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On deviations from theory of electron–atom elastic scattering cross sections
Nuclear Instruments and Methods in Physics Research B 279 (2012) 49–52
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
On deviations from theory of electron–atom elastic scattering cross sections R. Moreh ⇑ Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84120, Israel
a r t i c l e
i n f o
Article history: Received 15 July 2011 Received in revised form 13 September 2011 Available online 20 November 2011 Keywords: Electron elastic scattering Rutherford scattering Quantum entanglement H2 HD D2 He Ar H2O D2O CH4
a b s t r a c t In some recent studies it is claimed that the electron elastic scattering intensities at keV energies from atoms and molecules at high momentum transfers do not conform to the Rutherford relation. Huge reductions in the ratios of the electron scattering intensities were reported in the following binary gas mixtures: H2/D2, H2/He, H2/Ar, D2/Ar and He/Ar where the scattering intensities from the light partners were found to be lower than the heavier ones by 30% and higher. Of particular interest is the case of the H2/D2 isotopic mixture found to deviate from the Rutherford relation by 30%. Similar intensity reductions were reported in samples of solid polymers where the scattering intensity from H was compared to that of C; the strong anomalous H-scattering intensity reduction was attributed to short lived quantum entanglement of a proton pair in the solid sample. It was stated that a quantum mechanical treatment of this scattering process in the framework of the Born approximation could not yield an explanation for the above observation. Here a critical examination of the above reports is given with the conclusion that the origin of all above deviations is very likely instrumental and not due to any real deviation from the Rutherford formula. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In the literature [1–4] on quasi-elastic scattering of electrons from bound systems, several cases were reported where very large deviations from the Rutherford scattering relation were observed. In those cases the ratios of the elastic e-scattering intensities were determined rather than the absolute cross sections because of the much higher accuracies which can be obtained in ratio measurements. The electron energies involved were between 2 and 40 keV. Note that at such energies the electrons scatter from each atom in a bound system as if the atom was a free particle. The energy of the elastically scattered electrons follows a billiard ball type kinematics which is justified using the impulse approximation [3]. The term ‘‘electron Compton scattering (ECS)’’ was used to describe such an electron scattering process. Normally, the intensities of the elastically scattered electrons are expected to follow the Rutherford scattering formula which scales as Z2 where Z is the nuclear charge of the scattering atom. In the above studies however, huge deviations (of around 30%) of the scattering intensity ratios from the Z2 relation of Rutherford were reported. In one case [3,4] solid thin films of polyethylene ⇑ Tel.: +972 8 6461569; fax: +972 8 6472903. E-mail address: [email protected]. 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.10.036
(CH2)n, and formvar C8H14O2 were used as scattering samples. The electron scattering intensity ratios IH/IC from H and C or from H and C + O were compared at incident energies between 15 and 30 keV and a scattering angle of 44.3°. The deviations from the Rutherford ratios at such high momentum transfers were 20% at 15 keV increasing to 50% at 30 keV. Similar large deviations of the scattering intensity ratios from the Z2 relation were also reported in binary gas mixtures at much lower incident electron beam energies. The binary gas mixtures used in those measurements were taken from any two [1,2] of the following gases: H2, D2, He and Ar. The incident electron energies were around 2 keV at a scattering angle of 100°. Note that a quantum mechanical treatment of the above scattering process within the Born approximation did not provide any clue as to the reason for the above deviations [5]. At this point it must be emphasized that the available evaluations of the elastic scattering cross section for electron–atom interaction [6] slightly deviate from the Z2 relation. Such deviations are of the order of 1% for the experimental conditions considered in the present work and were ignored. Table 1 gives a summary of the samples used in the above measurements together with the magnitude of the deviations of the electron scattering intensities from the Rutherford ratios. In the following we discuss those deviations in an attempt to find the reason for the anomaly.
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R. Moreh / Nuclear Instruments and Methods in Physics Research B 279 (2012) 49–52
Table 1 List of measured elastic scattering intensity ratios for a solid polyethylene (CH2)n sample and for mixed binary gaseous samples taken from Refs. [1–4]. IX denotes the scattering intensity from atom X. The incident electron energies and the scattering angles are indicated. The Rutherford predicted ratio is denoted R. Numbers in parentheses indicate the error in the last digit. The last column shows the calculated p ratio after accounting for the velocity difference (which vary as 1/ M) of the binary gas mixture components. Sample
Energy (keV)
Angle (°)
Ratio
Measured ratio
Calc. ratio
(CH2)n solid (CH2)n solid H2 + D2 H2/He H2/Ar D2/Ar He/Ar
15 30 2.25 2.25 2.25 2.25 2.25
44.3 44.3 100 100 100 100 100
IH/IC IH/IC IH/ID IH2 /IHe IH2 /IAr ID2 /IAr IHe/IAr
0.80(8)R 0.50(5)R 0.70(3)R 0.52(6)R 0.37(6)R 0.55(8)R 0.65(8)R
0.71R 0.71R 0.22R 0.32R 0.32R
2. Experimental details The results of the measurements being discussed here require electron spectrometers of very high resolution [1–4,7,8] which determine not only the small energy separations of the scattered electron lines from the different atoms of the bound system but also the widths of the those lines. The electron energies used in the above measurements were between 1 and 40 keV and the scattering angles were between 44.3° and 135°. The instrumental resolution depends on the incident electron energy; in Ref. [1] it was 0.8 at 2.25 keV while in Ref. [7] it was 0.25 eV at 2.5 keV and 0.4 eV at 6 keV. It should be remarked that the line widths are caused not only by the instrumental resolution of the spectrometer but also by the Doppler broadening caused by the momentum distribution of the scattering atoms. This in turn is contributed by the various modes of motion of the atoms which include translation, libration (hindered rotations) and vibrations. In some molecules such as CH4 the Doppler broadening of the scattered electron lines from the H-atom and C-atoms can be calculated quite precisely as was done in detail in Ref. [9]. 