PH YSI CAL REVIEW VOL UME 99, NUMBER 4 AUGUST 15, 1955 Velocity Distributions in Potassium and Thallium Atomic Beams* R.. MILLERI AND P. KUSCH Colambia University, New York, Newo York (Received February 23, 1955) A high-resolution, high-intensity, sp en designed for the study of the velocity It has been found possible to design oven slits which closely approximate kinetic theory. An analysis has been made of the velocity distributions in beams er a range of velocity from 0.3 to 2.5 times the most probable velocity in the ween the observed distribution and that deduced on the basis of the Maxwellian and that the aperture is ideal, INTRODUCTION n in the oven and with velocity analysis of NUMBER of investigators have measured the atoms at equilibrium in an isothermal enclosure. VELOCITY SELECTOR velocity selector consists of a solid cylinder on ber of helical slots tion about the cylinder lector has inherent advantages of the A the Air 1954. Now at the Bell Hill, New Jersey. 1 J. Eldridge, Phys. on,may and con- Estermann, Simpson, and Ste sity of the 7I. Kofsky and H. Le be studied to be stuc . Bennett, Jr., and I. Es Institute of Technology 1953(un considerably greater than noise. The present selector Phys.rev.95,608(a)(1954) is similar to a high tr on, slow neutron velocity 1314
VELOCITY DISTRIBUTIONS IN K AND TI BEAMS 1315 lector. a similar helical velocity selector has also where v lies between the limits vmax and vmin. Over the been used in a study of the characteristics of thin limited range of o allowed in the present case, the admit metal films as a function of the incident velocity of the tance is very nearly triangular in terms of v. The resolu- deposited atoms. tion calculated in the usual way at A=0.5 is Av/vo=y The notation used in a discussion of the selector is If it is assumed the velocity distribution incident on shown in Fig. 1 as are the dimensions of the rotor used the rotor, Iodm, does not change significantly over the in the present work. The cylinder was made of Dural range of velocities admitted for a given a, then the because this material allowed the cutting of slots to the transmitted intensity is necessary depth and at close intervals without signi ficant distortion of adjacent slots. The ends of the slots I,=-Iovo In(1-y2EIovor occur at 0.5 intervals on the end faces. To allow ad- to powers of y through ?. Iodv is the modified Max- justment of the line connecting the source slit and the wellian distribution detector parallel to the cylinder axis, it is necessary to have at least one slot parallel to the axis. Actually, two such slots were cut to avoid dynamical imbalance where a=m/2kT, and the distribution has been nor- The resolution properties of the velocity selector will malized to unity. Then Eq. (4)becomes be derived under the assumption that both source and detector have infinitesmal dimensions. All molecules I,=2yaD exp(-a2vo?) which are ultimately detected then travel in a plane Equation(5)becomes exact as the slot width becomes ontaining the cylinder axis, oven slit, and detector. infinitesimal. It will be shown that this expression is The validity of this assumption will be discussed in sufficiently close to the much more cumbersome exact more detail later. A molecule of velocity vo which goes expression so that it can be used as the theoretical through the slot without changing its position relative distribution to the sides of the slot will satisfy the equation To obtain an exact expression for I the variation of (1)the incident distribution, Io over the admittance must in which o and L are fixed, and w is the angular velocity. I=(Y1-1)exp[-x2/(1-)2] Since the slots have a finite width, a range of veloci transmitted by the rotor for a fixed w. The limiting 21exp(-30)+x:01 velocities correspond to molecules which enter the slot at one wall and leave the slot at the opposite wall, dx max=20(1-y)-1 and vmi=0(1+y)2,(2) erey=l/(ropo)and ro is the mean radius. In the present apparatus, y=0.05033, a fixed property. A variation of the radius by a few percent will have a where avo=xo. This expression is unwieldy and gives no negligible effect on the limiting values of D and thus, insight into the general nature of Io. Table I contains the mean value, ro, may be used. The admittance A, TABLE I, A comparison of the intensity distribution after analysis the ratio of the effective slot aperture for the velocity y the rotor as calculated in several approximations. v to that for the velocity vo is A=1-|(0/)-1l/, Exact hs0. 318 cm FIG. 1. Schematic diagram of velocity selector 38909850 切=都 25.40cm
