J. Nig. Soc. Phys. Sci. 2 (2020) 55–60 Journal of the Nigerian Society of Physical Sciences Original Research Energy distribution of an ion cloud in a quadrupole Penning Trap B. M. Dyavappa∗ Department of Physics, Government College for Women, Kolar, India. Abstract Ions are confined in Penning trap by the combination of electric and magnetic fields, as the electric field confines ions in the axial direction through an electric potential minimum and the magnetic field applied along the axis of the trap confines the ions in the radial direction. In the high temperature limit Coulomb interaction of ions can be neglected and the total energy is due to the electrostatic potential energy of the charge of ions and kinetic energy due to thermal energy. However, in the low temperature limit the trapping potential created by the DC voltage applied between the end cap and ring electrodes is cancelled by Coulomb interaction of ions and the total energy is mainly kinetic energy of ions. The probability density of energy distribution of ions along axial direction, in radial plane and the total probability density of energy distribution due to resulting motion of both axial and radial motion of ions under high temperature and low temperature limits in a Quadrupole Penning trap are presented here. These results reveal the energy properties of ion cloud and are useful to carry out accurate measurement experiments on single stored particle, antiparticles with energy related parameters, under high temperature and low temperature limits in a Quadrupole Penning trap. Keywords: Quadrupole Penning trap, Probability of energy distribution function, Probability density of energy distribution function Article History : Received: 25 January 2020 Received in revised form: 09 March 2020 Accepted for publication: 20 March 2020 Published: 14 May 2020 c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: B. J. Falaye 1. Introduction Penning traps are used for high precision mass spectrome- try, time of flight detection of ion cyclotron and magnetron res- onances and for cooling of antiprotons, spectral intensity dis- tribution of fermions [1], quantum information processing [2] etc. A thorough knowledge of different properties of stored ions like stability parameters, energy distribution, potential depths, space charge effects, anharmonicities etc. in the trap are es- sential. The Penning trap uses a time-independent, spatially homogeneous uniform magnetic field along the symmetry axis of the trap. The end-cap electrodes are biased with a positive DC voltage for the confinement of positive ions or negative DC voltage for the confinement of negative ions with respect to the ∗Corresponding author tel. no: +919483113600 Email address: dyavappabm@gmail.com (B. M. Dyavappa) ring electrode; hence both electric and magnetic fields are su- perposed to trap ions, as the resultant of quadrupolar potential in a small region of space (see Figure 1). The quadrupole Pen- ning trap is made with three-electrodes; two are end-cap elec- trodes and a ring electrode. The equation of ring electrode is r2 r20 − z2 z20 = +1 and those of two similar end-cap electrodes is r2 r20 − z2 z20 = −1, where r0 is the inner radius of the ring electrode in the radial plane and z0 is half of the vertical distance between the two end-cap electrodes designed such that r0 = √ 2 z0 . The trap potential is [3, 4] VT (r, z) = V0 r20 + 2z 2 0 ( 2z2 − r2 ) (1) The potential well due to the electric field in the axial direction 55 Dyavappa / J. Nig. Soc. Phys. Sci. 2 (2020) 55–60 56 Figure 1: Quadrupole Penning trap with cloud of ions confined in trap space. and radial plane is given by [5] VT (r, z) = 1 4π�0r [ qe − ( r λD )] + md2ω2z 4q (2) where λD = ( �0kBT/nq2 ) is Debye length, r is the radial co- ordinate, n is density of particles, T is temperature and q is the charge of ion. The plasma parameter ( ND = 4πnλ3D/3 ) gives the average number of particles found in the Debye sphere of plasma. Penning traps are used in high precision mass spec- trometry [6, 7] for cooling antiprotons for measuring their grav- itational acceleration [8]. Ions are confined first in a catching trap and then transferred to high precision trap to carry out mea- surements. Time of flight technique is used to detect ion cy- clotron and magnetron resonances. A thorough knowledge of different properties of ions including energy distribution along axial direction, in radial plane and the total energy distribution is necessary to understand these processes. 