Electron-hydrogen phase shift below inelastic threshol

Temkin, A.

English

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TECHNICAL NOTE
D-1720
ELECTRON-HYDROGENPHASE SHIFTS
JUST BELOWTHE INELASTIC THRESHOLD
Aaron Temkin
Goddard Space Flight Center
Greenbelt, Maryland
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON May 1963
https://ntrs.nasa.gov/search.jsp?R=19630006415 2020-03-24T05:55:31+00:00Z
ELECTRON-HYDROGENPHASE SHIFTS
JUST BELOW THE INELASTIC THRESHOLD
by
Aaron Temkin
Goddard Space Flight Center
SUMMARY
The singlet S-wave phase shifts for the scattering of electrons
by atomic hydrogen just below the inelastic threshold (10.2 ev)
are calculated by using a recently developed nonadiabatic theory.
Previous calculations of Burke and Schey based on the nmch more
restricted close-coupling approximation yielded a resonance
characterized by the phase shift going through _, radians. In view
of a theorem of Wigner and a calculation of Hol¢ien this result is
indicative of a bound state of the compound system near the energy
9.4 ev. It is argued on the basis of a previous calculation thatthis
resonance must be distinct from another resonance near 10.1 ev.
And on this basis it is concluded that this partial cross-section
must exhibit two resonances below threshold.
ELECTRON-HYDROGENPHASE SHIFTS
JUST BELOWTHE INELASTICTHRESHOLD
by
Aaron Temkin
Goddard Space Flight Ccuh'r
INTRODUCTION
Burke and Schey (Reference 1) have recently calculated a resonance in the elastic scattering of
electrons by atomic hydrogen just below the threshold for inelastic scattering (10.203 ev). The basis
of their calculation was the close-coupling approximation with ls-2s-2p states. The nomenclature
refers to hydrogenic states, each of which is multiplied by an initially undetermined function. For
S-wave _ scattering, their wave function can be written:
r 1 r 2 _[fcc -- [t,(rl) RI_ (r2] + v(rl] R2_ (r2) + (1=21] Po (cos _'J12)
(1)
where".°;(r! is r times the n l radial wave function of hydrogen. It is clear from Equation 1 that
this function is approximate in two ways. First, it contains only 2 of an infinite number of relali_e
angular momenta P: (cos _2)" Second, the "coefficients" of the included P_ have a comparatively re-
stricted form (which, however, is manifestly symmetric with respect to the interchange of 1 and2
corresponding to the necessary symmetry of singlet scattering, with which we shall here be concerned).
Clearly the most general function containing Po and P_ can be written:
rx r2,tJ : ,0 (r, r2)Po (cOSUl2) ++]_,bt (r, r2)P_, (cos !:,2) (2)
where the two-dimensional functions c_o and ,_ are required to have the correct symmetry. Substitu-
tion of Equation 2 into the variational principle, _ ; ,p*(tl -F,) ,I_,tT : 0, yields the following coupled set
tLetters referring to angular momentum states are capitalized when the total angular momentum state (parti,d wave) is being con_,idcred.
Lo_er case _ refers to the relative angular momentum within a given total angular momentum state.
of partial differential equations* in the region r 1 _> r_with appropriate boundary conditions (Refer-
ences 2 and 3):
A 2 ) 2r2+-- + E _o = _ c_l , (3a)
12 r 2 _ rl
-- - + E q_o
+ + r 2 5r12 (hi _ r12 •
(3b)
It is the object of the present nonadiabatic theory (References 2 and 3) to identify and directly
attack these partial differential equations, and thus avoid the second category of approximation im-
plicit in the close-coupling method. The purpose of this note is to reporton resultsof calculations on
the lowest order approximation of the equations and to give some speculations concerning the next
higher correction.
The lowest order approximation is the so-called zeroth order problem and is given by the equation:
( 2) (4)A12 + _-2 + E ePo(°) = 0 .
