IL NUOVO CIMENTO VET.. 9 C, N. 2 ~Iarzo-Aprile 1986

Derivation of Muon Range Spectrum at Different Polar Angles in Deep Underground from Primary Nucleon Spectrum.

O. C. /~kLLKOFER and D. P. BHATTACHARYYA (*)

Z~stitut ]i~r Reine u~d A~gewa, n, dte Kerquphysik, der Ghristian Albrechts U~iversit~t Kio~ K@l, B.R.D.

PRATIBHi PAL

DeparSment o/ Theore~oat Physics, I~,dian Association /or the G~tivation o/ Science Jadavpur, Oalou~a, India

(rieevuto il 9 8ettembre 1985)

Snmmary . - - The energy spectra of sea-level mucus at different zenith angles have been estimated from the latest JACEE primary spectrum using ISR results modified for p-nucleus interactions. The theoretical results have been compared with the experimental results. Using the recent modified range-energy relation after Kobayakawa along with the fluc- tuation correction procedure after Kiraly, Kiraly and Osborne, the rouen range spectra in deep underground at different polar angles have been calculated. The derived results are well in agreement with the latest l~ont Blanc data after Bergamaseo st al., Battistoni et aL and Utah experiments after Bergesson et aL

PACS. 94.40. - Cosmic rays.

1. - Introduct ion .

The loss of rouen energy in s t a n d a r d rocks due to ion iza t ion , b remss t rah lung ,

pair p roduc t ion and nuclear in te rac t ions has been accoun ted following the

(*) Av. H. Fellow, on leave from Indian Association for the Cultivation of Science, Calcutta.

421

422 O.C. ALLKOFE~, D. P. BHATTACHA.RYYA an4 PRATIBHA PAL

recent ly modified fo rmula t ion of K o b a y a k a w a (~). The m u o n in tens i ty esti- m a t e d f rom the usual range ene rgy re la t ion has been corrected for range fluc- tua t ions in ca tas t roph ic processes b y using the m e t h o d of Ki ra ly , Ki ra ly and Osborne (2). The der ived corrected m u o n range spect ra unde r s t andard rock a t different polar angles have been com pa red wi th the recent resul ts of BER- GA~_~SCO et al. (3), N U S E X da ta (~) and also wi th the earl ier U t a h measure- m e n t s (~).

2. - Nuclear physics and Mnematics.

The al l-part icle p r i m a r y cosmic- ray nucleon spec t rum es t ima ted f rom the la tes t J A C E E da t a (6) has been found to follow this power law fit for nucleon energy up to 100 TeV

(1) iV(E) d E = AB-(v+~ dE (cm ~ s sr GeV)-~,

where A is t he spec t ra l amp l i t ude and ~ is t he in tegra l spec t ra l index. The pion and kaon p roduc t ion spect ra in t he a tmosphe re have the fo rm

(2)

(3) K(E) d_E : (Z~K§ -~ Z~K_)IV(E ) d E -~ 4 E - ( r + ~ d E .

The f rac t ional hadronic-energy m o m e n t s Z~o in hA--> cX inclusive react ion processes have been considered f rom ~_n~ORIKAWA and ~r (7) who have replaced A-A collisions b y the sum of nueleon-A collisions.

