VNU JOURNAL OF SCIENCE Mathematics - Physics, t.XVIII n°l - 2002 C E R E N K O V R A D IA TIO N SIM ULA TIO N IN T H E A U G ER W ATER G R O U N D D E T E C T O R L e V a n N g o c , V o V a n T h u a n a n d D a n g Q u a iig T h ie u Institute o f N uclear Science and Technique A b s t r a c t The sim ulation of response o f the A uger water C crcnkov ground dctcctof ft) atmospheric shower muons is practically needed f o r the experimental research of cosmic rays at extreme energies We consider herd a sim ulation model f o r thr p ro a ss of ('.mission and diffusion of Cerenkov photons concerned with muons moving tliwugh the detector volume with the velocity greater than the phase velocity o f light in the water on purpose to define photons producing signal ill the detector I I n t r o d u c t io n In 1962 one observed cosm ic rays w ith energy approxim ate to 10~° eV In 30 subse quent yciiirs extensive atm ospheric showers vvil.il energy exceeding 1()~° eV wen! observed How is it possible to explain the existen ce of these extraordinarily energetic cosmic rays T his is a scientific m istery which there have not been reliable schemes for explaining If understood th e source and nature o f extrem ely high-energy cosm ic rays are we will lead to new discoveries or ill th e fundam ental physics or in the astrophysics In recent years the interest ill cosm ic rays at extrem e energies has increased rapidly The experim ental study o f such cosm ic rays will be carried out w ithin the framework o f the international Pierre Auger project w ith Auger observatories o f a hybrid design including fluorescence detectors used to observe th e longitudinal developm ent of showers ill the atm osphere and water Cerenkov ground detector arrays to sam ple th e lateral density distribut ion OI1 the ground level 11, 21 T h e sim ulation of response of the water Cerenkov ground detector to atm ospheric shower muons is pract ically needed for th e experim ental research o f extrem ely high energy cosm ic rays In th is paper we develop a sim ulation m odel for the process o f emission and diffusion of Cererikov photons concerned with m uons passing through th e detector volume with the velocity greater than the phase velocity of light in the water on purpose to define photons producing signal in th e detector II S im u la tio n m o d e l fo r g e n e r a t io n a n d r a y -tr a c e o f C e r e n k o v p h o t o n s in th e a u g e r w a te r g r o u n d d e t e c t o r T he Auger water Cerenkov ground detector considered in our m odel is a 10 1112 < 1.2m deep cylindrical volum e of water, lined w ith a diffusely reflective w h ite material, and viewed vertically from above by photom ultiplier tubes (pints) 200m m in diam eter T he detector geom etry is illustrated oil the figure T y p e s e t by v fS -T £ X 28 Cerankov radiation sim u la tio n in the For sinm lntion I hr p i n t s arc approxim ated as circular areas in the plane of the detector top surface witli area equal to the effective area of 200nim pints (5 n n 2) v z F k j m v Illustration of the Auger water ground detector geometry W hen an atm ospheric shower muon strikes the top surface of the detector and move's through its volume w ith the velocity greater than the phase velocity of light, in the water the Corenkov photons are em itted T hese Cerenkov photons are the optical photons Passing through ihv water th ey undergo throe kinds of interaction: R ayleigh scattering, absorption and water boundary interaction (absorption and reflection) However ễt\n tho water contained ill tilt* detector volum e is purified, it may be considered as an optically hom ogenous medium and therefore tile Rayleigh scat tering is negligible T he motion of 1he* atm ospheric shower muon through th e detector volum e ac com panied by the pn xrsses: the energy loss for ionization and atom ic excitation, the hrcm sstrahlung the direct pair v ¥e production T he M onte-Carlo algorithm s for sim u lât ion of th e hrcmsst-rahluug and direct pair fã' ( production processes have Ixvn analyzed in detail by IIS in [3| However, at