the hybrid scheme of simulations of the electron- photon and electron-hadron cascades in a dense...
DESCRIPTION
GOALS Simulations of cascades at ultra-high energies Acoustical (radio) signals production Transport of acoustical (radio) signals in the real matter Detections of signalsTRANSCRIPT
The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies
L.G. DedenkoM.V. Lomonosov Moscow State University,119992 Moscow, Russia
Content
Introduction Hybrid multilevel scheme The 5-level scheme for the atmosphere Examples Conclusion
GOALS
Simulations of cascades at ultra-high energies
Acoustical (radio) signals production Transport of acoustical (radio) signals in
the real matter Detections 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 E÷E+dE and depth bin x÷x+dx; λk(E) – interaction length; Bk – decay constant; Wik(E′,E) – energy spectra of hadrons of type k produced by hadrons of type i.
),(/),(),(
)/(),()(/),(),(
1
EEEWxEPEd
xExEPBExEPxxEP
iik
m
ii
kkkkk
dEdxxEPk ),(
The integral form:
here
E0 – energy of the primary particle; Pb (E,xb) – boundary condition; xb – point of interaction of the primary particle.
),,())/ln()/()(/)(exp(
))/ln()/()(/)(exp(),(),(
EfxEBExd
xxEBExxxEPxEPx
x
bbbbk
b
0
)(/),(),(),(1
E
Eiiki
m
i
EEEWEPEdEf
The decay products of neutral pions are regarded as a source function Sγ(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΄
.0),(
/)),(2),(0
0
xES
EEdExEPxES
e
E
E
EdxEP ),(0
The basic cascade equations for electrons and photons can be written as follows:
where Г(E,t), P(E,t) – the energy spectra of photons and electrons at the depth t; β – the ionization losses; μe, μγ – the absorption coefficients; Wb, Wp – the bremsstrahlung and the pair production cross-sections; Se, Sγ – the source terms for electrons and photons.
EdГWEdPWSEPPtP pbee //
'/ dEPWStГ b
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.
)])((exp(),(),( 00 ttEtEГtEГ
,)],(),(),()][)(([exp0
EEWEPEdEStEd b
t
t
,)](,[),( EdtEEWEPA be
t
te dtEtttEPtEP
0
))]([exp(]),([),( 00
t
t
t
eeee BAtEStdttEd0
]]),([[)]([exp(
EdtEEWEГB pe )](,[),(
Source functions for low energy electrons and gamma quanta
x=min(E0;E/ε)
)),(),(),(),((),(
),(),(),(
0
EEWtEEEWtEPEdtES
EEWtEPEdtES
p
E
Ebe
x
Eb
For the various energies Emin≤ Ei ≤ Eth (Emin=1 MeV, Eth=10 GeV)
and starting points of cascades0≤Xk≤X0 (X0=1020 g∙cm-2)
simulations of ~ 2·108 cascades in the atmosphere with help of CORSIKA code and responses (signals) of the scintillator detectors using GEANT 4 code
SIGNγ(Rj,Ei,Xk)SIGNγ(Rj,Ei,Xk)10m≤Rj≤2000m
have been calculated
SIGNAL ESTIMATION
s
th
E
E
x
e rxxEQGSrxxEQExESdxdEQ0
000 ,,,,,
Responses of scintillator detectors at distance Rj from the shower core (signals S(Rj))
Eth=10 GeV
Emin=1 MeV
)),,(),(),,(),(()(min
0
ERSIGNESERSIGNESdEdRS jee
E
Ej
x
xj
th
b
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
EXAMPLES
tQ
Ctp
cp
p
2
2
2
1
tqQ
2
2
2
2
2
1tq
Ctp
cp
p
pqCc
p
2
or
012
2
2 tp
cp
ctRS
dSRq
rtzyxp
4,,,
The Poisson formulae
00,,,`00,,,
zyxpzyxp
0t zyxqzyxp ,,0,,, ss hd
58 107107
I.C.:
It is possible at time
because
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
Conclusion
The hybrid multilevel scheme has been suggested to estimate acoustical (radio) signals produced by eγ and eh cascades in dense medium.
Acknowledgements
We 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,Eμ) – decay function; limits Emin(Eμ), Emax(Eμ); W(Eμ,Ethr,x,x0) – probability to survive.
1 1 max
min
)(
)(0 ),,(),(),,,(
k
k
i
i
x
x
E
E
EE
EEthr xEPEED
EdExxEEWdE
xdxN
here p0=0.2 ГэВ/с.
,/)/exp()( 200 pdppppdppf
Transverse impulse distribution:
here hk= hk(xk) – production height for hadrons.
,// Ecphrtg kjj
The angle α:
Direction of muon velocity is defined by directional cosines:
All muons are defined in groups with bins of energy Ei÷Ei+ΔE; angles αj÷αj+Δαj,
δm÷ δm+Δ δm and height production hk÷ hk +Δhk. The average values have been used: , , and . Number of muons and were regarded as some weights.
cossinsincoscos;sinsincoscossinsincoscossinsin;sinsinsincossincoscoscoscossin
E jm kh
N N
The relativistic equation:
here mμ – muon mass; e – charge; γ – lorentz factor; t – time; – geomagnetic field.
,BVedtVdm
B
The explicit 2-d order scheme:
here ; Ethr , E – threshold energy and muon energy.
);5.0()(2/1ty
nzz
ny
nnx
nx hBVBVCHEVV )5.0(2/1
tnx
nn hVxx
);5.0()(2/1tz
nxx
nz
nny
ny hBVBVCHEVV
);5.0()(2/1tx
nyy
nx
nnz
nz hBVBVCHEVV
;)( 2/12/12/11ty
nzz
ny
nnx
nx hBVBVCHEVV
;)( 2/12/12/11tz
nxx
nz
nny
ny hBVBVCHEVV
;)( 2/12/12/11tx
nyy
nx
nnz
nz hBVBVCHEVV
)5.0(2/1t
ny
nn hVyy
)5.0(2/1t
nz
nn hVzz
tnx
nn hVxx 2/11
tny
nn hVyy 2/11
,2/11t
nz
nn hVzz
)/( EEeCHE thr
Ratio with to without magnetic field