3. Discussion 3.1. Solid samples We first discuss the case of solid samples [3,4]. It was suggested that the reduction of the electron scattering intensity ratio IH/IC in (CH2)n and IH/IC+O in C8H14O2 is due to quantum entanglement (QE) whereby the incident electron interacts with a quantum entangled proton pair of the sample. This process was suggested to be similar to the case of neutron scattering from liquid H2O/ D2O samples where a strong deficiency in the scattering intensity from the H-atom (as compared to that from the O-atom) was reported [10]. The anomaly in the neutron case was thought to have the same origin as that occurring when sub-thermal neutrons scatter from two protons in liquid molecular H2 at 10 K. In this process predicted theoretically in Ref. [11] the n-scattering intensity from a quantum-entangled (QE) pair of protons in H2 was found to drop by 50% compared to scattering from two independent free protons. Here the de-Broglie wave length of the incident sub-thermal neutron kn 2 Å interacts with the two protons of H2 (separated by dHH 0.76 Å) giving rise to the strong drop in IH. The QE explanation do not seem to be applicable to neutrons of much higher incident energies of 20 eV used in [10] because of the much lower value of kn 0.064 Å << dHH. It was however suggested in Ref. [12] that because of the special scattering technique used in [10], the main factor governing the occurrence of the interference
between exchange correlated protons is not the magnitude of kn but rather the coherence length given by lc = knEr/DEr which must be larger than dHH, namely the condition lc P dHH must be fulfilled. Here Er is the energy of the scattered neutrons being detected using a resonance gold filter [10] whose resonance energy Er = 4.9 eV while DEr = 0.3 eV is the effective width of the resonance. This yields lc = 2.3 Å which is larger than the HAH distance in H2O being dHH 1.6 Å. The above process used for explaining the anomalous scattering intensity in the neutron scattering from two exchange correlated protons in H2O/D2O was adopted in [3] for explaining the drop in the elastic scattering intensity of electrons from proton pairs contained in the (CH2)n and C8H14O2 solid samples. For those samples the coherence length of the electrons defined as: lc = keEe/ DEe may be calculated to be: lc = 3400 Å [4] which is far larger than the distance dHH 1.8 Å of the solid polymer samples. Note that lc was calculated for electrons having Ee = 20 keV whose de-Broglie wave length ke = 0.085 Å and DEe 0.5 eV is the resolution of the e-spectrometer [4]. This would mean that QE can occur in the scattering process of 20 keV electrons from solid polyethylene. Note however that the validity of the criterion, lc P dHH, suggested in Ref. [12] for the occurrence of quantum entanglement in electron scattering or in neutron scattering has never been verified experimentally and there are some good reasons to believe that it is not applicable to the above scattering processes. In fact, it turned out [7,13] that the drop in the electron scattering intensity ratio IH/IC in solid samples was not caused by the mechanism suggested above but was due to the radiation damage to the sample caused by the impinging e-beam. In this process, the electrons tend to break the CAH bonds of the solid sample causing desorption of H-atoms and deposition of the C-atoms thus reducing the amount of H/C ratio in the solid film and also the measured IH/IC ratio. This explanation is supported by the fact that the magnitude of IH/IC was found to decrease substantially with increasing e-beam energy Ee which tends to increase the radiation damage. In fact, the deficiency from the Z2 relation of IH/IC increased enormously between 15 and 30 keV (see Table 1). This effect of increasing deviation from the Rutherford ratio with increasing e-energy was attributed in the original work [3] to the increase in momentum transfer of the scattering electrons. Note that the effect of radiation damage in solid samples was discussed in Refs. [13–17]. In [13] it was stated that the damage was minimized by working at low electron energies (1.5 keV), reducing electron beam currents (<1 lA/cm2) and using short running times of 1 min. It was also noted that the use of samples with high-H content polymers such as polyethylene (CH2) was avoided because such samples are known to release H easily upon electron bombardment. With such precautions it was possible to get the correct electron scattering intensity ratios IH/IC where the Z2 relation was obeyed to within 10% and no anomalies were observed. As another test of the validity of the radiation damage explanation, a gaseous sample (such as CH4) was used instead of a solid sample. The advantage of using a gaseous sample is that it could be made very thin and be continuously replenished [7,13], thus the effect of radiation damage and of multiple scattering can be practically eliminated; it also leads to an improved signal to noise ratio in the electron scattering spectra. The result of such measurements on a gaseous CH4 sample was striking: the electron scattering intensity ratios IH/IC were found to conform to those of the Rutherford relation and the anomaly in the intensity ratio IH/IC disappeared [7,13]. This shows beyond any doubt that the radiation damage was responsible for the deviation from the Rutherford relation ratios and that there is no need to invoke the quantum entanglement mechanism to explain the anomaly. At this point it should be emphasized that quantum entanglement (QE) is far more likely to occur in a gaseous sample such as CH4 (using Ee = 6 keV and a scattering angle of 135°) than in a solid
R. Moreh / Nuclear Instruments and Methods in Physics Research B 279 (2012) 49–52
(CH2)n sample (using Ee = 30 keV and a scattering angle of 44.3°). This may be seen as noted in [17] by considering the three criteria which in principle may govern the occurrence of QE: (1) The deBroglie wave length of the incident electrons; (2) the interaction or scattering time of the electron from the H-atom defined in [18,19] and (3) the de-coherence time (which is a measure of the time necessary to destroy the phases of the two exchange correlated protons) defined in [20]. Few remarks concerning the above three quantities are in order. (i) We used the magnitude of ke as the first criterion in our comparison and not le because ke has been experimentally shown to be a very important factor [11] for observing an interference effect of QE while the use of le was only speculated but never really proved. (ii) For observing the effect of QE the scattering time should be much shorter than the de-coherence time. (iii) The de-coherence time is expected to be longer in a gaseous isolated CH4 molecule than in a solid (CH2)n where the more frequent atomic collisions can change the relative phase of the proton pair. Table 2 compares the magnitude of the above three quantities in a gas and solid samples. It shows that it is far more likely that the QE may occur in CH4 rather than in solid (CH2)n. Since the ratio of the elastic scattering intensities IH/IC of electrons from CH4 at 6 keV did not reveal any anomaly [7], it is therefore very unlikely that the elastic scattering of electrons from a solid polymer (CH2)n can reveal an anomaly caused by QE.
