@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 The Study of A Improve Model For The D-D Nuclear Fusion Reaction Cross Section Raad H. Majeed Laheeb A. Mohammed Dapt. of Physics/College of Education for Pure Science /(Ibn-Al-Haitham)/ University of Baghdad Received in : 25 June 2012 , Adccepte in: 9 December 2012 Abstract The study of improved model for measuring the total nuclear fusion cross section characteristics the D-D reaction may play an important role in deciding or determining the hot plasma parameters such as mean free path , the reaction rate , reactivity and energy for emitted neutrons or protons in our work we see the it is necessary to modify the empirical formulas included the total cross section in order to arrive or achieve good agreement with the international publish result. Keyword: Dense plasma focus; the mean free path; the reactivity; D-D cross section; fusion reaction 120 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Introduction Dense Plasma Focus (DPF) machines are pulsed discharges in which microinstabilities and turbulence lead to generation of powerful beams of electron, ions, large emission of x- rays and of fusion neutrons when the filling gas is deuterium. The plasma Focus (PF) belongs to the family of the dynamic Z-pinch. The first systematic study of Z-pinches started in the 1950s in connection with the controlled thermonuclear fusion research. It is a non-cylindrical Z-pinch formed on the axis at the end of a coaxial electrode system of a plasma accelerator. They were originally developed in early 1960s independently in the former Soviet Union (Filippov type), and the USA (Mather type) as shown in fig (1). Most realizations belongs to one of two of the following geometrical types: ●Mather-type characterized by a small anode aspect ratio (diameter / length < 1) ● Filippov-type, characterized by a large aspect ratio. [1, 2, 3] Plasma Focus produces a short living, rather dense plasma, the properties of which are dominated by the occurrence of macroscopic and microscopic instabilities. At present, DPF is one of the most efficient sources of neutron emission. Also scaling laws of neutron yield prepared on the basis of long experience with different devices of this type are very promising. Unfortunately, it was found that neutron emission saturates at an energy level of several hundreds kJ. [1] In typical DPF the z- pinch is typically confined for ~100ns and intense x-ray, ion, and electron beams are generated though a complex combination of non-linear instabilities and other mechanisms that result in effective acceleration gradients of ~100MV/m. [5] When deuterium is used as a working gas, neutron is emitted from the pinch due to following nuclear fusion reactions: PF-type facilities emit both neutrons, and hard X-rays with quantum energies of tens and hundreds electron volts. [3] The dominant part of neutrons (70%) with energies in the range of (2.6-3.2 MeV) was produced by deuterons moving downstream. The deuterons generating these neutrons have the energy in the range of (20-200 keV) with most probable value of 50 keV. From these energies, we can calculate the mean free paths of D-D reaction for the convenient deuteron densities of (1018-1020cm-3). On the basis of these results, we can estimate the number of fast deuterons lower than 1018and the density of the deuteron target higher than 1019cm-3. The D-D reaction is tested for