- Email: [email protected]
Search for inelastic electrons scattered off ions in energetic ion-atom collisions
.-_ __ i!B
&3*H
Nuclear Instruments and Methods in Physics Research B 98 (19951371-374
NONil B
Beam Interactions with Materials 8 Atoms
ELSEVIER
Search for inelastic electrons scattered off ions in energetic ion-atom collisions T.J.M. Zouros a,*, C. Liao b, S. Hagmann b, G. Toth b, E.C. Montenegro E.P. Benis a
‘, P. Richard b,
A Physics Dept., University of Crete and Institute of Electronic Structure and Laser, P.O. Box 1527, Heraklion, 71110 Crete, Greece h J.R. Macdonald Laboratory, Kansas State Unicersity, Manhattan, KS 66506, USA ’ Dept. de Fisica, Pontificia Unirersidade Cat6lica do Rio de Janeiro, Caixa Postal 38071, 22453 Rio de Janeiro, Rio de Janeiro, Brazil Abstract
The collision of target electrons with ionic projectiles, in highly asymmetric ion-atom collisions, can be simulated within the impulse approximation (IA) by a beam of electrons scattering off the projectile ion, with a collision energy broadened by the momentum distribution (Compton profile) of the target. This description works particularly well for 180” elastic scattering of target electrons off the ion (Phys. Rev. A 41 (1990) 4816 [l]) appearing in the laboratory as the well known binary encounter electron peak. Inelastic processes due to target-electron-projectile-electron interactions (e-e) such as e-e excitation (Phys. Rev. Lett. 62 (1989) 2261 [2]) and e-e ionization (Phys. Rev. Lett. 63 (1990) 1938; J. Phys. B 24 (1991) 977; Phys. Rev. A 46 (1992) 1374; Phys. Rev. Len. 69 (1992) 3033, 72 (19941 3170, 72 (1994) 3166 [3-811 are also well documented. However, for these processes, the target electron undergoing inelastic scattering has never been explicitly identified and measured in ion-atom collisions. A simple theoretical Born-IA treatment gives the expected double differential cross sections and their angular distributions for H-like ions in collision with H, targets. The viability of observing these effects in ion-atom collisions by electron spectroscopy is investigated.
1. Introduction
Recent investigations of excitation and ionization in collisions of fast highly-charged projectiles with Ha and He targets have shown that effects due to projectile-electron-target-electron interactions can be clearly identified [2-lo]. The interacting electrons, traditionally assigned only a passive screening role, are now seen to acquire a new dynamic role, contributing a significant part of the excitation cross section, measuring up to 60%, as in the case of K-shell projectile ionization at collision energies of a few MeV/u [3], or up to almost 100% for particular states, as in the case of e-e excitation with spin-exchange [2] in collisions of Li-like ions. Theoretically, both the Born approximation and the impulse approximation (IA) have provided significant new insight into the collision mechanisms involved. Within the IA framework, applicable for fast collisions, as seen from the projectile frame, the perturbing target electrons are found to act as an incident beam of free electrons with an energy broadened by the momentum distribution due to their orbital motion around the target.
* Corresponding
author. E-mail: [email protected].
This simplifying realization has allowed for significant progress to be made in separating the e-e interaction from the normally competing e-n interaction. Thus, experimental signatures for e-e excitation have been found in the observation of excitation thresholds and electron exchange [2] processes traditionally associated with excitation of ions by free electrons. Within the Born approximation [ll] it can be shown that e-e excitation (as well as ionization considered as a special case of excitation to the continuum) must in fact involve a single correlated double-excitation event. Thus, as a consequence of energy and momentum conservation, the excitation of the projectile ion by the target electron necessitates the simultaneous excitation of that target electron. In the case of a heavy ion colliding with H, or He target, di-electronic excitation of the projectile predominatly leads to the ionization of the interacting target electron [12,13,6]. Using this feature, projectile e-e ionization has been successfully isolated from competing projectile e-n ionization (ionization due to interaction with the target nucleus) by detecting the ionized ion in coincidence with the ionized recoiling target [6-81. In this paper, we use both features, i.e. the quasi-free character of the target electrons within the IA framework and the correlated-double excitation process within the
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00149-2
3.2. COLLISIONS
WITH HEAVY PARTICLES
T.J.M. Zouros et al. / Nucl. lnstr. and M&h. in Phys. Rex 6 98 (1995) 371-374
372
Born approximation to explore the fate of the target electron following e-e excitation. Our goal is to find new signatures for identifying projectile e-e excitation in the double-differential electron spectrum of ion-atom collisions. Similar signatures have been sought for e-e ionization in collisions of neutral H + He [14].