1316 R, C. MILLER AND P. KUSCH hamber and the other two chambers are the slits shown in fig. 2 A large liquid nitrogen cooled surface is provided in each chamber, to increase the pumping speed of the cold traps for condensable gases. Pressures of 1. 2X10-7 and 1.3X10-7 mm of mercury were obtainable in the detector and rotor chambers respectively, with the rotor at rest The velocity selector was provided with a four-to-one tepup gear built into the mount. Power was trans- mitted to the rotor from the outside of the vacuum envelope by a shaft which rotates in a long, lubricated phosphor bronze bearing. As shown in Fig.2,two holes were drilled perpendicular to the axis of the bearing, one on the high pressure side so the lubricant FIG. 2. Schematic diagram of the apparatus designed could be replenished and one in the middle which was pumped by a mechanical pump. The vacuum seal some values of I, calculated from Eqs. (5)and(6) functioned very well, though there was some tendency Both sets of values have been multiplied by a constant The velocity selector was seldom run over 4000 rpm Eq.(5)equal to 20.00. Table I demonstrates that the at which speed the vacuum in the rotor chamber simple expression, Eg.(5), can be used for calculating creased to about 10-6 mm of mercury. theoretical distributions to be compared to the experi OVENS mental data in the present work, as the average experi mental uncertainty is about one percent of Iu(max) The ovens used in these experiments were much like The intensity at the detector, Io, may be expanded as conventional molecular beam ovens, the main differ greater than or equal to two. Then the leading terms are Since the interpretation of velocity distribution data requires that the beam produced at a wn equilib- I=2yxo4exp(-x02)[1+2 rium temperature, the ovens must be made of high con 137xo /6+730 / 3+...](7) ductivity material to avoid significant temperature gradients. Thus the present ovens were constructed than 2 ae he x terta in values af te catenated rgen of oxygen-free, high-conductivity copper, instead of the customary iron or nickel. Preliminary experiments Finite source and detect in the present work have shown that in the beam not all of whose elements are parallel to the axis hood of 900K, temperature differences of 30Coccur of the rotor. The effect of the finite vertical dimension oven. This temperature difference is roughly six times of the beam is negligible since for an angle of elevation, the estimated accuracy of the measured temperatures B, of a beam element with respect to the axis, the When the same measurements were repeated with a analysis of v cose is made and coso differs only trivially copper oven, the temperature difference was reduced from 1. The center of the admittance curve for non- to 3.5C parallel beam elements which results from the finite Kinetic theory considerations indicate that the horizontal dimensions, or widths, of the oven slit and width of the slit in a direction parallel to the direction detector wire is at either higher or lower velocities of the propagation of the beam must be much smaller than the center of the curve for parallel beam elements. than the mean free paths of the atoms in order to The solution of this problem has been discussed else- eliminate scattering in the neighborhood of the slit where. In the present apparatus, the detector and The defining slits were made of 0.0038 or 0.0025cm source widths are sufficiently small, 2.5X10- cm, so thick steel strips held on the face of the oven with copper hat the widths have a negligible effect on the shape of strips, whose edge extended to within about 0.25 mm the velocity distributio of the slit itself so that the orifice was determined b GENERAL DESCRIPTION OF APPARA the thin steel strips. At a nominal temperature of 900K, a thermocouple inserted in one of the copper strips The vacuum envelope consists of the oven, rotor and indicated a temperature 0.7 C lower than the tenlfthe detector chambers. Other than the pumping connec- ture measured with a thermocouple mounted tions, through which a free flow of gas between chambers front part of the oven as shown in Fig 3. This tempera- cannot occur, the only openings between the rotor ture difference is small compared to the estimated
VELOCITY DISTRIBUTIONS IN K AND TI BEAMS 1317 + 0.5 percent accuracy of the temperature measure- ment and has no observable effect on the results The oven temperatures were measured with Chromel P-Alumel thermocouples peened directly into the oven O RUN 57 as shown in F all available information and auxiliary checks against a Pt, Pt-Rh thermocouple indicate the temperature measurement to be accurate absolute. The beam intensity is very sensitive to the i temperature of the oven, so a temperature control unit, 2 essentially an on-off switch to control a portion of the oven heater current, was employed to maintain the oven within 0. 25.C of the nominal desired temperature The beam was detected on a tungsten surface ionization detector. In the case of thallium, oxygen as sprayed on the detector wire to increase the detec- ion efficiency. The filament was conditioned so that the detected beam was relatively insensitive to the filament temperature in the neighborhood of the operating temperature. The detection efficiency fc potassium can be made greater than 99 perce while for thallium this may not be true. Preliminary of the velocity distribution as long as the over-all detec- with tres in the ovens are given in itx detection efficiency has no observable effect on the shape press nd 60 with thin tion efficiency remains constant during the run. This shows that if there is a velocity-dependent detection in which the dimensionless variable V, the reduced efficiency, it is not very sensitive to the over-all detec- velocity, is the atom velocity vo, divided by the velocity tion efficiency of the intensity maximum of the distribution. This is RESULTS a very convenient expression for examining the agree- Differentiation of Eq. (5)shows that a?v, or., ment between theoretical and experimental curves since equals two at the intensity maximum, so that Eq. (5) will be the same for all velocity distributions which esult from a single molecular species. To take into I,=8yV4 exp(-2v) (8) account the effect of finite rotor slot width, the actual theoretical distribution used in this work was a < uni- obtained from Eq.