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Monte Carlo simulation of the high energy cosmic muon background in a resonant gravitational wave antenna
Nuclear Instruments and Methods in Physics Research A260 (1987) 491-500 North-Holland, Amsterdam
491
MONTE CARLO SIMULATION OF THE HIGH ENERGY COSMIC MUON BACKGROUND IN A RESONANT GRAVITATIONAL WAVE ANTENNA Fulvio RICCI Dtparttmento dt Ftstca, Untoersttà "La Sapienza" Roma, Italy and Istttuto Naztonale di Ftsica Nucleare, Seztone dt Roma, Italy Received 9 March 1987
We present the results of a Monte Carlo simulation for the energy loss distribution of high energy cosrruc muons crossing a Weber-type gravitational wave (g .w.) antenna. The number of events per day of energy greater than an assigned value, generated in the antenna by the muons, is deduced. The simulation shows that a rate of 60 events per week due to the cosmic background is expected in a sea-level g.w . detector with an energy sensitivity ten times greater than that of the present antennas . With the sensitivity of the resonant detectors approaching the quantum limit value the rate will increase to 5 X 10 ° events per days . Therefore it seems unavoidable to carry on the experiment in an underground laboratory.
1. Introduction The generation of mechanical vibrations in solids by pulses of cosmic radiation has already been considered in connection with the detection of very large cosmic ray events, and, more recently, even as a possible mechanism for revealing very high energy neutrinos [1], monopoles or more exotic particles [2]. Our interest in the problem of the mechanical response of a metal bar to the impact of charged particles is due to the generation of the background signals for a gravitational radiation detector of Weber's type . The first contribution on this subject was given by the pioneering work of Beron and Hofstader who used high energy electrons to induce oscillations in solids . Later Weber et al. showed experimentally that a gravitational radiation detector does not produce a signal when hit by showers of particle density of the order of 100 particles/m2. In 1979 a group of the university of Milan [3] investigated the mechanical response of a small aluminium alloy bar to pulses of 30 MeV protons, and they related quantitatively with simple theoretical considerations the energy loss of the particles in the bar with the amplitudes of its natural vibration modes. At that time, from these results it was possible to conclude that any effect due to cosmic rays was below the sensitivity of the available detectors [4]. However, in consideration of the much higher sensitivity reached today [5] and even more in view of the sensitivity which has to be reached in order to detect events due to supernova explosions in the Virgo cluster, the problem of the influence of cosmic rays on gravitational-wave antennas has been re-examined by Amaldi and Pizzella [6]. In their paper they estimate the rate of the energy lost by the high energy muons impinging in the vertical direction, perpendicularly to the axis of the cylindrical antenna. In order to derive the number of events due to this kind of process which give a signal in the antenna higher than an assigned value, the same approximation of Grassi-Strini et al ., [3] was applied. This numerical computation was done assuming also that the electromagnetic shower generated inside the bar by the muon interaction with matter is completely absorbed into the bar itself and neglecting the effect of the secondary products produced in the thick wall of the cryostat surrounding the antenna. Under these approximations the authors show that at sea level the secondaries generated in the bar by the interaction of high energy muons produce signals with a rate much higher than that expected from supernova explosions in the Virgo cluster. 0168-9002/87/$03 .50 © Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
492
F Rica / Monte Carlo simulation of high energy cosmic muon background
This interesting results suggests the opportunity of trying an improvement in the evaluation of possible disturbances due to the cosmic muons . We present here the results of a Monte Carlo calculation for the energy loss distribution released in the antenna by the muons and by the fraction of the e .m. shower absorbed in the bar . From this energy loss spectrum it is easy to deduce, in the Grassi-Strini et al. approximation, the rate of events which give signals in the antenna greater than an assigned value . The contribution to the energy loss distribution of the secondary products generated in the thick iron shell which surrounds the antenna is also taken into account . 2. The initial parameters of the Monte Carlo simulation The simulation of the interaction of the cosmic muons with the antenna has been carried on by using the software package GEANT3 developed at CERN . GEANT3 allows to generate simulated events from standard Monte Carlo generators and to control the "tracking" of particles through various regions of an experimental setup, taking into account the geometrical volume boundaries and all the physical effects due to the nature of the particles themselves and to their interaction with matter. The "tracking" of a particle through matter consists in predicting the spatial coordinates of a set of points which define the trajectory and in computing the components of the momentum at these points . The hypothetical apparatus in our case consists of two coaxial cylinders : the inner one is the g.w. antenna, a cylindrical bulk made of aluminum of 60 cm diameter and 3 m length; the second one is a tube, made of iron, 1 cm thick, with an external diameter of 2.3 m and 5 m long, which simulates the external shell of a cryostat of a g.w. detector. These are the dimensions of the Rome group g.w. antenna installed at CERN. The sensitive volume inside which the tracking of a particle is performed is a vacuum box surrounding the experimental setup. Following a flat probability distribution, the coordinates of the origin of the muon trajectories are randomly generated on the upper plane of the boundary box . In principle the cylinder can be hit by particles travelling through one half of the total solid angle . On the other hand the finite dimensions of the upper plane of the box along which the particles are randomly generated determine a limitation on the acceptance angle of the cylindrical antenna . In order to minimize this effect the dimensions of the external vacuum box are chosen as follows : 16.7 m in the horizontal plane along the diameter of the antenna, 18.5 m along the axis of the bar, and 2.32 m along the vertical direction . By assuming that the intensity of the cosmic ray muons decreases as the square of the cosine of the zenith angle, we evaluate that, with this geometry, the numbers of particles which are not included in the acceptance angle are of the order of 0.5% of the total number. In order to randomly extract the initial momentum of the particle, the integral distribution function of probability of the cosmic muon energy at sea level has been derived from formula (6) of ref . [6] : I,( ( >-
(
E
Ei ,
EJ = Io exp - al 1n2 E ` - a 2 In E` - a 3 , o
o
where to = 1 (s sr cm2 )-1, Eo = 1 GeV and al = 0.1594, a 2 = 0.6718, a 3 = 4.826 . The relation is a good fit to the experimental data given by Menon and Ramana [8] for E,, >_ 20 GeV and by Allkofer and Tokisch for lower energies [9] . From eq. (1) we derive the integral probability function of having an incident cosmic muon with an energy smaller than E,, : (
P(EiJ=1-expl-a
l In2 Eo - a2InEo 1 . E
E l
(2)
F. Ricci / Monte Carlo simulation of high energy cosmic muon background
493
The random choice of E,, is now performed by means of a random extraction of a P(E,,) value, with a flat probability distribution, and using the inverse relation of eq . (3) E~ - E° exp
-a 2 + az-41n(1-P) 2a1
The direction of the momentum of the incident particle is defined by a random choice of the azimuth and zenith angle. The zenith angle is assigned by a random extraction weighted by a cosine square dependence of the muon intensity, while a flat distribution is assumed for the azimuth angle. Finally, for the random choice of the charge of the incident muon, a constant value for the charge ratio is considered [10] : Nl, ./NI,-
=
1 .3 .
