Improved 𝑫 − 𝑯𝒆 𝟑 Fusion Reaction Characteristics Parameters Ali K. Taki Raad H. Majeed Mustafa K. Jassim Dept. of Physics / College of Education for pure science (Ibn al-Haitham)/ University of Baghdad Received in :12 September 2012 , Accepted in: 17 March 2013 Abstract The most likely fusion reaction to be practical is Deuterium and Helium-3 (𝐷 − 𝐻𝑒 3 ), which is highly desirable because both Helium-3 and Deuterium are stable and the reaction produces a 14 𝑀𝑒𝑉 proton instead of a neutron and the proton can be shielded by magnetic fields. The strongly dependency of the basically hot plasma parameters such as reactivity, reaction rate, and energy for the emitted protons, upon the total cross section, make the problems for choosing the desirable formula for the cross section, the main goal for our present work. Key words:- 𝐷 − 𝐻𝑒 3 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛, proton energy, reactivity, reaction rate, fusion cross section , hot plasma 132 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Introduction The most likely fusion reaction to be practical is Deuterium and Helium-3 (𝐷 − 𝐻𝑒 3 ), which is highly desirable because both Helium-3 and Deuterium are stable and the reaction produces a 14 𝑀𝑒𝑉 proton instead of a neutron and the proton can be shielded by magnetic fields. This means it is possible to make a fusion reactor using 𝐷 − 𝐻𝑒 3 reaction where the fuels and the the reactions produce no radioactivity and the reaction occurs in a magnetized plasma and can be engineered to release its energy directly into electricity. This means a very compact and efficient fusion power plant, suitable for aircraft and spacecraft is possible using − 𝐻𝑒 3 . Because of the lack of radioactivity for direct power conversion to electricity, the practical use of 𝐷 − 𝐻𝑒 3 fusion reaction for power is the long-range goal of most fusion researchers [1]. The difficulty of 𝐷 − 𝐻𝑒 3 fusion relative to 𝐷 − 𝑇 stems from the scarcity of Helium- 3 on earth because the Helium-3 nuclei have two positive electric charges instead of the one in hydrogen isotopes, therefore, it requires higher collision energy for a Deuterium nuclei and Helium-3 nuclei to approach each other closely to fuse. This means 𝐷 − 𝐻𝑒 3 requires higher plasma temperature and pressures than 𝐷 − 𝑇 fusion so it is more technologically challenging. Also Helium-3 is found only in great abundance on the sun, so it is space rather than terrestrial resource. Therefore, 𝐷 − 𝐻𝑒 3 fusion is most often associated with space propulsion and power and a future economy that includes ready acess to space resource [1]. One of the possible techniques to decrease neutron load on plasma facing components and superconducting coils in fusion reactor is to use fuel cycle based on 𝐷 − 𝐻𝑒 3 as alternative to 𝐷 − 𝑇 [2]. Taking into account that the thermal