The Structure of +He Ions in Cryogenic Helium Vapor

In this research we have studied the structure of +He ions in He vapor at temperatures between l.32K 4.22K and at saturation ratios between 0.05 ~ 1. Classical macroscopic thermo~namics predicts formation of a liquid drop around the ion, and the drop radius is gtven by the Thomson equation. In the above temperature and pressure ranges the radius of the drop varies between 6A 9A. An experimental verification of the Thomson equatton shows the validity of the macroscopic thermodynamics when it is applied to microscopic systems and also gives information about the drop structure. To show the existence of the drops and to determfne their sizes experimen· tally, we have measured the mobilities of +He ions in He vapor in the above temperature and pressure ranges. The mobility is related to the radius of the drop through the momentum transfer cross section. Hence the drop size can be determined from the mobility data if the interaction potential and the nature of coll is ions between the charged drop and the neutral vapor atom is known. We have assumed that the interaction potential is the sum of the polarization potential between the central ion and the neutral vapor atom, and the van der Waals interactions between the vapor atom and each of the liquid atoms in the drop'. With the above potential the "experimental" drop radius is calculated in the elastic and "inelastic" models. Quantum corrections are made for the elastic model. The Thomson equation predi"ctions were compared with the "experimental 11 radii and a good agreement was found. This comparison also showed the existence of a solid core within the ' liquid drop. The . classical macroscopic thermodynamics was applied successfully to calculate the solid core radius. Finally, the temperature dependencies of the "experimental" radii showed slight variations from the predictions of the Thomson equation at T<2.3K. The deviations reach their maximum at ~l.9K. The existence of superfluid transition in the liquid helium layer of the ion-solid-liquid complex is suggested as an explanation of the temperature dependencies of the "experimental" radii. The proposed transition temperature is ~l.9K and it is broadened up to ~2.3K. The transition starts when the liquid thickness becomes more