3. The tracking control of the primary particle The production of secondary particles by muons travelling in matter is due to four different processes: fotonuclear interaction, bremsstrahlung, direct pair production and knockon electron production . In the computer simulation, during the tracking of the primary particle through the experimental setup, these four fundamental processes are taken into account and the final state after the interaction is generated by sampling the corresponding differential cross section, while the total cross section of each process is used to evaluate the probability of occurrence of the process. It is useful to recall that knockon electron production is more probable than the other mechanisms and its different cross section for production of electrons with energy between T and T + d T, valid for incident particles with spin 1/2, is : dT -
2aZe2m RZ
2 (1 - /32
f'
+ T max
2É2
~~
where T = kinetic energy of the scattered electron, Z = charge number of the medium, re = classical electron radius, m = electron mass, M = muon mass, and T ,-
_
2m
(Y2
- 1)
1 + 2y (m/M) + (m/M)2
. is the maximum energy transferable to the electron by a p-meson of energy E,, = yM and '82 = 1 - 1/Y2 The generation of bremsstrahlung photons by a high energy muon is treated as a discrete process below a defined cutoff energy . Its differential cross section is given by the well-known formula of Petrukhin and Shetakov [11] 2 1 du . 3(2ZX,m 4 a ~i_V M ) V[_~
- 4v
+ v2
where V is the fraction of energy transferred to the photon by the muon, a is the fine structure constant, A e the Compton wavelength of the electron, and, for Z > 10, O(S) = In
2
189( M/m) Z- 2/3
1+
189,% ____2
8Z-1/3
494
F. Ricci / Monte Carlo simulation
of high energy cosmic muon background
with e = 2.71828 and 8
-
M2V
2E (1 - V)
The direct pair production by muons is taken into account by considering the differential cross section of the process as reported in ref. [12]. The expression of the cross section is a convenient parametrization due to Kokoulin and Petrukhin based on a QED calculation performed by Kelner and Kotov. Finally, the generation of secondary particles by the interaction of muons with the nuclei of the tracking materials is also considered, assuming as differential cross section of the process: d'a _ T(a,+ eas ), d12dW -
(6)
with ( 2 e=1+2I1+(Q) ) tan2
2,
(7)
and Q2=2(ENW-Ip11p'Icos0
v=Ev -W.
M2),
(8)
(9)
Here OT and as are the photoabsorption cross section for transverse and longitudinal photons respectively . CT is assumed to be constant and equal to 0.12 mb, while for as we have : as = 0.3 1-
1 OT . 1 .868Q2/v )
(10)
4. The electromagnetic shower control The interaction of the primary cosmic muon with matter can generate secondary particles with energies high enough to produce tertiary particles with rather high probabilities. If the incident particle has high energy, these processes continue to take place successively and the number of particles produced thereby increases. This multiplicative sequence, called an e.m. shower, begins to decline as some particles receive energies too low for producing further particles effectively. The longitudinal development of an electromagnetic shower in matter is determined by the radiation length Xo . Its lateral spread is mainly due to the multiple scattering of electrons that have enough energy to travel far away from the axis. In order to eliminate the material dependence, the lateral spread of an electron beam of energy e after traversing a thickness X0 is expressed in Molière units, RM= Es Xo> E
where Es = 21 MeV is a typical constant appearing in multiple scattering theory and e is the critical energy of an electron that loses as much energy in collisions as in radiation. The equations giving the radiation unit Xo and the Molière unit RM are rather complicated. To give an estimate of the extension of the
F. Ricc / Monte Carlo simulation of high energy cosmic muon background
495
shower we can follow the suggestion of ref . [13], in which the quantities mentioned above are approximated by: X°
_ 180A ZZ
e- 5 RM
g , cm2
(12) (13)
MeV,
7Z
(14)
cmz
These estimates are accurate within 10% (20% for Xo ) for 13 < Z < 92. The longitudinal distribution of the track length, defined as the sum of the tracks of all the particles in the shower, is proportional to the longitudinal energy deposition which is well represented by a parametrized formula [13]. Its knowledge allows us to compute the length T, expressed in Xo units, along which a fixed fraction of the incident energy is deposited . It can be shown that for an absorption of 98% of the incident energy E T(98%) =
3I In( l
E)
+ a],
(15)