reactivity of 𝐷 − 𝐻𝑒 3 is much lower than that of 𝐷 − 𝑇, the approach such as ICRF catalyzed fusion should be developed. The main idea of this technique is to modify reagent distribution function in order to achieve favorable reaction rate for nuclear fusion energy production [3]. Recent experimental results show high efficiency of ICRH (Ion Cyclotron Resonance Heating) acceleration of He-3 minority in D plasma in order to increase fusion reaction rate, from the other hand this technique could be used to achieve the favorable distribution of T ions in D plasma and hence to reduce the amount tritium needed for sustainable fusion plasma burning [4]. The effect of transition to non-Maxwellain plasma is essential for reactor aspects studies both in tokamaks and heliotrons, and the study of this effect is done by means of numerical code, based on test-particle approach, this code solves the guiding center equation of a general vector form. To simulate the coulomb collisions of test-particle with the other species, the discretized collision operator based on binomial distribution is used. The possibility of increasing the average reactivity by modification of distribution function of He- 3 and T minorities in D plasma due to selective ICRF (Ion Cyclotron Rang of Frequency) heating [5]. Theoretical models By definition, the reaction rate 𝑹 for reaction 𝒊 in a space independent problem is given by 𝑹 = 𝒏𝑫 𝒏𝑯𝒆 ∭𝒇(𝒗��⃗ 𝑫)𝑭(𝒗��⃗ 𝑯𝒆) 𝒈 𝒅𝝈𝒊 𝒅𝜴 𝒅𝜴𝒅𝒗��⃗ 𝑫𝒅𝒗��⃗ 𝑯𝒆 (1) Where 𝑛𝐷 𝑎𝑛𝑑 𝑛𝐻𝑒 are the beam and target ions densities with unit-normalized distribution functions 𝑓 𝑎𝑛𝑑 𝐹 respectively; g is the modulus of the relative velocity between beam and target ions and 𝑑𝜎𝑖 𝑑𝛺 is the doubly differential cross section. In the Bernstein – Comisar formalism, one can introduce the dummy variable 𝐸 by use of a Dirac Delta distribution, so that 𝑹 = 𝒏𝑫 𝒏𝑯𝒆 ∭𝒇(𝒗��⃗ 𝑫)𝑭(𝒗��⃗ 𝑯𝒆) 𝒈 𝒅𝝈𝒊 𝒅𝜴 𝜹�𝑬 − 𝑬𝒑�𝒅𝜴𝒅𝒗��⃗ 𝑫𝒅𝒗��⃗ 𝑯𝒆 𝒅𝑬 (2) 133 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Where 𝐸𝑝 𝑖𝑠 the emitted proton energy. This is given, in the case of axially moving target ions. By the laws of conservation of energy and momentum, and from the general basic fundamental equation of calculating the energy for the emitted charge particles from any nuclear reaction is given below [6]. �𝑬𝟑 𝟐 = 𝒗 ± √𝒗𝟐 + 𝝎 𝟐 (3) Where 𝒗 = � 𝑴𝟏 𝑴𝟑 𝑬𝟏 𝟐 𝑴𝟑+𝑴𝟒 cos𝜽 𝒂𝒏𝒅 𝝎 = 𝑴𝟒 𝑸+𝑬𝟏 (𝑴𝟒−𝑴𝟏) 𝑴𝟑+𝑴𝟒 (4) For the case of the binary 𝐷 − 𝐻𝑒 3 fusion reaction, 𝑫 + 𝑯𝒆 𝟑 → 𝑯𝒆 𝟒 (𝟑. 𝟕𝟏𝟐 𝑴𝒆𝑽) + 𝑷 (𝟏𝟒. 𝟔𝟒𝟏 𝑴𝒆𝑽) Where 𝑬𝟑 = 𝑬𝒑 , 𝑴𝟏 = 𝑴𝒅 , 𝑴𝟑 = 𝑴𝒑 , 𝑴𝟒 = 𝑴 𝑯𝒆 𝟒 , 𝑬𝟏 = 𝑬𝒅 And 𝑄 = 18.353 𝑀𝑒𝑉 is the Q-value of the above D-He fusion reaction, 𝐸 is the deuteron bombarding energy. Substituting the values for the quantities 𝑣, 𝜔, 𝑀1, 𝑀2, 𝑀3, 𝑀4, 𝐸1, 𝑄 as it is described above in equation (3), and taken into account some mathematical analysis steps to get or deduced a formula for evaluating the energy of the emitted neutrons that is given below. 