THE JOURNAL OF CHEMICAL PHYSICS 142, 084301 (2015)
Investigation of Compton profiles of molecular methane and ethane Xiao-Li Zhao,1,2 Ke Yang,3,a) Long-Quan Xu,1,2 Xu Kang,1,2 Ya-Wei Liu,1,2 Yong-Peng Ma,3 Shuai Yan,3 Dong-Dong Ni,1,2 and Lin-Fan Zhu1,2,b) 1
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 2 Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 3 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201204, People’s Republic of China
(Received 10 December 2014; accepted 5 February 2015; published online 23 February 2015) The Compton profiles of methane and ethane molecules have been determined at an incident photon energy of 20 keV based on the third generation synchrotron radiation, and the statistical accuracy of 0.2% is achieved near pz = 0. The density functional theory with aug-cc-pVTZ basis set was used to calculate the Compton profiles of methane and ethane. The present experimental Compton profiles are in better agreement with the theoretical calculations in the whole pz region than the previous experimental results, which indicates that the present experimental Compton profiles are accurate enough to serve as the benchmark data for methane and ethane molecules. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4908534]
I. INTRODUCTION
Compton profiles of atoms, molecules, liquids, and solids with high accuracy can not only explore the electronic structures, Fermi surfaces, and the bonding properties but also test the theoretical modes, calculational codes, as well as the wave functions of the target strictly, because they reveal the information of the electron momentum density of the ground states and are highly sensitive to the change of the wave functions. Now, Compton scattering has become a powerful tool to study the electronic structure of the ground state of the target. For a long time, Compton scattering has been used to explore the character of solids and liquids and there are few Compton profile studies on gas samples, because the photon flux of the traditional x-ray machine, the target’s density of gas, and the scattering cross section are all low. With the dramatical improvement of the experimental techniques, especially the third generation synchrotron radiation light source, the x-ray scattering has been extended to study the electronic structures of the ground and excited states of atoms and molecules.1–8 Compton profiles of some gaseous atoms and molecules, such as He, Ne, Ar, Kr, Xe, H2, N2, O2, and a number of hydrocarbon molecules, were measured using the x-ray tube equipped with LiF analyzing crystal9–12 or γ-ray source with Li-drifted-germanium detector,13–16 or synchrotron radiation light source.1,2,17 In this work, the Compton profiles of molecular methane and ethane are studied. The Compton profile of methane has been measured by Eisenberger and Marra10 using x-ray tube, Paakkari and Merisalo16 using γ-rays, and, more recently, Sternemann et al.17 using synchrotron radiation. In addition, Klapthor and Lee18 and Lahmam-Bennani et al.19 have also studied the a)[email protected] b)[email protected]
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Compton profile of methane with electron scattering. These experimental results deviate from the theoretical calculations at that time in the region of low momenta, as noticed by Kaijser et al.20 and Sternemann et al.17 Furthermore, Sternemann et al. surmised that the difference mentioned above is derived from a deficiency of the calculation by the Kohn-Sham (KS) density functional theory (DFT) in predicting quantitatively the peak height of the absolute Compton profiles. However, the real reason has not been identified. Because of the fundamental nature of the accurate knowledge of the Compton profiles of simple systems, further and independent measurement and calculation of Compton profile of methane are of crucial importance to judge which one results in the difference. The Compton profile of ethane has hardly been studied except the work of Eisenberger et al.10 using x-ray tube. But their result is only up to pz = 2 a.u. and seems to have the same discrepancy problem as methane. In order to achieve the better accuracy of the Compton profile data in wider range, the fresh high-accurate experimental measurement of Compton profile of ethane is required. With these motivations, we present the joint experimental and theoretical investigations of the Compton profiles of methane and ethane in gas phase. The paper is organized as follows. In Sec. II, the theoretical approach for the calculation of the Compton profiles is introduced. The experimental method and the data analysis are described in detail in Sec. III. In Sec. IV, the experimental Compton profiles are compared with the theoretical calculations, and a detailed discussion is performed. Finally, a short summary is given. II. THEORETICAL BACKGROUND AND CALCULATIONS
It is well known that the differential cross section of Compton scattering can be described by the following equation 142, 084301-1
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(atomic units are used in the whole paper): dσ dσ ω f 1 =( )Th J(pz ). dωdΩ dΩ ωi k
