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X-ray inelastic scattering from small particles of graphite
Volume 85A, number 2
PHYSICS LETTERS
14 September 1981
X-RAY INIILASTIC SCATFERING FROM SMALL PARTICLES OF GRAPHITE G.D. PRIFIFIS and J. BOVIATSIS Solid State 1~hysicsLaboratory, University of Patras, Patras, Greece Received 23 March 1981
The spec~trumof inelastically scattered X-ray radiation from small particles of graphite shows energy losses which are due to vo1ui~ieand surface plasmons, as in the foil case. Surface plasmons from spherical particles (SPS) have not been observed. It is ~hownthat graphite is inappropriate in the study of SPS using X-ray scattering experiments.
It has been shown experimentally as well as theoretically, first by Fujimoto et al. [1,2], that fast electrons can excite Surface plasmons, when they are scattered from small~spherical metal particles, with energy hc~,given by th~relation [1]1/2 (1) = ,~, / r 1 + q + f~/P I
p
L
a~
/
I
‘
where = (41r$e2/m)1/2 is the plasma frequency, 6a is the dielectric Øonstant of the medium surrounding the sphere and 1 1, 2, 3, indicates the different oscifiation modes. The existence of such excitation which is a follow, up of the spheric~alsurface boundaries called hereafter for brevity SPS (~SurfacePlasmon from Spheres) was confirmed by other investigators using either fast electron spectrometi~y[3] and/or optical measurements [3—5]. From eq. (1),~when 1 goes to infinity w~equals ~ and eq. (1) becomes = ~ ,r~+ ~a)~1/2 ~2’ S pI’ ‘~ ‘ ...
‘
which is the ene~gyfor the well-known surface plasma oscillation for a plane boundary (that is from a sphere of infinite radiu~),called hereafter SP. X-ray photons scattered by small metal particles can in principle excite SPS as in the case of electrons, Ashley et al. [6] have calculated the differential prob. ability for such ~lasmon excitation using the hydrodynamical appro~rimation.Lushnikov and Simonov [7], using RPA theory, have calculated the differential cross section~for electrons as well as for X-ray
scattering and they found similar results with those of Fujimoto et al. [1] and Ashley et al. [6]. So far the contribution from the experimental X-ray scattering has been the work of Kokkinakis and Alexopoulos [8] who observed the light emitted from the decay of X-ray excited plasmons from small partides of Ag, of Koumelis et a!. [9] who reported the existence of surface plasmon excitation from small particles of graphite at k = 0 and k = 0.526 A—’ and of Priftis et al. [10] who have established that there is no measureable intensity corresponding to either surface or volume plasmon excitation at k = 0. In this paper, in order to investigate the existence of SPS, the spectrum of X-ray scattering from small particles of graphite is reported for k = 0.789 A—’ and the various plasmon excitations which are observed are discussed. The experimental apparatus has been described previously [11]. The Cr K~is used as primary radiation and a crystal. quartz crystal cut along isthe 1340 planes analyzing The experiment performed in a as fixed scattering angle p = 150 and the transferred momentum k, defined by the relation k=(4ir/X)sin(ço/2)
(3)
where A is the wave length of the incident photons, was 0.789 A—’, which is smaller than the critical wave vector kc for plasmon excitation in graphite. The 1 mm thick samples were prepared from colloidal Acheson graphite. The mean size of the particles was found by• electron microscopy to be less than 250 A. Electron
0 031—9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
115
Volume 85A, number 2
PHYSICS LETTERS
diffraction patterns showed that the small particles
were crystalline, The spectrum from the X-ray scattering from graphite is shown in fig. 1. The arrow R indicates the position of the elastically scattered radiation (Rayleigh line). The energy loss indicated by the arrow P~and corresponding to an energy value of 9.5 eV is attributed to the excitation of the volume plasmon of the ir electrons of graphite (1 per atom). This is concluded from the following: The plasmons according to RPA theory follow the dispersion relation ~ + ~2 ~“p
=
~“k
‘
From this relation, taking into account that /lwk = 9.5 eV and the values of the dispersion coefficient a given by refs. [12] and [13], it is obtained that flw~ 7 eV. This value, thus obtained, good agreement with the values reported for theis irinplasmon of graphite as it is found from electron scattering experiments [12,13] as well as optical measurements [14]. The peak indicated by the arrow Pa corresponds to an energy loss of 25.4 eV and it is attributed to the excitation of the volume plasmon of the ii plus a electrons of graphite (4 per2 atom). ThisInvalue includes of eq. (4). order to find the dispersion 1iw~(k = 0) theterm ak2 a/c term is subtracted using the ap-
~
> —
1
2
z
R
wi
I
SP 0
0
0 0
I
0
I
I
10 20 E N E R G ‘(
I
I
I
30 (eV)
Fig. 1. Spectrwn of X-ray scattered radiation from small spherical particles of graphite. The arrow R indicates the position of elastically scattered radiation. The arrows P~,P and SP indicate the energy loss due to their plasmon, (71 + a)plasmon and surface plasmon excitation, respectively. The vertical lines indicate the statistical error.
