An Analytical Formula for the Continuous Absorption Coefficient and the Scattering Amplitude of the Electron Detachment for the Negative Hydrogen Ion by Electron Impact.код для вставкиСкачать
ANNALEN D E R PHYSIK 7.FOLGE BAND23, * HEFT7/8 * 1969 An Analytical Formula for the Continuous Absorption Coefficient and the Scattering Amplitude of the Electron Detachment for the Negative Hydrogen Ion by Electron Impact By T.TIETZ Abstract I n this paper we derive an approximate analytical formula for the continuous absorption coefficient of the hydrogen negative ion. The numerical results of this formula are compared with the results of CHANDRASEKH~R and ELBERT who used many-parameter groundstate function with the dipole-velocity matrix element. Here we derive also an analytical formula for the scattering amplitude for the detachment of electrons by electron impact from hydrogen negative ion in its ground '8,state. This paper contains numerical tables for the continuous absorption coefficient and other results. 1. Introduction It is well known that the continuous absorption coefficient of H- is of great astrophysical interest. This coefficient has been calculated by several authors [ l -131. The accurate numerical calculations being those of GIANDRASEEHAR and of JOHN. The previous theoretical calculations have been carried out with many-parameter ground state wave functions e. g. CHANDRASEEHAR and ELBERT  used a HARTand HERZBERG 20-parameter ground-state wave function and a HARTREE approximation for the continuum wave function. The simplest calculations concerning the continuous absorption coefficient for H are given by GELTMANand the author. GELTMAN'S considerations do not give an analytical formula for the continuous absorption coefficient for the hydrogen negative ion. The purpose of this note is to derive without much trouble, by a simple assumption an approximate analytical formula for the continuous absorption coefficient of H - and an analytical formula for the scattering amplitude of the electron detachment for H - by electron impact. 2. The approximate ground-state function for HI n order to obtain an analytical function for qo(r)we must solve the following SCHR~DINGER equation written in atomic units [A - 2V(r) + 2E01 %(r) = 0 (2.1) where V ( r )is the exact spherical potential and 214, I = k;; E, is the bound state having an energy equal i n magnitude to the accepted affinity of the hydrogen 20 Ann. Phye. 'I. Folge, Bd. 23 306 Annalen der Phyaik * 7.Folge & Band 23, Heft 718 * 1969 atom. The exact potential appearing in the last equation is given by: As known the SCHRODINCER equation for this potential cannot be solved exactly therefore we have approximated V ( r )given by eq. (2.2)as follows: where a, b, c and d are constants. Substituting the approximate potential V(r) given in the last formula in the SCHR~DINOER equation we see that po(r)is I n this case the constants b and d appearing in eq. (2.3)are given by the independent constants a, c as follows + a) U b =(1 - c) (2k0 2 a= + a) (1 - 4 (2kO + 4 w (2.5) o No appearing in vo(r)given by (2.4)is the radail normalization constant kO(2kO + a ) PIC, + (ko + 2a) 1.(2.6) The eigenfunction To(?+)for the ground-state corresponds t o an eigenvalue Eo where I n the physical literature there are known in atomic units some numerical values for k:. The HEINRICH'S value of kg is kg = 0.05466. The JOHN'S values of k; is kg = 0.055289 and the PEEERIS values of k% is kg = 0.0555027. Eq.(2.5) shows that if the constants a and c are given numerically then the approximate potential V ( r )given by eq. (2.3) and vo(r)by eq. (2.4)are determined. The constants a and c will be determined by help of variational method. 3. The variational method for hydrogen negative ion I n order t o determine the constants a and c appearing in the previous chapter we write the SCHRODINGER equation for H - in atomic units as follows: [- +dl - -12A 2 1 1 1 -TI - r2 + , - E l @ ( , , r r 2) = O (34 where E is the energy of the hydrogen negative ion in its groundstate and is the corresponding normalized wave function. The indices appearing in the last equation reffer t o the first and second electron. Since @(rl, r2) is normalized so that 1 E = JJ@*(r,, r,) sd2 - 1- 1 L ] @ ( r l , r 2 )dt, dt,. (3.2) ri ra Putting (3.3) @;P(rvr2) = vo(rd vo(r2) where cpo(r)is given by (2.4)and substituting the last equation in eq. (3.2) we see that after some simple but long calciilations we obtain for E the following @ ( r l , r,) [- - + 307 T. TIETZ: Scattering Amplitude of