(1)
2 Here, the Thomson cross section ( dσ dΩ )Th is equal to r 0 because the scattering plane is perpendicular to the polarization plane for this work. r 0 is the classical electron radius. ωi and ω f are the initial and final energies of the photon, while ω = ωi − ω f . ⃗k = k⃗1 − k⃗2 is the momentum transfer to the system, where k⃗1 and k⃗2 are the initial and final photon momenta, while k is the ⃗ modulus of ⃗k. J(pz ) is the Compton profile. pz = kk· p⃗ can be understood as the component of the electron momentum of the initial state in the direction of the photon momentum transfer. According to the energy conservation, the relationship between pz and ω can be described by
ω k − . (2) k 2 It should be pointed out that the formulas mentioned above are valid only at the limit of the impulse approximation (IA), i.e., the binding energy E B of the electron in the target is much less than its recoil energy E R .21 When the IA is satisfied, Eqs. (1) and (2) can be used to convert the energy spectrum into the experimental Compton profile. If the system is composed of independent particles described by a single wavefunction ψ j (⃗r ), we can write its wavefunction in momentum space ψ j (⃗ p) by Fourier transform 1 3 ψ j (⃗r ) exp(−i⃗ p · ⃗r )d⃗r (3) ψ j (⃗ p) = ( √ ) 2π and the electron density distribution in momentum space of the system dΩ|ψ j (⃗ p)|2, (4) n(p) = pz =
j
where n(p) is the probability of the electrons with a momentum magnitude p. The integral in the Eq. (4) is known as the spherically averaged one-electron momentum distribution or electron momentum profile,22 and the sum is over all occupied electron orbitals. Finally, the Compton profile can be written as ∞ J(pz ) = 2π n(p)pdp. (5) |p z |
It is clear that we can calculate the Compton profiles of methane and ethane by Eqs. (3)–(5) when their wavefunctions of the ground states are prepared. For this purpose, the neutral molecular geometric configuration of methane or ethane was optimized using the second-order Møllet-Plesset perturbation (MP2) method with aug-cc-pVTZ basis set at first. Based on the optimized molecular structure, the corresponding orbital wavefunctions in the position space were calculated by GAUSSIAN 03 program23 using density functional theory (DFT-B3LYP) with aug-cc-pVTZ basis set. The acronym B3LYP refers to Becke’s three-parameter hybrid functional combined with Lee-Yang-Parr correlation functional.24,25 Then, the Fourier transform and the calculation of n(p) in Eqs. (3) and (4) were carried out, while the Compton profiles were calculated by integrating the momentum distribution of the target electrons by Eq. (5). The present calculation
of methane by B3LYP/aug-cc-pVTZ is in good agreement with the recent theoretical calculation of Hu et al.26 In order to compare the calculated Compton profiles with the experimental ones, some corrections are necessarily needed. The kinematics of the Compton scattering process dictates that enough energy must be transferred to the bound electron to ionize it. So, this places a kinematic limit on the highest scattered photon energy. This limit is27 ωk = ωi − EB .
(6)
Here, E B is the binding energy of the corresponding shell. Therefore, the calculated Compton profile in the region of energies more than ωk should be removed. The Compton profiles of the five orbits of methane and those of the nine orbits of ethane should be corrected, then the respective corrected results of methane or ethane were added to obtain a total curve, which is hereafter called the corrected Compton profile. We directly use the corrected Compton profile of methane or ethane to compare with the respective experimental Compton profile, which will be discussed in Sec. III.
III. EXPERIMENTAL APPARATUS AND PROCEDURES
Compton profile measurements were carried out at the beamline BL15U1 of the Shanghai Synchrotron Radiation Facility (SSRF). Fig. 1 shows the experimental setup. The measurements are performed at an incident photon energy of 20 keV with an energy spread of 3 eV achieved by a Si(111) monochromator. The direction of the incident photon is along the z axis and its polarization direction is along the x axis. The scattered photon is detected by a silicon drift xray detector (SDD). The Compton profiles of gaseous samples (99.99% pure methane and 99.999% pure ethane from Nanjing Shangyuan Industrial Gas Company) were measured at room temperature and a gas pressure of 20 atm using two same gas cells, which are made of aluminium alloy. Each of the gas cell has a volume of 3 cm3, and the portholes for incident and scattered radiation are covered by Kapton film with 150 µm thickness. The diameters of the aperture 1 and 2 are both 1 mm while the distance between them is 33 mm, which limits a visual angle of 3.5◦ that results in a momentum resolution of 0.1 a.u. The SDD, gas cell, as well as the region between
FIG. 1. Schematic diagram of the Compton scattering spectrometer.
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FIG. 2. A Compton scattering spectrum measured at 90◦ for methane.