In the above temperature and pressure ranges the radius of the drop varies between 6A -9A. An experimental verification of the Thomson equatton shows the validity of the macroscopic thermodynamics when it is applied to microscopic systems and also gives information about the drop structure. To show the existence of the drops and to determfne their sizes experimen· tally, we have measured the mobilities of +He ions in He vapor in the above temperature and pressure ranges. The mobility is related to the radius of the drop through the momentum transfer cross section. Hence the drop size can be determined from the mobility data if the interaction potential and the nature of coll is ions between the charged drop and the neutral vapor atom is known. We have assumed that the interaction potential is the sum of the polarization potential between the central ion and the neutral vapor atom, and the van der Waals interactions between the vapor atom and each of the liquid atoms in the drop'. With the above potential the "experimental" drop radius is calculated in the elastic and "inelastic" models.
Quantum corrections are made for the elastic model. The Thomson equation predi"ctions were compared with the "experimental 11 radii and a good agreement was found. This comparison also showed the existence of a solid core within the ' liquid drop. The . classical macroscopic thermodynamics was applied successfully to calculate the solid core radius.
Finally, the temperature dependencies of the "experimental" radii showed slight variations from the predictions of the    Classical macroscopic thermodynamics is often used to explain both the homogeneous and the inhomogeneous nucleation (1)(2)(3) of microscopic clusters in gases It has also been applied + (4,5) to understand the structure of He ions in liquid helium In both cases the classical theory of nucleation is based on the f homson equation which was derived about a hundred years (6) ago by J.J. Thomson  where R 1 is the liquid drop radius, cr 1 v is the surface tension of the liquid-vapor interface, n 1 is the density of the bulk liquid, a. is the atomic polarizability and P t is the saturated sa vapor pressure at temperature T.
In the derivation one assumes an ideal vapor and an incompressible liquid with nearly unity dielectric constant and a sharp liquid-vapor interface with zero thickness.
There are several reasons to check the validity of the Thomson equation. The main criticism is directed to its application to microscopic systems. For the systems to which the Thomson equation is applied the drop sizes are of the order of angstroms.
The application of macroscopic thermodynamics to such small systems is questionable. Also, perhaps the drop structure is not as simple as it is assumed to be in the Thomson  such microscopic droplets will present a valuable opportunity to study the effect of finite size on the thermodynamical properties of the fluids.
The predictions of the Thomson equation have been tested experimentally for various systems. In these studies different experimental methods have been applied. The measurements which are done with a mass spectrometer are used to obtain the enthalpies and the entropies of successive clustering reactions for \ (7,8) water and ammonia about various positively charged ions Since helium is an inert gas this possibility is minimized in our system. Also, because of its unusual phase diagram the helium drop is expected to remain liquid even at absolute 4ero. Furthermore, bulk liquid helium makes a transition from superfluid to normal fluid as the temperature increases above 2.17K.
It is hard to believe that a superfluid transition will occur in such a microscopic liquid helium drop, and the Thomson equation predicts no effect of the transition on the drop radius even if it does~ Nonetheless, a liquid helium drop has the potential to give us a chance to understand if superfluidity exists in such a microscopic system. As a final practical point, the temperature range of this experiment, l.32K -4.22K, is a definite advantage in assuring the cleanliness of the experimental cell. For experiments operating at higher temperatures, undesired vapors and ions in the system are difficult to eleminate and their existence can create problems in analyzing the data.
While it does not provide the most direct information about the structure of the ion, we have nonetheless chosen to study experimentally the ion mobility. We have done so for two principal reasons: First, because the technique is simple and direct, and capable of quite high accuracy, and second, because the ion remains in thermal equilibrium with the vapor throughout the measurement, as assumed in the Thomson model. 5. The zero field mobility, µ ' of an ion in a gas is defined as; lim vd ( 2)  during the measurement. For example in a mass spectroscopy experiment the correction has to be made to compensate the effect of evaporation of the liquid atoms from the surface of the drop when the drops are placed in vacuum. (11)(12)(13)(14) In addition to our previous reports , there are + two more measurements of the mobilities of -He ions in He vapor which indicate droplet formation. The first is the a.c. mobility ( 15)  In our preliminary publications we have reported some samples (11)(12)(13)(14) of mobilities at temperatures 1 .32K -4.22K The existence of the drops were shown qualitatively even at low densities.
The development of our quantitative analysis can be found in our previous publications. The final form of this analysis will be discussed in section IV of this report. The "experimental" 7. radii obtained in each model were compared with the predictions of the Thomson equation. Good agreement was found over the entire pressure and temperature range of the experiment.
In this report we extend our calculations of 11 experimental 11 radii to include quantum effects. Together with the previous results, these calculations provide us with a deeper insight into the structure of the liquid helium drop. Here, we would like to report the details of the experiment and the results that we ~ave obtained. In the second section the details of the experimental set up and the procedure followed for measurement and the analysis of the raw data will be given. In section III a complete list of the mobilities that we have measured will be given. Their dependence on temperature and pressure will be displayed. The details of the calculations of the theoretical and the experimental radii will be discussed. The last section is reserved for the discussion of the results that we have obtained from the mobility data and from the comparison of the theoretical and the 11 experimental 11 radii. 8. The grid assembly is placed in a copper can which is filled with helium gas, and surrounded by a liquid helium bath. Schematic of the experimental cell. 10.

II. EXPERIMENT
The tritium-8 source is a tritiated titanium foil with activity 600 m Ci, maximum energy 18 KeV, active area l.2~x 2 cm 2 .
The 5.23 cm long source region is long compared to the track length so that ionization is produced only in the source region.
To obtain a constant electric field, Ed' the drift space is divided by 9 equally spaced electrodes with 3 cm inner diameter. Since the electric field in the collector region is the same as Ed, the field penetration due to the grid spacing on G3 is zero. All the electrodes, grids, and the collector are connected by lexan spacers. The total distance from G2 to C for which the time of flight is measured is 10.50 cm at helium temperatures (G2-G3 = 10.00 cm, G3-C = 0.50 cm). This length is calculated from room temperature measurements of the same length and the known expansion coefficient of the spacers. To prevent oxidation, the stainless steel electrodes, grids, collector and source support are gold plated. To assure -4 cleanliness the system is evacuated up to ~1 x 10 torr before l l. each run. A small amount of pure helium exchange gas is admitted into the can through a liquid nitrogen cooled charcoal trap at the beginning of the run. Otherwise during the experiment only the boil off from liquid helium in the bath is used to increase the can pressure.