2. Inelastic electron scattering and e-e excitation Within the IA, the double differential cross section for inelastic electron scattering (target ionization) due to the e-e excitation of the projectile by an energy A Ei, is given in the center-of-mass (CM) frame by [I]:
(1) where the CM energy of the impinging electron is [l]: Ei(P)=;pV*+VI) Z 7p
P
2
t&-z
quasi-free
target
._
,
,
T
(
,
where q is the momentum transfered in the collision, with q = ki - k, and kF/2~ = ljr ki, k, being the initial and final electron momenta in the CM, for which [16]: q2 = k” + k; - 2kik, cos 0,.
(4)
For a hydrogenic ion, expressions of 1ht;,(q) 1 for Is + 2s, 2p, 3s, 3p and 3d excitation are given analytically in Bates and Griffing [17]. In Fig. 1 the single differential cross sections du/d0 are plotted vs er for the case of
2E1
14
Here VP is the velocity of the ion in the laboratory frame, p = mM&m + M,) is the reduced electron-projectile ion mass, pZ is the projection of the target electron momentum along the collision axis taken to lie in the z-direction and I, is the target ionization potential. The final energy of the electrons following excitation is given by energy conservation, i.e. lf = E, - A&,. The single differential cross section for inelastic scattering of a beam of monoenergetic electrons of energy ei through a scattering angle 0, following excitation of an ion by AE, appearing in Eq. (1) can be expressed within the first Born approximation in terms of the inelastic form factor _Yfi(q) [1.5]:
r
9,=180°
AE(l+z)=367.35eV AE(r+3)=435.38eV
12
4
10
0
100
200
em+ C5+(nl)
300
Energy of scattered electron (eV)
e- +
?+(ls)-
em+ C5+(nl)
400
Energy of scattered electron (eV)
Fig. 1. Calculated single differential cross sections for inelastic electron scattering through 0, = 180” and 6’:= 0” for electron impact excitation of e-+ C’+(ls) -+ em+ C5+(nl), for principal quantum number n = 2,3 and orbital angular momentum I = 0, 1.. , n ~ 1. The cross section is plotted as a function of the electron energy following the collision, E,, which is just equal to the impinging energy t, - AE,,.
T.J.M. Zouros et al. /Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) 371-374
BEe
IO'
1oo
1
LL
0
1000
2000
3000
4000
Electron energy (eV)
Fig. 2. 18 MeV C5’(1s)+Hz-)C5+(nl)+e-. Lines: inelastic double differential cross sections for scattering into zero degrees in the laboratory frame. The binary encounter electron peak corresponding to elastic electron scattering through 0, = 180” is also shown for comparison. Dark circles: experimental data normalized to theory at the maximum of the BEe peak.
C5+ ions. In this case, AE,,, 2s,2P= 367.35 eV and AE 1s + 3s,3p,3d = 435.38 eV. It is interesting to note that the single differential cross sections for inelastic scattering through 6, = 0” are seen to increase with collision energy, within the Born approximation, clearly an unphysical result. However, the total excitation cross section does remain finite. This behavior needs to be investigated more thoroughly to check whether single differential cross sections at 0, = 0” given by first-order theories are physically meaningful. The double differential cross sections (DDCS) computed in Eq. (1) can be readily transformed to the laboratory frame using the relevant kinematic transformations [l]. These DDCS are shown in Fig. 2 for the case of 18 MeV Csc+ H, collisions for a laboratory observation angle, 8, = 0”. Note that for 0” laboratory observation angle, both 0, = 0” and 0, = 180” contribute to different parts of the DDCS electron spectrum on either side of the cusp position at 1/2mVpz, the more intense inelastic peak appearing at the smaller laboratory energies corresponding to 0, = 0” and small values of momentum transfer 4.