3.2. Effect of bond breaking It is important to note that in [2] it is claimed that the anomalous drop in the electron scattering ratio IH/IC from the solid (CH2)n sample is analogous to the anomaly reported for the case of neutron scattering from the same sample and have the same origin. It appears however that this is not always the case because in a neutron–proton scattering study [21] on two different samples: HD and a mixture of H2 + D2, the IH/ID measured ratio revealed the same drop (30%) in the scattering intensity from the H-atom in both samples. This result is in striking contrast to the case of electron scattering where an anomalous drop in the IH/ID ratio was found only for the H2 + D2 gas mixture while no anomaly was found in the HD sample. Another point was raised in [2] in which an attempt was made to relate the difference between electron and neutron scattering on HD to the issue of bond breaking. It was speculated that the difference between e-scattering and n-scattering from HD is caused by the difference in the energy transferred to HD. In the n-scattering process the energy transfer can be higher than the dissociation energy of the HD molecule (depending on the scattering angle) while in e-scattering the energy transfer is not enough to disrupt the HAD bond in HD. This issue of relating the HAD bond breaking to the absence of an electron scattering anomaly in HD is very likely irrelevant. This is because an anomalous drop in the IH/IY ratio was also observed in the neutron scattering from a YH2 sample Table 2 Factors influencing the occurrence of short lived quantum entanglement in proton pairs in samples of CH4 gas and solid polyethylene (CH2)n where ke denotes the deBroglie wave length of the incident electrons. The de-coherence time is the time required to destroy the relative phase of protons [17]. Sample
CH4 (gas)
(CH2)n (solid)
Energy Scattering angle dHH DeBroglie (ke) Scattering time De-coherence time
6 keV 135° 1.8 Å 0.025 Å 3.2 10-16 s 1012 s
30 keV 44.3° 1.6 Å 0.011 Å 3.5 1016 s 1014 s
51
[22] at a scattering angle of 45° when a resonance Rh foil was used instead of a gold resonance foil. The resonance energy in Rh is 1.23 eV which is far less than that of gold Er = 4.9 eV (normally used for neutron scattering studies [10]). Note that the neutron energy transfer at 45° for Rh is less than the bond breaking energy of 3.1 eV in YH2. Thus HAD bond breaking do not seem to be a relevant factor for explaining the absence of an anomaly in the electron scattering from HD. 3.3. Mixed binary gas samples More large deviations from the Rutherford predicted electron scattering intensity ratios were reported in binary gas mixtures [1] when two of any of the following gases were used as a sample: H2, D2, He, and Ar. A reduction in the electron elastic scattering intensity from the light partner was always revealed in those studies as may be seen in Table 1. Thus the reduction in the scattering intensity of the H signal was highest in the binary mixture H2/Ar in which the scattering intensity ratio IH2 /IAr from the lightest and heaviest partners were compared [1]. In Ref. [1] it is stated that the reductions in the electron elastic scattering intensities are not caused by the screening of the nuclear charge by the atomic electrons. Such a screening process would tend to reduce the effective nuclear charge of the heavy partner and thus to enhance the above scattering intensity ratios and not to reduce them. Moreover, it was emphasized [1] that the reduction in the scattering intensity ratio could not be explained neither classically nor by a quantum mechanical treatment within the Born approximation [1,5]. An explanation of the H2/D2 anomaly of the scattering intensity ratio was proposed in a recent paper [17] as due to the different velocities of the gas components in the binary mixture in thermal equilibrium. The velocity difference would mean that the heavier D2 gas would spend more time crossing the interaction region of the electron beam than the lighter H2 component. This mass factor p of the form M (where M is the molecular mass of the gas) was calculated to cause an intensity reduction of 29% in the electron scattering signal from H in the H2/D2 mixture which is in excellent agreement with the measured value of (30 ± 3)% reported in [1,2]. Note that when the same correction was applied to the other binary mixtures, the corrected intensity ratios did not agree with the measured ones but reduced substantially the deviation from the Rutherford relation reported in [1] as may be seen in Table 1. However the explanation proposed in [17] was rejected in [1] claiming that they used a gas cell where the composition of the gas mixture in the interaction region with the electron beam was checked experimentally using an electron–ion coincidence TOF mass spectrometer; the number of molecules of each component was found to be the same because a gas cell was used and not an effusive jet. Moreover it is claimed [1] that this anomaly is a more general one appearing not only in the H2/D2 case but also when the e-scattering intensity from a light gas is compared to a heavy one such as in a He/Ar mixture. It was also emphasized [1] that the intensity relationships between all binary gas mixtures are internally consistent. In addition the anomaly could not be explained [5] using any theoretical argument. Thus an important question remained if the anomaly is real or instrumental. In order to critically check this point one must compare the ratio of the scattering intensities after ensuring that the two components of any binary gas mixture must have the same atomic speed in the interaction region with the electron beam. This crucial condition is naturally achieved if a pure molecular gas such as HD or CH4 is used as a sample and not a gas mixture. In a pure gas, both the light and heavy components: H and D in HD, and H and C in CH4 move with the same speed as that of the entire molecule in the interaction region with the electrons. Thus all difficulties and
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R. Moreh / Nuclear Instruments and Methods in Physics Research B 279 (2012) 49–52