(1) K. KOBAYAKAWA: private communication (1984); Proceedings o] the X I I I Inter- national on Cosmic Rays Con]erence, Vol. 3 (Denver, 1973), p. 1740. (2) E. KIRALY, P. KInALY and J .L . 0SB01~Nv,: J. Phys. A, 5, 444 (1972). (a) L. BEROA~ASCO, A. CASTELLINA, B. D 'ETTo~E PIAZZOLI, G. ~NNOCCHI, P. PICCm, S. VERNETTO and H. BILOKON: NUOVO Cimento G, 6, 569 (1983). (4) G. BATTISTONI, E. BELLOTTI, G. BOLOGNA, P. CAMFANA, C, CASTAGNOLI, A. CAST]~L- LINA, V. CHIARELLA, D. C. CIYNDY, B. D'ETTORRE PIAZZOLI, E. FIORINI, P. GALEOTTI, E. IAROCCI, C. LIGUORI, G. ~/[ANNOCCHI, G.P. ~r P. NEGRI, G. NICOLETTI, P. PICCHI, A. PULLIA, M. PRICE, S. I~AGAZZl, i~. I~OLLIEI~, 0. SAAVEDRA, i. SATTA, L. TRASATTI, S. VEI~NETTO and L. ZANOTTI: Proceedings o] the X V I I I International on Cosmic Eays Con]ercnce, Vol. l l (Bangalore, 1983), p. 466. (5) H.E. BERGESSON, G. L. BOLINGBROKE, G. CARLSON, D. E. GRooM, J. W. KEUFFEL, J . L . )s and J .L . 0SBORNE: Phys. ~ev. T~ett., 27, 160 (1971). (o) T.H. BURNETT, S. DAKE, I~. FUKI, J.C. GREGOI~Y, T. IIAYASIII, R. HOLYNSKI, J. IwAI, W.V. JONES, h. JUt~AK, J . J . LORD, C. MEEGAN, O. ]~IYAM~RA, H. ODA, T. OGADA, T.A. PARNELL, T. TABUKI, Y. TA~AHASm, T. TOMINAGA, J. WATTS, B. WILEZVNSKA, R . J . WILKES, W. WOLTE~ and B. WosIEx: I X European Cosmic Ray Symposium, Kosice, Czccholsovakia, August 20-25, 1984, HE4. (D Y. ~'VIINORIKAWA an4 K. MITSW: T~ett. s Gimento, 41, 333 (1984).

DEI%IVATION OF M~5OI~ I~ANGE SPECTIC~M )~T DIFFEI%ENT POLAIr ANGLES ETC. 423

The m u o n flux f rom the decay of charged pions, kaons and b y secondary pious has been calculated b y (s)

(4) Mi(E, Yo) dE = M~:(E, Yo) dE + M~K(J~ , Yo) d_E + M=,~(.E, Yo) d ~ ,

where M = ( E , Yo), M~K(E, Yo) and M=.(E, Yo) represen t the muon flux with energy be tween E and E + dE at an a tmospher ic dep th Yo aris ing f rom p p - > ~• p p - > K • and = p - + ~• inclusive reac t ion channels.

The solution of the usual kinet ic equa t ion for charged pious genera ted via p p - + =• inclusive reac t ion channel in the a tmosphere has been considered for m n o n flux calculat ion a t the hr-th genera t ion of pa ren t s and the resul t follows :

(5)

where ~ is the elast ici ty,

prck ~ ] )

1 N Do~(F,~) = N(B~) <z#~>~ A~(a~/,L) ~ ( _ ~ _ ~ ) !

. ~ (1 - - 2,~/A,)'~-~m(m + 1) ... (m -t- N - - 2) m=( l ~ ( m ~ ~ - i)R~</R,~(0*) W(~ , 0, V, Vo),

0* is the local zenith angle near the p roduc t ion he ight y along the s t ra ight p a t h of muons and 0 is the zeni th angle of the muon reaching sea-level wi th respect to the vert ical , W(E, O, y~ Yo) represents surv iva l p robabi l i ty t h a t a muon produced a t an a tmospher ic dep th y should be observed a t a dep th Yo g cm -2 and is t a k e n a f te r iV[AED~ (9), iV is the n u m b e r of generat ion of parents , yV(J~) is the differential p r i m a r y spec t rum of energy E~, for i = 7:~ or K~

1 - ~,2(7+1) - i

A, = (1-- r~)(7 + 1)'

(m[ + mD rt - - 2 m ~ '

R , ---- (7 § 2 ) ( 1 - r~ (~+') (7 -~- 1)(l - - r~ (~+~) '

Hi(0*) = m, c~H/(cvi)

is the cri t ical energy for meson decay wi th scale height of the a tmosphere H, m i and v t are the mass and l i fe t ime of i meson at rest~ i ~-- 7: • or K • I n a s imilar manne r one can calculate the m u o n energy flux a t different generat ion 7q f rom pp -~ K • inclusive reac t ion channels as

(6) N

(s) D. :P. BHATTACHAI~YYA and PRATIBHA PAL: N~tovo Cimento G, 5, 287 (1982); Eizika (Zagreb), 15, 283 (1983). (9) K. ~AEDA: Fortsehr. Phys., 21, 113 (1973).

