Calculation of muonic fields around large high energy proton accelerators

Nuclear Instruments and Methods 217 (1983) 419-424 North-Holland Publishing Company

CALCULATION OF MUONIC ACCELERATORS M.A. MASLOV, N.V. MOKHOV

FIELDS AROUND

419

LARGE HIGH ENERGY PROTON

a n d A.V. U Z U N I A N

Institute for High Energy Physics, Serpukhov, USSR Received 20 December 1982 and in revised form 14 April 1983

The transport of muons and hadrons, arising from beam losses, in the magnetic structure of a proton accelerator is investigated. A method of statistical simulation of this transport has been developed and incorporated into the RING program. Muonic fields in the UNK and Tevatron soil shield are calculated for some model cases. The essential role of magnet elements and of prompt muons in the production of the muon field is emphasized, The radiation shield outside large accelerator rings is shown to be defined completely by muons.

1. Introduction The problem of calculating muon fields around accelerator rings is an essential one for the design of the new generation of proton accelerators [ i,2]. Muons being the most penetrating particles essentially determine the size of the shield. M u o n production is mainly caused by proton losses on structural elements of the ring accelerator. The muon flux distribution in the side shield of proton accelerator tunnels has been examined in refs. 3 and 4. The results obtained are very rough. Prompt m u o n production and the influence of the magnetic structure of the ring were not taken into account. Muon distributions inside and above the accelerator ring were not calculated. All these factors are taken into consideration in the present work. The R I N G F O R T R A N program for calculating the production and transport of muons and hadrons in the real structure of accelerators has been created. Muon fields in the U N K and Tevatron soil shield are calculated for some model cases.

2. Calculational method The algorithm us%l in the Monte Carlo simulation of muon production and transport allows the selection of only relevant secondary particles in hadron-nucleus interactions, each secondary being given a statistical weight based on the mathematical expectation for its production probability. The scheme of muon production is the same as that realized in the M U T R A N program [5]. Six hadrons (p, n, ~r +, K+) with statistical weights, and their mathematical expectations coinciding with the average multi0167-5087/83/0000-0000/$03.00 © 1983 North-Holland

plicity, are generated in each stage of proton-nucleus interaction [6]. The trajectory tree is examined up to the third generation since hadrons of higher generations practically do not contribute to the muon field [7]. Muon production is realized through a compulsory decay of all ¢r and K-mesons in the path up to their next nuclear interaction but with a correspondingly reduced statistical weight. In a d d i t i o n p r o m p t muon production in hadron-nucleus interactions is included in the present work. We have used a parametrization for the relation R = (/x++ #-)/(Tr+ + ¢r-) in the form [8]

R=A(S)B(XF)×IO-', with A(S)=

/0.5

v@-~< 15.44 GeV,

- 1.91 + 0.88 In v@-, f l --2XF,

e(xp

= \ (I - x v ) / 3 ,

v ~ > 15.44 GeV,

0~XF~<0.4 ,

x v > 0.4,

where S is the centre-of-mass energy squared and x F is the Feynman scaling variable. Hadron and muon transport is realized with a step method. The following requirements should be fulfilled when choosing the step length h i) A E ( I ) < < E is the condition for the small angle approximation. AE(I) is the energy loss in the step l, E is the current energy. 2) The variation of the magnetic field B must be small over the step l. 3) The value of l must not be more than the hadron mean free path for nuclear interaction or not more than the muon mean free path for bremsstrahlung or photo-nuclear interaction.