For future reference note that this basic relation can be derived:
fO _ _0 rl r2sin (b(l)-5o) k_3 drl dr2 qb°(°)-_12 r}l ' (5)
where 6 o is the phase shift of Equation 4, and _(1) is the phase shift of Equations 3.
In practice, Equation 4 has been solved by expanding in exact separable solutions (Reference 3),
the program for which has been extended to double precision arithmetic (16 significant figures). For this
discussion the resultant values for C}o_°) and _0 can be considered correct to the significance given in
Table 1. A close- coupling approximation of Equation 4 gives results in good accord with _ o (Reference 4).
CONJECTURESON THE SOLUTION OF EQUATIONS3*
An obvious first step in an iterative attempt to solve Equations 3 would be to use the ¢po(°> inthe
right-hand side of Equation 3b to get an approximate _1 which we can call rpL(0) •
E IAI2 + -- ÷ + _pl( o ) : CPo(O)12 r_ r_ 5r13 }/_ r12 (6)
*Energy is in rydbergs (1 ryd = 13.605 ev) and length is in Bohr radii.
tThe quantitative speculations of this section are quite different from what was calculated in Reference 5. The author is particularly
indebted to Dr. T. F. O'Malley for pointing out the usual increase of 7r of phase shifts at resonances and the analytical derivation of
this behavior for some simple nonloeal potentials. This discussion led, directly, to a re-examination of 60 (Reference 5) and the dis-
covery that it does indeed go through 7r. Dr. O'Malley also brought Reference 6 to the author's attention.
This neglects the coupling of _o to ,_, (but
not the reverse). Figure 1 gives the close-
coupling results, labeled _cc, as well as the
results for 00. A possible explanation for
the shift in energy at which the resonance
in Occ occurs as compared with _0 is that
it is due to the coupling of _o to '_i, which
is included in some approximate way in the
close-coupling approximation. However,
the previous calculation (Reference 5) was
based on the assumption that all resonances
arise directly from the resonance in _o(°).
The calculated results give absolutely no
indication of such a shift. The reason for
taking the close coupling seriously is ex-
plained below. If it is assumed that there
must be a resonance at or below the energy
corresponding to the close-coupling reso-
nance, a more plausible explanation of the
lower resonance would seem to be that it
is due to a resonance (change in sign) of '_1
in Equation 3b which is fairly independent
of the right-hand side.* If this is so then
a resonance very close to k=0.84 should
also come from Equation 6. Hopefully, so-
lutions of this equation will be obtained by
using a noniterative numerical technique
for solving partial differential equations
(Reference 7). As another test of this hy-
pothesis a close-coupling expansion with
only ls-2p states would also be expectedto
show a resonance near the lower energy, t
Mathematically a resonance in
manifests itself in (o)
I
01(o)
Table 1
Zeroth Order Phases Shifts Below Threshold.
k(ryd ''2)
O. 85
O.855
0.860
0.861
O.862
O.863
0.864
O.8645
0.865
O.8656
0.866
]<2
(impacting
energy in ev)
9.830
9.944
10.061
t O.085
10.108
10.132
10.155
10.169
10.180
10.194
10. 203
S 0 (radians)
0.721
0.729
0.7545
0.768
0.793
0.853
1.25
3.14
3.628
3.738
3.781
4.0
3.5
3.0
.9 2.5
?
2.o-
.-r
1.5-
<
I
1.0
0.5-
0.80
| ,I I I I I
0.81 0.82 0.83 0.84 0.85 0.86 0.866
k (ryd yz)
Figure 1--1-he ls-2s-2p close-coupling phase shift (6c¢) and
S0 as a function of k.
, the first approximation of _ (,), whereby in the expression
-2fo_ _o _' r2
*Mr..John Cooper pointed out the importance of the energy shift of the lower resonance.
*Schey anJ i_urke are now examining this point.
a change in sign of _o) would mean a change in sign of the integral, since ¢o_o) is varying very
slowly at the lower energy. This could (see below) mean an increase of 0_o) by _ which wouldbe the
mathematical statement of the resonance.