4 2 4 o . c . ALLKOFE~R, I). P. BHATT&CHAI%YYA a n d PICATIBHA PAL

where ~ is the to t a l e las t ic i ty ,

1 D~(E~) • bKv N(I~,~) (Z~}~AK(2~/2,) ~ (_,V-- 1)!"

. ~ (1 - - 2K/Ap)'*-lm(m + 1) ... (m + _hr-- 2) ~1 1 + (m + ~ - 1)R~B~/H~(O*) W(B~, O, y, Yo) ,

bx~ is the b ranch ing ra t io for K--~ ~ decay mode. The muon flux genera ted b y secondary pious a t first genera t ion of pa ren t s

f rom ~p--> 7:• inclusive react ions is calculated b y the following expression:

o, v , Vo) . (7) M~(E~) = A_~ 2.(1 ~- 2 R , E/H~(O*)) 1 - - b~:= ~ r ~ ]

The integral muon spec t rum above ene rgy ~ h~s been calculated b y the ex- press ion

co

(8) /~(> E) : t'M~(E, Yo) dE ~ KE-rV, Q

The cont r ibut ion of p r o m p t muons to t o t a l muon spec t rum has been calculated b y following the dynamica l m e t hod of Inazaw~ and K o b a y a k ~ w a (~0) who solved the diffusion equat ions b y account ing differeutial cross-section with respect to heavy f lavour part icles. The m u o n flux f rom the decays of D-mesons to p r o m p t muons is p r o b a b l y negligible for muons of energies up to 20 TeV as

(9) #p . . . . t(/~) d E -~ 1 .1236.10-6E -~.4a (cm 2 s sr GeV) -1 .

The corrected ~bsolute m u o n spectra a t different polar angles have been con- ve r t ed to depth in tens i ty spec t rum b y using the modified muon energy loss fo rmula t ion a f te r ](OB&YAKAWA (1):

(10) -- dE/dh[to,~-~ a~o~ + (be -iv bB + bN)E' ~ a~o. + bTE GeV g-~ em 2.

a) Loss due to ionization and exoitation:

(11) a~o~(E) ~ -- dE/dh[~o~ : (Z[A}(3.6995 + 0.1535 In (]~J#)) M e V g - ' em ~ ,

where Z, A are the a tomic n u m b e r and weight of the rock, E is the m a x i m u m t ransfe rab le energy in )/[eV b y a muon of mass # to an e lect ron ~nd E is the

(10) H. INAZAWA and K. KOBAYAKAWA: Prog. Theor. Phys., 69, 1195 (1983).

DERIVATION OF MUON :RANGE SPECT/%~/M AT DIFFE:RENT POLA:R ANGLES ETC. 4 2 5

t o t a l energy of the muon E ---- E~/[E ~- (m~v~)2/m~ ~] with m u o n and electron res t energies # ~ - m , c ~ and m.c ~, respect ively .

b) Loss due to pair production. The final f o rm of the m u o u energy loss

due to puir p roduc t ion is

(12) b..(~) = 0.451 Z(Z § 1) Ira (2,/~) -- 5.4~] A " in (El#) - - 4 ~ ] " 10-6 g-1 cm 2

for muon energies above 100 GeV, # represents the muon mass in GeV.

c) Loss due to bremsstrahlung. KOBAYAKAWA (privnte communica t ion) has again corrected his earlier fo rmula t ion (') for muon energy loss es t imat ion due to b remss t r ah lung b y s impl i fy ing the following formula t ion of :Rozental (11) :

(13) bs(.E) : d[10.168 - - (2/3) In Z § ( 1 - - B ) -~.

�9 (-- 1/6 -}- ( 1 / 3 ) ( 1 - - B ) -1 § ( l n B ) ( 1 - - ( 1 / 3 ) ( 1 - - B ) -1 + ( l / 3 ) ( 1 - - B ) - 2 ) } ] ,

where d : 3.1 Z(Z ~- 1)/A.IO -s g-1 cm 2, B : ZllsE/3.405 with E in TeV.

d) Loss due to nuclear interactions, bN(E ) has been calcula ted following the me thod of Cassiday (1,) b y replacing ah, ---- 1.25.10 -s cm2/nucleon as follows :

(14) b~(E) ---- (0.413 § 0.01 In (B/#)) .10 -6 g-~ cm ~ .

The absolute dep th- in tens i ty re la t ion is:

(15) 0

f d E 1 In (1 § bo,~E/ao~),

where a~u and bef f are the effective values of a,o n and bT(/~ ) ---- be(E ) § b~(E) q- + bA~).

