Cosmic ray muon spectra at sea-level up to 10 TeV

Nuclear Physics B259 (1985) 1-18 © North-Holland Publishing Company

C O S M I C RAY M U O N SPECTRA AT S E A - L E V E L U P T O 10 TeV o.c. ALLKOFER, G. BELLA*,W.D. DAU tt, H. JOKISCH, G. KLEMKE, Y. OREN t, and R. UHR lnstitut fiir Reine und A ngewandte Kernphysik, University of Kie[

Received 27 December 1984 (Revised 20 March 1985)

The momentum spectra of muons in cosmic radiation have been measured with high statistics at sea-level with the DEIS spectrometer by a Kiel-Tel Aviv collaboration. The apparatus consists of solid iron magnets and spark chambers with magnetostrictive readout. The momentum spectra and the charge ratio of muons are obtained in the momentum range from 10 GeV/c to 10 TeV/c in the angular range 780-90 °. The production spectrum was derived from the measured muon intensities.

1. Introduction T h e m o m e n t u m spectra of cosmic ray m u o n s at sea-level have o b t a i n e d interest p a r t i c u l a r l y in the high m o m e n t u m region because of several reasons: (i) The s p e c t r a a n d the c h a r g e r a t i o are i m p o r t a n t geophysical s t a n d a r d s b y themselves a n d are not p r e c i s e l y k n o w n in the high energy region. (ii) Because m u o n s at sea-level are s e c o n d a r y p r o d u c t s from d e c a y i n g hadrons, i n f o r m a t i o n a b o u t i n t e r a c t i o n processes c a n b e d e r i v e d f r o m the m e a s u r e d values. (iii) The p r i m a r y energy s p e c t r u m a n d the m a s s c o m p o s i t i o n are o n l y directly m e a s u r e d up to a b o u t 1 TeV. O n l y one m e a s u r e m e n t at higher energies exists. Beyond this energy limit results can be r e c e i v e d f r o m m u o n d a t a a n d E A S data. (iv) There is f u r t h e r m o r e s o m e c h a n c e to d e t e c t l o w - e n e r g y cosmic ray sources if m u o n s have b e e n a c c u m u l a t e d with high statistics. S t u d y i n g c o s m i c ray m u o n spectra has been d o n e with the techniques of (i) m a g n e t i c s p e c t r o m e t e r , (ii) b u r s t size spectra, a n d (iii) u n d e r g r o u n d m e a s u r e m e n t s . By far the m o s t accurate m e a s u r e m e n t is achieved b y the direct m e a s u r e m e n t in a spectrometer. I n this p a p e r the a p p a r a t u s a n d the d a t a analysis will be d e s c r i b e d a n d the sea-level s p e c t r a will be p r e s e n t e d from which the p r o d u c t i o n s p e c t r a are derived. P r e l i m i n a r y results have been p u b l i s h e d [1]. The analysis a b o u t the p r i m a r y spect Department of Physics and Astronomy, Tel-Aviv University. t~-CERN, Geneva.

2

O.C. Allkofer et aL / Cosmic ray muon spectra

trum, mass composition, point source results and geophysical modifications in the low m o m e n t u m range are published elsewhere [2-5].

2. Experimental Features 2.1. A P P A R A T U S

The experiment was performed on the Tel-Aviv University Campus. The DEIS apparatus (fig. 1) was a large aperture, high resolution spectrometer, which detected charged cosmic ray muon transversals. This section will review the salient features. The parameters of the apparatus are shown in table 1. Spectrometer type. The spectrometer (axis pointing nearly to the west, azimuth 86 °, at a zenith angle of 85 °) was set up from 6 solid iron magnets, 10 scintillation counters and 30 wire spark chambers. The main deflection for charged particles was in the vertical x-z plane. The spectrometer was a combination of a multilayer and conventional type. The multilayer part (magnets interspaced with spark chambers) had the advantage of tracing unambiguously the muon track out of the spark signals which were caused by the muon and accompanying secondaries. By adding additional outer spark chambers to the multilayer spectrometer part the resolution was enhanced by a factor 2.7 with no restriction on the aperture for high energy particles. Magnets. The magnets were of rectangular shape (outer dimensions 0.5 m x 1.8 m x 3.7 m) with a hole (0.3 m x 2.2 m x 0.5 m). The current coils were mounted on b o t h long limbs of each magnet. Only the central 2 m of the long limbs were used for particle bending, thus being sure to have a field homogeneity to better than 1%.

(a)

0 (b)

]

2,rn "

"

'

,~,L__J~| /

ii~i

--SPARK CHAMBER

SCALE

MODULE

l

m TRIGGER COUNTER SHOWER COUNTER M MAGNET (c)

. . . . . .

I

Fig. 1. Schematic view of the DEIS spectrometer (a) side view, (b) top view, (c) cross section of a magnet.

