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An analytic calculation of the critical screening parameter for the exponential cosine screened Coulomb potential
Volume 77A, number 4
PHYSICS LETTERS
26 May 1980
AN ANALYTIC CALCULATION OF THE CRITICAL SCREENING PARAMETER FOR THE EXPONENTIAL COSINE SCREENED COULOMB POTENTIAL Ranabir DUTT Department of Physics, Visva-Bharati University, Santiniketan- 731235, West Bengal, India Received 10 December 1979 Revised manuscript received 12 March 1980
The critical screening parameter for the exponential cosine screened Coulomb potential is obtained analytically following a method proposed by Sedov. The result is in agreement with the values obtained by numerical and other approximate methods.
Bound s-state energies as a function of the screening parameter for the exponential cosine screened Coulomb (ECSC) potential, V(r) = —e2(e~’7r)cos(5r) have been obtained recently by numerical as well as various approximate methods [1—3].The ECSC potential is found to be very important in the study of electron—positron interaction in a positronium atom in solids [4} and hence attempts have been made to determine very precisely the critical screening coefficient ~c’ which is the cut-off value of the parameter beyond which no bound state with negative energy exists for a given quantum state. Although ~c for the ECSC potential had been determined numerically by Bouch-Bruevich and Glasko [1] and also by approximate methods [2,3], no analytic calculation for it exists so far. For the Yukawa potential, Sedov [5] first proposed an analytic method of determining the critical screening coefficient from an infinite series representation for the bound state wave function. Although no serious argument for the convergence of the series was given, it was shown that the value of the critical screening parameter obtained from the first few terms in the series is in good agreement with the numerical result [6]. In this letter, we want to emphasize that the technique proposed by Sedov may be extended to deal with more complex problems. For illustration, we work with the ECSC potential and determine 6c in the is state analytically. For the ECSC potential, the radial part of the Schrodinger equation for a bound electron in the s-state is given by ,
d,2+h2[E~rc05)]~~)0. Writing e~T cos(~r)= ~(e~” + e_?~*1~) A = ~(i ,
(1) (2)
+ i)
eq. (1) becomes dr2 132~(r)= ~ (e~’ + e_?~*T)p(r), in which ~32= —2pE/h2, a = pe2/h2. The solution of eq. (3) powers of a, —
(3)
—
may
be written in the form of an infinite series in
229
Volume 77A, number 4
PHYSICS LETTERS
+7 7+1) ~9_][(7 I )
~r)e~+a[(f
+7 )~1[(7+j~ 2][(f 7)
e~k])
)~-r~]2{[(f
~
+a3~[(
~
—q
~
/3
—q
q+X
2
i3
q+x*
~
~
+
~
26 May 1980
—q
q+X
+
k+X
—
dP])
~
k+?~*
/3
q+x*
/3
—
+....
(4)
—~
One may easily check by direct substitution that eq. (4) is a solution of eq. (3). Each term of the expression (4) beginning with the second when substituted into the left hand side of eq. (3), gh~esan expression which is obtained upon substituting the preceding term into the right hand side of eq. (3). For bound states, we now insert the boundary condition ~(0) = 0 which guarantees that the radial wave function t,(1(r) = ~(r)/r is finite at r = 0. Further, we require that 3 = 0 (i.e., E = 0) when the screening parameter attains its maximum (critical) value 6~ However, strictly speaking, i~i(r)is unnormalized in2the is no rigorous proof that the series on the r2 sense dr < that forthere j3 = 0). right hand side of eq. (4) converges f~”I’,c(r)1r1 Imposing these requirements and(i.e., applying the change of variables q = 6 cq’~k = 6 c1~’,etc., eq. (4) becomes
1-
~[(L÷[~)~~(~)2 {[( L~)~]RL1+11i)~J) W3 ~
(5)
The dimensionless integrals in eq. (5) are obtained in fairly standard forms [7] and may be evaluated in closed form. We give the results here up to order (a/6c)3, 1
—
(cr16 c)ji
+
(a/6~)21
4)
31
2
—
(a/6c)
3 +
=
0
(6)
,
O((a/6c)
with I~= 1
12
=
(1
—
ir/4)
,
13
=
[1
—
7ir/i6
+
~log~3 ~log~5 ~tan~(~)] —
—
Using the numerical values of 1~,‘2 and 13, eq. (6) gives (‘~c1°~)~ (6c/a)2 + 0.214 60 (6~/a) 0.00702 which has only one real positive root, —
—
=
0,
(7)
6~/a=O.7l27. (8) This is in excellent agreement with the value 6 ckX = 0.72 obtained by Bouch-Bruevich and Glasko using a numerical method [1] and with 6~/a= 0.7 115 obtained by Lam and Varshni who used a variational technique [21.The success of the result indicates that applications of the present method to other short-range potential problems will be worth persuing. I feel happy to thank Professor S.N. Biswas for suggesting the problem. [1] [2] [3] [4]
V.L. Bouch-Bruevich and V.B. Glasko, Soy. Phys. Dokl. 4 (1959) 147. C.S. Lam and Y.P. Varshni, Phys. Rev. A6 (1972) 1391. R. Dutt, Phys. Lett. 73A (1979) 370. E.P. Prokepev, Soy. Phys. Solid State 9 (1967) 993. [5] V.L. Sedov, Soy. Phys. Doki. 14(1970) 871. [6] F.J. Rogers, H.C. Graboske Jr. and DJ. Harwood, Phys. Rev. Al (1970) 1577. [7] I.S. Gradshteyn and I.W. Ryzhik, Tables of integrals, series and products (Academic Press, 1965).