By assuming tha t the energy loss of muons in rock is continuous, t he muon in tens i ty a t a dep th h follows:

(16) ao. __ 1)]-,~ M ( > h) --~ K [ - - (exp [be,,. h] ~ . [b~

For h igh-energy muons the f luctuat ions in range arise f rom ca tas t rophic processes, e.g. bremss t rah lung and nuclear inter~ct ioas in rocks. Several

(11) I . L . ROZENTAL: SOY. Phys. Usp., 11, 4 9 (1968) . (12) G.L. CASSIDAY: Phys. Rev. D, 3, 1109 (1971).

426 O . C . ALLKOFER, D. P. BHATTACHA~YYA ~ n d P~ATIBHA PAL

methods have been used in the past for the est imation of range fluctuations (~s.~7). In the present invest igat ion we choose the numerical me thod of Kiraly, Kiraly

~nd Osborne (2) who used an analyt ic solution to lqishimura's (~s) differential equat ion for the calculation of range f luctuation R~, In an earlier survey IqG and WOLFENDALE (19) have successfully used their procedure to in terpret dif-

ferent depth- intensi ty results.

3. - R e s u l t s and d iscuss ion .

The all-particle p r imary nucleon spectrum es t imated from the direct measurements of J A C E E (6) and indirect est imate f rom D E I S magnet ic spectro-

graph data by ALLKOFEtr et al. (,0) has been found to follow this power law fit:

(17) N(E) dE --~ 2.026E -~'? dE (era 2 s sr GeV) -~ .

Using this nucleon spectrum as hadron source along with the modified hadronic- energy moments for p-air collisions from pp collision ISI~ data after MINORIKAWA

and 1V[ITSUI (?) (shown in table I) the pion and kaon product ion spectra in air

TABLE I . - - The ]factional hadronic-energy moments ]or p-air collisions alter MINORI- KAWA and MITSUI (7).

Z~o For spectral index y = 1.7

Z,p 0.199 Z,~ 0.0718 Zp~+ 0.040 99 Z~- 0.028 3 ZpK+ 0.006 239 ZpK- 0.002 43 Zr:+z:- 0.15 Z~+~- 0.05

(iS) E. D. ~IKHALCHI and G. T. ZATSEPIN : Proceedings o/the I X International on Cosmic Rays Con]erenee, Vol. 2 (London, 1965), p. 997. (14) S. MIYAKE, V . S . NAI~ASIMItAM and P.V. I~AMANAMUICTEu Nnovo Cimento, 32, 1524 (1964). (15) p . j . I-IAYI~AN, N.S. PALMER and A. W, WOLF~DALE: Proc. R. Soc. London, Ser. A, 275, 391 (1963). (16) V.I . GCBE~TS0V, G.T. ZATSEPI~ and E.D. tV[IKHAL'CHI: SOY. J. Nucl. Phys., 6, 527 (1976). (17) L. BERGAMASCO and P. PzCeHI: ffuovo Cimento B, 3, 134 (1971). (lS) J. NISYIIMURA: Proceedings o/ the I X International on Cosmic Rays Con/erence, Vol. 2 (London, 1965), p. 1003. (1~) L .K. IqG an4 A.W. WOLF]~DALE: IYUOVO Cimento B, 20, 163 (1974). (2o) 0. C. ALLKOFER, D.P. BHATTACrrAI~YYA and J .N. CAI~DEWELLE: Lett. 2Fuovo Cimento, 44, 227 (1985).

D E R I V A T I O N OF 1VfUON R A N G E S P E C T R U M AT D I F F E R E N T P O L A R A N G L E S ETC. 427

have been es t imated as follows:

( i s )

(19)

~z(E) dE = 0.140 43 E -~.? d.E,

K(~) dE = 0.017 7 /~-~.~ d E .

We have considered the following parametr ic values for the calculation of sea-level muon spectra at different polar angles:

t / ,(0") =

H~(0*) =

A K

r .

rK

R~

buu

interact ion mean free pa th of nucleons in the air --~ 85 g c m - ~

absorpt ion mean free pa th of nucleons in the air = :[10 g em -2,

in teract ion mean free pa th of pions in the air = ]20 g e m -~,

interact ion mean free pa th of kaons in ~he air = :[50 g cm -~,

critical energy for pion decay at local zenith angle 0"~

critical energy for kaon decay at local zenith angle 0",

kinetic constant ~ 0.6963,

kinetic constant = 0.4928,

energy degradation factor for ~-~ decay ~ 0.78~

energy degradat ion factor for K-~ decay ~ 0.52~

~-- ].2157~ /~K ~ 1.3409~ a~= ~ 0.421~ a=~_ = 0.579~

--~ 0.384~ b,~_ = 0.27:[~ bKa ~ branching rat io for K-Ez decay ~ 0.63~

W(E, O, y, Yo) represents the survival probabi l i ty t h a t muons at an atmospheric depth y should be observed at a depth Y0 g c m -2 with energy E along the direc- t ion of zenith angle 0 and is es t imated b y following the usual expression after ~ i E D A (9)

Y0

g ( ~ , 0, y, yo) = exp - b ~ ( B , Yo, 0 )0(x ' )J ' y

with bv = m~o*/(e~), the decay cons tant for muons ; @(x') represents the at-

mospheric density (g cm -2) at a depth x'. By using these interact ion parameters and relations (1)-(9) the muon energy spectra at sea-level tha t have been esti- mated and compared with the direct measurements (21.24) for vert ical incidence

(21) O.C. A_LLKOFER, K. CARSTENSEN and W.D. DAu: Phys. •ett. B, 36, 425 (1971). (32) T.H. BURNETT, G.E. ~SEK, T. ]IIAUNG, E. S. i~[ILLV, R, H. RUD~RMAN and W. VERNON: Proceedings o] the X I I I International Con]erence on Cosmic Rays, Vol. 3 (Denver, 1973), p. 1764. (~s) B.C. RASTIN: J. Phys. G, 10, 1609 (1984). (~a) I~. G. THOMPSON, R. THOICNLEY, !~. l~. WHALLEY and A. W. WOLFENDA.LE: Pro- ceedings o] the X V International on Cosmic Rays Con]erenee, Vol. 10 (Plovdiv, 1977), p. 35.

~ 2 ~ 0 . C. A L L K O F E R , D. P . BHATTACI tARYYA a n d PICATIBtIA PAL

T

10 a

,-s

(j

10 -1

"d

10 -2 10 2

, I , 10 3 I 0 ~

muon m o m e n t u m (OeV/c)

Fig. 1. - Differen t ia l m u o n m o m e n t u m s p e c t r u m a t sea-level . The fu l l Line is t he de r i ved s p e c t r u m . E x p e r i m e n t a l d a t a : v A~LKOFER et al. (~1), <> B v ~ T et aL (~), �9 RASTIN (~a), �9 THOMrSON et al. (~).

10 ~

CD

T lo_2

7

T E

"~ lo-3

? \ B , k ~ ' ~ - i ~ e = 72 ~

~ " - ' ~ "I" -'~. L " ~ v

i0 -a I I 10 2 10 3 10 a 10 5

E (GeV)

Fig. 2. - Di f ferent ia l m u o n ene rgy s pec t r a a t po la r angles 45 ~ a n d 72 ~ U p p e r a n d lower cu rves r e p r e s e n t t he de r i ved re su l t s for po la r angles 72 ~ a n d 45 ~ respec t ive ly . E x p e r i m e n t a l d a t a : u THO~PSO~ et al. (24) for 0 = 0~ �9 J o x I s c H et al. (2~) for 0 = ---- 75 ~ A, �9 IVA~OVA et at. (~) for 0 ~ 45 ~ a n d 72 ~ respec t ive ly .

DERIVATION OF MUON RANGE SPECTRUM AT DIFFERENT POLAR ANGLES ETC. 429

and for oblique incidence a t polar angles 45 ~ and 72 o they agree wi th the da ta (25,2e) in fig. 1 and 2, respect ively . I t is ev iden t f rom the figure t h a t for ver t ica l incidence the calculated resul t agrees wi th the measu remen t s of D u r h a m group (,4) f avourab ly for m u o a energies beyond 500 GeV. l~or oblique incident m u o n flux it is found f rom the plot t h a t the magne t i c spec t rograph measure- men t s of gokisch et al. (25) and mul t ip l a t e X - r a y emuls ion chambe r da ta of I v a n o v a et al. (~6) a t t he respect ive zeni th angles, viz. a t 75 ~ 42 ~ and 72 ~ are well in ag reement wi th the der ived spectra . The calcula ted m u o n energy

E o 3

7 Ol

~c 2 o

I l

10 2 10 3 10 ~ 10 5 E (OeV)

Fig. 3. - Muon energy loss parameters for standard rocks as a function of energy due to collision, excitation and knock-on electrons a~ter KO~AYAKAWA (1). Full and broken curves represent the effeotive and exact values viz. by a, ff and a~o~, respectively.

spectra a t different zeni th angles have been found to be

(20) it(> ~, 0 ~ : 2.92/~ -2"6

(21) #(> E, 45 ~ : 4.01~ -~.s

(22) /~(> E, 60 ~ = 5 . 6 2 ~ -~.6 (era 2 s sr) -I .