them are surrounded by a lead leather with a 0.5 mm thickness to shield the stray photons. The two cells were put on the experimental platform for the measurement in turn. The same setup has been successfully used in our previous measurement of Compton profile of hydrogen gas.28 Fig. 2 shows a x-ray scattering spectrum of methane with the elastic scattering and the Compton scattering assigned. The background signals from the scattering by Kapton windows and the wall of the gas cell were measured with the evacuated gas cell, and the result was subtracted point by point from the raw spectra. The elastic scattering fitted by a Gaussian function was also deducted. Then the low-energy side of the Compton scattering is used to determine the Compton profile since the elastic scattering in the high-energy side is close to the Compton scattering and its contribution is fairly great for these polyatomic molecules. Then, relative Compton profile J(pz ) was determined according to Eqs. (1) and (2). The experimental momentum resolution is mainly from the energy resolution of the spectrometer and the definite acceptance angle of the detector. The theoretical corrected Compton profile should be convoluted with the experimental momentum resolution function to enable a direct comparison with the experimental data. The momentum resolution caused by the detector’s visual angle is 0.1 a.u., which is obtained by simulating the known geometric arrangement of the spectrometer and used to convolve the corrected Compton profile at first. Subsequently, the momentum resolution caused by the energy resolution, i.e., 1.5 a.u.,28 is convolved. By normalizing the first two experimental points near pz = 0 to the corresponding theoretical values calculated by this work, the absolute Compton profile of methane was obtained. It should be pointed out that the data processing of ethane is exactly the same as methane, so we no longer introduce more about that. Further details concerning the experimental resolution and data processing were described in our previous paper.28
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FIG. 3. Compton profiles for methane. The present calculation is convolved with the experimental momentum resolution.
previous theoretical ones reported by Kaijser et al.20 and Sternemann et al.,17 although the result of Sternemann et al. is slightly lower than the present one, by about 1%, which is far less than the largest differences between the experimental results and calculations that will be discussed below. So we will compare the experimental Compton profiles with the present calculation by B3LYP/aug-cc-pVTZ as follows. It can be seen from Fig. 3 that overall good agreement between the present experimental result and calculation is found. Careful examination, however, reveals some subtle differences. The calculated curve predicts slightly higher values for the Compton profile in 0.8 a.u. 6 pz 6 4.4 a.u. These discrepancies are beyond the statistical accuracy of the measurement and they become more apparent if we use the deviation profile, J (p )−Jcal(p z ) △J(pz ) = expJ z(p z =0) , to express them in Fig. 4. Here, cal Jexp(pz ), Jcal(pz ), and Jcal(pz = 0) denote the experimental Compton profile, the present calculated one, and the value at pz = 0, respectively. It is clear that the present experimental results are lower than the calculation in 0.5 a.u. < pz < 4.7 a.u., and there is a trough at pz = 1.2 a.u. where the maximum deviation is about 1%. In general, the differences between
IV. RESULTS AND DISCUSSIONS
The presently measured Compton profile of methane is shown in Fig. 3 along with the theoretical calculation. It should be mentioned that our calculation is in agreement with the
FIG. 4. The differences between the experimental Compton profiles Jexp(p z ) and the present calculated profile Jcal(p z ) for methane. The present calculation is convoluted with the corresponding experimental resolutions.
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our experimental and theoretical Compton profiles are not very prominent, and the present measurement gives a good statistical accuracy, which is 0.2% near pz = 0. In order to compare the present results with the previous experimental ones, we give their deviation profiles in Fig. 4. The previous Compton profiles were measured by Sternemann et al.17 using 56.4 keV synchrotron radiation, Eisenberger and Marra10 using silver and molybdenum radiations produced by x-ray tube, Klapthor and Lee18 using 25 keV electrons, and Lahmam-Bennani et al.19 using 35 keV electron beam. We can see that compared with the theoretical calculation, the electron scattering result by Lahmam-Bennani et al. gives the worst consistency. The difference even reaches 5% at pz = 0.5 a.u., although their results agree with our calculation well around pz = 0 and in the region of pz > 1.6 a.u. The other electron scattering result by Klapthor and Lee gives the same trend, i.e., it is consistent with our calculation within the experimental error around pz = 0 and pz > 1.2 a.u. but significantly deviates from the theory near pz = 0.5 a.u. The Compton profile reported by Klapthor and Lee is slightly lower than that of Lahmam-Bennani et al. at pz = 0, but most noticeable is the increase in the region 0.2 a.u. 6 pz 6 1.2 a.u., i.e., an improvement towards a better agreement with theory. For the x-ray scattering result of Eisenberger et al., we can see that it gives too low values in the region pz < 1 a.u. and too high values at larger pz . Its tendency does not agree with that of the calculation at all. As for the experimental Compton profile measured by Sternemann et al.17 with the synchrotron radiation, it is lower than the calculated one seriously in the vicinity of the Compton peak but reverses in pz = 0.6 a.u. Besides the four previous experimental Compton profiles mentioned above, there is still one needed to analyze, which is the γray experimental measurement by Paakkari and Merisalo,16 not shown in Fig. 4. However, referring to the Fig. 3 given by Paakkari and Merisalo,16 we can know that the measurement by Paakkari and Merisalo is lower than the electron scattering result of Klapthor and Lee around the Compton peak and the discrepancy is beyond 2%. Of course, the γ-ray experimental result must show the same differences with our calculation in the same region. Based on those facts above, we can know that the present experimental Compton profile of methane agrees with the present calculation best, while the previous experimental Compton profiles of methane show different levels of deviations, respectively, in the low pz region. It should be mentioned that these deviations are almost exactly the same when comparing with the previous calculations.17,20 Therefore, we believe the present experimental Compton profile of methane is more accurate than before and is more suitable for serving as the benchmark data. Fig. 5 shows the present experimental Compton profile of ethane and the calculated one by B3LYP/aug-cc-pVTZ. The present experimental result agrees with our calculation very well in the whole range at a first glance. But similar to methane, the experimental Compton profile of ethane is slightly lower than the calculated curve in the 1 a.u.6 pz 6 4 a.u. region. Fig. 6 shows the corresponding deviation profile of ethane. It is clear that the largest deviation is less than 1% at pz = 1.2 a.u., and a good consistency is found in other region. The
J. Chem. Phys. 142, 084301 (2015)
FIG. 5. Compton profiles for ethane. The present calculation is convolved with the experimental momentum resolution.