Along this mathematical line, it might be expected that as the energy increases toward k = 0.8645,
%(o) would undergo a change of sign and that there the integral would again change sign, corresponding
to an increase of 50 into the same quadrant as 5(0) . This would be in accord with olc0> exhibiting
only one increase of _ corresponding precisely to what Burke and Schey calculated. Howe_,er, here
Equation 6 must be considered more carefully; it has been presumed to have a resonance due to the
structure of the left-hand side at the lower energy. At the higher energy, however, the right-hand
side has been shown here to exhibit a resonance and this could cause a second resonance in _p(o).
Mathematically, this would mean that at the same energy at which %(0) undergoes a change of sign
¢,co) undergoes a second change in sign. Thus, the integral on the right-hand side of Equation 7 under-
goes no change of sign because of the simultaneous changes in sign of ¢0(°) and _o) . This could
correspond to a second increase of _0) bye. It is problematical whether or notthis second phenom-
enon will correspond, in fact, to a second increase of the phase shift by _ (see below). However, it is
quite likely that the phase shift will have a sufficiently uneven behavior to appear in the cross-section
5.0
4.5
4.0
"o
._2 :3.5
.Q
3.0
2.5
2.0
1.5
1.0
.5
0
7.0
6.5
6.0
5.5-
I I
.81 .82
I I I l
.83 .84 .85 .86 .866
k (ryd ½ )
Figure 2--Conjecture of the first order (Equat;ons3) phase shifts
according to the arguments in the text.
as a resonance. (It is only a matter of
definition whether a resonance is called
a sudden increase of the phase shift by
or a sudden up and down behavior in the
cross- section. )
In Figure 2 the conjectured behavior
of _(0) showingtwo increases of 77 is given.
The actual values were gotten by super-
posing the _0 resonance (modulo 77)on _
at the second resonant energy. In Figure 2
it has been assumed that resonances cor-
respond to increases of the phase shifts
by 77 (in the case of _0 this has been cal-
culated to be the case). This is based on
a theorem of Wigner and Eisenbud (Ref-
erence 6) which says
d8
d-k _ - R .
This statement is derived from the fact
that 2(d:_/'dk) is the delay time of a particle
striking a target. Although there is no upper
limit on this quantity, its negative value,
which is the time advancement of the
scattered wave, is optimal if there is a
4
hard core in the region of interaction. Thus, under any reasonable process of the formation of the
compound state corresponding to a sticking of the incoming particle to the atom, the phase shift must
exhibit a sharp rise. A calculation by Hol6ien gives an upper limit of E = 9.4 ev on such a state of
the compound H- ion (Reference 8). A more direct argument that there must be a resonance at or
below the close-coupling resonance is afforded by theorems on bounds for phase shifts which have
recently been proved (Reference 9).* As it applies in the present case it says that ?, > '_0uP to the
inelastic threshold. +
Figure 3 shows the plot of the cross-section corresponding to the phase shift curve in Figure 2.
This is meant to apply only to the S-wave part of the singlet scattering. In view of the fact that the
cross-section contains two resonances, it is felt that the physical reasons for such a behavior which
were included in Reference 5 still apply. That argument assumes a correspondence of each state of
the target with a state of the compound system (with possible shifts in energy). The main reason for
thinking that this correspondence does hold in this case is the large energy difference of the lower
resonance from the upper one. The previous calculation (Reference 5) indicates that otherwise this
energy shift would be much smaller.
¢,1oo
_.G.s
I I I I I I I
8.5 8.6 8.7 8.8 8.9 9 9.1
. I I I I
9.2 9.3 9.4 9.5
ENERGY (ev)
I
9.6 9.7 9.8 9.9 10 10.1 10.2
Figure 3--STnglet S-wave cross-section corresponding to the phase shifts Tn Figure 2. The
value a c is the Bohr radius.