where a = 0.4 for electrons and a = 1.2 for y-rays . The multiple scattering for the central part and the propagation of photons for the peripheral part determine the lateral spread of the shower, with no simple dependence upon A and Z. In any case, it can be shown that 95% of the total energy of the shower is contained in a cylinder of radius 2 R M . Using these relations it is easy to see that the complete absorption of the shower generated by the muon inside the antenna is a crude approximation because its longitudinal development in aluminum is of the same order of magnitude as the diameter of the antenna . Therefore, the fraction of the shower completely absorbed inside the aluminum bar is strongly dependent on the point where the primary particle interacts inside the antenna. On the other hand, the lateral spread of a shower crossing a thickness of matter of one radiation length is of the order of one centimeter for both aluminum and iron. This means that all the processes are developed mainly in the forward direction. In order to evaluate the fraction of the energy lost from the shower in the bar, in the Monte Carlo simulation each secondary particle is "tracked" through the whole experimental apparatus . The greater part of the computing time of the simulation is spent in the tracking phase ; for this reason, and considering the small lateral spread of the shower in the iron thickness, the tracking is started only if the initial direction of the primary particle crosses the boundary of the antenna . In practice, when the muon generates secondary products the nature and the initial position and momentum of the new particles are stored temporarily in a memory stack, and the tracking of the primary particle, if it survives after the interaction, is completed . So it is possible to come back at the interaction point and follow the secondary products through the whole experimental apparatus . These new particles, which are essentially electrons, positrons and y-rays, further interact with the matter until its energy degrades to the level of atomic ionization and excitation. For each particle, depending on its nature, some dominant processes are considered for their interaction with matter. In particular, the tracking of y-rays is affected by pair conversion, Compton collision and photoelectric effects, while as concerns e +/e - interactions multiple scattering, ionization and delta ray production, bremsstrahlung and positron annihilation are considered. As for the primary particle, the total cross section of these physical mechanisms is used to evaluate the probability of occurrence of the discrete processes ; the corresponding differential cross-section is sampled to generate the final state after the interaction [7] . The further products stored in a temporary stack, are tracked at the end of the tracking of the corresponding generating particles.
496
F. Ricc3
/ Monte Carlo simulation of high energy cosmic muon background
When the kinetic energy of the particles is below a defined cutoff value, which can be different for each mechanism, for the corresponding process only a continuous energy loss of the particle along its trajectory is considered . In this way we are able to keep track of the values of the energy of each particle which crosses the boundary of the antenna at its input and at its output . The difference between these two values gives us the information about the energy loss inside the antenna by the muon and by the corresponding e.m . shower which in principle can be generated not only in the aluminum bar but also in the iron cylinder surrounding the gravitational wave antenna.
5. Results and conclusions The Monte Carlo program was run on a VAX 8600 generating 58 496 165 muons. The energy distribution obtained for the incident particles is shown in fig. 1 . In order to check the procedure of random generation, the integral vertical spectrum of cosmic muons is numerically deduced, under the hypothesis that the differential spectrum decreases as the square of the
lo'
1 63
Z 10 6 0 ó w
E
m
lo
d N
a
Y U'
w AI
lo
s
w
z
1
10
ENERGY (GeV)
100
1000
Fig. 1 . Energy spectrum of the incident muons obtained by Monte Carlo generation for 58496165 random extractions.
I 10
I 100
E[, (GeV)
L 1000
Fig. 2 . Integral vertical spectrum derived from the Monte Carlo generation, normalized as described in the text. Open circles : experimental results from ref . [81 ; full circles : experimental results from ref . [9] .
F. Rica / Monte Carlo simulation of high energy cosmic muon background
497
cosine of the zenith angle. The normalization of the integral spectrum is performed by assuming as value for the vertical intensity : I~( >_ 1 GeV) = 80 .2 m-2 s -1 sr. In fig . 2 we report the integral vertical spectrum obtained from the random generation and the experimental data of refs. [8] and [9] . The agreement seems satisfactory. By retaining the information of the total energy loss of muons and its consequent shower in the antenna, we obtained the diagram of fig . 3 ; it represents the integral energy loss spectrum due to the 150000 muons which have hit the antenna among all the incident particles generated on the upper plane of the boundary box . We tried successfully to fit the integral loss spectrum -Y(>_ Elost) in two different energy ranges with the following parametrization : T(> E l ,,,,) =kEF t
(16)
obtaining k = (1799 ± 1) GeVY,
= 2 .879 ± 0.001,
Y
(17)
for 0.3< E,,s, (GeV) _< 1 .0 and k = (1921 ± 10) GeVY, Y = 2 .178 ± 0 .006, for
1 .0< E, os ,(GeV)
(18)
<_ 1000.