A. Drift time measurements
The lo-10 -lo-11 A/cm 2 current pulses, received by the collector are amplified by a current amplifier in which an Analog Device 41J operational amplifier is used with 10 9 n feedback resistor, Re. The signal to noise ratio (~7) of the current amplifier is low enough so that the amplifier did not create any problem in finding a recognizable pulse, and after the signal averaging, the uncertainity due to amplifier noise was always negligible compared to the errors due to diffusion, space charge etc. The 0.36 msec time constant, <, of the amplifier is uncertain by about± 0.10 msec. To increase the accuracy a delayed trigger was used for the signal averager, and only the trailing edge of the amplifier output was averaged. The digital signal averager was operated so that its resolution was not a limiting factor. Its output was stored in a computer to be analyzed later. The drift time of the ions was found by the following procedure: The output voltage, Vvt), is related to the input current, i(t), for an idealized current amplifier by i(t) = l/Rf ( V(t) + < dV/dt ) where • is the feedback time constant.   The deconvoluted pulses with four different amplitudes. 14. -.
Four different pulse amplitudes were chosen for each drift field Ed to determine the effect of space charge on the transit time. In Fig. 2 the different values of the intercept , t 2 for each pulse amplitude show the magn i tude of this effect. We compensate for the space charge effect by the following argument: Each ion in the drift region experiences an additional electric field resulting from the other charges around it. This electric field is assumed to be proportional to the charge density n = j / ~eµEd), where j is the current in equilibrium andµ is the mobility . Hence the total electric field experienced by an ion at the trailing edge of the pulse is Ed-Kn, where K is the proportionality constant. For Kn <<Ed, one can find: If the above assumption and approximation are valid then a graph of EdTd vs. j /E~ is a straight line with intercept L/µ. In Fig.   3a, 3b, 3c we show three typical results that we obtained from our data. The first one corresponds to the small charge densities where Eq. 2 is valid. Fig. 3b shows the effect of higher order terms for high charge densities. Whenever there is an ambiguity due to lack of data at small charge densities, as shown in Fig. 3c, we took the value of EdTd at the mid-point of the intercept l and and the intercept 2 to calculate the mobility. For these cases we assign larger uncertainity to include both of the intercepts.     is provided by a mechanical pump and measured by a thermocouple -'to an accuracy of ~10 mtorr. In order to determine the pressure variation due to small leaks in the system, bath temperature drifts, drift· of the pressure gauge, etc, each can and bath pressure was measured both before and after the transit time measurements at this particular temperature and pressure. Fot the highest and the lowest can pressures the total uncertainities are 0.3% and 0.8% respectively.

17.
The temperature of the system is determined from the bath vapor pressure. The coarse temperature regulation is obtained by controlling the pumping speed on the bath. For fine temperature regulation an uncalibrated Ge resistor, in an a.c. bridge circuit, is epoxied on top of the can, and used with a ~ 100 n, ~4xlo-2 W heater in a feedback circuit. Below the superfluid transition temperature, the heater resistor is placed on top of the can.
Above this temperature, a chroma wire heater is wrapped around the can to maintain constant temperature through the experimental cell.
' Also a bellows manostat is used to control the bath pressure at higher temperatures. Above 3K only a needle valve was more practical to keep the bath pressure constant to within 1% accuracy. In this way we could keep the uncertainities in the temperature within 0.08% and 0.24% below and above the A-point temperature respectively.

C. Error analysis
Most of the uncertainty in the mobility is due to the time of flight measurements. It originates basically from two sources: up to 3.8% however.
Uncertainties in the pressure and the temperature are discussed in the previous section. Briefly, they are 0.3% -0.8% and 0.8% -0.24% in the pressure and in the temperature respectively.
In the above error analysis the systematic errors in the drift length arising from the uneven drift field, grid spacing etc. are not considered. These uncertainties were discussed in some detail (17)  The reduced mobility µr is defined as n n ref (4) where n f=2 . 69 x 10 19 cm-3 and n is the vapor density calculare (18) ted by the virial equation of state The reduced mobility is displayed as a function of saturation ratio for several temperat~res and as a function of temperature for several saturation ratios in Fig. 4 and Fig. 5 respectively.
The dependence of reduced mobility on the saturation ratio. Different symbols co-rr-espond to different temperatures.

