Investigation of microwave radiation from a compressed beam of ions using generalized Planck’s radiation law Sreeja Loho Choudhury1, R. K. Paul2 Department of Physics, Birla Institute of Technology, Mesra, Ranchi-835215, Jharkhand, India Abstract An ion-beam compressed by an external electric force is characterized by a unique non- equilibrium distribution function. This is a special case of Tsallis distribution with entropy index q=2, which allows the system to possess appreciably low thermal energy. The thermal radiation by such compressed ion-beam has been investigated in this work. As the system is non extensive, Planck’s law of radiation has been modified using Tsallis thermostatistics for the investigation of the system. The average energy of radiation has been derived by introducing the non extensive partition function in the statistical relation of internal energy. The spectral energy density, spectral radiation and total radiation power have also been computed. It is seen that a microwave radiation will be emitted by the compressed ion-beam. The fusion energy gain Q (ratio of the output fusion power to the power consumed by the system) according to the proposed scheme (R. K. Paul 2015) using compressed ion-beam by electric field will not change significantly as the radiated power is very small. Keywords: non extensive partition function, Tsallis distribution, generalized Planck’s radiation law, compressed ion-beam, microwave radiation, fusion. 1 Email address: [email protected] 2 Email address: [email protected] 1. Introduction Earlier, an attempt was made to generalize the Planck’s radiation law [1] for the explanation of the cosmic microwave background radiation [2] at a temperature of 2.725 K. There are some versions of generalized Planck’s law available in the existing literature [3] in this regard. There are also recent attempts to generalize the Planck’s radiation law using Kaniadakis approach [4, 5]. In 2015, distribution function of an ion-beam under compression by an external electric field was derived by R. K. Paul. It is known that an ion-beam under compression by an external electric field is characterized by a unique non-equilibrium distribution [6]. This is a special case of Tsallis distribution [7] with entropy index q=2. The characteristic feature of this distribution is that the system possesses a very small thermal energy [6, 7]. The main objective of this work is to investigate the thermal radiation from such a compressed ion-beam. The thermal radiation from an object at any equilibrium temperature obeys Planck’s radiation law [8]. However, since the system of ion-beam under compression by an external electric field is non extensive, explanation of its thermal radiation demands modification of Planck’s law of radiation. Here, we have derived the average energy of radiation by introducing the non extensive partition function [9] in the