uncertainties encountered in Refs. [1,2] as to the method used to account for the effect of the difference between the atomic speeds of the binary components are eliminated. Note that other molecular gases such as C2H4 or C2H2D2 were also used as samples [23] for testing the scattering intensity ratios of electrons from H, D and C; all scattering intensity ratios were found to conform to that of the Z2 relation of Rutherford. It should be stressed at this point that the internal atomic motion of e.g. H and C in CH4 has no influence on the e-scattering intensities from the two elements but affects only the Doppler widths of the scattering lines [9] of the H- and Catoms. Scattering measurements from each of the above three gaseous samples: CH4, C2H4 or C2H2D2 have already been performed at two different experimental facilities using different incident electron energies and scattering angles [7,13,19] where a careful comparison of the H and C scattering intensities were carried out. In Refs. [7,13] the scattering angle was set at 135° with 6 different incident electron energies between 0.75 and 6 keV. In Ref. [19], the incident energy was 2.25 keV at an angle of 100°. All these experiments have shown beyond any doubt that the scattering intensity ratios obey the Z2 relation of Rutherford and no reduction in the IH/IC ratio was observed. A similar situation was found using a pure HD gas [2,13] where the H/D e-scattering intensity ratio was found to be 1. It is important to note that in Ref. [13] the effect of the difference in atomic velocities was discussed in detail; the case of the H2/He binary gas mixture was studied using an effusive gas jet cell and a different experimental facility than that used in [1,2]. Again, the result revealed no reduction [13] of the H-scattering signal (relative to the He-signal) from that of the Rutherford relation. It should be emphasized that one cannot pin point the source of the reported anomaly because the fine details of the experimental system used in Refs. [1,2] are not known. It may be speculated however that at least part of the anomaly may be caused by the gas cell holes which can let electrons and ions, involved in scattering and coincidence measurements, to enter and leave the interaction region. The light atoms should leave the cell through these openings much faster than the heavier ones, leading to a deficiency in their signal. 4. Conclusions In conclusion, it appears that the origin of the huge deviations of the electron scattering intensity ratios from the Z2 relation of Rutherford are instrumental. In the case of solid samples the deviations were caused by the radiation damage caused by the incident electron beam. When this factor was removed by using thin gaseous samples, the scattering anomaly disappeared.
The deviation from the Z2 relation in the case of a binary gas mixture samples must also be instrumental. Here again the deviations disappeared when the problem of the different atomic speeds of the gases passing through the interaction region with the electrons was eliminated as found when pure gases such as HD or CH4 were used as samples and not a binary mixture. Roughly speaking the reduction in the ratio of the scattering intensities of Refs. [1,2] seem to indicate that the heavier gas component in any binary mixture spends on the average more time in the interaction region than the light partner. It is believed that this deviation from the Rutherford relation is not genuine and should have an instrumental origin. Acknowledgements The author is grateful to Dr. D. Nemirovsky for helpful discussions and to Dr. M. Vos from the Australian National University in Canberra for useful remarks. References [1] A.P. Hitchcock, G. Cooper, R.A. Bonham, C.A. Chatzidimitriou-Dreismann, J. Electron Spectrosc. Relat. Phenom. 181 (2010) 135. [2] G. Cooper, A.P. Hitchcock, C.A. Chatzidimitriou-Dreismann, Phys. Rev. Lett. 100 (2008) 043204. [3] C.A. Chatzidimitriou-Dreismann, M. Vos, C. Kleiner, T. Abdul-Redah, Phys. Rev. Lett. 91 (2003) 57403. [4] M. Vos, C.A. Chatzidimitriou-Dreismann, T. Abdul-Redah, J. Mayers, Nucl. Instrum. Methods Phys. Res. B 227 (2005) 233. [5] R.A. Bonham, G. Cooper, A.P. Hitchcock, J. Chem. Phys. 130 (2009) 144303. [6] F. Salvat, A. Jablonski, C.J. Powell, Comput. Phys. Commun. 165 (2005) 157. [7] M. Vos, J. Chem. Phys. 132 (2010) 074306. [8] D. Varga, K. Tokesi, Z. Berenyi, J. Toth, L. Kover, Surf. Interface Anal. 38 (2006) 544. [9] R. Moreh, D. Nemirovsky, J. Chem. Phys. 130 (2009) 174303. [10] C.A. Chatzidimitriou-Dreismann, T. Abdul-Redah, R.M.F. Streffer, J. Mayers, Phys. Rev. Lett. 79 (1997) 2839. [11] J. Schwinger, E. Teller, Phys. Rev. 52 (1937) 286. [12] E.B. Karlsson, J. Mayers, Phys. Rev. Lett. 92 (2004) 249601. [13] M. Vos, M.R. Went, J. Phys. B 42 (2009) 065204. [14] F. Yubero, V.J. Rico, J.P. Espinos, J. Cotrino, A.R. Gonzalez-Elipe, Appl. Phys. Lett. 87 (2005) 084101. [15] B. Lesiak, J. Zemek, J. Houdkova, Polymer 49 (2008) 4127. [16] M. Filippi, L. Calliari, C. Verona, G. Verona-Rinati, Surf. Sci. 603 (2009) 2082. [17] R. Moreh, D. Nemirovsky, J. Chem. Phys. 131 (2009) 054305. [18] V.F. Sears, Phys. Rev. B 30 (1984) 44. [19] G.I. Watson, J. Phys. Condens. Matter 8 (1996) 5955. [20] E.B. Karlsson, Phys. Rev. Lett. 90 (2003) 095301. [21] C.A. Chatzidimitriou-Dreismann, T. Abdul-Redah, M. Krzystyniak, Phys. Rev. B 72 (2005) 054123; R. Moreh, R.C. Block, Y. Danon, IBID 75 (2007) 057101. [22] O. Hartman, E.B. Karlsson, C.A. Chatzidimitriou-Dreismann, J. Phys.: Condens. Matter 20 (2008) 104248. [23] G. Cooper, E. Christensen, A.P. Hitchcock, J. Chem. Phys. 127 (2007) 084315.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
On deviations from theory of electron–atom elastic scattering cross sections R. Moreh ⇑ Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84120, Israel
a r t i c l e
i n f o
Article history: Received 15 July 2011 Received in revised form 13 September 2011 Available online 20 November 2011 Keywords: Electron elastic scattering Rutherford scattering Quantum entanglement H2 HD D2 He Ar H2O D2O CH4