Fig. 5
The temperature dependence of reduced mobility. The lines between the data points are guides to the eye. The numbers at the end of the lines show different saturation ratios. The dashed line shows the "boundary " betv1een the binary and multiple collision regimes. 26.

IV. THEORY AND CALCULATIONS
In the first part of this section we will explain the calculation of the theoretical drop radius predicted by the Thomson equation. Some modifications that we have made to this equation will be discussed. Also the possibility of formation of a solid core inside a liquid drop and its effect on the drop radius will be introduced. The theoretical determination of the solid core radius will be given.

'
In the second part of this section we will discuss the de- In this second part of this section, the determination of the 11 experimental 11 drop radius in two different scattering models and the quantum effects on these calculations will be discussed.
It should be emphasized that probably none of these model dependent 11 experimental 11 radii reflect the precise size of the drop, but nonetheless they are helpful in understanding some features ' of the drop structure.

A. Thomson equation's predictions
The theoretical drop radius, R 1 , is determined by the it is reasonable to argue that the liquid density increases over the bulk liquid density as the distance from the ion decreases. Also the surface tension of the liquid-vapor interface of the finite size drop is different from, probably (22) smaller than, the bulk surface tension Hence one can speculate that the(~vln1)ratio in Eq.l is smaller than its bulk value.
To understand how sensitive is the drop radius to decreasing the a 1 v;n 1 ratio we have calculated R 1 in Eq. 1 for crlv I n 1 = ' (8/10)(cr 1 v!n 1 )bulk" This choice of the a 1 v;n 1 ratio increased the drop radius for all temperatures and pressures by about (14) 5-6% , but the results did not change any of the conclusions that we will discuss in the following section. Therefore they will not be included in our discussion. The theoretical drop radius is then given by Eq. 1 with no modifications.
The pressure increase in the drop due to the polarization force should not only increase the liquid density but it should (4,5) also result in a solid core inside the liquid drop The and as 1 =0.l dyne/cm is the solid liquid surface tension In the above equation the solid is assumed to be incompressible with nearly unity dielectric constant. Equations l and 4 are used to calculate the liquid and the solid radii. The measured (20) values of bulk Pm(T) and ns(T) are used in these calculations The effects of the solid core on the drop size as liquid thickness, R 1 -Rs, decreases will be discussed in the following section.
B. Model dependent 11 experimental 11 radii As mentioned before the mobility is not directly related to the size of the drop but it is related to the momentum transfer cross section. In Chapman Enskog theory the relationship between the reduced mobility, µr' and a thermally averaged momentum (24) transfer cross section, ~{l,l), is given by 30.
where mr is the reduced mass, indistinguisable from the mass of a single helium atom in the present case. The above equation is valid only for elastic, binary collisions but in some cases (24,25) it can also be applied to inelastic collisions We used Eq. 5 to calculate the experimental cross sections n(l ,l) from exp the meas~red values of the mobility.
On the other hand, if the interaction potential between the charged drop and the neutral vapor atom is known for a given drop size, we can adopt a model describing the nature of scattering and calculate thermally averaged momentum transfer cross sections theoretically for any drop size. Then for each model an 11 experimental 11 drop radius for which n(l,l) = n(l,l) can be th exp found.
In the calculation of the theoretical cross sections we assumed that the interaction potential is the sum of the -C/r 4 polarization potential between the central charge on the vapor atom, and the van der Waals potential, -c 6 ;r 6 + c 12 ;r 12 , between the vapor atom and each of the neutral atoms in the liquid.
If the density of the liquid is assumed to be constant, integrating the latter over the volume of the drop gives 31. (7) where C=ae 2 /2 , A=4rrn 1 /3 , c 6 =1.139xl0 4 K-A 6 , c 12 =3.1778xl0 6 K-A 12 and R is the drop radius which is chosen as an adjustable para-(26) meter ( 1 ) To calculate the transport cross section, Q , from the above potential we have adopted two different scattering models.
1. 'Elastic scattering model: In this model we assumed that all the collisions are elastic. In this case the problem is well defined both in classical and in quantum mechanical approach.
In the classical case Q(l) is defined by (8) where b is the impact parameter, E is the relative energy of the incident atom. The scattering angle e is given in terms of b as where r , the distance of closest approach, is the outermost a root of l-b2/r~-V(ra)/E=O and V(r) is the interaction potential given by Eq. 7. The transport cross section Q(l) is calculated by numerical integration of Equations 8 and g.