the Electron Detachment formula : + 2(2k0 + a) (ko + 2a) c + (2ki -I- 3k0a + 2a2)C21 + +4 8(24 + (5+ w C2 +--(In (4+ 2a) + WkO 3 4 (k, (4ko 5a)' + 5a)+(41C,3 +~ 7 )4 ~ 2c In (4k0 >I + + c21n + 4(2k0 (4k0 7 4 2 where N2 appearing here is given by (2.6). The numerical values of the constants a and c we determine from the minimum value of E. It has been found that E has a minimum in the neighbourhood of a = 3k0 and c = 1.31. If we accept the PEKERIS' values of ko we get E = -0.4856. I n case if ko takes the JOHN'S value we obtain for E the following value E = -0.4855. We see that the JOHN'S and PEKERIS' value of ko give nearly the same values for E. It has been found too Table 1 A comparison of t h e a p p r o x i m a t e p o t e n t i a l g i v e n b y eq. (2.3) with t h e e x a c t p o t e n t i a l given b y eq. (2.2) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.7 1.0 2.0 -m 20* I I +m 99.00 49.00 32.33 24.00 19.00 15.67 13.29 11.50 10.12 9.01 4.02 2.38 1.57 0.60 0.27 0.028 0.000 +m I 90.89 45.44 30.28 22.70 18.14 15.09 12.92 11.28 10.01 8.98 4.33 2.73 1.92 0.84 0.053 0.33 0.000 308 Annden der Physik * 7.Folge * Band 23, Heft 718 * 1969 that E given by the last formula has another minimum in the neighbourhood of a = 8ko and c = -0.125. For the PERERIS’ and JOHN’Svalues of k, the numerical values of E are respectively E = -0.4853 and E = -0.4852. Thus we have determined the numerical values of the constants a and c appearing in all above given formulas. I n Table 1 we have compared our approximate potential V ( r )given by eq. (2.3) with the exact potential V ( r )given by eq. (2.2). The comparison has been carried out for the numerical values a = 8k, and c = -0.125. We have used here the PEKERIS’ value of k,. The constants b and d appearing in the approximate potential V ( r ) are calculated from eq. [2.5). Table 1 shows that for small values of r the approximate potential - V ( r ) given by eq. (2.3) fits well the exact potential - V ( r ) given by eq. (2.2). 4. The continuous absorption coefficient of HThe purpose of this chapter is to derive an analytical formula for the continuous absorption coefficient of H-. It is known that the total continuous absorption coefficient x, for H - is given by radius, [x is the fine structure constant andv,is the normalwhere a, is the BOHR ized ground-state function which in our case is given by eq. (2.4), k2 = 2kk where Ekis the continuum state energy. Between the frequency v, E,, and Ek as it is known there exists the relation E, - Ek = hv. The continuum wave function qkas we know may be expended in term of LEQENDRE polynomials as The partial wave xlin eq. (4.1) contributes to the dipole matrix element in the pwave. We do not make a great error if we adopt for x1 the solution of the SCHR~DINQER equation for V = 0. Which is known and can be written in the following form : sin kr %=-cos k r . (4.3) kr Substituting 9, given by eq. (2.4) ans xl given by eq. (4.3) in the formula for the total continuous absorption coefficient given by eq. (4.1) we obtain after some calculations the following simple analytical formula for x, in our case. In Table 2. We have compared the numerical values of x, given by eq. (4.4) for the HEINRICH’S value of k, with the corresponding numerical values given by CHANDRASERHAR and ELBERT for the constants a and c we have accepted the numerical values a = 8ko and c = -0.125 discussed in the previous chapter. Table 2 shows that our results for x, calculated from eq. (4.4) agree well with the corresponding numerical values of CHANDRASEEHAR and ELBERT. I n Table 2 we have some numerical values given by eq. (4.4)in case of a = 8ko and 309 T. TIETZ:Scattering Amplitude of the Electron Detachment Table 2 A comparison of o u r c o n t i n u o u s a b s o r p t i o n coefficient of H eq. (4.4) w i t h t h e c o r r e s p o n d i n g n u m e r i c a l r e s u l t s of CHANDRASEKAAR and ELBERT x, (1O-I’ cma) k2 Our results eq. (4I4) 0 0.010 0.020 0.030 0.035 0.045 0.050 0.055 0.060 0.070 0.080 0.090 0.100 0.125 0.175 0.250 0.500 8.000 16533 13994 12131 10706 10112 9 102 8 669 8275 7 916 7 283 6 744 6279 5 875 5 059 3960 2 987 1642 1066 CHANDRASEKHAR and results ELBERT’S 0 1.70 3.12 3.93 4.17 4.42 4.47 4.48 4.47 4.38 4.24 4.09 3.91 3.49 2.76 2.00 0.93 0.50 0 2.83 4.03 4.37 4.45 4.49 4.41 4.09 3.90 3.41 2.30 1.83 0.75 0.34 c = -0.125 for the PEKERIS’ and JOHN’Svalues for ko. I n Table 2 we have also calculated some numerical values of x, from eq. (4.4)for the PERERIS’ value of ko if a = 3k0 and c = 1.31. Table 3 Some