Communications in Physics, Vol 19, No (2009), pp 39-44 TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE DIODE-LASER SIDE-PUMPED SOLID-STATE LASER ROD MAI VAN LUU, DINH XUAN KHOA, AND VU NGOC SAU Vinh University HO QUANG QUY Academy of Military Science and Technology Abstract Based on the assumption that Gaussian pump power of diode laser bar is the same at any cross-section along the laser rod and its curvated surface plays as thin lens, the expression describing the pump intensity distribution inside laser rod was obtained by transfer matrix To have the cross-section of active volume or excited volume coincides with one of laser mode volume, the dependence of pump intensity distribution on location of outside pump beams is investigated by simulation I INTRODUCTION Recently, the diode laser-pumped solid-state lasers from the very small [1] to the kilowatt level of output power [2,3] are interested and developed by because of their efficient use in high technology Mode size optimization in laser-diode end-pumped lasers has been investigated [4,5] Side-pumping geometry can be used to achieve higher output powers [6] For analysis in above-mentioned work, there were the following assumptions made: Distribution of diode bar around the rod is assumed to produce an azimuthal uniform illumination; reflection and refraction effects are not considered; to separate the calculation of the absorption profile from how the pump light travels from the diode bar to the surface of the rod, one describes the pump beam from the diodes only after the beams have entered the rod; the pump beam is assumed to travel through the rod only once, i.e reintroduction of a pump beam through reflectors is not discussed; a single-absorption coefficient can be used to describe the absorption process Consequently, these assumptions lead to that: first, it is not suitable for optimality of pump stored energy in laser rod; second, one can not choose the optimal parameters for matching between the size of the pump volume and laser mode one; thirdly, it is still not assumed the laser rod as a focusing lens, which is a important fact influences on the pump energy distribution.To advoid above problems, we present a four-side-pumped structure for solid-state laser and a new cross-sectional geometry of the laser rod pumped by diode bar Pumped by four laser-diode bars, which has a Gaussian distribution in far field, so transverse intensity distribution in active rod of solid-state laser can be changed and influences on dimension of effective “pencil” and then on laser beam structure 40 TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE DIODE-LASER SIDE-PUMPED II PUMP INTENSITY DISTRIBUTION As shown in Keming’s work [7] and Carts’s work [8], the cross-sectional geometry of the laser rod pumped by four laser diode bars can be illustrated in Fig The diode sources are assumed to have a Gaussian emittance profile (transverse distribution) and are conditioned such that they are effectively arrayed uniformly around the rod, i.e they uniformly distribute along axis-z (a) (b) (x,y) point Reforming lens x Laser Rod with r0 and n W0 W in0 R’(y) R(y) y Laser Beam Diode laser y0 r0 Fig a- Cavity geometry for four sides-pumping module, b- Cross-sectional geometry of Gaussian beam outside and inside rod We assume that the laser rod has a radiusr0 and a refractive index n, the Gaussian beam of laser diode bar in cross-section of laser rod, which is placed at point y0 from outside surface of rod, has a complex amplitude [9]: x2 x2 W0 exp − exp −jky − jk + jξ(y) W (y) W (y) 2R(y) where, y is the proprating direction and x is expanding direction, U (x, y) = A0 R(y) = y + W0 = λb π 1/2 y b W (y) = W0 + b y is the radius at point y in propagating direction, (1) (2) is the wavefront radius of curvature, (3) 1/2 is the beam waist, ξ(y) = tan−1 (y/b) is the excess phase (i.e.,initial phase), (4) (5) and b is the Rayleigh range (see Fig 1b) Propagating through the rod from one side, the phase of this beam will be changed as well as after propagating through thin lens (see Fig 1a) with focal length [9] MAI VAN LUU et al 41 f = r0 /(n − 1) , (6) so that the complex amplitude transmittance of this lens is proportional toexp jkx /2f and then the phase of the transmitted wave is altered to [9] ky + k x2 x2 x2 − ξ (y) − k = ky + k − ξ(y) 2R(y) 2f 2R (y) (7) where 1/R (y) = 1/R(y) − 1/f (8) Using (3), (7) and (8), substituting into (1), we obtained the complex amplitude of pump beam inside laser rod, given by Uin (x, y) = A0 x2 x2 Win0 exp − exp −jky − jk + jξ(y) Win (y) 2R (y) Win (y) where Win0 = M W0 ; bin = M= √Mt ; 1+t t= M b; Win = Win0 + b y0 −r0 /(n−1) ; Mt = r0 y0 (n−1)−r0 y bin 1/2   ;    (9) (10) The waist of “inside” beam has the location at center of laser rod when following condition is satisfied r0 r0 r0= M y0 − + (11) n−1 n−1 Using (4), (10) and (11) we obtained the location of waist of “outside” beam r0 πW02 − (12) n−2 λ i.e., it depends on radius and refractive index of laser rod and waist and wavelength of pump beam From (9) we have the expression of the pump energy distribution inside the laser rod for single-side-pumping as following y0 = Iin (x, y) = I0 Win0 Win (y) exp − 2x2 (y) Win (13) We assume that pump intensity distribution (13) is symmetry for center of the rod, which seems to be an origin of co-ordinate system (x=0, y=0), i.e it is means that Iin (x, y) = I(−x, y) = I(x, −y) = I(−x, −y) (14) Really, every laser diode bar used as a pump lamp having narrow spectra to enhance conversion efficiency The coherent quality of laser beam is not important for this purpose Moreover, the absorption of active particles in laser rod to create population inversion is not stimulated, but is spontanuous (every pump photon reaches laser rod at different time after reflection from reflector [10]), so population inversion creating is statistical process and population inversion depends on total pump intensity (the sum of intensities integrated over pulse-duration time of all