statistical relation of internal energy. From the average energy, we have computed the spectral energy density and spectral radiance of the emitted radiation. It has been seen that this expression of average energy recovers the original Planck’s relation for q= 1 (extensive). The current paper shows that a microwave radiation will be emitted by the compressed ion- beam. The characteristic feature of this radiation in terms of peak wavelength, peak amplitude of radiance and the total radiation power are also computed. 2. Method Let us consider a cube of side L having conducting walls which is filled with electromagnetic radiation in thermal equilibrium at a temperature T. The radiation emitted from a small hole in one of the walls will be characteristic of a perfect black body. According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by: exp(−훽퐸 ) 푃 = 푛 (1) Z 1 where 퐸 = (푛 + ) ℎ휈, n=0,1,2,3,..... and partition function 푛 2 ∞ 푍 = ∑푛=0 exp⁡(− 훽퐸푛) (2) Ordinary statistical mechanics is derived by maximizing the Boltzmann Gibbs entropy given by: ⁡⁡푆 = −푘 ∑ 푝 ln 푝 (3) which is subjected to constraints, whereas in case of non extensive statistical mechanics the more general Tsallis entropy [10] given by: 푘(1−∑ 푝 푞) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푆 = (4) 푞 (푞−1) is maximized. Here⁡푝 are probabilities associated with the microstates of a physical system and q is the non extensive entropic index. The ordinary Boltzmann Gibbs entropy is obtained in the limit q→1. For some given set of probabilities 푝, one can proceed to another set of probabilities 푃⁡given as: 푞 푝 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푃 = 푞 (5) ∑ 푝 The probability⁡푃 coming out after maximizing 푆푞⁡under the energy constraint ∑ 푃 ∊ = 푈푞, where ∈ are the energy levels of the microstates is: −푞 (푞−1) (1+(푞−1)훽∊⁡) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푃 = (6) 푍푞 −푞 (푞−1) where 푍푞 = ∑(1 + (푞 − 1)훽 ∊⁡) is the partition function [7] in case of non extensive 1 statistical mechanics and 훽 = , 푘= Boltzmann constant. 푘푇 If we consider =n then, 푞 ∞ (1−푞) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푍푞 = ∑푛=0{1 − (1 − 푞)훽퐸푛} (7) 푞 1 ℎ휈 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푍 = ∑∞ {1 − (1 − 푞) (푛 + ) }(1−푞) (8) 푞 푛=0 2 푘푇 2.1 Calculation of Average energy The average energy in a mode can be expressed in terms of the partition function as: 휕 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡< 퐸 >= 푘푇2 ln 푍 (9) 휕푡 푞 Now 휕 1 푞 ∞ 1 ℎ휈 1 ℎ휈 ⁡⁡ ln 푍푞 = 푞 [ ∑푛=0{1 − (1 − 푞) (푛 + ) } (1 − 푞) (푛 + ) ] 휕푡 1 ℎ휈 (1−푞) 2 푘푇 2 푘푇2 ∑∞ {1−(1−푞)(푛+ ) }(1−푞) 푛=0 2 푘푇 Let 휕 푞ℎ휈 푎 ln 푍 = (10) 휕푡 푞 푘푇2 푏 Now, 푞 1 ℎ휈 푏 = ∑∞ {1 − (1 − 푞) (푛 + ) }(1−푞) (11) 푛=0 2 푘푇 푥 푛 From the definition of limit we have, lim (1 + ) = 푒푥. 