105 5
Table 1 Normalized E(>- El,,,) for events with an energy loss greater than Eros,
10~~-
Eios, [GeV]
5
10
ó
3
\\ k=1921±10 GeV 7 ~Y=1794±0006 E-Y =K Los
\ \
5
\\
\ 102,, 5
L
k=1799±1 GeV Y Y=2 .879±0 .001
~4~+
\
\
\
10
\
\
5
\ t \
\ \
\ \
\ 1
0.1
0.5
1
5
10
50
100
E dust (GeV)
Fig. 3. Energy loss spectrum of cosmic ray events inside the antenna. The dashed line gives power law fits E = kEl-os"t .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 2.0 3.0 4.0 5 .0 6.0 7.0 8.0 9.0 10 15 20 30 40 50 100
E( >_ E,os,) [event/day] 2.23 x 10 7 1.87 x 10 7 1.20 x 10 7 5.26X10 6 2.68 x 10 6 1.55 X 10 6 1.00 x 10 6 7.22 x 10 5 5.36x10 5 4.18 x 105 1.02 x 105 4 5.03 x 10 3.19x10 4 2.30x10 4 1.95X10 4 1 .57x10 4 1.32X10 4 1.08 X 10 4 9 .73X10 3 6.00 X 10 3 3.52x10 3 2.28 X 103 8.28 X 10 2 6.21 x 10 2 2.07 X 10 2
F. Ricci / Monte Carlo simulation of high energy cosmic muon background
498
The two fits are shown in fig. 3; in table 1 we report Z(>_ Elos,) normalized as the integral vertical spectrum to give the number of events per days with an energy loss greater than El ,,,,, . We can now derive the number of events which give signals in the antenna with energy higher than an assigned value kBTeff, where k B is the Boltzmann constant . Let us recall that a particle which interacts traversing the solid body of the antenna acts as a source of sonic waves, and let us recall also that in a gravitational wave detector of the Weber type the change in the amplitude of the first longitudinal mode of the cylinder is monitored. We then need to consider the projection of the sonic wave perturbation due to the incident particle on the first eigenmode of the bar. The exhaustive calculus for the resulting excitation of the vibration mode of the detector is in general very difficult, because of the complexity of the motion of the solid body in which the tensor of deformations and the boundary conditions are not related in a simple way. For this reason we follow the approach of the Grassi-Strini et al . group in which the vibration of the fundamental mode of the bar is originated by the conversion of the local thermal expansion caused by the sudden energy lost by particles into an acoustic wave . The consequent elastic energy of the first longitudinal vibration is given by [3] k BTeff -
v 2 1
2
( C ) Ma v
COS
2 ( 7rx )
L
(19)
2
Elost ,
where q is the linear thermal expansion, c,, the specific heat at constant volume of aluminum *, Ma the mass and L the length of the bar, x the distance between the center of the bar and the impact point and o, the velocity of sound in the material. In order to derive the number of events due to the muon background which give signals with energy greater than kBTeff, we should also consider that the event can take place for any value of x between ±L/2 . Using the parametrization of eq . (16) for 1, we obtain : N(> k B Teff )
_ 1 -
L
+L/ 2 k L/2
kBTefrMa \
U
1(7j/CJ
cos(9rx/L )
dx .
(20)
The sensitivity range of interest for the experimentalist is near or above the quantum limit of the experimental apparatus, Teff = 10 - ' K. Assuming as antenna parameters those of the aluminum bar of the Rome group, i.e. Ma = 2300 kg and L = 3.0 m, the most interesting region of the integral energy loss spectrum of the muons in the detector is from 1 GeV up . In fig. 4 we give the result of the numerical integration of eq. (20), using the parametrization of eq. (18) . The comparison with the numerical calculation performed in ref. [6] is also shown. The Monte Carlo calculation of the effect of the background due to the cosmic rays confirms the results obtained by Amaldi and Pizzella, indicating that the rate of background events is much larger than the rate of signals expected from supernova explosions in the Virgo cluster. The disagreement shown in fig. 4 is reasonably well understood by considering the more general conditions imposed on the Monte Carlo simulations, i.e. cosmic muons incident on the bar from any possible direction, the correct evaluation of the fraction of the electromagnetic shower absorbed in the bar, and the interaction of the primary particle with the external iron shell. On the other hand, it is important to realize that for energies of the incident muons of the order of 100 GeV the contribution of the stochastic radiative processes becomes important and eventually dominates the total energy loss . Under these conditions, the derivation of the energy loss by a single mean value may become inadequate while, using the detailed geometry of the detector, a Monte Carlo package for the statistical simulation of the muon energy loss gives correct information. From fig. 4 it appears that an improvement with a further order of magnitude in the energy sensitivity of the available gravitational wave detectors, at present in the range of Teff - 10 mK [5], will lead to * The ratio 71/c v , for aluminum, is assumed equal to 4 .75 X 10 -5 kg J -1 and independent of the temperature .
F.
Rfcci
/ Monte Carlo simulation of high energy cosmic muon background
499
10 5
]C _AI Z
10 2
10
10
8
10
1
10
6
10
Ten (K)
5
10
1
10
3
Fig. 4. Number of events per day of energy >- k B Teff produced at sea level by cosmic muons in a g.w . detector like that of the Rome group at CERN . The corresponding result of ref. [6] is shown (dotted line) for comparison .
conditions in which cosmic muons should give a number of the order of 60 events detected per week at sea level . Approaching the quantum limit = 10-7 K, the event rate due to the muon background is expected to be of the order of 5 X 10 4 events per day for a detector at sea level. Therefore it seems compulsary to carry on the experiment in an underground laboratory, where the rate of cosmic muons impinging on the antenna is drastically reduced and can easily be kept under electronic control. Teff
Acknowledgements I am indebted to Profs. E. Amaldi and G. Pizzella for the suggestions, the encouragement and support during this work. I also which to thank Dr. F. Carminati for useful discussions about the GEANT3 software package and Dr. M. Bassan for a critical reading of the manuscript. References [1] M. Levi et al., IEEE Trans. Nucl . Sci. NS-25 (1978) 325. [2] C. Bernard, A. De Rujula and B. Lautrup, Nucl . Phys B242 (1984) 93 .
500 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
F. Ricci / Monte Carlo simulation of high energy cosmic muon background A.M. Grassi-Strini, G. Strini and G. Tagliaferri, J. Appl. Phys. 51 (1980) 948. J. Weber, Phys. Rev. Lett . 22 (1969) 1320 . E. Amaldi et al . Nuovo Cim. 9C (1986) 829. E. Amaldi and G. Pizzella, Nuovo Cim. 9C (1986) 612. GEANT3 User's Guide DD/EE/84-1 (May 1986). M.G .K. Menon and P.V. Ramana Murthy, Progr. Elementary Part . Cosmic Ray Phys . 9 (1969) 220. O.C . Allkofer and H. Jokisch, Nuovo Cim. 15A (1973) 371. O.C . Allkofer, Proc. 16th Int. Cosmic Ray Conf. 14 (1979) 385. S. Hayakawa, Cosmic ray physics (Wiley, New York, 1969). W. Lohmann, R. Kopp and R. Voss, CERN 85-03 (21 March 1985). U. Amaldi, Phys . Scripta 23 (1981) 409.