32.
To be able to include this type of scattering in our model we assumed that every atom which "touches" the drop is absorbed by the drop and simultaneously another atom is emitted in a random direction but with .the same energy so as to maintain the thermal equilibrium. Furthermore we assumed that an atom "touches" the surface if its classical distance of closest approach is less that rm, the position of the minimum of the potential V(r). All the other atoms which do not "touch" \ the surface are assumed to be scattered elastically. We call this model the 11 inelastic 11 model. Only classical mechanics is used in these calculations. For elastically scattered atoms (ra>rm), q(l) are calculated from equations 8 and 9.
Because the scattering angle, e, is assumed to be random for atoms with ra~.rm• ~cose>=O. Hence 1-Cose=l is substituted into Eq.8 if ra<r .
-m As mentioned before to estimate the "experimental" radii within each model, the thermally averaged momentum transfer cross section, n(l~l), must be calculated. The relationship between n(l,l)and q(l) is defined as; where E is the relative energy of the incident atom. The transport cross sections calculated in each model are used in .
(l,l) Eq.11 to determine n for given drop radii, R. The value th of R for which n(l, 1 )= n(l, 1 ) is taken as the "experimental 11 exp th 34.
drop radius for each model. The uncertainty in the "experimental" radii due to the above calculations alone are 0.4%, 1% and o.4% for the elastic classical, the elastic quantum mechanical and the inelastic cases respectively. For the details of these calculations see Appendix C.
In Fig.6  The dependence of the "experimental" rad1i on saturation ratio. The solid lines through the data points are guides to the eye.

V. DISCUSSION
In this section we will first make a qualitative analysis of our data. This analysis is independent of any model adopted and it shows the existence of the drops. Then we will compare, quantitatively, the model dependent 11  transfer cross section depends on the pressure which means that the drops are indeed growing around the ions. It should be emphasized that this qualitative confirmation of the Thomson equation is independent of any scattering model adopted.
To be able to make quantitative comparisons, the model dependent "experimental" radii are determined by the model explained in the previous section. In Fig.7 and Fig.8  this experiment. The dependence of the 11 experimental" radii, Re , Ri and the theoretical radius R 1 on the temperature. The solia lines through the data points are guides to the eye. 41. The dependence of the "experimental" radii, R, R. and the theoretical radius R 1 , on the temperature. TMe s6lid lines through the data points are guides to the eye.

42.
The small difference between the theoretical and the 11 experimenta 1 11 radii shows interesting dependencies both on the size of the drop and the temperature. To be able to emphasize these points we define the "relative difference", o, between the theoretical and the "experimental" radii as; o · = (R · -Rn )/Rh. e,1 e,1 )I., 12) where subscripts have their usual meaning.
In ~ig.9 o is displayed as a function of the theoretical e radius, R , for all our measurements at temperatures higher than in it, macroscopic thermodynamics is still expected to be valid Hence the beginning of the change in o at R i~ 6.7 A in Fig.9 probably does not show the beginning of the failure of macros- (7) copic thermodynamics due to the decrease in the drop size. Also we do not think that this increase arises from the imperfections of our scattering models since both oe and oi show the same qualitative dependence on the theoretical radius at R£> 6.7 A.
We believe that the deviation of the "relative difference" from 44.   The dependence of the "relative difference" on the temperature shows an interesting behavior. In Fig.11  "Relative Differences", Se and bj , versus temperature.
The solid lines throu9h the data points are guides to the eye.