diode lasers), it means that total excited particles at defined 42 TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE DIODE-LASER SIDE-PUMPED time is proportional to sum of intensities of all pump lasers at that time With considering that the delay time between all lasers is less than lifetime of upper laser level (it means that phase mitmach between all laser beams can be negleted), so that the pump intensity distribution for two opposite sides-pumping is given by Itwoside (x, y) = Iin (x, y) + Iin (x, −y) (15) and similarly, for four sides-pumping is given by If ourside (x, y) = Itwoside (x, y) + Itwoside (y, x) (16) III SIMULATION AND DISCUSSION We assume that the parameters of pump beam chosen to be W0 = 1mm, λ = 860nm and the parameters of laser rod chosen to be r0 = 6mm, and a refractive index, n = 1.78 The location of waist of pump beam is calculated from (12) Now pump intensity distributions inside the laser rod for side-pumped solid-state laser with two sides and four sides can be obtained as shown in Fig.2a and Fig.2b, respectively (a) (b) Fig Pump intensity distribution for a two-side-pumped (a) and four-sidepumped (b) solid-state lasers In Fig can see overlap pump profile in x-axial plane for one-side-pumped (a) and for four side-pumped (b) laser In Fig can see overlap pump profile in y-axial plane for one side-pumped (a) and four-side-pumped (b) laser After comparison between all profiles in two figures (Fig and Fig 4), we can conclude that the overlap of the pump intensity distribution at the center of the laser rod closely resembles a Gaussian distribution for the four-side-pumped laser The waist (Wp) of the Gaussian overlap can be changed by changing the location of outside beam For example, in Fig 5, one can see that the cross section (πW2p) at level with the same energy in case of y0 =10 mm is larger than the one in the case of y0 =15mm This means that the effective cross-section defined as cross-section of total intensity distribution at level of IM AX /e can be chosen so that it coincides with cross-section of the laser mode MAI VAN LUU et al (a) 43 (b) Fig Overlap pump-intensity profile in the x-axial plane for one-side-pumped (a) and four-side-pumped (b) lasers (a) (b) Fig Overlap pump-intensity profile in the y-axial plane for one-side-pumped (a) and four-side-pumped (b) lasers volume (πW20L) by changing the location of outside beam (y0 ) as shown above, when other parameters as beam waist of pump beam (W0p) and pump wavelength (λ) are given IV CONCLUSION The expression of the pump intensity deposition inside the solid-state laser rod pumped by diode laser bars is introduced The pump intensity distribution, the pump volume are dependent not only on parameters of pump beam, but also on parameters of laser rod and the location of pump beam from the laser rod Since that obtained results are useful not only for optimization conversion efficiency, but also for reducing the thermal 44 TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE DIODE-LASER SIDE-PUMPED (a) (b) Fig Four-side-pump- intensity distribution inside the laser rod with two values of pump beam location: y0 = 15 mm (a) and y0 = 10 mm (b) effect in laser rod Moreover, the longitudinal distribution of pump intensity in laser mode volume of the side-pumped laser is important question, which will be investigated in the next article REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] B J Comaskey, et al., IEEE J Quantum Electron., 28 (1992) 992-996 N Hodgson, S Dong, and Q Lu, Opt Lett 18 (1993) 1727-1729 R J St Pierre et al., J Sel.Top Quantum Electron (1997) 53-58 T Y Fan, and R L Byer, IEEE J Quantum Electron 24 (1988) 895-912 Y F Chen et al., IEEE J Quantum Electron 33 (1997) 1424-1429 W Xie et al., Applied Optics 39 (2000) 5482-5487 Du Keming et al., Appl Optics 37 (1998) 2361-2364 Y A Carts, Diode Lasers, Nonlinear Optics, and Solid-State Lasers, 1992 B E A Saleh, and M.C Teich, Fundamentals of Photonics, A Wiley-Interscience Publication (1991) O Svelto, Principles of Lasers, Plenum Press, New York and London, 1979 Received 11 December 2008 Communications in Physics, Vol 19, No (2009), pp 45-52 INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY OF HPGE GAMMA SPECTROSCOPY USING MCNP TRUONG THI HONG LOAN, DANG NGUYEN PHUONG, DO PHAM HUU PHONG, AND TRAN AI KHANH Faculty of Physics, University of Natural Sciences, Vietnam National University, Ho Chi Minh City Abstract When determining radioactivities in environmental samples using low-level gamma spectroscopy, in order to raise detection limit, voluminous samples are used It takes in account for the self-absorption (self-attenuation) of gamma rays in samples The self-absorption effect is small or large depend on the sample shapes, matrices and densities In this paper, we investigated the effect of some regular matrices such as water, soil, epoxy resin on the detector efficiency Some analytical formulas for the correction of matrix and densities for soil sample was established and applied to calculate some activities from standard sample of IAEA-375 I INTRODUCTION One of the most important problems of radioactivity measurement is investigating the detection efficiency There are lots of factors can affect the efficiency such as: incident gamma ray energy, measuring geometry, electronic system, detector itself, other effects like coincidence summing or self-absorption Among them, self-absorption is the most interesting effect when investigating activities of environmental samples because of their large volumes One of the most regular geometries used in investigating activities of environmental samples is Marinelli beaker geometry, which has 3π measuring geometry, so the efficiency is very high Usually, Marinelli beaker samples have large volumes so the self-absorption effect of these samples is significant With the MCNP4C2 code [1], by simulating the measuring processes of environmental samples using the HPGe spectroscopy in Nuclear Physics Laboratory, we investigated the effect