A mathematical model is presented to describe the replacement process to make zeolite containing a particular species of alkali ions for the beam source. The fraction of the alkali ions trapped in zeolite lattice can be expressed as a function of the times of substitution reaction in a concise recursion formula. The formula is used for a simple estimation of efficiency for making alkali zeolite in terms of time and cost. A rough comparison between the model and a trial is presented

A. Fujisawa

A. Shimizu

H. Nakano

S. Ohshima

National Institute for Fusion Science (NIFS-Repository)

A RcciPctO Makc Zco】it ion Sourcc for Plasma Diagnostics BcamA FuuisaWa A ShiH zu s oh、htrna and H Nょに“n。〈Rcccived _ A、,= 22 2005)NIFS-816 ScP 2005                                     Inquiries about copyright should be addressed to the Research Information Center,    National Institute for Fusion Science, Oroshi-cho, Toki-shi, Gifu-ken 509-5292 Japan.    E-mail: bunken@nifs.ac.jp  <Notice about photocopying>  In order to photocopy any work from this publication, you or your organization must obtain permission from the following organizaion which has been delegated for copyright for clearance by the copyright owner of this publication.  Except in the USA  Japan Academic Association for Copyright Clearance (JAACC)  6-41 Akasaka 9-chome, Minato-ku, Tokyo 107-0052 Japan  Phone: 81-3-3475-5618  FAX: 81-3-3475-5619  E-mail: jaacc@mtd.biglobe.ne.jp  In the USA  Copyright Clearance Center, Inc.  222 Rosewood Drive, Danvers, MA 01923 USA  Phone: 1-978-750-8400   FAX: 1-978-646-8600  A Recipe to Make Zeolite Ion Source for Plasma DiagnosticsBeamA. Fujisawa, A. Shimizu, S. Ohshima, H. NakanoNational Institute for Fusion ScienceOroshi-cho, Toki, Japan, 509-5292(Dated: September 1, 2005)AbstractA mathematical model is presented to describe replacement process to make zeolite containinga particular species of alkali ions for the beam source. The fraction of the alkali ions trapped in azeolite lattice can be expressed as a function of the number of the times of substitution reaction ina concise recursion formula. The formula is used for a simple estimation of efficiency for makingalkali zeolite in terms of time and cost. A rough comparison between the model and an experimentis presented.keywords: alkali ion source, zeolite, substitution reaction, plasma diagnostics1Zeolite is a porous material containing (or trapping) alkali ions in its lattice, and thealkali ions can be extracted by heating up zeolite above an appropriate temperature. Owingto the property zeolite is often utilized as an alkali ion source such as sodium, cesium andthallium. In our laboratory, cesium zeolite is used for a heavy ion beam probe[1] to measurepotential, density and their fluctuations in high temperature plasmas.The cesium zeolite is produced by ourselves through a substitution reaction from molecu-lar sieves (or sodium zeolite), since the zeolite allows to exchange its own ions with other ions.This is a convenient way for a laboratory to make an ion source. However, the self-madecesium zeolite is contaminated with the remaining sodium unnecessary for our purpose. It isessential to increase the target alkali fraction in the zeolite in order to obtain higher current.This short note presents consideration on an efficient manner to refine a zeolite source tocontain more fraction of the favorable alkali element.In our laboratory, the cesium zeolite is made as follows. Cesium compound is solved intowater, then sodium zeolite is put into the solution. The solution is stirred for several hoursin temperature of approximately 70◦C until the substitution is completed. The solutionis percolated to weed the sodium compound out (although cesium compound is removedsimultaneously). The zeolite is put into the water again, in