Relations between photoionization cross sections and photon radius
Shan-Liang Liu
Shandong Key Laboratory of Optical Communication Science and Technology, School of Physical Science and Information Engineering, Liaocheng University, Shandong 252059, People’s Republic of China
The relations between photoionization cross sections and photon radius are obtained on basis of quantum mechanics and the particle-like properties of a photon. The photoionization cross sections of H atom and H-like ions, He atom and He like ions, alkali metal atoms, and Rydberg atoms are calculated using the relations. The calculation results are found to be good agreement with the known experimental data. The results show that the photoionization cross section is always smaller than the cross section of the photon to ionize the atom or ion and can be expressed as the product of the cross section of the photon and the probability that electron meets with the photon. These provide the intuitive understanding for the photoionization phenomena and open a new avenue of research on interaction between a photon and an atom or ion.
Keywords: photoionization; photon radius; Rydberg atoms; alkali metal atoms; He like ions; H-like ions sections and photon radius are obtained here on 1. Introduction basis of quantum mechanics and the particle-like An atom or ion is ionized by a single photon properties of a photon. The calculations from the whose energy hν is higher than the ionization relations are found to be in good agreement with energy Ui, and the kinetic energy of the the known experimental data. These provide the intuitive understanding for the photoionization photoelectrons is given by hν−Ui, which is the most obviously particle-like characteristic of a phenomena and open a new avenue of research on photon. Recently, it has been shown that a photon interaction between a photon and an atom or ion. has the shape of a cylinder with length of λ/2 and 2. Photoionization of atoms and ions its radius is proportional to square root of the wavelength [1]. However, the photoionization 2.1. Photoionization of H and H-like ions cross section is usually calculated by the dipole Radius of a photon is proportional to square root approximation which results from the combination of the wavelength and is given by [1], of quantum mechanics and wave theory of electromagnetic fields [2-4]. There is not intuitive 2 2r λ ρ = e , (1) understanding for some phenomena that the photo- 0 ( 2 -1)π ionization cross section monotonously decreases 2 2 with increase of the photon energy; the electrons in where re=e /(4πε0m0c ) is the classical radius of an inmost shell in the atoms is ionized easier than electron. Since the electron acts with the photon only if an electron is in the electromagnetic fields those in the outer shell for the same photon energy 2 which can ionize them, and so on. The dipole of a photon, the cross section of a photon S0=πρ0 approximation is theoretically no longer valid in is the interaction cross section between an electron and a photon. There is only one electron in an H some cases, but the calculations that make the 2 approximation are in agreement with the atom or H-like ion, the ionization energy Ui=Z UH experimental data [5, 6]. To reconcile this apparent at the ground state where Z is the proton number in inconsistency, one may argue that the photo- the atom or ion and UH=13.6 eV is the ionization ionization process occurs close to the nucleus in energy of the H atom. Substituting λ=hc/Ui into Eq. the atom [7, 8]. The measured values of the photo- (1) gives the maximum radius of the photon to ionization cross sections for potassium (K) atoms ionize the atom or ions in the 5d and 7s excited states are much larger than ρmax = 0.66r1 , (2) the theoretical values [9,10], whereas the measured where r1=a0/Z, a0 is the Bohr radius. Equation (2) values of the cross section for the Rydberg reveals that an H atom or H-like ion can be ionized rubidium (Rb) atom in the nd states are lower by only by the photon whose radius is shorter than r1. about a factor of 4 than the theoretical values [11]. The radial wavefunction of the electron in H The striking differences between the theoretical atom or H-like ion at the ground state is calculations and experimental data are still to be 3/ 2 R10 = 2(1/ r1 )()exp − r / r1 . (3) further investigated. 2 The relations between photoionization cross The probability density |R10| of the electron exponentially decreases with increase of r and is at
2 4 6 3 r=r1 only 1/e of the maximum at r=0. The which is proportional to Z and ρ0 or λ . For 5 5 7 probability of the electron in the cylinder region V0 ρ0<