O.C. Allkofer et al. / Cosmic"ray muon spectra TABLE 1 Parameters of the DEIS spectrometer magnets trigger system control system shower counter track detectors

cut-off energy mean mdm acceptance direction open angle measuring time trigger total usable tracks corrected muon rates

6 solid iron magnets: 0.5 × 1.8 x 3.7 m3, hole: 0.3 × 2.2 x 0.5 m 3 B = 16.31 kG; Bh = 49.9 kGm 2 × 3 scintillation counters: area 200 x 65 x 1.2 m3 3 scintillation counters: area 200 x 20 x 1.2 cm3 1 scintillation counter: area 80 x 80 x 1.2 cm3 30 wire spark chambers with effective area 2.26 × 2.26 m2; magnetostrictive readout with 20 MHz counters, space resolution: Ax = 0.69 ram; Ay = 0.71 mm 4.5 GeV 6.0 TeV 2 × 582.8 cm2 sr 85 ° zenith; 266" azimuth _+8° zenith; + 22 ° azimuth 6426.79 hours 13 x 106 upper telescope 6 x 106 lower telescope 2752410 > 100 GeV 223554events > 1 TeV 3612 events > 5 TeV 242 events > 6.3 TeV 195 events

R u n n i n g t h e s y s t e m at 50 A (9 k W ) g a v e an i n d u c t i o n B = 16.31 _+ 0.16 k G a n d a f i e l d l i n e i n t e g r a l f B d l = 50.0 _+ 0.50 k G m . Track detection. T r a c k i n g a n d m o m e n t u m

m e a s u r e m e n t s in the s p e c t r o m e t e r

u s e d 15 d o u b l e g a p m o d u l e s ( e a c h o n e w i t h 4 w i r e planes, m a t e r i a l t h i c k n e s s 2.4 g / c m 2 ~ 1 / 7 r.l., e f f e c t i v e a r e a 240 c m X 240 cm, 1 m m w i r e s p a c i n g , m a g n e t o s t r i c rive r e a d o u t ) i n s i d e a n d o u t s i d e the m a g n e t s . T h e wires w e r e o r i e n t a t e d in x- (24 r e a d o u t s ) , i n y - (30 r e a d o u t s ) , in v- (25 ° w i t h r e s p e c t to + y ; 3 r e a d o u t s ) a n d in w - d i r e c t i o n (25 ° w i t h r e s p e c t to - y ;

3 readouts). The drift velocity on each readout

w a s c o n t r o l l e d a n d m e a s u r e d in e v e r y e v e n t b y a f r o n t a n d r e a r f i d u c i a l w i r e signal at w e l l k n o w n

p o s i t i o n s . T h e t r a c k e f f i c i e n c y was { 7 ) =

(64_+ 15)%, the s p a t i a l

r e s o l u t i o n { a x ) = 0.69 +_ 0.20 m m a n d { o ~ , ) = 0.70 _+ 0.20 m m , a v e r a g e d o v e r all r e a d o u t s a n d t h e t o t a l r u n n i n g time. A p p l y i n g a p p r o p r i a t e c o n s t r a i n t s to a t r a c k the c h a r g e d p a r t i c l e r m s m o m e n t u m r e s o l u t i o n c o u l d be e x p r e s s e d as

8 p / p = (0.132 + 1.667 x 1 0 - 4 p ) 1/2,

(1)

w h e r e p is t h e m o m e n t u m in G e V / c . T h e first t e r m is d u e to the m u l t i p l e s c a t t e r i n g o f t h e m a g n e t s , t h e s e c o n d o n e is fixed b y the spatial r e s o l u t i o n . T h e 100% e r r o r is r e a c h e d at 6.00 T e V / c

Trigger system.

( m d m ) . T h i s c o r r e s p o n d s to a d e f l e c t i o n a n g l e o f 0.2 m r a d .

T h r e e v e r t i c a l p l a n e s of l a r g e a r e a s c i n t i l l a t i o n c o u n t e r s in front,

i n t h e m i d d l e a n d b e h i n d the s p e c t r o m e t e r d e f i n e d a p a r t i c l e event. T h e c o u n t e r s

O.C, Allkofer et a L / Cosmic ray muon spectra

b) N

-

104

lo4 r

103

103

102

102;

10 1 0

I 20

I I 40 60 time

[ 80 100 [ns]

~

'°i 1

•

0

I

20

i

I

40 60 time

i

80 100 I'ns]

Fig. 2. Distribution of the time-of-flight measurement. Note the log scale. The two peaks contain 99% of all events and correspond to m u o n s coming from west and east. (a) Upper telescope, (b) lower telescope.

were viewed at both ends by phototubes, providing signals for analog-, timing-, pattern- and meantimer units. The output signals were arranged to give a trigger from three muon telescopes. The acceptance of the top telescope (formed by the top counters of each of the 3 planes) and bottom telescope (formed by the bottom counters of the planes) covered the magnetic field area. The middle telescope triggered on particles passing the iron and field free region, thus providing straight line tracks for alignment procedures. For each trigger in one of the telescopes there were 3 independent time-of-flight information (TOF), thus it was unambiguously possible to determine the flight direction and the charge of the particle. The off-line analysis of the TOF system gave a time resolution of FWHM = 3.5 nsec (fig. 2) and allowed easy discrimination of events from west (95% of the muons) and east (4%) and shower initiated triggers. An additional counter gave hints on EAS showers. Data recording. Up to 8 spark chamber signals could be recorded by each magnetostrictive readout line and were digitized with 20 MHz. The timing information of the spark signals was derived by zero-cross technique. The mean error in the spark position due to the jitter in the signal timing and the digitizing electrons was 0.2 mm. The minimum distance for double hit recording on the same readout was 4 mm. A NOVA 1220 mini-computer read in the data from each event, wrote it to magnetic tape and provided on-line displays and diagnostics of chambers and apparatus performance. 2.2. A L I G N M E N T

The internal accuracy of each chamber with respect to external fiducials on the chamber was known to be less than 0.1 ram. At the beginning of data taking an optical alignment between trigger counters, spark chambers and magnets was performed by surveyor techniques with an error of + 0.04 mm in the x-coordinate, _ 0.1 mm in the z-coordinate, and + 0.16 mm in the y-coordinate. During the actual

O.C. Allkofer et al. / Cosmic ray muon spectra

5

run the final alignment was obtained from the straight line muon tracks through the field and iron free center part of the system; the residual error was reduced by 50% when going from optical to straight line track alignment. In addition we monitored the alignment by magnet-off runs, by the rms distribution of the reconstructed deflected muons and by another optical survey at the end of the experiment. 2.3. OPERATION The D E I S spectrometer has been operated between May 1977 and December 1978 during an effective running time of 6426.8 hours. The polarity of the field has been changed every second day to avoid systematic asymmetry in the measurement of the m u o n charge ratio.