116
propriate values of a as they are found from other experiments [12,15]. The final result is/lw1, = 23.5 eV. The values for the plasmon excitation of graphite, as they are found in the literature, varies and as Leder and Suddeth [16] showed depends on the specimen density. On the other hand Zeppenfeld [12] showed that plasmon loss depends on the direction which the transferred momentum k forms with the c axis, and varies from 19 eV fork parallel to c axis to 27eV for k perpendicular to c axis. Due to the randomness of the orientations, associated with the polyciystalline nature of the sample and the above mentioned ref. [16], the value of/lw~= 23.5 eV is considered most satisfactory. The energy loss peak indicated by the arrow in fig. 1 corresponds to the energy 16.5 eV.6a From = 1, eq.(l) that follows, for the principal mode 1 1 and 11w 1 = 1tw~,/s,/iSubstituting in this relation the value )1w1, = 23. 5 eV, obtained in this experiment, it is found that 11w1 = 13.6 eV. This value is too far away from that corresponding to the SP arrow, in order to interprete this peak as due to SPS excitation. The same conclusion is drawn even if hwk = 25.4 eV is substituted for/lw~ kw1,.=However, from (2)=follows for ~a = 1 that ~/\/~. Usingeq. 1tw~, 23.5 eV in the last relation, it follows that 11w 5 = 16.6 eV, in excellent agreement with the energy corresponding to the SP peak. Therefore, this peak is attributed to surface plasnions as they are known from the foil case. This is further supported from what follows in the next paragraph. As it has been shown by Fujimoto et al. [1] the maximum intensity for surface plasmon excitation shifts to modes of higher 1 with increasing particle radiusR, that is to higher energy values, and tends to surface plasmons, as in the foil case. This has also been confirmed experimentally by refs. [2] and [3].
0 —
14 September 1981
It is understood that the mfluence of the spherical boundaries of the particle on the plasma oscillations is important when the electron mean free path A is of the order of the particle size. For example in the case of Al the electron mean free path is found to be 160 A from the relation A = VFT, where ~F is the Fermi velocity and r the mean time between two consecutive collisions, calculated from the dc conductivity [17]. Fujunoto et al: [2] have measured the energy of surface plasmons m Al as a function of particle size and found that the lowest mode (1 = 1) is excited
Volume 85A, number 2
PHYSICS LETTERS
for particle size ~ 150 A while for particle size ~600 A the surface plasmon oscifiation is excited as in the foil case. Similar results have been reported by Kreibig and Zacharias [3] in the case of Ag. As it has been stated earlier the mean particle size in the present case is 250 A. The mean free path of the valence electtons of graphite which are assumed to form the electron gas is found to be 6 A using the dc conductivity l]181 and the plasma frequency wp, according to the relation r = ira/w~.From this small value of-the mean free path, in the case of graphite, which is a consequence of the small value of the conductivity (100 times smaller compared to that of Al) it is expected that only surface plasmons as in the foil case are excifed. Therefore, correctly the SP peak has been interpre~edas such. The results of~thepresent work show that only energy losses which~are due to volume and surface plasmons, as in the foil case, appear in the spectrum of inelastically scattered X-ray radiation from small partides of graphite, while tha lack of SPS is indicated. From the above discussion it is concluded also that graphite, although suitable for X-ray inelastic scattering experiments, is ii~appropriatefor the study of the SPS and samples fronf small metal particles should be used in order to verify the existing theories with X-ray scattering experiments.
