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Inter-edge magnetoplasma modes in two-dimensional sheets of positive ions trapped below the surface of superfluid helium
Physica B 284}288 (2000) 162}163
Inter-edge magnetoplasma modes in two-dimensional sheets of positive ions trapped below the surface of super#uid helium C.I. Pakes*, P.L. Elliott, W.F. Vinen School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
Abstract We report an experimental investigation of inter-edge magnetoplasma modes in circular 2D sheets of positive ions within which a step in ion density has been imposed. The modes were detected through a non-linear coupling with an axisymmetric plasma mode of the sheet, when both types of mode were driven simultaneously. Evidence is given for the existence of a possible new family of inter-edge modes. 2000 Elsevier Science B.V. All rights reserved. Keywords: Plasmon; Ions
1. Introduction Sheets of ions trapped below the surface of super#uid helium by the combination of an image potential and a uniform external electric "eld are proving ideal for the study of collective excitations of a bounded two-dimensional plasma, in both the #uid and crystal phases [1]. This paper considers the low magnetic "eld (B) behaviour of inter-edge magnetoplasma modes [2], which are localised at, and propagate along, the boundary between two regions of di!erent, non-zero areal ion density. Such modes were "rst observed in the system of electrons on helium in the limit of high B [3,4]; their frequencies are proportional to the density di!erence across the boundary and to B\.
2. Results and discussion The apparatus and methods of excitation and detection are similar to those reported by us earlier [1]. The ions are trapped typically 40 nm below the helium surface midway between two circular electrode assemblies,
* Corresponding author. National Physical Laboratory, Queens Road, Teddington, Middlesex TW11 0LW, UK. Fax: #44-181-9436939. E-mail address: [email protected] (C.I. Pakes)
separated by a distance 2h of 3 mm. Horizontal con"nement of a circular sheet of ions, of typical radius 12.5 mm, is achieved using a suitable ring electrode. A circular step in the ion density, concentric with the sheet, may be imposed by applying a steady potential to an appropriate electrode, so that the ion density n in the outer region of the pool is reduced and a boundary step to the inner region, of higher density n , is created at a radius R of about 5 mm. The width of the boundary region is presumably of order h. The ion densities were determined, to within an accuracy of about 10%, from the observed Shikin mode frequency [5]. The inter-edge modes were excited by an oscillating electric "eld and detected indirectly by their e!ect, through a non-linear coupling, on the response of the fundamental axisymmetric magnetoplasma mode, which was driven simultaneously at a frequency of typically 150 kHz. The observed spectrum of inter-edge modes is shown in Fig. 1 (䊉), along with the lowest frequency conventional edge magnetoplasmon (䊊), for a temperature such that both regions of the ion sheet are in a molten phase. The inter-edge modes are identi"ed as such through their dependence on the density di!erence n !n . The solid lines are a guide to the eye. Two of the inter-edge modes, indicated by bold lines, appear to behave in a di!erent manner to the rest. We have developed a simple analytical theory based on the propagation of edge magnetoplasma modes along an abrupt step in ion density which separates two
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 2 5 7 - 7
C.I. Pakes et al. / Physica B 284}288 (2000) 162}163
Fig. 1. Spectrum of inter-edge modes. n "2.0;10 m\, n "1.08;10 m\, ¹"107.5 mK.
semi-in"nite sheets of charge [5]. The theory may be applied to a circular step in ion density as long as its radius R is large compared to the distances s /u , s being the speed of propagation of the plasma waves each side of the boundary step. Normal modes are de"ned by wavenumbers, in the direction parallel to the boundary, given by q "m/R , where m is an integer. Our theory is K found to be in qualitative agreement with a more involved theory [2], in the low-"eld limit. Our theory suggests that there ought to be only two inter-edge modes (m"1 and 2) in the range of frequencies shown in Fig. 1. It seems possible that these are the two highlighted modes: we "nd that these modes behave qualitatively in the expected way, and, using values of n and n that are within our experimental uncertainties,
163
we obtain semi-quantitative agreement with the observed frequencies, except at the higher values of B when the modes become so strongly localised in the radial direction that the step function approximation in n !n is no longer adequate. If this interpretation is correct, there remains the task of identifying the other modes. Perhaps the unidenti"ed modes are multipole inter-edge modes, analogous to the Nazin}Shikin modes at an ordinary edge [1]; it is believed that such modes should exist [2]. A detailed theory of inter-edge multipole modes is now required.
3. Conclusions We have shown that 2D sheets of ions can support the propagation of edge magnetoplasma modes along an imposed step in ion density. Theoretical calculations suggest the possible existence of a new family of such modes.
References [1] [2] [3] [4] [5]
P.L. Elliott et al., Phys. Rev. B 56 (1997) 3447. S.A. Mikhailov et al., J. Phys. C 4 (1992) 6523. P.K.H. Sommerfeld et al., Phys. Rev. Lett. 74 (1995) 2559. O.I. Kirichek et al., Low. Temp. Phys. 21 (1995) 305. C.I. Pakes, Ph.D. Thesis, University of Birmingham, 1999.
