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
Trang 1VNU JOURNAL OF SCIENCE Mathematics - Physics, t.XVIII n°l - 2002
C E R E N K O V R A D IA T IO N SIM U LA TIO N IN
T H E A U G E R W A TER 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 i e 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 i o n
In 1962 one observed cosm ic rays w ith energy approxim ate to 10~° eV In 30 subse quent yciiirs 8 ex ten siv e atm ospheric showers vvil.il energy exceeding 1()~° eV wen! observed How is it possible to explain th e ex isten ce o f th ese extraordinarily energetic cosm ic rays
T h is is a scientific m istery w hich 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
T he experim ental study o f such cosm ic rays will be carried out w ithin the framework
o f the international Pierre A uger 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 osph ere and water Cerenkov ground d etector arrays to sam ple th e lateral density distribut ion OI1 th e ground level 11, 21
T h e sim ulation of response o f the w ater Cerenkov ground d etecto r to atm ospheric shower m uons is pract ically needed for th e experim ental research o f extrem ely high energy cosm ic rays In th is paper w e develop a sim ulation m odel for the process o f em ission and diffusion of Cererikov photons concerned w ith m uons passing through th e detector volum e with the velocity greater than the phase velocity o f light in the water on purpose to define photons producing signal in th e detector.
II S im u la t i o n m o d e l fo r g e n e r a t io n a n d r a y - t r 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 t e r g r o u n d d e t e c t o r
T he Auger water Cerenkov ground d etector considered in our m od el is a 10 1112 < 1.2m deep cylindrical volum e o f water, lined w ith a diffusely reflective w h ite m aterial, and viewed vertically from above by 3 ph otom u ltiplier tu bes (pints) 200m m in diam eter
T he detector geom etry is illustrated oil the figure 1.
T y p e s e t by 4 v fS -T £ X
28
Trang 2C erankov radiation s im u la tio n in the.
For sinm lntion I hr p i n t s arc approxim ated as circular areas in th e plane of the detector top surface witli area equal to the effective area o f 200nim pints (5 3 0 n n 2).
v z
F k j m v 1. Illustration o f the Auger water ground d etector geom etry.
W hen an atm ospheric shower muon strikes th e top surface o f the detector and move's through its volum e w ith the velocity greater than the phase velocity o f light, in the water the Corenkov photons are em itted T hese Cerenkov ph otons are th e optical photons Passing through ihv water th ey undergo throe kinds o f interaction: R ayleigh scattering, absorption and water boundary interaction (absorption and reflection) However ễt\n tho water contained ill tilt* d etecto r volum e is purified, it may be considered as an optically hom ogenous m edium and therefore tile R ayleigh scat tering is negligible.
T he m otion of 1 he* atm ospheric shower muon through th e d etector volum e ac com panied by the p n x rsses: th e energy loss for ionization and atom ic excitation , the hrcm sstrahlung the direct pair v ¥e production T h e M onte-C arlo 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 2 T eV (th e energies which most
o f atm ospheric shower m uons have on the ground level) the ionization and excitation of atom s are the main m echanism and both the hrem sstrahlung and direct pair r 4 e “ pro duction processes may be neglected T he sim ulation o f th e generation and diffusion of Ccrcnkov photons concerned w ith the muon m oving in the d etector volum e can he carried out then based 011 th e sim ulation algorithm s developed by US as follows:
1 C alculating the characteristics of the considered muon: coord inates o f its position
2 C alculating the threshold energy for Ceronkov radiation em ission according to the coherent condition: //.? > 1
where El) - th e union energy a t rest, n- th e refractive index o f the water, (3 = v /c
3 Checking w het lier th e m u on’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* th e calculation w ith the considered nmon is ceased and t hen the operations are passed to perform ing with
Trang 3L e V a n N g o e , V o V a n T h u a n a n d D a n g Q u a n y T h i e u
a new m uon (when a num ber o f given m uons N ụ , rem ains in the m em ory o f the com puter) by returning to 1 O therw ise, go to 4.
4 Advancing th e muon a rather sm all step A / along its m oving direction.
5 Verifying w hether th e final point o f A/ is inside th e d etector volum e or n o t by calcu lating its coordinates:
X fl = X qh + A / sin 0fl COS ộự,
= Xo/i + A / cos 0^.
