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Photodetachment of hydrogen negative ion: A many-body approach
Volume
34A, number
PHYSICS
7
LETTERS
19 April
P HOTODETACHMENT OF HYDROGEN NEGATIVE A MANY -BODY APPROACH
1971
ION:
N. C. DUTTA and C. M. DUTTA Department
of Physics,
University Received
of Utah,
Salt Lake
City,
Utah,
USA
19 March 1971
The Brueckner-Goldstone many-body perturbation theory is applied to calculation of the photodetachment cross section of H- . Various electron correlation effects associated with the initial and final states are discussed using diagrammatic techniques.
The theoretical evaluation of the photodetachment cross section of H- is of considerable importance in astrophysics [e.g. 11. The most accurate cross section to date has been provided by the close coupling calculations [ 21. However, such calculations are known to be very involved and extension of these techniques to more complicated systems with the same accuracy as in H- seems unlikely with the present computers. In this letter, we present our results on the application of the Brueckner -Goldstone many -body perturbation theory [3] to this problem. In our earlier work [4] we have obtained a very good result on the binding energy of H- by the Brueckner-Goldstone procedure. The same complete set of one electron states generated in the field of VNml potential of H’ is utilized here to compute the photodetachment cross section, a, for the process H- + hu + H + e-. In the dipolelength and velocity approximation, u is written as
utLT ‘) =Q7ra cz2(Z+e)l rntL9 ‘)I2 0
(cm2)
(1)
where Z is the ionization potential, E is the kinetic energy of the ejected electron, and ,(L)
= SPi(r)r Pf(~)dr
(2)
and
m(‘)=&
_/Pi(r)(+Pf(r)
+& Pf(y))d’.
(3)
Atomic units such that ti= 1, e2 = l/m = 2 are used. Pi(r) and Pf(y) are normalized reduced radial wavefunctions of the initial and final states. Utilizing the diagrammatic techniques, u is expressed as a sum of diagrams such as shown in fig. 1. (la) is the lowest zero-order diagram,
v$_ Ltf ___r (4
(b)
(4
-a-
(8)
(h)
(0
Fig. 1. Diagrams for the photodetachment cross section of H-. The dashed lines denote l/r interaction whereas the wiggly lines represent t+12 e photon operator (dipole velocity or length).
representing the Hartree-Fock contribution to Q The first-order diagrams (lb) and (lc) depict the correlation corrections to the initial and final states, respectively. (lb) may be interpreted as a short range correlation effect. The secondorder diagrams are shown in (Id)-(li). (ld) represents “hole-hole” interaction and performs the self energy correction, which was found to be substantial for the binding energy of H-. There are also corresponding “hole-particle” and “particle-particle” interaction terms. These are included to all orders by the shifted energy denominator technique [3,4]. (le)-(lf) are the higherorder counterparts of (lb) and (lc), respectively. (lg) represents a second-order correction to a, whereas (lh) gives contributions to u arising from transitions between the short-range corre369
Volume 34A, number 7
PHYSICS
Fig. 2. Computed photodetachment cross section for H-. Continuous and dashed curves are obtained in the dipole velocity and dipole length approximation, respectively. Circles denote the experimental points.
lation parts of Pi(r) and P&r). One of the dominant second-order diagrams in the present calculation is (li). This diagram takes account of the distortion of the neutral atom due to the ejected electron via polarization effecta The major contribution to such effects comes from the dipole polarization. Our final results for u are shown in fig. 2, together with the experimental points [ 51. The agreement with the experiment is good and within the quoted experimental error. It is noticed that the velocity formula gives better agreement. Our final a(L) and o(v) results are also close to each
*****
370
LETTERS
19 April 1971
other, testifying to the overall correctness of the procedure. It is demonstrated here that the complete set of states used in the framework of BruecknerGoldstone theory to calculate the energy of a negative ion is also adequate to describe the final state in photodetachment problems, resulting in a great reduction of the computing time. The diagrammatic technique utilized here makes calculations convenient, and points out the relative importance of various physical effects. This procedure can be easily applied to more complicated systems. All computations were performed in UNIVAC 1108, using the programs written by the authors. The entire calculation took about eight minutes in this machine.
References in: Physics of the one- and two111 L. M. Branscdmb, electron atoms (North H&and, 1969) p. 669. PI N.A. Doughty, P. A. Fraser and R. P.McEachran, Mon. Not. Roy. Astr. Sot. 132 (1966) 255. Proc. Roy. Jot. (London), Ser. A 239 [31 J.Goldstone, (1957) 267; ‘. H. P. Kelly, in: Perturbation theory and its application in quantum mechanics, ed. C. H. Wilcox (John Wiley and %me, Inc., 1966) p. 215. C.M.DuttaandT.P.Das, Phys.Rev. 141 N.C.Dutta, A 2 (1970) 2289. 151 S. J. Smith and D. J. Burch, Phys. Rev. 116 (1959) 1125.