3. Data analysis 3.1. EVENT RECONSTRUCTION The operating time was divided into 16 measuring periods, the data of each period having a different set of alignment parameters. The latter parameters were derived f r o m the straight line tracks in the middle telescope. Data reduction involved a selection stage to filter out nonreconstructable events, track finding in the chambers and trajectory fitting. The track finding and trajectory fitting procedures were done independently in the vertical bending plane ( x - z plane) and horizontal y - z plane. In b o t h planes the track was fitted by a parabola in the spectrometer part containing the magnets and two straight lines outside the magnets. The fitting procedure used weights wn = 1/aft ( G spatial chamber resolution at the respective detector level n) for the chambers outside and wn = 1 / ( G + c / E n ) 2 for the chambers in the magnets (c = 9.33 cm 2 GeV 2, E n = muon energy at the respective level). The latter equation took care of multiple scattering in the iron modules. The reconstructed particle tracks were demanded to give X 2 ~< 9 DOF. If this condition was not fulfilled, sparks with highest X 2 contribution were cancelled and the procedure repeated. If there were more than one spark in a gap, the one with lowest X 2 contributions was chosen. With the minimal requirement of at least 4 sparks in each projection plane 73% and 78% of all triggers were reconstructed from the top respectively bottom telescope, the losses were mostly due to shower initiated triggers. Using these selection limits we got 2.75 × 106 events from the first reconstruction stage. 3.2. SELECTION CRITERIA T o extract a clean sample the following cuts and criteria have been applied to each individual event: (i) Cuts were introduced in the effective area of the scintillation- and spark chamber counters to avoid edge effects and to achieve identical apertures for top

6

O.C. Allkofer et al. / Cosmic ray muon spectra

and b o t t o m telescope at high momenta. The geometrical acceptance is thus reduced by 14% to 1168 cm 2 sr, equally shared between top and bottom telescope. (ii) A cut of + 7 ns with respect to the mean value was applied to the time-of-flight distribution, to extract muons coming from the west and east direction. In this way a direction could be assigned to 98.9% of all reconstructed events. (iii) Only muons with a reduced x 2 / D O F ~< 9 for the trajectory fitting were accepted. (iv) The deflection angle in the non-bending y-z plane was required to be less or equal 4 times the mean multiple scattering angle. (v) Two data sets A and B have been extracted by means of a Monte Carlo optimized track selection criterion with minimal spark configuration in the different groups of the spark chambers. The minimum number of sparks contributing to the track had to be at least 6 for the x- as well the y- coordinate. Data set A (1.82 X 10 6 events) accumulated muons at high statistics. The sample included muons with m o m e n t a >/8 G e V / c . The high quality data set B, which was a subgroup (58%) of A, was aimed to give high m o m e n t u m resolution by demanding sparks in the outmost chambers. After introducing the above described cuts and selection criteria we were left for the high quality data set B with 5.99 x 105 muon-tracks: 2.23 x 105 muons had a m o m e n t u m p >~ 100 G e V / c , 3600 muons p >~ 1000 G e V / c and 242 events p >~ 5011 GeV/c.

3.3. CORRECTIONS The muons were binned according to their zenith-angle 0 (west: 78.15°-80°; 8 0 ° - 8 2 ° ; 8 2 ° - 8 4 ° ; 8 4 ° - 8 6 ° ; 8 6 ° - 8 7 ° ; 8 7 ° - 8 8 ° ; 8 8 ° - 8 9 ° ; 8 9 ° - 9 0 ° ; east: 8 8 ° - 8 9 ° ) , and m o m e n t u m p (10 logarithmic equally spaced intervals per momentum decade). To determine the charge ratio and the corrections the spectra were further subdivided according to types s (i, j, k) events (i = t o p / b o t t o m telescope, j = positive/negative magnetic field, k = positive/negative particle charge), which describes the event configuration. Thus for each zenith-angle we had 8 uncorrected, observed m o m e n t u m spectra Nob~(p, 0, i, j, k), which had to be corrected individually for apparatus effects. A detailed Monte Carlo calculation was used to simulate the transversal of particles through the spectrometer. In the correction the following individual effects were considered: Momentum acceptance. For each zenith angle bin the m o m e n t u m acceptance for particle tracks of the muons was calculated. .Reconstruction efficiency. The parabola approximation inside the magnet area, the experimental efficiencies and spatial resolutions of the spark chambers, and the cuts mentioned in the previous section lead to a loss of reconstructed events. The reconstruction efficiency which was simulated in the Monte Carlo, is m o m e n t u m dependent below 150 G e V / c because of the parabola approximation and the cuts on

