scholarly journals Some Series and Mathematic Constants Arising in Radioactive Decay

2019 ◽  
Vol 11 (6) ◽  
pp. 14
Author(s):  
Xun Zhou
Keyword(s):  
Infinite Series ◽  
Half Life ◽  
Radioactive Decay ◽  
Convergent Series ◽  
Divergent Series ◽  
Radioactive Isotopes ◽  
Initial Value ◽  
Related Series ◽  
Mathematical Relations ◽  
Mathematical Constants

In this paper we show the construction of 32 infinite series based on the law of decay of radioactive isotopes, which indicates that a radioactive parent isotope is reduced by 1/2 and 1/e of its initial value during each half-life and mean life, respectively. We found that the ratios among the values of the radioactive parent isotope and the radiogenic daughter isotope for each half-life’s and mean life’s decay can be used to construct 16 half-life related (or 2-related) and 16 mean life related (or e-related) infinite series. There are 8 divergent series, 4 previously known convergent series and 2 series converging to the Erdös-Borwein constant. The remaining 18 series are found to converge to 18 mathematical constants and the divergent and alternating mean life related series leads to another 2 mathematical constants. A few interesting mathematical relations exist among these convergent series and 5 sequences are also attained from the convergent half-life related series.

Rearrangements that Preserve Rates of Divergence

1982 ◽  
Vol 34 (4) ◽  
pp. 916-920
Author(s):  
Elgin H. Johnston
Keyword(s):  
Real Number ◽  
Infinite Series ◽  
Special Interest ◽  
Convergent Series ◽  
Divergent Series ◽  
Classical Theorem ◽  
Real Numbers ◽  
Conditionally Convergent Series ◽  
Positive Integers

Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak. A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.

Summation of slowly convergent series

1950 ◽  
Vol 46 (3) ◽  
pp. 436-449 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Infinite Series ◽  
Main Part ◽  
Asymptotic Series ◽  
Convergent Series ◽  
Divergent Series ◽  
Finite Form ◽  
Sum Formula ◽  
New Form ◽  
Practical Sense ◽  
Direct Summation

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

1. Identical outsides … different insides

2016 ◽  
Author(s):  
Rob Ellam
Keyword(s):  
Ionizing Radiation ◽  
Half Life ◽  
Spontaneous Fission ◽  
Radioactive Decay ◽  
Radioactive Isotopes ◽  
Carbon 14 ◽  
Alpha Beta

‘Identical outsides … different insides’ describes the isotopes of hydrogen (protium, deuterium, tritium) and carbon (carbon-12, carbon-13, carbon-14). The isotopes exist due to an extra particle (neutron) or two in the element’s nucleus, which adds extra mass to the atom. Tritium and carbon-14 are both unstable and undergo radioactive decay. There are four types of radioactive decay: α, β, γ (alpha, beta, gamma), and spontaneous fission. Radioactive decay is the process whereby the nucleus of an unstable atom loses energy by emission of ionizing radiation. The rate of the radioactive decay is measured by the ‘half-life’—the time needed for half the radioactive isotopes of a substance to decay.

scholarly journals In-Situ Cosmogenic 14C: Production and Examples of its Unique Applications in Studies of Terrestrial and Extraterrestrial Processes

Radiocarbon ◽  
2001 ◽  
Vol 43 (2B) ◽  
pp. 731-742 ◽  
Author(s):  
D Lal ◽  
A J T Jull
Keyword(s):  
Half Life ◽  
Earth Science ◽  
Chemical Properties ◽  
Radioactive Isotopes ◽  
Earth’S Atmosphere ◽  
The Earth ◽  
Wide Range ◽  
History Of ◽  
Earth's Atmosphere

Nuclear interactions of cosmic rays produce a number of stable and radioactive isotopes on the earth (Lai and Peters 1967). Two of these, 14C and 10Be, find applications as tracers in a wide variety of earth science problems by virtue of their special combination of attributes: 1) their source functions, 2) their half-lives, and 3) their chemical properties. The radioisotope, 14C (half-life = 5730 yr) produced in the earth's atmosphere was the first to be discovered (Anderson et al. 1947; Libby 1952). The next longer-lived isotope, also produced in the earth's atmosphere, 10Be (half-life = 1.5 myr) was discovered independently by two groups within a decade (Arnold 1956; Goel et al. 1957; Lal 1991a). Both the isotopes are produced efficiently in the earth's atmosphere, and also in solids on the earth's surface. Independently and jointly they serve as useful tracers for characterizing the evolutionary history of a wide range of materials and artifacts. Here, we specifically focus on the production of 14C in terrestrial solids, designated as in-situ-produced 14C (to differentiate it from atmospheric 14C, initially produced in the atmosphere). We also illustrate the application to several earth science problems. This is a relatively new area of investigations, using 14C as a tracer, which was made possible by the development of accelerator mass spectrometry (AMS). The availability of the in-situ 14C variety has enormously enhanced the overall scope of 14C as a tracer (singly or together with in-situ-produced 10Be), which eminently qualifies it as a unique tracer for studying earth sciences.

