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Hall effect on the surface of ice
Volume 126, number 1
PHYSICSLETTERSA
14 December 1987
HALL EFFECT ON T H E SURFACE OF ICE J.M. CARANTI and M.A. LAMFRI Faculty of Mathematics, Astronomy and Physics, National University of C6rdoba, Laprida 854, 5000 C6rdoba, Argentina Received I I June 1987;accepted for publication 20 October 1987 Communicated by D. Bloch
The Hall effect on the surface of ice has been detected using an ac technique. The measurements were carried out in the range - 1.3 to -20 °C. The Hall mobility shows a double Arrhenius dependence with slopes of 2.5 eV at high temperature and 0.3 eV at lowertemperature beingthe transition between - 5 and - 8°C. The net carrier concentration varies three orders of magnitude in this temperature range.
1. Introduction There is ample evidence showing that the surface of ice has a different behavior from that of the bulk. In particular ice presents a surface electrical conductivity [ 1,2]. This fact suggests that the surface structure might not be simply the continuation of the bulk structure. Caranti and Illingworth [ 3 ] and later Caranti and R6 [ 4 ] measured this conductivity as a function of frequency for polycrystaliine and monocrystalline samples respectively. Their measurements showed a remarkable similarity between the surface of ice and several amorphous electronic semiconductors. These results stimulated the idea of an amorphous surface [ 5 ] of ice in a similar way as put forward by Fletcher [6,7] long time ago. In order to achieve a better understanding of the ice surface further work is needed. In this sense Hall effect measurements are ideal because of the possibility of knowing separately the conductivity components. Since the early and controversial work by Bullemer and co-workers [8] later criticized by Engelhardt [ 9 ] and not supported by further experimental results by Kern (unpublished) it appeared that the Hall effect in ice was not worth investigating anymore. The main objections to the existence of this effect in ice comes from the special way ice conducts electricity. The carriers are the point defects L - , D+, O H - and H~" O. But when their motions are ex-
amined more closely it is always a proton that moves under the applied electric field irrespective of the sign of the defect. However, this motion is not free, since protons 'move only along and between close molecular bonds. This channeled conduction cannot be influenced by a magnetic field. So, the Hall effect is not observable in bulk crystalline ice, in agreement with theoretical calculations by Gosar [ 10]. Nevertheless, as mentioned above, the surface may have an amorphous structure so the restrictions that operate for the bulk do not necessarily apply to the surface. The conduction mechanism can be of a different nature in this structure. This work describes Hall effect measurements on the surface of polycrystalline ice as a function of temperature. In order to compare this with previous works on conductivity the samples were rubbed and contaminated. It must be noted that at this stage we were only interested in the detection of the effect and no attempt was made in determining the sign of the carriers. The set up for Hall measurements is the following: a sample-holder with four electrodes on the periphery of the sample surface is placed between the poles of an electromagnet with the magnetic field perpendicular to the sample surface. Two opposite electrodes carry the driving current to the sample and the other two sense the Hall voltage. The Hall electric field results from
Eh=vxB,
(l ) 47
Volume 126, number 1
where v is the carrier velocity and B the applied magnetic field. The velocity is related to the driving current through
v=j/ne=Rhj,
Vh=Ehd,
(4)
and o =p.E=pEo sin(t02 t),
(5)
where/t is the mobility and E the driving field. Hence, Eh=~/zEoBo[cOs(09t + t o 2 ) / - c o s ( t o l - c o z ) t ] .
/z= -2 ( E - h ) ~ 2-( V h )
( E) Bo
( V) Bo
(7)
and 2(Eh )
2( Vh )
(8)
Rh-- (j)Bo ~ (i)Bo The brackets express effective value and i is the driving current. The last members arise from the assumption of same length and width of the sample, which is true within 5% for the present geometry, and uniformity in all fields.