O.C. Allkofer et al. / Cosmic ray muon spectra

the spark configuration, however, reached 96% (84%) in data set A (B) at higher energies. The Monte Carlo X2 distribution fitted well the experimental one. M o m e n t u m correction. These corrections are discussed in terms of the deflection angle ~p = l / p , because of the nearly gaussian error distributions in this variable. The resolution function F ( + - ~p) gives the probability that a deflection angle 7Jl, is measured when an angle ~b is expected. The difference is due to spatial position resolution in the chambers (o+p = a = 1.667 × 10 _4 c/GeV), multiple scattering in the magnet blocs (o~ =/~+~, = 0.13qJp), parabola approximation in the trajectory fitting (O~a = s(i, j, k)o~p = - 1.84~bp for p >/200 G e V / c ) and systematic shift (A~'~p = s(i, j, k) C = 1.2 x 10 - 4 c/GeV). The contribution of the first two effects was determined from an experimental deflection error distribution, which was calculated from the error matrix of each individual event-track fit. As the number of sparks contributing to the fit is varying, the deflection error distribution is not a strict gaussian with a fixed standard error. So the commonly used term "maximum detectable momentum" mdm, defined by mdm x % = p h b p does not give an exact description of the resolution of the spectrometer. For events with p >/200 G e V / c our mdm ranged from 4 T e V / c up to 9 T e V / c with a mean value of 6.00 T e V / c and lead to the momentum resolution of eq. (1). The Monte Carlo calculation resembled the experimental error distribution and gave a mean mdm, which was 6% better than the experimental one. The third effect and its correction was derived from simulated tracks. The fourth effect may arise from different sources and can be regarded as an apparent residual alignment error. In the DEIS experiment the major contribution came from the temperature difference ( 3 - 5 ° C ) along the vertical magnetostrictive readout bars (near the magnets), caused by the heat dissipation of the powered magnets. The temperature coefficient of the signal velocity ( - 3 x 10 4 / ° C ) , the readout- and alignment procedures lead to an apparent negative displacement ( - 0 . 3 mm) in the top and bottom part in the chambers. This effe,ct was known, experimentally proofed and nearly completely accounted for by the above mentioned number of C = 1.2 X 10 - 4 c/GeV. Actually the correction was evaluated from the muon data by comparing the charge ratios of type s(top, B +, _+1), s(bottom, B ~, _+1) events (/~+: up deflected tracks; # : down deflected tracks) with type s(top, B ~:, -T-l), s(bottom, B-+,-T-l) events (/z+: down deflected tracks; tz : up deflected tracks). Normalization. A careful study on the normalization and absolute scale was performed by two nearly independent procedures. One method (sample with 2.1 x l0 s subsequent triggers) started from the extracted data set A, which was corrected by intense calculations on reconstruction efficiencies, detection-probabilities, acceptance and time-of-flight analysis. Following corrections were obtained: +0.5% ( + 2.3%) due to counter inefficiency in the top (bottom) telescope, + 6.0% due to losses from showers, +0.8% due to rejection by TOF cuts. The second approach (sample of 1.2 X 105 events) started from the measured trigger rate and the data from the first analysis stage, where all information of non-reconstructed and

O.C. Allkofer et aL / Cosmic ray muon spectra

reconstructed (>~ 4 sparks) were available. Triggers by noise and showers were recognized by the complete hit pattern of all scintillation counters and the manifold time-of-flight information. Both procedures agreed within 1%. The total data set A was then normalized to the corrected investigated sample. Finally data set B was normalized to data set A at p = 200 GeV/c. Charge Ratio. The observed charge ratio R o b s ( p ) = N + ( p ) / N (p) is most sensitive to any asymmetries in the top and bottom telescope with runs of positive and negative magnetic field. Asymmetries in aperture, alignment, efficiencies and running time show up and contribute to the observed charge ratio, which then is systematically different from the true value R true(P)" It was shown that by equalizing the running times at the two polarities, these effects apparently disappear and completely cancel when calculating the arithmetic or geometric mean value from the single charge ratios. From the data in this experiment we were able to derive several independent sets of charge ratios Robs(i , j ) from different constellations (i = t o p / b o t t o m telescope, j = pos./neg, magnetic field). This feature allowed to crosscheck the data, to find out the inconsistencies and corrections, and finally to present a self-consistent charge ratio. The expressions Robs(tOp, B +) R obs(top, B - ) '

Robs(bOt, B +) Robs(bOt, B ) '

Robs(tOp, B +) Robs(bot, B - ) '

Robs (bot, B - ) Robs(to p, B +) '

Robs(tO p, B +)/Robs (bot, B - ) Robs(bot, B+ )/Robs(tOp, B - )

should yield the value 1 after equalizing the B +, B running times. From the observed values we calculated an apparent residual misalignment factor C, which was already mentioned in the momentum error discussion. From the resolution and acceptance corrected spectra N ( p , 0, i, j, k) a self-consistent charge ratio R = (Robs(tOp, B+)Robs(bot, B+)Robs(tOp, B-)Robs(bot, B - ) ) 1/4 was obtained.

Geomagnetic effects. The low momenta muon tracks are modified by the geomagnetic field and multiple scattering in the atmosphere. Thus for the viewing direction west, the negative muons entering the apparatus will be suppressed because they have a longer path in the atmosphere than the positive ones. This severely effects the charge ratio. Corrections have been calculated with a trial incident muon spectrum for each zenith angle interval. The corrections are ,%<1% for the particle flux with p >~ 35 G e V / c and for the charge ratio with p >/250 G e V / c . Wide angle corrections. For spectrometers erected in horizontal directions the effective mean zenith angle differs from the spectrometer axis, because of the strong

O.C. A l l k o f e r et all /

Cosmic" ray m u o n spectra

3.0 2.6

~~

\

correction-functions

U3 Z

o_ I..,.-

2.2

Z kt.

1.8

Z O t-.-

1.4

¢,¢

1

............

8/,o-86

- -----....... .....

total efficiency rr~mentum ~solufion occep~once

.......

geomagnetic

°

.~..:~_.

