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Band structure effects in the heat capacity of adsorbed helium
Volume 76A, number 3,4
PHYSICS LETTERS
31 March 1980
BAND STRUCTURE EFFECTS IN THE HEAT CAPACITY OF ADSORBED HELIUM A.F. SILVA-MOREIRA1, Johanan CODONA2 and David GOODSTEIN3 california Institute of Technology, Low Temperature Physics 63-37, Pasadena, CA 91125, USA Received 11 January 1980
Departures from two-dimensional ideal gas behavior in the low density heat capacity of 3He and 4He adsorbed on graphite are shown to be due to two-diemsnional tunneling bands.
Films of 3He and 4He adsorbed on graphite substrates have proved to be systems remarkably rich in phenomena of interest [11. Underlying many of these phenomena is the fact that both isotopes are highly mobile on the graphite surface at low temperature. Dash [2] was the first to suggest that their mobility could be ascribed to tunneling bands analogous to those of electrons in metals. The purpose of this paper is to show for the first time that the effects of these tunneling bands are found directly in experimental data. In principle, it is not difficult to see how band structure effects ought to show up in measureable quantities. The isosteric heat capacity of the film, CN, depends directly on the single particle density of states, n (e). If the substrate were smooth, one would expect n (e) = constant for two-dimensional motion, leading to CN/NkB = 1 in the limit of low density at low temperature. On a real, periodic substrate, however, the energies e are those of states in the bands, with the result that CN/NkB departs from its ideal value. Unfortunately, if the coverage N is not small, CN also depends on the effects of He—He interactions and quantum degeneracy, while at very low N, CN is difficult to measure, and in any case comes to be dominated by the influence of substrate inhomogeneities which are inevitably present. The result of these factors has in the past been to obscure the influence of band structure on CN.
2
On leave from the University of Campinas, Brazil. Supported by the Brazilian agency FAPESP, contract 78/0303. Physics Department Scholar.
~ Supported by NSF grant no. DMR77-00036.
324
Elgin and Goodstein [3] have shown, however, how a knowledge of the distribution of inhomogeneities on the surface, together with sufficiently complete thermodynamic data, may be used to correct CN to find the values it would have on an ideally homogeneous graphite surface. We have reeexamined their 4He data and, performed the necessary analysis for (previously unpublished) 3He data, to obtain values of C~(T),where C~is the N -÷ 0 limit of CN on an ideal graphite substrate. Values of C~/NkB are obtained from the corrected data by extrapolating to N = 0 plots of CN/NkB versus p, the two-dimensional density. This procedure is standard for analyzing a virial gas. The plots are expected to yield straight lines intercepting at 1. Instead we found intercepts at the values of C~/Nk presented below. We were unable to understand the discrepancy until a preprint of band structure calculations by Carlos and Cole [4] fortuitously arrived. Results for CJ~/NkB are shown in fig. 1 where they are compared to the band structure predictions received from Carlos and Cole [4]. The effects of band structure are strikingly evident in the data, particularly in the case of 3He, where more complete data are available. Earlier band structure calculations based on much less complete data about the He—graphite system were presented by Hagen et al. [5]. The qualitative shape of the curves in fig. 1 is not difficult to understand. The two-dimensional periodiof the substrate gives rise to a gap of forbidden states at an energy (according to Carlos and Cole) about 12 K above the ground state. Excess states pile up just before and after the gap. The heat capacity rises as the city
Volume 76A, number 3,4
I
PHYSICS LETTERS
I
31 March 1980
calorimeter I~tgas= 11~’(P
3He
1,T) C
where Pgas is known as a function of the total amount of gas adsorbed and the temperature. Operationally, we divide the system into a large but finite number of subsystems with binding energies and numbers of sites designed to closely reproduce eq. (1). It would be easy to deduce the equilibrium amount adsorbed on each subsytem, N1 = p1N’, if the homowere known. In practice, use forpothis purpose geneous surface chemical we potential = ~(p0, T)the measured chemical potential, ~ 11gas + e~,modified pected to occur if the overall coverage is not near monoto eliminate second layer formation which is not ex-
‘~~,~•!_-
NkB I
0
I
I
T (K) (a) I
I
4He • NkB 07
I
0
•
.