1 10 known as the screening constant and arises from interaction between two electrons in the atom or 0 10 ion. The parameters s and r1 for all the He-like ions
-1 10 are calculated from Eqs. (8) and (9) using the measured values of Ud in Ref [12]. The screening -2 10 constant of the He-like ion at Z<13 has almost the
-3 10 same value of s=0.3 as that of He and obeys Slater’s rule, but the screening constant slowly Cross Section (Mb) Section Cross -4 10 decreases with increase of Z for Z>13, and s=0.2 +24 -5 for Fe is obviously smaller than 0.3 from 10 Slater’s rule. It means that Slater’ rule is no longer
-6 10 accurate for Z>13 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Photon Radius /r ρ0 1 The Substituting λ=hc/U1 into Eq. (1) gives the maximum radius of the photon to ionize the atom FIG. 1. Photoionization cross sections of H atom and H-like ions as function of photon radius. The solid or ions −10 −1/ 2 curves are plotted by Eq. (5), the experimental data: ρ max = 1.286×10 U1 , (10) ● for H, ◊ for Li+2, ○ for Na+10; ∆ for Fe+25 ) are taken from Ref. [12]. where U1 is the ionization threshold of the He atom or He-like ions in the unit of eV. The maximum The cross sections of H atom and all H-like ions radius of the photon is obtained by substituting the of the Opacity Project (OP) elements are calculated measured values of U1 into Eq. (10). The using Eq. (5). Figure 1 shows the cross sections of maximum ratio ρmax/r1 monotonously decreases the H atom and H-like ions as function of the from 0.83 to 0.67 as Z increases from 2 to 26 and photon radius in unit of r1, where r1 varies with Z is smaller than that of H atom or H-like ions. The and has different values for different ions. The larger Z is, the closer ρmax/r1 of the He-like ion is calculation results (solid curve) agree with the to that of the H-like ion. This shows that the effect experimental data (signs) very well for ρ0>0.1r1, as of interaction between two electrons on their states shown in Fig. 1. The calculation results are in good becomes the weaker and weaker as Z increases; the agreement with the experimental data for the He atom or He-like ion at the ground state can be photon energy from the ionization threshold to 50 ionized only by the photon whose radius is smaller kev if the probability that the electron meets with than the orbital radius. the photon is given by There are two identical electrons in the He atom 4 or He-like ion, the wavefunction and P0 have the ⎛ ρ 0 ⎞ ⎡ ⎛ 12ρ0 ⎞⎤ P = ⎜ ⎟ ⎢1− exp⎜− ⎟⎥ . (6) same form as Eqs. (3) and (4) for each of them if ⎝ r1 ⎠ ⎣ ⎝ r1 ⎠⎦ the independent particle approximation is made, 4 respectively, and the photoionization cross sections For ρ0>0.1r1, P≈(ρ0/r1) , being in good agreement 2 ρ >0.1r is given by with P0=(ρ0/r1) obtained from Eq. (4), and the 0 1 2 4 cross section is given by σ = 2πρ0 (r0 / r1) . (11) 2 4 σ = πρ0 (ρ0 / r1) , (7) The cross sections of He atom and all He-like ions
of the Opacity Project (OP) elements are calculated ρ0>r1, as shown in Fig. 3. using Eq. (11). The calculation results deviate from Since the probability that electron meets with the experimental data for the He atom near the the photon is equal to P0 in the case of ρ0>r1, 2 ionization threshold due to interaction between two instead of P0(ρ0/r1) , and the photoionization cross electrons, especially. With increase of Z, the section ρ0>r1 is given by deviation as well as the interaction effect becomes σ = πρ 2P . (17) smaller and smaller, the calculation results for 0 0 +24 The calculation results (solid curves) for the Li and Fe are in good agreement with the experimental Na atoms at the ground state using Eqs. (12)-(17) data (Δ), as shown in Fig. 2. for ρ0>r1 are in agreement with experimental data 2 10 except near the ionization threshold, as shown in
1 10 Fig. 3. The differences between the theoretical
0 calculations and the experimental data near the 10 ionization threshold can be attributed to the core -1 10 polarization [10]. The cross section increases with
-2 10 the photon radius for ρ0>r1.
-3 2 10 10 Cross Section (Mb)
-4 0 10 10
-5 10 -2 10
-6 10 -4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 Photon Radius ρ /r 0 1 -6 10 FIG. 2. Photoionization cross section of He and CrossSection (Mb) -8 He-like as function of photon radius. The curves 10 are the calculations results, the experimental data -10 (● for He, ◊for Be+2, ○ for Na+9; Δ for Fe+24 ) are 10