of matrices and densities on the efficiency Based on that, a correction method was presented to calculate detection efficiencies for environmental samples II CONFIGURATION OF SPECTROSCOPY - SAMPLE USED IN SIMULATION AND EXPERIMENT II.1 HPGe spectroscopy The HPGe detector in Department of Nuclear Physics, model GC2018, is a coaxial detector with configuration showed in Fig.1, including a germanium cylinder crystal with 52 mm outer diameter, 49.5 mm height Inside the crystal, there is a hole with mm 46 INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY diameter, 35 mm depth There are outer n-type contact layer (lithium layer), inner p-type contact layer (boron layer) of the crystal The detector is hold in an aluminium box with 1.5 mm thickness [3] There is a lead shield outside detector to absorb gamma rays from environment and suppress spectrum background The interactions between gamma rays and lead shield layer produce X-rays with energies in the range 7388 keV These X-rays can be detected by detector and effect on the gamma spectrum To limit this problem, the copper and tin liners were lined covering the lead shield with the thickness of 1.6 mm and mm respectively The X-rays emitted by lead will be absorbed by the tin, and X-rays from the tin (about 2530 keV) will be absorbed by cooper Finally, the cooper emits low energy X-rays (about keV) which does not present on the spectrum Fig The configuration of HPGe detector (in milimeter) II.2 Samples The samples were contained in Marinelli beakers, which sizes were shown in Fig These beakers were put on detector to make the 3π measuring geometry Fig The configuration of Marinelli sample (in centimeter) TRUONG THI HONG LOAN et al 47 III SIMULATION OF PEAK EFFICIENCY CURVES OF HPGE DETECTOR WITH MATRICES AND DENSITIES III.1 Matrices used in simulation To investigate the effect of matrices on detection efficiency, we need to simulate the efficiencies with and without matrices There were three types of matrices to simulate: soil, water and epoxy resin The simulated volumes were the same with all types, the simulated densities were 0.5 g/cm3 , 1.0 g/cm3 and 2.0 g/cm3 Three types of matrices [3]: Soil (% mass of atom in molecular): hydrogen 2.2%, oxygen 57.5%, aluminium 8.5%, silicon 26.2%, iron 5.6%; Epoxy resin (% mass of atom in molecular): hydrogen 6.0%, oxygen 21.9%, carbon 72.1%; Water (% mass of atom in molecular): hydrogen 11.11%, oxygen 88.89% To obtain the efficiency without matrix, simulated sample was chosen is air sample with density 0.00129 g/cm3 , includes 79% nitrogen and 21% oxygen The size and volume of this sample is the same as soil, water and resin samples The simulated results of air matrix (efficiencies without self-absorption) were presented in Table Table Detection efficiencies with air matrix (ε0 ) Radionuclide 241 Am 238 U 109 Cd 228 Ac 57 Co 214 Pb 137 54 60 Cs Mn Co Energy (keV) 59.6 63.3 88.2 93.3 122.0 295.0 352.0 661.6 834.8 1173.3 1332.5 Detection efficiency (ε0 ) 0.0186080 0.0225394 0.0423534 0.0446783 0.0508223 0.0316577 0.0268676 0.0151516 0.0124248 0.0094232 0.0085116 By presenting the dependence of efficiency on energy as a logarithmic function [3] by fitting, we have: ln(ε) = 0.0221(ln E)5 − 0.7226(ln E)4 + 9.4711(ln E)3 − 62.158(ln E)2 + 203.16 ln E − 266.2 (1) Fig 3, Fig 4, and Fig presented the simulated efficiencies with different matrices and densities There are some comments based on the above results: - The difference between soil, water and epoxy resin in compare with air samples increases when densities of matrices increase This can be explained when we know that if 48 INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY Fig Efficiencies at density 0.5 g/cm3 Fig Efficiencies at density 1.0 g/cm3 Fig Efficiencies at density 2.0 g/cm3 the density increases, the number of gamma rays can reach detector will decrease (because of losing more energy by interacting with matrix), so the efficiency will decrease - In the energy range below 100 keV, the effect of matrix is more significant than in the energy range above 100 keV 50 INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY Table Self-absorption correction factor of soil sample E (keV) 59.6 63.3 88.2 93.3 122.0 295.0 352.0 661.2 834.8 1173.3 1332.5 Self-absorption correction factors f at densities ρ 0.5 g/cm3 0.8 g/cm3 1.0 g/cm3 1.2 g/cm3 2.0 g/cm3 0.89 0.83 0.79 0.76 0.64 0.89 0.83 0.80 0.77 0.65 0.92 0.87 0.84 0.81 0.72 0.92 0.87 0.85 0.82 0.72 0.93 0.89 0.86 0.84 0.75 0.95 0.92 0.90 0.88 0.81 0.95 0.92 0.91 0.89 0.83 0.96 0.94 0.93 0.91 0.86 0.97 0.95 0.93 0.92 0.88 0.97 0.95 0.94 0.93 0.89 0.97 0.96 0.95 0.94 0.90 Fig The dependence of factor f on energy and density of soil matrix Marinelli beaker with the same geometry as the simulation (Fig 2) The sample density ρ = 1.503 g/cm3 , sample was measured for days with HPGe detector Activities of long-lived radionuclides were calculated by absolute method: A= S ε(E).θ.m.tm (7) A is the source activity at the time of acquisition (Bq/kg), S is the net peak area of the concerned peak, ε(E) is the efficiency at energy E, m is sample weight (kg), θ is the branching ratio of the observed nuclide at this energy E(%), tm : the live time of the measurement (s) TRUONG THI HONG LOAN et al 51 Using formulas (4), (5), and (6) to calculate three parameters a, b, and c: a = −0.0071 × 1.503 − 0.0054 = −0.01607 b = 0.1144 × 1.503 + 0.0710 = 0.24294 c = −0.5067 × 1.503 + 0.7622 = 0.00063 After that, using formula (3) to obtain self-absorption correction factor f Applying formula (1) to calculate the detection efficiencies without self-absorption ε0 The actual efficiencies were calculated by formula (2) Calculated results were presented in Table Table Detection efficiencies at some investigated energies of standard sample IAEA-375 Radionuclide 137 212 Pb 214 ( Cs 232 Th Pb (226 Ra) 40 E (keV) Correction factor f K ) 661.7 238.6 338.3 583.2 911.6 295.2 351.9 609.3 1460.8 0.900569 0.848982 0.870405 0.895998 0.909874 0.862510 0.872578 0.897642 