O (..)

0.6

0.2 10

100

1000 10000 MUON MOMENTUM [OeV/c]

Fig. 3. Correction function for the measured muon spectrum in the zenith interval 840-86 ° due to momentum resolution, acceptance,geomagneticand efficiencyeffects.

angular dependence of cosmic rays. To present a full statistical spectrum at a mean zenith angle 0 = 85 ° we calculated correction features Cw.a.(p), which have to be applied to the mean acceptance weighted measured spectra from the different contributions of the total interval 78.15 -%<0 < 90 °. The ratio of expected to accepted frequencies gives the correction factor G(p, O) by which the measured frequency per m o m e n t u m and angle bin have to be multiplied (fig. 3).

4. Spectrummodeling The model calculation used conventional muon-production from meson decay (~r, K) and is a similar way as in previous papers [6]. The differential muon intensity N~(E°, x °, 0 °) at the atmospheric depth x ° (measured vertical in g cm -2) of the observation level is derived from the integral contribution of the pion spectrum N=(E,, x, O) by x0

[i°

X £r~ E~

.

XD~( E~, x, O) X N~( E~, x, O) d E ~ d x ] ,

10

o.c. Allkoferet al. / Cosmicray muonspectra

where the muon energy E~, the pion energy E~ and the local zenith angle 0 are taken at the respective atmospheric depth x. The local zenith angle 0 differs from the zenith angle 0 ° at the level of observation and has to be computed for every x-value. W~(E~, x, 0, x °) is the muon survival probability from the level x to x °, D=(E~, x, O) gives the pion decay probability, P,(E~, E~) is the muon energy distribution of a decaying pion of E~ from units r~ = (mJm~) 2. All variables are defined in the lab system. The pion spectrum N,(E~, x, O) in turn is given from the main term in the solution of the pion diffusion equation:

N.(E.,x,O)= [fo W.(<,x',O',x) , ×AE'~

secO'

(

cx secO"

v~ ~kN exp - J 0 - ~ d x

,,]

)dx'

]

with the nucleon-nucleus interaction length k N and the pion attenuation lengths A~ in air. The pion flux is a convolution of the produced pion flux in the atmosphere (pion production spectrum, second factor) and the respective survival probability (first factor) W=. The energy independent power index % reflects the constant steepness of the primary nucleon spectrum. The contribution from kaons to the flux N~,(E°, x °, 0 °) is described by similar equations by replacing pion terms by the respective kaon expression. In addition one has to take into account the K/Tr ratio = 0.15 at particle production and the branching ratio for the Ku2decay mode. The equations of our Maeda-type model have been explicitly quoted because in this way the handling of the pion production spectrum is improved over previous expressions [8] by taking into account contributions from ~r-, K-decay at every atmospheric level and different interaction and attenuation lengths for every type of nucleons and mesons. The constraints on the model are one dimensional particle propagation, strict scaling for the inclusive ( ~ , K ) cross sections and in every independent power index of the primary nucleon spectrum. It follows that % = 7K = % where ~, is now defined as power index of the meson production spectrum. Under the assumptions made in this paper 7 is identical with the index for the primary spectrum of nucleons. A would then be the amplitude. We know that our assumptions on 7r, K, # production/decay kinematics and the derived muon spectrum are only approximative. Therefore the calculation in this paper should be called a phenomenological model. In this context the power index y is an effective power index. In this experiment the model has been used at one hand as trial spectrum to calculate geomagnetic and multiple scattering corrections in the atmosphere, momentum corrections, normalization corrections, wide angle corrections, and on the other hand for fitting procedures to derive a muon production spectrum from the data.

O.C. Allkofer et al. / Cosmic ray muon spectra

11

TABLE 2 T h e m e a s u r e d c o r r e c t e d m u o n f l u x e s N ( pj,, 0 )

1 1

p,~

78.15 °

78 °

80 °

82 °

84 °

86 °

87 °

88 °

89 °

Po

90°

80°

- 82°

84°

- 86°

87°

88°

89°

- 90°

6.09 1.01 4.98 0.88 4.46 0.85 3.47 0.88 3.04 0.93 2.63 0.99 2.27 1.07 2.04 1.14 1.79 1.28 1.62 1.43 1.42 1.66 1.34 1.91 1.18 2.32 1.13 2.75 1.11 3.27 1.06 4.04 0.88 5.41 0.92 6.60 0,96 8,07 0.86 10,61 1.02 12.71 1.04 16.67 0.58 22.93 0.50 37.82 0.78 44.79 1.19 50.00

3.35 1.14 3.00 0.91 3.02 0.80 2.57 0.74 2.31 0.71 2.07 0.69 1.85 0.70 1.70 0.73 1.60 0.77 1.46 0.86 1.35 0.97 1.26 1.12 1.20 1.32 1.16 1.54 1.07 1.88 1.01 2.34 1.00 2.88 0.92 3.74 0,95 4.58 0,93 5,79 1.00 7.27 0.84 10.54 0.81 11.04 1.00 15.30 0.86 24.32 1.06 30.18

2.03 1.64 1.87 1.13 1.89 0.86 1.71 0.72 1.61 0.65 1.53 0.62 1.45 0.61 1.37 0.63 1.31 0.67 1.25 0.72 1.21 0.81 1.15 0.92 1.11 1.07 1.07 1.26 1.04 1.50 1.04 1.81 1.00 2,25 1.02 2,77 0,97 3.54 1.05 4.26 0.88 6.06 0.94 7.77 0.84 8,50 0,91 12.49 0.71 20.78 0.77 27.62