energies sm aller than TeV (th e energies which most o f atm ospheric shower muons have on the ground level) the ionization and excitation of atom s are the main mechanism and both the hrem sstrahlung and direct pair r e “ pro duction processes may be neglected T he sim ulation o f the generation and diffusion of Ccrcnkov photons concerned w ith the muon m oving in the detector volum e can he carried out then based 011 the sim ulation algorithm s developed by US as follows: C alculating the characteristics of the considered muon: coordinates o f its position (A'n,, the m oving direction energy ( £ ,,) C alculating the threshold energy for Ceronkov radiation em ission according to the coherent condition: //.? > lires = - r1 (1) where El) - the union energy at rest, n- the refractive index of the water, (3 = v /c Checking whet lier th e m uon’s energy E fi is greater than th e thresold energy for Cerenkov radiation em ission JE7|lires» or H°t- If E fl < z?thres* the calculation w ith the considered nmon is ceased and t hen the operations are passed to performing with Le V an N goe, Vo Van T huan and D ang Q u a n y T h ie u a new muon (when a number o f given m uons N ụ , rem ains in the m em ory o f the com puter) by returning to O therw ise, go to 4 Advancing th e muon a rather sm all step A / along its m oving direction Verifying w hether the final point o f A/ is inside the detector volum e or n o t by calculating its coordinates: X fl = X qh + A / sin 0fl COS ộự , Yịi = Yoịầ + A /sin ớ;4sin (2) = Xo/i + A / cos 0^ If the final point of A / Is outside th e detector volum e, the calculation is finished with the given muon O therwise, go to 6 C alculating Cerenkov radiation energy oil A / dE A Ev = = dl xAI, (3) here d E r _ n - z 2e'2 di c1 ( _ _ /r W f \ "V " J Ui C2 _ [ l \ n 20 ) 2 “,ax and in -r I/max i‘s th e frequency band w ith in which the detector is sen sitive C alculating th e number o f Cerenkov ph otons em itted on A / T h e average number of Cerenkov photons produced on A / is defined by: Ne = y x A i , (4) here, djV22^2 I'E, Mmx / d t = H W J r 1\ _ / 1\ ) d E c ~ 37022 V1 “ ^ ^ l 1" ^ ) (£rm ax ■ E r,il", ) - T hus the number of Cerenkov ph otons given ofl’ on A l is calculated according to the Poisson distribution P {n ) = e~K X Nn - f n! (5) by tile sim ulating form ula ft = fc , here Ả: is th e least w hole number taken so as to satisfy th e inequality: = ! iVf with the random numbers a , uniform ly distributed on ( 0,1) Each Cerenkov photon generated OI1 A / is sim ulated then, in turn, as follows: a C alculating the em ission angle of photons produced on A / cos“ = ả H and X 4* Y~ < R th e photon hits the detector b o tto m surface -f If z > H and X -f Y > R then the treatm ent is m ade in such a similar way as for the case z < and X *f Y > R ■f If < z < H aiìd X + Y > R 2, th e photon strikes th e detector sideward surface Now wo shall consider th e separate concrete cases where th e photon h its a detector surface T he first case is associated w ith th e fact th at th e photon strikes the top detector surface Then, we must define the coordinates of the h ittin g point ( X , Y, , Z r ) and verify the following conditions: ( x c - x imUi) + (Y c - Y p7ntl) < R % n l, ( X c - x pml2f + (Y c - Yprnh) < Iif „nt, (13) ( X c - ' X pm, f + (Yc - YJ)inl, f < K ị n t , where X prntị , Ypintị, x pmt2, Ypmi2, x pm tị' Ypmt:i are th e coordinates o f the center o f pints, respectively, and Rpmt is their radius If one of three abovem entioned conditions is realized, th e photon hits a p m t and is absorbed here O therwise, it is necessary to verify the condition Pa > tt, here p a is the probability th at the photon is absorbed in the detector wall, a is a random number uniformly distributed on (0.1) W hen th is condition is satisfied, th e photon is absorbed ill the detector top surface s wall and th e operations are passed to perform ing w ith a new C e r e n k o v r a d ia tio n s im u la tio n in th e photon l>y n't lim ing I), or ftoin^ to if there is not any photon in the memory of the» com puter l or the* rase w Ik w th