0.917563 Detecting ε0 0.013684 0.036071 0.026255 0.015454 0.010112 0.029861 0.025279 0.014813 0.006670 efficiency ε 0.012323 0.030624 0.022853 0.013847 0.009201 0.025755 0.022058 0.013297 0.006120 Using formula (7) to calculate activities of radionuclides after background subtraction, results are presented in Table 4: Table Activities of investigated radionuclides Radionuclide 137 Cs 212 Pb ( 214 232 Th ) Pb (226 Ra) 40 K E (keV) Peak area S Emission probability (%) 661.667 7,308,484 (0.04) 0.8499 238.632 52,289 (2.70) 0.436 338.320 10,554 (7.52) 0.1127 583.187 16,239 (2.64) 0.845 911.204 11,134 (1.16) 0.258 Mean activity of 232Th : A = 21 ± Bq/kg 295.224 19,546 (4.23) 0.18414 351.932 34,019 (2.36) 0.356 609.316 22,391 (2.80) 0.4642 Mean activity of 226Ra : A = 20.2 ± 0.7 Bq/kg 1460.822 54,922 (0.42) 0.1066 Activity A (Bq/kg) 5.190 ± 260 20 ± 21 ±2 19.6 ± 1.1 23.9 ± 1.2 21.0 ± 1.4 22.1 ± 1.2 18.5 ± 1.0 429.4 ± 21.6 Note: The number in parentheses is the relative standard deviation (%) due to counting statistics Finally, comparing calculated results with values of IAEA: 52 INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY Table Activity comparison of investigated radionuclides of standard sample IAEA-375 Radionuclide 137 Cs Ra 232 Th 40 K 226 Activity A (Bq/kg) (95% Confidence Interval) Our results IAEA [4] 4680 – 5700 5200 – 5360 17.2 – 23.2 18 – 22 17.8 – 24.2 19.2 – 21.9 387 – 472 417 – 432 From Table 5, the calculated activities of three radionuclides 137 Cs, 226 Ra, 232 Th and K agreed with given values of IAEA-375 In brief, we can accept this calculation method in calculating detection efficiency of environmental samples by using self-absorption correction factor with varied density 40 IV CONCLUSION In this paper, the MCNP4C2 code was used to investigate the effect of matrices on detection efficiency of HPGe detector of Nuclear Physics Department, University of Natural Sciences, HCMC The results showed that with regular densities (from 0.5 to 2.0 g/cm3 ), the effect of matrices can be neglected when investigating gamma rays with energies higher than 100 keV Then the MCNP4C2 code was continued to establish the relation between self-absorption correction factor and sample density The simulation results showed that the correction factor changes linear with the change of sample density, and we also established the analytic formulas for correction factor With the obtained analytic formulas, we carried out correcting detection efficiency with standard sample IAEA-375 The agreement between calculated activities with self-absorption correction and values from IAEA showed that the correction is quite exact Therefore, the simulation method with MCNP4C2 code can help us in investigating and correcting the effect of matrices and density on detection efficiency of gamma spectroscopy REFERENCES [1] J.F Briesmeister, MCNP4C2- Monte Carlo N-particle Transport Code System, LA-13709-M (June 2001) [2] Truong thi Hong Loan, Tran Ai Khanh, Dang Nguyen Phuong, Do Pham Huu Phong, Efficiency calibration for HPGe detector with voluminous sample geometry using Monte Carlo method, Summarization Report, Vietnam National University – HCMC Scientific Project, Code number: B2007-18-08, The University of Natural Sciences - HCMC (2008) [3] http://www.canberra.com [4] http://www-naweb.iaea.org/naml/AQCS.asp Received 20 August 2008 Communications in Physics, Vol 19, No (2009), pp 53-58 NEUTRON YIELD FROM (γ, n) AND (γ , 2n) REACTIONS FOLLOWING 100 MeV BREMSSTRAHLUNG IN A TUNGSTEN TARGET NGUYEN TUAN KHAI, TRAN DUC THIEP, TRUONG THI AN, PHAN VIET CUONG, AND NGUYEN THE VINH Institute of Physics, VAST Abstract The photonuclear reactions of (γ, xn) or (γ, xnp) types can be used to produce highintensity neutron sources for research and applied purposes In this work a Monte-Carlo calculation has been used to evaluate the production yield of neutrons from the (γ, n) and (γ, 2n) reactions following the bremsstrahlung produced by a 100 MeV electron beam on a tungsten target I INTRODUCTION The bremsstrahlung emissions produced by accelerated electron beams are intense and high-energy photon sources They are widely used in photonuclear reaction research and applied nuclear physics In electron accelerators, tungsten (W) is often used as a target because it has a large cross section for bremsstrahlung production, a high melting temperature and good heat conductivity [1] Moreover, for the tungsten isotopes, 180,182,183,184,186W, the cross sections of the (γ, n) and (γ, 2n) reactions are relatively high [2] Therefore, these reactions can be used to produce secondary neutrons for research purposes during accelerator operation In this paper we show that such reactions provide a high-intensity neutron source by evaluating the expected neutron yield in the case of the 100 MeV electron beam of the Linear Electron Accelerator (LUE-100 Linac) at the Joint Institute for Nuclear Research (JINR), Dubna, using a 1.5 mm thick tungsten target [3] In a first step the energy and angular distributions of the bremsstrahlung photons are evaluated, using both theoretical models and experimental data, especially those related to non-zero emission angles [4, 5] In a second step a folding with the reaction cross sections gives an estimation of the production yield of neutrons from the photonuclear reactions of interest II TOTAL NEUTRON YIELD Bremsstrahlung photons can be emitted whenever a charged particle experiences a change in momentum under the influence of the Coulomb field of a nucleus The rate of energy loss due to bremsstrahlung and the cross-section for its production are inversely 54 NEUTRON YIELD FROM (γ, n) AND (γ, 2n) REACTIONS proportional to the square of the mass of the incident particle [6]: dEb/dt ∼ Z Zt2 /m2 σb ∼ Zt2 (e2 /mc2 )2 (1) (2) where m and Z are, respectively, the mass and the charge of the beam particle, and Zt is the atomic number of the target Bremsstrahlung emission is, therefore, a major energy loss mechanism for electrons, the lightest charged