Calculations on the Cascade Theory of Showers

Chakrabarty, S. K.; Gupta, Manash Ranjan

doi:10.1103/physrev.101.813

Cited by 7 publications

(5 citation statements)

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“…Однако им не удалось получить физически новых результатов. Физически новые результаты были получены в работе 31 , к изложению основных резуль-татов которой мы сейчас переходим.…”

Section: ζ(ζ-1)(ζ-2)(3ζ-7) 7 ζ(ζ-4 ο (ζ) =unclassified

“…Далее, как и в работе 33 , проводя преобразова-ние Лапласа по t уравнения (2.25), авторы получают: 31 , нетрудно написать решение разностного уравнения в форме…”

Section: метод чакрабарти и гун таunclassified

“…В пре-дельном случае (s + ρ) -> 0, что соответствует Ε -> 0, получаемые выра-жения совпадают с соответствующими формулами Снайдера. Когда 31 . Кривые 1 и 2 достаточно близки между собой, исключая об-ласть £^-0, 1β, где различие достигает 20%.…”

Section: метод чакрабарти и гун таunclassified

“…На рис. 5 приведена зависимость числа частиц с энергией больше Ε от глубины и ливне, вызванном первичным электроном энергии г/,, = 6 по данным 31 (кривые 1) и по данным работы 2 (кривые 2). Из рисунка видно, что кривые 1 и 2 очень хорошо совпадают друг с другом.…”

Section: метод чакрабарти и гун таunclassified

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Cascade theory of showers

Belen'kii¹,

Ivanenko²

1959

Успехи физических наук

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Section: ζ(ζ-1)(ζ-2)(3ζ-7) 7 ζ(ζ-4 ο (ζ) =unclassified

Section: метод чакрабарти и гун таunclassified

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Cascade theory of showers

Belen'kii¹,

Ivanenko²

1959

Успехи физических наук

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“…Previously, the solution of the diffusion equation under Approximation B was given by an analytic continuation of the infinite series solution obtained by means of a perturbation technique. 3),4) As a result, the solution for arbitrary values of the energy has a double complex integral, 3), 6) and it is difficult to evaluate the numerical values accurately.…”

Section: §1 Introductionmentioning

confidence: 99%

An Alternative Method for Solving the Diffusion Equation under Approximation B in Electron-Photon Cascade Theory

Nii

1998

Progress of Theoretical Physics

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A new analytical formulation is proposed to solve the diffusion equation under Approximation B in electron-photon cascade theory. The Suzuki-Trotter formula, analytical continuation of the hypergeometric function, and product integration are introduced. By using these methods the usual series solutions are obtained, and summation of the infinite series for arbitrary values of the energy E is performed by using the method of Prony's interpolation. As E~ 0, the infinite sum for the electron component turns out to be the function of Kl(8, -8) used in the usual cascade theory, and a logarithmic divergence arises for the photon component. Use of Prony's method makes it possible to derive the energy spectra as well as the track length distributions and the transition curves. Our numerical results agree well with previous authors' as expected. Our analytical approach provides a general framework for solving other diffusion equations containing non-commutative operators in different contexts. at University of North Dakota on May 29, 2015 http://ptp.oxfordjournals.org/ Downloaded from 350 N. Nii agreement with previous results within the limits of the approximation ( § §3 and 4). The incomplete part of our previous formalism has thus been totally completed.Applying the method of Prony's interpolation 11) to our infinite series solution, we can obtain a simple expression of the cascade function, as shown in (1·1), in which the electron energy E is meaningful from 0 to Eo -€t:As a result, it becomes possible to easily obtain with high accuracy the value of the integral energy spectra for any value of energy E ( §5). Our numerical results on the cascade functions, i.e., the electron transition curves, JI(Eo, 0, t) and JI(Wo, 0, t), and the electron track length distributions, Zrr(E o , E) and Zrr(W o , E), are compared P = (-t,' --=:0) (radiation operator), Q = (€(8~8E) ~) (ionization loss operator), and, P and Q do not commute (PQ -QP -=I 0).

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Theory of Cascade Showers

Nishimura¹

1967

Kosmische Strahlung II / Cosmic Rays II

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Calculations on the Cascade Theory of Showers

Cited by 7 publications

References 9 publications

Cascade theory of showers

Cascade theory of showers

An Alternative Method for Solving the Diffusion Equation under Approximation B in Electron-Photon Cascade Theory

Theory of Cascade Showers

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