which appropriate amount ofcesium compound is newly soluted. The same process can be repeated for several times.Finally, zeolite with a certain amount of cesium fraction is produced.In the first place, the process of substitution can be described in the following model, asdN1dt= −αN1 + βN2 + γ(N −N1 −N2), (1)dN2dt= +αN1 − βN2 + δ(N −N1 −N2), (2)where N , N1 and N2 are the total number of the holes to trap alkali ions, the number of theholes to be occupied by the first chemical element, and that by the second chemical element,respectively. The parameters α and β are the rates of replacement, and the γ and δ are therates of trapping. Summation of these two equations proves that the total number of theholes is kept constant.In equilibrium, the solution is described asN1 =βα+ βN, N2 =αα+ βN (3).Note that the number of occupation for the first chemical element N1 is identical with the2average value predicted from a Bernoulli distribution when the probability for the first alkalielement to occupy a hole is assumed to be β/(α+ β).The coefficient α (or β) is supposed to increase as the total number of the second (or first)chemical element increases. The simplest assumption is α = α0M2 , β = β0M1, where thevariables M1 and M2 are the concentrations of the first and second elements in the water.Then, the formula can be rewritten intoN1 =M1M1 + (1 + ε)M2N, N2 =(1 + ε)M2M1 + (1 + ε)M2N (4)where the ratio of α0/β0 = 1 + ε (−1 < ε <∞) is used for simplicity.Next, we assume that the newly added amount of the first alkali compound is ∆ · N inevery substitution process, and that all the compounds are removed with the percolationprocess. Under this assumption, the alkali fractions can be expressed as a function ofthe number of times of substitution processes in a recursion formula. By introducing newvariables p(n) = N1/N and q(n) = N2/N which represent the fractions of the first andsecond alkali ions after the n-th time substition, respectively, the concentrations of alkaliions in solution for the n-th substitution are found to be (p(n)+∆)N and q(n)N for the firstand the second components, respectively. According to Eqs. (3,4), the variables p(n + 1)and q(n+ 1) can be connected to the fraction of n-th time, p(n) and q(n), asp(n+ 1) =p(n) + ∆p(n) + ∆ + (1 + ε)q(n), q(n+ 1) =(1 + ε)q(n)p(n) + ∆ + (1 + ε)q(n). (5)This recursion formula can be solved to make the ratio p(n)/q(n)(≡ a(n)) with utilizingthe relation of p(n) + q(n) = 1; the recursion formula for a(n) is quite simplified intoa(n+1) = Ξa(n) +∆(1+ ε)−1, where Ξ = (1+∆)/(1 + ε). Then the recursion formula canbe easily solved.The solution is explicitly written asp(n) =Ξn[a(1)(1− ξ) + 1]− 1Ξn[a(1)(1− ξ) + 1]− ξ , q(n) =1− ξΞn[a(0)(1− ξ) + 1]− ξ , (∆ 6= ε) (6)orp(n) =n∆+ a(0)(1 + ∆)n∆+ (a(0) + 1)(1 + ∆), q(n) =1 + ∆n∆+ (a(0) + 1)(1 + ∆), (∆ = ε) (7)where ξ = ε/∆ and a(0) = p(0)/q(0). In the case of a(0) = p(0) = 0, the formula issimplified into the following form,p(n) =Ξn − 1Ξn − ξ , q(n) =1− ξΞn − ξ (∆ 6= ε), (8)3orp(n) =n∆n∆+ (1 + ∆), q(n) =1 + ∆n∆+ (1 + ∆)(∆ = ε). (9)According to the analysis, two characteristic regions are identified by the sign of ε. Whenε < 0, the complete replacement (p→ 1 for n→∞), is possible for any small amount of thesubstitution compound. When ε > 0, the final fraction of the first element is controlled bythe amount of the compound. In the case of the latter, the complete replacement is possiblefor ∆ > ε, while the maximum fraction is limited below ∆/ε(< 1) for ∆ < ε. When ∆ = ε,the complete replacement is possible but in the slowest manner. Figure 1 shows changesof the first element fraction as a function of the number of times of substitution processesfor four cases; i) ε = −0.5, ∆ = 0.5, ii) ε = 0.5, ∆ = 1.0, iii) ε = 0.5, ∆ = 0.5, and iv)ε = 1.0, ∆ = 0.5.The efficiency can be evaluated in terms