statistical accuracy of the present experimental result of ethane is also reached 0.2% near pz = 0. The deviation profile of the previous x-ray scattering result by Eisenberger and Marra10 is also presented in Fig. 6. Its variation trend is almost identical with that of methane shown in Fig. 4 and there exists a serious discrepancy with our present calculation by B3LYP/aug-ccpVTZ. From Figs. 4 and 6, it can be seen that the present experimental Compton profiles are in better agreement with the latest theoretical calculations than the previous experimental ones, which may infer that the present experimental results are more accurate. It should be emphasized that we not only use the advanced synchrotron radiation but also minimize the experimental background. For the early x-ray scattering experiment,10 the low photon flux obstructs them to monochromatize the x-rays produced by x-ray tube. So, some inevitable uncertainties may be introduced. For the electron scattering results, the validity of the binary encounter approximation in electron spectroscopy may be doubtful, which was pointed out by Paakkari and Merisalo.16 To elucidate the minor differences between the present experimental results and the calculations, more experimental and theoretical works are greatly recommended.
FIG. 6. The differences between the experimental Compton profiles Jexp(p z ) and the present calculated profile Jcal(p z ) for ethane.
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V. SUMMARY AND CONCLUSION
In summary, based on the third generation synchrotron radiation light source, the Compton profiles of methane and ethane molecules were measured with an incident photon energy of 20 keV, and the statistical accuracy of 0.2% is achieved near pz = 0. The DFT method has also been used to calculate their Compton profiles. The present high-precision experimental Compton profiles of methane and ethane are more consistent with the present theoretical curves than the previous experiments, which indicates that the present Compton profiles of methane and ethane measured by synchrotron radiation are more accurate and can serve as the experimental benchmark. ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant Nos. U1332204, 11274291, 11104309, and 11320101003) and the Fundamental Research Funds for the Central Universities, China. The experiment was carried out in a beam time approved by Shanghai Synchrotron Radiation Facility (SSRF) (Proposal Nos. 11sr0210, 12sr0009, and 13SRBL15U15487). 1H.
Sakurai, H. Ota, N. Tsuji, M. Itou, and Y. Sakurai, J. Phys. B: At., Mol. Opt. Phys. 44, 065001 (2011). 2K. Kobayashi, M. Itou, T. Hosoya, N. Tsuji, Y. Sakurai, and H. Sakurai, J. Phys. B: At., Mol. Opt. Phys. 44, 115102 (2011). 3B. P. Xie, L. F. Zhu, K. Yang, B. Zhou, N. Hiraoka, Y. Q. Cai, Y. Yao, C. Q. Wu, E. L. Wang, and D. L. Feng, Phys. Rev. A 82, 032501 (2010). 4L. F. Zhu, L. S. Wang, B. P. Xie, K. Yang, N. Hiraoka, Y. Q. Cai, and D. L. Feng, J. Phys. B: At., Mol. Opt. Phys. 44, 025203 (2011).
