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Theory of ionization of atomic hydrogen by electron impact and threshold law
Volume
25A.
number
PHYSICS
1
THEORY OF IONIZATION ELECTRON IMPACT
Hasbrouck
LETTERS
17 July 1967
OF ATOMIC HYDROGEN AND THRESHOLD LAW *
IK-JU KANG and W. D. FOLAND Laboratoyr , University of Massachusetts, Amherst, Received
Massachusetts,
BY
USA
22 May 1967
Ionization of hydrogen atoms by electron impact is treated by employing the time-independent Green’s function method in such a wav that the interaction energy of the two electrons is the only perturbation. The nearly linear threshold l”aw is obtained.
The theory of ionization of hydrogen atoms by electron impact has been studied by many authors [1], employing a variety of approximations. There is, however, no convincing theory explaining the experimental threshold law [ 21 that the nearthreshold total ionization cross section is a nearly linear function of the total energy of the system. In this paper, we attempt to formulate a theory which gives the correct threshold behavior. Consider a system of an electron and a hydrogen atom. The total Hamiltonian, H, is given by H = Ka + K,, + va + vb + va,, = Ho + vab,
(1)
where K, denotes the kinetic energy of a-electron, vb the potential energy of b-electron in the field of the atomic nucleus, and vab the interaction of the two electrons. It can be shown [3] that the scattering matrix S of ionization processes (ko,no) - (kl , lq) can be written as sfi = bfi - 2xi (Ef - Ei) (MD f MR)fi
(2)
+ for spin singlet state - for spin triplet states with (3) and MR = ($)(k2)xb_)(kl)(
vabl It/k))*
(4)
xk)(fcl) and $@) in the above refer, respectively, to the “out”-state Coulomb wave function of aelectron with momentum kl and to the “in”-state * Supported in part by N.S.F. Faculty Research chusetts.
Grant GY-503, and by Grant of the University of Massa-
total wave function with a-electron in the continuum state. In this formalism, the Peterkop relation [4] MR(kl,k2)
(5)
= MD(k2,kI)
is evident without any phase requirement. Also, the question of “prior” and “post” interaction does not exist, Born-Oppenheimer and Born-exchange approximations become identical. Furthermore, vab is regarded as a perturbation, and the exchange effect is fully taken into account to the same accuracy as that of the direct scattering. To the lowest order of the perturbation interaction, one should approximate @b) by the first term of the Lippmann-Schwinger equation v ab q(+), qk’ =$)(kohb(no) +E_io+iE a
(6)
with nb(tio) denoting the b-electron’s initial hydrogenie wave function with a set of quantum numbers no. This choice gives consistency within the perturbation theory. It also gives orthogonality between the initial and the final states. Thus the boundary condition that any constant potential should lead to no transition is satisfied. In the limit of the total energy Etot(=fk,2++k!$ approaching zero, the lowest order matrix element for the direct scattering, iVZ&), becomes lim Etot-0
&&) = N(kl)N(k2)
X a certtin function of
k, only;
(7)
with N(k) denoting the normalization constant of an attractive Coulomb wave function normalized to a volume V; 2dk bV)i2 = V(l-exp[-2r/k])
’
(8)
Following the usual procedure [e.g.51 of getting the 11
Volume 25A.
number
1
PHYSICS
LETTERS
total cross sections from the scattering matrix, one finds in the limit kl, k2 - 0 that
perimental work of McGowan et al. [2]. Details will be reported later elsewhere.
Qtot&q (cer tain function of k, only) x Etot (9) which exhibits the (nearly) linear threshold law (the function of k, is sensitive to small variations of k,, so that the slope does not remain constant). It should be remarked that this threshold law is closely related to the long-range nature of the Coulomb potential and is consistent with Wigner’s [6] finding that the threshold power law depends on the asymptotic behavior of potentials involved. The iV&) in eq. (7) has been evaluated, and the numerical calculation has been carried out for the threshold region to give total cross sections which exhibit good agreement with the latest ex-
A REMARK
17 July 1967
References
1. see references
cited in K. Omidvar: Phys.Rev. Letters 18 (1967) 153. Proc.4th Int.Conf. 2’ J.W.McGowan and M.A.Fineman. on Atomic collisions. Quebec, Canada (1965): J. W. McGowan. M.A. Fineman. E. M. Clark and H. P. Hanson. to be published. and references therein. Phys. 3. See for an approach, I.J.Kang and J.Suchers. 4 Letters 20 (1966) 20. Proc. Phys.Soc. (London) 77 (1961) . R. K.Peterkop, 1220. quantum 5. J. D. Bjorken and S. D. Drell. Relativistic mechanics (McGraw-Hill Book Co. Inc.. New York, 1964) p. 101. Phys.Rev. 73 (1948) 1002. 6. E.P.Wigner.