𝑬𝒑 = 𝟐𝟎 𝑸 + 𝟏𝟐 𝑬𝒅 𝟐𝟓 ��𝟏 − 𝜸𝟐 𝐬𝐢𝐧𝜽 𝟐 + 𝜸𝐜𝐨𝐬𝜽� 𝟐 (5) Where 𝜸𝟐 = 𝟐 𝑬𝒅 𝟐𝟎 𝑸+𝟏𝟐 𝑬𝒅 Equation (5), explains the relationship between the energy of the emitted proton from the 𝐷 − 𝐻𝑒 3 fusion reaction with the energy of bombarding deuterons and reaction angle 𝐸𝑑 , 𝜃, respectively, and it's predominate recommended value is equal to 14.641. A most important quantity for the analysis of nuclear reaction is the cross section, which measures the probability per pair of particles for the occurrence of the reaction. The cross section 𝜎12(𝑣1) is defined as the number of reactions per target nucleus per unit time when the target is hit by a unit flux of projectile particles that is by one particle per unit target area per unit time. Actually, the above definition applies in general to particles with relative velocity 𝑣 , and is therefore symmetric in the two particles, since we have 𝜎12(𝑣) = 𝜎21(𝑣). Cross section can also express in terms of the center of mass energy, and we have 𝜎12(𝜖) = 𝜎21(𝜖) . In most cases, however, the cross section are measured in experiments in which a beam of particles with energy , measured in the laboratory frame, hits a target at rest. The corresponding beam-target cross section 𝜎12 𝑏𝑡(𝜖1) is related to the center-of-mass cross section 𝜎12(𝜖) by 𝝈𝟏𝟐(𝝐) = 𝝈𝟏𝟐 𝒃𝒕 (𝝐𝟏) With 𝝐𝟏 = 𝝐 ( 𝒎𝟏 + 𝒎𝟐)/𝒎𝟐 Where 𝒎𝟏 represents 𝑫 𝑚𝑎𝑠𝑠, (𝒎𝟏 = 2) , 𝒎𝟐 represents 𝑯𝒆 𝟑 𝑚𝑎𝑠𝑠, (𝒎𝟐 = 3) If the target nuclei have density 𝑛 and are at rest or all move with the same velocity, and the relative velocity is the same for all pairs of projectile-target nuclei, then the probability of reaction per unit times is obtained by multiplying the probability per unit path times the distance travelled in the unit time, which gives 𝑛2 𝜎(𝑣)𝑣. Another important quantity is the reactivity, defined as the probability of reaction per unit time density of target nuclei. It is just given by the product 𝜎 𝑣. In general, target nuclei moves, so that the relative velocity 𝑣 is different for each pair of interacting nuclei. In this case, we compute an averaged reactivity [7]. 〈𝝈 𝒗〉 = ∫𝝈(𝒗)𝒗 𝒇(𝒗)𝒅𝒗, (6) 134 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Where 𝑓(𝑣) is the distribution function of the relative velocities, normalized in such a way that ∫ 𝑓(𝑣)𝑑𝑣 = 1 ∞ 0 . Both controlled fusion fuels and stellar media are usually mixtures of elements where species `1` and `2` have number densities 𝑛1 , 𝑛2 , 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦. the volumetric reaction rate, that is the number of reactions per unit time and per unit volume is then given by [7]: 𝑹 = 𝒏𝟏 𝒏𝟐 𝟏+𝜹𝟏𝟐 〈𝝈 𝒗〉 = 𝒇𝟏 𝒇𝟐 𝟏+𝜹𝟏𝟐 𝒏𝟐 〈𝝈 𝒗〉 (7) Here 𝑛 is the total nuclei number density and 𝑓1 𝑓2, are the atomic fraction of species "1" and "2",respectively.The Kronecker symbol 𝛿12(with 𝛿12 = 1, 𝑖𝑓 𝑖 = 𝑗 𝑎𝑛𝑑 𝛿12 = 0 𝑒𝑙𝑒𝑠𝑤ℎ𝑒𝑟𝑒) is introduced to properly take into account the case of reaction between like particles. Equation 7 shows a very important feature for fusion energy research: the volumetric reaction rate is proportional to the square of the density of the mixture. For feature reference, it is also useful to recast it in terms of the mass density 𝜌 of the reacting fuel 𝑹𝟏𝟐 = 𝒇𝟏 𝒇𝟐 𝟏+𝜹𝟏𝟐 𝝆 𝟐 𝒎� 𝟐 〈𝝈 𝒗〉 (8 ) Where 𝑚� is the average nuclear mass. Here, the mass density is computed as = ∑ 𝑛 𝑚 = 𝑛𝑚� 𝑗 , where the sum is over all species. We also immediately see that the specific reaction rate is proportional to the mass density, again the role of the density of the fuel is achieving efficient release of fusion energy [7]. Main controlled fusion reaction The most important main controlled fusion reactions necessary as power source applications that are recently used in many interested countries which supported huge efforts are listed in Table (1). In the present work, we concentrate on the study of the reaction between the hydrogen isotopes, namely, the deuterium and helium, which are the most important fuels for controlled fusion research. 𝑫 + 𝑯𝒆 𝟑 → 𝑯𝒆 𝟒 (𝟑. 𝟕𝟏𝟐 𝑴𝒆𝑽) + 𝑷 (𝟏𝟒. 𝟔𝟒𝟏 𝑴𝒆𝑽) The above reaction has a cross section, which reaches its maximum (about 0.9 barns) at the relatively modest energy of 250 𝑘𝑒𝑉 (see fig. 1). Its 𝑄𝑣𝑎𝑙𝑢𝑒 = 18.353 𝑀𝑒𝑉 is one of the largest of this family of reactions [7]. Calculation and results Our calculations concern the 𝐷 − 𝐻𝑒 3 fusion reaction, because of its huge applications in power sources due to their high energy release, such as fusion reactors, (Tokomak), and other small systems like the dense plasma focus devices (DPF). Clearly, as it is described previously, in order to calculate such hot parameters, i.e., reactivity, reaction rate, and energy emitted of neutrons are all controlled by the cross section, here represents the essential factor in the calculations. A widely used parameterization of fusion reaction cross section is [10]: 𝝈 ≈ 𝝈𝒈𝒐𝒆𝒎 × 𝓣 × 𝓡 (9) Where 𝜎𝑔𝑜𝑒𝑚 is a geometrical cross section, 𝒯 is the barrier transparency, and is the probability that nuclei come into contact fuse. The first quantity is of the order of the square of the de-Broglie wavelength of the system [10]: 𝝈𝒈𝒐𝒆𝒎 ≈ ג 𝟐 = � ħ 𝒎𝒓𝒗 � 𝟐 ∝ 𝟏 𝝐 , (10) Where is the reduced Planck constant and 𝑚𝑟 is the reduced mass. Equation (9) concerning the barrier transparency, and its often well approximated by 135 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 𝓣 ≈ 𝓣 = 𝐞𝐱𝐩�−� 𝝐 𝝐𝑮 𝟐 � (11) Which is known as the Gamow factor (after the scientist who first computed it) [10]: where 𝝐𝑮 = �𝝅𝜶𝒇𝒁𝟏𝒁𝟐�𝟐𝒎𝒓𝒄𝟐 = 𝟗𝟖𝟔. 