I
T (K)
—
I
layer. Thus for each subsystem, at each N and T, we read off the density p~at which the chemical potential would differ from the measured value by the amount e 0.
to
(1,)
—
Fig. 1. Plots of CRJ/NkB for 3He and 4He, where CJ~Jis the N —* 0 limit of CN on an ideal graphite substrate. The dots represent our data (typical low and high temperature error bars are shown) and the full curves are taken from a preprint by Carlos and Cole [4]
This procedure is illustrated schematically in fig. 2. The coverage on the inactivated subsystem, N 0, is then given by the equation N= i=0
excess states below the gap are populated with increasing temperature, falls as the missing states in the gap are encountered, then rises again as the next band begins to be populated. We wish to describe briefly our method of correcting the CN data. The distribution of binding energies for helium on the grafoil substrate used for the measurements has been shown [3] to be fitted by the form 3] (1) (N)=—Eb[l+(l+N/N0Y gy substantially different from the ideal binding enerwhere N 0, crudely the number of sites of binding energy, Eb, corresponds to approximately 0.025 mono4He [3],and l36Kfor layers, andEb/kB = l42Kfor 3He [6]. Consider a region of the substrate consisting of N’ sites of binding energy e~.If Ie,I >Eb then in equilibrium the density of helium adsorbed on this subsystem, p~,should be higher than the ideal density p 0 on
a subsystem with binding energy e~= —Eb. The equilibrium density is governed by the fact that in equilibrium the chemical potential of the adsorbed helium on all subsystems is equal to the (measured) chemical potential of the three-dimensional gas in the
I
~,,
200
-
~He(T
= 5.0K)
—
—~(K) 00
\.
20
0.0
0.5
‘‘‘‘i’
-
‘1
N
1.0
N (mono layers) Fig. 2. Schematic illustration of the procedure utilized in this paper to extract N0, the coverage on the inactivated subsystern. Nt is any chosen coverage, N1 is the coverage on the subsystem of binding energy e~.The dotted line represents the modification that eliminates second layer formation.
325
Volume 76A, number 3,4
PHYSICS LETTERS
The picture, then, is of a large inactivated subsystem, whose density p0 ‘N0/N°is lower than average, in equilibrium with a known distribution of activated subsystems with higher densities. Each subsytem is assumed to have the thermodynamic properties that the entire film is measured to have at the same temperature when the overall density is equal to the subsystem density. For example, an inactivated subsystem in a twodimensional gas state might be in equilibrium with highly activated subsystems on which the helium is a two-dimensional solid. The solid is assumed to have the heat capacity of a solid measured at that temperature. For the system as a whole, the energy U may be written as dU = TdS + pdN; for the subsystems, one has dU1 = TdS1 + p1dN1. Expressing dU as ~
dU = f-~-~-I dT + \aT J-[N~)
~
~ (—j dN~ \aN~/T, ~N1—i}
where in (aU/aN1)T ~N1~i} one N1 is changed while the others are kept fixed, leads to /~p-\ 13N-\ CN = ~CN. T (2) \I3TI7v~~—‘-j aT/N —
where Civ, inis quantities
326
I~i—’-~
,
.
the (2) heatexcept capacity eq. Cjv of each subsystem. All 0 are known by the
31 March 1980
procedure outlined above. Thus we are able to deduce Cjv0 at the correct coverage N0 using eq. (2). The entire procedure breaks down at very low total coverage, because the corrections then become sensitive to the way in which second layer formation has been prevented (see fig. 2) and to the fact that we have used a discrete rather than continuous distribution of subsystems. To test this point, the entire analysis was repeated varying the second layer cutoff and the nurnber of subsystems used. The resulting values of Cjv0 were insensitive to those parameters of coverages above about 0.2 layers, and so the remainder of our analysis was restricted to those data. References [1] Cf. the review article by: J.G. Dash and M. Schick, in: The physics of liquid and solid helium, Part II, eds. K.H. Benneman and J.B. Ketterson (Wiley, New York, 1978) [2] J.G.Dash,J. Chem. Phys. 48 (1968) 2820. [3] R.L. Elgin and D.L. Goodstein, Phys. Rev. A9 (1974)
2657. [4] [51W.E.CarlosandM.W.Cole,preprint. D.E. Flagen, A.D. Novaco and F.J. Milford, in: Adsorption—desorption phenomena, 99. ed. F. Ricca (Academic Press, New York, 1972) p. [6] R.L. Elgin, J.M. Greif and D.L. Goodstein, Phys. Rev. Lett. 41(1978)1723.