230
PHYSICS LETTERS
26 May 1980
AN ANALYTIC CALCULATION OF THE CRITICAL SCREENING PARAMETER FOR THE EXPONENTIAL COSINE SCREENED COULOMB POTENTIAL Ranabir DUTT Department of Physics, Visva-Bharati University, Santiniketan- 731235, West Bengal, India Received 10 December 1979 Revised manuscript received 12 March 1980
The critical screening parameter for the exponential cosine screened Coulomb potential is obtained analytically following a method proposed by Sedov. The result is in agreement with the values obtained by numerical and other approximate methods.
Bound s-state energies as a function of the screening parameter for the exponential cosine screened Coulomb (ECSC) potential, V(r) = —e2(e~’7r)cos(5r) have been obtained recently by numerical as well as various approximate methods [1—3].The ECSC potential is found to be very important in the study of electron—positron interaction in a positronium atom in solids [4} and hence attempts have been made to determine very precisely the critical screening coefficient ~c’ which is the cut-off value of the parameter beyond which no bound state with negative energy exists for a given quantum state. Although ~c for the ECSC potential had been determined numerically by Bouch-Bruevich and Glasko [1] and also by approximate methods [2,3], no analytic calculation for it exists so far. For the Yukawa potential, Sedov [5] first proposed an analytic method of determining the critical screening coefficient from an infinite series representation for the bound state wave function. Although no serious argument for the convergence of the series was given, it was shown that the value of the critical screening parameter obtained from the first few terms in the series is in good agreement with the numerical result [6]. In this letter, we want to emphasize that the technique proposed by Sedov may be extended to deal with more complex problems. For illustration, we work with the ECSC potential and determine 6c in the is state analytically. For the ECSC potential, the radial part of the Schrodinger equation for a bound electron in the s-state is given by ,
d,2+h2[E~rc05)]~~)0. Writing e~T cos(~r)= ~(e~” + e_?~*1~) A = ~(i ,
(1) (2)
+ i)
eq. (1) becomes dr2 132~(r)= ~ (e~’ + e_?~*T)p(r), in which ~32= —2pE/h2, a = pe2/h2. The solution of eq. (3) powers of a, —
(3)
—
may
be written in the form of an infinite series in
229
Volume 77A, number 4
PHYSICS LETTERS
+7 7+1) ~9_][(7 I )
~r)e~+a[(f
+7 )~1[(7+j~ 2][(f 7)
e~k])
)~-r~]2{[(f
~
+a3~[(
~
—q
~
/3
—q
q+X
2
i3
q+x*
~
~
+
~
26 May 1980
—q
q+X
+
k+X
—
dP])
~
k+?~*
/3
q+x*
/3
—
+....
(4)
—~
One may easily check by direct substitution that eq. (4) is a solution of eq. (3). Each term of the expression (4) beginning with the second when substituted into the left hand side of eq. (3), gh~esan expression which is obtained upon substituting the preceding term into the right hand side of eq. (3). For bound states, we now insert the boundary condition ~(0) = 0 which guarantees that the radial wave function t,(1(r) = ~(r)/r is finite at r = 0. Further, we require that 3 = 0 (i.e., E = 0) when the screening parameter attains its maximum (critical) value 6~ However, strictly speaking, i~i(r)is unnormalized in2the is no rigorous proof that the series on the r2 sense dr < that forthere j3 = 0). right hand side of eq. (4) converges f~”I’,c(r)1r1 Imposing these requirements and(i.e., applying the change of variables q = 6 cq’~k = 6 c1~’,etc., eq. (4) becomes
1-
~[(L÷[~)~~(~)2 {[( L~)~]RL1+11i)~J) W3 ~
(5)
The dimensionless integrals in eq. (5) are obtained in fairly standard forms [7] and may be evaluated in closed form. We give the results here up to order (a/6c)3, 1
—
(cr16 c)ji
+
(a/6~)21
4)
31
2
—
(a/6c)
3 +
=
0
(6)
,
O((a/6c)
with I~= 1
12
=
(1
—
ir/4)
,
13
=
[1
—
7ir/i6
+
~log~3 ~log~5 ~tan~(~)] —
—
Using the numerical values of 1~,‘2 and 13, eq. (6) gives (‘~c1°~)~ (6c/a)2 + 0.214 60 (6~/a) 0.00702 which has only one real positive root, —
—
=
0,
(7)
6~/a=O.7l27. (8) This is in excellent agreement with the value 6 ckX = 0.72 obtained by Bouch-Bruevich and Glasko using a numerical method [1] and with 6~/a= 0.7 115 obtained by Lam and Varshni who used a variational technique [21.The success of the result indicates that applications of the present method to other short-range potential problems will be worth persuing. I feel happy to thank Professor S.N. Biswas for suggesting the problem. [1] [2] [3] [4]
V.L. Bouch-Bruevich and V.B. Glasko, Soy. Phys. Dokl. 4 (1959) 147. C.S. Lam and Y.P. Varshni, Phys. Rev. A6 (1972) 1391. R. Dutt, Phys. Lett. 73A (1979) 370. E.P. Prokepev, Soy. Phys. Solid State 9 (1967) 993. [5] V.L. Sedov, Soy. Phys. Doki. 14(1970) 871. [6] F.J. Rogers, H.C. Graboske Jr. and DJ. Harwood, Phys. Rev. Al (1970) 1577. [7] I.S. Gradshteyn and I.W. Ryzhik, Tables of integrals, series and products (Academic Press, 1965).
230