𝟏 𝒁𝟏 𝟐𝒁𝟐 𝟐𝑨𝒓 𝒌𝒆𝑽 (12) Is the Gamow energy, 𝛼𝑓 = 𝑒 ℏ𝑐 = 1 137.04 , is the fine structure constant commonly used in quantum mechanics, 𝐴𝑟 = 𝑚𝑟/𝑚𝑝 The reaction characteristics contains essentially all the nuclear physics of the specific reaction [7]. it takes substantially different values depending on the nature of the interaction characterizing the reaction. It is the largest for reaction due to strong nuclear interactions; it is smaller by several orders of magnitude for electromagnetic nuclear interactions; it is still smaller by as many as 20 orders of magnitude for weak interactions. For most reactions, the variation of ℛ(𝜖) is small compared to the strong variation due to the Gamow factor. In conclusion the cross section is often written as 𝝈(𝝐) = 𝑺(𝝐) 𝝐 𝒆𝒙𝒑�−� 𝝐 𝝐𝑮 𝟐 � (13) Where the function 𝑆(𝜖) is called the astrophysical 𝑆 factor, which for many important reactions is a weakly varying function of the energy [9]. Plasma reactivity calculations require reaction cross sections for energies well below those at which direct measurement are practicable, so it is necessary to extrapolate downwards using the theoretical formula [10]. A more convents or suitable formula for the calculation of the total cross section for D-T or others main controlled fusion reactions (listed below), which has a compatible agreement results with the really experimental results is given by. 𝑫 + 𝑫 → 𝑻 (𝟏. 𝟎𝟏𝟏) + 𝑷 (𝟑. 𝟎𝟐𝟐) 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟓𝟎%) 𝑫 + 𝑫 → 𝑯𝒆 (𝟎. 𝟖𝟐𝟎) + 𝒏 (𝟐. 𝟒𝟒𝟗) 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟓𝟎%) 𝑷 + 𝑻 → 𝑯𝒆 𝟑 + 𝒏 − 𝟎. 𝟕𝟔𝟒 𝑫 + 𝑻 → 𝑯𝒆 𝟒 (𝟑. 𝟓𝟔𝟏 𝑴𝒆𝑽) + 𝒏 (𝟏𝟒. 𝟎𝟐𝟗 𝑴𝒆𝑽 ) 𝑻 + 𝑻 → 𝑯𝒆 𝟒 + 𝟐𝒏 + 𝟏𝟏. 𝟑𝟑𝟐 𝑴𝒆𝑽 𝑫 + 𝑯𝒆 𝟑 → 𝑯𝒆 𝟒 (𝟑. 𝟕𝟏𝟐 𝑴𝒆𝑽) + 𝑷 (𝟏𝟒. 𝟔𝟒𝟏 𝑴𝒆𝑽) 𝑻 + 𝑯𝒆 𝟑 → 𝑯𝒆 𝟒 + 𝒏 + 𝑷 + 𝟏𝟐. 𝟎𝟗𝟔 𝑴𝒆𝑽 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟓𝟗%) 𝑻 + 𝑯𝒆 𝟑 → 𝑯𝒆 𝟒 (𝟒. 𝟖𝟎𝟎 𝑴𝒆𝑽) + 𝑫(𝟗. 𝟓𝟐𝟎 𝑴𝒆𝑽) 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚(𝟒𝟏%) 𝑯𝒆 𝟑 + 𝑯𝒆 𝟑 → 𝑯𝒆 𝟒 (𝟎. 𝟖𝟐𝟎 𝑴𝒆𝑽) + 𝟐𝒑 + 𝟏𝟐. 𝟖𝟔𝟎 𝑴𝒆𝑽 𝝈(𝝐) = 𝑺(𝝐) 𝝐 𝒆𝒙𝒑�− 𝑹 √∈𝟐 � 𝒘𝒊𝒕𝒉 𝑹 = 𝝅� 𝒆𝟐 ħ𝒄 � �𝟐𝒎𝒄𝟐 𝟐 𝒁𝟏𝒁𝟐 (14) Where the cross section is expressed in centre of mass units, ∈= 1 2 𝑚𝑣 2, 𝑚 = 𝑚1𝑚2/(,𝑚1 + 𝑚2) and 𝑣 is the relative velocity of the interacting particles which have masses 𝑚1𝑎𝑛𝑑 𝑚2 and charges 𝑍1𝑎𝑛𝑑 𝑍2 respectively.the constants 𝑒, ħ 𝑎𝑛𝑑 𝑐 have their usual meaning. 𝑆(𝜖) = 𝐴 exp(−𝛽 ∈) 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝐴, 𝛽 𝑎𝑛𝑑 𝑅 are given in Table (2). Note that laboratory energies may be used if the substitution ∈= � 𝑚 𝑚1 � ∈𝑙𝑎𝑏 [7]. By testing equation 14, for the 𝐷 − 𝐻𝑒 3 fusion reaction and in order to arrive to results which gives high agreement with the corresponding experimental published results, we find that it is very necessary to introduce a correction factor related to each deuterons energy ( ) in the above equation, and in this case is completely described in Table (3). Theoretically, the energy of the emitted protons from the 𝐷 − 𝐻𝑒 3 fusion reaction can be exactly determined from equation (5) as a function of both the incident deuteron energy 𝐸𝑑and the reaction angle 𝜃. The calculated results are completely described in table (4) and completely described in figure (2). 