a b s t r a c t In some recent studies it is claimed that the electron elastic scattering intensities at keV energies from atoms and molecules at high momentum transfers do not conform to the Rutherford relation. Huge reductions in the ratios of the electron scattering intensities were reported in the following binary gas mixtures: H2/D2, H2/He, H2/Ar, D2/Ar and He/Ar where the scattering intensities from the light partners were found to be lower than the heavier ones by 30% and higher. Of particular interest is the case of the H2/D2 isotopic mixture found to deviate from the Rutherford relation by 30%. Similar intensity reductions were reported in samples of solid polymers where the scattering intensity from H was compared to that of C; the strong anomalous H-scattering intensity reduction was attributed to short lived quantum entanglement of a proton pair in the solid sample. It was stated that a quantum mechanical treatment of this scattering process in the framework of the Born approximation could not yield an explanation for the above observation. Here a critical examination of the above reports is given with the conclusion that the origin of all above deviations is very likely instrumental and not due to any real deviation from the Rutherford formula. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In the literature [1–4] on quasi-elastic scattering of electrons from bound systems, several cases were reported where very large deviations from the Rutherford scattering relation were observed. In those cases the ratios of the elastic e-scattering intensities were determined rather than the absolute cross sections because of the much higher accuracies which can be obtained in ratio measurements. The electron energies involved were between 2 and 40 keV. Note that at such energies the electrons scatter from each atom in a bound system as if the atom was a free particle. The energy of the elastically scattered electrons follows a billiard ball type kinematics which is justified using the impulse approximation [3]. The term ‘‘electron Compton scattering (ECS)’’ was used to describe such an electron scattering process. Normally, the intensities of the elastically scattered electrons are expected to follow the Rutherford scattering formula which scales as Z2 where Z is the nuclear charge of the scattering atom. In the above studies however, huge deviations (of around 30%) of the scattering intensity ratios from the Z2 relation of Rutherford were reported. In one case [3,4] solid thin films of polyethylene ⇑ Tel.: +972 8 6461569; fax: +972 8 6472903. E-mail address: [email protected]. 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.10.036
(CH2)n, and formvar C8H14O2 were used as scattering samples. The electron scattering intensity ratios IH/IC from H and C or from H and C + O were compared at incident energies between 15 and 30 keV and a scattering angle of 44.3°. The deviations from the Rutherford ratios at such high momentum transfers were 20% at 15 keV increasing to 50% at 30 keV. Similar large deviations of the scattering intensity ratios from the Z2 relation were also reported in binary gas mixtures at much lower incident electron beam energies. The binary gas mixtures used in those measurements were taken from any two [1,2] of the following gases: H2, D2, He and Ar. The incident electron energies were around 2 keV at a scattering angle of 100°. Note that a quantum mechanical treatment of the above scattering process within the Born approximation did not provide any clue as to the reason for the above deviations [5]. At this point it must be emphasized that the available evaluations of the elastic scattering cross section for electron–atom interaction [6] slightly deviate from the Z2 relation. Such deviations are of the order of 1% for the experimental conditions considered in the present work and were ignored. Table 1 gives a summary of the samples used in the above measurements together with the magnitude of the deviations of the electron scattering intensities from the Rutherford ratios. In the following we discuss those deviations in an attempt to find the reason for the anomaly.
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R. Moreh / Nuclear Instruments and Methods in Physics Research B 279 (2012) 49–52
Table 1 List of measured elastic scattering intensity ratios for a solid polyethylene (CH2)n sample and for mixed binary gaseous samples taken from Refs. [1–4]. IX denotes the scattering intensity from atom X. The incident electron energies and the scattering angles are indicated. The Rutherford predicted ratio is denoted R. Numbers in parentheses indicate the error in the last digit. The last column shows the calculated p ratio after accounting for the velocity difference (which vary as 1/ M) of the binary gas mixture components. Sample
Energy (keV)
Angle (°)
Ratio
Measured ratio
Calc. ratio
(CH2)n solid (CH2)n solid H2 + D2 H2/He H2/Ar D2/Ar He/Ar
15 30 2.25 2.25 2.25 2.25 2.25
44.3 44.3 100 100 100 100 100
IH/IC IH/IC IH/ID IH2 /IHe IH2 /IAr ID2 /IAr IHe/IAr
0.80(8)R 0.50(5)R 0.70(3)R 0.52(6)R 0.37(6)R 0.55(8)R 0.65(8)R
0.71R 0.71R 0.22R 0.32R 0.32R
2. Experimental details The results of the measurements being discussed here require electron spectrometers of very high resolution [1–4,7,8] which determine not only the small energy separations of the scattered electron lines from the different atoms of the bound system but also the widths of the those lines. The electron energies used in the above measurements were between 1 and 40 keV and the scattering angles were between 44.3° and 135°. The instrumental resolution depends on the incident electron energy; in Ref. [1] it was 0.8 at 2.25 keV while in Ref. [7] it was 0.25 eV at 2.5 keV and 0.4 eV at 6 keV. It should be remarked that the line widths are caused not only by the instrumental resolution of the spectrometer but also by the Doppler broadening caused by the momentum distribution of the scattering atoms. This in turn is contributed by the various modes of motion of the atoms which include translation, libration (hindered rotations) and vibrations. In some molecules such as CH4 the Doppler broadening of the scattered electron lines from the H-atom and C-atoms can be calculated quite precisely as was done in detail in Ref. [9]. 3. Discussion 3.1. Solid samples We first discuss the case of solid samples [3,4]. It was suggested that the reduction of the electron scattering intensity ratio IH/IC in (CH2)n and IH/IC+O in C8H14O2 is due to quantum entanglement (QE) whereby the incident electron interacts with a quantum entangled proton pair of the sample. This process was suggested to be similar to the case of neutron scattering from liquid H2O/ D2O samples where a strong deficiency in the scattering intensity from the H-atom (as compared to that from the O-atom) was reported [10]. The anomaly in the neutron case was thought to have the same origin as that occurring when sub-thermal neutrons scatter from two protons in liquid molecular H2 at 10 K. In this process predicted theoretically in Ref. [11] the n-scattering intensity from a quantum-entangled (QE) pair of protons in H2 was found to drop by 50% compared to scattering from two independent free protons. Here the de-Broglie wave length of the incident sub-thermal neutron kn 2 Å interacts with the two protons of H2 (separated by dHH 0.76 Å) giving rise to the strong drop in IH. The QE explanation do not seem to be applicable to neutrons of much higher incident energies of 20 eV used in [10] because of the much lower value of kn 0.064 Å << dHH. It was however suggested in Ref. [12] that because of the special scattering technique used in [10], the main factor governing the occurrence of the interference