14 September 1981
References [1] F. Fujimoto and K. Komaki, J. Phys. Soc. Japan 25 (1968) 1679.
[2] F. Fujimoto, K. Komaki and K. Ishida, J. Phys. Soc. Japan 23 (1967) 1186. [3] U. Kreibig and P. Zacharias, Z. Phys. 231 (1970) 128. [4] U. Kreiblg, J. Phys. F4 (1974) 999. [5] GAl Papavassiliou and Th. Kokkinakis, I. Phys. F4 (1974) L67. [61 J.C. Ashley, T.L. Ferrel and R.H. Ritchie, Phys. Rev. BlO (1974) 554. [7] A.A. Lushnlkov and AJ. Simonov, Z. Phys. 270 (1974)
17. [8] Th. Kokkinakis and K. Alexopoulos, Phys. Rev. Lett. 28 (1972) 1632.
[9] C. Koumelis, D. Leventouri and K. Alexopoulos, Phys. Stat. S0L (b) 46 (1971) K89; C. Koumelis and D. Leventouri, Phys. Rev. B7 (1973)
181. [10] G.D. Priftis, J. Boviatsis, A. Vradis and K. Kountouris, Phys. Stat. SoL 95 (1979) 301. [11] G.D. Priftis, J. Boviatsis and A. Vradis, Phys. Lett. 68A (1978) 462. [12] K. Zeppenfeld, Z. Phys. 211 (1968) 399. [13] H. Venghaus, Phys. Stat. SoL (b) 66 (1974) 145. [141E.A. Taft and H.R. Philipp, Phys. Rev. 138 (1965) A197. [15] P. Eisenberger and P.M. Platzman, Phys. Rev. B13 (1976) 934. [16] L.B. Leder and Y.A. Suddeth, J. AppL Phys. 31 (1960) 1422. [17] N. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976) pp. 20, 38. [18] W. PrilTiak and L.H. Fuchs, Phys. Rev. 95 (1954) 22.
117
PHYSICS LETTERS
14 September 1981
X-RAY INIILASTIC SCATFERING FROM SMALL PARTICLES OF GRAPHITE G.D. PRIFIFIS and J. BOVIATSIS Solid State 1~hysicsLaboratory, University of Patras, Patras, Greece Received 23 March 1981
The spec~trumof inelastically scattered X-ray radiation from small particles of graphite shows energy losses which are due to vo1ui~ieand surface plasmons, as in the foil case. Surface plasmons from spherical particles (SPS) have not been observed. It is ~hownthat graphite is inappropriate in the study of SPS using X-ray scattering experiments.
It has been shown experimentally as well as theoretically, first by Fujimoto et al. [1,2], that fast electrons can excite Surface plasmons, when they are scattered from small~spherical metal particles, with energy hc~,given by th~relation [1]1/2 (1) = ,~, / r 1 + q + f~/P I
p
L
a~
/
I
‘
where = (41r$e2/m)1/2 is the plasma frequency, 6a is the dielectric Øonstant of the medium surrounding the sphere and 1 1, 2, 3, indicates the different oscifiation modes. The existence of such excitation which is a follow, up of the spheric~alsurface boundaries called hereafter for brevity SPS (~SurfacePlasmon from Spheres) was confirmed by other investigators using either fast electron spectrometi~y[3] and/or optical measurements [3—5]. From eq. (1),~when 1 goes to infinity w~equals ~ and eq. (1) becomes = ~ ,r~+ ~a)~1/2 ~2’ S pI’ ‘~ ‘ ...