n u m e r i c a l v a l u e s f o r x, given b y eq. (4.4) f o r a = 3k0, c = 1.31 i f k, is given b y PEKERIS’ a n d a = 8 4 , c = -0.125 i f k, is given b y PEEERIS a n d JOHN cm2) a = 8kO c = -0,125 for PEKERIS’ value of k, a = 8ko c = -0,125 for JOHN’S 0 1.64 3.04 3.84 4.08 4.34 4.40 4.42 4.41 4.33 4.20 4.04 3.88 3.46 2.75 2.00 0.93 0.50 0 1.66 3.06 3.86 4.10 4.36 4.41 4.43 4.42 4.34 4.21 4.05 3.88 3.46 2.75 2.00 0.93 0.50 x, (1O-l’ ka 0 0.010 0.020 0.030 0.035 0.045 0.050 0.055 0.060 0.070 0.080 0.090 0.100 0.125 0.175 0.250 0.500 0.800 a 34, c = 1.31 for PEKERIS’ value of k, A 0 1.61 2.97 3.81 4.00 4.27 4.32 4.34 4.33 4.25 4.13 3.99 3.83 3.43 2.73 2.01 0.95 0.54 value of k, 310 Annalen der Physik * 7.Folge * Band 23, Heft 7/8 * 1969 Table 3 shows that the PERERIS’ value of ko for the case a = 8k0 and = -0.125 gives for x, a little lower values than the values obtained for this case for the JOHN’S values of ko. The case a = 3k0 and c = 1.31 as visible from Table 3 gives lower values for x, than the case a = 8ko and c = -0.125. If we G compare the numerical values for x,, given in Table 2 and 3 with the numerical of x, given by SMITH and BURCH  we see that our numerical values of x, are in a good accordance with the experimental values. 6. An analyticaI formuIa for the scattering amplitude of the electron detachment for € by Ielectron impact As known the total cross-section  for the detachment of an electron by an electron impact in its ground ISo state in the BORN-OPPENHEIMER approximation depends on the following scattering amplitudes where k,, k, and x are the incident, scattered and ejected wave vectors respectively, K = k, - k, - x and oi, A = j dz yOqH, J ( K ) = drrqo sin Kr. (5.2) 0 In the last formula yo(r)is given by eq. (2.4) and rpH(r)= 1 e-T. Substituting G qo(r)given by eq. (2.4) in the formula for d and J ( K ) we obtain the following formulas : and values for ko we obtain for A the value d = 0.8949. I n Using the PERERIS’ Table 4 we have some numerical values for J ( K ) / K calculated from eq. (5.4) for the PEIIERIS’ value of k;. Table 4 Some numerical values of J ( K ) / K given by eq. (5.4) K 10.1 1 0 . 2 10.3 10.4 10.5 10.6 10.7 10.8 / 0 . 9 11.0 J ( K ) / K 1 04.12 3.51 2.37 1.54 1.04 0.706 0.506 0.374 0.284 0.220 0.174 1 Since J ( E ) / Kis given analytically so that &l), = gkl) are also given analytically then we can derive an approximate formula for the total crosssection for the detachment of an electron by electron impact in its ground ISo state in the BORN-OPPENHEIMER approximation., These calculations are in progress. T. !I!IETZ: Scattering Amplitude of the Electron Detachment 311 6. Consequences and discussion The importance of stellar atmospheres of the continuous absorption of radiation by free-free transitions of electrons in the field of hydrogen atoms has long been known. All physical problems connected with H - cannot be done exactly since this would require a knowledge of the exact wave function of the hydrogen negative ion, therefore these problems can be solved only approximately. The numerical values of x, and J ( K ) / K show that the approximate simple formula for Q ) ~ ( T )given by eq. (2.4) allows us to calculate integrals appearing in different problems e.g. the scattering amplitudes of the electron detachment for H - by electron impact, the total cross-section for detachment of an electron by an electron impact in its ground 'So state in the BORN-OPPENHEIMER approximation, detachment from the negative hydrogen ion by electron or positron impact  and other problems. References [l] JEN,C.K., Phys. Rev. 48 (1933) 640.  lliZ4ssEY, H.S.W., and R.A.SYITH,Proc. Roy. SOC.,Lond. A. 17,166 (1936) 47. , W., and D.R.BATEs, Astrophys. J. 91 (1940) 202.  M ~ S S E YH.S.  WILLIAMS, R.E., Aetrophys. J. 96 (1942) 438.  HEINRICH, L. S., Astrophys. J. 99 (1943) 59.  CHANDRASEEHAR, S., Revs. Modern Ph 8.16 (1944) 301; Astrophys. J. 100 (1944) 176.  CHANDRASEKHAR, S., Astrophys. J. lOi(1945 )223; 102 (1945) 395. [S] GELTMAN, S., Phys. Rev. 104 (1956) 346.  CEANDRASERHAR, S., Astrophys. J. 128 (1958) 114. [lo] CHANDRASEKHAR, S., and D. D. ELBERT, Astrophys. J. 128 (1958) 633. [ll] SMITH,S.J., and D.S.BURCH,Phys. Rev. 116 (1959) 1125.  JOHN,T.L., Monthly Notices Roy. Astron. SOC.121 (1960) 41.  TIETZ,T., Phys. Rev. 124 (1961) 493.  see ref. [lo]. r151 see ref. r111. [16j see ref. r8]. RUDQE,M.R.N., Proc. phys. SOC.88 (1964) 419. L6di (Poland), Department of Theoretical Physics, University of U d i . Bei der Redaktion eingegangen am 3.Dezember 1968.