Neutron Thermalization: A Fractional Calculus. Theoretical Approach

Transcription

1 It. Joural of Math. Aalysis, Vol. 6,, o. 9, Neutro Thermalizatio: A Fractioal Calculus Theoretical Approach * Abdul-Wali Ajloui, Ami Al-Okour, Abdullah Ajloui ad 3 Sheri A. Sareireh Miistry of Eergy ad mieral resources, Amma, Jorda Departmet of Applied scieces, Husu College, Balqa Uiversity, Husu, Jorda 3 Physics Departmet, Al-Hussei Bi Talal Uiversity, Ma'a, Jorda. *Correspodig author. awajloui@hotmail.com Abstract Oe of the techologically most importat iteractios of eutros with matter is their loss of eergy ( slowig dow ) by a series of elastic collisios. These ca be treated by the methods of classical mechaics, assumig the iteractig particles as perfectly elastic spheres. The eergy loss is a importat subject, ad is discussed i several books where umerical tables ad graphs are preseted. Formulas are foud semiempirically with several correctio coefficiets. Despite all efforts, o direct, exact formula has so far bee obtaied aalytically. The purpose of this paper is to itroduce just such a direct formula of the eergy loss aalytically by usig a recetly itroduced method of the quatizatio of ocoservative systems based o fractioal calculus.

2 438 Abdul-Wali Ajloui et al. Itroductio The distace that a fast eutro will travel, betwee its itroductio ito the slowigdow medium (moderator) ad thermalizatio, is depedet o the umber of collisios ad the distace betwee collisios. Though the actual path of the eutro slowig dow is tortuous because of collisios, the average straight-lie distace ca be determied; this distace is called the fast diffusio legth or slowig-dow legth. The distace traveled, oce thermalized, util the eutro is absorbed, is called the thermal diffusio legth. Fast eutros rapidly degrade i eergy by elastic collisios whe they iteract with low atomic umber materials. As eutros reach thermal eergy, or ear thermal eergies, the likelihood of capture icreases. I preset day reactor facilities the thermalized eutro cotiues to scatter elastically with the moderator util it is absorbed by fuel or o-fuel material, or util it leaks from the core. Most of these iteractios idividually trasfer oly miute fractios of the eutro s kietic eergy, ad it is coveiet to thik of the eutro as loosig its kietic eergy gradually, ofte referred to as cotiuous slowig-dow approximatio. Because of the multitude of iteractios udergoe by each eutro i slowig dow, its path legth teds to approach the expectatio value that would be observed as a mea for a very large populatio of idetical particles. That expectatio value is called the rage. The purpose of this paper is to itroduce a direct aalytical formula related the eergy loss with the rage aalytically by usig Ajloui's theory of quatizatio of ocoservative systems depedig o fractioal calculus (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6).. Neutro Slowig Dow ad Thermalizatio Fissio eutros are produced at a average eergy level of MeV ad immediately begi to slow dow as the result of umerous scatterig reactios with a variety of target uclei. After a umber of collisios with uclei, the speed of a eutro is reduced to such a extet that it has approximately the same average kietic eergy as the atoms (or molecules) of the medium i which the eutro is udergoig elastic scatterig. This eergy, which is oly a small fractio of a electro volt at ordiary temperatures,.5 ev at (C), is frequetly referred to as the thermal eergy, sice it depeds upo the temperature. Neutros whose eergies have bee reduced to values i this regio (< ev) are desigated thermal eutros. The process of reducig the eergy of a eutro to the thermal regio by elastic scatterig is referred

3 Neutro thermalizatio 439 to as thermalizatio, slowig dow, or moderatio. The material used for the purpose of thermalizig the eutros is called a moderator. A good moderator reduces the speed of eutros i a small umber of collisios, but does ot absorb them to ay great extet. Slowig the eutros i as few collisios as possible is desirable i order to reduce the amout of eutro leakage from the core ad also to reduce the umber of resoace absorptios i o-fuel materials. The ideal moderatig material (moderator) should have the followig uclear properties (DEO, 993): -large scatterig cross sectio -small absorptio cross sectio 3-large eergy loss per collisio A coveiet measure of eergy loss per collisio is the logarithmic eergy decremet. The average logarithmic eergy decremet is the average decrease per collisio i the logarithm of the eutro eergy, represeted by the symbol ξ (DEO, 993): Ei ζ = l Ei - l E f = l () E f where E i ad E f are the average iitial ad fial eutro eergy, respectively. Sice the fractio of eergy retaied by a eutro i a sigle elastic collisio is a costat for a give material, ξ is also a costat. Because it is a costat for each type of material ad does ot deped upo the iitial eutro eergy, ξ is a coveiet quatity for assessig the moderatig ability of a material. The total umber of collisios ecessary for a eutro to lose a give amout of eergy may be determied by dividig ξ ito the differece of the atural logarithms of the eergy rage i questio. The umber of collisios (N) to travel from ay eergy, E high, to ay lower eergy, E low, ca be calculated as (DEO, 993): N Ehigh l ( ) l Ehigh - l Elow Elow = = () ζ ζ Sometimes it is coveiet, based upo iformatio kow, to work with a average fractioal eergy loss per collisio as opposed to a average logarithmic fractio. If the iitial eutro eergy is E, ad the average fractioal eergy loss per collisio is