(3)
where d is the separation of the electrodes. Actually there are corrections to be made [ 11 ], but they are within a few percents. During this work a complete set up was designed and constructed to measure with a large dc electromagnetic and ac driving current (to avoid polarization effects). The first problem arises from the fact that it is impossible to align the sensitive electrodes on an equipotential of the driving field. So, in absence of the magnetic field there will be an unwanted resistive potential difference across them. A buck-out system was built to cancel the resistive drop with great accuracy. But, because the ice sample evolved very quickly it proved to be extremely difficult to maintain the balance for long periods, making impracticable any reliable measurements. Thus, the dc, ac arrangement was abandoned in favor of an ac field, ac driving current method. In this case it is assumed that
B=Bo sin( co~t)
the effective values of the Hall field (or Hall voltage) and driving field (or driving voltage):
(2)
where j is the current density, n the carrier density, e is the elementary charge and R h is the so called Hall coefficient. It is supposed at this stage that there is only one kind of carrier. Because b o t h j and B in general are nonuniform, the Hall electric field is not uniform. However, as a first approximation it will be considered that the field Eh is uniform, thus the potential difference at the sense electrodes will be
(6)
The system will select the sum of frequencies because, as it will become apparent later, the difference can be close to one of the driving frequencies. It is convenient to express the mobility and Hall coefficient in terms of the peak magnetic field Bo and 48
14 December 1987
PHYSICS LETTERS A
2. Apparatus It has been suggested that for high resistivity samples like ice it is not advisable to use ac because parasitic capacitances introduce inaccuracies [ 12 ]. But, as mentioned before, we must use an ac technique. So, a method had to be devised in order to overcome these parasitic capacitances. Guarded coaxial cables were first tried, driving the inner shield with a voltage follower but this proved to be unstable and oscillations were unavoidable. The negative input capacitance amplifier was a different method that worked well in ac. The sample can be replaced by a Thevenin equivalent with the Hall voltage as the source and a large resistance in series. The parasitic capacitance such as that of the coaxial cable is connected from this resistance to ground and the voltage drop on this capacitance is the measured Hall voltage. This drop may be a small fraction of the source one. But if we measure the potential difference with an amplifier presenting a negative adjustable capacitance we can restore the full Hall voltage. The circuit used (as in ref. [ 13 ], with modifications) can be adjusted to achieve zero total capacitance. Each electrode is connected to one of these input circuits and the potential difference between them is obtained using an instrumentation amplifier with unity gain. As mentioned above we are interested in selecting the signals with the sum of frequencies and rejecting the others, therefore a set of filters was included in the circuit. The electromagnet derived its
Volume 126, number 1
PHYSICS LETTERS A
current from the 50 Hz mains and the driving current was obtained from a sinusoidal wave generator at 107 Hz. Our main concern was to eliminate the 107 Hz component which was present as a resistive potential drop between the electrodes and was orders of magnitude larger than the Hall signal (157 Hz) thus avoiding saturation of the measurement stages. The filtered signal was then applied to a lock-in amplifier with a reference generated by an analog multiplier operating on a 50 Hz signal from a transformer connected to the mains and the same driving voltage applied to the sample. The lock-in amplifier was also used to measure the driving current through the sample. This was accomplished by taking the voltage drop on a resistor in series with the sample, using the driving voltage as reference. The electromagnet was specially built in laminated iron with a square section of 6.4 cm X 6.4 cm and 3.5 cm of gap. The peak magnetic field was 0.38 T, with a uniformity of 1% within the sample area. The sample-holder was built in perspex. The electrodes were pushed by springs into contact with the sample. The electrode assembly was fixed on a perspex extension of the box containing the front end electronics. This rigid mounting avoid long coaxial cables and changes in the geometry. The electronics was in turn shielded from the intense magnetic field. To maintain a constant temperature the sample was placed in a perspex reservoir between the poles of the electromagnet and connected with a liquid circulator kept at constant temperature containing ethyleneglycol. The temperature range was - 1.3 to -20°C.