the determination of basic parameters of hot plasma- temperature, density and duration of high value of these parameters. This reaction is tested mainly in Tokomaks [6], High power lasers [7] and Z-pinches (DPF) [8]. The D-D reaction is realized in 2 branches with the probability of 50%. These qualities make possible the usage of neutrons as a convenient tool for the diagnostics of fast deuterons with energies above 20 keV. In this paper we refer about the determinations of velocities and number of fast deuterons producing neutrons in D-D reaction and the estimations of densities of the targets in plasma focus PF-1000 facility. [9] Fast neutrons from deuteron –deuteron fusion reaction were used for a study of fast deuterons in the PF-1000 plasma focus device. Neutron energy-distribution enabled the determination of axial and radial components of energy of deuterons producing the fusion neutrons, as well as )01.1()02.3( )82.0()45.2( 3 1 1 1 2 1 2 1 3 2 1 0 2 1 2 1 MeVTMeVHDD MeVHeMeVnDD +→+ +→+ 121 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 a rough evaluation of the total energy distribution of all fast deuterons in the pinch. It was found that the total deuteron energy-distribution function decreases with the deuteron energy more slowly than the tail of the maxwellian distribution for (1-2 keV) [10]. The most important quantity for the analysis of nuclear reactions is the cross section, which measures the probability per pair of particles for the occurrence of the reaction. Another important quantity is the reactivity, defined as the probability of reaction per unit time per unit density of target nuclei. In the present simple case, it is just given by the product συ. In general, target nuclei move, so that the relative velocity υ is different for each pair of interacting nuclei. In this case, we compute an averaged reactivity <συ> = ∫ ∞ 0 )()( υυυυσ df Where f (υ) is the distribution function of the relative velocities, normalized in such a way that ∫ ∞ 0 )( υυ df = 1. It is to be observed that when projectile and target particles are of the same species, each reaction is counted twice. [11] And in turn the reaction rate can be evaluated by using the following: 〉〈 + = συ δ 12 21 12 1 nnR The Kronecker symbol δij (with δij = 1, if i = j and δij = 0 elsewhere) R12=n1n2<συ> If the particles are dissimilar. Where R is the reaction rate [12]. Theory The energy of fast deuterons producing neutrons can be calculated from the known neutron energy using the equilibrium of the momentum and energy. The required energy of deuterons is about 200keV. The dominant part of neutrons with the energy of 2.6-2.8 MeV is produced with deuterons in the energy range between 20 and 80 keV. The deuterons with the energy below 20 keV have very low and cross-section for the D-D fusion reaction and we do not take them into account. [9] The hot plasma parameters such as the mean free path of D-D reaction (λD-D), the number of fast deuterons (ND) and the probability of D-D reaction are strongly depending on the total cross-section of the D-D reaction (σ(ED)). The total cross-section is calculated by some empirical formulas given below [12]: is the cross section in barns [12]. σ E D and Here ED is energy for deutereron in keV, In order to get suitable values for hot plasma parameters as in following: Where ni is the ion density of the target (per m3 ). BarnsEe EE D D D )1( 288 ) 8.45 ( −−−−−= − σ )2( 1 −−−−−= σ λ E n D i DD )3(/* −−−−−−−= LN DDxD N λ 122 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Where ND is the number of fast deuterons Nx Is the total neutron yield= 1011 in the target with length mean free path [9]. λDD L=1cm, It is necessary to note that the above empirical formulas for the total cross-section given in equation (1) are not compatible with a given agreement between the recently calculated parameters and the published experimental values. For this reason and to avoid the error factor in calculating the hot plasma parameter, we must find to need of inserting a correction factor for all the above formulas which described the hot plasma parameters. A good approximation for the reactivity for the two main branches of the DD reaction are provided by slightly modified Gamow expressions (Hively 1977) [11]: 〉〈συ DDP = 2˟10 -14 T T 3/2 949.0 00577.01+ exp(- T 3/1 31.19 ) (cm3/s) …… (4) and 〉〈συ DDn = 2.72˟10 -14 T T 3/2 917.0 00539.01+ exp (- T 3/1 80.19 ) (cm3 /s) …… (5) Here the subscripts DDp and DDn indicate the reaction branches (4) and (5), releasing a proton and a neutron respectively, T is the temperature in keV [11]. Results Our calculated results deal with the total cross-section for the D-D reaction , the mean free path of D-D reaction (λDD) for various values of ion densities and the number of the fast deuterons (ND) before inserting the correction factor are listed in table(1) and these results are plotted in figures(2-4). We can conclude from the figures that there is a shift between the calculated results and the acceptable published experimental results, thus it is necessary to make an agreement between the two values by inserting a correction factor about (1.86) in the formulas previously described and for this case we found results yield a good agreement as it is presented in table (2) and plotted in figures (5-7). The variation of reactivity for temperatures ranging from (0 - 100 keV) are completely described in fig (8). Discussion and Conclusions The reason for inserting a correction factor is related to the fact that the empirical formula for the total cross-section of D-D reaction present an old and applied to sum facilities that are designed for previous decades and we found that it cannot be applied for the recent facilities such as the modern (PF-devices) so that we conclude that it is very important to modify these formulas by testing them for calculating sum fundamental hot plasma parameters. We have got a compatible formula by inserting a correction factor of values (1.86) which therefore makes a good agreement between the calculated and experimental data. Finally, from fig (8) we observe agreement with the published result [11] for the D-D reactivity. 123 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 References 1- Marek Scholz ,Leslaw karpinski ; Marian paduch ; Krzysztof Tomaszewski;Ryszard Miklaszewski and Adam Szydlowski; (2001),Recent progress in 1 MJ plasma – focus research, NUKLEONIKA, 46:35-39. 2- Klir,D. ; Kravarik, J.; Kubes , P.; Rezac , K. ; Cikhardt , J.; Litseva , E.; Hyhlik , T.; Ananev ,S.S.; Bakshaev , Yu. L; Bryzgunov ,V. A.; Chernenko ,A S.;Kalinin ,Yu .G; E, D, Kazakov ,E .D; Korolev ,V .D; Ustroev ,G. I; AZelenin ,A. A; Juha , L; Krasa, J; Velyhan ,A; Vysin ,L; Sonsky ,J. and Volobuev I, V. (2010), plasma phy . Control. Fusion, Efficent production of 100 keV deuterons in deuterium gas puff Z-pinches at MA current, 52: 17. 