Inter-edge magnetoplasma modes in two-dimensional sheets of positive ions trapped below the surface of super#uid helium C.I. Pakes*, P.L. Elliott, W.F. Vinen School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
Abstract We report an experimental investigation of inter-edge magnetoplasma modes in circular 2D sheets of positive ions within which a step in ion density has been imposed. The modes were detected through a non-linear coupling with an axisymmetric plasma mode of the sheet, when both types of mode were driven simultaneously. Evidence is given for the existence of a possible new family of inter-edge modes. 2000 Elsevier Science B.V. All rights reserved. Keywords: Plasmon; Ions
1. Introduction Sheets of ions trapped below the surface of super#uid helium by the combination of an image potential and a uniform external electric "eld are proving ideal for the study of collective excitations of a bounded two-dimensional plasma, in both the #uid and crystal phases [1]. This paper considers the low magnetic "eld (B) behaviour of inter-edge magnetoplasma modes [2], which are localised at, and propagate along, the boundary between two regions of di!erent, non-zero areal ion density. Such modes were "rst observed in the system of electrons on helium in the limit of high B [3,4]; their frequencies are proportional to the density di!erence across the boundary and to B\.
2. Results and discussion The apparatus and methods of excitation and detection are similar to those reported by us earlier [1]. The ions are trapped typically 40 nm below the helium surface midway between two circular electrode assemblies,
* Corresponding author. National Physical Laboratory, Queens Road, Teddington, Middlesex TW11 0LW, UK. Fax: #44-181-9436939. E-mail address: [email protected] (C.I. Pakes)
separated by a distance 2h of 3 mm. Horizontal con"nement of a circular sheet of ions, of typical radius 12.5 mm, is achieved using a suitable ring electrode. A circular step in the ion density, concentric with the sheet, may be imposed by applying a steady potential to an appropriate electrode, so that the ion density n in the outer region of the pool is reduced and a boundary step to the inner region, of higher density n , is created at a radius R of about 5 mm. The width of the boundary region is presumably of order h. The ion densities were determined, to within an accuracy of about 10%, from the observed Shikin mode frequency [5]. The inter-edge modes were excited by an oscillating electric "eld and detected indirectly by their e!ect, through a non-linear coupling, on the response of the fundamental axisymmetric magnetoplasma mode, which was driven simultaneously at a frequency of typically 150 kHz. The observed spectrum of inter-edge modes is shown in Fig. 1 (䊉), along with the lowest frequency conventional edge magnetoplasmon (䊊), for a temperature such that both regions of the ion sheet are in a molten phase. The inter-edge modes are identi"ed as such through their dependence on the density di!erence n !n . The solid lines are a guide to the eye. Two of the inter-edge modes, indicated by bold lines, appear to behave in a di!erent manner to the rest. We have developed a simple analytical theory based on the propagation of edge magnetoplasma modes along an abrupt step in ion density which separates two
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 2 5 7 - 7
C.I. Pakes et al. / Physica B 284}288 (2000) 162}163
Fig. 1. Spectrum of inter-edge modes. n "2.0;10 m\, n "1.08;10 m\, ¹"107.5 mK.
semi-in"nite sheets of charge [5]. The theory may be applied to a circular step in ion density as long as its radius R is large compared to the distances s /u , s being the speed of propagation of the plasma waves each side of the boundary step. Normal modes are de"ned by wavenumbers, in the direction parallel to the boundary, given by q "m/R , where m is an integer. Our theory is K found to be in qualitative agreement with a more involved theory [2], in the low-"eld limit. Our theory suggests that there ought to be only two inter-edge modes (m"1 and 2) in the range of frequencies shown in Fig. 1. It seems possible that these are the two highlighted modes: we "nd that these modes behave qualitatively in the expected way, and, using values of n and n that are within our experimental uncertainties,
163
we obtain semi-quantitative agreement with the observed frequencies, except at the higher values of B when the modes become so strongly localised in the radial direction that the step function approximation in n !n is no longer adequate. If this interpretation is correct, there remains the task of identifying the other modes. Perhaps the unidenti"ed modes are multipole inter-edge modes, analogous to the Nazin}Shikin modes at an ordinary edge [1]; it is believed that such modes should exist [2]. A detailed theory of inter-edge multipole modes is now required.
3. Conclusions We have shown that 2D sheets of ions can support the propagation of edge magnetoplasma modes along an imposed step in ion density. Theoretical calculations suggest the possible existence of a new family of such modes.
References [1] [2] [3] [4] [5]
P.L. Elliott et al., Phys. Rev. B 56 (1997) 3447. S.A. Mikhailov et al., J. Phys. C 4 (1992) 6523. P.K.H. Sommerfeld et al., Phys. Rev. Lett. 74 (1995) 2559. O.I. Kirichek et al., Low. Temp. Phys. 21 (1995) 305. C.I. Pakes, Ph.D. Thesis, University of Birmingham, 1999.