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

ContentIntroductionHybrid multilevel schemeThe 5-level scheme for the atmosphereExamplesConclusion

GOALS Simulations of cascades at ultra-high energiesAcoustical (radio) signals productionTransport of acoustical (radio) signals in the real matterDetections of signals

ENERGY SCALE

SPACE SCALE

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.

The integral form: here

E0 energy of the primary particle; Pb (E,xb) boundary condition; xb point of interaction of the primary particle.

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

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.

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.

Source functions for low energy electrons and gamma quanta

x=min(E0;E/)

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

SIGNAL ESTIMATION

Responses of scintillator detectors at distance Rj from the shower core (signals S(Rj))

Eth=10 GeVEmin=1 MeV

ENERGY DEPOSITION

POSITIVE CHARGE (GEANT4)

NEGATIVE CHARGE (GEANT4)

FOR HADRON CASCADESFLUCTUATIONS ARE OF IMPORTANCE

CHARGE EXCESS (GEANT4)

THIS FUNCTIONS SHOULD BE ESTIMATED WITH THE GEANT4 CODE WITH STATISTICS OF 10**6

FOR E=10**12 GEV NEARLY10**12 PARTICLES SHOULD BETAKEN INTO ACCOUNT

FOR ELECTRON-PHOTON CASCADES FLUCTUATIONS ARE VERY IMPORTANT DUE TO THE LPM-EFFECT

EXAMPLESor

The Poisson formulae

Energy deposition Q=dE/dV in water

Energy deposition in water

Energy deposition in water

Energy deposition in water

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

ENERGY DEPOSITION IN WATER

Charge excess

Lateral distributions of gammas, electrons and positrons

ENERGY DEPOSITION in detector

Energy distributions of gammas, electrons, positrons

Ratio of a signal to a charge particle density

el_ed.jpg

ga_ed.jpg

pos_ed.jpg

ConclusionThe hybrid multilevel scheme has been suggested to estimate acoustical (radio) signals produced by e and eh cascades in dense medium.

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.

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.

Transverse impulse distribution:

here p0=0.2 /.

The angle :

here hk= hk(xk) production height for hadrons.

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.

The relativistic equation:

here m muon mass; e charge; lorentz factor; t time; geomagnetic field.

The explicit 2-d order scheme:

here ; Ethr , E threshold energy and muon energy.

Ratio with to without magnetic field