373
Singles experimental results are also shown for comparison. It is clear that the predicted features are much smaller than the observed electron spectrum, the wings of the cusp and binary encounter peaks contributing substantially to the spectrum. Thus, it seems that these inelastic structures will probably not be readily observable in the measured singles double differential electron spectrum. However, a more judicious choice of collision system and observation angle, or the use of coincidence measurements, could possibly reduce other contributions enough to allow for these structures to be observed. Also, going to lower 2 projectiles, for which 1s + 2p excitation becomes even stronger, while the BEe cross section becomes weaker, would probably be helpful. More theoretical analysis of the importance of the different collisions parameters is clearly needed. It bears emphasis that these structures should disappear at collision energies for which the electrons have kinetic energies below the excitation threshold. This should constitute a further test for the e-e excitation signature. Another possibility is to also look for the e-e ionization channel alone which is expected to be of the same magnitude as the e-e excitation channel. Finally, our theoretical approach is of questionable validity at energies close to the ion excitation threshold and thus improved predictions can only be obtained using more rigorous theoretical treatments.
3. Summary and conclusion We have discussed the possibility of observing projectile excitation due to two-center e-e interactions by searching for the inelastically scattered target electrons resulting from this d&electronic process. Using the impulse approximation, double differential electron emission cross sections based on single differential inelastic electron scattering cross sections, computed in the Born approximation, were obtained for collisions of hydrogenic ions with H, targets at an observation angle of zero degrees. Two inelastic peaks of different magnitudes are predicted to lie on either side of the cusp peak corresponding to scattering through 0” and 180” in the CM frame. These predictions could not be readily identified in singles electron spectra from 18 MeV C’++ H, collisions. We hope that this attempt will stimulate more sophisticated calculations and experimentation allowing for a more rigorous treatment of target ionization due to di-electronic processes, eventually providing for a new signature through which e-e excitation and ionization might be identified in the laboratory.
Acknowledgements We would like to acknowledge stimulating conversations with Dievad BelkiC, Bob Dubois, Horst SchmidtB&king and Nice Stolterfoht. One of us (T.J.M.Z.) would
3.2. COLLISIONS WITH HEAVY PARTICLES
374
T.J.M. Zouros et al. /Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) 371-374
like to acknowledge partial support from NATO grant CRG-910567. Supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy.
References
[ll D.H. Lee et al., Phys. Rev. A 41 (1990) 4816. RI T.J.M. Zouros, D.H. Lee and P. Richard, Phys. Rev. Lett. 62 (1989) 2261. [31 H.P. HiilskGtter, W.E. Meyerhof, E. Dillard and N. Guardala, Phys. Rev. Lett. 63 (1990) 1938. I41 M.B. Shah and H.B. Gilbody, J. Phys. B 24 (1991) 977. 151D.H. Lee et al.. Phys. Rev. A 46 (1992) 1374.
[61 E.C. Montenegro.
W.S. Melo, W.E. Meyerhof and A.G. de Pinho. Phys. Rev. Lett. 69 (1992) 3033. [71 W. Wu et al., Phys. Rev. Lett. 72 (1994) 3170. [81 R. Diirncr et al., Phys. Rev. Lett. 72 (1994) 3166. 191 W.E. Mcyerhof et al.. Phys. Rev. A 43 (1991) 5907. [lOI T.J.M. Zouros et al., Phys. Rev. A 49 (1994) 3155. [I 11 E.C. Montenegro. W.E. Meyerhof and J.H. McGuire, Adv. At. Mol. Opt. Phys. 34 (1994) 249. [I21 D. BelkiC and R. Gayet, J. Phys. B 9 (1976) Lll I. [131 H.M. Hartley and H.R. Walters, J. Phys. B 20 (1987) 1983. 1141 R.D. Dubois and S.T. Manson, Nucl. Instr. and Meth. B 79 (1993) 93. Ml B.C. Bransdcn and C.J. Joachain, Physics of Atoms and Molecules (Longman, England, 1983). M.R.C. McDowell and J.P. Coleman. Introduction to the [161 Theory of Ion-Atom Collisions (North-Holland, New York. 1970). [171 D.R. Bates and G. Griffing, Proc. Phys. Sot. London, A 66 (1953) 961.
