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The resonance annihilation of slow positrons on molecules
Volume 135, number 1,2
CHEMICAL PHYSICS LETTERS
THE RESONANCE ANNIHILATION
27 March 1987
OF SLOW POSITRONS ON MOLECULES
G.K. IVANOV Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow V-334, USSR Received 16 December 1986
The resonance mechanism for the interaction of slow positrons with molecules and the effect of a strong electromagnetic field on this mechanism are studied. It is shown that when the field is in resonance with the transition between the vibrational levels of the intermediate complex e+XY, the annihilation characteristics of the system e+ +XY are changed substantially.
There are two fundamental mechanisms for the interaction of slow positrons with molecules: resonance and direct. The resonance mechanism in the theory of positron annihilation was first discussed in ref. [ 11. For a molecule XY this mechanism assumes the formation of an intermediate complex e+XY with a lifetime of the order of the molecular vibrational period r or greater. The characteristic time of the direct mechanism is much less than r. A description of the latter mechanism involves detailed interaction potentials and is a complicated quantum-mechanical problem. All characteristics of a direct process are weak functions of the energy. A combined description of these two mechanisms is possible if one can extract the strong (resonance) dependence from the background of the weak dependence created by the direct processes. In atom autoionization theory this problem was solved by Fano [ 21 (see also refs. [ 3-51) by utilizing a “prediagonalization” of the Hamiltonian with the subspace of closed channels. The method proposed in the present work is a generalization of the Fano method and is the first time such a method has been applied to the problem of positron annihilation. We assume that in the annihilation process
e++XY--+
+2y+xy+
(1)
a population of different interacting quasistationary states of the complex (e+XY and e+XY’) is possible
with their decay occurring in the continua corresponding to the different particles: e+ +XY and Ps + XY+. Note that under normal conditions overlap of resonance configurations is unusual; however, such an overlap is common when the system interacts with a strong electromagnetic field. The calculation of the annihilation amplitude A, reduces to the determination of the wavefunction Yy, of the system e+ + XY in the main transition region: A,=
YU;;v+exp(ik,-r+)
YY:,dr,...dr, dr+ .
(2)
Here Yxy + is the wavefunction of the ion XY +, cy is its vibrational quantum number (the molecular orientation is assumed to be fixed), k7 is the total momentum of the two photons, Y ;= YY,(r,, . ... r,, rN+,, r + = rN+, ) , with r + the positron coordinate and r,, .... r,, rN+, the coordinates of the electrons. Four main groups of states may be introduced for the process under consideration: e+ + XY, Ps + XY+, e+XY and e+XY’. We call these configurations and denote the corresponding states by q, 8, p and v. Let the operator V include the configuration interaction (CI) and that part of the interaction in the contigurations e+ +XY and Ps+XY+ which remains after displacing the potentials k$‘+,, and p&+xu away from the equilibrium interatomic distances in XY and XY+. Thus for the system without CI we propose to fulfil the “prediagonalisation” condition for the isolated configurations e+XY and e+XY’ only. This enables us to fix the vibrational states 0, cr of 89
Volume
135, number
CHEMICAL
1,2
PHYSICS
27 March
LETTERS
XY and XY+ in the wavefunctions I q) and I/3 ) for the open channels q and j?. These functions are normalized with respect to energy as follows:
Vqp=(4lW)
9 Vs,=(PIVlP)
V,=(4VIPL)
3
(qE lqr > =x&E-E’)
bv=(41t01Q,
, (BE WE. > =n@E-W
.
To find the wavefunction Yq of the system e+ + XY consider the e+-XY scattering due to the operator potential V. For the scattering operator T we use the Lippman-Schwinger equations (e= fi = m, = 1) T=t+t(G-G,)T,
t= V+ VG,t,
(3)
where G is the Green’s function of the Hamiltonian excluding V. In the function GO it is convenient to take into account contributions from the configurations e+ +XY, Ps+XY+ and e+XY calculating the integrals over energy in the sense of the principal value. Using transformations which are analogous to those used in refs. [ 6,7] we get T=t-it
C ]4)(4]T-itF 4
(j)(flIT
(4)
1987
3
v~v=<~lt”l~>,
V4P=(41t01B> 9 &,=(41f014’)
9 t&3~=(BIt”lP’>
& = ( Pl tol V’) .