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F. Zhu, W. Q. Xu, K. Yang, Z. Jiang, X. Kang, B. P. Xie, D. L. Feng, N. Hiraoka, and K. D. Tsuei, Phys. Rev. A 85, 030501(R) (2012). 6X. Kang, K. Yang, Y. W. Liu, W. Q. Xu, N. Hiraoka, K. D. Tsuei, P. F. Zhang, and L. F. Zhu, Phys. Rev. A 86, 022509 (2012). 7Y. W. Liu, X. X. Mei, X. Kang, K. Yang, W. Q. Xu, Y. G. Peng, N. Hiraoka, K. D. Tsuei, P. F. Zhang, and L. F. Zhu, Phys. Rev. A 89, 014502 (2014). 8Y. G. Peng, X. Kang, K. Yang, X. L. Zhao, Y. W. Liu, X. X. Mei, W. Q. Xu, N. Hiraoka, K. D. Tsuei, and L. F. Zhu, Phys. Rev. A 89, 032512 (2014). 9P. Eisenberger, Phys. Rev. A 2, 1678 (1970). 10P. Eisenberger and W. C. Marra, Phys. Rev. Lett. 27, 1413 (1971). 11P. Eisenberger, Phys. Rev. A 5, 628 (1972). 12P. Eisenberger, J. Chem. Phys. 56, 1207 (1972). 13P. Eisenberger and W. A. Reed, Phys. Rev. A 5, 2085 (1972). 14P. Eisenberger and W. A. Reed, Phys. Rev. B 9, 3237 (1974). 15T. Paakkari and M. Merisalo, Chem. Phys. Lett. 33, 432 (1975). 16T. Paakkari and M. Merisalo, Chem. Phys. Lett. 53, 313 (1978). 17C. Sternemann, S. Huotari, M. Hakala, M. Paulus, M. Volmer, C. Gutt, T. Buslaps, N. Hiraoka, D. D. Klug, K. Hämäläinen et al., Phys. Rev. B 73, 195104 (2006). 18R. W. Klapthor and J. S. Lee, Chem. Phys. Lett. 45, 513 (1977). 19A. Lahmam-Bennani, H. F. Wellenstein, A. Duguet, B. Nguyen, and A. D. Barlas, Chem. Phys. Lett. 41, 470 (1976). 20P. Kaijser, V. H. Smith, Jr., A. N. Tripathi, and G. H. F. Diercksen, Phys. Rev. A 35, 4074 (1987). 21P. Eisenberger and P. M. Platzman, Phys. Rev. A 2, 415 (1970). 22M. Yan, X. Shan, F. Wu, X. X. Xia, K. D. Wang, K. Z. Xu, and X. J. Chen, J. Phys. Chem. A 113, 507 (2009). 23M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., 03, Revision B. 04, Gaussian, Inc., Pittsburgh, PA, 2003. 24A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 25C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). 26X. L. Hu, Y. Z. Qu, S. B. Zhang, and Y. Zhang, Chin. Phys. B 21, 103401 (2012). 27P. M. Bergstrom, Jr. and R. H. Pratt, Radiat. Phys. Chem. 50, 3 (1997). 28X. L. Zhao, K. Yang, L. Q. Xu, Y. P. Ma, S. Yan, D. D. Ni, X. Kang, Y. W. Liu, and L. F. Zhu, Chin. Phys. B 24, 033301 (2015).
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Investigation of Compton profiles of molecular methane and ethane Xiao-Li Zhao,1,2 Ke Yang,3,a) Long-Quan Xu,1,2 Xu Kang,1,2 Ya-Wei Liu,1,2 Yong-Peng Ma,3 Shuai Yan,3 Dong-Dong Ni,1,2 and Lin-Fan Zhu1,2,b) 1
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 2 Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 3 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201204, People’s Republic of China
(Received 10 December 2014; accepted 5 February 2015; published online 23 February 2015) The Compton profiles of methane and ethane molecules have been determined at an incident photon energy of 20 keV based on the third generation synchrotron radiation, and the statistical accuracy of 0.2% is achieved near pz = 0. The density functional theory with aug-cc-pVTZ basis set was used to calculate the Compton profiles of methane and ethane. The present experimental Compton profiles are in better agreement with the theoretical calculations in the whole pz region than the previous experimental results, which indicates that the present experimental Compton profiles are accurate enough to serve as the benchmark data for methane and ethane molecules. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4908534]
I. INTRODUCTION
Compton profiles of atoms, molecules, liquids, and solids with high accuracy can not only explore the electronic structures, Fermi surfaces, and the bonding properties but also test the theoretical modes, calculational codes, as well as the wave functions of the target strictly, because they reveal the information of the electron momentum density of the ground states and are highly sensitive to the change of the wave functions. Now, Compton scattering has become a powerful tool to study the electronic structure of the ground state of the target. For a long time, Compton scattering has been used to explore the character of solids and liquids and there are few Compton profile studies on gas samples, because the photon flux of the traditional x-ray machine, the target’s density of gas, and the scattering cross section are all low. With the dramatical improvement of the experimental techniques, especially the third generation synchrotron radiation light source, the x-ray scattering has been extended to study the electronic structures of the ground and excited states of atoms and molecules.1–8 Compton profiles of some gaseous atoms and molecules, such as He, Ne, Ar, Kr, Xe, H2, N2, O2, and a number of hydrocarbon molecules, were measured using the x-ray tube equipped with LiF analyzing crystal9–12 or γ-ray source with Li-drifted-germanium detector,13–16 or synchrotron radiation light source.1,2,17 In this work, the Compton profiles of molecular methane and ethane are studied. The Compton profile of methane has been measured by Eisenberger and Marra10 using x-ray tube, Paakkari and Merisalo16 using γ-rays, and, more recently, Sternemann et al.17 using synchrotron radiation. In addition, Klapthor and Lee18 and Lahmam-Bennani et al.19 have also studied the a)[email protected] b)[email protected]
0021-9606/2015/142(8)/084301/5/$30.00