&3*H
Nuclear Instruments and Methods in Physics Research B 98 (19951371-374
NONil B
Beam Interactions with Materials 8 Atoms
ELSEVIER
Search for inelastic electrons scattered off ions in energetic ion-atom collisions T.J.M. Zouros a,*, C. Liao b, S. Hagmann b, G. Toth b, E.C. Montenegro E.P. Benis a
‘, P. Richard b,
A Physics Dept., University of Crete and Institute of Electronic Structure and Laser, P.O. Box 1527, Heraklion, 71110 Crete, Greece h J.R. Macdonald Laboratory, Kansas State Unicersity, Manhattan, KS 66506, USA ’ Dept. de Fisica, Pontificia Unirersidade Cat6lica do Rio de Janeiro, Caixa Postal 38071, 22453 Rio de Janeiro, Rio de Janeiro, Brazil Abstract
The collision of target electrons with ionic projectiles, in highly asymmetric ion-atom collisions, can be simulated within the impulse approximation (IA) by a beam of electrons scattering off the projectile ion, with a collision energy broadened by the momentum distribution (Compton profile) of the target. This description works particularly well for 180” elastic scattering of target electrons off the ion (Phys. Rev. A 41 (1990) 4816 [l]) appearing in the laboratory as the well known binary encounter electron peak. Inelastic processes due to target-electron-projectile-electron interactions (e-e) such as e-e excitation (Phys. Rev. Lett. 62 (1989) 2261 [2]) and e-e ionization (Phys. Rev. Lett. 63 (1990) 1938; J. Phys. B 24 (1991) 977; Phys. Rev. A 46 (1992) 1374; Phys. Rev. Len. 69 (1992) 3033, 72 (19941 3170, 72 (1994) 3166 [3-811 are also well documented. However, for these processes, the target electron undergoing inelastic scattering has never been explicitly identified and measured in ion-atom collisions. A simple theoretical Born-IA treatment gives the expected double differential cross sections and their angular distributions for H-like ions in collision with H, targets. The viability of observing these effects in ion-atom collisions by electron spectroscopy is investigated.
1. Introduction
Recent investigations of excitation and ionization in collisions of fast highly-charged projectiles with Ha and He targets have shown that effects due to projectile-electron-target-electron interactions can be clearly identified [2-lo]. The interacting electrons, traditionally assigned only a passive screening role, are now seen to acquire a new dynamic role, contributing a significant part of the excitation cross section, measuring up to 60%, as in the case of K-shell projectile ionization at collision energies of a few MeV/u [3], or up to almost 100% for particular states, as in the case of e-e excitation with spin-exchange [2] in collisions of Li-like ions. Theoretically, both the Born approximation and the impulse approximation (IA) have provided significant new insight into the collision mechanisms involved. Within the IA framework, applicable for fast collisions, as seen from the projectile frame, the perturbing target electrons are found to act as an incident beam of free electrons with an energy broadened by the momentum distribution due to their orbital motion around the target.
* Corresponding
author. E-mail: [email protected].
This simplifying realization has allowed for significant progress to be made in separating the e-e interaction from the normally competing e-n interaction. Thus, experimental signatures for e-e excitation have been found in the observation of excitation thresholds and electron exchange [2] processes traditionally associated with excitation of ions by free electrons. Within the Born approximation [ll] it can be shown that e-e excitation (as well as ionization considered as a special case of excitation to the continuum) must in fact involve a single correlated double-excitation event. Thus, as a consequence of energy and momentum conservation, the excitation of the projectile ion by the target electron necessitates the simultaneous excitation of that target electron. In the case of a heavy ion colliding with H, or He target, di-electronic excitation of the projectile predominatly leads to the ionization of the interacting target electron [12,13,6]. Using this feature, projectile e-e ionization has been successfully isolated from competing projectile e-n ionization (ionization due to interaction with the target nucleus) by detecting the ionized ion in coincidence with the ionized recoiling target [6-81. In this paper, we use both features, i.e. the quasi-free character of the target electrons within the IA framework and the correlated-double excitation process within the
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00149-2
3.2. COLLISIONS
WITH HEAVY PARTICLES
T.J.M. Zouros et al. / Nucl. lnstr. and M&h. in Phys. Rex 6 98 (1995) 371-374
372
Born approximation to explore the fate of the target electron following e-e excitation. Our goal is to find new signatures for identifying projectile e-e excitation in the double-differential electron spectrum of ion-atom collisions. Similar signatures have been sought for e-e ionization in collisions of neutral H + He [14].