(6)
Due to the fact that the amplitude of the zero vibration is much smaller than the interatomic distances in XY in the case of weak CI ( I VI * < 1) the diagonal elements toqq, t& are negligibly small. Nondiagonal elements tiqs, tjp, describing direct “intraconfiguration” transitions are more important and should normally be taken into account. With the help of eq. (4) the wavefunctions of the system may be obtained if a functional separation of the contributions from the resonance and the direct mechanism is made. We first define the operators Q and Q. according to the relation t=vi2,
sz=Q)+
pgv P
(5)
and substitute (7) into eq. (4). Considering the action of the various operators on the unperturbed wavefunction 14) and taking into account the delinition TI 4) = WY, we have Yq=Oq-i
1 OqpTq.q-i 4’
1 QsT,, B
(8) value, E is the total energy, of the nuclear subsystem in the states q, /3, ,u, Y. The functions lq), ID), Ip) and the energies Es are detemined taking into account the interaction of the configurations represented in t as the expression for t has the form of the exact solution of the second equation in ( 3). The functions I q) , Ifi ) , I v ) , the values I?” and energies eq, ta of particles e+ and Ps are taken for V= 0. Eq. (4) provides an algebraic method of obtaining the unitary scattering matrix S= (1/2i)( 1 - T). The T-matrix is expressed through the matrix elements of Vand to Here PP is the principal
Eq, ED, E,, E, are energies
90
where
14)=Qol4) 7 lP>=52,lB)
7 lv>=fioI@
* (9)
Here the strong dependence
on the energy is con-
Volume 135, number 1,2
tained in the terms with the poles and in elements of the T-matrix. The weak dependence is represented by the functions Iq), 18), Ip), Iv) and matrix elements of V and to. If we exclude the configurations e+XY and Ps + XY + ( 8 ) can be simplied to yg=
C (l-iT,,,)@,, 4’
27 March 1987
CHEMICAL PHYSICS LETTERS
,
(10)
where the T-matrix is represented in this case by T= a( ^i+it^, (the ranks of the matrices i and ^i are equal to the number of open channels q). It can be shown that for I VI 24 1 eq. (10) coincides with eq. (27) inref. [3]. Eqs. (4), (8) and (9) are used below to study the resonance mechanism in annihilation processes and the effect on this mechanism of a strong electromagnetic field in resonance with the transition between the vibrational levels of the complex e+XY. It is known [B] that if the field depends periodically on time the quasi-energy representation can be used. Then the same conditions hold as for conservative systems taking into account the terms differing from the ground state by energy + mo, where CIIis the field frequency and m is an integer. The interaction between the terms e+XY and e+XY’ with m =O and m= 1 is proportional to the electric dipole moment D and the field strength f: Vvpa Df: If f,5 IO’ V cm-’ then the conditions I VvflI 2Q I V, 1, yr, yy (yp=.EqV~/:q+&V~, yy=I.4V~q+&V~p) are more typical. Here every quasi-energy state ( k m) of the complex e+XY has its own set of continua into which it decays. Besides if the parameters
tqq= vt9 E_E,
tqq,
-0 -
=tq*q,
tw+,
>
P
t4’“=v@“)
V2 2 E-E,
t,,=&+
and substituting the T-matrix obtained into (8) and (2), we get
(11) where c,=E-E,,, c,=E-E,=E-&--tzv, yr= V&, yv = VZq,; aq,/l are annihilation amplitudes for the states q and p ac =
s
!P$&+
exp(ik,*r+)
jr)’ dr,...dr,
dr+ ,
(12)
C=4,P
(~,,=a,, =O since they belong to states with m = 1, U,%U,). The amplitude (11) depends on the field characteristics as V,af; E,= E,_ 1+ w (it can be shown that (11) still holds if the Stark shift of the levels is taken into account). For A = )E, - E, I ti I V, I in the vicinity of E, there is a simple resonance ( ~Aq~2=y,~u,~2(~~+y~)-‘) with intensity Z,= A I a, (‘. At the same time for A= 0 the resonance part of the spectrum is described by a more complex curve
(E=Ey=$)
(p, A, are the velocity and wavelength of the positron, c, ;1,are the velocity of ligt and the wavelength of the radiation) the coupling between states q and B with different m is not significant [ 91. A special case arises when the resonance level E,, merging in the presence of the electromagnetic field with the level E,= Ep_ 1+ w , lies below the threshold for vibrational excitation of the molecule in the region where the channels for Ps formation are closed. Here we have two non-interacting continua: q (v=O, m=O) for the state p and q’ (v=O, m=l) for the state v. Solving eq. (4) with
2
(13)
)
with I,=
s
=xlu,12-
lAgI
de
YP
Y”+Ya
(
1+
2
yv Y”YP+I% > .
