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![Page 1: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/1.jpg)
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
![Page 2: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/2.jpg)
Content
Introduction Hybrid multilevel scheme The 5-level scheme for the atmosphere Examples Conclusion
![Page 3: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/3.jpg)
GOALS
Simulations of cascades at ultra-high energies
Acoustical (radio) signals production Transport of acoustical (radio) signals in
the real matter Detections of signals
![Page 4: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/4.jpg)
ENERGY SCALE
![Page 5: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/5.jpg)
SPACE SCALE
![Page 6: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/6.jpg)
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 ),(
![Page 7: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/7.jpg)
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
![Page 8: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/8.jpg)
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
![Page 9: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/9.jpg)
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
![Page 10: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/10.jpg)
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 )](,[),(
![Page 11: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/11.jpg)
Source functions for low energy electrons and gamma quanta
x=min(E0;E/ε)
)),(),(),(),((),(
),(),(),(
0
EEWtEEEWtEPEdtES
EEWtEPEdtES
p
E
Ebe
x
Eb
![Page 12: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/12.jpg)
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
![Page 13: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/13.jpg)
![Page 14: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/14.jpg)
SIGNAL ESTIMATION
s
th
E
E
x
e rxxEQGSrxxEQExESdxdEQ0
000 ,,,,,
![Page 15: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/15.jpg)
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
![Page 16: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/16.jpg)
ENERGY DEPOSITION
![Page 17: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/17.jpg)
POSITIVE CHARGE (GEANT4)
![Page 18: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/18.jpg)
NEGATIVE CHARGE (GEANT4)
![Page 19: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/19.jpg)
FOR HADRON CASCADESFLUCTUATIONS ARE OF IMPORTANCE
![Page 20: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/20.jpg)
CHARGE EXCESS (GEANT4)
![Page 21: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/21.jpg)
THIS FUNCTIONS SHOULD BE ESTIMATED WITH THE GEANT4 CODE WITH STATISTICS OF 10**6
![Page 22: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/22.jpg)
FOR E=10**12 GEV NEARLY10**12 PARTICLES SHOULD BETAKEN INTO ACCOUNT
![Page 23: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/23.jpg)
FOR ELECTRON-PHOTON CASCADES FLUCTUATIONS ARE VERY IMPORTANT DUE TO THE LPM-EFFECT
![Page 24: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/24.jpg)
EXAMPLES
tQ
Ctp
cp
p
2
2
2
1
tqQ
2
2
2
2
2
1tq
Ctp
cp
p
pqCc
p
2
or
![Page 25: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/25.jpg)
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
![Page 26: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/26.jpg)
Energy deposition Q=dE/dV in water
![Page 27: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/27.jpg)
Energy deposition in water
![Page 28: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/28.jpg)
Energy deposition in water
![Page 29: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/29.jpg)
Energy deposition in water
![Page 30: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/30.jpg)
ENERGY DEPOSITION IN WATER
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ENERGY DEPOSITION IN WATER
![Page 32: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/32.jpg)
ENERGY DEPOSITION IN WATER
![Page 33: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/33.jpg)
ENERGY DEPOSITION IN WATER
![Page 34: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/34.jpg)
ENERGY DEPOSITION IN WATER
![Page 35: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/35.jpg)
![Page 36: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/36.jpg)
Charge excess
![Page 37: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/37.jpg)
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Lateral distributions of gammas, electrons and positrons
![Page 39: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/39.jpg)
ENERGY DEPOSITION in detector
![Page 40: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/40.jpg)
Energy distributions of gammas, electrons, positrons
![Page 41: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/41.jpg)
Ratio of a signal to a charge particle density
![Page 42: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/42.jpg)
![Page 43: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/43.jpg)
el_ed.jpg
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ga_ed.jpg
![Page 45: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/45.jpg)
pos_ed.jpg
![Page 46: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/46.jpg)
Conclusion
The hybrid multilevel scheme has been suggested to estimate acoustical (radio) signals produced by eγ and eh cascades in dense medium.
![Page 47: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/47.jpg)
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.
![Page 48: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/48.jpg)
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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,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
![Page 51: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/51.jpg)
here p0=0.2 ГэВ/с.
,/)/exp()( 200 pdppppdppf
Transverse impulse distribution:
![Page 52: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/52.jpg)
here hk= hk(xk) – production height for hadrons.
,// Ecphrtg kjj
The angle α:
![Page 53: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/53.jpg)
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
![Page 54: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/54.jpg)
The relativistic equation:
here mμ – muon mass; e – charge; γ – lorentz factor; t – time; – geomagnetic field.
,BVedtVdm
B
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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
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![Page 57: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov.](https://reader033.fdocuments.in/reader033/viewer/2022052607/5a4d1b497f8b9ab0599a4e30/html5/thumbnails/57.jpg)
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Ratio with to without magnetic field