Compton profile of methane with electron scattering. These experimental results deviate from the theoretical calculations at that time in the region of low momenta, as noticed by Kaijser et al.20 and Sternemann et al.17 Furthermore, Sternemann et al. surmised that the difference mentioned above is derived from a deficiency of the calculation by the Kohn-Sham (KS) density functional theory (DFT) in predicting quantitatively the peak height of the absolute Compton profiles. However, the real reason has not been identified. Because of the fundamental nature of the accurate knowledge of the Compton profiles of simple systems, further and independent measurement and calculation of Compton profile of methane are of crucial importance to judge which one results in the difference. The Compton profile of ethane has hardly been studied except the work of Eisenberger et al.10 using x-ray tube. But their result is only up to pz = 2 a.u. and seems to have the same discrepancy problem as methane. In order to achieve the better accuracy of the Compton profile data in wider range, the fresh high-accurate experimental measurement of Compton profile of ethane is required. With these motivations, we present the joint experimental and theoretical investigations of the Compton profiles of methane and ethane in gas phase. The paper is organized as follows. In Sec. II, the theoretical approach for the calculation of the Compton profiles is introduced. The experimental method and the data analysis are described in detail in Sec. III. In Sec. IV, the experimental Compton profiles are compared with the theoretical calculations, and a detailed discussion is performed. Finally, a short summary is given. II. THEORETICAL BACKGROUND AND CALCULATIONS
It is well known that the differential cross section of Compton scattering can be described by the following equation 142, 084301-1
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J. Chem. Phys. 142, 084301 (2015)
(atomic units are used in the whole paper): dσ dσ ω f 1 =( )Th J(pz ). dωdΩ dΩ ωi k
(1)
2 Here, the Thomson cross section ( dσ dΩ )Th is equal to r 0 because the scattering plane is perpendicular to the polarization plane for this work. r 0 is the classical electron radius. ωi and ω f are the initial and final energies of the photon, while ω = ωi − ω f . ⃗k = k⃗1 − k⃗2 is the momentum transfer to the system, where k⃗1 and k⃗2 are the initial and final photon momenta, while k is the ⃗ modulus of ⃗k. J(pz ) is the Compton profile. pz = kk· p⃗ can be understood as the component of the electron momentum of the initial state in the direction of the photon momentum transfer. According to the energy conservation, the relationship between pz and ω can be described by
ω k − . (2) k 2 It should be pointed out that the formulas mentioned above are valid only at the limit of the impulse approximation (IA), i.e., the binding energy E B of the electron in the target is much less than its recoil energy E R .21 When the IA is satisfied, Eqs. (1) and (2) can be used to convert the energy spectrum into the experimental Compton profile. If the system is composed of independent particles described by a single wavefunction ψ j (⃗r ), we can write its wavefunction in momentum space ψ j (⃗ p) by Fourier transform 1 3 ψ j (⃗r ) exp(−i⃗ p · ⃗r )d⃗r (3) ψ j (⃗ p) = ( √ ) 2π and the electron density distribution in momentum space of the system dΩ|ψ j (⃗ p)|2, (4) n(p) = pz =
j
where n(p) is the probability of the electrons with a momentum magnitude p. The integral in the Eq. (4) is known as the spherically averaged one-electron momentum distribution or electron momentum profile,22 and the sum is over all occupied electron orbitals. Finally, the Compton profile can be written as ∞ J(pz ) = 2π n(p)pdp. (5) |p z |
It is clear that we can calculate the Compton profiles of methane and ethane by Eqs. (3)–(5) when their wavefunctions of the ground states are prepared. For this purpose, the neutral molecular geometric configuration of methane or ethane was optimized using the second-order Møllet-Plesset perturbation (MP2) method with aug-cc-pVTZ basis set at first. Based on the optimized molecular structure, the corresponding orbital wavefunctions in the position space were calculated by GAUSSIAN 03 program23 using density functional theory (DFT-B3LYP) with aug-cc-pVTZ basis set. The acronym B3LYP refers to Becke’s three-parameter hybrid functional combined with Lee-Yang-Parr correlation functional.24,25 Then, the Fourier transform and the calculation of n(p) in Eqs. (3) and (4) were carried out, while the Compton profiles were calculated by integrating the momentum distribution of the target electrons by Eq. (5). The present calculation
of methane by B3LYP/aug-cc-pVTZ is in good agreement with the recent theoretical calculation of Hu et al.26 In order to compare the calculated Compton profiles with the experimental ones, some corrections are necessarily needed. The kinematics of the Compton scattering process dictates that enough energy must be transferred to the bound electron to ionize it. So, this places a kinematic limit on the highest scattered photon energy. This limit is27 ωk = ωi − EB .
(6)
Here, E B is the binding energy of the corresponding shell. Therefore, the calculated Compton profile in the region of energies more than ωk should be removed. The Compton profiles of the five orbits of methane and those of the nine orbits of ethane should be corrected, then the respective corrected results of methane or ethane were added to obtain a total curve, which is hereafter called the corrected Compton profile. We directly use the corrected Compton profile of methane or ethane to compare with the respective experimental Compton profile, which will be discussed in Sec. III.