3- Siahpoush,V. Mohammadi, .M.A.; Khorram ,S.; , Shabani,I.; Brohanina, J.;Ashrafi ,S.; Naghshara ,H.; Sobhanian ,S.; (April 27-30 ,2004) , 2nd International conference on nuclear science and technology in Iran , Shiraz University, preliminary results of neutron production in sahand plasma focus device, 1-4. 4- Berard;A.; Cloth,P.; Conrads,H.; Coudeville,A.; Gourlan,G.; Jolas,A.; Maisonnier,Ch. and Rager,J.P.(1977), Nuclear Instruments and Methods , The Dense Plasma Focus – A High Intensity Neutron Source , 145 :191-218. 5- Vincent Tang and Brain Rusnak ; (February 29 , 2008), Lawrence Livermore National Laboratory (LLNL– TR – 401875 ) ) , Review of Dense Plasma Focus Technology for Intense and Directional Neutron Sources , 1-6 6- Giacomelli,L.;Conroy,S. ;Ericsson,G. ; Gorini,G. ; Heriksson,H.; Hjalmarsson,A.; Kallne,J. and Tardocchi,M .(2005), Comparison of neutron emission spectra for D and DT plasmas with auxiliary heating ”, the European physical Journal D-Atomic , Molecular , optical and plasma physics, 33 , number 2 , 235-241. 7-Mima,K.K.A.;Tanaka,K.A.;Kodama,R.;Johzaki,T.;Nagatoma,H.;Shiraga,H.; Sentoku,Y.;Miyanaga ,N.;Azechi,H.; Nakai,M.; Norimatu,T.; Nagai,K.; Sunanara,J.; Nishihara,K; Taguchi,T and Sakagami,H; (2004),plasma Science and Technology 6 :2179. 8-.Velikovich,A.L.;Clark,R.W;Davis,J.;Chong,Y.K.;.Deeney,C.;Coverdale,C.A.; C.LRuiz.C.L.; Cooper,G.W.; Nelson,A.J.; Franklin,J. and Rudakov, L.I. (2007) ,physics of plasmas 14 , 022701. 9- Kubes,P.;Klir,D.; Kravarik,J.; Rezac,K.; Bienkowaka,B.; Ivanova-Stanik,I.; Karpinski,L.; Paduch,M.; Schmidt,H.; Sadowski,M.J.andTomaszewski.K; (July 15-20,2007 ), D-D reaction and fast deuterons in plasma focus facility” 28th ICPIG ,Prague ,Czech Republic(The 28th International Conference on Phenomena in Ionized Gases,Topic number:17. 10- Kubes,P.;Kravarik,J.;D.Klir,D.;Rezac,K.;Bohata,M.; Scholz,M.; Paduch,M.;Tamaszewski,K.; Ivanova-Stanik,I.; Karpinski,L.and Sadowski,M.J. (Jan 2009) , Determination of Deuteron Energy Distribution from Neutron Diagnostics in a Plasma– Focus Device ” , plasma Science , IEEE Transactions, 37,Issue(1):83-87. 11-Stefano Atzeni and Jurgen Meyer-Ter-Vehn; , (2004) ,The physics of Intertial: Beam plasma interaction,Hydrodynamics ,Hot ,Dense matter , Clarenon press –Oxford . 12- David, J.Rose and Melville ,Clark ; JR. (1961), plasmas and controlled fusion, John Wiley and sons Inc., New York , London . 13-Heltemes,T.A.; Moses,G.A. and Santarius,J.F.; (April 2005), Fusion Technology institute university of Wisconsin Madison Wisconsin, Analysis of an Improved Fusion Reaction Rate model for use in fusion plasma simulations. 124 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Table (1): dependence of the total cross-section, mean free path, and the number of fast deuterons for the D-D fusion reaction on incident deuteron. ni=1026 (m-3 ) ni =1025 (m-3) ni =1024(m-3) σ (E D ) E -2 8 (m 2 ) ke V ( E d) *** ND (m- 3) ** ND (m- 3) * ND (m- 3) *** λDD (m) ** λDD (m) * λDD (m) *** ND (m- 3) ** ND (m- 3) * ND (m- 3) *** λDD (m) ** λDD (m) * λDD (m) *** ND (m- 3) ** ND (m- 3) * ND (m-3) *** λDD (m) ** λDD (m) * λDD (m) 3.61 E18 1.94 E18 3.6 E19 3.61 E5 1.94 E5 3.6 E5 3.61 E19 1.94 E19 3.6 E20 3.61 E6 1.94 E6 3.6 E6 3.61 E20 1.94 E20 --------- 3.61 E7 1.94 E-7 3.6 E-7 5.13 E-4 20 2.09 E17 1.12 E17 2.2 E18 2.09 E4 1.11 E4 2.2 E4 2.09 E18 1.12 E18 2.2 E19 2.09 E5 1.11 E5 2.2 E5 2.09 