2. Theory We assume that the ion cloud is in thermal equilibrium through Coulomb interaction between ions. The axial component of an- gular momentum of the ions is conserved as the Penning trap is symmetric along its axis in the absence of external torques. This causes the rotation of the ion cloud as a whole [9] and the rotation in presence of magnetic field provides radial confining force to balance the electrostatic force of expansion. There- fore the rotation in magnetic field is equivalent to neutralization by opposite charge of ions and the distribution of magnetically confined ions in thermal equilibrium without rotation can be treated as ions confined and neutralized by a cylinder of oppo- site charge. The axial, pure cyclotron, reduced cyclotron and magnetron frequencies are given respectively by [10, 11, 12, 13] fz = 1 2π √ 4eV0 md2 , fc = eB 2πm , f ′ c = fc + √ f 2c − 2 f 2z 2 , fm = fc− √ f 2c − 2 f 2z 2 , ∴ fm < fφ < fc − fm. The probability of energy distribution function in thermal-equilibrium is [14] dP ( Er,φ,z,Pr,Pφ,Pz ) = 2π (πkBT ) 3 2 E 1 2 exp [ − ( E−ωφPφ kB T )] · drdφdzdPr dPφdPz (3) where ωφ is the rotational frequency of ion cloud as a whole determined by total angular momentum and energy determined by the temperature of the ion cloud. Energy of single ion is [14] E = 12m [ P2r + ( Pφ r − qBr 2c )2 + P2z ] + q [ VT (r, z) + Vq (r, z) ] (4) The momentums in radial plane, azimuthal and axial directions are [14] Pr = mvr = m dr dt , Pφ = mr 2 dφ dt + qB 2c r2, = mrvφ − ωc 2 mr2, Pz = m dz dt = mvz  (5) Coulomb interaction potential of ions is [14] Vq (r, z) = ∫ ∞ 0 nq ( r ′ , z ′ ) |r − r′ | r ′ dr ′ dφ ′ dz ′ (6) 3. Energy distribution of an ion cloud at the high tempera- ture limit The Penning trap is designed with its dimension to be 20 mm and magnetic field applied is 6 T , when the DC voltage applied is 3000 V for an ion cloud density of 1014 m−3 with Debye length 2.4 × 10−3 m , the energy and temperature of ions at high temperature limit considered are 10 eV , 116045.05 K respectively. In high temperature limit [14] qVq (r, z) � kBT � qVT (r, z) + m 2 ωφ ( ωc −ωφ ) r2 (7) If we neglect Coulomb interaction Vq (r, z) then the probability of energy distribution function is [14] dP ( Er,φ,z,Pr,Pφ,Pz ) = A exp [ − ( Ez +E ′ r kB T )] · drdφdzdPr dPφdPz (8) The energy along Z-direction is [14] Ez = 1 2m P2z + kzz 2 (9) The energy in the radial plane is [14] Er = 1 2m ( P2r + P2φ r2 ) + ( mω2c 8 + qV0 d2 ) r2 + ωc 2 Pφ (10) 56 Dyavappa / J. Nig. Soc. Phys. Sci. 2 (2020) 55–60 57 E ′ r = Er −ωφPφ = 12m ( P2r + P2φ r2 ) + ( mω2c 8 + qV0 d2 ) r2 +( ωc 2 −ωφ ) Pφ (11) The energy distribution in Z-direction is [14] dP ( Ez,Pz ) = Bex p [ − ( Ez kBT )] dzdPz (12) The probability density of energy distribution in Z-direction is [14] ρz (Ez) = 1 kBT exp ( − Ez kBT ) (13) The average energy in Z-direction is [14] 〈Ez〉 = ∫ ∞ 0 Ezρz (Ez) dEz = kBT (14) The energy distribution in the radial direction is [14] dP ( Er,φ,Pr,Pφ ) = Cex p ( − E ′ r kBT ) drdφdPr dPφ (15) dP ( E ′ r ) = ( 1 kBT )2 exp ( − E ′ r kBT ) E ′ r dE ′ r (16) The probability density of energy distribution in the high tem- perature limit with ωφ � ωc 2 is [14] ρr ( E ′ r ) = ( 1 kBT )2 exp ( − E ′ r kBT ) E ′ r (17) ρr (Er ) = ( 1 kBT )2 exp ( − Er kB T ) Er{ ∵ Er ≈ E ′ r } (18) The average energy in the radial plane is [14] 〈Er〉 = ∫ ∞ 0 Erρr (Er ) dEr = 2kBT (19) The total probability of energy distribution is [14] dP (E) = ρz (Ez) ρr (Er ) dEzdEr = ( 1 kBT )3 exp ( − E kB T ) Er dEzdEr (20) The total probability density of energy distribution is [14] ρ ( E kBT ) = 1 2 ( 1 kBT )3 exp ( − E kBT ) E2 (21) The total average energy is [14] 〈E〉 = ∫ ∞ 0 Eρ (E) dE = 〈Ez〉 + 〈Er〉 = kBT + 2kBT = 3kBT (22) The total energy is [14] E = Ez + Er (23) The probability density of energy distribution in axial direction increases sharply