III. EXPERIMENTAL APPARATUS AND PROCEDURES
Compton profile measurements were carried out at the beamline BL15U1 of the Shanghai Synchrotron Radiation Facility (SSRF). Fig. 1 shows the experimental setup. The measurements are performed at an incident photon energy of 20 keV with an energy spread of 3 eV achieved by a Si(111) monochromator. The direction of the incident photon is along the z axis and its polarization direction is along the x axis. The scattered photon is detected by a silicon drift xray detector (SDD). The Compton profiles of gaseous samples (99.99% pure methane and 99.999% pure ethane from Nanjing Shangyuan Industrial Gas Company) were measured at room temperature and a gas pressure of 20 atm using two same gas cells, which are made of aluminium alloy. Each of the gas cell has a volume of 3 cm3, and the portholes for incident and scattered radiation are covered by Kapton film with 150 µm thickness. The diameters of the aperture 1 and 2 are both 1 mm while the distance between them is 33 mm, which limits a visual angle of 3.5◦ that results in a momentum resolution of 0.1 a.u. The SDD, gas cell, as well as the region between
FIG. 1. Schematic diagram of the Compton scattering spectrometer.
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FIG. 2. A Compton scattering spectrum measured at 90◦ for methane.
them are surrounded by a lead leather with a 0.5 mm thickness to shield the stray photons. The two cells were put on the experimental platform for the measurement in turn. The same setup has been successfully used in our previous measurement of Compton profile of hydrogen gas.28 Fig. 2 shows a x-ray scattering spectrum of methane with the elastic scattering and the Compton scattering assigned. The background signals from the scattering by Kapton windows and the wall of the gas cell were measured with the evacuated gas cell, and the result was subtracted point by point from the raw spectra. The elastic scattering fitted by a Gaussian function was also deducted. Then the low-energy side of the Compton scattering is used to determine the Compton profile since the elastic scattering in the high-energy side is close to the Compton scattering and its contribution is fairly great for these polyatomic molecules. Then, relative Compton profile J(pz ) was determined according to Eqs. (1) and (2). The experimental momentum resolution is mainly from the energy resolution of the spectrometer and the definite acceptance angle of the detector. The theoretical corrected Compton profile should be convoluted with the experimental momentum resolution function to enable a direct comparison with the experimental data. The momentum resolution caused by the detector’s visual angle is 0.1 a.u., which is obtained by simulating the known geometric arrangement of the spectrometer and used to convolve the corrected Compton profile at first. Subsequently, the momentum resolution caused by the energy resolution, i.e., 1.5 a.u.,28 is convolved. By normalizing the first two experimental points near pz = 0 to the corresponding theoretical values calculated by this work, the absolute Compton profile of methane was obtained. It should be pointed out that the data processing of ethane is exactly the same as methane, so we no longer introduce more about that. Further details concerning the experimental resolution and data processing were described in our previous paper.28
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FIG. 3. Compton profiles for methane. The present calculation is convolved with the experimental momentum resolution.
previous theoretical ones reported by Kaijser et al.20 and Sternemann et al.,17 although the result of Sternemann et al. is slightly lower than the present one, by about 1%, which is far less than the largest differences between the experimental results and calculations that will be discussed below. So we will compare the experimental Compton profiles with the present calculation by B3LYP/aug-cc-pVTZ as follows. It can be seen from Fig. 3 that overall good agreement between the present experimental result and calculation is found. Careful examination, however, reveals some subtle differences. The calculated curve predicts slightly higher values for the Compton profile in 0.8 a.u. 6 pz 6 4.4 a.u. These discrepancies are beyond the statistical accuracy of the measurement and they become more apparent if we use the deviation profile, J (p )−Jcal(p z ) △J(pz ) = expJ z(p z =0) , to express them in Fig. 4. Here, cal Jexp(pz ), Jcal(pz ), and Jcal(pz = 0) denote the experimental Compton profile, the present calculated one, and the value at pz = 0, respectively. It is clear that the present experimental results are lower than the calculation in 0.5 a.u. < pz < 4.7 a.u., and there is a trough at pz = 1.2 a.u. where the maximum deviation is about 1%. In general, the differences between
IV. RESULTS AND DISCUSSIONS
The presently measured Compton profile of methane is shown in Fig. 3 along with the theoretical calculation. It should be mentioned that our calculation is in agreement with the
FIG. 4. The differences between the experimental Compton profiles Jexp(p z ) and the present calculated profile Jcal(p z ) for methane. The present calculation is convoluted with the corresponding experimental resolutions.