491
MONTE CARLO SIMULATION OF THE HIGH ENERGY COSMIC MUON BACKGROUND IN A RESONANT GRAVITATIONAL WAVE ANTENNA Fulvio RICCI Dtparttmento dt Ftstca, Untoersttà "La Sapienza" Roma, Italy and Istttuto Naztonale di Ftsica Nucleare, Seztone dt Roma, Italy Received 9 March 1987
We present the results of a Monte Carlo simulation for the energy loss distribution of high energy cosrruc muons crossing a Weber-type gravitational wave (g .w.) antenna. The number of events per day of energy greater than an assigned value, generated in the antenna by the muons, is deduced. The simulation shows that a rate of 60 events per week due to the cosmic background is expected in a sea-level g.w . detector with an energy sensitivity ten times greater than that of the present antennas . With the sensitivity of the resonant detectors approaching the quantum limit value the rate will increase to 5 X 10 ° events per days . Therefore it seems unavoidable to carry on the experiment in an underground laboratory.
1. Introduction The generation of mechanical vibrations in solids by pulses of cosmic radiation has already been considered in connection with the detection of very large cosmic ray events, and, more recently, even as a possible mechanism for revealing very high energy neutrinos [1], monopoles or more exotic particles [2]. Our interest in the problem of the mechanical response of a metal bar to the impact of charged particles is due to the generation of the background signals for a gravitational radiation detector of Weber's type . The first contribution on this subject was given by the pioneering work of Beron and Hofstader who used high energy electrons to induce oscillations in solids . Later Weber et al. showed experimentally that a gravitational radiation detector does not produce a signal when hit by showers of particle density of the order of 100 particles/m2. In 1979 a group of the university of Milan [3] investigated the mechanical response of a small aluminium alloy bar to pulses of 30 MeV protons, and they related quantitatively with simple theoretical considerations the energy loss of the particles in the bar with the amplitudes of its natural vibration modes. At that time, from these results it was possible to conclude that any effect due to cosmic rays was below the sensitivity of the available detectors [4]. However, in consideration of the much higher sensitivity reached today [5] and even more in view of the sensitivity which has to be reached in order to detect events due to supernova explosions in the Virgo cluster, the problem of the influence of cosmic rays on gravitational-wave antennas has been re-examined by Amaldi and Pizzella [6]. In their paper they estimate the rate of the energy lost by the high energy muons impinging in the vertical direction, perpendicularly to the axis of the cylindrical antenna. In order to derive the number of events due to this kind of process which give a signal in the antenna higher than an assigned value, the same approximation of Grassi-Strini et al ., [3] was applied. This numerical computation was done assuming also that the electromagnetic shower generated inside the bar by the muon interaction with matter is completely absorbed into the bar itself and neglecting the effect of the secondary products produced in the thick wall of the cryostat surrounding the antenna. Under these approximations the authors show that at sea level the secondaries generated in the bar by the interaction of high energy muons produce signals with a rate much higher than that expected from supernova explosions in the Virgo cluster. 0168-9002/87/$03 .50 © Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
492
F Rica / Monte Carlo simulation of high energy cosmic muon background
This interesting results suggests the opportunity of trying an improvement in the evaluation of possible disturbances due to the cosmic muons . We present here the results of a Monte Carlo calculation for the energy loss distribution released in the antenna by the muons and by the fraction of the e .m. shower absorbed in the bar . From this energy loss spectrum it is easy to deduce, in the Grassi-Strini et al. approximation, the rate of events which give signals in the antenna greater than an assigned value . The contribution to the energy loss distribution of the secondary products generated in the thick iron shell which surrounds the antenna is also taken into account . 2. The initial parameters of the Monte Carlo simulation The simulation of the interaction of the cosmic muons with the antenna has been carried on by using the software package GEANT3 developed at CERN . GEANT3 allows to generate simulated events from standard Monte Carlo generators and to control the "tracking" of particles through various regions of an experimental setup, taking into account the geometrical volume boundaries and all the physical effects due to the nature of the particles themselves and to their interaction with matter. The "tracking" of a particle through matter consists in predicting the spatial coordinates of a set of points which define the trajectory and in computing the components of the momentum at these points . The hypothetical apparatus in our case consists of two coaxial cylinders : the inner one is the g.w. antenna, a cylindrical bulk made of aluminum of 60 cm diameter and 3 m length; the second one is a tube, made of iron, 1 cm thick, with an external diameter of 2.3 m and 5 m long, which simulates the external shell of a cryostat of a g.w. detector. These are the dimensions of the Rome group g.w. antenna installed at CERN. The sensitive volume inside which the tracking of a particle is performed is a vacuum box surrounding the experimental setup. Following a flat probability distribution, the coordinates of the origin of the muon trajectories are randomly generated on the upper plane of the boundary box . In principle the cylinder can be hit by particles travelling through one half of the total solid angle . On the other hand the finite dimensions of the upper plane of the box along which the particles are randomly generated determine a limitation on the acceptance angle of the cylindrical antenna . In order to minimize this effect the dimensions of the external vacuum box are chosen as follows : 16.7 m in the horizontal plane along the diameter of the antenna, 18.5 m along the axis of the bar, and 2.32 m along the vertical direction . By assuming that the intensity of the cosmic ray muons decreases as the square of the cosine of the zenith angle, we evaluate that, with this geometry, the numbers of particles which are not included in the acceptance angle are of the order of 0.5% of the total number. In order to randomly extract the initial momentum of the particle, the integral distribution function of probability of the cosmic muon energy at sea level has been derived from formula (6) of ref . [6] : I,( ( >-
(
E
Ei ,
EJ = Io exp - al 1n2 E ` - a 2 In E` - a 3 , o
o
where to = 1 (s sr cm2 )-1, Eo = 1 GeV and al = 0.1594, a 2 = 0.6718, a 3 = 4.826 . The relation is a good fit to the experimental data given by Menon and Ramana [8] for E,, >_ 20 GeV and by Allkofer and Tokisch for lower energies [9] . From eq. (1) we derive the integral probability function of having an incident cosmic muon with an energy smaller than E,, : (
P(EiJ=1-expl-a
l In2 Eo - a2InEo 1 . E
E l
(2)
F. Ricci / Monte Carlo simulation of high energy cosmic muon background
493
The random choice of E,, is now performed by means of a random extraction of a P(E,,) value, with a flat probability distribution, and using the inverse relation of eq . (3) E~ - E° exp
-a 2 + az-41n(1-P) 2a1
The direction of the momentum of the incident particle is defined by a random choice of the azimuth and zenith angle. The zenith angle is assigned by a random extraction weighted by a cosine square dependence of the muon intensity, while a flat distribution is assumed for the azimuth angle. Finally, for the random choice of the charge of the incident muon, a constant value for the charge ratio is considered [10] : Nl, ./NI,-
=
1 .3 .