ON FULIGSKI’S PAPER “ON OF GENERALIZED MASTER
THE “MEMORY” EQUATIONS”
PROPERTIES
T. GESZTI Institute foor Technical
Physics Received
of the Hungarian
d& dt
t
=
LdTktdP^l(t-d + A(t)&(O)
(1)
where k(T) and A(t) are expressed through the Liouville operator L . The kernel k(7) describes memory effects, while A(t)&(O) (the “destruction term”) describes the relaxation of initial “non-relevant” fluctuations. In a recent note Fuliliski [2] has shown that an 12
of Sciences
22 May 1967
Physical arguments are put forward in favour of master the single-time equation proposed by FUiliski.
On investigating the foundation of non-equilibrium statistical mechanics on the basis of the exact equations of motion of a many-particle system, several authors have arrived at master equations, containing an integral operator in time, interpreted as describing memory. In Zwanzig’s notations (11, see his references for relevant papers) if B is the density operator and D is a projector separating j? into a “relevant” part 61 = Dp^and an “irrelevant” part 62 = (l-D@, then fil(t ) obeys the equation
Academy
equations
containing
a memory
term,
and against
equation of the form dp/dt = K(t)p(t) + Q( t)p(O)
(2)
can be given that is mathematically equivalent to eq. (1) but does not contain the memory integral. On the basis of eq. (2) he argues that the memory effects appearing in eq. (1) are spurious and can be avoided by proper mathematics. The mathematical equivalence of eqs. (1) and (2) is fairly obvious, and mrtny other (differential or integral) equations can be constructed which are obeyed by some projection of a(t) = exp(-it L)$(O). Still we think that physical arguments decide rather uniquely in favour of eq. (1) with the memory and destruction terms, and against Fulitiski’s eq. (2). As a matter of fact, for the most typical systems dealt with in transport theory, the projection D may be chosen in such a way that fi,(Oj should include all macroscopic information about the initial state [ 31, while the extra microscopic information contained in 32(O) is smeared out by dissipative
25A.
number
PHYSICS
1
THEORY OF IONIZATION ELECTRON IMPACT
Hasbrouck
LETTERS
17 July 1967
OF ATOMIC HYDROGEN AND THRESHOLD LAW *
IK-JU KANG and W. D. FOLAND Laboratoyr , University of Massachusetts, Amherst, Received
Massachusetts,
BY
USA
22 May 1967
Ionization of hydrogen atoms by electron impact is treated by employing the time-independent Green’s function method in such a wav that the interaction energy of the two electrons is the only perturbation. The nearly linear threshold l”aw is obtained.
The theory of ionization of hydrogen atoms by electron impact has been studied by many authors [1], employing a variety of approximations. There is, however, no convincing theory explaining the experimental threshold law [ 21 that the nearthreshold total ionization cross section is a nearly linear function of the total energy of the system. In this paper, we attempt to formulate a theory which gives the correct threshold behavior. Consider a system of an electron and a hydrogen atom. The total Hamiltonian, H, is given by H = Ka + K,, + va + vb + va,, = Ho + vab,
(1)
where K, denotes the kinetic energy of a-electron, vb the potential energy of b-electron in the field of the atomic nucleus, and vab the interaction of the two electrons. It can be shown [3] that the scattering matrix S of ionization processes (ko,no) - (kl , lq) can be written as sfi = bfi - 2xi (Ef - Ei) (MD f MR)fi
(2)
+ for spin singlet state - for spin triplet states with (3) and MR = ($)(k2)xb_)(kl)(
vabl It/k))*
(4)
xk)(fcl) and $@) in the above refer, respectively, to the “out”-state Coulomb wave function of aelectron with momentum kl and to the “in”-state * Supported in part by N.S.F. Faculty Research chusetts.