particle, especially at relativistic energies greater than a few MeV At very low electron energies the angular distribution of bremsstrahlung is maximum in the direction perpendicular to the incident beam [6, 7] However, as the energy is increased, the maximum occurs at increasingly forward angles and in the limit of very high energies, the emission of bremsstrahlung essentially occurs in a narrow cone in the forward direction The root-mean-square (rms) angle of emission is then given by [6]: θγ ≈ me c2 /Ee (3) with Ee being the total energy of the incident electron and me – its rest mass Calculations of the spectral characteristics of bremsstrahlung photons and scattered electrons when the relativistic incident electron beam hits a target have been described earlier [4, 5] Amongst the secondary interactions induced by bremsstrahlung photons with the target material, photonuclear reactions become possible at energies larger than the reaction thresholds Fig shows the spectra of bremsstrahlung photons emitted in the angular range of 0˚ – 20˚ at the incident electron energy of 100 MeV on a 1.5 mm thick tungsten target [5] The resulting energy distribution is used in the Monte Carlo calculation to generate randomly photons having the proper energy spectrum The reaction yield is expressed by the relation: Emax Y = Nt σ(E)I(E)dE (4) Eth where σ(E) is the photonuclear reaction cross section and I(E) is the bremsstrahlungspectral intensity, Nt being the number of the target nuclei per cm2 : Nt = ζ(NAvog /A)ρt (5) Here, ζ is the isotopic enrichment, NAvog is Avogadro number, ρ(g / cm3 ) and t (cm) are the density and thickness of the target, respectively Eth and Emax are the reaction threshold and the maximal energy of the bremsstrahlung spectrum For each tungsten isotope the yields of (γ, n) and (γ, 2n) reactions are determined by using the simulated bremsstrahlung spectrum (curve in Fig 1) and the proper reaction cross sections (Fig for 186 W isotope) The neutron yield, Y(γ,xn) , from the two types of reactions is the sum of the individual yields The total yield is obtained by adding the yields of each isotope properly weighted by their fractional abundance NGUYEN TUAN KHAI et al 55 Fig Bremsstrahlung emission at different angles for the case of using the 100 MeV electron beam incident on 1.5 mm tungsten [5]: (1) total spectrum, (2) from to 5˚, (3) from 5˚ to 10˚, (4) from 10˚ to 15˚, (5) from 15˚ to 20˚ Photon Energy, MeV Fig Excitation functions [2]: (1)186 W(γ, n)185W reaction (2)186 W(γ, 2n)184 W reaction The uncertainty of the neutron yield was determined on the basis of the uncertainty of the cross section data and the error in determining the photon intensities from simulated bremsstrahlung spectrum Fig shows the simulation results for the production yields of the secondary particles, Y(e,xγ) for the bremsstrahlung photons and Y(e,xe’) for the emitted electrons as a function of the tungsten target thickness, i.e number of these secondary particles per one incident electron [5] For example, at 1.5 mm thickness of the tungsten target the yield values are, respectively, 4.14 and 1.14 for the bremsstrahlung photons and emitted electrons This consideration is necessary to determine the neutron yields, Y(e,xn) , directly from 56 NEUTRON YIELD FROM (γ, n) AND (γ, 2n) REACTIONS Fig Production yields of secondary particles as a function of the target thickness [5] information on the electron current used in accelerator operation: Y(e,xn) = Y(e,xγ) xY(γ,xn) (6) The obtained results of the neutron yields Y(e,xn) are summarized in Table As a result, a total yield of about (1.01 ± 0.09)10−3 n / electron was determined for neutron production from the above mentioned photonuclear reactions For example, at a typical electron current 100 µA, i.e corresponding to the beam intensity of about 6.2 × 1014 electron / s, we can estimate two following results for the secondary neutron emission: i) A total neutron intensity of about (6.26 ± 0.56)1011 n / s is able to be produced during the accelerator operation ii) If it is supposed that the neutron measurement is performed at a distance 10 m from the target by using a detector with 30 cm radius The solid angle covered by this detector is about 0.25 mrad Therefore, the neutron amount which is able to reach the detector is evaluated as: N = (6.26 ± 0.56)1011 × 0.25 × 10−3 /4π = (1.25 ± 0.11)107n/s III NEUTRON ENERGY AND ANGULAR DISTRIBUTIONS Besides evaluating the total neutron yield, the Monte Carlo method makes it possible to calculate the energy and angular distributions of the produced neutrons once the angular dependence of the photonuclear reaction cross-section is known In case of the (γ, n) reaction, the energy – momentum conservation relates the neutron energy En to its production angle θn via: √ M2r = (∆E + Mn )2 - 2∆E(En + Mn ) + 2Eγ (∆E - En – cosθn E2n – M2n ) (7) where Mr is the mass of the final state nucleus (c = 1), ∆E = (Mt - Mr – Mn )c2 with Mt the mass of the target nucleus, Mn that of the neutron and Eγ the incident photon energy – Note that the neutron kinetic energy is Tn = En – Mn The angular distribution of the photonuclear cross section is taken from [8, 9] It has form P(θn ) = A + B* sin2 θn with B / A = 2.0 ± 0.5 [9] We justify this choice by remarking that the photons which are active in producing neutrons have energies concentrated above threshold whatever the electron energy NGUYEN TUAN KHAI et al 57 Table Neutron yields and nuclear data used for yield determination Abundance (%) 186 W (28.60) Reaction Threshold energy (MeV) 186 W(γ, n) 185 W 7.19 Yield (n / electron) (1.91 ± 0.13)*10−4 186 (1.00 ± 0.09)*10−4 W(γ, W 184 W(γ, n) 2n) 12.95 184 184 W (30.70) 184 W(γ, 183 W 7.41 (2.21 ± 0.14)*10−4 2n) 13.60 (1.11 ± 0.09)*10−4 182 W 183 W (14.28) 183 W(γ, n) 183 W(γ, W 182 W(γ, n) 182 W 6.19 (0.95 ± 0.08)*10−4 2n) 14.26 (0.38 ± 0.04)*10−4 181 182 W (26.30) 182 W(γ, 181 W 8.07 (1.65 ± 0.12)*10−4 2n) 14.75 (0.92 ± 0.07)*10−4 180 W 180 W (0.12) 180 W (γ, 179 W 180 W(γ, 178 W n) 