2. Inelastic electron scattering and e-e excitation Within the IA, the double differential cross section for inelastic electron scattering (target ionization) due to the e-e excitation of the projectile by an energy A Ei, is given in the center-of-mass (CM) frame by [I]:
(1) where the CM energy of the impinging electron is [l]: Ei(P)=;pV*+VI) Z 7p
P
2
t&-z
quasi-free
target
._
,
,
T
(
,
where q is the momentum transfered in the collision, with q = ki - k, and kF/2~ = ljr ki, k, being the initial and final electron momenta in the CM, for which [16]: q2 = k” + k; - 2kik, cos 0,.
(4)
For a hydrogenic ion, expressions of 1ht;,(q) 1 for Is + 2s, 2p, 3s, 3p and 3d excitation are given analytically in Bates and Griffing [17]. In Fig. 1 the single differential cross sections du/d0 are plotted vs er for the case of
2E1
14
Here VP is the velocity of the ion in the laboratory frame, p = mM&m + M,) is the reduced electron-projectile ion mass, pZ is the projection of the target electron momentum along the collision axis taken to lie in the z-direction and I, is the target ionization potential. The final energy of the electrons following excitation is given by energy conservation, i.e. lf = E, - A&,. The single differential cross section for inelastic scattering of a beam of monoenergetic electrons of energy ei through a scattering angle 0, following excitation of an ion by AE, appearing in Eq. (1) can be expressed within the first Born approximation in terms of the inelastic form factor _Yfi(q) [1.5]:
r
9,=180°
AE(l+z)=367.35eV AE(r+3)=435.38eV
12
4
10
0
100
200
em+ C5+(nl)
300
Energy of scattered electron (eV)
e- +
?+(ls)-
em+ C5+(nl)
400
Energy of scattered electron (eV)
Fig. 1. Calculated single differential cross sections for inelastic electron scattering through 0, = 180” and 6’:= 0” for electron impact excitation of e-+ C’+(ls) -+ em+ C5+(nl), for principal quantum number n = 2,3 and orbital angular momentum I = 0, 1.. , n ~ 1. The cross section is plotted as a function of the electron energy following the collision, E,, which is just equal to the impinging energy t, - AE,,.
T.J.M. Zouros et al. /Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) 371-374
BEe
IO'
1oo
1
LL
0
1000
2000
3000
4000
Electron energy (eV)
Fig. 2. 18 MeV C5’(1s)+Hz-)C5+(nl)+e-. Lines: inelastic double differential cross sections for scattering into zero degrees in the laboratory frame. The binary encounter electron peak corresponding to elastic electron scattering through 0, = 180” is also shown for comparison. Dark circles: experimental data normalized to theory at the maximum of the BEe peak.
C5+ ions. In this case, AE,,, 2s,2P= 367.35 eV and AE 1s + 3s,3p,3d = 435.38 eV. It is interesting to note that the single differential cross sections for inelastic scattering through 6, = 0” are seen to increase with collision energy, within the Born approximation, clearly an unphysical result. However, the total excitation cross section does remain finite. This behavior needs to be investigated more thoroughly to check whether single differential cross sections at 0, = 0” given by first-order theories are physically meaningful. The double differential cross sections (DDCS) computed in Eq. (1) can be readily transformed to the laboratory frame using the relevant kinematic transformations [l]. These DDCS are shown in Fig. 2 for the case of 18 MeV Csc+ H, collisions for a laboratory observation angle, 8, = 0”. Note that for 0” laboratory observation angle, both 0, = 0” and 0, = 180” contribute to different parts of the DDCS electron spectrum on either side of the cusp position at 1/2mVpz, the more intense inelastic peak appearing at the smaller laboratory energies corresponding to 0, = 0” and small values of momentum transfer 4.