‘
which is the ene~gyfor the well-known surface plasma oscillation for a plane boundary (that is from a sphere of infinite radiu~),called hereafter SP. X-ray photons scattered by small metal particles can in principle excite SPS as in the case of electrons, Ashley et al. [6] have calculated the differential prob. ability for such ~lasmon excitation using the hydrodynamical appro~rimation.Lushnikov and Simonov [7], using RPA theory, have calculated the differential cross section~for electrons as well as for X-ray
scattering and they found similar results with those of Fujimoto et al. [1] and Ashley et al. [6]. So far the contribution from the experimental X-ray scattering has been the work of Kokkinakis and Alexopoulos [8] who observed the light emitted from the decay of X-ray excited plasmons from small partides of Ag, of Koumelis et a!. [9] who reported the existence of surface plasmon excitation from small particles of graphite at k = 0 and k = 0.526 A—’ and of Priftis et al. [10] who have established that there is no measureable intensity corresponding to either surface or volume plasmon excitation at k = 0. In this paper, in order to investigate the existence of SPS, the spectrum of X-ray scattering from small particles of graphite is reported for k = 0.789 A—’ and the various plasmon excitations which are observed are discussed. The experimental apparatus has been described previously [11]. The Cr K~is used as primary radiation and a crystal. quartz crystal cut along isthe 1340 planes analyzing The experiment performed in a as fixed scattering angle p = 150 and the transferred momentum k, defined by the relation k=(4ir/X)sin(ço/2)
(3)
where A is the wave length of the incident photons, was 0.789 A—’, which is smaller than the critical wave vector kc for plasmon excitation in graphite. The 1 mm thick samples were prepared from colloidal Acheson graphite. The mean size of the particles was found by• electron microscopy to be less than 250 A. Electron
0 031—9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
115
Volume 85A, number 2
PHYSICS LETTERS
diffraction patterns showed that the small particles
were crystalline, The spectrum from the X-ray scattering from graphite is shown in fig. 1. The arrow R indicates the position of the elastically scattered radiation (Rayleigh line). The energy loss indicated by the arrow P~and corresponding to an energy value of 9.5 eV is attributed to the excitation of the volume plasmon of the ir electrons of graphite (1 per atom). This is concluded from the following: The plasmons according to RPA theory follow the dispersion relation ~ + ~2 ~“p
=
~“k
‘
From this relation, taking into account that /lwk = 9.5 eV and the values of the dispersion coefficient a given by refs. [12] and [13], it is obtained that flw~ 7 eV. This value, thus obtained, good agreement with the values reported for theis irinplasmon of graphite as it is found from electron scattering experiments [12,13] as well as optical measurements [14]. The peak indicated by the arrow Pa corresponds to an energy loss of 25.4 eV and it is attributed to the excitation of the volume plasmon of the ii plus a electrons of graphite (4 per2 atom). ThisInvalue includes of eq. (4). order to find the dispersion 1iw~(k = 0) theterm ak2 a/c term is subtracted using the ap-
~
> —
1
2
z
R
wi
I
SP 0
0
0 0
I
0
I
I
10 20 E N E R G ‘(
I
I
I
30 (eV)
Fig. 1. Spectrwn of X-ray scattered radiation from small spherical particles of graphite. The arrow R indicates the position of elastically scattered radiation. The arrows P~,P and SP indicate the energy loss due to their plasmon, (71 + a)plasmon and surface plasmon excitation, respectively. The vertical lines indicate the statistical error.
116
propriate values of a as they are found from other experiments [12,15]. The final result is/lw1, = 23.5 eV. The values for the plasmon excitation of graphite, as they are found in the literature, varies and as Leder and Suddeth [16] showed depends on the specimen density. On the other hand Zeppenfeld [12] showed that plasmon loss depends on the direction which the transferred momentum k forms with the c axis, and varies from 19 eV fork parallel to c axis to 27eV for k perpendicular to c axis. Due to the randomness of the orientations, associated with the polyciystalline nature of the sample and the above mentioned ref. [16], the value of/lw~= 23.5 eV is considered most satisfactory. The energy loss peak indicated by the arrow in fig. 1 corresponds to the energy 16.5 eV.6a From = 1, eq.(l) that follows, for the principal mode 1 1 and 11w 1 = 1tw~,/s,/iSubstituting in this relation the value )1w1, = 23. 5 eV, obtained in this experiment, it is found that 11w1 = 13.6 eV. This value is too far away from that corresponding to the SP arrow, in order to interprete this peak as due to SPS excitation. The same conclusion is drawn even if hwk = 25.4 eV is substituted for/lw~ kw1,.=However, from (2)=follows for ~a = 1 that ~/\/~. Usingeq. 1tw~, 23.5 eV in the last relation, it follows that 11w 5 = 16.6 eV, in excellent agreement with the energy corresponding to the SP peak. Therefore, this peak is attributed to surface plasnions as they are known from the foil case. This is further supported from what follows in the next paragraph. As it has been shown by Fujimoto et al. [1] the maximum intensity for surface plasmon excitation shifts to modes of higher 1 with increasing particle radiusR, that is to higher energy values, and tends to surface plasmons, as in the foil case. This has also been confirmed experimentally by refs. [2] and [3].