3. Results and discussion
Several preliminary runs were carded out in which the various parts of the measuring protocol and the amplifier sensitivities were tested. In particular the way measurement of the current through the sample was changed as a consequence of these runs. A differential method was originally used. The currents with and without connection to the sample were taken and the difference was interpreted as the current through the sample. This method gave large er-
14 December 1987
Vm
-7.0
.v
4'm
~V
~V VV
V
~.0 o
~.0 3.6
3.7
3.0
a.a
4.0
1000IT
Fig. 1. Log of the ac surface conductivity a (f~-~) of a normal and rubbed polycrystalline sample at 107 Hz as function of ! O00/T
(K-l).
rors because it was the result of a difference between large currents. A better method was obtained by taking into account that the current we are interested in is only resistive as has been shown in previous works on ice surface [ 4 ]. Therefore, using the internal phase shifter the lock-in amplifier was adjusted to measure only the resistive component. Fig. l shows the ac surface conductivity of a polycrystalline sample at 107 Hz as function of 1000/T. The graph is not a perfect straight line but rather an S-shape with an average slope of 0.78 eV. These values of conductivity and activation energy are similar to those reported previously for slightly impure polycrystalline samples [ 3 ]. Fig. 2 shows the Hall mobility for the same sample. The slopes are 2.5 eV and 0.3 eV at high and low temperatures respectively. When th~ surface was rubbed with a clean sharp blade, a great reduction in mobility was observed from two to five times depending on temperature, as may be seen in the same figure. Fig. 3 shows the calculated concentration using (8) and assuming a single kind of carders. Comparing with the concentration values predicted from Fletcher's model [ 6 ], which are in the range 1 0 1 6 - 1 0 Is m -3, we can interpret from the re49
Volume 126, number 1
PHYSICS LETTERS A
V IIUI~
-3.0
,t I~qt~t.
...-4"
4"
V 4,
~
VVV
4"
"4.0
o
4,
V
S
V
V
"-5.0 3.6
3.7
3.e
3.0
4~
IO00/T
Fig. 2. Log of the Hall mobility/t ( m 2 V - *s - ~) at 157 Hz for the sample in fig. I as a function of 1000/T ( K - ' ) . The Bo field was 0.38 T and o9, was 50 Hz.
suits that only a fraction of the carriers are in movement at a given instant. For example at - 10°C one in one thousand carriers contribute to the conduction. The measured mobilities are several orders of magnitude larger than the corresponding drift mo-
v BtlIBBED
17.0
14 December 1987
bility of the bulk. The latter is around 5 × 10- s m 2 / V s at - 2 0 ° C [ 14]. As we lack any measurement of the drift mobility for the surface carriers, comparisons are impossible. It is our intention to complement the present measurements with those of surface drift mobility. One interesting behavior of the observed mobility is the change in slope (activation energy) at a definite temperature suggesting that there might be intrinsic and extrinsic mechanisms driving the transport at each range. Work on doped ice is in progress at this moment and results will be published elsewhere. Kunst and Warman [ 15 ] found a similar double slope for bulk drift mobility, but again, the large difference in magnitude should be noted. Another way of interpreting the results is to allow two kinds of carriers of opposite signs. The second one appears at high temperature and produces the apparent lowering in mobility. But its concentration is not enough to become the dominant carrier. We do not know if the point defects recognized for the bulk are also operating on the surface but we expect some comprehension from the doped ice experiment. The uniform decrease in fig. 2 when ice is rubbed suggests that the drag forces on the carriers have increased in a factor approximately independent of temperature, although they leave the charge generation mechanism invariable
~,, MOlIM/d.
V V
References
16.0 4' 4' 4'
'-
15.0
4"
V
o
V
v v v 4"e,,
14.0
4"4" 4,
13.0 3.6
3.7
3.8
3.o
4.o
IO00/T
Fig. 3. Log of the calculated concentration n (m -3) assuming only one kind of carrier.