The dependence of Z, on f and w suggests that the radiation effects can be studied using existing experimental techniques with positron beams. Let F(E) be a normalized function of the positron distribution with characteristic width AE, such that w >AE% 91
Volume 135, number 1,2
CHEMICAL PHYSICS LETTERS
1I’,[, yp+ yV. Then the cross section integrated over the energy assumes the form Q,~~:IQ,~*+xF(E~)Z,IU,~*.
(14)
In the energy region where F(E,) $1 the second term in (14) is dominant. The gamma-quantum yield observed here must be strongly dependent on the field frequency and strength. Finally note that a similar dependence also holds for an arbitrary number of open channels q and /I. In this case the T matrix and the annihilation amplitude A, may be expressed analytically in the general form (neglecting the direct transitions; t$ = tip = t;p = 0) 7-c VIp)
y
A,-_a,+
v,
x
a,-i
(
92
(PI
VP
9 1
4’
Vw,a,, -i C V,ufl , > B
(15)
27 March 1987
where the indices q, q’, j3 and ,u belong to the states with m=O, a, is the amplitude (12) for [=/I and d is the denominator of eq. (11) in which 2y, and 2y, are the complete level widths.
References [ 1 ] V.I. Goldanskii and Yu.S. Sayasov, Phys. Letters 13 (1964) 300. [2] U. Fano, Phys. Rev. 124 (1961) 1866. [3] F. Mies, Phys. Rev. 175 (1968) 164. [4] A.F. Starace, Phys. Rev. B5 (1972) 1733. [ 51 F. Combet-Famoux, Phys. Rev. A25 (1982) 287. [6] G.V. Golubkov and G.K. Ivanov, Chem. Phys. Letters 81 (1981) 110. [ 71 G.K. Ivanov and G.V. Golubkov, Z. Physik Dl (1986) 199. [8] Ja.B. Zeldovich, Usp. Fiz. Nauk 110 (1973) 139. [ 91 L. Rosenberg, Advan. At. Mol. Phys. 18 (1982) 48.
CHEMICAL PHYSICS LETTERS
THE RESONANCE ANNIHILATION
27 March 1987
OF SLOW POSITRONS ON MOLECULES
G.K. IVANOV Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow V-334, USSR Received 16 December 1986
The resonance mechanism for the interaction of slow positrons with molecules and the effect of a strong electromagnetic field on this mechanism are studied. It is shown that when the field is in resonance with the transition between the vibrational levels of the intermediate complex e+XY, the annihilation characteristics of the system e+ +XY are changed substantially.
There are two fundamental mechanisms for the interaction of slow positrons with molecules: resonance and direct. The resonance mechanism in the theory of positron annihilation was first discussed in ref. [ 11. For a molecule XY this mechanism assumes the formation of an intermediate complex e+XY with a lifetime of the order of the molecular vibrational period r or greater. The characteristic time of the direct mechanism is much less than r. A description of the latter mechanism involves detailed interaction potentials and is a complicated quantum-mechanical problem. All characteristics of a direct process are weak functions of the energy. A combined description of these two mechanisms is possible if one can extract the strong (resonance) dependence from the background of the weak dependence created by the direct processes. In atom autoionization theory this problem was solved by Fano [ 21 (see also refs. [ 3-51) by utilizing a “prediagonalization” of the Hamiltonian with the subspace of closed channels. The method proposed in the present work is a generalization of the Fano method and is the first time such a method has been applied to the problem of positron annihilation. We assume that in the annihilation process
e++XY--+
+2y+xy+
(1)
a population of different interacting quasistationary states of the complex (e+XY and e+XY’) is possible
with their decay occurring in the continua corresponding to the different particles: e+ +XY and Ps + XY+. Note that under normal conditions overlap of resonance configurations is unusual; however, such an overlap is common when the system interacts with a strong electromagnetic field. The calculation of the annihilation amplitude A, reduces to the determination of the wavefunction Yy, of the system e+ + XY in the main transition region: A,=
YU;;v+exp(ik,-r+)
YY:,dr,...dr, dr+ .