3. The tracking control of the primary particle The production of secondary particles by muons travelling in matter is due to four different processes: fotonuclear interaction, bremsstrahlung, direct pair production and knockon electron production . In the computer simulation, during the tracking of the primary particle through the experimental setup, these four fundamental processes are taken into account and the final state after the interaction is generated by sampling the corresponding differential cross section, while the total cross section of each process is used to evaluate the probability of occurrence of the process. It is useful to recall that knockon electron production is more probable than the other mechanisms and its different cross section for production of electrons with energy between T and T + d T, valid for incident particles with spin 1/2, is : dT -
2aZe2m RZ
2 (1 - /32
f'
+ T max
2É2
~~
where T = kinetic energy of the scattered electron, Z = charge number of the medium, re = classical electron radius, m = electron mass, M = muon mass, and T ,-
_
2m
(Y2
- 1)
1 + 2y (m/M) + (m/M)2
. is the maximum energy transferable to the electron by a p-meson of energy E,, = yM and '82 = 1 - 1/Y2 The generation of bremsstrahlung photons by a high energy muon is treated as a discrete process below a defined cutoff energy . Its differential cross section is given by the well-known formula of Petrukhin and Shetakov [11] 2 1 du . 3(2ZX,m 4 a ~i_V M ) V[_~
- 4v
+ v2
where V is the fraction of energy transferred to the photon by the muon, a is the fine structure constant, A e the Compton wavelength of the electron, and, for Z > 10, O(S) = In
2
189( M/m) Z- 2/3
1+
189,% ____2
8Z-1/3
494
F. Ricci / Monte Carlo simulation
of high energy cosmic muon background
with e = 2.71828 and 8
-
M2V
2E (1 - V)
The direct pair production by muons is taken into account by considering the differential cross section of the process as reported in ref. [12]. The expression of the cross section is a convenient parametrization due to Kokoulin and Petrukhin based on a QED calculation performed by Kelner and Kotov. Finally, the generation of secondary particles by the interaction of muons with the nuclei of the tracking materials is also considered, assuming as differential cross section of the process: d'a _ T(a,+ eas ), d12dW -
(6)
with ( 2 e=1+2I1+(Q) ) tan2
2,
(7)
and Q2=2(ENW-Ip11p'Icos0
v=Ev -W.
M2),
(8)
(9)
Here OT and as are the photoabsorption cross section for transverse and longitudinal photons respectively . CT is assumed to be constant and equal to 0.12 mb, while for as we have : as = 0.3 1-
1 OT . 1 .868Q2/v )
(10)
4. The electromagnetic shower control The interaction of the primary cosmic muon with matter can generate secondary particles with energies high enough to produce tertiary particles with rather high probabilities. If the incident particle has high energy, these processes continue to take place successively and the number of particles produced thereby increases. This multiplicative sequence, called an e.m. shower, begins to decline as some particles receive energies too low for producing further particles effectively. The longitudinal development of an electromagnetic shower in matter is determined by the radiation length Xo . Its lateral spread is mainly due to the multiple scattering of electrons that have enough energy to travel far away from the axis. In order to eliminate the material dependence, the lateral spread of an electron beam of energy e after traversing a thickness X0 is expressed in Molière units, RM= Es Xo> E
where Es = 21 MeV is a typical constant appearing in multiple scattering theory and e is the critical energy of an electron that loses as much energy in collisions as in radiation. The equations giving the radiation unit Xo and the Molière unit RM are rather complicated. To give an estimate of the extension of the
F. Ricc / Monte Carlo simulation of high energy cosmic muon background
495
shower we can follow the suggestion of ref . [13], in which the quantities mentioned above are approximated by: X°
_ 180A ZZ
e- 5 RM
g , cm2
(12) (13)
MeV,
7Z
(14)
cmz
These estimates are accurate within 10% (20% for Xo ) for 13 < Z < 92. The longitudinal distribution of the track length, defined as the sum of the tracks of all the particles in the shower, is proportional to the longitudinal energy deposition which is well represented by a parametrized formula [13]. Its knowledge allows us to compute the length T, expressed in Xo units, along which a fixed fraction of the incident energy is deposited . It can be shown that for an absorption of 98% of the incident energy E T(98%) =
3I In( l
E)
+ a],
(15)