scholarly journals Application of the radionuclide 210Pb in glaciology – an overview

2020 ◽  
Vol 66 (257) ◽  
pp. 447-456 ◽  
Author(s):  
Heinz W. Gäggeler ◽  
Leonhard Tobler ◽  
Margit Schwikowski ◽  
Theo M. Jenk
Keyword(s):  
Half Life ◽  
Arid Regions ◽  
Noble Gas ◽  
Radioactive Decay ◽  
Ice Cores ◽  
Snow Accumulation ◽  
The Earth ◽  
Special Cases ◽  
Glacier Flow

Abstract210Pb is an environmental radionuclide with a half-life of 22.3 years, formed in the atmosphere via radioactive decay of radon (222Rn). 222Rn itself is a noble gas with a half-life of 3.8 days and is formed via radioactive decay of uranium (238U) contained in the Earth crust from where it constantly emanates into the atmosphere. 210Pb atoms attach to aerosol particles, which are then deposited on glaciers via scavenging with fresh snow. Due to its half-life, ice cores can be dated with this radionuclide over roughly one century, depending on the initial 210Pb activity concentration. Optimum 210Pb dating is achieved for cold glaciers with no – or little – influence by percolating meltwater. This paper presents an overview which not only includes dating of cold glaciers but also some special cases of 210Pb applications in glaciology addressing temperate glaciers, glaciers with negative mass balance, sublimation processes on glaciers in arid regions, determination of annual net snow accumulation as well as glacier flow rates.

scholarly journals Modeling and sensitivity analysis of transport and deposition of radionuclides from the Fukushima Dai-ichi accident

2014 ◽  
Vol 14 (20) ◽  
pp. 11065-11092 ◽  
Author(s):  
X. Hu ◽  
D. Li ◽  
H. Huang ◽  
S. Shen ◽  
E. Bou-Zeid
Keyword(s):  
Size Distribution ◽  
Dry Deposition ◽  
Wet Deposition ◽  
Emission Rate ◽  
Radioactive Decay ◽  
Radioactive Isotopes ◽  
Wind Fields ◽  
Total Deposition ◽  
Horizontal Diffusion ◽  
Ground Deposition

Abstract. The atmospheric transport and ground deposition of radioactive isotopes 131I and 137Cs during and after the Fukushima Dai-ichi Nuclear Power Plant (FDNPP) accident (March 2011) are investigated using the Weather Research and Forecasting-Chemistry (WRF-Chem) model. The aim is to assess the skill of WRF in simulating these processes and the sensitivity of the model's performance to various parameterizations of unresolved physics. The WRF-Chem model is first upgraded by implementing a radioactive decay term into the advection–diffusion solver and adding three parameterizations for dry deposition and two parameterizations for wet deposition. Different microphysics and horizontal turbulent diffusion schemes are then tested for their ability to reproduce observed meteorological conditions. Subsequently, the influence of emission characteristics (including the emission rate, the gas partitioning of 131I and the size distribution of 137Cs) on the simulated transport and deposition is examined. The results show that the model can predict the wind fields and rainfall realistically and that the ground deposition of the radionuclides can also be captured reasonably well. The modeled precipitation is largely influenced by the microphysics schemes, while the influence of the horizontal diffusion schemes on the wind fields is subtle. However, the ground deposition of radionuclides is sensitive to both horizontal diffusion schemes and microphysical schemes. Wet deposition dominated over dry deposition at most of the observation stations, but not at all locations in the simulated domain. To assess the sensitivity of the total daily deposition to all of the model physics and inputs, the averaged absolute value of the difference (AAD) is proposed. Based on AAD, the total deposition is mainly influenced by the emission rate for both 131I and 137Cs; while it is not sensitive to the dry deposition parameterizations since the dry deposition is just a minor fraction of the total deposition. Moreover, for 131I, the deposition is moderately sensitive (AAD between 10 and 40% between different runs) to the microphysics schemes, the horizontal diffusion schemes, gas-partitioning and wet deposition parameterizations. For 137Cs, the deposition is very sensitive (AAD exceeding 40% between different runs) to the microphysics schemes and wet deposition parameterizations, but moderately sensitive to the horizontal diffusion schemes and the size distribution.

scholarly journals On rearrangements of infinite series

1958 ◽  
Vol 3 (4) ◽  
pp. 182-193 ◽  
Author(s):  
A. P. Robertson
Keyword(s):  
Infinite Series ◽  
Convergent Series ◽  
Complex Numbers ◽  
Famous Theorem

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.(i) What is the condition on a given series for every rearrangement to converge?(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).

Sign in / Sign up

Export Citation Format

Share Document