373
Singles experimental results are also shown for comparison. It is clear that the predicted features are much smaller than the observed electron spectrum, the wings of the cusp and binary encounter peaks contributing substantially to the spectrum. Thus, it seems that these inelastic structures will probably not be readily observable in the measured singles double differential electron spectrum. However, a more judicious choice of collision system and observation angle, or the use of coincidence measurements, could possibly reduce other contributions enough to allow for these structures to be observed. Also, going to lower 2 projectiles, for which 1s + 2p excitation becomes even stronger, while the BEe cross section becomes weaker, would probably be helpful. More theoretical analysis of the importance of the different collisions parameters is clearly needed. It bears emphasis that these structures should disappear at collision energies for which the electrons have kinetic energies below the excitation threshold. This should constitute a further test for the e-e excitation signature. Another possibility is to also look for the e-e ionization channel alone which is expected to be of the same magnitude as the e-e excitation channel. Finally, our theoretical approach is of questionable validity at energies close to the ion excitation threshold and thus improved predictions can only be obtained using more rigorous theoretical treatments.
3. Summary and conclusion We have discussed the possibility of observing projectile excitation due to two-center e-e interactions by searching for the inelastically scattered target electrons resulting from this d&electronic process. Using the impulse approximation, double differential electron emission cross sections based on single differential inelastic electron scattering cross sections, computed in the Born approximation, were obtained for collisions of hydrogenic ions with H, targets at an observation angle of zero degrees. Two inelastic peaks of different magnitudes are predicted to lie on either side of the cusp peak corresponding to scattering through 0” and 180” in the CM frame. These predictions could not be readily identified in singles electron spectra from 18 MeV C’++ H, collisions. We hope that this attempt will stimulate more sophisticated calculations and experimentation allowing for a more rigorous treatment of target ionization due to di-electronic processes, eventually providing for a new signature through which e-e excitation and ionization might be identified in the laboratory.
Acknowledgements We would like to acknowledge stimulating conversations with Dievad BelkiC, Bob Dubois, Horst SchmidtB&king and Nice Stolterfoht. One of us (T.J.M.Z.) would
3.2. COLLISIONS WITH HEAVY PARTICLES
374
T.J.M. Zouros et al. /Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) 371-374
like to acknowledge partial support from NATO grant CRG-910567. Supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy.
References
[ll D.H. Lee et al., Phys. Rev. A 41 (1990) 4816. RI T.J.M. Zouros, D.H. Lee and P. Richard, Phys. Rev. Lett. 62 (1989) 2261. [31 H.P. HiilskGtter, W.E. Meyerhof, E. Dillard and N. Guardala, Phys. Rev. Lett. 63 (1990) 1938. I41 M.B. Shah and H.B. Gilbody, J. Phys. B 24 (1991) 977. 151D.H. Lee et al.. Phys. Rev. A 46 (1992) 1374.
[61 E.C. Montenegro.
W.S. Melo, W.E. Meyerhof and A.G. de Pinho. Phys. Rev. Lett. 69 (1992) 3033. [71 W. Wu et al., Phys. Rev. Lett. 72 (1994) 3170. [81 R. Diirncr et al., Phys. Rev. Lett. 72 (1994) 3166. 191 W.E. Mcyerhof et al.. Phys. Rev. A 43 (1991) 5907. [lOI T.J.M. Zouros et al., Phys. Rev. A 49 (1994) 3155. [I 11 E.C. Montenegro. W.E. Meyerhof and J.H. McGuire, Adv. At. Mol. Opt. Phys. 34 (1994) 249. [I21 D. BelkiC and R. Gayet, J. Phys. B 9 (1976) Lll I. [131 H.M. Hartley and H.R. Walters, J. Phys. B 20 (1987) 1983. 1141 R.D. Dubois and S.T. Manson, Nucl. Instr. and Meth. B 79 (1993) 93. Ml B.C. Bransdcn and C.J. Joachain, Physics of Atoms and Molecules (Longman, England, 1983). M.R.C. McDowell and J.P. Coleman. Introduction to the [161 Theory of Ion-Atom Collisions (North-Holland, New York. 1970). [171 D.R. Bates and G. Griffing, Proc. Phys. Sot. London, A 66 (1953) 961.