136 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Finally, the reactivity as a function of the temperature, obtained by numerical integration of the following equation with the best available cross section for the reactions of interest to controlled fusion. 〈𝝈𝒗〉 = �� 𝒎𝟏 + 𝒎𝟐 𝟐𝝅𝒌𝑩𝑻 � 𝟑/𝟐 �𝒅𝑽𝒄 𝒆𝒙𝒑�− 𝒎𝟏 + 𝒎𝟐 𝟐𝒌𝑩𝑻 𝑽𝒄𝟐�� × �� 𝒎𝒓 𝟐𝝅𝒌𝑩𝑻 � 𝟑/𝟐 �𝒅𝑽𝒄 𝒆𝒙𝒑�− 𝒎𝒓 𝟐𝒌𝑩𝑻 𝑽𝟐�𝝈(𝒗)𝒗 � The term in square bracket is unity, being the integral of a normalized Maxwellain, and we left the integral over the relative velocity. By writing the volume element in velocity space as = 4𝜋𝑣2𝑑𝑣 , and using the definition of center of mass energy 𝜖 , we finally get 〈𝝈𝒗〉 = 𝟒𝝅 (𝟐𝝅𝒎𝒓)𝟏/𝟐 𝟏 (𝒌𝑩𝑻)𝟑/𝟐 � 𝝈(∈) ∈ 𝒆𝒙𝒑(−∈/𝒌𝑩𝑻)𝒅 ∞ 𝟎 For the 𝐷 − 𝐻𝑒 3 fusion reaction, which is by far the most important one for present fusion research, the following expression is used to calculate the reactivity [7]. And the present calculated results are completely described in figure 3. 〈𝝈𝒗〉 = 𝟒. 𝟗𝟖 × 𝟏𝟎−𝟏𝟔 𝒆𝒙𝒑�−𝟎. 𝟏𝟓𝟐�𝒍𝒏 𝑻 𝟖𝟎𝟐.𝟔 � 𝟐.𝟔𝟓 � 𝒄𝒎𝟑/𝒔 (15) Which is 10% accurate for temperature in the range 0.5-100 keV Discussion and conclusion Clearly, from the results about the total cross section for the 𝐷 − 𝐻𝑒 3 fusion reaction calculated by equation 14, it appears a common shift from the published experimental results and one can interpret that by some physical reasons that directly correlated with the fundamental parameters deal with the system or device designing, geometrical dimension for the cathode and anode, and the operating factors, such as the fuel pressure, initial power, and we can add another reason which deals with the construction time for building the experiment, in which that any system exactly differs in all covering physical conditions with the recent ones. In other words, it is necessary to give empirical formula for each system (experimental devices). Therefore, we concluded that it is important and necessary to modify the formula for the total cross section for the 𝐷 − 𝐻𝑒 3 fusion reaction by introducing a fixed correction factor for certain deuteron energy/or energy intervals to avoid the disagreement between the theoretical and experimental results. From Table (1) and Table (3), we see that our calculated results about the total reaction cross section, after introducing the correction factors are more compatible with the published results. Also, from Fig. 3, the calculated results about the 𝐷 − 𝐻𝑒 3 fusion reaction reactivity by using equation (15) give or reflect a physical behavior that is more suitable with the corresponding published results, and this case can be interpreted for the reason of the cross section data which are available in the reactivates equation. Finally it's useful to