푛→∞ 푛 푞 푞 푞 ℎ휈 3ℎ휈 5ℎ휈 푏 = {1 − (1 − 푞) }(1−푞) + {1 − (1 − 푞) }(1−푞) + {1 − (1 − 푞) }(1−푞) + ⋯ ∞ (12) 2푘푇 2푘푇 2푘푇 푞 as q→1, (1-q)→0, and → ∞ then (1−푞) −푞ℎ휈 −3푞ℎ휈 −5푞ℎ휈 푏 = 푒 2푘푇 + 푒 2푘푇 + 푒 2푘푇 + ⋯ ∞⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ (13) Therefore, 푞ℎ휈 푒2푘푇 푏 = 푞ℎ휈 (14) (푒 푘푇⁡ −1) From (10) we have, 2푞−1 1 1 ℎ휈 푎 = ∑∞ (푛 + ) {1 − (1 − 푞) (푛 + ) }(1−푞) (15) 푛=0 2 2 푘푇 Let 푎 = 푐 + 푑 2푞−1 1 1 ℎ휈 where 푑 = ∑∞ {1 − (1 − 푞) (푛 + ) }(1−푞) (16) 2 푛=0 2 푘푇 2푞−1 2푞−1 2푞−1 1 ℎ휈 3ℎ휈 5ℎ휈 푑 = [{1 − (1 − 푞) }(1−푞) + {1 − (1 − 푞) }(1−푞) + {1 − (1 − 푞) }(1−푞) + ⋯ ∞] 2 2푘푇 2푘푇 2푘푇 Therefore, ℎ휈 (2푞−1) 1 푒 2푘푇 푑 = [ ℎ휈 ] (17) 2 (2푞−1) {푒 푘푇−1} Now, 2푞−1 1 ℎ휈 푐 = ∑∞ 푛{1 − (1 − 푞) (푛 + ) }(1−푞) (18) 푛=0 2 푘푇 ℎ휈 (2푞−1) 푒 2푘푇 푐 = ℎ휈 (19) (2푞−1) {푒 푘푇−1}2 Hence, ℎ휈 (2푞−1) 푒 2푘푇 1 1 푎 = ℎ휈 ( + ℎ휈 ) (20) (2푞−1) 2 (2푞−1) (푒 푘푇−1) 푒 푘푇−1 Therefore, ℎ휈 ℎ휈 (2푞−1) (푞−1) 푎 (푒 푘푇−푒 푘푇) 1 1 = ℎ휈 ( + ℎ휈 ) (21) 푏 (2푞−1) 2 (2푞−1) (푒 푘푇−1) 푒 푘푇−1 From (9) we have the average energy given as: ℎ휈 ℎ휈 (2푞−1) (푞−1) (푒 푘푇 − 푒 푘푇) 1 1 < 퐸 >= ℎ휈 ( + ℎ휈 ) (2푞−1) 2 (2푞−1) (푒 푘푇 − 1) 푒 푘푇 − 1 For q=1 (extensive), we have ℎ휈 ℎ휈 < 퐸 >= + (22) 2 ℎ휈 (푒푘푇−1) ℎ휈 Neglecting the vacuum energy term , 2 ℎ휈 < 퐸 >= ℎ휈 (23) (푒푘푇−1) For q=2, 3ℎ휈 ℎ휈 (푒 푘푇 −푒푘푇) 1 1 < 퐸 >= 2ℎ휈 ( + ) (24) 3ℎ휈 2 3ℎ휈 (푒 푘푇 −1) 푒 푘푇 −1 Neglecting the vacuum term, 3ℎ휈 ℎ휈 (푒 푘푇 −푒푘푇) < 퐸 >= 2ℎ휈 (25) 3ℎ휈 2 (푒 푘푇 −1) Here ℎ휈 ≫ 푘푇⁡and hence we have, 3ℎ휈 ℎ휈 −6ℎ휈 < 퐸 >= 2ℎ휈(푒 푘푇 − 푒푘푇)푒 푘푇 (26) −3ℎ휈 −5ℎ휈 < 퐸 >= 2ℎ휈(푒 푘푇 − 푒 푘푇 ) (27) Similarly, we can have the average energy for other q values such as q=0.95, q=1.5, etc. 2.2 Energy Density The energy density per unit frequency is given by: 8휋휈2 푢 (푇) = < 퐸 > (28) 휈 푐3 Therefore, ℎ휈 ℎ휈 (2푞−1) (푞−1) 8휋ℎ휈3푞 (푒 푘푇−푒 푘푇) 푢휈(푇) = ℎ휈 (29) 푐3 (2푞−1) (푒 푘푇−1)2 Hence the generalized energy density per unit frequency is given by: ℎ휈 ℎ휈 (2푞−1) (푞−1) 8휋ℎ휈3푞 {푒 푘푇 − 푒 푘푇} 푢휈(푇) = 3 ℎ휈 2 푐 (2푞−1) {푒 푘푇 − 1} For q=1, 8휋ℎ휈3 푢휈(푇) = ℎ휈 (30) 푐3(푒푘푇−1) For q=0.95, 0.9ℎ휈 −0.05ℎ휈 (푒 푘푇 −푒 푘푇 ) 7.6휋ℎ휈3 푢 (푇) = (31) 휈 푐3 0.9ℎ휈 2 (푒 푘푇 −1) For q=1.5, 12휋ℎ휈3 −2ℎ휈 −3.5ℎ휈 푢 (푇) = (푒 푘푇 − 푒 푘푇 ) (32) 휈 푐3 Similarly for q=2, 16휋ℎ휈3 −3ℎ휈 −5ℎ휈 푢 (푇) = (푒 푘푇 − 푒 푘푇 ) (33) 휈 푐3 The variation of energy density with frequency for different q values has been shown in the following figure 1 for the temperature T=2.725K. FIGURE 1. The plot of energy density versus frequency for different q values at a temperature T= 2.725 K. 2.3 Spectral Radiance The spectral radiance is given by: 푢 (푇)푐 퐼 (푇) = 휈 (34) 휈 4 푐 We have, frequency of radiation 휈 = 휆 푑휈 −푐 Therefore, = 푑휆 휆2 So we get, 푑휈 퐼 (푇) = 퐼 (푇) | | (35) 휆 휈 푑휆 ℎ푐 ℎ푐 (2푞−1) (푞−1) {푒 휆푘푇−푒 휆푘푇} 2휋ℎ푐2푞 퐼 (푇) = (36) 휆 5 ℎ푐 2 휆 (2푞−1) {푒 휆푘푇−1} Therfore, for q=1 we have, 2휋ℎ푐2 ⁡퐼휆(푇) = ℎ푐 (37) 휆5(푒휆푘푇−1) For q=2, 4휋ℎ푐2 −3ℎ푐 −5ℎ푐 퐼 (푇) = (푒 휆푘푇 − 푒 휆푘푇 ) (38) 휆 휆5 Similarly for q=1.5, 3휋ℎ푐2 −2ℎ푐 −3.5ℎ푐 퐼 (푇) = (푒 휆푘푇 − 푒 휆푘푇 ) (39) 휆 휆5 For q=0.95, 0.9ℎ푐 −0.05ℎ푐 (푒 휆푘푇 −푒 휆푘푇 ) 1.9휋ℎ푐2 퐼 (푇) = (40) 휆 휆5 0.9ℎ푐 (푒 휆푘푇 −1)2 Figure 2 is the plot of spectral radiance versus wavelength for a particular temperature say T=2.725K.