_,,
To be sure that the apparent temperature dependence of the 11 relative difference 11 reflects a physical reality, and is not an artifact of our m,ethod of determining o from the measured mobility, we must consider the possible sources or error in our calculations. One might think that the temperature dependence arises from the imperfections in our scattering models. To cross section, n , from the measured mobility. Fig.5 shows exp the estimated "boundary" between the binary and 11111tiple collisions. It is clear that this "boundary" does not pass through a constant temperature and definitly not through l.9K or 2.3K. Furthermore, as mentioned before, most of our data is in the binary collision regime. Hence this possibility 50.
is also ruled out in explaining the temperature dependence of o.
As a result we conclude that even though our method of determining oe and oi ts indirect and probably imperfect, the results displayed in Figures 11 and 12 have physical significance.
Before we draw any conclusion from the temperature dependence of o, it might be helpful to see the variation of the "relative difference" with the liquid thickness at low temperatures. In  Fig.12 shows that the deviation starts as soon as the liquid thickness becomes more than a monolayer (~2.2 A), and it increases with the increasing thickness of the liquid helium.
Because of the lack of data for Ri -Rs<2.5 A at low temperatures, the effect of the solid core can not be seen in this figure. But the effect is observed for some intermediate temperatures.
No difference is found between the high and the low temperature o's for R -R <2.5 A. Since the ion-solid-liquid complex is  (20) systems is known Qualitatively, ~l.9K for transition temperature and ~o.4K broadening above the transition temperature is consistent with this fact.

54.
One needs to have more information to be able to explain the details of the variation of the cross section with the temperature.
But it is not unreasonable to expect a change in the scattering process as the liquid layer passes through a transition from the normal fluid to superfluid as the temperature decreases. This We feel that with the present data any further comment on the temperature dependence of the momentum transfer cross section, ( l 'l) n , can not be more than a speculation. Perhaps more information is needed to give a satisfactory explanation of this point. 15. A. Dahm and T.M.Sanders Jr., J.Low Temp.Phys., _g_, 199 (1970).
27. T.G. Wang et al., Phys.Rev.Lett. 30, 485 (1973 When an ion is placed in a polarizable vapor, the vapor atoms will be polarized in the electric field of the ion.
Hence an attractive polarization force will be created between the ion and the neutral vapor atoms. The effect of the polarization potential will be an increase in the Gibb's free energy of the system. To reduce the free energy, or equivalently, to reduce the effect of the ion, a liquid drop will form around the ion. Above reasoning is valid also for an ion in a liquid. Therefore it is reasonable to expect a solid core within the liquid drop. But in the absence of the ion the liquification and the condensation are not favorable at pressures less than the saturated vapor pressure because the free energies of the liquid and the solid phases are more than the free energy of the gas phase at p<psat" Hence the total change, ~G, in the free energy, G, of the system due to the formation ~f the solid-liquid complex around the ion can be written as; where ~Ge is the decrease in the free energy due to the reduction in the electric field energy, and, ~Gs and ~G t are the increasesin G due to the condensation and the liquification respectively. At the thermal equilibrium ~G is minimized.
~Ge is the difference between the electric field energies of an ion with charge e and radius R 0 surrounded by the two concentric dielectric spheres of solid and liquid with radii rs and r t , and of the same ion in a vapor. The energy density, u, in the presence of a dielectric is given by; where D is the displacement vector with magnitude e/r 2 and

If the vapor is ideal and if vg>>vt ' the equations A -5 and
A -6 will give ; Ni =n£4/3n(r1 -r~) is the number of atoms which condense to form a homogeneous liquid drop with a solid core in it.
ni and ~£ are the density and the radius of the liquid drop and rs is'the radius of the solid core. Then; where cr is the surface tension of the liquid-vapor interface.

£V
The second term on the right hand side of the Eq.A -8 is the surface energy which can not be neglected for small drops with large surface/volume ratio. Similarly for the solid core inside the liquid; ( A -9) and (A-10) where cr is the surface tension of the solid-liquid interface s t and N =n 4/3nr 2 . In Eq.A -10 an incompressible solid with s s s density ns> >ng is assumed.

