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
VELOCITY DISTRIBUTIONS IN K AND TI BEAMS 1319 BEAM PURITY Ke is negligible in the present work. A search for TI The vendor of the potassium reported that it was made with a molecular beam apparatus. It was ined 2 percent sodium and 0.5 percent oncluded that there was less than 0.05 percent Tl2 Because of the low vapor pressure of these ty an amount wholly negligible in the present analysis stituents compared to that of potassium, no meas amount of sodium or lithium is expected to be present COMPARISON WITH OTHER RESULTS in the potassium beam with a magnetic resonance The experimental techniques used in two early atomic beam apparatus, but no sodium transitions were attempts to measure velocity distributions.were not sufficiently refined to permit an accurate test of the make it unlikely that an amount of sodium sufficient Maxwell velocity distribution law to be made. The to give a observable distortion of the velocity distribu- work of both Eldridge and Lammert shows deviations The thallium metal was reported to be 99.95 percent greater than the experimental uncertainties. Zartmand thallium,exclusive of the oxide on the surface. The investigated the velocity distributions in bismuth remaining 0.05 percent was largely lead, copper, and beams. Unfortunately, no critical test of the Maxwell cadmium, all of which are undetectable. Since no beam distribution could be made since bismuth vapor con- as detected with an unoxygenated detector wire, it is tains significant amounts of molecular species other certain that there were no measurable amounts of than atomic Bi. Thus zartman actually measured the esIum, rubidium, or potassium in the thallium beam. superpositions of distributions of Bi, Big, and Big e There is the possibility that the beam may also When he adjusted the amount of Big so that the agree- contain dimers of the atomic species. In the case of ment was good on the low-velocity side, there was a thallium and potassium the beam in question is in- pronounced excess of molecules on the high-velocity vestigated with an atomic beam apparatus that will side. The amount of Bis was small so that it little atoms with magnetic moments of the order of a effect on the major part of the distribution. Bohr magneton, which characterize the atomic states Ko used essentially the same apparatus as Zartman of potassium and thallium. Molecules of K, are in to measure the fraction of Big in bismuth beams as a a 2 states as are, presumably, those of the Tl, if they function of the oven temperature and pressure.From exist. They have magnetic moments of the order of a his data, he was able to calculate the dissociation nuclear magneton and are not significantly deflected energy of the Bis molecule. When the amount of dimerization was adjusted to give good agreement on by inhomogeneous magnetic fields which deflect atoms the low-velocity side of the distribution, there were too rough great excursions The molecule of potassium, K,, is known to be many molecules on the high-velocity side. He also found present in a potassium beam, but the data on the frac- evidence that Bis was present in the beam tion of molecules is inconsistent. In a molecular beam Cohen and Ellett measured velocity distributions in experimentl in which the nuclear magnetic moment of sodium and potassium beams. As a velocity selector potassium was measured, it was found that the ion current produced at the detector by the molecular perpendicular to the direction of propagation of the component may be as great as one percent of that beam. Since the deflection of beam atoms is proportional to the gradient of the magnetic field, it is important favorable experimental conditions, Rosenbergl4 ob- ent to be constant if the observed distribution could be tained 0.25 percent k, in a potassium beam at 5o0%k. fitted to the theoretical distribution after the adjust The fraction of K, may be shown from thermodynamic ment of constants not determinable from a priori considerations to increase with increasing oven tem- considerations. The nature of the constants is such peratures. Thus the distortion of the velocity distribu- that an independent check of the observed maximum and that deduced from the temperature cannot be ture cannot be the result of a larger fraction of K, made. Subject to these limitations, they observed no as the dimer would shift the velocity distribution to the systematic deviation from Maxwells theory at low low velocity side. If 0.25 percent K, is assumed to be oven pressures. At high oven pressures, they observed present in the beam, the maximum increase in the serious deviations from theory, notably a deficiency experimental velocity distribution at any velocity at knife-edge slits would give better results than their which observations have been made would be only 2-mm thick rectangular slits. However, on the basis of 0.5 percent of the maximum intensity. The effect of the that the 1s Kusch, Millman, and Rabi, Phys. Rev. 55, 1176(1939) ical distributions coul IR. Rosenberg, Phys. Re even at low pressures
R. C. MILLER AND P. KUSCH In 1947, Estermann, Simpson, and Stern measured of the beam by placing a 0.005-cm slit at each end of velocity distributions in cesium beams by use of the velocity selector; but it was impossible to observe the free fall due to gravity as a velocity selector. They a velocity distribution with this arrangement, since observed a pronounced deficiency of atoms on the low- vibration caused by the rotating velocity selector pro- velocity side and noted that the discrepancy increased duced serious unsteadiness in the beam. Sufficient en- deficiency to collisions near the oven slit. An order of t gement of the collimation slits to reduce variations the beam intensity consequent to vibration reduces magnitude calculation is given which indicates that the collimation of the beam to the point where the the deficiency might be explained on the basis of slits no longer serve to prevent the detection of atoms Cs- Cs collisions in the neighborhood of the slit. which may have undergone reflections from the wal The most recently reported results on velocity of the slots and the slits were eventually removed distributions are those of Bennett and Estermann who However, before the slits were removed, an attempt measured potassium velocity distributions. Their ex- was made to reflect potassium atoms from the wall of perimental velocity distributions were in good agree- one of the two straight slots when