taken from Ref. [12]. -12 10 -2 -1 0 1 10 10 10 10 Photon Radius /r 2.3. Photoionization of alkali metal atoms ρ0 1 The probability that the valence electron at alkali metal atom occurs within the cylinder region FIG. 3. Photoionization cross section of Li and Na V0 has the maximum atoms as function of photon radius. The solid curves ∞ ρ 0 2 are the calculations result from Eq. (17), the dotted P0 = 4π |ψ nl0 | ρdρdz . (12) curve from Eq. (5). The signs denote the ∫∫00 experimental data of Li atom (●) and Na atom (□) where ψnlm=Rnl(r)Yl0(cosθ) is the normalized wave from Ref. [12]. function, Y (cosθ) is the spherical harmonic l0 The photoionization cross sections of K atoms in function, and Rnl(r) are Slater radial wavefunction of the valence electron [13,14], respectively. The the 4p, 5d, and 7s excited states as function of the Slater radial wavefunctions are given by photon radius are calculated using Eqs. (12)-(17) for ρ >r . The calculated value of K atom in the 4p ⎡ n−1 ⎤ 0 1 n−1 , (13) Rnl = Nnl ⎢r exp(−r / rn ) + ∑ Anlj Rjl ⎥ excited state for the 355-nm photon is 8 Mb and ⎣ j=1 ⎦ agrees with the measured value of 7.2±1.1 Mb [9]. ⎡ n−1 ⎤ The calculated value of K atom in the 7s excited N = (2n)!(r / 2)2n+1 − A2 ,n > l , (14) nl ⎢ n ∑ nlj ⎥ state for the 660.6-nm photon is 0.67 Mb and ⎣ j=l +1 ⎦ agrees with the measured value of 0.61±0.09 Mb ⎡ j−1 ⎤ (n + j)! , (15) [9], which are larger by two orders of magnitude Anlj = N jl ⎢∑ Anlk Ajlk − n+ j+1 ⎥ ⎣⎢ k=1 (rj + rn ) ⎦⎥ than the previous calculation [10]. However, the na / Z , Z >1 calculated value of K atom in the 5d excited state ⎧ 0 n n (16) rn = ⎨ n*a , Z =1 for the 662.6-nm photon is 10 Mb, about the twice ⎩ 0 n the previous calculation value [10], and is where Zn is the effective atomic number and is obviously smaller than the measured value of determined by Slater’s rule, n*=n−δl is the 28.9±4.3 Mb [9]. effective quantum number, and δl is the quantum The probability density of valence electron in K defect. For Li atom at the ground state, r1=a0/2.7, atoms in the ns, np, and nd states is calculated the calculation results (black dotted curve) using using the Slater wavefunction. Unlike the ns state, Eq. (5) and the Slater wavefunction are agreement the probability density is zero at r=0 in the np and with experimental data (red dots) for ρ0 The probability density (red plus) at r=1.78a0 (the 3. Photoionization of Rydberg atoms photon radius at 662.6 nm) is only about 1/3 of the The radial wavefunction of Rydberg atoms can maximum, as shown in Fig. 4 (a). The probability be given by [15,16] in the cylinder with the cross section S0 between ρ r 3n !r'l* exp(−r')Lµ (r') n r nr , (19) and ρ+ρ in the 5d state slowly decreases with Rnl = 0 3 increase of ρ and is still more than 1/3 of the (2nr + µ +1)Γ (nr + µ +1) maximum at ρ=6a0, and the valence electron is in where r′=r/rn, rn=n*a0, µ=2l*+1, l*=l−δl>0, µ the region of ρ>ρ0 with large probability, as shown nr=n−l−1, and L nr is the generalized Laguerre in Fig. 4(b). Since ρ0 25 cross section is given by 10 2 2 2 nd 1 1 σ = S (S / S ) = S / S . (18) R 10 10 eff 0 eff 0 eff 24 10 Equation (18) gives σ=27 Mb for K atom in the 5d Cross Section (Mb) Cross Section (Mb) 23 state and the 662.6-nm photon if the effective 10 radius reff=5.7a0 is taken, which is in good 22 16d Rb 16d 20s 10 0 0 16d H 10 20d 10 20p agreement with the measured value. It is seen in 5d K 30d 20d 21 10 Fig. 4 that reff>rmax>ρ0, the probability density 0 10 20 0 10 20 0 5 10 15 r/a ρ /a ρ /a (blue circle) at r=reff is obviously larger than that 0 0 0 0 0 at r=ρ0, and the averaged probability at ρ=reff is about 1/3 of the maximum at ρ=0. For alkali metal FIG. 5. (a) Probability density distributions of the different atoms in the nd states. The photoionization atoms, n*>1, ρmax=0.66n*a0 is obviously shorter 2 cross section as function of photon radius: (a) for than the atom radius n* a0. Rydberg Rb atoms with different values of n and (c) for Rydberg H atoms with different values of l. 28 10 0.04 For Rydberg atom n*>>1, ρmax=0.66rn>>rmax, 27 10 0.035 and the photoionization cross section can be calculated using Eq. (17). The photoionization 26 10 0.03 cross sections of Rydberg Rb atoms in the nd states are calculated using the relation (17) and the 25 10 0.025 wavefunction (19) from the ionization threshold to Probability 1 eV. The cross section decreases with increase of 24 Probability Density 10 0.02 n for the given photon energy and increase with the 23 photon radius for the given value of n, as shown in 10 0.015 Fig. 5(b). For the 10.6-µm photon, ρ0=7.1a0 is 22 10 0.01 obviously larger than rmax=a0, the calculations for 0 5 10 15 0 2 4 6 r/a /a the Rb atoms in the nd states (16≤n≤20) give 32 0 ρ 0 Mb, 26 Mb, 21.1 Mb, 17.3 Mb, and 14.5 Mb, FIG. 4. (a) Probability density distribution of the respectively. 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