4 44 Abdul-Wali Ajloui et al kow, X, the fial eergy for a give umber of collisios, E N, may be computed usig the followig formula (DEO, 993): E = (3) N N E ( - X ) Material ξ Number of Collisios to Thermalize H O.97 9 D O.5 35 Helium.47 4 Beryllium.7 86 Boro.7 5 Carbo Quatizatio of Nocoservative Systems: Free Particle i a Dissipative Medium Accordig to Ajloui's theory of the quatizatio of ocoservative systems, the Hamiltoia ca be writte as follows (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6): N s( i ) s( i) d H = + q, ( ) p, ( ) L, i N- s( i ) s( i) r s i r s i + d( t b) i= N = qr, s( i+ ) pr, s( i) L, (4) i= ad the Schrödiger equatio reads [6] HΨ = ih Ψ. (5) t Cosider a free particle movig i a dissipative medium where dissipatio is proportioal to velocity (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6), i.e., F = γq, (6)

5 Neutro thermalizatio 44 γ beig a positive costat. The potetial related to this dissipatio is (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6) iγ U = q. (7) The Lagragia is (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6) iγ L = mq q, (8) where dx d x q = x, q =, q = ; (9) dt d( t b) The caoical mometa are (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6) p L = q d + i d( t a ) 3 L q = i γ q + imq ; () ad L p = q = mq. () Makig use of Eq.(3), we have for the Hamiltoia (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6) ( p ) γ H = + q p + m i q. () Here p ad p are the caoical cojugate mometa to q ad q, respectively. Schrödiger's equatio reads [, 5] h h ih Ψ = + q + γq Ψ. (3) t m q i q i which is Schrödiger's equatio for a dissipated free particle, has the followig solutio (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6)

6 44 Abdul-Wali Ajloui et al where Ψ mγ q 4 mγ i i q i q exp h exp Ex exp E t. (4) ih h q h = AH H are Hermite polyomials. 5. Neutro thermalizatio I this problem we assume that the free particle represets the eutro, the dissipative medium is the matter where the particle passes, ad the dissipative effects result from a drag force iflueces the eutro motio iside the matter, similar to that take place, whe a object movig through a fluid, ad will geerally experiece a drag force proportioal to its velocity. Despite the fact that, may processes may occur durig particle passage, but, what we cocer about, is the over all eergy loss of the particle i matter. The dissipatio force ad potetial are foud by velocity proportioal law, i.e., F = γq (5) The potetial correspodig to this dissipatio is iγ U = q. (6) which leads to Schrödiger's equatio for a dissipated free particle, Eq. (3), has the solutio, (Ajloui, 4, Ajloui,, Ajloui,, Rabei et al., 6) mγ q 4 mγ ih i q i Ψ = AH q exp exp Ex exp Et. (4) ih h q h Eergy expectatio value is: E x * = HΨ x= Ψ dydx (7) Isertig Eq. (4) i Eq. (7), ad usig the relatios (Dass, ad Sharma, 998, Arfke, 985): H ( y ) = H ( y ), (8) -

7 Neutro thermalizatio 443 ad, e y H ( y )H r y y e H ( y )H + p( y )dy = m ( y )dy = π! δ, (9) π, m ( + r)!, p = r p > r that makes all the o- x terms of Eq. (4), costats or zeros. The remaider part is the x -depedet terms, appears as: () E x α y i x ie x x Ex x ie x x H Exp Exp Ex 3 y y y y x = = α + h h h h H Exp α y i x Exp E x dydx h y Usig mathematical idetities (Dass, ad Sharma, 998, Arfke, 985): E 3 = bx ax cx ; () where a, b, ad c are costats. () Figure : A schematic represetatio of average eutro eergy loss as a fuctio of distace through moderator.

8 444 Abdul-Wali Ajloui et al 6. Coclusio Quatizatio of ocoservative free particle system, accordig to Ajloui's recet theory is applied o eutro thermalizatio whe passed through matter. The formula of eergy loss versus distace traveled iside matter has bee itroduced ad plotted. The figure agrees widely with the experimetal results. Refereces [] Ajloui, A-W. Ph.D. thesis: Quatizatio of Nocoservative Systems, Jorda Uiversity, Amma, Jorda. 4. [] Ajloui, A-W. "Quatizatio Of Nocoservative Systems". ISBN-NR: LAP LAMBERT Academic Publishig AG & Co. KG, Dudweiler Ladstr. 99, 663 Saarbrücke Germay.. [3] Ajloui, A-W. Fractioal Calculus Tools i Physics. Applied Mathematics & Iformatio Scieces. 5 (3) (), 53S-6S.. [4] Arfke, G. Mathematical Methods for Physical Scieces. 3 rd ed., Academic Press, INC [5] Dass, T., ad, Sharma, S. Mathematical Methods i Classical ad Quatum physics. st ed., Uiversity Press, Hayderabad, Idia [6] Rabei, E., Ajloui, A-W., ad Ghassib, H. "Quatizatio of a Free Particle i A Dissipative Medium: Iteractio of Charged Particles with Matter". ASME 5 Iteratioal Desig Egieerig Tech. Co.(5 th It. Co. O Multibody systems, oliear Dyamics, ad Cotrol MSNDC), DETC5-84, Sep. 5, Log Beach, Ca, USA.

9 Neutro thermalizatio 445 [7] Rabei, E., Ajloui, A-W., ad Ghassib, H. "Quatizatio of Nocoservative Systems Usig Fractioal Calculus". Wseas Trasactios o Mathematics, p , Issue 7, Volume 5, ISSN Received: July,