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our experimental and theoretical Compton profiles are not very prominent, and the present measurement gives a good statistical accuracy, which is 0.2% near pz = 0. In order to compare the present results with the previous experimental ones, we give their deviation profiles in Fig. 4. The previous Compton profiles were measured by Sternemann et al.17 using 56.4 keV synchrotron radiation, Eisenberger and Marra10 using silver and molybdenum radiations produced by x-ray tube, Klapthor and Lee18 using 25 keV electrons, and Lahmam-Bennani et al.19 using 35 keV electron beam. We can see that compared with the theoretical calculation, the electron scattering result by Lahmam-Bennani et al. gives the worst consistency. The difference even reaches 5% at pz = 0.5 a.u., although their results agree with our calculation well around pz = 0 and in the region of pz > 1.6 a.u. The other electron scattering result by Klapthor and Lee gives the same trend, i.e., it is consistent with our calculation within the experimental error around pz = 0 and pz > 1.2 a.u. but significantly deviates from the theory near pz = 0.5 a.u. The Compton profile reported by Klapthor and Lee is slightly lower than that of Lahmam-Bennani et al. at pz = 0, but most noticeable is the increase in the region 0.2 a.u. 6 pz 6 1.2 a.u., i.e., an improvement towards a better agreement with theory. For the x-ray scattering result of Eisenberger et al., we can see that it gives too low values in the region pz < 1 a.u. and too high values at larger pz . Its tendency does not agree with that of the calculation at all. As for the experimental Compton profile measured by Sternemann et al.17 with the synchrotron radiation, it is lower than the calculated one seriously in the vicinity of the Compton peak but reverses in pz = 0.6 a.u. Besides the four previous experimental Compton profiles mentioned above, there is still one needed to analyze, which is the γray experimental measurement by Paakkari and Merisalo,16 not shown in Fig. 4. However, referring to the Fig. 3 given by Paakkari and Merisalo,16 we can know that the measurement by Paakkari and Merisalo is lower than the electron scattering result of Klapthor and Lee around the Compton peak and the discrepancy is beyond 2%. Of course, the γ-ray experimental result must show the same differences with our calculation in the same region. Based on those facts above, we can know that the present experimental Compton profile of methane agrees with the present calculation best, while the previous experimental Compton profiles of methane show different levels of deviations, respectively, in the low pz region. It should be mentioned that these deviations are almost exactly the same when comparing with the previous calculations.17,20 Therefore, we believe the present experimental Compton profile of methane is more accurate than before and is more suitable for serving as the benchmark data. Fig. 5 shows the present experimental Compton profile of ethane and the calculated one by B3LYP/aug-cc-pVTZ. The present experimental result agrees with our calculation very well in the whole range at a first glance. But similar to methane, the experimental Compton profile of ethane is slightly lower than the calculated curve in the 1 a.u.6 pz 6 4 a.u. region. Fig. 6 shows the corresponding deviation profile of ethane. It is clear that the largest deviation is less than 1% at pz = 1.2 a.u., and a good consistency is found in other region. The
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FIG. 5. Compton profiles for ethane. The present calculation is convolved with the experimental momentum resolution.
statistical accuracy of the present experimental result of ethane is also reached 0.2% near pz = 0. The deviation profile of the previous x-ray scattering result by Eisenberger and Marra10 is also presented in Fig. 6. Its variation trend is almost identical with that of methane shown in Fig. 4 and there exists a serious discrepancy with our present calculation by B3LYP/aug-ccpVTZ. From Figs. 4 and 6, it can be seen that the present experimental Compton profiles are in better agreement with the latest theoretical calculations than the previous experimental ones, which may infer that the present experimental results are more accurate. It should be emphasized that we not only use the advanced synchrotron radiation but also minimize the experimental background. For the early x-ray scattering experiment,10 the low photon flux obstructs them to monochromatize the x-rays produced by x-ray tube. So, some inevitable uncertainties may be introduced. For the electron scattering results, the validity of the binary encounter approximation in electron spectroscopy may be doubtful, which was pointed out by Paakkari and Merisalo.16 To elucidate the minor differences between the present experimental results and the calculations, more experimental and theoretical works are greatly recommended.
FIG. 6. The differences between the experimental Compton profiles Jexp(p z ) and the present calculated profile Jcal(p z ) for ethane.
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V. SUMMARY AND CONCLUSION
In summary, based on the third generation synchrotron radiation light source, the Compton profiles of methane and ethane molecules were measured with an incident photon energy of 20 keV, and the statistical accuracy of 0.2% is achieved near pz = 0. The DFT method has also been used to calculate their Compton profiles. The present high-precision experimental Compton profiles of methane and ethane are more consistent with the present theoretical curves than the previous experiments, which indicates that the present Compton profiles of methane and ethane measured by synchrotron radiation are more accurate and can serve as the experimental benchmark. ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant Nos. U1332204, 11274291, 11104309, and 11320101003) and the Fundamental Research Funds for the Central Universities, China. The experiment was carried out in a beam time approved by Shanghai Synchrotron Radiation Facility (SSRF) (Proposal Nos. 11sr0210, 12sr0009, and 13SRBL15U15487). 1H.
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