PHYSICS LETTERS
31 March 1980
BAND STRUCTURE EFFECTS IN THE HEAT CAPACITY OF ADSORBED HELIUM A.F. SILVA-MOREIRA1, Johanan CODONA2 and David GOODSTEIN3 california Institute of Technology, Low Temperature Physics 63-37, Pasadena, CA 91125, USA Received 11 January 1980
Departures from two-dimensional ideal gas behavior in the low density heat capacity of 3He and 4He adsorbed on graphite are shown to be due to two-diemsnional tunneling bands.
Films of 3He and 4He adsorbed on graphite substrates have proved to be systems remarkably rich in phenomena of interest [11. Underlying many of these phenomena is the fact that both isotopes are highly mobile on the graphite surface at low temperature. Dash [2] was the first to suggest that their mobility could be ascribed to tunneling bands analogous to those of electrons in metals. The purpose of this paper is to show for the first time that the effects of these tunneling bands are found directly in experimental data. In principle, it is not difficult to see how band structure effects ought to show up in measureable quantities. The isosteric heat capacity of the film, CN, depends directly on the single particle density of states, n (e). If the substrate were smooth, one would expect n (e) = constant for two-dimensional motion, leading to CN/NkB = 1 in the limit of low density at low temperature. On a real, periodic substrate, however, the energies e are those of states in the bands, with the result that CN/NkB departs from its ideal value. Unfortunately, if the coverage N is not small, CN also depends on the effects of He—He interactions and quantum degeneracy, while at very low N, CN is difficult to measure, and in any case comes to be dominated by the influence of substrate inhomogeneities which are inevitably present. The result of these factors has in the past been to obscure the influence of band structure on CN.
2
On leave from the University of Campinas, Brazil. Supported by the Brazilian agency FAPESP, contract 78/0303. Physics Department Scholar.
~ Supported by NSF grant no. DMR77-00036.
324
Elgin and Goodstein [3] have shown, however, how a knowledge of the distribution of inhomogeneities on the surface, together with sufficiently complete thermodynamic data, may be used to correct CN to find the values it would have on an ideally homogeneous graphite surface. We have reeexamined their 4He data and, performed the necessary analysis for (previously unpublished) 3He data, to obtain values of C~(T),where C~is the N -÷ 0 limit of CN on an ideal graphite substrate. Values of C~/NkB are obtained from the corrected data by extrapolating to N = 0 plots of CN/NkB versus p, the two-dimensional density. This procedure is standard for analyzing a virial gas. The plots are expected to yield straight lines intercepting at 1. Instead we found intercepts at the values of C~/Nk presented below. We were unable to understand the discrepancy until a preprint of band structure calculations by Carlos and Cole [4] fortuitously arrived. Results for CJ~/NkB are shown in fig. 1 where they are compared to the band structure predictions received from Carlos and Cole [4]. The effects of band structure are strikingly evident in the data, particularly in the case of 3He, where more complete data are available. Earlier band structure calculations based on much less complete data about the He—graphite system were presented by Hagen et al. [5]. The qualitative shape of the curves in fig. 1 is not difficult to understand. The two-dimensional periodiof the substrate gives rise to a gap of forbidden states at an energy (according to Carlos and Cole) about 12 K above the ground state. Excess states pile up just before and after the gap. The heat capacity rises as the city
Volume 76A, number 3,4
I
PHYSICS LETTERS
I
31 March 1980
calorimeter I~tgas= 11~’(P
3He
1,T) C
where Pgas is known as a function of the total amount of gas adsorbed and the temperature. Operationally, we divide the system into a large but finite number of subsystems with binding energies and numbers of sites designed to closely reproduce eq. (1). It would be easy to deduce the equilibrium amount adsorbed on each subsytem, N1 = p1N’, if the homowere known. In practice, use forpothis purpose geneous surface chemical we potential = ~(p0, T)the measured chemical potential, ~ 11gas + e~,modified pected to occur if the overall coverage is not near monoto eliminate second layer formation which is not ex-
‘~~,~•!_-
NkB I
0
I
I
T (K) (a) I
I
4He • NkB 07
I
0
•
.
