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1) Mando and Sona (1953); 2) Zatsepin and Mikhalichi (1965); 3) Nishimura (1964); 4) Bollinger (1951); 5) Hayman et al. (1962, 1963); 6) Miyake et al. (1964). For full references see end of section.

1) Mando and Sona (1953); 2) Zatsepin and Mikhalichi (1965); 3) Nishimura (1964); 4) Bollinger (1951); 5) Hayman et al. (1962, 1963); 6) Miyake et al. (1964). For full references see end of section.

4.2.5 Average Depth-Intensity Relation of Muons

By expressing the high energy portion of the integral energy spectrum of muons at sea level as

Figure 4.8: Fluctuation correction factor in sea water for different exponents 7 of the integral energy spectrum of muons at sea level (Inazawa and Kobayakawa, 1985).

Figure 4.8: Fluctuation correction factor in sea water for different exponents 7 of the integral energy spectrum of muons at sea level (Inazawa and Kobayakawa, 1985).

where A is a constant and 7 the spectral exponent, and using the range-energy relation, eq. 4.12, the vertical intensity I(X) at depth X can be written as

Here a and b are the energy loss parameters defined earlier. For ebX » 1, and using

the following depth-intensity relation is obtained:

If the intensity is measured in an inclined direction under a zenith angle 9 > 0° and a flat surface topography at a vertical depth X, the depth-intensity relation must be modified by the factor sec(0) because of the muon enhancement in the atmosphere, outlined in Section 3.6. Moreover, another factor of sec(0) must be introduced in the exponent because of the slant depth, X3, which is larger than X for an inclined trajectory underground. Thus

For small depths in the vertical direction, a series expansion of the previous equation (eq. 4.19) yields

where A' is a constant. This equation shows that the shape of the muon spectrum is reproduced. For large depths one gets

Thus the intensity decreases exponentially with depth. The principal energy loss process at small depths (and low energies) is ionization, represented by the factor a, whereas at large depths (and high energies) the electromagnetic processes, accounted for by b, are more important. Energy losses due to ionization exhibit only small fluctuations because the loss is due to many single processes. In contrast, as pointed out above, the bremsstrahlung mechanism which is dominating at high energies, is a single process in which a large fraction of the energy can get lost. Consequently large fluctuations are produced in the depth-intensity relation at larger depths.

The consequence resulting from the difference between the screening functions of Petrukhin and Shestakov (1968) and Rozental (1968) for the theoretical depth-intensity relation, as pointed out before, are shown in Fig. 4.9 (Lipari and Stanev, 1993).

Experimental data of muons underground and underwater are presented in Sections 4.3 and 4.4.

4.2.6 Indirect Determination of the Energy Spectrum Underground and Average Energy

The vertical muon intensity at depth X, I(X, 0°), is related to the muon differential energy spectrum at the surface,

Figure 4.9: Depth-intensity relation calculated with the Petrukhin and Shes-takov (1968) (curve 1, solid) and the Rozental (1968) (curve 2, dashed) screening functions in standard rock using an atmospheric differential muon energy spectrum of the form E~3,7 (after Lipari and Stanev, 1993).

Figure 4.9: Depth-intensity relation calculated with the Petrukhin and Shes-takov (1968) (curve 1, solid) and the Rozental (1968) (curve 2, dashed) screening functions in standard rock using an atmospheric differential muon energy spectrum of the form E~3,7 (after Lipari and Stanev, 1993).

by the relation

JEth where 7 is the spectral index of the integral muon energy spectrum at the surface, P(E,X) the survival probability that a muon of energy E reaches depth X, and Eth is the energy threshold corresponding to P(Eth,X) = 0. The energy spectrum is then calculated with the help of a minimizing procedure.

If we are neglecting fluctuations that are embedded in the survival probability, P(E,X), we can easily derive the vertical energy spectrum at depth X using eqs. 4.1 and 4.19, which yields j(E,X) = Be■ (E + - e-6X))_(7+1) . (4.25)

In this equation the first exponent describes the attenuation of the high energy muon flux, whereas the expression in parenthesis determines the shape of the local energy spectrum underground.

The average energy of the muons at depth X is given by the relation

Some important facts that result from the mathematical treatment above should be remembered (Castagnoli et al., 1995):

1. The energy spectrum underground is flat at energies E « Eave.

2. The slope of the spectrum at the surface is reproduced for E » Eave.

3. At shallow depths where bX « 1 the average muon energy is nearly proportional to X.

4. At great depths where bX > 1 (X > 2 • 103 hgcm-2) the spectral slope varies slowly with depth.

5. The average energy approaches the asymptotic limit a/[b(7 — 1)].

Rhode (1993) computed the muon energy spectra for different levels underground using the energy loss dependencies shown earlier in Fig. 4.3. The results of his calculations are illustrated in Fig. 4.10.

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