420

M.A. Maslov et al. / Calculatu_m

4) Ar +z t, where Ar is the transverse

displacement of a particle due to multiple Coulomb scattering (MCS) and the magnetic field, t is the smallest lateral dimension of an element of the magnetic structure. Let Ar = 1Acp= const., where AT = l/R is the change of particle direction caused by the magnetic field on the step I. The radius of the trajectory curvature R is proportional to the particle energy E, then I = C@. The value of C is determined by conditions (l)-(4). Then the convergence of results is provided within 10%. A change of particle coordinates and angles is presented in the form of two components: one of them is the MCS result, the second is a result of trajectory integration in the magnetic field [9]. The magnetic field distribution is presented in the following form: dipole: r(CR,;

B,=B,,B,=O, B, = B,(l

B(r,+)=

- t) - 0.45B,

sin +,

B,=O,4t&cos+,

I

1

cos $J,

0<+<2n;

quadrupole:

B,= B(x,y)

+Gy,B,=

k-

$Gx,r$R,;

GR:Y r4

=

r>R,;

* By=

!=I

where wkr is the statistical weight of muons having a step I,, in the volume V,. The threshold energy is 3 GeV for hadrons and 0.1 GeV for muons. Calculations have revealed that the contribution of muons with E < 0.1 GeV to $(V) is less than - 5%. The algorithm described is realized in the RING FORTRAN program. The computational time is - 4-6 h on the computer BESM-6 for protons of energy l-3 TeV. The statistical errors of the calculation are less than lo-30% in this case.

3. Results

R,
+,=+-W((P-a),

(B,=

N

I

where

t=(r-R,)/(R,-R,),

The muon energy losses are simulated as continuous processes for the direct production of e’e- pairs and for inelastic collisions with atomic electrons, but as discrete processes for bremstrahlung and nuclear interactions. The quantity of interest is the muon fluence which is calculated from a track length estimation:

$ R,;

sin $I,

B, = -0,4B, B,, = 0,4B,

R-,gr

of muonicfields

k r4

and G is the field gradient in where r= Jz’ x +y quadrupole aperture, B, = const. The regions are fined in fig. 1: the vacuum chamber wall; R, < r < R, R, < r < R, - the magnet winding and bandage; R 4 < r < R 5 - the iron screen. For quadrupoles we do not take into account thickness of the superconducting windings and the fluence of the iron screen on the field distribution

the de-

The results of the calculation are presented in the form of muon fluence distributions in the soil shield of accelerator rings. The soil density is 1.8 g. cme3. Fig. 1 presents the calculation geometry and cross-section of the model magnets which are used in the present work. The values of magnetic induction B, and the field gradient in the quadrupole aperture G and the geometrical dimensions used in our calculations are given in table 1. All dimensions are given in centimeters. E, is the proton energy in TeV. Figs. 2a,b show projections of muon trajectories on the YZ plane in the magnetic structure of the UNK (2nd stage). The trajectory distortion due to the magnetic field results in a considerable increase of particle path in the magnet material. The muon may lose up to 30% of the initial energy in the magnets. Let us consider the case of uniform proton losses along the accelerator ring perimeter. The calculations show that changing the angle of incidence in the interval

the in*. MAGNET

SYSTEM

* Results of calculations using a real field map of the Tevatron magnet structure [2] have shown that the accuracy of the model description

B( r,cp) is not worse than

- 20% for muon

fluence in the ring plane and - 100% for lx/> 1 m This is not so bad for practical applications.

(fig.

1).

Fig. 1. Approximations to the geometry used in the calculations. (a) Ring tunnel; (b) magnet cross sections.

M.A. Maslov et al. / Calculation of muonic fields

421

Table 1 Parameters used in the calculations. Accelerator

B0(T )

G(T/cm)

Ep(TeV)

Ra a)

RT

Oa t

RI

R2

R3

R4

R5

UNK, 1st stage

0.667

0.107

0.4

2.62+5

250

150

3.5

3.65

6.7

10

20

UNK, 2nd stage

5

0.8

3

2.62+5

150

150

3.5

3.65

4.5

9

17

Tevatron

4.4

1

1

1.0 + 5

152

95

3.1

3.25

3.85

9.3

15

a) 2.62+5 = 2.62× 105.