I
T (K)
—
I
layer. Thus for each subsystem, at each N and T, we read off the density p~at which the chemical potential would differ from the measured value by the amount e 0.
to
(1,)
—
Fig. 1. Plots of CRJ/NkB for 3He and 4He, where CJ~Jis the N —* 0 limit of CN on an ideal graphite substrate. The dots represent our data (typical low and high temperature error bars are shown) and the full curves are taken from a preprint by Carlos and Cole [4]
This procedure is illustrated schematically in fig. 2. The coverage on the inactivated subsystem, N 0, is then given by the equation N= i=0
excess states below the gap are populated with increasing temperature, falls as the missing states in the gap are encountered, then rises again as the next band begins to be populated. We wish to describe briefly our method of correcting the CN data. The distribution of binding energies for helium on the grafoil substrate used for the measurements has been shown [3] to be fitted by the form 3] (1) (N)=—Eb[l+(l+N/N0Y gy substantially different from the ideal binding enerwhere N 0, crudely the number of sites of binding energy, Eb, corresponds to approximately 0.025 mono4He [3],and l36Kfor layers, andEb/kB = l42Kfor 3He [6]. Consider a region of the substrate consisting of N’ sites of binding energy e~.If Ie,I >Eb then in equilibrium the density of helium adsorbed on this subsystem, p~,should be higher than the ideal density p 0 on
a subsystem with binding energy e~= —Eb. The equilibrium density is governed by the fact that in equilibrium the chemical potential of the adsorbed helium on all subsystems is equal to the (measured) chemical potential of the three-dimensional gas in the
I
~,,
200
-
~He(T
= 5.0K)
—
—~(K) 00
\.
20
0.0
0.5
‘‘‘‘i’
-
‘1
N
1.0
N (mono layers) Fig. 2. Schematic illustration of the procedure utilized in this paper to extract N0, the coverage on the inactivated subsystern. Nt is any chosen coverage, N1 is the coverage on the subsystem of binding energy e~.The dotted line represents the modification that eliminates second layer formation.
325
Volume 76A, number 3,4
PHYSICS LETTERS
The picture, then, is of a large inactivated subsystem, whose density p0 ‘N0/N°is lower than average, in equilibrium with a known distribution of activated subsystems with higher densities. Each subsytem is assumed to have the thermodynamic properties that the entire film is measured to have at the same temperature when the overall density is equal to the subsystem density. For example, an inactivated subsystem in a twodimensional gas state might be in equilibrium with highly activated subsystems on which the helium is a two-dimensional solid. The solid is assumed to have the heat capacity of a solid measured at that temperature. For the system as a whole, the energy U may be written as dU = TdS + pdN; for the subsystems, one has dU1 = TdS1 + p1dN1. Expressing dU as ~
dU = f-~-~-I dT + \aT J-[N~)
~
~ (—j dN~ \aN~/T, ~N1—i}
where in (aU/aN1)T ~N1~i} one N1 is changed while the others are kept fixed, leads to /~p-\ 13N-\ CN = ~CN. T (2) \I3TI7v~~—‘-j aT/N —
where Civ, inis quantities
326
I~i—’-~
,
.
the (2) heatexcept capacity eq. Cjv of each subsystem. All 0 are known by the
31 March 1980
procedure outlined above. Thus we are able to deduce Cjv0 at the correct coverage N0 using eq. (2). The entire procedure breaks down at very low total coverage, because the corrections then become sensitive to the way in which second layer formation has been prevented (see fig. 2) and to the fact that we have used a discrete rather than continuous distribution of subsystems. To test this point, the entire analysis was repeated varying the second layer cutoff and the nurnber of subsystems used. The resulting values of Cjv0 were insensitive to those parameters of coverages above about 0.2 layers, and so the remainder of our analysis was restricted to those data. References [1] Cf. the review article by: J.G. Dash and M. Schick, in: The physics of liquid and solid helium, Part II, eds. K.H. Benneman and J.B. Ketterson (Wiley, New York, 1978) [2] J.G.Dash,J. Chem. Phys. 48 (1968) 2820. [3] R.L. Elgin and D.L. Goodstein, Phys. Rev. A9 (1974)
2657. [4] [51W.E.CarlosandM.W.Cole,preprint. D.E. Flagen, A.D. Novaco and F.J. Milford, in: Adsorption—desorption phenomena, 99. ed. F. Ricca (Academic Press, New York, 1972) p. [6] R.L. Elgin, J.M. Greif and D.L. Goodstein, Phys. Rev. Lett. 41(1978)1723.