where a = 0.4 for electrons and a = 1.2 for y-rays . The multiple scattering for the central part and the propagation of photons for the peripheral part determine the lateral spread of the shower, with no simple dependence upon A and Z. In any case, it can be shown that 95% of the total energy of the shower is contained in a cylinder of radius 2 R M . Using these relations it is easy to see that the complete absorption of the shower generated by the muon inside the antenna is a crude approximation because its longitudinal development in aluminum is of the same order of magnitude as the diameter of the antenna . Therefore, the fraction of the shower completely absorbed inside the aluminum bar is strongly dependent on the point where the primary particle interacts inside the antenna. On the other hand, the lateral spread of a shower crossing a thickness of matter of one radiation length is of the order of one centimeter for both aluminum and iron. This means that all the processes are developed mainly in the forward direction. In order to evaluate the fraction of the energy lost from the shower in the bar, in the Monte Carlo simulation each secondary particle is "tracked" through the whole experimental apparatus . The greater part of the computing time of the simulation is spent in the tracking phase ; for this reason, and considering the small lateral spread of the shower in the iron thickness, the tracking is started only if the initial direction of the primary particle crosses the boundary of the antenna . In practice, when the muon generates secondary products the nature and the initial position and momentum of the new particles are stored temporarily in a memory stack, and the tracking of the primary particle, if it survives after the interaction, is completed . So it is possible to come back at the interaction point and follow the secondary products through the whole experimental apparatus . These new particles, which are essentially electrons, positrons and y-rays, further interact with the matter until its energy degrades to the level of atomic ionization and excitation. For each particle, depending on its nature, some dominant processes are considered for their interaction with matter. In particular, the tracking of y-rays is affected by pair conversion, Compton collision and photoelectric effects, while as concerns e +/e - interactions multiple scattering, ionization and delta ray production, bremsstrahlung and positron annihilation are considered. As for the primary particle, the total cross section of these physical mechanisms is used to evaluate the probability of occurrence of the discrete processes ; the corresponding differential cross-section is sampled to generate the final state after the interaction [7] . The further products stored in a temporary stack, are tracked at the end of the tracking of the corresponding generating particles.
496
F. Ricc3
/ Monte Carlo simulation of high energy cosmic muon background
When the kinetic energy of the particles is below a defined cutoff value, which can be different for each mechanism, for the corresponding process only a continuous energy loss of the particle along its trajectory is considered . In this way we are able to keep track of the values of the energy of each particle which crosses the boundary of the antenna at its input and at its output . The difference between these two values gives us the information about the energy loss inside the antenna by the muon and by the corresponding e.m . shower which in principle can be generated not only in the aluminum bar but also in the iron cylinder surrounding the gravitational wave antenna.
5. Results and conclusions The Monte Carlo program was run on a VAX 8600 generating 58 496 165 muons. The energy distribution obtained for the incident particles is shown in fig. 1 . In order to check the procedure of random generation, the integral vertical spectrum of cosmic muons is numerically deduced, under the hypothesis that the differential spectrum decreases as the square of the
lo'
1 63
Z 10 6 0 ó w
E
m
lo
d N
a
Y U'
w AI
lo
s
w
z
1
10
ENERGY (GeV)
100
1000
Fig. 1 . Energy spectrum of the incident muons obtained by Monte Carlo generation for 58496165 random extractions.
I 10
I 100
E[, (GeV)
L 1000
Fig. 2 . Integral vertical spectrum derived from the Monte Carlo generation, normalized as described in the text. Open circles : experimental results from ref . [81 ; full circles : experimental results from ref . [9] .
F. Rica / Monte Carlo simulation of high energy cosmic muon background
497
cosine of the zenith angle. The normalization of the integral spectrum is performed by assuming as value for the vertical intensity : I~( >_ 1 GeV) = 80 .2 m-2 s -1 sr. In fig . 2 we report the integral vertical spectrum obtained from the random generation and the experimental data of refs. [8] and [9] . The agreement seems satisfactory. By retaining the information of the total energy loss of muons and its consequent shower in the antenna, we obtained the diagram of fig . 3 ; it represents the integral energy loss spectrum due to the 150000 muons which have hit the antenna among all the incident particles generated on the upper plane of the boundary box . We tried successfully to fit the integral loss spectrum -Y(>_ Elost) in two different energy ranges with the following parametrization : T(> E l ,,,,) =kEF t
(16)
obtaining k = (1799 ± 1) GeVY,
= 2 .879 ± 0.001,
Y
(17)
for 0.3< E,,s, (GeV) _< 1 .0 and k = (1921 ± 10) GeVY, Y = 2 .178 ± 0 .006, for
1 .0< E, os ,(GeV)
(18)
<_ 1000.