50
[ 1 ] C. Jaccard, Physics of snow and ice (1967) p. 173 [ 2 ] N. Maeno, in: Physics and chemistry of ice, eds. E. Whalley, S.J. Jones and L.W. Gold (Royal Society o f Canada, Ottawa, 1973) 13. 140. [3] J. Caranti and A. Illingworth, J. Phys. Chem. 87 (1983) 4078. [4] J. Caranti and M. R6, in: Proc. of 13 Reunion Cientifica de Geografia y Geodesia, eds. M. Ornstein and M.L. Altinger (Asociacion Argentina de Geofisicos y Geodestas, Buenos Aires, 1985). [5l I. Goleeki and C. Jaccard, Phys. Lett. A 63 (1977) 374. [6] N. Fletcher, Philos. Mag. 18 (1968) 1287. [ 7 ] N. Fletcher, in: Physics and chemistry of ice, eds. E. Whalley, S.J. Jones and L.W. Gold (Royal Society o f Canada, Ottawa, 1973) p. 132. [8] B. Bullemer and N. Riehl, Phys. Lett. 22 (1966) 411.
Volume 126, number 1
PHYSICS LETTERS A
[9] H. Engelhardt, in: Physics and chemistry of ice, eds. E. Whalley, S.J. Jones and L.W. Gold (Royal Society of Canada, Otawa, 1973) p. 226. [ 10] P. Gosar, Phys. Condens. Matter 17 (1974) 183. [ l l ] W. Flanagan, P. Flin and B. Averbach, Rev. Sci. Intrum. 25 0 9 5 4 ) 593.
14 December 1987
[ 12] D. Marioli and V. Varioli, Rev. Sci. Instrum. 16 (1983) 892. [ 13] J. Mattauch, Rev. Sci. Instrum. 41 (1970) 592. [ 14] V.F. Petrenko and I.A. Ryzhkin, Sov. Phys. JETP 60 (1984) 320. [ 15 ] M. Kunst and J. Warman, J. Phys. Chem. 87 ( 1983 ) 4093.
51
PHYSICSLETTERSA
14 December 1987
HALL EFFECT ON T H E SURFACE OF ICE J.M. CARANTI and M.A. LAMFRI Faculty of Mathematics, Astronomy and Physics, National University of C6rdoba, Laprida 854, 5000 C6rdoba, Argentina Received I I June 1987;accepted for publication 20 October 1987 Communicated by D. Bloch
The Hall effect on the surface of ice has been detected using an ac technique. The measurements were carried out in the range - 1.3 to -20 °C. The Hall mobility shows a double Arrhenius dependence with slopes of 2.5 eV at high temperature and 0.3 eV at lowertemperature beingthe transition between - 5 and - 8°C. The net carrier concentration varies three orders of magnitude in this temperature range.
1. Introduction There is ample evidence showing that the surface of ice has a different behavior from that of the bulk. In particular ice presents a surface electrical conductivity [ 1,2]. This fact suggests that the surface structure might not be simply the continuation of the bulk structure. Caranti and Illingworth [ 3 ] and later Caranti and R6 [ 4 ] measured this conductivity as a function of frequency for polycrystaliine and monocrystalline samples respectively. Their measurements showed a remarkable similarity between the surface of ice and several amorphous electronic semiconductors. These results stimulated the idea of an amorphous surface [ 5 ] of ice in a similar way as put forward by Fletcher [6,7] long time ago. In order to achieve a better understanding of the ice surface further work is needed. In this sense Hall effect measurements are ideal because of the possibility of knowing separately the conductivity components. Since the early and controversial work by Bullemer and co-workers [8] later criticized by Engelhardt [ 9 ] and not supported by further experimental results by Kern (unpublished) it appeared that the Hall effect in ice was not worth investigating anymore. The main objections to the existence of this effect in ice comes from the special way ice conducts electricity. The carriers are the point defects L - , D+, O H - and H~" O. But when their motions are ex-
amined more closely it is always a proton that moves under the applied electric field irrespective of the sign of the defect. However, this motion is not free, since protons 'move only along and between close molecular bonds. This channeled conduction cannot be influenced by a magnetic field. So, the Hall effect is not observable in bulk crystalline ice, in agreement with theoretical calculations by Gosar [ 10]. Nevertheless, as mentioned above, the surface may have an amorphous structure so the restrictions that operate for the bulk do not necessarily apply to the surface. The conduction mechanism can be of a different nature in this structure. This work describes Hall effect measurements