the rotor was at rest ment with theoretical distributions on the high-velocity There was no evidence at all of specular reflection. The side of the intensity maxima, but there was always a diffused reflection and scattering of atoms was so small deficiency of atoms on the low-velocity side, a deficiency that the intensity intercepted on the detector when which increased with increasing oven pressure and with not in the direct line of the beam was less than 0.1 a decreasing velocity. Even at the lowest oven pressure percent of the beam and hence not important in the at which experimental results were obtained, 1.8X10-3 present work. mm of mercury, the intensity was about 90 percent of the theoretical intensity for atoms at a velocity one-half SCATTERING FROM RESIDUAL GAS the velocity of the intensity maximum. The deficiency The theoretical velocity distribution to be observed of atoms on the low-velocity side was greater if the oven at the detector has been derived under the assumption slits were rectangular than if they were knife edges. that a Maxwellian distribution occurs in the oven and The velocities at which the experimental intensity that the slit of the oven is an ideal aperture. It is inter maxima occurred were usually found to agree within esting to investigate the occurrence of a possible distor tion of the velocity distribution due to velocity de- In the present work, experimental velocity distribu- pendent scattering of the beam atoms by the residual gas tions of both potassium and thallium were found to be in good agreement with theoretical velocity distribu- velocity b in a single Maxwellian gas islmolecule with in the apparatus. The mean free path of tions. It has been shown that the experimental and theoretical distributions, for small beams, do not differ L,=a1/nIS by more than one percent of the maximum intensity over the entire range of the measured distributions extending from as low as 0.2 to as high as 1.8 times ep(-)+(2+x)Je(-y)y,⑨ the velocity of the intensity maximum. Furthermore the velocities at which the intensity maxima occur where a1=(1/2)m/(kTi)in which the subscript 1 agree within 0.5 percent with velocities calculated refers to the scattering gas, n is the density of the gas from the measured oven temperature. These results a is equal to a1v and S is the collision cross-sectional area provided the best agreement to date between observed To make any calculations from Eq. (9), one has to distributions and theoretical distributions calculated assign a value to an and the cross-sectional scattering from the assumption that the beam is a consequence area. In the case at hand, the scattering gas probably of a Maxwellian distribution within the oven which consists of air plus all sorts of condensable vapors including pump oil dissociation products. The tempera- ture of the residual gases is not defined but an"effective REFLECTION FROM THE ROTOR SLOTS temperature may range from slightly above room tem- It has been assumed that all molecules which strike perature to liquid nitrogen temperatures. Since the the wall of the slots in the velocity selector are removed from the beam. If the beam were very narrowly colli- increasing rotor speed, there may be a change in com- mated so that the detector would intercept only atoms postion of the gas at different values of xo. Thus it is virtually impossible to calculate Ly for this situation which arrive along a line parallel to the rotor axis and To estimate the magnitude of the effects due to scatter which leave the oven along the same line, the assump- ing, it will be assumed that the residual gas approxi tion would be justified. However, no collimation other mates air at room temperature and that the cross than that provided by a source and detector of small sectional scatter a is 4.0X10-14 cm". This area width was provided in the original design of the appa-I6Eh.Ke ratus. An attempt was made to improve the collimation Company, hanard. kinetic ases(McGraw-Hill Book
VELOCITY DISTRIBUTIONS IN K AND TI BEAMS 1321 close to the value found by Mais and Rosenbergl4 for the scattering of potassium beams by nitrogen. cath the property that a non-Maxwellian distribution using through a nonideal aperture gives a distribution The probability that an atom will travel from the experimentally indistinguishable from that arising ven slit to the detector wire without a collision under our assumptions. An ideal aperture is one in which and the detector wire, 60 cm. Then Pn can be calculated precisely to meet this condition within the range of from Eqs. (9)and (10). If Eq.(5)is multiplied by the experimental feasibility. The agreement between theore- appropriate value of Pu when account is taken of the tical prediction and observed distributions is there- variation of the pressure with rotor velocity, a dis- fore, not fortuitous and the result of the present work tribution is obtained whose shape is only trivially demonstrates within relatively narrow limits of error distorted on the scale of Figs. 4 and 5. In view of the that a gas in thermal equilibrium with its surroundings uncertainty in the composition of the gas, the gas possesses a Maxwellian velocity distribution temperature, and the related cross sections, the result In view of the success of this work, a study of the is subject to a corresponding uncertainty. However, the velocity distributions of alkali halide molecular beams excellent results for low and moderate beams of potas- has been made, in which it has been assumed that the sium and thallium make it clear that any distortion due experimental velocity spectrum is the superposition to scattering by the residual gas in the apparatus must of the Maxwellian velocity distribution of the molecular components which are in thermal and chemical equil brium within the oven chamber. The results of this DISCUSSION AND CONCLUSIONS work will be published elsewhere The results presented in this work demonstrate that he experimental velocity distribution can be predicted ACKNOWLEDGMENTS velocity spectrum in the oven is Maxwellian and that many helpful discussions. The authors also take this effusion occurs through an ideal aperture. It may be opportunity to express their appreciation to Mr Joseph thought that the excellent agreement is somewhat Haas of the Columbia Radiation Laboratory for the fortuitous, i. e, that an oven slit design was discovered construction of the cylindrical velocity selector witl 16W. H. Mais, Phys, Rev. 45, 773(1934) the 702 perfectly milled slots