suggest the recommendation of the modified formulas instead of the previously described ones to be applied in the recent systems. From Table (4), it is clear that the energies of emitted proton, which are calculated by our expressed formula at incident reaction angle of 9Oo degree are of quite agreement with the recommended value of (14.029 𝑀𝑒𝑉), and this case can be interpreted as, there exists a small percentage of incident deuterons which are scattered from its original direction, and all the really occurring physical experimental phenomenon can be explained at this angle instead of other angles. 137 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 References 1- John E. Brandenburg, (2012),The hybrid fusion-fission reactor-The energy crisis is solved, STAIF II(The Space Technology and Applications International Form),13-15. 2- Stott P. E. , (2005), Numerical simulations for fusion reactivity enhancement in D-3He and D-T plasmas due to 3He and T minorities heating, Plasma Physics. Control. Fusion, 47, 1305. 3- Stix,T. H. (1975), fast – wave heating of a two-component plasma Nucl. Fusion , 15, 737. 4- Yanagi N. et al,( 2011), Feasibility of Reduced Tritium Circulation in the Heliotron Reactor by Enhancing Fusion Reactivity Using ICRF, Plasma and Fus. Res., 6, 2405046. 5- SHYSHKIN Oleg,( 2012),24th IAEA fusion energy conference-IAEA CN-197. 6- Robly D.Evans , (1955), The atomic nucleus, Mcgraw-Hill book company, inc. 7- Stefano Atzeni, and Juroen M., (2004), The physics of inertial, Oxford University press,. 8- Marc Decrton, Vincent Massaut, Inge Uytdenhouwen, Johan Braet, Frank Druyts, and Erik Laes, (2007), Controlled nuclear fusion:The energy of the stars on earth, Open Report SCK.CEN-BLG-1049. 9- . Heitemes, T.A, . Moses, G.A,and. Santarius, J.F,( 2005), Anaylsis of an improved fusion reaction rate model for use in fusion plasma simulation, fusion technology institute, university of Wisconsin, 1500 engineering drive, Madison, WI 53706, UWFDM- Report. 10- BOSCH, H.S, and HALE, G.M. (1992), Improved formulas for fusion cross-sections and thermal reactivities", nuclear fusion. 32 (4): 611-631. Table No.(1). Main controlled fusion reactions [7] Reaction 𝝈(𝟏𝟎 𝒌𝒆𝑽) 𝒃𝒂𝒓𝒏 𝝈(𝟏𝟎𝟎 𝒌𝒆𝑽) 𝒃𝒂𝒓𝒏 𝛔𝐦𝐚𝐱 𝒃𝒂𝒓𝒏 ∈𝒎𝒂𝒙 𝒌𝒆𝑽 𝐷 + 𝑇 → 𝛼 + 𝑛 2.72 × 10−2 3.43 5.0 64 𝐷 + 𝐷 → 𝑇 + 𝑝 2.81 × 10−4 3.3 × 10−2 0.096 1250 𝐷 + 𝐷 → 𝐻𝑒 3 + 𝑛 2.78 × 10−4 3.7 × 10−2 0.11 1750 𝑇 + 𝑇 → 𝛼 + 2𝑛 7.90 × 10−4 3.4 × 10−2 0.16 1000 𝐷 + 𝐻𝑒 3 → 𝛼 + 𝑝 2.20 × 10−7 0.1 0.9 250 Table No. (2). Low energy cross section parameterization [7] Reaction 𝑨( 𝒃𝒂𝒓𝒏𝒔 − 𝒌𝒆𝑽) 𝜷 (𝒌𝒆𝑽−𝟏) 𝑹 𝒌𝒆𝑽𝟏/𝟐) 𝐷 − 𝐷𝑝 52.6 −5.8 × 10−3 31.39 𝐷 − 𝐷𝑛 52.6 −5.8 × 10−3 31.39 𝐷 − 𝑇 9821 −2.9 × 10−2 34.37 𝑇 − 𝑇 175 9.6 × 10−3 38.41 𝐷 − 𝐻𝑒 3 5666 −5.1 × 10−3 68.74 𝐻𝑒 3 − 𝐻𝑒 3 5500 −5.6 × 10−3 153.70 138 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Table No.