on the surface of polycrystalline ice as a function of temperature. In order to compare this with previous works on conductivity the samples were rubbed and contaminated. It must be noted that at this stage we were only interested in the detection of the effect and no attempt was made in determining the sign of the carriers. The set up for Hall measurements is the following: a sample-holder with four electrodes on the periphery of the sample surface is placed between the poles of an electromagnet with the magnetic field perpendicular to the sample surface. Two opposite electrodes carry the driving current to the sample and the other two sense the Hall voltage. The Hall electric field results from
Eh=vxB,
(l ) 47
Volume 126, number 1
where v is the carrier velocity and B the applied magnetic field. The velocity is related to the driving current through
v=j/ne=Rhj,
Vh=Ehd,
(4)
and o =p.E=pEo sin(t02 t),
(5)
where/t is the mobility and E the driving field. Hence, Eh=~/zEoBo[cOs(09t + t o 2 ) / - c o s ( t o l - c o z ) t ] .
/z= -2 ( E - h ) ~ 2-( V h )
( E) Bo
( V) Bo
(7)
and 2(Eh )
2( Vh )
(8)
Rh-- (j)Bo ~ (i)Bo The brackets express effective value and i is the driving current. The last members arise from the assumption of same length and width of the sample, which is true within 5% for the present geometry, and uniformity in all fields.
(3)
where d is the separation of the electrodes. Actually there are corrections to be made [ 11 ], but they are within a few percents. During this work a complete set up was designed and constructed to measure with a large dc electromagnetic and ac driving current (to avoid polarization effects). The first problem arises from the fact that it is impossible to align the sensitive electrodes on an equipotential of the driving field. So, in absence of the magnetic field there will be an unwanted resistive potential difference across them. A buck-out system was built to cancel the resistive drop with great accuracy. But, because the ice sample evolved very quickly it proved to be extremely difficult to maintain the balance for long periods, making impracticable any reliable measurements. Thus, the dc, ac arrangement was abandoned in favor of an ac field, ac driving current method. In this case it is assumed that
B=Bo sin( co~t)
the effective values of the Hall field (or Hall voltage) and driving field (or driving voltage):
(2)
where j is the current density, n the carrier density, e is the elementary charge and R h is the so called Hall coefficient. It is supposed at this stage that there is only one kind of carrier. Because b o t h j and B in general are nonuniform, the Hall electric field is not uniform. However, as a first approximation it will be considered that the field Eh is uniform, thus the potential difference at the sense electrodes will be
(6)
The system will select the sum of frequencies because, as it will become apparent later, the difference can be close to one of the driving frequencies. It is convenient to express the mobility and Hall coefficient in terms of the peak magnetic field Bo and 48
14 December 1987
PHYSICS LETTERS A
2. Apparatus It has been suggested that for high resistivity samples like ice it is not advisable to use ac because parasitic capacitances introduce inaccuracies [ 12 ]. But, as mentioned before, we must use an ac technique. So, a method had to be devised in order to overcome these parasitic capacitances. Guarded coaxial cables were first tried, driving the inner shield with a voltage follower but this proved to be unstable and oscillations were unavoidable. The negative input capacitance amplifier was a different method that worked well in ac. The sample can be replaced by a Thevenin equivalent with the Hall voltage as the source and a large resistance in series. The parasitic capacitance such as that of the coaxial cable is connected from this resistance to ground and the voltage drop on this capacitance is the measured Hall voltage. This drop may be a small fraction of the source one. But if we measure the potential difference with an amplifier presenting a negative adjustable capacitance we can restore the full Hall voltage. The circuit used (as in ref. [ 13 ], with modifications) can be adjusted to achieve zero total capacitance. Each electrode is connected to one of these input circuits and the potential difference between them is obtained using an instrumentation amplifier with unity gain. As mentioned above we are interested in selecting the signals with the sum of frequencies and rejecting the others, therefore a set of filters was included in the circuit. The electromagnet derived its