1 0 - 5 < O < 10 -3 rad essentially does not influence the m u o n fluence distribution. Proceeding from this the case of the most p r o b a b l e losses has been chosen [11], i.e. p r o t o n losses are considered in the b e a m circulation plane; the angle of incidence is equal to 0 = 10 -3 rad. T h e radial distributions of the m u o n fluence are averaged over the range - 1 m < x < 1 m (fig. 1). Fig. 3 shows the m u o n fluence radial distributions calculated with (curve 1) a n d without (curve 3) magnet material. Results o b t a i n e d using the algorithm of ref. 3 are also presented in fig. 3 (curve 2). Fig. 4 shows the m u o n fluence distributions in soil outside a n d inside the U N K ring (2nd stage). It also

shows m u o n c o n t r i b u t i o n s from various sources: p r o m p t muons, m u o n s from ~r ± a n d K ±-decays. The analysis of the results leads to the conclusion that the m u o n field in the shield inside a n d above the accelerator ring is defined by m u o n s from ~r ±-decay, b u t in the shield outside the ring at large distances it is d o m i n a t e d by p r o m p t m u o n s of the first generation. Table 2 gives the average n u m b e r ~ N ) a n d average energy ( E ) of m u o n s at the p r o d u c t i o n point for five tunnel regions of the U N K 2nd stage. The effective ~r (K)-meson m e a n free path for decay is - 60 cm in this case. It is clear from table 2 that the d o m i n a n t role of p r o m p t m u o n s at large distances outside the ring is a consequence of its high energy. Otherwise the absolute p r o m p t m u o n s yield is essentially less t h a n the total m u o n yield for rr a n d K-mesons decays. Regions in table 2 are: (1) 0 < r ~ < R l , ( 2 ) R l < r ~ < R 2 , ( 3 ) R 2 < r ~ <

-IS -0 I

-5

f

I

.5

~o !TERS

5

d~

i-~ ~04 cb

Z O

15

O +

o-

I~C'~ r.cV

-15

-0

6~ Z O

I

-5

I

I

I

i

I00

200

300

>~ 0 5 o ~5

Fig. 2. Projections on the horizontal Y Z plane of muon trajectories in the magnet structure of the UNK 2nd stage. F, D, are focusing and defocusing quadrupoles, respectively. B is a dipole magnet. (a) Muon energy E = 0.5 TeV; (b) E = 1.5 TeV.

dour,

METERS

Fig. 3. Radial distribution of muon fluence in the soil outside the UNK ring for Ep = 3 TeV. (1) Our calculation for the actual magnet structure with prompt muons. (2,3) Calculations for an infinite line source of proton losses, placed on the axis of the ring tunnel and when the magnet structure is absent. (2) Calculation following the algorithm from ref. 3; (3) our calculations without prompt muons.

422

M.A. Maslov et al. / Calculation of muonic fields i05

I

I

I

40 J._.

I

I ~ - ~ 7"ev_l

I

I

!

O4 -:2 -1

Z

"",,"

/o*

-

,J o_ Q -~.

0

,3 / J

Ioo

oz

%

l,s'

16i I

I

5

0

6

I

9

15

12

08

011n,METERS

ih

I

I

I

I

.,.,,.r-# o

b

i ~,oo

i 2ao

i 3<9,o

/ix 1 ~au

I soo

Fig. 5. Longitudinal distributions of muon fluence due to 3 TeV protons stopping in iron absorber. (1) Actual calculation; (2) calculation [5] without prompt muons. 1 - Statistical errors of the calculation.

d~

z O

-[l//

L

Q

. , ~

. . . . . .

CD

L.J I...........l

Z O Z;

-I!

i0

I

L-.__.....

0

I00

200

t

500

400

i00

CI~,METERS

Fig. 4. Radial distribution of muon fluence in soil inside (a) and outside (b) the ring due to proton losses on the vacuum chamber walls of the UNK 2nd stage. Ep = 3 TeV. --All muons; ...... muons from pion and kaon decays;..- first generation prompt muons; . . . . . muons from kaon decays only.