0.97 2.65 0.99 1.50 1.06 1.08 0.99 0.89 0.94 0.79 0.98 0.72 0.99 0.68 0.98 0.67 0.99 0.68 0.99 0.73 0.97 0.79 0.97 0.88 0.98 1.00 (I.98 1.16 0.98 1.36 1.00 1.62 0.98 2.00 1.00 2.45 1.00 3.07 0.98 3.86 0.96 5,12 1.01 6,59 1.03 6,72 0.93 10.92 1.35 13.36 0.88 22.91

0.45 4.68 0.51 2.74 0.54 2.07 0.55 1.66 0.59 1.43 0.64 1.30 0.68 1.22 0.71 1.18 0.72 1.20 0.75 1.25 0.78 1.34 0.82 1.46 0.85 1.63 0.88 1.87 0.91 2.14 0.88 2.62 0.93 3.11 0.91 3,92 0.90 4.91 0.93 6,03 0,99 7.62 0,88 1027 1.13 9.79 1.00 16.06 0.73 27.64 1.51 26.76

0.24 5.40 0.30 3.38 0.34 2.60 0.37 2.05 0.38 1.86 0.43, 1.67 0.47 1.60 0.51 1.52 0.54 1.51 0.60 1.54 0.64 1.62 0.70 1.74 0.72 1.96 0.75 2.21 0.81 2.49 0.84 2.95 0.94 3.42 0.93 4.27 0.93 5.31 0,91 6.72 0.95 8,57 1.02 11.00 1.13 10.79 1.15 16.45 0.82 28.97 0.79 40.81

0.13 6.18 0.17 4.12 0.19 3.47 020 2.95 0.23 2.57 0.26 2.37 0.30 2.15 0.34 2.07 0.38 2.02 0.43 2.03 0.47 2.14 0.54 2.22 0.60 2.40 0.63 2.71 0.68 3.06 0.77 3.45 0.81 4.15 0.78 5.23 0.84 6,27 0.79 8,10 0.92 9.79 1.14 11.63 0.99 12.91 0.98 19.96 0.61 37.69 1.00 40.85

0.07 7.17 0.08 5.57 0.10 4.78 0.10 4.39 0.12 3.94 0.14 3.57 0.18 3.13 0.20 3.04 0.23 2.97 0.27 2.95 0.34 2.91 0.37 3.11 0.44 3.25 0.48 3.61 0.54 3.99 0.58 4.62 0.68 5.26 0.80 5.97 0.82 7.37 0.84 9.08 0.95 11.20 0.77 16.43 0.85 16,23 1,02 22.9l 1.53 27.74 0.68 57.81

10.8 12.6 2 12.6 15.8 3 15.8 20.0 4 20.0 25.1 5 25.1 31.6 6 31.6 39.8 7 39.8 50.1 8 50.1 63.1 9 63.1 79.4 10 79.4 100.0 11 100.0 125.9 12 125.9 158.5 13 158.5 199.5 14 199.5 251.2 15 251.2 316.2 16 316.2 398.1 17 398.1 501.2 18 501.2 631.0 19 631.0 794.3 20 794.3 1000.0 21 1000.0 1258.9 22 1258.9 1584.9 23 1584.9 2511,9 24 2511.9 3981.1 25 3981.1 6309.6 26 6309.6 10000.0

1.37E

06 0.65 1.30E 06 0.50 1.15E 06 0.42 1.08E 06 0.38 8.76E 07 0.35 6.99E 07 0.34 5.34E - 07 0.33 3.90E 07 0.34 2.72E - 07 0.35 1.84E - 07 0.38 1.19E -- 07 0.41 7.44E - 08 0.46 4 . 4 5 E - 08 0.53 2.62E 08 0.61 1.50E 08 0.72 8.16E 09 0.87 4.31E 09 1.06 2 . 2 0 E - 09 1,32 1 . 1 2 E - 09 1,64 5 . 6 7 E - 10 2.05 2 , 6 3 E - 10 2.67 1 , 1 8 E - 10 3.54 3 , 8 0 E - 11 3,66 9.31E 12 5.61 2.14E 12 8.27 4 . 6 9 E - 13 11.47

The data for each momentum bin index 1 is contained in two consecutive lines. The second column for each bin I contains the lower p. and upper Pn momentum bin limit. The third column gives in the respective first line the differential intensity (in units of ( Ge V/c ) i sec- 1 sr 1) in the total zenith range 7 8 . 1 5 ° - 9 0 7 . The following columns give a scaling factor, which has to be multiplied to the (78.15 ° 90 °) intensity to get the differential intensity for the zenith angle bin, specified in the heads of the columns. The line following the intensities gives the error in percentage.

O.C. Allkofer et al. / Cosmic ray muon spectra

12

5. R e s u l t s

5.1. E R R O R S

The errors quoted in table 2 and the following figures are point-to-point errors which are due to statistical fluctuations. An uncertainty in the magnetic field B of 1% leads to m o m e n t u m dependent errors in the spectrum: 1.3% at 50 G e V / c ; 1.9% at 100 G e V / c ; and 3.3% at 1 TeV/c. From the normalization procedure for the top and b o t t o m telescopes of the spectrometer for positive and negative magnetic field strengths corrections for the absolute flux can be obtained with 3.1% leading to a total error of 4.5% at 1 TeV/c. Below p~ = 60 G e V / c the error increased because of multiple scattering and energy losses dE/dx. The estimate of this error is 1% at p~ = 45 G e V / c , 5% at p~ = 20 G e V / c , and 20% at p~ = 10 G e V / c . Concerning the charge ratio the

10-3

'EU 'T I-

10-4

U

"~ 10 -5 i

(,~ i

o.5 1.0

IX. lO-e

1.5

Z 2.0

13.. >,, .~ U~ t"

1 0 -7

c

10 - e

~

b2.0 13.0 t3.5

10 -9

10 2

P

muon

momentum

10 3

10 4

EGeV/c]

Fig. 4. D i f f e r e n t i a l m o m e n t u m spectra N(p~,O)p~ versus m u o n m o m e n t u m w i t h z e n i t h angle 0 as p a r a m e t e r . The c u r v e s are d r a w n displaced in the o r d i n a t e a c c o r d i n g to the q u o t e d factor o n the right scale.