between exchange correlated protons is not the magnitude of kn but rather the coherence length given by lc = knEr/DEr which must be larger than dHH, namely the condition lc P dHH must be fulfilled. Here Er is the energy of the scattered neutrons being detected using a resonance gold filter [10] whose resonance energy Er = 4.9 eV while DEr = 0.3 eV is the effective width of the resonance. This yields lc = 2.3 Å which is larger than the HAH distance in H2O being dHH 1.6 Å. The above process used for explaining the anomalous scattering intensity in the neutron scattering from two exchange correlated protons in H2O/D2O was adopted in [3] for explaining the drop in the elastic scattering intensity of electrons from proton pairs contained in the (CH2)n and C8H14O2 solid samples. For those samples the coherence length of the electrons defined as: lc = keEe/ DEe may be calculated to be: lc = 3400 Å [4] which is far larger than the distance dHH 1.8 Å of the solid polymer samples. Note that lc was calculated for electrons having Ee = 20 keV whose de-Broglie wave length ke = 0.085 Å and DEe 0.5 eV is the resolution of the e-spectrometer [4]. This would mean that QE can occur in the scattering process of 20 keV electrons from solid polyethylene. Note however that the validity of the criterion, lc P dHH, suggested in Ref. [12] for the occurrence of quantum entanglement in electron scattering or in neutron scattering has never been verified experimentally and there are some good reasons to believe that it is not applicable to the above scattering processes. In fact, it turned out [7,13] that the drop in the electron scattering intensity ratio IH/IC in solid samples was not caused by the mechanism suggested above but was due to the radiation damage to the sample caused by the impinging e-beam. In this process, the electrons tend to break the CAH bonds of the solid sample causing desorption of H-atoms and deposition of the C-atoms thus reducing the amount of H/C ratio in the solid film and also the measured IH/IC ratio. This explanation is supported by the fact that the magnitude of IH/IC was found to decrease substantially with increasing e-beam energy Ee which tends to increase the radiation damage. In fact, the deficiency from the Z2 relation of IH/IC increased enormously between 15 and 30 keV (see Table 1). This effect of increasing deviation from the Rutherford ratio with increasing e-energy was attributed in the original work [3] to the increase in momentum transfer of the scattering electrons. Note that the effect of radiation damage in solid samples was discussed in Refs. [13–17]. In [13] it was stated that the damage was minimized by working at low electron energies (1.5 keV), reducing electron beam currents (<1 lA/cm2) and using short running times of 1 min. It was also noted that the use of samples with high-H content polymers such as polyethylene (CH2) was avoided because such samples are known to release H easily upon electron bombardment. With such precautions it was possible to get the correct electron scattering intensity ratios IH/IC where the Z2 relation was obeyed to within 10% and no anomalies were observed. As another test of the validity of the radiation damage explanation, a gaseous sample (such as CH4) was used instead of a solid sample. The advantage of using a gaseous sample is that it could be made very thin and be continuously replenished [7,13], thus the effect of radiation damage and of multiple scattering can be practically eliminated; it also leads to an improved signal to noise ratio in the electron scattering spectra. The result of such measurements on a gaseous CH4 sample was striking: the electron scattering intensity ratios IH/IC were found to conform to those of the Rutherford relation and the anomaly in the intensity ratio IH/IC disappeared [7,13]. This shows beyond any doubt that the radiation damage was responsible for the deviation from the Rutherford relation ratios and that there is no need to invoke the quantum entanglement mechanism to explain the anomaly. At this point it should be emphasized that quantum entanglement (QE) is far more likely to occur in a gaseous sample such as CH4 (using Ee = 6 keV and a scattering angle of 135°) than in a solid
R. Moreh / Nuclear Instruments and Methods in Physics Research B 279 (2012) 49–52
(CH2)n sample (using Ee = 30 keV and a scattering angle of 44.3°). This may be seen as noted in [17] by considering the three criteria which in principle may govern the occurrence of QE: (1) The deBroglie wave length of the incident electrons; (2) the interaction or scattering time of the electron from the H-atom defined in [18,19] and (3) the de-coherence time (which is a measure of the time necessary to destroy the phases of the two exchange correlated protons) defined in [20]. Few remarks concerning the above three quantities are in order. (i) We used the magnitude of ke as the first criterion in our comparison and not le because ke has been experimentally shown to be a very important factor [11] for observing an interference effect of QE while the use of le was only speculated but never really proved. (ii) For observing the effect of QE the scattering time should be much shorter than the de-coherence time. (iii) The de-coherence time is expected to be longer in a gaseous isolated CH4 molecule than in a solid (CH2)n where the more frequent atomic collisions can change the relative phase of the proton pair. Table 2 compares the magnitude of the above three quantities in a gas and solid samples. It shows that it is far more likely that the QE may occur in CH4 rather than in solid (CH2)n. Since the ratio of the elastic scattering intensities IH/IC of electrons from CH4 at 6 keV did not reveal any anomaly [7], it is therefore very unlikely that the elastic scattering of electrons from a solid polymer (CH2)n can reveal an anomaly caused by QE.