Volume 126, number 1
PHYSICS LETTERS A
current from the 50 Hz mains and the driving current was obtained from a sinusoidal wave generator at 107 Hz. Our main concern was to eliminate the 107 Hz component which was present as a resistive potential drop between the electrodes and was orders of magnitude larger than the Hall signal (157 Hz) thus avoiding saturation of the measurement stages. The filtered signal was then applied to a lock-in amplifier with a reference generated by an analog multiplier operating on a 50 Hz signal from a transformer connected to the mains and the same driving voltage applied to the sample. The lock-in amplifier was also used to measure the driving current through the sample. This was accomplished by taking the voltage drop on a resistor in series with the sample, using the driving voltage as reference. The electromagnet was specially built in laminated iron with a square section of 6.4 cm X 6.4 cm and 3.5 cm of gap. The peak magnetic field was 0.38 T, with a uniformity of 1% within the sample area. The sample-holder was built in perspex. The electrodes were pushed by springs into contact with the sample. The electrode assembly was fixed on a perspex extension of the box containing the front end electronics. This rigid mounting avoid long coaxial cables and changes in the geometry. The electronics was in turn shielded from the intense magnetic field. To maintain a constant temperature the sample was placed in a perspex reservoir between the poles of the electromagnet and connected with a liquid circulator kept at constant temperature containing ethyleneglycol. The temperature range was - 1.3 to -20°C.
3. Results and discussion
Several preliminary runs were carded out in which the various parts of the measuring protocol and the amplifier sensitivities were tested. In particular the way measurement of the current through the sample was changed as a consequence of these runs. A differential method was originally used. The currents with and without connection to the sample were taken and the difference was interpreted as the current through the sample. This method gave large er-
14 December 1987
Vm
-7.0
.v
4'm
~V
~V VV
V
~.0 o
~.0 3.6
3.7
3.0
a.a
4.0
1000IT
Fig. 1. Log of the ac surface conductivity a (f~-~) of a normal and rubbed polycrystalline sample at 107 Hz as function of ! O00/T
(K-l).
rors because it was the result of a difference between large currents. A better method was obtained by taking into account that the current we are interested in is only resistive as has been shown in previous works on ice surface [ 4 ]. Therefore, using the internal phase shifter the lock-in amplifier was adjusted to measure only the resistive component. Fig. l shows the ac surface conductivity of a polycrystalline sample at 107 Hz as function of 1000/T. The graph is not a perfect straight line but rather an S-shape with an average slope of 0.78 eV. These values of conductivity and activation energy are similar to those reported previously for slightly impure polycrystalline samples [ 3 ]. Fig. 2 shows the Hall mobility for the same sample. The slopes are 2.5 eV and 0.3 eV at high and low temperatures respectively. When th~ surface was rubbed with a clean sharp blade, a great reduction in mobility was observed from two to five times depending on temperature, as may be seen in the same figure. Fig. 3 shows the calculated concentration using (8) and assuming a single kind of carders. Comparing with the concentration values predicted from Fletcher's model [ 6 ], which are in the range 1 0 1 6 - 1 0 Is m -3, we can interpret from the re49
Volume 126, number 1
PHYSICS LETTERS A
V IIUI~
-3.0
,t I~qt~t.