E19 1.12 E19 --------- 2.09 E6 1.11 E6 2.2 E6 8.860 E-3 50 9.59 E16 5.15 E16 ---- 9.59 E3 5.15 E3 ---- 9.59 E17 5.15 E17 ---- 9.59 E4 5.15 E14 ---- 9.59 E18 5.15 E18 -------- 9.59 E5 5.15 E5 ---- 0.01938 75 6.29 E16 3.38 E16 6.2 E17 6.29 E3 3.38 E3 6.2 E3 6.29 E17 3.38 E17 6.2 E18 6.29 E4 3.38 E4 6.2 E4 6.29 E18 3.38 E18 -------- 6.29 E5 3.38 E5 6.2 E5 0.02953 100 4.85 E18 2.61 E16 ---- 4.85 E3 2.61 E3 ---- 4.85 E17 2.61 E17 ---- 4.85 E4 2.61 E4 ---- 4.85 E18 2.61 E18 --------- 4.85 E5 2.61 E5 ---- 0.03831 125 4.07 E16 2.19 E16 3.7 E16 4.07 E3 2.19 E3 3.7 E3 4.07 E17 2.19 E17 3.7 E18 4.07 E4 2.19 E4 3.7 E4 4.07 E18 2.19 E18 --------- 4.07 E5 2.19 E5 3.7 E5 0.04562 150 3.60 E16 1.93 E16 ---- 3.60 E3 1.93 E3 ---- 3.60 E17 1.93 E17 ---- 3.60 E4 1.93 E4 ---- 3.60 E18 1.93 E18 --------- 3.60 E5 1.93 E5 ---- 0.05159 175 3.29 E16 1.77 E16 2.8 E15 3.29 E3 1.77 E3 2.8 E3 3.29 E17 1.77 E17 2.8 E18 3.29 E4 1.77 E4 2.8 E4 3.29 E18 1.77 E18 ---------- 3.29 E5 1.77 E5 2.8 E5 0.05647 200 * Published [9],** before using correct factor,*** after using correct factor, E#=10# Table (2): variation the D-D fusion Reactivity with deuteron temperature. <συ>DDn(cm 3/s) ˟10-14 <συ>DDp (cm 3/s) ˟10-14 T(keV) 0 0.0003 0.0019 0.0047 0.0084 0.0128 0.0176 0.0226 0.0279 0.0333 0.0388 0 0.0003 0.0016 0.0041 0.0072 0.0108 0.0147 0.0188 0.0231 0.0274 0.0319 0 10 20 30 40 50 60 70 80 90 100 Fig.(1,A ): Schematic view of PF machine- Filippov type: 1- anode;2- cathode; 3- insulator; 4- vacuum frame; C0-capactive bank, L0-external inductance, S- spark gap switch.[3] 125 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Fig. (1,B): Coaxial gun of the Mather type.[4] Fig.(2 ): Mean free path for D-D reaction versus incident deuteron energy for ni=1024 m-3 Fig.(3 ):Mean free path for D-D reaction versus incident deuteron energy for ni=1025 0 20 40 60 80 100 120 140 160 180 200 10 5 10 6 10 7 10 8 Ref [9] Befor correction After correction 0 20 40 60 80 100 120 140 160 180 200 10 4 10 5 10 6 10 7 Ref [9] Befor correction After correction 126 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Fig.(4 ):Mean free path for D-D reaction versus incident deuteron energy for ni=1026 Fig (5): Reactivity versus temperature for DD reaction Fig.(6):Mawell-averaged reaction reactivity versus temperature For the interest controlled fusion reactions [Ref 11]. 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 -16 Temperature [keV] D Dp -D Dn R EA CT IV IT YT [c m3 /s] reactivity DDp reactivity DDn 0 20 40 60 80 100 120 140 160 180 200 10 3 10 4 10 5 10 6 Ref [9] Befor correction After correction 127 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 نوع للتفاعل النووي االندماجي العرضیةنموذج تجریبي لقیاس المقاطع دراسة أ D-D رعد حمید مجید لھیب احمد محمد جامعة بغداد/ )ابن الھیثم( للعلوم الصرفھكلیة التربیھ /الفیزیاء علوم قسم 2012كانون االول 9، قبل البحث في 2012حزیران 25ستلم البحث في :ا الخالصة مھم�ھ الن�ھ لھ�ا ت�أثیر D-Dللتفاع�ل الن�ووي االن�دماجي ن�وع وذج تجریبي لقیاس المقاطع العرضیة الكلی�ةدراسة أنم نیترون�ات او البروتون�ات التفاعلی�ھ وطاق�ة ال،مثل معدل المسار الحرو معدل التفاع�ل ومھم في تحدید عوامل البالزما الحاره ط�ویر او تح�ویر العالق�ات التجریبی�ة المتض�منة حس�اب المق�اطع المنبعثھ .لوحظ في البحث الح�الي ان م�ن الض�روري ج�دا ت المنشوره. توافق مقبول مع النتأئج العالمیة لبلوغ او تحقیق العرضیة الكلیة ،االندماج النووي. D-Dنوع ر،التفاعلیة،المقاطع العرضیةفة، معدل المسار الح:بؤرة البالزما الكثیالكلمات المفتاحیة 128 | Physics