105 5
Table 1 Normalized E(>- El,,,) for events with an energy loss greater than Eros,
10~~-
Eios, [GeV]
5
10
ó
3
\\ k=1921±10 GeV 7 ~Y=1794±0006 E-Y =K Los
\ \
5
\\
\ 102,, 5
L
k=1799±1 GeV Y Y=2 .879±0 .001
~4~+
\
\
\
10
\
\
5
\ t \
\ \
\ \
\ 1
0.1
0.5
1
5
10
50
100
E dust (GeV)
Fig. 3. Energy loss spectrum of cosmic ray events inside the antenna. The dashed line gives power law fits E = kEl-os"t .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 2.0 3.0 4.0 5 .0 6.0 7.0 8.0 9.0 10 15 20 30 40 50 100
E( >_ E,os,) [event/day] 2.23 x 10 7 1.87 x 10 7 1.20 x 10 7 5.26X10 6 2.68 x 10 6 1.55 X 10 6 1.00 x 10 6 7.22 x 10 5 5.36x10 5 4.18 x 105 1.02 x 105 4 5.03 x 10 3.19x10 4 2.30x10 4 1.95X10 4 1 .57x10 4 1.32X10 4 1.08 X 10 4 9 .73X10 3 6.00 X 10 3 3.52x10 3 2.28 X 103 8.28 X 10 2 6.21 x 10 2 2.07 X 10 2
F. Ricci / Monte Carlo simulation of high energy cosmic muon background
498
The two fits are shown in fig. 3; in table 1 we report Z(>_ Elos,) normalized as the integral vertical spectrum to give the number of events per days with an energy loss greater than El ,,,,, . We can now derive the number of events which give signals in the antenna with energy higher than an assigned value kBTeff, where k B is the Boltzmann constant . Let us recall that a particle which interacts traversing the solid body of the antenna acts as a source of sonic waves, and let us recall also that in a gravitational wave detector of the Weber type the change in the amplitude of the first longitudinal mode of the cylinder is monitored. We then need to consider the projection of the sonic wave perturbation due to the incident particle on the first eigenmode of the bar. The exhaustive calculus for the resulting excitation of the vibration mode of the detector is in general very difficult, because of the complexity of the motion of the solid body in which the tensor of deformations and the boundary conditions are not related in a simple way. For this reason we follow the approach of the Grassi-Strini et al . group in which the vibration of the fundamental mode of the bar is originated by the conversion of the local thermal expansion caused by the sudden energy lost by particles into an acoustic wave . The consequent elastic energy of the first longitudinal vibration is given by [3] k BTeff -
v 2 1
2
( C ) Ma v
COS
2 ( 7rx )
L
(19)
2
Elost ,
where q is the linear thermal expansion, c,, the specific heat at constant volume of aluminum *, Ma the mass and L the length of the bar, x the distance between the center of the bar and the impact point and o, the velocity of sound in the material. In order to derive the number of events due to the muon background which give signals with energy greater than kBTeff, we should also consider that the event can take place for any value of x between ±L/2 . Using the parametrization of eq . (16) for 1, we obtain : N(> k B Teff )
_ 1 -
L
+L/ 2 k L/2
kBTefrMa \
U
1(7j/CJ
cos(9rx/L )
dx .
(20)
The sensitivity range of interest for the experimentalist is near or above the quantum limit of the experimental apparatus, Teff = 10 - ' K. Assuming as antenna parameters those of the aluminum bar of the Rome group, i.e. Ma = 2300 kg and L = 3.0 m, the most interesting region of the integral energy loss spectrum of the muons in the detector is from 1 GeV up . In fig. 4 we give the result of the numerical integration of eq. (20), using the parametrization of eq. (18) . The comparison with the numerical calculation performed in ref. [6] is also shown. The Monte Carlo calculation of the effect of the background due to the cosmic rays confirms the results obtained by Amaldi and Pizzella, indicating that the rate of background events is much larger than the rate of signals expected from supernova explosions in the Virgo cluster. The disagreement shown in fig. 4 is reasonably well understood by considering the more general conditions imposed on the Monte Carlo simulations, i.e. cosmic muons incident on the bar from any possible direction, the correct evaluation of the fraction of the electromagnetic shower absorbed in the bar, and the interaction of the primary particle with the external iron shell. On the other hand, it is important to realize that for energies of the incident muons of the order of 100 GeV the contribution of the stochastic radiative processes becomes important and eventually dominates the total energy loss . Under these conditions, the derivation of the energy loss by a single mean value may become inadequate while, using the detailed geometry of the detector, a Monte Carlo package for the statistical simulation of the muon energy loss gives correct information. From fig. 4 it appears that an improvement with a further order of magnitude in the energy sensitivity of the available gravitational wave detectors, at present in the range of Teff - 10 mK [5], will lead to * The ratio 71/c v , for aluminum, is assumed equal to 4 .75 X 10 -5 kg J -1 and independent of the temperature .
F.
Rfcci
/ Monte Carlo simulation of high energy cosmic muon background
499
10 5
]C _AI Z
10 2
10
10
8
10
1
10
6
10
Ten (K)
5
10
1
10
3
Fig. 4. Number of events per day of energy >- k B Teff produced at sea level by cosmic muons in a g.w . detector like that of the Rome group at CERN . The corresponding result of ref. [6] is shown (dotted line) for comparison .
conditions in which cosmic muons should give a number of the order of 60 events detected per week at sea level . Approaching the quantum limit = 10-7 K, the event rate due to the muon background is expected to be of the order of 5 X 10 4 events per day for a detector at sea level. Therefore it seems compulsary to carry on the experiment in an underground laboratory, where the rate of cosmic muons impinging on the antenna is drastically reduced and can easily be kept under electronic control. Teff
Acknowledgements I am indebted to Profs. E. Amaldi and G. Pizzella for the suggestions, the encouragement and support during this work. I also which to thank Dr. F. Carminati for useful discussions about the GEANT3 software package and Dr. M. Bassan for a critical reading of the manuscript. References [1] M. Levi et al., IEEE Trans. Nucl . Sci. NS-25 (1978) 325. [2] C. Bernard, A. De Rujula and B. Lautrup, Nucl . Phys B242 (1984) 93 .
500 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
F. Ricci / Monte Carlo simulation of high energy cosmic muon background A.M. Grassi-Strini, G. Strini and G. Tagliaferri, J. Appl. Phys. 51 (1980) 948. J. Weber, Phys. Rev. Lett . 22 (1969) 1320 . E. Amaldi et al . Nuovo Cim. 9C (1986) 829. E. Amaldi and G. Pizzella, Nuovo Cim. 9C (1986) 612. GEANT3 User's Guide DD/EE/84-1 (May 1986). M.G .K. Menon and P.V. Ramana Murthy, Progr. Elementary Part . Cosmic Ray Phys . 9 (1969) 220. O.C . Allkofer and H. Jokisch, Nuovo Cim. 15A (1973) 371. O.C . Allkofer, Proc. 16th Int. Cosmic Ray Conf. 14 (1979) 385. S. Hayakawa, Cosmic ray physics (Wiley, New York, 1969). W. Lohmann, R. Kopp and R. Voss, CERN 85-03 (21 March 1985). U. Amaldi, Phys . Scripta 23 (1981) 409.