O.C. Allkofer et al. / Cosmic' ray muon spectra

13

systematic errors are 15% at p, = 10 G e V / c , 2% at p, = 20 G e V / c , 1.5% at p, = 100 G e V / c , and 1.3% for p, >/150 G e V / c . The statistical error, however, is superior for p, >~ 300 G e V / c .

5.2. M U O N F L U X

In table 2 the corrected absolute differential muon fluxes for different zenith angles are presented. The third column gives the flux N(p,,(O)= 85 °) which is received for the complete zenith aperture 78.15 ° ~< 0 ~< 90 o. From the quoted statistical error the number of events per zenith angle- and m o m e n t u m bin can be recalculated. The measured intensities from table 2 are plotted in fig. 4, the ordinates being displaced for reasons of clearness. The curves in fig. 4 have been obtained f r o m least square fit technique of our model to the data, where the intensity factor A

,o o

/

T

/

1/1

'V •

lJ--

I

~

P[GeV/c]

,t'rJ / ~ ¢ ' ~ ~ " ~

7

o

J ' - ~ i ~ ' " " ' ~

Zenith- ongle

--"-"

•

706

o

22/.

•

dependence

"1/~ of Ipdiff.

in 6b ~ ~d3,1s ~iTs 8~.5 1

10

p~.s 100

o see G

zenith angle Fig. 5. The m u o n flux as function of zenith angle with muon m o m e n t u m as parameter. The plot shows the relative deviation to a reference spectrum at 0 °. The curves again are from the parametrization we found from fig. 4. This parametrization has been used to calculate the reference spectrum for zenith angle 0=0".

14

O.C. Allkofer et al. / Cosmic ray muon spectra

a n d the s p e c t r a l i n d e x ~{ in the p o w e r s p e c t r u m A E v were d e t e r m i n e d . T h e fit is d o n e in the c o m p l e t e energy range 1 0 - 1 0 000 GeV. T h e r e is a n excellent m a t c h i n g of d a t a p o i n t s a n d calculation over three d e c a d e s of m u o n m o m e n t u m . T h e m u o n flux as function of the zenith angle is d e m o n s t r a t e d in fig. 5. T h e a b o v e m e n t i o n e d p a r a m e t r i z a t i o n allows to calculate a vertical reference s p e c t r u m ( 0 = 0 ° ) . T h e curves show for 0 < 75 ° the well k n o w n s e c 0 e n h a n c e m e n t for inclined intensities; the s e c 0 d e p e n d e n c e is m o d i f i e d for 0 > 75 ° b y the c u r v e d e a r t h surface.

6.

Comparison

C o m p a r a b l e results in the energy region b e y o n d 1 TeV the M U T R O N - p r o j e c t [7]. In this project a m d m = 20 a n g u l a r r a n g e is 87 ° ~< 0 ~< 90 ° with the m e a n zenith angle o u r zenith angle intervals 8 8 ° - 8 9 ° a n d 8 9 ° - 9 0 ° we reach

have been o b t a i n e d with TeV is c l a i m e d a n d the ~0)~ = 88.8 °. C o m b i n i n g a m e a n angle of 88.9 °, a

2.0 1.8 1.6

1.~,

I

1.2 '4"-' .~

ul r4) t--

1.0 0.8

I

0.6 4)

0.4

MUTRON

o 4) t..

I

1.4

DEIS

I

I

DIF

I

It

I

1.2 1.0

0.8 0.6

0.~

I I

°1~Oo ° ° ° ~

.L I I

I I

10 2

10 3

* 104

MUON MOMENTUM EGeVIc'I Fig. 6. Comparison of the integral and differential muon intensities as measured with the M U T R O N and DEIS spectrometer.

15

O.C. Allkofer et al. / Cosmic ray muon spectra I

0.4

I

1 t~lt;tj

=,D D E I S • KIELx KIEL

[

I [11[]

I

[I

I

DESY

• SAN DIEGO

v,. =

I

-

I

=

]l

~

|

79.1

=

-0.4 0.4~

T l

0 I~

¢-

*5

I I

{{

¢-

8 .o

-0.4 t 0.4" 0 I

x

t'''i

1

I t IL~ll

8&O

fl

L

-0.4

I 10 2

[

I I I II~,J

I

I

10 3

IlL

lO 4

P muon momentum EGeV/c] Fig. 7. Comparison of muon spectra in the momentum range up to 1 TeV. The reference spectrum I fit has been obtained from the parametrization of the DEIS data as mentioned in a previous chapter.

value very similar to M U T R O N . The DEIS events in this particular zenith range represent only 11% of the complete data set for p >~ 300 G e V / c . As M U T R O N and D E I S use the same m o m e n t u m binning a comparison on a histogram basis was performed (fig 6). From fig. 6 can be seen that the integral and differential intensities of both spectrometers are in the whole m o m e n t u m range roughly the same. We conclude that absolute intensities and the shape of the m o m e n t u m spectra are for both measurements in agreeement. U p to around 1 TeV further measurements exist [8]. In fig. 7 the DEIS, M U T R O N , K I E L - D E S Y , and SAN D I E G O data are compared for the mean angles 0 = 79.1 °, 81.0 ° and 83.0 ° . With the model the fluxes for near vertical direction have been calculated and c o m p a r e d with measured data [9]. The results are shown in fig. 8. Using the data in

O.C. Allkofer et aL / Cosmic ray muon spectra

16

102-

103

104

4-J .w q..