3.2. Effect of bond breaking It is important to note that in [2] it is claimed that the anomalous drop in the electron scattering ratio IH/IC from the solid (CH2)n sample is analogous to the anomaly reported for the case of neutron scattering from the same sample and have the same origin. It appears however that this is not always the case because in a neutron–proton scattering study [21] on two different samples: HD and a mixture of H2 + D2, the IH/ID measured ratio revealed the same drop (30%) in the scattering intensity from the H-atom in both samples. This result is in striking contrast to the case of electron scattering where an anomalous drop in the IH/ID ratio was found only for the H2 + D2 gas mixture while no anomaly was found in the HD sample. Another point was raised in [2] in which an attempt was made to relate the difference between electron and neutron scattering on HD to the issue of bond breaking. It was speculated that the difference between e-scattering and n-scattering from HD is caused by the difference in the energy transferred to HD. In the n-scattering process the energy transfer can be higher than the dissociation energy of the HD molecule (depending on the scattering angle) while in e-scattering the energy transfer is not enough to disrupt the HAD bond in HD. This issue of relating the HAD bond breaking to the absence of an electron scattering anomaly in HD is very likely irrelevant. This is because an anomalous drop in the IH/IY ratio was also observed in the neutron scattering from a YH2 sample Table 2 Factors influencing the occurrence of short lived quantum entanglement in proton pairs in samples of CH4 gas and solid polyethylene (CH2)n where ke denotes the deBroglie wave length of the incident electrons. The de-coherence time is the time required to destroy the relative phase of protons [17]. Sample
CH4 (gas)
(CH2)n (solid)
Energy Scattering angle dHH DeBroglie (ke) Scattering time De-coherence time
6 keV 135° 1.8 Å 0.025 Å 3.2 10-16 s 1012 s
30 keV 44.3° 1.6 Å 0.011 Å 3.5 1016 s 1014 s
51
[22] at a scattering angle of 45° when a resonance Rh foil was used instead of a gold resonance foil. The resonance energy in Rh is 1.23 eV which is far less than that of gold Er = 4.9 eV (normally used for neutron scattering studies [10]). Note that the neutron energy transfer at 45° for Rh is less than the bond breaking energy of 3.1 eV in YH2. Thus HAD bond breaking do not seem to be a relevant factor for explaining the absence of an anomaly in the electron scattering from HD. 3.3. Mixed binary gas samples More large deviations from the Rutherford predicted electron scattering intensity ratios were reported in binary gas mixtures [1] when two of any of the following gases were used as a sample: H2, D2, He, and Ar. A reduction in the electron elastic scattering intensity from the light partner was always revealed in those studies as may be seen in Table 1. Thus the reduction in the scattering intensity of the H signal was highest in the binary mixture H2/Ar in which the scattering intensity ratio IH2 /IAr from the lightest and heaviest partners were compared [1]. In Ref. [1] it is stated that the reductions in the electron elastic scattering intensities are not caused by the screening of the nuclear charge by the atomic electrons. Such a screening process would tend to reduce the effective nuclear charge of the heavy partner and thus to enhance the above scattering intensity ratios and not to reduce them. Moreover, it was emphasized [1] that the reduction in the scattering intensity ratio could not be explained neither classically nor by a quantum mechanical treatment within the Born approximation [1,5]. An explanation of the H2/D2 anomaly of the scattering intensity ratio was proposed in a recent paper [17] as due to the different velocities of the gas components in the binary mixture in thermal equilibrium. The velocity difference would mean that the heavier D2 gas would spend more time crossing the interaction region of the electron beam than the lighter H2 component. This mass factor p of the form M (where M is the molecular mass of the gas) was calculated to cause an intensity reduction of 29% in the electron scattering signal from H in the H2/D2 mixture which is in excellent agreement with the measured value of (30 ± 3)% reported in [1,2]. Note that when the same correction was applied to the other binary mixtures, the corrected intensity ratios did not agree with the measured ones but reduced substantially the deviation from the Rutherford relation reported in [1] as may be seen in Table 1. However the explanation proposed in [17] was rejected in [1] claiming that they used a gas cell where the composition of the gas mixture in the interaction region with the electron beam was checked experimentally using an electron–ion coincidence TOF mass spectrometer; the number of molecules of each component was found to be the same because a gas cell was used and not an effusive jet. Moreover it is claimed [1] that this anomaly is a more general one appearing not only in the H2/D2 case but also when the e-scattering intensity from a light gas is compared to a heavy one such as in a He/Ar mixture. It was also emphasized [1] that the intensity relationships between all binary gas mixtures are internally consistent. In addition the anomaly could not be explained [5] using any theoretical argument. Thus an important question remained if the anomaly is real or instrumental. In order to critically check this point one must compare the ratio of the scattering intensities after ensuring that the two components of any binary gas mixture must have the same atomic speed in the interaction region with the electron beam. This crucial condition is naturally achieved if a pure molecular gas such as HD or CH4 is used as a sample and not a gas mixture. In a pure gas, both the light and heavy components: H and D in HD, and H and C in CH4 move with the same speed as that of the entire molecule in the interaction region with the electrons. Thus all difficulties and