34A, number
PHYSICS
7
LETTERS
19 April
P HOTODETACHMENT OF HYDROGEN NEGATIVE A MANY -BODY APPROACH
1971
ION:
N. C. DUTTA and C. M. DUTTA Department
of Physics,
University Received
of Utah,
Salt Lake
City,
Utah,
USA
19 March 1971
The Brueckner-Goldstone many-body perturbation theory is applied to calculation of the photodetachment cross section of H- . Various electron correlation effects associated with the initial and final states are discussed using diagrammatic techniques.
The theoretical evaluation of the photodetachment cross section of H- is of considerable importance in astrophysics [e.g. 11. The most accurate cross section to date has been provided by the close coupling calculations [ 21. However, such calculations are known to be very involved and extension of these techniques to more complicated systems with the same accuracy as in H- seems unlikely with the present computers. In this letter, we present our results on the application of the Brueckner -Goldstone many -body perturbation theory [3] to this problem. In our earlier work [4] we have obtained a very good result on the binding energy of H- by the Brueckner-Goldstone procedure. The same complete set of one electron states generated in the field of VNml potential of H’ is utilized here to compute the photodetachment cross section, a, for the process H- + hu + H + e-. In the dipolelength and velocity approximation, u is written as
utLT ‘) =Q7ra cz2(Z+e)l rntL9 ‘)I2 0
(cm2)
(1)
where Z is the ionization potential, E is the kinetic energy of the ejected electron, and ,(L)
= SPi(r)r Pf(~)dr
(2)
and
m(‘)=&
_/Pi(r)(+Pf(r)
+& Pf(y))d’.
(3)
Atomic units such that ti= 1, e2 = l/m = 2 are used. Pi(r) and Pf(y) are normalized reduced radial wavefunctions of the initial and final states. Utilizing the diagrammatic techniques, u is expressed as a sum of diagrams such as shown in fig. 1. (la) is the lowest zero-order diagram,
v$_ Ltf ___r (4
(b)
(4
-a-
(8)
(h)
(0
Fig. 1. Diagrams for the photodetachment cross section of H-. The dashed lines denote l/r interaction whereas the wiggly lines represent t+12 e photon operator (dipole velocity or length).
representing the Hartree-Fock contribution to Q The first-order diagrams (lb) and (lc) depict the correlation corrections to the initial and final states, respectively. (lb) may be interpreted as a short range correlation effect. The secondorder diagrams are shown in (Id)-(li). (ld) represents “hole-hole” interaction and performs the self energy correction, which was found to be substantial for the binding energy of H-. There are also corresponding “hole-particle” and “particle-particle” interaction terms. These are included to all orders by the shifted energy denominator technique [3,4]. (le)-(lf) are the higherorder counterparts of (lb) and (lc), respectively. (lg) represents a second-order correction to a, whereas (lh) gives contributions to u arising from transitions between the short-range corre369
Volume 34A, number 7
PHYSICS
Fig. 2. Computed photodetachment cross section for H-. Continuous and dashed curves are obtained in the dipole velocity and dipole length approximation, respectively. Circles denote the experimental points.
lation parts of Pi(r) and P&r). One of the dominant second-order diagrams in the present calculation is (li). This diagram takes account of the distortion of the neutral atom due to the ejected electron via polarization effecta The major contribution to such effects comes from the dipole polarization. Our final results for u are shown in fig. 2, together with the experimental points [ 51. The agreement with the experiment is good and within the quoted experimental error. It is noticed that the velocity formula gives better agreement. Our final a(L) and o(v) results are also close to each
*****
370
LETTERS
19 April 1971
other, testifying to the overall correctness of the procedure. It is demonstrated here that the complete set of states used in the framework of BruecknerGoldstone theory to calculate the energy of a negative ion is also adequate to describe the final state in photodetachment problems, resulting in a great reduction of the computing time. The diagrammatic technique utilized here makes calculations convenient, and points out the relative importance of various physical effects. This procedure can be easily applied to more complicated systems. All computations were performed in UNIVAC 1108, using the programs written by the authors. The entire calculation took about eight minutes in this machine.
References in: Physics of the one- and two111 L. M. Branscdmb, electron atoms (North H&and, 1969) p. 669. PI N.A. Doughty, P. A. Fraser and R. P.McEachran, Mon. Not. Roy. Astr. Sot. 132 (1966) 255. Proc. Roy. Jot. (London), Ser. A 239 [31 J.Goldstone, (1957) 267; ‘. H. P. Kelly, in: Perturbation theory and its application in quantum mechanics, ed. C. H. Wilcox (John Wiley and %me, Inc., 1966) p. 215. C.M.DuttaandT.P.Das, Phys.Rev. 141 N.C.Dutta, A 2 (1970) 2289. 151 S. J. Smith and D. J. Burch, Phys. Rev. 116 (1959) 1125.