I-!

!

0.4

75.0

=_ "O

1,.-I

-0,4 0,4

1

'"

0.0 c

_>

-04 • DURHAM (Thompson) • SAN DIEGO • KIEL

O

10 =

P

muon

103

momentum

104

CGeV/c'I

Fig. 8. Comparison of muon spectra in the near vertical direction.

,

u

v

!

'

' ' ' I

,

,

.

!

v

,.w|

v

i

mode( colculotion. (best fit for 0EIS "300 GeV/c) • Durham 77

% ~

* Son Diego 73

(J:)

01

0 ° spectrum

x

001100

I

I

I

I

t

,

~ tl

1000

t

i

I

~ i

ill

,

10000 p mu0n momentum [6eV/c]

Fig. 9. The vertical muon momentum spectrum.

A

17

O.C. A l l k o f e r et al. / C o s m i c rc~v m u o n spectra

2.9

I

I

PARENT MESONS SPECTRUM EXPONENT

~

eDEIS 78°-90 ° x

T?--"

x KIEL-DESY v MUTRON (PRL) 87 ° - 9 0 °

2.8

~ 1 ~ t

LU E3 Z _J

2.7

<

n; i-.

u 2.6

e-*-

LLI O.. U3

I lOO

2.5

2.4

,

I = AE -~ ,

I

,

,

I

I

1000 10000 EGeV/c "1 primary nucLeon momentum ,

10

~

,

I

,

,

l

i

100 1000 MUON MOMENTUM EGeV/c]

Fig. 10. The spectral e x p o n e n t y in the p r o d u c t i o n s p e c t r u m I = A E momentum.

,

10000

v as a function of the m u o n

the angle range 0 ~ 0 ~< 30 ° the vertical spectra are compared with the results in fig. 9.

7. The productionspectrum The model was used to determine the parameters A and y in the spectrum - ~ for different momenta. The higher the energy, the more the approximations in the model are valid. Taking only data with E >/Ecu t and varying Ecu t the dependence from m o m e n t u m is derived as shown in fig. 10.

I = AE

We wish to thank Profs. Y. Ne'eman, Y. Yeivin, A. Godsman and A. Seidman for encouraging and helping to start the DEIS project. Also we like to acknowledge the help from Mr. and Mrs. Carstensen for data processing. We wish furthermore to express our thanks to the technical staffs at both universities. This work was supported by the Deutsche Forschungsgemeinschaft, Bonn, the D e p a r t m e n t of Physics and Astronomy of the Tel-Aviv University, and the Kiel University.

18

O.C. Allkofer et aZ / Cosmic ray muon spectra

References [1] G. Klemke, G. Bella, W.D. Dau, H. Jokisch, Y. Oren, A. Liland, K. Carstensen, and O.C. Allkofer, 17th Int. Cosmic Ray Conf., Paris, 9 (1981) 150; O.C. Allkofer, K. Carstensen, G. Bella, W.D. Dan, H. Jokisch, G. Klemke, Y. Oren, and R.C. Uhr, 17th Int. Cosmic Ray Conf., Paris, 10 (1981) 321 [2] O.C. Allkofer, G. Bella, W.D. Dau, H. Jokisch, G. Klemke, Y. Oren, and R.C. Uhr, The primary cosmic ray energy spectrum up to 50 TeV Derived from sea-level muon measurements, Phys. Rev., in print [3] O.C. Allkofer, G. Bella, W.D. Dau, H. Jokisch, G. Klemke, Y. Oren, and R.C. Uhr, The mass composition of the primary cosmic rays in the low TeV range derived from ground level muon measurements, Nuovo Cim. Lett., in print [4] O.C. Allkofer, G. Bella, W.D. Dan, H. Jokisch, G. Klemke, Y. Oren, and R.C. Uhr, Searches for narrow-angle anisotropies in the primary energy range 0.1-10 TeV, Ap. J., in print [5] O.C. Allkofer, G. Bella, W.D. Dan, H. Jokisch, G. Klemke, Y. Oren, and R.C. Uhr, Geomagnetic Influence on the Muon Charge Ratio of Cosmic Rays, J. Geophys. Res., in print [6] K. Maeda, Fortschr. Phys. 21 (1973) 113 [7] Y. Muraki et al., Phys. Rev. D28 (1983) 40; S. Matsumo et al., Phys. Rev. D29 (1984) 1; data communicated prior to pubfication by T. Kitamura and Y. Muraki [8] H. Jokisch et al., Phys. Rev. D19 (1979) 1368, KIEL-DESY measurement; T.H. Burnett et al., 13th Int. Cosmic Ray Conf., Denver, 3 (1973) 1764; San Diego measurement; R.G. Kellogg et al., Phys. Rev. D17 (1978) 98, Yale measurement [9] M.G. Thompson et al., J. Phys. G3 (1977) 97; 15th Int. Cosmic Ray Conf., Plovdiv, 6 (1977) 21; O.C. Allkofer et al., Phys. Lett. 36B (1971) 425; C.A. Ayre et al., J. Phys. G1 (1975) 584