8.41 (0.009 ± 0.003)*10−4 2n) 15.35 (0.002 ± 0.001)*10−4 En (MeV) Fig The energy spectrum of neutrons emitted by the 186 W(γ, n)185 W reaction The bremsstrahlung radiation is taken to be exactly forward The angular distribution of the produced neutrons is therefore the same as [8, 9] while the neutron energy spectrum is displayed in Fig 58 NEUTRON YIELD FROM (γ, n) AND (γ, 2n) REACTIONS IV CONCLUSION We have used a Monte-Carlo calculation to evaluate the total neutron yield from photonuclear reactions (γ, n) and (γ, 2n) induced by bremsstrahlung photons radiated by a 100 MeV electron beam incident on a 1.5 mm tungsten target The bremsstrahlung spectrum was calculated and folded with the cross sections of the (γ, n) and (γ, 2n) reactions for the various tungsten isotopes present in the target The energy and angular distributions of the produced neutrons were calculated under the assumptions that the bremsstrahlung radiation is exactly forward and the direct interaction model can be used to consider the neutron emission In this work we determined the total neutron yield which is about (1.01 ± 0.09)10−3 n / electron This value makes us possible to evaluate the neutron emission as a secondary source produced when accelerator is operated at a given electron current In reality there may be additional contributions to the neutron production from other types of nuclear reactions induced by 100 MeV bremsstrahlung photons such as (γ, np) and (γ, xn) reactions with high neutron multiplicity as well as spallation processes However, their contribution to the total neutron yield should not exceed a few percents as they imply higher energy incident photons associated with lower bremsstrahlung photon intensities ACKNOWLEDGEMENT This work has been performed with the financial support by the National Research Program on Natural Science under grant No 403806 We would like to express sincere thanks for this precious assistance We are grateful to Prof Pierre Darrulat for useful discussions to improve the quality of the work REFERENCES [1] C P Kapisa and V N Melekhin,Microtron, Publisher Nauka, Moscow, 1969 [2] IAEA Photonuclear Data Library: http://www-nds.iaea.org/photonuclear/ http://cdfe.sinp.msu.ru/exfor/index.php [3] A V Belushkin, Report on Scientific Programme of the Frank Laboratory of Neutron Physics, Dubna 2006 [4] N T Khai and T D Thiep, Comm in Phys 13 (2003) 149 [5] GEANT4: http://www.slac.stanford.edu/comp/physics/geant4/geant4.html [6] P Marmier and E Sheldon, Physics of Nuclei and Particles, Vol 1, Academic Press, New York and London, 1969 [7] W R Leo, Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag Berlin Heidelberg 1987, 1994 [8] F Tagliabue and J Goldemberg, Nucl Phys 23 (1961) 144 [9] G E Price et al., Phys Rev 93 (1954) 1279 [10] F R Allum et al., Nucl Phys 53 (1964) 545 [11] G C Reinhardt et al., Nucl Phys 30 (1962) 201 Received 20 August 2008 Communications in Physics, Vol 19, No (2009), pp 1-6 A SUPERLUMINAL FORMALISM FOR MAJORANA-LIKE LEPTON VO VAN THUAN Vietnam-Auger Cosmic Ray Laboratory (VATLY) Institute for Nuclear Science and Technology (INST) 179 Hoang Quoc Viet Street, Nghia Do, Hanoi, Vietnam ∗ E-mail: vvthuan@vaec.gov.vn Abstract This work deals with the nature of Majorana particles by applying the timelike formalism of the superluminal Lorentz transformation (SLT) It is proposed that along with the SLT of the space-time coordinates, the Dirac equation should be treated simultaneously by a Majorana-like representation to be invariant This formalism leads to a natural understanding of Majorana physics I INTRODUCTION The paper of Majorana published more than 70 years ago [1], at the beginning was applied for a symmetrical view on electron and positron However, it was found later that the represented formalism is not for electron-positron, but describes neutral leptons, probably, a new kind of hypothetical neutrinos, which differ from Dirac neutrinos by the identical symmetry between particle and anti-particle While massless Dirac and Majorana neutrinos seem to be indistinguishable and well described by the Standard Model (SM) where only their left-handed eigenstates can interact with the gauge fields, the neutrino oscillation implies that neutrino should have non-zero mass Along with the traditional tendency of developing the SM to generate a normal mass and let the right-handed neutrinos show up, we proposed a model of space-time symmetry as an alternative approach which considers neutrino as time-like leptons, traveling in the flat 3D-time while twisting in the 3D-space [2] The first step in the present study is to formulate a formalism to understand the physics of massive superluminal leptons in the frame of which the superluminal leptons would formally satisfy Majorana physics II FORMALISM OF THE SUPERLUMINAL LORENTZ TRANSFORMATION (SLT) The {1,3} Minkowski time-space (with geometrical unit c = 1) corresponds to the flat pseudo-Euclid geometry as follows: ds2 = dt2 − dx2 − dy − dz (1) Let’s consider a material point moving in Minkowski time-space The superluminal Lorentz transformation suggested by Recami [3] seems to keep a real 3D-Euclid space (x , y , z ), VO VAN THUAN however, the two transverse coordinates are, indeed, imaginary in opposite to the real longitudinal space axis Therefore, in application of the formal SLT we link these two transverse axes with the (longitudinal) time axis to form a 3D-Euclid time Following Recami [3] we introduce the SLT from a subluminal reference frame K to a superluminal reference frame K as: z = γ(t − βz ); x = i.x = v t = γ(z − βt ); y = i.y = w (2) Recami [3] suggested that for a material point (or a particle) moving faster than light (β > 1), there are two operations needed in the SLT Firstly, it turns up the relative speed (β = 1/β ), which is equivalent to turn time axes to spatial ones and vice-versa; then γ = − β Secondly, it converts all imaginary variables into real ones to meet the physical reality At variance with [3] we propose to replace the imaginary “space” coordinates (x , y ) in (2) by the real time-like coordinates (v , w ) Such a SLT converts a {1,3}-Minkowski time-space {t, x, y, z} with the geometry (1) to a {3,1} time-space {v , w , t , z } with the following quadratic equation: ds2 = dz − dv − dw − dt (3) ds2 = dx + dy + dz − dt = dz − dx2 − dy − dt (4) p2z − E = m2 > (5) Et2 − p2x − p2y − p2z = Et2 − (i.px)2 − (i.py )2 − p2z = µ2 = (i.m)2 < (6) For the tachyon in according to the transformation (2) for an economic version we may imply a dual role to the transverse time axes (v , w ), namely, the same axes play a role of transverse times for tachyon and simultaneously, a role of transverse real space (x, y) for bradyon and for us, as subluminal observers Consequently, the equation (3) can be rewritten as: The geometry in (3) and (4) is identical to the SLT in (2) The corresponding energymomentum relation of tachyon is: It is to emphasize that in (5) the momentum is single directional, while the energy is three dimensional in according to our definition of the superluminal space and time The equation (5) may be rewritten conventionally with a formal 3D-momentum presentation as in [3]: However, such formal 3D-momenta px , py , pz can not form a real 3D-Euclid momentum space, because the “transverse momenta” are imaginary III REPRESENTATION OF DIRAC EQUATION FOR ELECTRON-POSITRON Let’s recall the traditional Dirac theory of free electron where the wave functions are complex and derive from a system of two equations, the primary Dirac equation and its conjugate: γ4 Eψ = i.γk pk ψ + mψ A SUPERLUMINAL FORMALISM FOR MAJORANA-LIKE LEPTON −Eψγ4 = i.pk ψγk + mψ (7) Originally, from the traditional Dirac electron-positron theory, each of the equations has a general solution, the four-component wave functions ψ or ψ associated with both positive and negative time-energy sub-solutions, correspondingly The time-energy dependence of a wave function is often expressed by a term of the form: ψ ∼ e−iΩ.t where Ω is de-Broglie frequency of electron In the momentum representation, instead of changing the sign of energy and time, the later is kept positive but the frequency Ω adopts both positive and negative signs Namely, ψ = ψ+ + ψ− , where ψ+ is a function of positive frequency, while ψ− is of the negative one In fact, the reality seems to need only four eigenstates: two states of electron and positron, each of which has two sub-states of opposite spin projections, while the above equation system gives twice more solutions For a reduction, it was assumed that only the positive energy is realistic, then instead of the sub-solution of the negative frequency, the second-conjugated equation in (7) is treated under C-operator which partly produces positron of positive energy, replacing the electron solution of negative energy Such a combined operation is called reinterpretation principle (RIP) Consequently, the traditional Dirac formalism leads to a realistic solution p p e as ψ = ψ+ + ψ− ⇒ (ψ+ + ψ+ ); where ψ− is replaced by ψ+ = Cψ − , a positron solution with positive energy However, the RIP solution is not identical to the general solution of Dirac equation because ψ and ψ are not solutions of the same equation In the present study, for an alternative discussion relating to the superluminal formalism, we propose to replace the Dirac’s RIP by the time reversion as following: We assume, firstly, that electron has a time-like spin or t-spin equal 1/2 of which the projection on the longitudinal time axis (t-helicity) should correlate strictly with the sign of the frequency Ω and then, with the electrical charge Secondly, for a more natural consideration, we operate the time reversion T which converts the time axis (and the sign of energy) and simultaneously changing the electrical sign to the opposite, i.e equivalent to C-operator in RIP, then T ψ− = Cψ − getting now an eigenstate of positron evolving toward the future together with electron Indeed, if ψ ∼ e−iΩ.t is the unique form of time evolution, the complex conjugation ψ equivalent to a reflection of time ψ(t) ↔ ψ(−t), as well as energy Acting the T-operator on Dirac equation (7) in combining with matrix transposition should lead to the corresponding conjugate equation of the (T ψ−) function as follows: T [γ4 Eψ− = i.γk pk ψ− + mψ− ] → [−E(T ψ−)γ4 = i.pk (T ψ− )γk + m(T ψ−)] where T = γ1 γ2 γ3 In this consideration we imply that the sign of electrical charge links with the projection of t-spin of electron Indeed, during the T-operation while time −t changes to +t, the projection of t-spin as an axial vector should flip back relatively to +t Instead of the solution ψ− of equation for an electron with negative energy and evolving to the past, we have the solution ψ p = T ψ− of Dirac equation for positron with positive energy and evolving toward the future We conclude that the Dirac equation and its conjugate for the positive solution describe the motion of electron; while the same pair of Dirac equations for the negative solution describe the motion of positron Each of the two solutions in Equations (7) corresponding to electron and positron, has only two linear independent subsolutions of two opposite spin’s projections of electron or VO VAN THUAN positron Therefore, in case of a maximal mixing it is applied to electron and positron as follows: e e ψ e = √ (|ψ+1/2 + |ψ−1/2 ) p p ψ p = √ (|ψ+1/2 + |ψ−1/2 ) (8) The combinations (8) describe the most natural stable states of lepton beam (electron or positron) in a dynamic equilibrium with the interacting medium IV REPRESENTATION OF DIRAC EQUATION FOR SUPERLUMINAL LEPTON Based on the superluminal geometry (4) and equation (6) we write a “formal” Dirac equation for tachyon in the momentum representation as follows: γ4 Et ψ = (−γ1 px − γ2 py + i.γ3pz )ψ + µψ (9) This expression differs from the subluminal Dirac equation by the imaginary transverse “momenta” and the mass terms Now applying a Majorana-like representation ψ = UM ψ , where: 1 I −i.σ3 UM = √ (γ4 + γ3 ) = √ (10) 2 i.