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The Hybrid Scheme of Simulations of the Electron- photon and
Electron-hadron Cascades In a Dense Medium at Ultra-high
EnergiesL.G. Dedenko M.V. Lomonosov Moscow State University, 119992
Moscow, Russia
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ContentIntroductionHybrid multilevel schemeThe 5-level scheme
for the atmosphereExamplesConclusion
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GOALS Simulations of cascades at ultra-high energiesAcoustical
(radio) signals productionTransport of acoustical (radio) signals
in the real matterDetections of signals
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ENERGY SCALE
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SPACE SCALE
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Transport equations for hadrons:here k=1,2,....m number of
hadron types; - number of hadrons k in bin EE+dE and depth bin
xx+dx; k(E) interaction length; Bk decay constant; Wik(E,E) energy
spectra of hadrons of type k produced by hadrons of type i.
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The integral form: here
E0 energy of the primary particle; Pb (E,xb) boundary condition;
xb point of interaction of the primary particle.
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The decay products of neutral pions are regarded as a source
function S(E,x) of gamma quanta which give origins of
electron-photon cascades in the atmosphere:
Here a number of neutral pions decayed at depth x+ dx with
energies E+dE
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The basic cascade equations for electrons and photons can be
written as follows:
where (E,t), P(E,t) the energy spectra of photons and electrons
at the depth t; the ionization losses; e, the absorption
coefficients; Wb, Wp the bremsstrahlung and the pair production
cross-sections; Se, S the source terms for electrons and
photons.
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The integral form:
where
At last the solution of equations can be found by the method of
subsequent approximations. It is possible to take into account the
Compton effect and other physical processes.
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Source functions for low energy electrons and gamma quanta
x=min(E0;E/)
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For the various energies Emin Ei Eth (Emin=1 MeV, Eth=10
GeV)
and starting points of cascades0XkX0 (X0=1020 gcm-2)
simulations of ~ 2108 cascades in the atmosphere with help of
CORSIKA code and responses (signals) of the scintillator detectors
using GEANT 4 code SIGN(Rj,Ei,Xk)SIGN(Rj,Ei,Xk)10mRj2000mhave been
calculated
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SIGNAL ESTIMATION
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Responses of scintillator detectors at distance Rj from the
shower core (signals S(Rj))
Eth=10 GeVEmin=1 MeV
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ENERGY DEPOSITION
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POSITIVE CHARGE (GEANT4)
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NEGATIVE CHARGE (GEANT4)
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FOR HADRON CASCADESFLUCTUATIONS ARE OF IMPORTANCE
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CHARGE EXCESS (GEANT4)
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THIS FUNCTIONS SHOULD BE ESTIMATED WITH THE GEANT4 CODE WITH
STATISTICS OF 10**6
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FOR E=10**12 GEV NEARLY10**12 PARTICLES SHOULD BETAKEN INTO
ACCOUNT
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FOR ELECTRON-PHOTON CASCADES FLUCTUATIONS ARE VERY IMPORTANT DUE
TO THE LPM-EFFECT
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EXAMPLESor
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The Poisson formulae
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Energy deposition Q=dE/dV in water
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Energy deposition in water
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Energy deposition in water
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Energy deposition in water
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ENERGY DEPOSITION IN WATER
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ENERGY DEPOSITION IN WATER
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ENERGY DEPOSITION IN WATER
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ENERGY DEPOSITION IN WATER
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ENERGY DEPOSITION IN WATER
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Charge excess
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Lateral distributions of gammas, electrons and positrons
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ENERGY DEPOSITION in detector
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Energy distributions of gammas, electrons, positrons
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Ratio of a signal to a charge particle density
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el_ed.jpg
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ga_ed.jpg
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pos_ed.jpg
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ConclusionThe hybrid multilevel scheme has been suggested to
estimate acoustical (radio) signals produced by e and eh cascades
in dense medium.
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AcknowledgementsWe thank G.T. Zatsepin for useful discussions,
the RFFI (grant 03-02-16290), INTAS (grant 03-51-5112) and
LSS-1782.2003.2 for financial support.
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Number of muons in a group with hk(xk) and Ei :
here P(E,x) from equations for hadrons; D(E,E) decay function;
limits Emin(E), Emax(E); W(E,Ethr,x,x0) probability to survive.
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Transverse impulse distribution:
here p0=0.2 /.
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The angle :
here hk= hk(xk) production height for hadrons.
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Direction of muon velocity is defined by directional
cosines:
All muons are defined in groups with bins of energy EiEi+E;
angles jj+j, m m+ m and height production hk hk +hk. The average
values have been used: , , and . Number of muons and were regarded
as some weights.
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The relativistic equation:
here m muon mass; e charge; lorentz factor; t time; geomagnetic
field.
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The explicit 2-d order scheme:
here ; Ethr , E threshold energy and muon energy.
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Ratio with to without magnetic field