...-4"
4"
V 4,
~
VVV
4"
"4.0
o
4,
V
S
V
V
"-5.0 3.6
3.7
3.e
3.0
4~
IO00/T
Fig. 2. Log of the Hall mobility/t ( m 2 V - *s - ~) at 157 Hz for the sample in fig. I as a function of 1000/T ( K - ' ) . The Bo field was 0.38 T and o9, was 50 Hz.
suits that only a fraction of the carriers are in movement at a given instant. For example at - 10°C one in one thousand carriers contribute to the conduction. The measured mobilities are several orders of magnitude larger than the corresponding drift mo-
v BtlIBBED
17.0
14 December 1987
bility of the bulk. The latter is around 5 × 10- s m 2 / V s at - 2 0 ° C [ 14]. As we lack any measurement of the drift mobility for the surface carriers, comparisons are impossible. It is our intention to complement the present measurements with those of surface drift mobility. One interesting behavior of the observed mobility is the change in slope (activation energy) at a definite temperature suggesting that there might be intrinsic and extrinsic mechanisms driving the transport at each range. Work on doped ice is in progress at this moment and results will be published elsewhere. Kunst and Warman [ 15 ] found a similar double slope for bulk drift mobility, but again, the large difference in magnitude should be noted. Another way of interpreting the results is to allow two kinds of carriers of opposite signs. The second one appears at high temperature and produces the apparent lowering in mobility. But its concentration is not enough to become the dominant carrier. We do not know if the point defects recognized for the bulk are also operating on the surface but we expect some comprehension from the doped ice experiment. The uniform decrease in fig. 2 when ice is rubbed suggests that the drag forces on the carriers have increased in a factor approximately independent of temperature, although they leave the charge generation mechanism invariable
~,, MOlIM/d.
V V
References
16.0 4' 4' 4'
'-
15.0
4"
V
o
V
v v v 4"e,,
14.0
4"4" 4,
13.0 3.6
3.7
3.8
3.o
4.o
IO00/T
Fig. 3. Log of the calculated concentration n (m -3) assuming only one kind of carrier.
50
[ 1 ] C. Jaccard, Physics of snow and ice (1967) p. 173 [ 2 ] N. Maeno, in: Physics and chemistry of ice, eds. E. Whalley, S.J. Jones and L.W. Gold (Royal Society o f Canada, Ottawa, 1973) 13. 140. [3] J. Caranti and A. Illingworth, J. Phys. Chem. 87 (1983) 4078. [4] J. Caranti and M. R6, in: Proc. of 13 Reunion Cientifica de Geografia y Geodesia, eds. M. Ornstein and M.L. Altinger (Asociacion Argentina de Geofisicos y Geodestas, Buenos Aires, 1985). [5l I. Goleeki and C. Jaccard, Phys. Lett. A 63 (1977) 374. [6] N. Fletcher, Philos. Mag. 18 (1968) 1287. [ 7 ] N. Fletcher, in: Physics and chemistry of ice, eds. E. Whalley, S.J. Jones and L.W. Gold (Royal Society o f Canada, Ottawa, 1973) p. 132. [8] B. Bullemer and N. Riehl, Phys. Lett. 22 (1966) 411.
Volume 126, number 1
PHYSICS LETTERS A
[9] H. Engelhardt, in: Physics and chemistry of ice, eds. E. Whalley, S.J. Jones and L.W. Gold (Royal Society of Canada, Otawa, 1973) p. 226. [ 10] P. Gosar, Phys. Condens. Matter 17 (1974) 183. [ l l ] W. Flanagan, P. Flin and B. Averbach, Rev. Sci. Intrum. 25 0 9 5 4 ) 593.
14 December 1987
[ 12] D. Marioli and V. Varioli, Rev. Sci. Instrum. 16 (1983) 892. [ 13] J. Mattauch, Rev. Sci. Instrum. 41 (1970) 592. [ 14] V.F. Petrenko and I.A. Ryzhkin, Sov. Phys. JETP 60 (1984) 320. [ 15 ] M. Kunst and J. Warman, J. Phys. Chem. 87 ( 1983 ) 4093.
51