variation of vf m with σm is the same as it was for the case of a round beam. Fig. 11 shows the difference in the vf distributions of H + cations in the horizontal and vertical slit, named X and Y respectively. It can be seen that both distributions evolve differently when ξ increases. The results for the other cations are similar. The histograms of the hitting positions at the detectors, after the cations have been bent by the magnetic field, have the same aspect ratio.
24
Figure 11: Horizontal (X) and vertical (Y) vf distributions for three values of ξ. Simulations carried out for H+ ; (σx + σy )/2 = 0.17 cm.
To measure the ξ value from the results in the horizontal and vertical detectors, we need to be able to relate the differences between both distributions with ξ. We have simulated the evaluation of this difference by taking the rate in the distribution maxima, coc = maxY /maxX as a variable. In Fig. 12 this variable is plotted as a function of ξ. The dependence is linear at low ξ values and saturates for ξ ≥ 3.0. The errors in the determination of ξ from the fitted function for ξ < 3.0 are of the order of 5 %. The width of the distribution is another characteristic feature which can be used as a method to determine ξ. We have looked at the full width at half maximum, F W HM, from the results in both detectors, F W HMx for XD (horizontal 25
Figure 12: Ratio of the maximum of the vf distributions, coc = maxY /maxX, as a function of ξ. Values calculated are for H+ . detector) and F W HMy for Y D (vertical detector). The evolution with ξ is shown in Fig. 13. The saturation is clearly seen in the F W HMy variable. The growing of the y distribution for the higher velocities from the x one is the process provoking this evolution. On the other hand, this increasing distribution has to be limited, as it is not an infinitive process. From all this, we can say that an analysis of the distributions can give us a measure of the flatness of the beam, for ξ less than some limiting values. For the expected parameters of the LHC, it seems to be possible to perform such an analysis with a good accuracy.
26
Figure 13: Full width half maximum of the vf distributions, and their ratio, as a function of ξ. Values calculated are for H+ .
7 Nitrogen as an optional cation We have seen that for very small beams the measurement at the detector is in some cases distorted. The use of He+ as the detected cation overcomes this problem. However, He is difficult to pump. Taking this into account, the use of a heavier and easier to manage gas such as N2 , could be an option to consider. The N+ mass is approximately 14 times that of H+ , giving a final velocity at the collector which is approximately 3.7 times smaller. As the cation moves more slowly, the number of kicks given to it by the beam increases. As a consequence, as happens with He+ when compared to H+ , the slope of the distribution at the highest velocities is more vertical and hence the determination of the Dmax is more accurate. The magnetic field range required will increase by the same factor as the velocity decreases, as B is inversely proportional to the velocity3. 3
B is proportional to the cation momentum. For the same beam parameters, the velocity ac-
27
The transverse beam dimensions which can be measured in this way decreases to σr (or σM ) ≥ 0.1 mm, due to the slower cation velocity. There is, nevertheless, an inherent problem with the use of N+ , which is that many more bunches are needed for the cation to reach the slit. The number of bunches is inversely proportional to the cation velocity, thus almost 4 times the number of kicks are required for N+ compared to H+ . This reduces the counting rates for N+ , as was explained in section 4.1.2. When nominal conditions are not attained, and the beam intensity is far below the expected one, the number of kicks felt by the cation can increase to up to 80 or more. In this case a measurement without distortions due to the empty bunches is not possible. The N+ cation is created by primary ionization, but for N secondary ionization is stronger than for the lighter atoms H and He. Although the probability of creating the N++ cations is not negligible in a gas target, it comes from the consecutive ionization process. In our case, the singly ionized cations are quickly accelerated, and the probability of a second ionization process is low. Nevertheless, the N++ cation evolution has been studied. We have seen that the N+ and N++ have distributions at the detector which are very well separated. The possible N++ distribution will therefore not disturb the N+ distribution.
8 Conclusions The detector described above is measuring the energy spectrum of ions produced by the beam in the residual gas of the machine and accelerated in the beam potential. From the top edge of the ion spectrum, the average transverse size: (σx +σy )/2, for an elliptic cross section, can be determined with great accuracy (of the order of 1 %). With a thin slit of 2 mm x 20 mm, a counting of 30000 ions will take one second. The detector will consist of a set of micro-channel plates arranged in a fixed geometry, with a minimum length of 4 cm and viewed by a CCD. The magnetic field in the spectrometer can be changed in order to place the top edge of the spectrum at the point Dm where the focussing is most efficient. This type of detector works independent of the beam energy (from 450 GeV to 7 TeV). When the rms beam size is smaller than 0.5 mm, the fast ions H+ show quired by a given cation is inversely proportional to the square root of its mass. Therefore B is inversely proportional to the cation velocity.
28
a spectrum distorted by individual bunch kicks, and consequently the resolution is not enough. Therefore it might be necessary to use heavier ions like He+ or N2 which are slower and less sensitive to the beam bunching. This will be possible with the injection of a local pressure bump of the order of 10−10 torr. In order to measure the beam ellipticity: σx /σy , it will be necessary to have two detectors collecting ions in the two transverse directions x and y. Having two independent detectors will give an interesting redundancy for tracking the beam blow-up.
References [1] J. Buon et al., A Beam Size Monitor for the Final Focus Test Beam, Nucl. Instrum. Meth. A306 (1991) 93-111. [2] P. Puzo et al., Results of a Gas-Ionization Beam Size Monitor at the Final Focus Test Beam, Nucl. Instrum. Meth. A425 (1999) 415-430. [3] LHC Yellow Book, CERN/AC/95-05(LHC) (1995). [4] P. Puzo, PhD Thesis, Orsay University (1994). [5] F.F. Rieke and W. Prepejchal, Ionization Cross Sections of Gaseous Atoms and Molecules for High-Energy Electrons and Positrons, Phys. Rev. A 6, 1507 (1972). [6] M. Inokuti, Inelastic Collisions of Fast Charged Particles with Atoms and Molecules-The Bethe Theory Revisited Rev. Mod. Phys. 43, 297 (1971). [7] U. Fano, Ionizing Collisions of Very Fast Particles and the Dipole Strength of Optical Transitions, Phys. Rev. 95, 1198 (1954). [8] The calculations of the space charge field have been done without taking into account the image field due to the presence of the vacuum chamber. For a centred round proton beam, this image field vanishes. In the case of a centred elliptic proton beam, the influence of the vacuum chamber is a second-order effect that can be neglected at an accuracy of 0.1%. [9] M. Bassetti and G.A. Erskine, Closed expression for the Electrical Field of a two-dimensional gaussian charge, CERN/ISR-TH/80-06.
29
[10] W. Gautschi, Efficient Computation of the Complex Error Function, SIAM J. Numer. Anal. 7, 187 (1970).
30