R3, (4)R 3 < r ~< Rs, (5) the region between magnets and tunnel wall (fig. 1) (1,3,5 - vacuum). Thus consideration of prompt muon production is important. Significant errors can arise if prompt muons are not taken into account at Ep >~ 1 TeV both in this situation and in a geometry of the " b e a m dump" type. Fig. 5 shows the longitudinal distributions of muon fluence in the iron absorber exposed to 3 TeV protons. Results with [formula (1)] and without prompt muon generation [5] are presented. The contribution of muons from K-meson decay to the total flux is conditioned by the correlation of the ~rand K-mdson spectrum in the high energy tail, decay kinematics and calculational geometry. In our case the relation N (Tr)/N~(K) is equal to ~ 2 in the point of

Table 2 Production of muons in the UNK 2nd stage. Region

( N ) ( × 10 -4)

number

1

2

3

4

5

( E ) (GeV) 1

2

3

4

5

~(~+) #(~-) ~(K +) #(K-) Prompt muons

116 60 58 28 0

13 6 9 4 8

41 22 24 10 0

58 34 38 20 0.6

45 21 24 17 0

16 12 13 9 0

16 16 10 9 86

10 9 8 7 0

6 6 5 4 16

5 3 1 2 0

424

M.A. Maslov et al. / Calculation o f muonic fields

¢0

,,2o, p.. -~

¢d

/D e

¢0 ~

10 ~z

500

/ 0 ¢°

/0"

¢0 '2

/2

4DD IJ

~u

300

% ¢,

//

>g

/O ~u

J

7'

lOO

_

_

_

¢0 ?

/0 e

/D j

/0 ~°

/0 7

/08

¢0

/0 ¢:

I, : .~-fs-: Fig. 10. Nomographs for the estimation of soil shielding outside and inside the UNK ring due to uniform (z31) and local (A10) proton losses on the vacuum chamber wall. The muon control level is 0.1 muon/(cm 2.s).

tual value of losses which could be significant. The configuration of a muon shield around a ring accelerator is determined by the sum of uniform and local losses. The nomographs given in fig. 10 allow one to estimate the dimensions of soil shielding for the U N K ring (inside and outside). In this case the muon fluence is attenuated down to ~ 0 = 0.1 muon/(cm2s).

The authors would like to express their sincere gratitude to V.N. Lebedev and S.I. Striganov for fruitful discussions. We are especially thankful to Dr. G. Stevenson for his reading of the text and his suggestions for improving the contents.

References 4. Conclusions (1) The dispersion due to magnetic fields, the attenuation by the construction material of the accelerator ring and the production of prompt muons must be taken into account in defining the configuration and dimensions of the shield. (2) The muon field in the shield inside the ring of the accelerator is defined by muons from ~r -~- decay, but that outside the ring at large distances is defined by p r o m p t muons. (3) The radiation shield outside the ring tunnels is defined completely by the need to attenuate the muon fluence to an acceptable level. These conclusions must be considered in relation to a real accelarator site, e.g. the actual density of the earth (in this work assumed to be 1.8 g/cm3), the presence of underground halls, geological inhomogeneities, etc. The results presented here can, however, be used to aid in the selection of a site having a suitable relief, in designing the halls and in solving some environmental problems.