If the final point of A / Is ou tsid e th e d etecto r volum e, the calcu lation is finished with the given m uon O therw ise, go to 6.
6 C alcu lating Cerenkov radiation energy oil A /
d E.
dl
here.
d E r _ 4 n - z 2e'2 ( _ 1 \ _ 2 / r W f _ 1 \ 2 2
d i c 1 J Ui "V " C2 [ l n 20 2 ) “,ax
and in -r I/max i‘s th e frequency band w ith in which the d etector is sen sitiv e.
7 C alculating th e number o f Cerenkov p h otons em itted on A / T h e average num ber
o f Cerenkov photons produced on A / is defined by:
here,
d t = H W J r ^ l 1 " ^ ) d E c ~ 37022 V1 “ ^ ) (£rm ax ■ E r,il", )
-T hus th e num ber o f Cerenkov p h oton s given ofl’ on A l is calcu lated accordin g to the Poisson distribution
N n
n!
by tile sim u lating form ula ft = fc , here Ả: is th e least w hole num ber taken so as to satisfy th e inequality:
.= ! iVf
w ith the random num bers a , uniform ly d istrib u ted on ( 0,1).
Each Cerenkov photon generated OI1 A / is sim u lated th en , in turn, as follows:
a C alcu lating th e em ission angle o f photons produced on A /
Trang 4I) C alcu lating th e azim uth al em ission angle o f a photon e m itted oil A / It is chosen
at random from t lie uniform d istrib u tion on (0 2 tt) according to th e form ula Ộ = 27TO.O
- a random num ber uniform ly distrib u ted on (0.1).
(' C alculating the flight direction o f a Cerenkov photon ill the C artesiean perpen dicular fixc.nl coord inate system
C e r e n k o v r a d i a t i o n s i m u l a t i o n i n t h e 'M
I/, — Uịt cosu; - (ưfl s in ọ + u tlwfl CQiỳậ) 1 — co s2 U) \
-/ • / 1 - c o s2 lj \ *
J/C = V COS ú; -f (Up sin 0 - VnWu COS0) — - — , (7)
V 1 - t n j J
-u\ = cos u + ( 1 - wft ) COS 0 — -~7j
V I - W / I /
Here ur <vr , Wr are the com pon ents o f the unit vector directing th e p h o to n ’s flight T hey are defined as follows:
= sill#, COS0,., //r = sin ớ ,:s i n 0 < wr — co s6 r (8)
A lso for th e muon:
Ufi — s i l l c o s ộ Ịly vỊt = s i n ớ/f cos (f)fJ, w;/t = C O S (8)
cl C alcu lating the energy o f an em itted photon As the energy o f generated photons has the distribu tion density function
/ ( £ , ) = 1 - n 20 21 ’
it m ay b e sim ulated according to th e formula E c = J5f,nin + ữ ( £ nnax - min)- here a is
a random num ber uniform ly d istrib u ted on (0,1).
e C alculating the absorption free pathlength o f a Cerenkov photon.
Each generated Cerenkov p h oton m ay travel a free path len gth before it is absorbed
As this free path length has th e probability distribution d ensity function
it m ay b e selected at random by solving the equation
./0 M
or Zci = - I A ln (l — a ). Here, IA is th e m ean absorption free p ath len g th dependen t on the energy o f the photon Of is a random number uniform ly d istribu ted on (0,1).
Trang 5L e V a n N i j o c, V o V a n T h u a n a n d D a n g Q u a n g T h i e u
A fter th e random selection o f th e considered photon s absorption free pathlen gth it
is needed to verify w hether its filial point is inside th e d etector volum e or not by ca lcu la tin g the coordinates o f this final point:
X = Xịt + la sill 0C COS 4>c
z = Zf, -h la COS <pr •
If the final point o f th e absorption free path length is insid e th e detector volum e, the photon is absorbed in th e water O therw ise, th e ph oton h its a d etector surface, we have to consider then, in turn, th e following situ a tio n s possible to occu r :
-f If z < 0 and X 2 -f Y 2 < R 2 the photon strikes the d etector top surface.
-f If z < 0 and X 2 -f- Y 2 > R 2 then givin g Zc = 0 and calcu latin g
X c = X{) -f If. sin 0t. cos ộc, Y( = V() + l c sin Or sin ậ r
Here X (.,Yr , Z c are th e coord inates o f th e intersectin g point o f th e ray traced a lon g th e photon's direction w ith th e plane including th e d etecto r top surface /(; is the d istan ce from th e initial possition o f th e photon to th is intersectin g point.