(2)
Here Yxy + is the wavefunction of the ion XY +, cy is its vibrational quantum number (the molecular orientation is assumed to be fixed), k7 is the total momentum of the two photons, Y ;= YY,(r,, . ... r,, rN+,, r + = rN+, ) , with r + the positron coordinate and r,, .... r,, rN+, the coordinates of the electrons. Four main groups of states may be introduced for the process under consideration: e+ + XY, Ps + XY+, e+XY and e+XY’. We call these configurations and denote the corresponding states by q, 8, p and v. Let the operator V include the configuration interaction (CI) and that part of the interaction in the contigurations e+ +XY and Ps+XY+ which remains after displacing the potentials k$‘+,, and p&+xu away from the equilibrium interatomic distances in XY and XY+. Thus for the system without CI we propose to fulfil the “prediagonalisation” condition for the isolated configurations e+XY and e+XY’ only. This enables us to fix the vibrational states 0, cr of 89
Volume
135, number
CHEMICAL
1,2
PHYSICS
27 March
LETTERS
XY and XY+ in the wavefunctions I q) and I/3 ) for the open channels q and j?. These functions are normalized with respect to energy as follows:
Vqp=(4lW)
9 Vs,=(PIVlP)
V,=(4VIPL)
3
(qE lqr > =x&E-E’)
bv=(41t01Q,
, (BE WE. > =n@E-W
.
To find the wavefunction Yq of the system e+ + XY consider the e+-XY scattering due to the operator potential V. For the scattering operator T we use the Lippman-Schwinger equations (e= fi = m, = 1) T=t+t(G-G,)T,
t= V+ VG,t,
(3)
where G is the Green’s function of the Hamiltonian excluding V. In the function GO it is convenient to take into account contributions from the configurations e+ +XY, Ps+XY+ and e+XY calculating the integrals over energy in the sense of the principal value. Using transformations which are analogous to those used in refs. [ 6,7] we get T=t-it
C ]4)(4]T-itF 4
(j)(flIT
(4)
1987
3
v~v=<~lt”l~>,
V4P=(41t01B> 9 &,=(41f014’)
9 t&3~=(BIt”lP’>
& = ( Pl tol V’) .
(6)
Due to the fact that the amplitude of the zero vibration is much smaller than the interatomic distances in XY in the case of weak CI ( I VI * < 1) the diagonal elements toqq, t& are negligibly small. Nondiagonal elements tiqs, tjp, describing direct “intraconfiguration” transitions are more important and should normally be taken into account. With the help of eq. (4) the wavefunctions of the system may be obtained if a functional separation of the contributions from the resonance and the direct mechanism is made. We first define the operators Q and Q. according to the relation t=vi2,
sz=Q)+
pgv P
(5)
and substitute (7) into eq. (4). Considering the action of the various operators on the unperturbed wavefunction 14) and taking into account the delinition TI 4) = WY, we have Yq=Oq-i
1 OqpTq.q-i 4’
1 QsT,, B
(8) value, E is the total energy, of the nuclear subsystem in the states q, /3, ,u, Y. The functions lq), ID), Ip) and the energies Es are detemined taking into account the interaction of the configurations represented in t as the expression for t has the form of the exact solution of the second equation in ( 3). The functions I q) , Ifi ) , I v ) , the values I?” and energies eq, ta of particles e+ and Ps are taken for V= 0. Eq. (4) provides an algebraic method of obtaining the unitary scattering matrix S= (1/2i)( 1 - T). The T-matrix is expressed through the matrix elements of Vand to Here PP is the principal
Eq, ED, E,, E, are energies
90
where
14)=Qol4) 7 lP>=52,lB)
7 lv>=fioI@
* (9)
Here the strong dependence
on the energy is con-
Volume 135, number 1,2
tained in the terms with the poles and in elements of the T-matrix. The weak dependence is represented by the functions Iq), 18), Ip), Iv) and matrix elements of V and to. If we exclude the configurations e+XY and Ps + XY + ( 8 ) can be simplied to yg=
C (l-iT,,,)@,, 4’
27 March 1987
CHEMICAL PHYSICS LETTERS
,
(10)