64.
The total free energy change, ~G, is the summation of the Equations A -4, A -8 and A -10. At thermal equilibrium both a~G/ a rt and a~G/ a rs must be zero. The first of these conditions gives the Thomson equation with the drop radius Ri ; (A-11) The second condition of the thermal equilibrium gives Eq. 5 of theman'-!script if p«p is satisfied. m (A-12) where R is the solid core radius at thermal equilibrium. To obtain a uniform field in the drift region, the equally spaced electrodes between G2 and G3 are seperated by lOKn precission resistors, R 1 s. The distance between the electrodes is l.OOcm which is twice as much as the distance between G3 and C.
To make the electric field in the collector region equal to the field in the region B, two resistances, R 1 s, each of which is lOKn, are connected parallel between G3 and C as shown in Fig.B-1.
The uniform electric field, Ed, is changed by changing Vd, which \ is the potential difference between G2 and C. Vs is the potential difference between S and G2. The field distribution in the source region can be arranged by the variable resistors Rl, R2, R3 and R4. Another variable resistor R5 is used to change the potential barrier between Gl and G2. During the actual run of the experiment the variable resistors Rl through R4 and R5 are preset to give the maximum ion current through the gate when the pulse is on. Vs and/or the amplitude of the Gl pulse from the pulse generator, P.G., are changed to obtain the different ion pulse amplitudes.

Electronic connections for the time of flight measurements
The schematic of the electronic connections to measure the time of flight of the ion pulses is shown in Fig.B-2  The two time intervals, t 1 and t 2 . 72.

Cryogenic system
The temperature range of this system is 1 .32K-4.22K.
To obtain and maintain such low temperatures require a special technique. Part of the details of this technique is discussed in the manuscript. In this section we will give a diagram of the cryogenic system and explain the procedure that we have followed to obtain the helium temperature (4.2K). The necessary vacuum and electronic connections to measure and to regulate ' the pressure and the temperature in the cryogenic system will be di . scussed in the following two sections.
The schematic of the cryogenic system is shown in Fig.B-4.
The copper can in which the grid assembly is placed is filled with helium gas at a desired pressure. The low temperatures  thermal contact between the nitrogen and helium dewars. Later when the system is at helium temperatures the exchange gas, nitrogen, freezes and the vacuum is achieved in the isolation vacuum again to maintain the thermal isolation of the helium bath. The helium dewar is also filled by the nitrogen gas up to l At. pressure to make the thermal contact between the isolation vacuum and the copper can. He gas can not be used for this purpose since it diffuses through the glass and enters in\o the isolation vacuum causing a poor thermal isolation of the helium bath at the helium temperatures. Finally, the nitrogen dewar is filled by the liquid nitrogen to cool down the whole system except the grid assembly up to 77K.
The grid assembly is cooled to nitrogen temperature by letting pure helium gas into the can through a charcoal trap which is surrounded by a liquid nitrogen cold trap. To reduce the temperature of the system further, down to 4.2K, the nitrogen exchange gas is pumped from the helium dewar, and the dewar is filled with the liquid helium.

Vacuum system
The schematic of the vacuum system is shown in Fig.B sion pump which is placed in the leak detector through valves A and B and 6, 7 for two or three days to achieve a pressure of ~1x10-4 torr. This pressure is measured by the ion gauge, IG.
As mentioned in the previous section, to cool down the grid assembly, pure helium gas is let into the can through the charcoal trap and the valves ll and A.
The vacuum system is mounted on a table which sits on vibration absorbers.

5.Temperature control system
The  Therefore, except a few cases, the temperature control system is only used to record the relative temperature on the chart recorder. 81.

CALCULATIONS OF MOMENTUM CROSS SECTIONS
To be able to calculate the momentum cross section, ~(l ,l), the interaction potential between the charged drop and the neutral vapor atom must be known. We assumed that this potential is the where A=4nn t /3, nt is the number density of the liquid, C=ae 2 /2, a is the atomic polarizability, and R is the radius of the drop.
In above equation first term represents the polarization interactions, second and third terms are the total of repulsive and attractive parts of van der Waals interactions respectively .
Above potential can be written in terms of reduced variables as· ' * * * p where -s is the minimum of interaction potential V(r) and r m * is the position of this minimum. P in above equations is the ratio of polarization term to the value of total potential at * r=r . Therefore for P =O Eq.C-2 gives the interaction potential m between the neutral drop and the vapor atom.