0 —
14 September 1981
It is understood that the mfluence of the spherical boundaries of the particle on the plasma oscillations is important when the electron mean free path A is of the order of the particle size. For example in the case of Al the electron mean free path is found to be 160 A from the relation A = VFT, where ~F is the Fermi velocity and r the mean time between two consecutive collisions, calculated from the dc conductivity [17]. Fujunoto et al: [2] have measured the energy of surface plasmons m Al as a function of particle size and found that the lowest mode (1 = 1) is excited
Volume 85A, number 2
PHYSICS LETTERS
for particle size ~ 150 A while for particle size ~600 A the surface plasmon oscifiation is excited as in the foil case. Similar results have been reported by Kreibig and Zacharias [3] in the case of Ag. As it has been stated earlier the mean particle size in the present case is 250 A. The mean free path of the valence electtons of graphite which are assumed to form the electron gas is found to be 6 A using the dc conductivity l]181 and the plasma frequency wp, according to the relation r = ira/w~.From this small value of-the mean free path, in the case of graphite, which is a consequence of the small value of the conductivity (100 times smaller compared to that of Al) it is expected that only surface plasmons as in the foil case are excifed. Therefore, correctly the SP peak has been interpre~edas such. The results of~thepresent work show that only energy losses which~are due to volume and surface plasmons, as in the foil case, appear in the spectrum of inelastically scattered X-ray radiation from small partides of graphite, while tha lack of SPS is indicated. From the above discussion it is concluded also that graphite, although suitable for X-ray inelastic scattering experiments, is ii~appropriatefor the study of the SPS and samples fronf small metal particles should be used in order to verify the existing theories with X-ray scattering experiments.
14 September 1981
References [1] F. Fujimoto and K. Komaki, J. Phys. Soc. Japan 25 (1968) 1679.
[2] F. Fujimoto, K. Komaki and K. Ishida, J. Phys. Soc. Japan 23 (1967) 1186. [3] U. Kreibig and P. Zacharias, Z. Phys. 231 (1970) 128. [4] U. Kreiblg, J. Phys. F4 (1974) 999. [5] GAl Papavassiliou and Th. Kokkinakis, I. Phys. F4 (1974) L67. [61 J.C. Ashley, T.L. Ferrel and R.H. Ritchie, Phys. Rev. BlO (1974) 554. [7] A.A. Lushnlkov and AJ. Simonov, Z. Phys. 270 (1974)
17. [8] Th. Kokkinakis and K. Alexopoulos, Phys. Rev. Lett. 28 (1972) 1632.
[9] C. Koumelis, D. Leventouri and K. Alexopoulos, Phys. Stat. S0L (b) 46 (1971) K89; C. Koumelis and D. Leventouri, Phys. Rev. B7 (1973)
181. [10] G.D. Priftis, J. Boviatsis, A. Vradis and K. Kountouris, Phys. Stat. SoL 95 (1979) 301. [11] G.D. Priftis, J. Boviatsis and A. Vradis, Phys. Lett. 68A (1978) 462. [12] K. Zeppenfeld, Z. Phys. 211 (1968) 399. [13] H. Venghaus, Phys. Stat. SoL (b) 66 (1974) 145. [141E.A. Taft and H.R. Philipp, Phys. Rev. 138 (1965) A197. [15] P. Eisenberger and P.M. Platzman, Phys. Rev. B13 (1976) 934. [16] L.B. Leder and Y.A. Suddeth, J. AppL Phys. 31 (1960) 1422. [17] N. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976) pp. 20, 38. [18] W. PrilTiak and L.H. Fuchs, Phys. Rev. 95 (1954) 22.
117