(6) with ao replace by Vv2 Figure 4 shows universal plots of typical velocity distributions for potassium; each distribution corre- COPPER sponds to different experimental conditions. To compute he reduced velocity V, the experimental velocities were divided by the theoretical velocity of the intensity maximum calculated from the measured oven tempera ture. Since the uncertainty in temperature measurement HOLes does not exceed +0.5 percent, the velocity correspo dis g to the tribution will be accurate to =0.25 percent. The ob served intensity measurements have been multiplied, in ll cases, by an appropriate factor to give coincidence maximum TAPERED GROOVES Some of the important results are tabulated in Table FOR SUPPORTING PINS II. The velocity at which the maximum intensity occurs oven used for potassium showing can be determined directly from the experimental velocity distribution and may also be calculated from (1948. Cogin and G. E.Kimball, J. Chem. Phys. 16, 1035 the oven temperature. The agreement of the two velocities is one criterion which must be satisfied if the
R. C. MILLER AND P. KUSCH TABLE II. Experimental conditions and results for measured velocity distribution distributions were shifted slightly to the high velocity side of the theoretical curves. The shift corresponds to re of the order of three percent higher the knife-edge copper slits was estimated to be 0 to 0.012 cm K 4662 628士2630土3 The potassium distributions were not extended 644-42 682*3 extension requires what appeared, at the time, to be a 9445 392=1 395+2 dangerously high speed of revolution. In the case of thallium, more complete distributions were obtained present work is to be a critical test of the Maxwellian The thallium distributions are quite similar to the distribution. The vapor pressures given in Table II potassium distributions already discussed. Runs 99 were obtained from the literature and represent the between experimental points and the theoretical curves question screpa ancies occur on the high-velocity side of the Of the three potassium distributions, Run 57 shown maximum, where there is a small excess of atoms in the in Fig. 4 provides the best agreement with the theo- retical curve. In this case, the vapor pressure in the experimental distribution. It should be noted that the oven was as low as wasexperimentally feasible. It was not experimental points could be plotted with the high possible to obtain velocity distributions for markedly tensities at the maximum velocity would no longer lower oven pressures since the beam intensity which coincide and the experimental distribution would then depends directly on the oven pressure, would then be appear to be deficient of atoms on the low-velocity so low that the several sources of noise would give data side. The observed discrepancies are again more of limited value long before statistical fluctuations pronounced at higher oven pressures ay a significant role. The experimental points are When thallium beams were observed, the tungsten seen to be in excellent agreement with the theoretical detector wire was constantly sprayed with oxygen to curve over the major part of the distribution. However, maintain a good oxidized surface on the wire. The velocity side of the maximum. The value of vr calculated from the oven temperature agrees with kept so small that the increase in pressure due to the experimental value It has been observed that the deficiency of atoms q ygen was about 5X10-8mm of mercury in the tector chamber and less in the other chamber on the low velocity side increases with increasing oven 2( pressures and with increasing slit depth. The deficiency 如如小m ▲RUN97 vapor pressure was markedly increased. The experi mental points of Run 60 are in good agreement with the theoretical curve in the neighborhood of the intensity maximum, but there is a pronounced defi- ciency of atoms on the low-velocity side and an ob- servable excess of atoms on the high-velocity side. It if the velocity distribution had undergo a velocity-dependent scattering which was not serious enough to shift the maximum of the distribution Run 31 illustrates the effect of a deep slit on the velocity distribution. These slits were made of copper trips which were 0.317 cm thick. In this case, the entire distribution has been shifted in the high velocity direction. It was also found that when the 0.003-cm or 0. 004-cm steel shims which formed the orifice from which the beams effused from ovens like those shown in Fig 3 were omitted, so that the orifice was determined by the knife edges of the copper strips, the velocity FIG. 5. Typical thallium velocity distributions. The data were taken with thin oven slits at vapor pressures given in Table II