We need to verify th e condition X f + Y(2 < R 2. If th is con d ition is realized, th e photon hits the detector to p surface O therw ise, th e photon strikes the d etector sidew ard surface.
-b If z > H and X 2 4* Y~ < R 2 th e ph oton h its the d etector b o tto m surface.
-f If z > H and X 2 -f Y 2 > R 2 th en the treatm en t is m ade in such a sim ilar way
as for the case z < 0 and X 2 *f Y 2 > R 2.
■f If 0 < z < H aiìd X 2 + Y 2 > R 2, th e photon strikes th e d etector sidew ard surface.
Now wo shall consider th e separate concrete cases w here th e ph oton h its a detector surface.
T h e first case is associated w ith th e fact th at th e p hoton strikes the top d etector surface Then, we must define th e coord inates o f th e h ittin g point ( X , Y, , Z r ) and verify the following conditions:
( x c - x imUi) 2 + (Y c - Y p7ntl) 2 < R % n l, ( X c - x pml2f + (Y c - Yprnh) 2 < Iif„n t, (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 coord in ates o f the center o f pints, respectively, and Rpmt is their radius.
If one o f three abovem entioned co n d itio n s is realized, th e p h oton hits a p m t and
is absorbed here O therw ise, it is necessary to verify the co n d itio n Pa > tt, here p a is the probability th a t the ph oton is absorbed in the d etector wall, a is a random num ber uniformly distribu ted on (0 1 ) W hen th is condition is satisfied, th e ph oton is absorbed ill th e detector top surface s wall and th e op era tio n s are passed to perform ing w ith a new
Trang 6C e r e n k o v r a d i a t i o n s i m u l a t i o n i n t h e
ph oton l>y n't lim in g I), or ftoin^ to 8 if th ere is not any photon in th e m emory o f the» com puter.
l or the* rase w Ik w th<‘ photon is not absorbed it will 1)0 reflected Imck into tin*
<let<‘etor volume» T hr ph oton’s (lirretion alter the reflection may be dofined by tli<‘ ỊHTÍírt iviU'ction condition: COS f - — COS tì' s in 6rrt.j = sill Oc'.Qcrtf — ộ c
I ho second cast' corm sponds to th e fact that the photon h its the detertor bottom surface* T his rasr is treat0(1 ill such a sim ilar way a*s th e first case.
I lie third cast1 is role»ted to th e fact th a t the photon strikes the flett'ctor sidrw m d Mil face T hr p h o t o n ' s fritc is treated here as well as ill th e first a n d second eases Howrvor.
I Ilf <•<>1H fliiiiit< > o f I lit* h illin g point in this case art* calculated by solvin g the system o f r<ịiiai i< )iis:
Thi' solu tion o f the a hove m entioned system o f equations gives
whorr c -Yu cos Ở, + y 0 SÌIỊ .
A* lar as the p h oton ’s direction is concerned upon b eing reflected bark into tlì(' (k’tcctoi volum e cl! Ỉ he lilt ting p o in t it may b e calc ulated according to the perfect reflect ion condit ion:
Thus, passing through the d etecto r volum e o f water the Cerenkov photon may hit tlie* d etector walls m any a tim e and be reflected from them T h e process o f reflection proceeds until th e photon will be absorbed or in th e water or ill th e d etector walls T he sim ulation is passed then to perform ing w ith subsequent photons.
After all t he photons em itted on a ste p A / are sim ulated, th e op erations are passed
to 8
JS C alculating the energy losses for ionization by the m uon on A /
(14)
Orrrf = Or
COSCVrr/ = ~ỹxF í ^ ‘2 ~ y,2)cos<t>r + 2 X r Yc sin Or] , sill Orr f = ^ 7 l ( X ? - Y '2) s i \ i ộ r - 2 X t Yr co sé c]
(16)
Hero.
I
= ()Z 2 -^ rA B u + Ü.69 + 2 111/Ì7 - 2 /i - Ổ )
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2 íi2j
1
-E,
1 + HI,, + -in _J*_
21
Ổ - { 2 / / 111 10 4- c + a(y\ — y ) h if i/o < ?y < 2/1 where, y = lg .
