Further Calculations on the Cascade Theory

Bhabha, H. J.; Chakrabarty, S. K.

doi:10.1103/physrev.74.1352

Cited by 44 publications

(3 citation statements)

References 7 publications

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“…7 The work of Bhabha and Heitler was carried out without including ionization loss and is thus limited to the high energy portion of the spectrum. The calculations of Carlson and Oppenheimer did include ionization loss, but used simplified asymptotic forms for the high energy cross sections.…”

Section: Introductionmentioning

confidence: 99%

The Uranium 234 Content of Natural Uranium and the Specific Alpha-Activities of the Isotopes

Kienberger¹

1949

Phys. Rev.

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The uranium 234 content of natural uranium was determined, by a combination of alpha-counting and mass spectrometer techniques, to be 0.005481 ±0.000012 weight percent, the precision being expressed on the basis of 95 percent probability. This corresponds to a half-life of (2.522±0.008) X10 5 years. Considering possibilities of bias, these figures are believed to lie within one percent of the true values. In conjunction with this determination, the half-lives of the other natural uranium isotopes were also measured. P REVIOUS determinations of the weight percent of uranium 234 in natural or "normal" uranium have all involved measurement by the mass spectrometer. Nier first used such an instrument to determine the relative abundance of uranium 238 and uranium 234 in normal uranium. 1 Chamberlain, Williams, and Yuster used two related methods. 2 The first method was identical in principle to Nier's method in that they used the mass spectrometer for a direct measurement of normal uranium. In the second method the specific activity of uranium 234 was determined from the total alpha-activity and relative isotopic abundance of enriched uranium samples, assuming the specific activities of uranium 238 and 235 from the data of others. From their data on the alpha-activity of enriched samples, together with a specific activity figure for uranium 238, 3 it is possible to deduce a value for the content of uranium 234 in natural uranium by computation of the uranium 234 alpha-emission enrichment factor.By using pure samples of uranium 234 grown from UX h Knight, Goldin, P. A. Macklin, and R. L. Macklin of this laboratory have determined the specific activity of uranium 234 from which they calculated, using the specific activity of uranium 238 given in this report, the weight percent of uranium 234 in normal uranium. Their normal abundance figures of 0.005805±0.00008 weight percent is significantly different from that reported in this paper, but the cause of the difference is not yet known.This report is concerned with a re-determination of the weight percent of uranium 234 in normal uranium by a method similar to the second method used by Chamberlain, Williams, and Yuster, and also an independent re-determination of the specific alpha-activities of the three isotopes. The uranium 234 content of uranium highly enriched in the 234 isotope can be determined with greater accuracy by the use of the mass spectrometer than that achieved in measurements on normal uranium, since the ratio of the uranium 234 to the sum of uranium 235 and 238 in greatly enriched uranium has been increased many times. The alpha- Nier, Phys. Rev. 55, 150 (1939). 2 Chamberlain, Williams, and Yuster, Phys. Rev. 70, 580 (1946). 3 G. T. Seaborg and I. Perlman, Rev. Mod. Phys. 20, 637 (1948). activities of the enriched and normal uranium can then be measured by the use of counting equipment.

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Section: Introductionmentioning

confidence: 99%

The Uranium 234 Content of Natural Uranium and the Specific Alpha-Activities of the Isotopes

Kienberger¹

1949

Phys. Rev.

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“…4),15) at University of North Dakota on May 29, 2015 http://ptp.oxfordjournals.org/Downloaded from §6. Transition curves, integral energy spectra and track length distributions for electronThe integral number spectrum of electrons initiated by an electron with energy Eo is calculated from (5·3·4),…”

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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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“…Early models for these were proposed independently by Bhabha and Heitler [8], and by Carlson and Oppenheimer [20]. Later developments have been made by a variety of people, e.g., Rossi and Greisen [96], Rossi [97], Greisen [46,47], Snyder [107], Serber [103], Heisenberg [54], Bhabha and Chakrabarty [9,10,11]. General reviews of the theory and properties of these extensive air showers (EAS) may be found in Rossi and Greisen [96], Rossi [97], Greisen [46,47], Hayakawa [52]; and more recently in Gaisser [41], Halzen [48].…”

Section: Extensive Air Showersmentioning

confidence: 99%

The properties of Very High Energy gamma ray sources observed using the air Cherenkov technique

Mohanty¹

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Further Calculations on the Cascade Theory

Cited by 44 publications

References 7 publications

The Uranium 234 Content of Natural Uranium and the Specific Alpha-Activities of the Isotopes

The Uranium 234 Content of Natural Uranium and the Specific Alpha-Activities of the Isotopes

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

The properties of Very High Energy gamma ray sources observed using the air Cherenkov technique

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