*Burke and Schey have obtaLned very narrow resonances (private communication) at low energies (:_ 1 ev), whereas careful calculations by
Schwartz (private communication)show no resonances. This is not necessarily a contradiction to the theorem of Hahn, O'Malley, and
Spruch (Reference 9), since the numerical method of l]urke and .q,chey may occasionally show some aberrations, llov, ever, other logical
possibilities, including a contradiction to the theorem in Reference 3, cannot be dismissed at this point.
tl'rivate communication from "1". F. O'Malley.
However, the present heuristic considerations give a more explicit indication of what the physical
argument means in terms of the nonadiabatic formalism. The ¢0<°_ equation (4) is the two-dimensional
analogue of the one-particle 2s equation. To say that the one-particle 2s equation corresponds to a
resonance in the compound system means that the ¢o (°) inherently contains zero resonance near the
2s energy. The present calculation for 30 explicitly yields this resonance. By the same token the
left-hand side of Equation 3b or Equation 6 is a two-particle analogue of the single particle 2p equa-
tion; therefore, by the same kind of correspondence, this left-hand side is expected also to contain a
spontaneous resonance. It should be emphasized that this latter resonance has not as yet been found
in the author's calculations. However, the author believes that Burke and Schey have found it (Refer-
ence 1), but that they misinterpreted it to be the shifted _0_°) resonance. The author further believes
that if the ls-2s-2p close-coupling calculation is extended even closer to the 2s threshold, it will re-
veal the second resonance. In this context it should be noted that Hol_6ien's calculation gives evidence
of two compound states. (Reference 8).
It may be inquired whether or not the higher L equations, of which Equations 3 are the first two,
might also reveal spontaneous resonances. They probably do; however, the one-particle target state
energies to which they correspond are higher than o states. Therefore, their principal quantum num-
ber is n? 3 and, thus, their energies should be significantly above the n: 2 threshold.
,
REFERENCES
Burke, P. G., and Schey, H. M., "Elastic Scattering of Low-Energy Electrons by Atomic Hydrogen,"
Phys. Rev. 126(1):147-162, April 1, 1962.
2. Temkin, A., "Nonadiabatic Theory of the Scattering of Electrons from Hydrogen," Phys. Rev.
Lelters 4(11):566-568, June 1, 1960.
3. Temkin, A., "Nonadiabatic Theory of Electron-Hydrogen Scattering," Phys. Rev. 126(1):130-142,
April 1, 1962; also NASA Technical Note D-1343.
4, Smith, K., McEachran, R. P., and Fraser, P. A., "Effect of Virtual Excitation of the 2s State on
the Elastic Scattering of Electrons by Atomic Hydrogen," Fhys. Rev. 125(2):553-558, January 15,
1962.
.
6,
Temkin, A., and Pohle, R., "Electron-Hydrogen Phase Shifts Just Below the Inelastic Threshold,"
Phys. Re_l. Letlers 10(1):22-25, January 1, 1963.
Wigner, E. P., "Lower Limit for the Energy Derivative of the Scattering Phase Shift," F'hys. Rev.
98(1):145-147, April 1, 1955.
7. Temkin, A., and Sullivan, E., "Nonadiabatic Theory of Electron-Hydrogen Scattering. II," Phys.
Rev. (to be published); also NASA Technical Note D-1702 (to be published).
8. Hol¢ien, E., "The (2s) 2 i S State Solution of the Non-relativistic Schro'dinger Equation for Helium
and the Negative Hydrogen Ion," Proc. Phys. Soc. (London) 71(3):357-368, March 1958.
9. Hahn, Y., O'Malley, T. F., and Spruch, L., "Static Approximations and Bounds on Single Channel
Phase Shifts," Phys. Rev. 128(2):932-943, October 15, 1962.
6 NASA-Langley, 1963


Electron-Hydrogen Phase Shifts Just Below the Inelastic Threshold

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