2y111 10 + c if y > y 0
For t he w ater we have [4]
/ = 74.1f vr, /, = 0 0 8 5 3 M e V c m 2 /g , - c = 3.47,
a = 0.519, 6 = 2.69, 2/1 = 2, y _ 0 ~ = 0.23.
9 C alculating the rem aining energy o f th e muon at th e end o f A /
E\ t = £ „ - AZ?, - AE-ion
and tIll'll return to 3.
If should be noted th at th e calcu lation o f Cerenkov p h otons has a cyclic character and leads to frequently rep eatin g som e basic blocks A fter a ph oton is calcu lated , the operations are repeated for th e other p h otons produced on the considered muon step, and thou for ph otons generated at the subsequent muon step s until th e final point o f a union step will be outsid e th e d etector volum e or th e m uon's energy at the end o f its step will fall below the threshold energy for Cerenkov radiation em ission This process
o f calculation can be repeated many a tim e w ith a num ber o f m uons for reducing th e statistical fluctuations of th e calculated qu antities Schem atically, it is illustrated on the figure 2.
III C o n c lu s io n
In this paper a sim ulation m odel for th e process o f em ission and diffusion of Cerenkov photons concerned w ith m uons m oving through th e volum e o f the Auger w ater ground d e tector with th e velocity greater than the phase velocity of light in th e w ater is developed
In the considered m odel th e m otion o f each m uon is tracked, th e num ber o f Cerenkov pho tons generated on a muon step a t an angle w ith the m uon's m ovin g direction is selected
at random from the Poisson ditribution Each ph oton is ray-traced as it passing through the water, and reflected from th e d etector walls W hen ph otons strike th e photom utiplier tubes, their arrival tim es are recorded T h e develop ed m odel is app licable to calculating photons producing signal in th e d etector and their characteristics of diffusion B ased on this m odel we have designed the com pu ter softw are to carry out th eoretical sim ulations The results o f numerical calcu lation s will be analized in com parision w ith th e experim ental data collected on th e water Cerenkov d etecto r in sta lla tio n o f th e Auger training laboratory
at INST and puhlised in a separated paper.
Trang 8C e r e n k o v r a d i a t i o n s i m u l a t i o n i n t h e
i i g i u e 2 Illu s tr a tio n o f ih e s im u la tio n p ro c e s s o f C e re n k o v r a d ia tio n in the A u g e r w a te r
g ro u n d d e te c to r c o n c e rn e d w ith the m o tio n o f m u o n s
Trang 9M) L e V a n N g o e, V o V a n T h u a n a n d D a n g Q u a n g T h i e u
A c k n o w le d g e m e n t The aut hors would like to exten d m any th an k s to Prof Pierre P a r- nilat for his interest in this work and useful discussions.
References
1 The Auger' collaboration, the P ie rre A uger observatory design report second ed itio n Fermilab (U S A ), 1997, app endix B.
2 K.G G ibbs Cosm ic rays at extreme energies. Enrico Fermi in stitu te , th e U niversity
o f C hicago 1997.
3 Le Van N goc Vo Van T huan N guyen Hao Quang P rop agation o f cosm ic rays in the atm osphere Internal Report., I N S T 1999.
1 A N K alinovxki, N V M ok hov, Yu P.N ikitin P assage o f high-energy particles through m atter A/ A to m izd a t 1985.
TAP CHI KHOA HOC ĐHQGHN Toán - Lý t XVIII, n°l - 2002
MO PHONG BÚC XẠ CERENKOV TRONG DETECTOR AUGER NUỠC MẬT DAT
Lê Vãn Ngọc, Võ Văn Thuận, Đặng Quang Thiệu
V iện K h o a h ọ c và K ỹ thuật Hạt nhản
Mõ phỏng đặc trung của detector Auger nước Cerenkov mặt đất đối với các muon mưa rào khí quyển trên thực tế là cần thiết cho nghiên cứu thực nghiêm các tia vũ trụ
có năng lượng cực lớn Chúng tôi nghiên cứu ờ đây một mô hình mồ phỏng quá trình phát x ạ v à k h u y ế c h tá n c ủ a c á c p h o to n C e r e n k o v g ắ n v ớ i m u o n c h u y ê n đ ộ n g q u a th ê
tích deiector với tốc độ lớn lìĩì tốc độ pha của ánh sáng trong nước nhằm xác định các photon tạo tín hiệu trong detector.