*
The value of P =O can also be used as an approximation for very large charged drops where the polarization term is small compared to van der * Waals interactions. Because this value of P might have more 83.
applications we have included it in our calculations even though it is not useful for us. Th 11 .
( 1 '1 ) erma y averaged momentum transfer cross section n is defined as 00 (1,1) 1 -5) where Eis the relative energy of incident atom and Q(l)(E) is the transport cross section. In terms of the reduced quantities; (1) n ' can be calculated from Eq.C-6. Q depends on both the nature and the mechanics of scattering between the drop and the incident atom. We have calculated the transport cross section both in elastic and in inelastic scattering models.
In elastic model the quantum effects are also included. * where ra=ra/R, the minimum approach, ra, is the outmost n*(l ,l) is calculated by numerical integration of three collision integrals given by Equations C-6, C-8 and C-9. The * * * * ranges and increments of parameters chosen for R ,P ,E and T are given in Table C-1. The estimated error in q*(l) and n*(l ,l) are within 0.05% and 0.1 % respectively. The results obtained for diffusion cross sections are listed in Table C For quantum mechanical case n*(l,l) can still be calculated by Eq.C-6. The main difference is the occurence of an extra parameter corresponding to the de Broglie wavelength.

85.
The reduced wavelength is defined as ( C-la) where mr is the reduced mass of drop-atom system.
t +l ( C-11 ) where ot is the phase shift for quantum number t . oi 's are calculated by solving the Schoredinger equation numerically.
Effective reduced potential *2 * * * * A v veff(r ) = v (r ) + - -12) has an horizontal inflection point at t =t a which corresponds to * * * * * Veff(r )=Ea. For t>ta Veff(r ) has no dip. Quantum effects * * become increasingly more important for E < Ea for which the wavelength of the incident atom becomes comparable to the characteristic length of the potential. We have made use of two different methods, namely JWKB and Noumerov methods, to calculate the phase * * shifts for E < Ea. In this range of energy there is a value of * * * * * * t , i c' for which Veff(r ) = Ec is a maximum, with aveff(r )nr =a. JWKB method is a semiclassical method which ignores the reflection and the transfer of the incident atom from and through the energy * * * * * barrier of Veff for E >Ee and E < Ec respectively. Hence this approximate method is not good for t near t . Noumerov method c gives almost exact phase shifts for all values of t . For each * incident energy, E , we have calculated the phase shifts around t c in Numerov method and rest of the phase shifts are calculated in JWKB method. The error in Q*(l) is estimated to be within 0.3%. For E*> E~ the values of Q*(l)•s, calculated by classical and the ~uantum mechanical approaches, are found to be the same within the error limits of each calculation. Therefore classical o*(l).s are used to calculate n*(l ,l) in this energy range.
The values of Q*(l),s are listed in Table C-3. * * * * The ranges and increments chosen for R , P , A and E are tabulated in Table C-1.

2.
Inelastic Model In this model we assumed that every atom which does 11 touch 11 the drop is absorbed and then reemitted randomly with the same energy to maintain the thermal equilibrium of the system. This corresponds to 1-Cose=l in Eq.C-8. Furthermore we assumed that an atom 11 touches 11 the drop if its closest distance of approach is smaller than the position of the minimum of the potential V(r).
The atoms which do not 11 touch 11 the drop are assumed to be scattered elastically. The method of calculation is the same as explained in elastic classical model. Calculated Q*(l) values are listed in Table C-4. 87.

Calculation of experimental radii
To be able to make use of the results of above calculations, *(1,1) namely Q 1 s; for each experimental temperature, the drop radius has changed from 6A to 15A by lA steps. The parameters * * * * Rexp' Pexp' ixp and Texp were calculated for each radius .
Successive interpolations of Q *(l, 1) over T*, p* ( and A* in quantum mechanical case) are used to calculate Q(l,l) as function of R for every experimental temperature. Comparison of these * ( 1 1 ) Q ' 1 s with experimental cross sections obtained directly from ' the mobilities gives the 11 experimental 11 drop radii.

88.
, TABLE C-1 The ranges and the increments chosen for the parameters * * * * * R , P , A , E and T in classical and quantum mechanical cal-. (1,1) culat1ons of n .