Grant GY-503, and by Grant of the University of Massa-
total wave function with a-electron in the continuum state. In this formalism, the Peterkop relation [4] MR(kl,k2)
(5)
= MD(k2,kI)
is evident without any phase requirement. Also, the question of “prior” and “post” interaction does not exist, Born-Oppenheimer and Born-exchange approximations become identical. Furthermore, vab is regarded as a perturbation, and the exchange effect is fully taken into account to the same accuracy as that of the direct scattering. To the lowest order of the perturbation interaction, one should approximate @b) by the first term of the Lippmann-Schwinger equation v ab q(+), qk’ =$)(kohb(no) +E_io+iE a
(6)
with nb(tio) denoting the b-electron’s initial hydrogenie wave function with a set of quantum numbers no. This choice gives consistency within the perturbation theory. It also gives orthogonality between the initial and the final states. Thus the boundary condition that any constant potential should lead to no transition is satisfied. In the limit of the total energy Etot(=fk,2++k!$ approaching zero, the lowest order matrix element for the direct scattering, iVZ&), becomes lim Etot-0
&&) = N(kl)N(k2)
X a certtin function of
k, only;
(7)
with N(k) denoting the normalization constant of an attractive Coulomb wave function normalized to a volume V; 2dk bV)i2 = V(l-exp[-2r/k])
’
(8)
Following the usual procedure [e.g.51 of getting the 11
Volume 25A.
number
1
PHYSICS
LETTERS
total cross sections from the scattering matrix, one finds in the limit kl, k2 - 0 that
perimental work of McGowan et al. [2]. Details will be reported later elsewhere.
Qtot&q (cer tain function of k, only) x Etot (9) which exhibits the (nearly) linear threshold law (the function of k, is sensitive to small variations of k,, so that the slope does not remain constant). It should be remarked that this threshold law is closely related to the long-range nature of the Coulomb potential and is consistent with Wigner’s [6] finding that the threshold power law depends on the asymptotic behavior of potentials involved. The iV&) in eq. (7) has been evaluated, and the numerical calculation has been carried out for the threshold region to give total cross sections which exhibit good agreement with the latest ex-
A REMARK
17 July 1967
References
1. see references
cited in K. Omidvar: Phys.Rev. Letters 18 (1967) 153. Proc.4th Int.Conf. 2’ J.W.McGowan and M.A.Fineman. on Atomic collisions. Quebec, Canada (1965): J. W. McGowan. M.A. Fineman. E. M. Clark and H. P. Hanson. to be published. and references therein. Phys. 3. See for an approach, I.J.Kang and J.Suchers. 4 Letters 20 (1966) 20. Proc. Phys.Soc. (London) 77 (1961) . R. K.Peterkop, 1220. quantum 5. J. D. Bjorken and S. D. Drell. Relativistic mechanics (McGraw-Hill Book Co. Inc.. New York, 1964) p. 101. Phys.Rev. 73 (1948) 1002. 6. E.P.Wigner.
ON FULIGSKI’S PAPER “ON OF GENERALIZED MASTER
THE “MEMORY” EQUATIONS”
PROPERTIES
T. GESZTI Institute foor Technical
Physics Received
of the Hungarian
d& dt
t
=
LdTktdP^l(t-d + A(t)&(O)
(1)
where k(T) and A(t) are expressed through the Liouville operator L . The kernel k(7) describes memory effects, while A(t)&(O) (the “destruction term”) describes the relaxation of initial “non-relevant” fluctuations. In a recent note Fuliliski [2] has shown that an 12
of Sciences
22 May 1967
Physical arguments are put forward in favour of master the single-time equation proposed by FUiliski.
On investigating the foundation of non-equilibrium statistical mechanics on the basis of the exact equations of motion of a many-particle system, several authors have arrived at master equations, containing an integral operator in time, interpreted as describing memory. In Zwanzig’s notations (11, see his references for relevant papers) if B is the density operator and D is a projector separating j? into a “relevant” part 61 = Dp^and an “irrelevant” part 62 = (l-D@, then fil(t ) obeys the equation
Academy
equations
containing
a memory
term,
and against
equation of the form dp/dt = K(t)p(t) + Q( t)p(O)
(2)
can be given that is mathematically equivalent to eq. (1) but does not contain the memory integral. On the basis of eq. (2) he argues that the memory effects appearing in eq. (1) are spurious and can be avoided by proper mathematics. The mathematical equivalence of eqs. (1) and (2) is fairly obvious, and mrtny other (differential or integral) equations can be constructed which are obeyed by some projection of a(t) = exp(-it L)$(O). Still we think that physical arguments decide rather uniquely in favour of eq. (1) with the memory and destruction terms, and against Fulitiski’s eq. (2). As a matter of fact, for the most typical systems dealt with in transport theory, the projection D may be chosen in such a way that fi,(Oj should include all macroscopic information about the initial state [ 31, while the extra microscopic information contained in 32(O) is smeared out by dissipative