up to 24.3559 × 1016 at E = 1.3375kBT , decreases abruptly and remains almost a constant for high tem- perature limit as shown in Figure 2(a). The probability density of energy distribution in the radial plane increases sharply up to 34.4544 × 1016 at E = 2.11487kBT , decreases abruptly and re- mains almost a constant for the high temperature limit as shown in the Figure 2(b). The total probability density of energy distri- bution increases sharply up to 42.5566×1016 at E = 3.0728kBT , decreases abruptly and remains almost a constant for the high temperature limit as shown in the Figure 2(c). The axial, radial and total probability densities of energy distribution increase sharply up to 24.3559×1016, 34.4544×1016, 42.5566×1016 at E = 1.3375kBT , 2.1149kBT , 3.0728 kBT respectively, decrease abruptly and remain almost constant as k ′ r r 2/kBT ≈ 5.4 × 102 and kBT/qVq (r, z) ≈ (λD/r) 2 ≈ 6 for the high temperature limit as shown in Figure 2(d). The axial probability density of en- ergy distribution is less than the radial probability density of energy distribution, which in turn less than the total probabil- ity density of energy distribution. This analysis helps to design trap that consists of appropriate structure trap electrodes to min- imize anharmonicities to improve storage time. 4. Energy distribution of an ion cloud at the low tempera- ture limit In a Penning trap of dimension 10 mm , magnetic field 2 T , when the DC voltage applied is 10 V for an ion cloud density 2× 1014 m−3 and Debye length 10−5 m , the energy and temperature of ions at low temperature limit considered are 3.6 × 10−4 eV , 4.2 K respectively. In the low temperature limit [14] qVq (r, z) � kBT (24) qVT (r, z) + qVq (r, z) + m 2 ωφ ( ωc −ωφ ) r2 = 0 (25) If we neglect Coulomb interaction potential Vq (r, z) then probability of energy distribution distribution function is [14] dP ( EPr,Pφ,Pz ) = A′ exp [ − ( Ez +E ′ r kB T )] · dPr d ( Pφ r ) dPz (26) The energy in Z-direction is [14] Ez = 1 2m P2z (27) The energy in the radial plane is [14] Er = 1 2m P2r + P2φr2  + ( mω2c 8 ) r2 + ωc 2 Pφ (28) E ′ r = Er −ωφPφ = 1 2m ( P2r + P2φ r2 ) +( mω2c 8 ) r2 + ( ωc 2 −ωφ ) Pφ (29) 57 Dyavappa / J. Nig. Soc. Phys. Sci. 2 (2020) 55–60 58 Table 1: The values of axial, radial and total probability density of energy distribution in the high temperature limit. E kB T e− E kB T T (K) kBT ( 10−19 J ) ρz ( Ez kB T ) ( 1016 ) ρr ( Er kB T ) ( 1016 ) ρ ( E kB T ) ( 1016 ) 0 1 ∞ ∞ 0 0 0 1 0.367879 115942 15.999996 22.9924 22.9924 11.4962 2 0.135335 57971 7.999998 16.9169 33.8338 33.8338 3 0.049787 38647.3 5.3333274 9.33507 28.0052 42.0078 4 0.018316 28985.5 3.999999 4.579 18.316 36.632 5 0.006738 23188.4 3.1999992 2.1056 10.528 26.32 6 0.00247875 19323.7 2.6666706 0.92953 5.57718 16.73154 7 0.00091188 16563.1 2.2857078 0.39895 2.79265 9.774275 8 0.00033546 14492.75 1.9999995 0.16773 1.34184 5.36736 9 0.00012341 12882.4 1.7777712 0.06942 0.62478 2.8115 10 0.0000454 11594.2 1.5999996 0.028375 0.28375 1.41875 Table 2: The values of axial, radial and total probability density of energy distribution in the low temperature limit. E kB T e− E kB T T (K) kBT ( 10−23 J ) ρz ( Ez kB T ) ( 1020 ) ρr ( Er kBT ) ( 1020 ) ρ ( E kB T ) ( 1020 ) 0 1 ∞ ∞ 0 0 0 1 0.367879 4.2 5.796 35.8098 63.471187 71.6195545 2 0.135335 2.1 2.898 18.63038 46.699448 74.521537 3 0.049787 1.4 1.932 8.39408 25.7696687 50.3644996 4 0.018316 1.05 1.449 5.166848 12.640442 28.5264222 5 0.006738 0.84 1.1592 1.4666034 5.812693 14.666034 6 0.00247875 0.7 0.966 0.591024 2.565994 7.0922868 7 0.00091188 0.6 0.828 0.234846 1.101304 3.2878457 8 0.00033546 0.525 0.7245 0.09235968 0.4630228 1.4777549 9 0.00012341 0.4667 0.644 0.01325787 0.1916304 0.6486954 10 0.0000454 0.42 0.5796 0.013975 0.07832988 0.2795 The probability of energy distribution in Z-direction is [14] dP ( EPz ) = B′ exp [ − ( Ez kBT )] dPz (30) The probability density of energy distribution in Z-direction is [14] ρz ( Ez kBT ) = √ 1 πkBT [ exp ( − Ez kBT )] 1√ Ez (31) The average energy in Z-direction is [14] 〈Ez〉 = ∫ ∞ 0 Ezρz (Ez) dEz = 1 2 kBT (32) The probability of energy distribution in the radial plane is [14] dP ( EPr,Pφ ) = C′ exp ( − E ′ r kBT ) dPr dPφ (33) dP ( E ′ r kBT ) = ( 1 kBT ) exp ( − E ′ r kBT ) dE ′ r (34) The probability density of energy distribution in radial plane in the low temperature limit with ωφ � ωc/2 is [14] ρr ( E ′ r kBT ) = ( 1 kBT ) exp ( − E ′ r kBT ) (35) ρr ( Er kBT ) = ( 1 kB T ) exp ( − Er kB T ) { ∵ Er ≈ E ′ r } (36) The average energy in radial plane is [14] 〈Er〉 = ∫ ∞ 0 Erρr (Er ) dEr = kBT (37) The total probability density of energy distribution is [14] ρ (E) = ( 4 π ) 1 2 ( 1 kBT ) 3 2 exp ( − E kBT ) √ E (38) The total average energy is [14] 〈E〉 = 〈Ez〉 + 〈Er〉 = 1 2 kBT + kBT = 3 2 kBT (39) The probability density of energy distribution in axial direc- tion increases sharply up to 36.6166 × 1020 at E = 1.1466kB, decreases abruptly and remains almost a constant for the low temperature limit as shown in Figure 3(a). The total probabil- ity density of energy distribution in the radial plane increases sharply up to 63.59 × 1020 at E = 0.99kBT , decreases abruptly and remains almost a constant for the low temperature limit as shown in the Figure 3(b). The total probability density of en- ergy distribution increases sharply up to 83.39 × 1020 at E = 1.674kBT , decreases abruptly and remains almost a constant 58 Dyavappa / J. Nig. Soc. Phys. Sci. 2 (2020) 55–60 59 Figure 2: (a) The probability density of energy distribution in axial direction in the high temperature limit; (b) The probability density of energy distribution in the radial plane in the high temperature limit; (c) The total probability density of energy distribution in the high temperature limit; (d) The axial, radial and total probability densities of energy distribution in the high temperature limit. Figure 3: (a) The probability density of energy distribution in axial direction in the low temperature limit; (b) The probability density of energy distribution in the radial plane in the low temperature limit; (c) The total probability density of energy distribution in the low temperature limit; (d) The axial, radial and total probability density of energy distribution in the low temperature limit. 59 Dyavappa / J. Nig. Soc. Phys. Sci. 2 (2020) 55–60 60 for the low temperature limit as shown in the Figure 3(c). The axial, radial and total probability densities of energy distribu- tion increase sharply up to 36.6166×1020, 63.59×1020, 83.39× 1020 at E = 1.1466kBT , 0.99kBT , 1.674kBT respectively, de- crease abruptly, remain almost constant as λD/r � 1, λD/z � 1 for the low temperature limit as shown in the Figure 3(d). The axial probability density of energy distribution is less than the radial probability density of energy distribution, which in turn less than the total probability density of energy distribution. The total probability densities of energy distributions in- crease sharply up to 42.5566 × 1016 , 83.39 × 1020 at E = 3.0728kBT, 1.674kBT decrease abruptly and remain almost con- stants under high and low temperature limits respectively as shown in the Figure 4. 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 ρ(1016) : High temperature limit ρ(1020) : Low temperature limit ρ ( E / k B T ) E / k B T Figure 4: The total probability density of energy distribution when fφ � fc/2 in the high temperature and low temperature limits. 5. Conclusion The probability density of energy distribution of ions along axial direction and in radial plane together results total prob- ability density of energy distribution under high temperature limit and similarly at low temperature limit measured in a Quad- rupole Penning trap. The total probability density of energy distribution in the low temperature limit is much sharper and greater than that at the high temperature limit. These results reveal the energy properties of an ion cloud and are useful to design and carry out experiments on stored single particle, an- tiparticles with energy related parameters, under high tempera- ture and low temperature limits in Quadrupole Penning trap. Acknowledgments We thank the referees for the positive enlightening com- ments and suggestions, which have greatly helped us in making improvements to this paper. The author also acknowledges Dr. Satyajith K. 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