σ3 −I we turn equation (9) into: γ3 Et ψ = (γ1 px + γ2 py )ψ + i.γ4 pz ψ + µψ (11) Applying the version of the physical geometry (3), we turn back: px = Ev = E1 , py = Ew = E2 , Et = −E3 and pz = pz , then rewrite (11) as: γ4 pz ψ = i.γk Ek ψ − mψ (12) We found that the Majorana-like equation (12) and its conjugate are exact as the form of Dirac equations(7) only with an exchange of the roles of energy and momentum which proves that instead of treating subluminal electron-positron the new equations govern superluminal leptons V MAJORANA PHYSICS IN THE SUPERLUMINAL FRAMES We are extending a similar analysis of the subluminal Dirac equations for electronpositron (7) now to solving the equation (12), for superluminal leptons Similar to the action of T-operator at the subluminal frame on Dirac equation, here an action by Poperator (the space convertor) in combination with matrix transposition is applied on the solution with negative momentum, which coverts this solution into a right-handed one, but with positive momentum Therefore, we can write the eigenstates of a free superluminal A SUPERLUMINAL FORMALISM FOR MAJORANA-LIKE LEPTON lepton as follows: L L ψL = √ (|ψ+ + |ψ− ) R R ψR = √ (|ψ+ + |ψ− ) (13) in which ψL is a wave function for the left-handed helicity and ψR is another wave function for the right-handed helicity They are two different superpositions of maximal mixing of two states ψ+ and ψ− which are regarded, formally as the eigenstates of particle and antiparticle, respectively For SLT formalism, superluminal leptons exist in a flat 3D-time being adopted as a realistic time-like 3D space, in which again we propose that a time-like spin (or simply t-spin, 1/2) of the superluminal lepton is able to rotate (in analogue to the s-spin of electron or positron orientable in 3D space) In case the maximal mixing (13) is keeping invariant, as the most stable states, ψL and ψR are the wave functions of superluminal Majorana particles, because they are identical in the relation between particle and anti-particle with well-conserved helicity In a complete similarity to (8), superpositions (13) can be also considered as the states of Majorana-like particle evolving to the future and back to the past, without Dirac’s RIP A half of Majorana-like leptons evolving to the past are almost sterile from subluminal observations Similar to electrons which may be polarized due to interaction with a polarizer and able to change their mixing in (8) between the two states ψ+1/2 and ψ−1/2 with opposite helicities, Majorana particles may also oscillate between eigenstates ψ+ and ψ− and send a part of them to the past as sterile particles For a total t-spin polarization we get a pure L R left-handed particle ψ+ (or right-handed anti-particle ψ− ), which seems to be nothing else as a Dirac eigenstates with both labels: helicity and lepton charge Therefore, it implies that the notions of Dirac neutrino and Majorana neutrino are relative, because they may oscillate to each other, depending on the mixing proportion in (13) However, such a definition of Dirac neutrino is not complete because the eigenstates in (13) are superluminal, while Dirac neutrinos, in their origin, should exist in 3D-space of the subluminal frames As a result, we found that the superluminal leptons are identified by their helicity Formally, we can assume in according to Parker [4] that instead of electric charge the superluminal lepton should have a magnetic monopole We assume further that the sign of monopole should be well correlated with helicity of Majorana-like particle VI CONCLUSION We found that along with the superluminal Lorentz transformation (SLT) of time and space coordinates, the quantum mechanical equations for leptons should be treated simultaneously by Majorana-like reinterpretation (10) to convert to an appropriate form of Dirac equation for superluminal lepton The later is shown up as Majorana-like particle, which conserves strictly its helicity even small real masses can be produced, as neutrino oscillation implies recently The proposed formalism is not yet realistic, as there is an obvious asymmetry between electron and neutrino which demands the next step of the study to understand the nature VO VAN THUAN of the mass of Majorana leptons and a suitable mechanism of violation of the spacetime symmetry As expected, the proposed model would shed light on those mysterious particles and the origin of their P-nonconservation This would also set new constraints on neutrinoless double beta decay ACKNOWLEDGMENTS The author is indebted to Pierre Darriulat (VATLY, INST) and Nguyen Anh Ky (IOP) for useful discussions The deep thanks is extended to VATLY members for their cooperation The research has been funded by the National Basic Research Program for Physics of the Ministry of Science and Technology (MOST) of Vietnam REFERENCES [1] E Majorana, Nuovo Cim 14(1937) 171 [2] Vo Van Thuan, Proc of the Osaka-Hanoi Forum 2005 on Frontiers of Basic Science, 2005, Hanoi, Vietnam, Sept 27-29, 2005 Ed H Takabe et al., Osaka University Press 2006, p.98 [3] E Recami and R Mignani, Lett Nuovo Cim (1972) 144 [4] L Parker, Phys Rev 188(1969) 2287 Received 18 August 2008 [...]... density on detection efficiency of gamma spectroscopy REFERENCES [1] J.F Briesmeister, MCNP4C2- Monte Carlo N-particle Transport Code System, LA-13709-M (June 2001) [2] Truong thi Hong Loan, Tran Ai Khanh, Dang Nguyen Phuong, Do Pham Huu Phong, Efficiency calibration for HPGe detector with voluminous sample geometry using Monte Carlo method, Summarization Report, Vietnam National University – HCMC Scientific... particles and the origin of their P-nonconservation This would also set new constraints on neutrinoless double beta decay ACKNOWLEDGMENTS The author is indebted to Pierre Darriulat (VATLY, INST) and Nguyen Anh Ky (IOP) for useful discussions The deep thanks is extended to VATLY members for their cooperation The research has been funded by the National Basic Research Program for Physics of the Ministry of