where the T-matrix is represented in this case by T= a( ^i+it^, (the ranks of the matrices i and ^i are equal to the number of open channels q). It can be shown that for I VI 24 1 eq. (10) coincides with eq. (27) inref. [3]. Eqs. (4), (8) and (9) are used below to study the resonance mechanism in annihilation processes and the effect on this mechanism of a strong electromagnetic field in resonance with the transition between the vibrational levels of the complex e+XY. It is known [B] that if the field depends periodically on time the quasi-energy representation can be used. Then the same conditions hold as for conservative systems taking into account the terms differing from the ground state by energy + mo, where CIIis the field frequency and m is an integer. The interaction between the terms e+XY and e+XY’ with m =O and m= 1 is proportional to the electric dipole moment D and the field strength f: Vvpa Df: If f,5 IO’ V cm-’ then the conditions I VvflI 2Q I V, 1, yr, yy (yp=.EqV~/:q+&V~, yy=I.4V~q+&V~p) are more typical. Here every quasi-energy state ( k m) of the complex e+XY has its own set of continua into which it decays. Besides if the parameters
tqq= vt9 E_E,
tqq,
-0 -
=tq*q,
tw+,
>
P
t4’“=v@“)
V2 2 E-E,
t,,=&+
and substituting the T-matrix obtained into (8) and (2), we get
(11) where c,=E-E,,, c,=E-E,=E-&--tzv, yr= V&, yv = VZq,; aq,/l are annihilation amplitudes for the states q and p ac =
s
!P$&+
exp(ik,*r+)
jr)’ dr,...dr,
dr+ ,
(12)
C=4,P
(~,,=a,, =O since they belong to states with m = 1, U,%U,). The amplitude (11) depends on the field characteristics as V,af; E,= E,_ 1+ w (it can be shown that (11) still holds if the Stark shift of the levels is taken into account). For A = )E, - E, I ti I V, I in the vicinity of E, there is a simple resonance ( ~Aq~2=y,~u,~2(~~+y~)-‘) with intensity Z,= A I a, (‘. At the same time for A= 0 the resonance part of the spectrum is described by a more complex curve
(E=Ey=$)
(p, A, are the velocity and wavelength of the positron, c, ;1,are the velocity of ligt and the wavelength of the radiation) the coupling between states q and B with different m is not significant [ 91. A special case arises when the resonance level E,, merging in the presence of the electromagnetic field with the level E,= Ep_ 1+ w , lies below the threshold for vibrational excitation of the molecule in the region where the channels for Ps formation are closed. Here we have two non-interacting continua: q (v=O, m=O) for the state p and q’ (v=O, m=l) for the state v. Solving eq. (4) with
2
(13)
)
with I,=
s
=xlu,12-
lAgI
de
YP
Y”+Ya
(
1+
2
yv Y”YP+I% > .
The dependence of Z, on f and w suggests that the radiation effects can be studied using existing experimental techniques with positron beams. Let F(E) be a normalized function of the positron distribution with characteristic width AE, such that w >AE% 91
Volume 135, number 1,2
CHEMICAL PHYSICS LETTERS
1I’,[, yp+ yV. Then the cross section integrated over the energy assumes the form Q,~~:IQ,~*+xF(E~)Z,IU,~*.
(14)
In the energy region where F(E,) $1 the second term in (14) is dominant. The gamma-quantum yield observed here must be strongly dependent on the field frequency and strength. Finally note that a similar dependence also holds for an arbitrary number of open channels q and /I. In this case the T matrix and the annihilation amplitude A, may be expressed analytically in the general form (neglecting the direct transitions; t$ = tip = t;p = 0) 7-c VIp)
y
A,-_a,+
v,
x
a,-i
(
92
(PI
VP
9 1
4’
Vw,a,, -i C V,ufl , > B
(15)
27 March 1987
where the indices q, q’, j3 and ,u belong to the states with m=O, a, is the amplitude (12) for [=/I and d is the denominator of eq. (11) in which 2y, and 2y, are the complete level widths.
References [ 1 ] V.I. Goldanskii and Yu.S. Sayasov, Phys. Letters 13 (1964) 300. [2] U. Fano, Phys. Rev. 124 (1961) 1866. [3] F. Mies, Phys. Rev. 175 (1968) 164. [4] A.F. Starace, Phys. Rev. B5 (1972) 1733. [ 51 F. Combet-Famoux, Phys. Rev. A25 (1982) 287. [6] G.V. Golubkov and G.K. Ivanov, Chem. Phys. Letters 81 (1981) 110. [ 71 G.K. Ivanov and G.V. Golubkov, Z. Physik Dl (1986) 199. [8] Ja.B. Zeldovich, Usp. Fiz. Nauk 110 (1973) 139. [ 91 L. Rosenberg, Advan. At. Mol. Phys. 18 (1982) 48.