(23) ~ ( > ~ , 75 o) = 3 . 5 8 ~ -~.'

(24) /~(> ~ , 80 o) : 5 . 2 0 ~ -2.5

The cont r ibu t ion to the m u o n flux a t g round level by p r o m p t muons f rom the decays of D-mesons below 20 T e u ( f rom rela t ion (9)) is a lmos t negligible when

(25) I~. JOKISCH, K. CARSTENSEN, W. I). DAT:, H . J . ]YIErER and 0. C. ALLXOFE~: Phys. ~ev. D, 19, 1368 (1979). (26) M.A. IVANOVA, L .A . KVZMICHEV, K .V. ~NDRITSKAYA, E .A. 0sIPOVA, I .V . ~AI~:OBOLSI~:AYA, N .V . SOII:OLSKAYA, A. YA. ~A:RKOVITS~AYA, G.T . ZATSEPI~ and V.I . ZATSEPIN: Proceedings o/ the X V I Iq~ternationat on Cosmic l~ays Co~]ereqlce, Vol. 10 (Kyoto, 1979), p. 35.

~ 3 0 O. C. A L L K O F E R , D. P. BHATTACt tARYYA a n d P R A T I B I I A PAL

compa t ib l e t o m u o n i n t e n s i t y cMcula ted f r o m t h e decays of c h a r g e d p ious

a n d kaons . The d e p t h - i n t e n s i t y re la t ion of m u o n s can be cMcula ted f r o m t h e m u o n

e n e r g y s p e c t r u m at d i f ferent zen i th angles b y u s ing t h e q u a n t u m e lec t rody- n a m i c M m e t h o d s r e c e n t l y modi f ied b y KOB),YAKAWA (~). B y p u t t i n g Z/A = 0.5, Z~/A = 5.5, Z = 11, A = 22 for s t a n d a r d rocks a n d us ing r e l a t ion (10)

~ 3 E o

T

I I

b T

~ ~,,. ~ -"'~ " " " " " " " beff"

bp

I o - b s

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b N

~I~o ~ ' , I , , I , , 10 3 10 ~' 10 5

E (GeV)

Fig. 4. - The energy loss co-efficient of muons in standard rocks vs. energy: . . . . . bv for pair production, -- . . . . . . b n for bremsstrahhmg and . - . b~ for nuclear inter- actions in standard rocks after KOBAYAYAWA (1). Broken and full curves represent the b T and its effective values bo~f, respectively.

t h e m u o n e n e r g y loss due t o ion iza t ion aio.(E ) a n d i ts effect ive va lues a.ff(E) h a v e been e s t i m a t e d a n d a r e d i sp layed in fig. 3. The coefficient of m u o n e n e r g y

loss p a r a m e t e r s for pa i r p roduc t ion , b r e m s s t r a h l u n g a n d nuc lear in teract io i l , viz. bp(E), b~(]~) a nd b~(E) h a v e been cMcula ted f r o m t h e re la t ions (12)-(14) a n d

d i sp l ayed in fig. 4 Mong wi th t h e t o t a l va lues bv(J~) a n d its effect ive values

D:ERIVATION OF MUON R A N G E 8PECTRUI~ AT RIFFle, R E N T P O L A R ANGLES ETC. 431

10

8

?E O

%

I , , I ,

10 3 10 4. E(GeV)

Fig. 5. - Depth-energy relation expected from the latest modified formulation after K O B A Y A K A W A (1).

10 -8

lo-9

I "C

U3

u E 10-1o

.E

10 -11

1 0 - 1 2

[]

I I I I I I I 4- 5 6 7 8 9 IO

rcLnge (10 3 hg c m -2)

Fig. 6. - Vertical depth-intensity spectrum of muons under standard rocks: the full line is the derived spectrum corrected for range fluctuations. Experimental data under Mont Blanc tunnel: ~ B~RGA~ASCO et al. (3) and o BA~TIS'rO~I et al. (4).