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uncertainties encountered in Refs. [1,2] as to the method used to account for the effect of the difference between the atomic speeds of the binary components are eliminated. Note that other molecular gases such as C2H4 or C2H2D2 were also used as samples [23] for testing the scattering intensity ratios of electrons from H, D and C; all scattering intensity ratios were found to conform to that of the Z2 relation of Rutherford. It should be stressed at this point that the internal atomic motion of e.g. H and C in CH4 has no influence on the e-scattering intensities from the two elements but affects only the Doppler widths of the scattering lines [9] of the H- and Catoms. Scattering measurements from each of the above three gaseous samples: CH4, C2H4 or C2H2D2 have already been performed at two different experimental facilities using different incident electron energies and scattering angles [7,13,19] where a careful comparison of the H and C scattering intensities were carried out. In Refs. [7,13] the scattering angle was set at 135° with 6 different incident electron energies between 0.75 and 6 keV. In Ref. [19], the incident energy was 2.25 keV at an angle of 100°. All these experiments have shown beyond any doubt that the scattering intensity ratios obey the Z2 relation of Rutherford and no reduction in the IH/IC ratio was observed. A similar situation was found using a pure HD gas [2,13] where the H/D e-scattering intensity ratio was found to be 1. It is important to note that in Ref. [13] the effect of the difference in atomic velocities was discussed in detail; the case of the H2/He binary gas mixture was studied using an effusive gas jet cell and a different experimental facility than that used in [1,2]. Again, the result revealed no reduction [13] of the H-scattering signal (relative to the He-signal) from that of the Rutherford relation. It should be emphasized that one cannot pin point the source of the reported anomaly because the fine details of the experimental system used in Refs. [1,2] are not known. It may be speculated however that at least part of the anomaly may be caused by the gas cell holes which can let electrons and ions, involved in scattering and coincidence measurements, to enter and leave the interaction region. The light atoms should leave the cell through these openings much faster than the heavier ones, leading to a deficiency in their signal. 4. Conclusions In conclusion, it appears that the origin of the huge deviations of the electron scattering intensity ratios from the Z2 relation of Rutherford are instrumental. In the case of solid samples the deviations were caused by the radiation damage caused by the incident electron beam. When this factor was removed by using thin gaseous samples, the scattering anomaly disappeared.
The deviation from the Z2 relation in the case of a binary gas mixture samples must also be instrumental. Here again the deviations disappeared when the problem of the different atomic speeds of the gases passing through the interaction region with the electrons was eliminated as found when pure gases such as HD or CH4 were used as samples and not a binary mixture. Roughly speaking the reduction in the ratio of the scattering intensities of Refs. [1,2] seem to indicate that the heavier gas component in any binary mixture spends on the average more time in the interaction region than the light partner. It is believed that this deviation from the Rutherford relation is not genuine and should have an instrumental origin. Acknowledgements The author is grateful to Dr. D. Nemirovsky for helpful discussions and to Dr. M. Vos from the Australian National University in Canberra for useful remarks. References [1] A.P. Hitchcock, G. Cooper, R.A. Bonham, C.A. Chatzidimitriou-Dreismann, J. Electron Spectrosc. Relat. Phenom. 181 (2010) 135. [2] G. Cooper, A.P. Hitchcock, C.A. Chatzidimitriou-Dreismann, Phys. Rev. Lett. 100 (2008) 043204. [3] C.A. Chatzidimitriou-Dreismann, M. Vos, C. Kleiner, T. Abdul-Redah, Phys. Rev. Lett. 91 (2003) 57403. [4] M. Vos, C.A. Chatzidimitriou-Dreismann, T. Abdul-Redah, J. Mayers, Nucl. Instrum. Methods Phys. Res. B 227 (2005) 233. [5] R.A. Bonham, G. Cooper, A.P. Hitchcock, J. Chem. Phys. 130 (2009) 144303. [6] F. Salvat, A. Jablonski, C.J. Powell, Comput. Phys. Commun. 165 (2005) 157. [7] M. Vos, J. Chem. Phys. 132 (2010) 074306. [8] D. Varga, K. Tokesi, Z. Berenyi, J. Toth, L. Kover, Surf. Interface Anal. 38 (2006) 544. [9] R. Moreh, D. Nemirovsky, J. Chem. Phys. 130 (2009) 174303. [10] C.A. Chatzidimitriou-Dreismann, T. Abdul-Redah, R.M.F. Streffer, J. Mayers, Phys. Rev. Lett. 79 (1997) 2839. [11] J. Schwinger, E. Teller, Phys. Rev. 52 (1937) 286. [12] E.B. Karlsson, J. Mayers, Phys. Rev. Lett. 92 (2004) 249601. [13] M. Vos, M.R. Went, J. Phys. B 42 (2009) 065204. [14] F. Yubero, V.J. Rico, J.P. Espinos, J. Cotrino, A.R. Gonzalez-Elipe, Appl. Phys. Lett. 87 (2005) 084101. [15] B. Lesiak, J. Zemek, J. Houdkova, Polymer 49 (2008) 4127. [16] M. Filippi, L. Calliari, C. Verona, G. Verona-Rinati, Surf. Sci. 603 (2009) 2082. [17] R. Moreh, D. Nemirovsky, J. Chem. Phys. 131 (2009) 054305. [18] V.F. Sears, Phys. Rev. B 30 (1984) 44. [19] G.I. Watson, J. Phys. Condens. Matter 8 (1996) 5955. [20] E.B. Karlsson, Phys. Rev. Lett. 90 (2003) 095301. [21] C.A. Chatzidimitriou-Dreismann, T. Abdul-Redah, M. Krzystyniak, Phys. Rev. B 72 (2005) 054123; R. Moreh, R.C. Block, Y. Danon, IBID 75 (2007) 057101. [22] O. Hartman, E.B. Karlsson, C.A. Chatzidimitriou-Dreismann, J. Phys.: Condens. Matter 20 (2008) 104248. [23] G. Cooper, E. Christensen, A.P. Hitchcock, J. Chem. Phys. 127 (2007) 084315.