[1] V.I. Balbekov et al., Proc. 6th Meeting on Charged particle accelerators, Dubna (1978) p. 1. [2] Report on the design of the FNAL superconducting accelerator (May, 1979). [3] A.Ya. Serov and B.S. Sichev, Proc. Radiotechnical Institute AN USSR, no. 9 (1971) p. 127. [4] A.J. Stevens and A.M. Thorndike, BNL 50540 (1976). [5] N.V. Mold0ov, G.I. Semenova and A.V. Uzunian, Preprint IHEP 79-101, Serpukhov (1979); Nucl. Instr. and Meth. 180 (1981) 469. [6] N.V. Mokhov, Preprint IHEP 76-64, Serpukhov (1976). [7] G.I. Britvich, N.V. Mokhov and A.V. Uzunian, Preprint IHEP 76-66, Serpukhov (1976). [8] J.L. Ritchie, A. Bodek and R.N. Coleman, Phyg. Rev. Lett. 44 (1980) 23. [9] M.A. Maslov, N.V. Mokhov and A.V. Uzunian, Preprint IHEP 78-153, Serpukhov (1978). [10] N.V. Mokhov, S.I. Striganov and A.V. Uzunian, Preprint IHEP 80-65, Serpukhov (1980). [11] M.A. Maslov and N.V. Mokhov, Preprint IHEP 79-135, Serpukhov (1979); Part. Accel. 11 (1980) 91.

M.A. Maslov et al. / Calculation of muonic fields ,

i

i

/

423

~,METERS

io4 _

2

T~

:[ '

o~- 102 , I I

-

,

-

i

I0 ~

d,., METERs :~i,

l ........

-I I0

i

4

i

I00

0

I

i

.300

dlmMETERS

500

Clout ~ METERS

Fig. 6. Radial fluence distributions as in fig. 4. (1) Proton losses on the UNK 2nd stage vacuum chamber walls (Ep = 3 TeV). (2) The same but without magnetic field. (3) Proton losses on the UNK 1st stage vacuum chamber walls (Ep = 0.4 TeV).

'°5

o

"

I

:

TUNNEL

Fig. 8. Muon isofluence contours in soil around the UNK ring due to uniform proton losses in the UNK second stage, Ep = 3 TeV. (1) 10 8 muon.cm ~/proton.m -I. (2) 10 _9 muon. c m - 2 / p r o t o n . m - 1 (3) 10 10 m u o n . c m - 2 / p r o t o n . m - 1.

i

IL-; ;..J 8

d0~+,~ETERS RING

,

,o°

L

I

t

q

f

I

I

-J

decay, then increases to 7 in the first meters of the shield a n d decreases to 5 - 6 at large shield thicknesses. It must b e n o t e d that the accelerator m a g n e t structure influences essentially the m u o n i c field production. T h e calculation without the magnetic field leads to a 10-15 times decrease of the m u o n fluence in the soil inside the ring (fig. 6). The radial distributions of m u o n fluence are also presented here for uniform p r o t o n losses in the rign of the U N K 1st stage. Fig. 7 shows the calculations of m u o n fluence distrib u t i o n s above the ring due to p r o t o n losses in the U N K 2 n d stage. Analysis of similar distributions at different distances from the ring gives a possibility to o b t a i n m u o n isofluence contours in the soil a r o u n d the U N K a n d T e v a t r o n rings. Figs. 8 and 9 show results for levels 10 8, 10-9, 10-10 m u o n c m - ~ / p r o t o n . m - I , respectively. In addition the case of local losses has been considered; this described the situation a r o u n d targets, collimators, electrostatic and magnetic septa. The shield thickness along the m u o n " t o r c h " depends on the ac-

h, METER8 10

,o £ :E to~ fOj -

[

,o°

.

,- I

I

6

h, METRES Fig. 7. The distribution of muon fluence in the direction perpendicular to the ring plane due to proton losses in the UNK 2nd stage (Ep = 3 TeV): d~n = 0-1.5 m o dout=0-10mzx - A ,

o,

3-4.5 m O 50-100mA--*.

O;

//

, ~

5o

I00

150

2bo

RING TUNNEL Fig. 9. Muon isofluence contours in soil around the Tevatron ring due to uniform proton losses, Ep = l TeV. (1) 10-s muon. c m - 2 / p r o t o n - m -1. (2) 10 -9 muon.cm 2/proton.m-l. (3) 1 0 - l0 muon- c m - 2/proton. m i.