28 - I l Nuovo OimenLo C.

~32 o.c. ALLKOF:ER, D. P. BHATTACHARYYA and PRATIBHA PAL

beff(.E), respect ively. F igure 5 shows the dep th-energy relat ion expec ted f rom re la t ion (15) wi th t he use of aof~ and b ~ values f rom fig. 3 and 4.

B y using the ver t ica l sea-level energy spec t rum and the range-energy re]at ions the depth- in tens i ty re la t ion in s t andard rock has been es t imated . We have used the ana ly t ica l m e t h o d of Ki ra ly , K i r a l y ~nd Osborne (8) to ac- count for the loss of m u o n in tens i ty in deep underg round due to r~nge fluctua- t ions. F igure 6 shows the calcula ted corrected in tens i ty for ranges be tween (4000--10000) hg em -*. The recent ~ o n t Blanc exper iments of Bergamasco

~E u 3 ,--.,

2

0 0.8 1.8

X10-9

a)

I I I I 1.0 1.2 1.4 1.6

sec e

Fig. 7. - ~uon intensities as a function of zenith angle at depths a) 4300 hg cm -2, b) 4700 hg cm -~, c) 5300 hg cm -2, d) 5900 hg cm-~: full lines represent the calculated results at the respective depths. Experimental data under Mont Blanc tunnel: o B~gGA~ASCO vt al. (17).

et al. (3) and N U S E X expe r imen t of Ba t t i s ton i et al. (4) have been compared with our calculated resul t for ve r t i ca l incident muons . I t is ev ident f rom the plot t h a t the calculated corrected in tens i ty for ve r t i ca l incident muons is in good agreement wi th the recent Moa t B]ane da ta for depths up to 8000 hg em -~. ~qo c o m m e n t can be made for the ag reement of the calcula ted resul t wi th exper iments beyond 8000 hg cm -~ depths due to the large s ta t is t ical uncer ta in ty in t he N U S E X da ta (4).

DERIVATION OF MI.ION RANGE SP:ECTI{'CM AT :I)II~FEI%E1NT :POLAR ANGLES ~]TC. ~ 3 ~

The muon energy spectra at different zeni th angles have been taken from re-

lations (20)-(24) and conver ted to depth in tensi ty spectrum. The derived muon

intensi ty at different depths has been plot ted as a funct ion of sec 0. Figure 7 shows tha t the calculated results at depths 4300, 4700, 5300 and 5900 hg cm -2 are in good agreement with the receipt angular dis tr ibut ion measurements of Ber-

16

12

u~

7

L_~J6

X 10-'0

~)

I I I I I r

x 10 ~!~ + lz+ b)

I I I I I i 1 2 3 4 5 6

sec 8

Fig. 8. - Muon intensities as a function of zenith angle at great depths, a) 6400 hg cm -~ and b) 7200 hg cm -~ : full lines represent the calculated results. Experimental data : o B~ReESSO~ et al. (5) under Utah rocks.

gamasco et al. (8). For great depths, viz. at 6400 and 7200 hg c m - ' under Utah

rocks~ the calculated results have been compared with the well-known Utah measurements of Keuffel group (5) in fig. 8. I t is evident from the plots

t ha t the calculated absolute mnon spectra at different polar angles are in ap- proximate agreement with the deep underground measurements.

~3~ O . C . ALLKOFER, D. P. BHATTACHARYYA and PRATIBttA PAL

�9 RIASSUNTO (*)

Si stimano gli spettri d'energia muoniea a livello del mare a differenti angoli zenitali dal pifi recente spettro delle primarie JACEE usando risulta~i ISR modificati per inte- razioni p-nueleo. I r isultat i teoriei sono stati confrontati con i r isultat i sperimentali. Usando la reeente relazione modifieata intervallo-energia di Kobayakawa oltre alla proeedura di correzione delle fluttuazioni di Kiraly, Kiraly e Osborne sono stati cal- eolati gli spettri dell ' intervallo muonieo in profonditt~ per differenti angoli polari. I r isultati derivati sono in buon aceordo con i dat i pift reeenti del Monte Bianeo di Bergamaseo et aL, Battistoni et a L e gli esperimenti Utah di Bergesson et aL

(*) Tra~zione a oura della Reda~ione.

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