of time and cost. The number of times ofsubstitution processes n necessary to obtain a fraction of the first element, pT, is describedas a function of the amount of the compound, ∆, asn(pT,∆) = log[(∆− pTε)/∆(1− pT)]/ log[(1 + ∆)/(1 + ε)], (∆ 6= ε), (10)n(pT,∆) = pT(1 + ∆)/∆(1− pT), (∆ = ε). (11)The total amount of the necessary compound, ζ = n×∆, is expressed asζ(pT,∆) = ∆ log[(∆− pTε)/∆(1− pT)]/ log[(1 + ∆)/(1 + ε)], (∆ 6= ε), (12)ζ(pT,∆) = pT(1 + ∆)/(1− pT), (∆ = ε). (13)The parameter ζ can be used as an indicator to evaluate the cost efficiency. The minimumamount of the compound, or minimum cost, can be found for a fixed fraction of the favorablealkali element. Note that Eqs. (10)-(13) are evaluated with Eqs. (8) and (9).Figure 2 illustrates examples of the times and cost efficiencies for three cases with pT = 0.8for three different values of ε = −0.2, 0.5 and 1.5. In all cases, as is expected, the necessarytimes of substitution decreases as the amount of the compound increases. In the first case,the total amount of the compound (or the cost) monotonically increases; a replacementprocess is enough if the parameter ∆ > pT(ε+1)/(1− pT) is chosen. In the other cases, themaximum of the cost efficiency (or minimum cost) exists for the choice of the parameter ∆.In the second case (ε = 0.5), the cost minimum is achieved for the four times replacementsn=4 at ∆ ' 1, while seven times replacements n = 7 are necessary for the third case(ε = 1.5) at the choice of ∆ ' 3.4A rough comparison between the model and an experiment is shown in Fig. 3.Here, the experiment was performed with the condition of ∆ = 1 using zeolite(Na12[(AlO2)12(SiO2)12]·27H2O) and cesium chloride (CsCl). Corresponding to the numberof the sodium atom in the chemical formula, the zeolite is assumed to have holes of 12 timesthe Avogadro number per a mole. Hence, the amount of cesium corresponding to ∆ = 1 is12 moles for a mole of the zeolite. The ratio of cesium to the total alkali ions (sodium andcesium) in zeolite increases obviously, as is shown in Fig. 3, according to the curve basedon Eq. (8) when ε = 1.22 is assumed. In this rough comparison, the present model can wellreproduce the real process of the replacement.[1] A. Fujisawa, H. Iguchi, S. Lee, Y. Hamada, Rev. Sci. Instrum. 68 3393(1997).5FIG. 1: Change of alkali fractions in zeolite through substitution reactions. Four cases are shownwith difference in parameters ε and ∆; i) ε = −0.5, ∆ = 0.5, ii) ε = 0.5, ∆ = 1.0, iii) ε = 0.5, ∆ =0.5, and iv) ε = 1.0, ∆ = 0.5.FIG. 2: Efficiency of alkali replacement in time and cost for three cases. The necessary times ofreplacement processes and the total amount of the alkali compound are shown as a function of theamount of compound for each process. (a) The case of ε = −0.2, (b) ε = 0.5 and (c) ε = 1.5. Thelines with circles represent the necessary times of substitution.FIG. 3: A comparison between the model and an experiment. The circles represent the fractionof cesium to the total alkali ions (sodium and cesium). The ratios are estimated from the mea-surements of atomic ingredients by using scanning electron microscopy. The line shows the curvebased on Eq. (8) with the parameter ε = 1.22.6Figure 1      A. Fujisawa et al.00.20.40.60.810 5 10 15 20Fraction of the first elementNumber of times of substitution procesε=0.5, ∆= 1ε=0.5, ∆=0.5ε=-0.5, ∆=0.5ε=1, ∆=0.5Figure 2      A. Fujisawa et al.(a) (b) (c)02468100 1 2 3 4 5 6 7 8n and ζ∆ε=-0.202468100 1 2 3 4 5 6 7 8n and ζ∆ε=0.502468100 1 2 3 4 5 6 7 8n and ζ∆ε=1.5Figure 3      A. Fujisawa et al.00.20.40.60.810 1 2 3 4 5Fraction of cesium (to sodium+cesium)number of replacement∆=1

A Recipe to Make Zeolite Ion Source for Plasma Diagnostics Beam

Abstract

Similar works

Full text

Available Versions

National Institute for Fusion Science (NIFS-Repository)