(3): The recommended correction factors necessarily for calculations the D- 𝑯𝒆 𝟑 fusion reaction cross section Deutron Energy keV Cross section Barn Deutron Energy keV Cross section Barn Deutron Energy keV Cross section Barn 10 2.162 x10-7 100 9.759 x10-2 190 0.536 20 6.624 x10-5 110 0.128 200 0.608 30 7.801 x10-4 120 0.164 210 0.686 40 3.308 x10-3 130 0.204 220 0.768 50 8.773 x10-3 140 0.249 230 0.856 60 1.794 x10-2 150 0.296 240 0.950 70 3.126 x10-2 160 0.349 250 1.049 80 4.894 x10-2 170 0.407 90 7.105 x10-2 180 0.469 Table No.(4): The calculated energy of emitted neutrons as a function of the reaction angle 𝜃 = 0 𝑑𝑒𝑔𝑟𝑒𝑒 𝜃 = 45 𝑑𝑒𝑔𝑟𝑒𝑒 𝜃 = 60 𝑑𝑒𝑔𝑟𝑒𝑒 𝜃 = 90 𝑑𝑒𝑔𝑟𝑒𝑒 𝐸𝑑 𝑘𝑒𝑉 𝐸𝑛 𝑀𝑒𝑉 𝐸𝑑 𝑘𝑒𝑉 𝐸𝑛 𝑀𝑒𝑉 𝐸𝑑 𝑘𝑒𝑉 𝐸𝑛 𝑀𝑒𝑉 𝐸𝑑 𝑘𝑒𝑉 𝐸𝑛 𝑀𝑒𝑉 20 15.000 20 14.909 20 14.845 20 14.691 30 15.075 30 14.962 30 14.884 30 14.695 40 15.138 40 15.008 40 14.917 40 14.699 50 15.195 50 15.049 50 14.947 50 14.703 60 15.247 60 15.087 60 14.975 60 14.707 70 15.296 70 15.122 70 15.000 70 14.711 80 15.341 80 15.155 80 15.025 80 14.715 90 15.384 90 15.186 90 15.048 90 14.719 100 15.425 100 15.216 100 15.069 100 14.723 139 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Figure No.(1) : Measured cross sections for different fusion reactions as a function of the averaged centre of mass energy. Reaction cross sections are measured in barn (1 barn= 10-28 m2) [8]. Figure No.( 2) : Variation of the emitted proton energy versus the incident deuteron energy 101 103 102 104 Energy [keV] C ro ss s ec tio ns σ [b ar n] D-T D -3He D-D 10 1 10-1 10-2 10-3 10-4 140 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Figure No.(3) : Variation of the D-He Reactivity versus the incident deuteron temperature 141 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 𝑫المعلمات الممیزة للتفاعل النووي االندماجي نوع − 𝑯𝒆 𝟑 علي كاظم تقي رعد حمید مجید مصطفى كامل جاسم جامعة بغدادقسم الفیزیاء / كلیة التربیة للعلوم الصرفة (ابن الھیثم) / 2013اذار 17قبل البحث في ، 2012ایلول 12استلم البحث في الخالصة من التفاعالت التجریبیة المرغوبة والمفضلة لدى الدول المھتمة بانتاج D-3Heالتفاعل النووي االندماجي نوع یعد وتفاعلھما االندماجي ینتج عد من المواد المستقره یالطاقة والبحوث العلمیة وذلك لكون كال من الدیتریوم والھلیوم امكانیة درع او ال عنفض، D-Tبدال من انتاج النیوترونات للتفاعل من نوع MeV 14البروتونات وبطاقة تصل الى التفاعلیة، مثل طة المجال المغناطیسي. االعتمادیة العالیة الدرجة لعوامل البالزما الحاره احصر البروتون الناتج بوس جعل خصوصیة بحث ت للتفاعل النووي االندماجي معدل التفاعل، وطاقة البروتونات المنبعثة على المقطع العرضي الكليو الجل الوصول الى امكانیة تحویرھا نظریا ومن ثم لھذه المقاطع العرضیة تجریبیة مالئمة قدر االمكان او اختیار عالقات .من اھداف البحث الحالي توافق عال جدا مع نظیرتھا العملیة المنشورة حدیثا، یعد ، التفاعلیة ، معدل التفاعل ، طاقة البروتون ، البالزما الحاره ، المقطع العرضي D-3He: التفاعل كلمات مفتاحیةال االندماجي 142 | Physics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013