- Email: [email protected]
Na2 molecular beams
ChemicalPhysics 124 (1988) 131-142 North-Holland, Amsterdam
CROSS SECTIONS FOR TRANSLATIONAL AND ROTATIONAL IN Li/L& AND Na/Na, MOLECULAR BEAMS
RELAXATION
G. GUNDLACH, E.L. KNUTH ‘, H.-G. RUBAHN and J.P. TOENNIES Max-Planck-Institutfdr Striimungsforschung, Bunsenstrasse IO, D-3400 Gijttingen, FRG Received 3 March 1988
Translational and rotational relaxation processes in alkali monomer-dimer free-jet expansions are investigated both theoretically and experimentally. Based on a sudden-freeze model, simple formulas are derived for the extraction of effective translational relaxation cross sections from terminal Mach numbers and effective rotational relaxation cross sections from measured terminal rotational populations. These formulas are verified in a state-resolved laser-molecular beam study. The magnitude of the rotational cross section is found to be strongly state dependent and to be essentially the same for Na, and Liz. The terminal dimer velocity exceeds that for the monomers in Na/NaZ but not in Li/L&. An earlier sudden-freeze analysis for rare gases is used to correlate terminal dimer mole fractions measured for Li/L& beams with dimensionless parameters. From this correlation and applying energy conservation to the measured terminal velocities we found that at least 60% of the binding energy released in dimer formation in Li/L& beams resides terminally in internal degrees of freedom.
1. Introduction
During the early 1970’s nozzle beams of the heavier alkalis were used extensively since they provided, together with Langmuir-Taylor detectors, a relatively simple tool for the investigation of chemical reactions under single-collision conditions. These beams have been characterized in terms of velocity distributions [ 1 ] and polymer concentrations [ 2 1. First insight into the internal state distributions of the expanding alkali dimers was made possible by analysis of the fluorescence following excitation by the light from high-pressure lamps or coherent light from an Ar+ laser [ 31. The advent of tunable visible cw dye lasers in the late 1970’s and the accessibility of the first electronic excited states of most of the alkalis by these lasers renewed interest in a more detailed analysis of the properties of these beams. For Na/Naz beams, several relaxation processes during the beam expansion have been investigated state selectively [ 4-8 1. Due to the significantly greater difficulties in generating stable Li/LiZ beams, the in’ Permanent address: Chemical Engineering Department, University of California at Los Angeles, Los Angeles, CA 90024, USA.
vestigation of properties of these beams has not reached the degree of accuracy of the Naz studies [ 913 1. For (v, J, m,) state-selective investigations of elementary chemical interactions, however, Liz has two advantages compared to Naz. For one, it is possible to prepare a small number of high vibrational states of the electronic ground state (up to 75O%1 of the dissociation energy) via optical pumping [ 14-18 1. Secondly, the ab initio calculation of Liz interactions is much more feasible due to the simpler electronic structure compared with Na,. In section 2 we use the sudden-freeze model [ 1922] to develop a procedure for extracting from terminal values of free-jet properties (e.g., the Mach number of the monomers and the rotational population of the dimers) the magnitudes of the associated effective cross sections. Relative to methods which require integration of a relaxation equation along the free-jet axis from the source to some far-downstream location [ 81, the present method has two obvious advantages. First, the extracted cross section is related clearly to the local temperature at the sudden-freeze location in agreement with the fact that the values of the cross section at upstream locations where thermodynamic equilibrium exists have no direct bearing on the terminal values. Second, the simplicity of
0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
132
G. Gundlach et al. /Relaxation in alkali dimer-monomer
the calculation procedure allows one to identify the physically important parameters. In the analysis of rotational relaxations, a procedure for handling the important case in which the terminal rotational population is non-Boltzmann is suggested. In a closely related analysis, the terminal dimer concentrations in Li free jets are correlated using the relation developed previously for rare gases [ 2 1 ] and changing only the value of the multiplying coefficient. In section 3 the experimental setup and procedures used for timeof-flight (TOF), laser-induced-fluorescence (LIF) and mass-spectrometer measurements on Li/L& and Na/Na2 beams are described. In section 4, the theoretical procedures developed previously are applied to the experimental results to extract values of effective translational and rotational cross sections and to adapt the dimer-mole-fraction relation to Li free jets. Section 5 includes a relatively simple scaling rule for the free-jet density at the sudden-freeze point as a function of source conditions, an analysis of the terminal distribution of the binding energy released in dimer formation and a discussion of some of the differences between Li/L& and Na/Na* expansions.
2. Sudden-freeze analysis The simplest useful method for extracting cross sections from terminal values of free-jet properties uses the sudden-freeze model [ 19-221. This model is motivated by the observation that, in a typical freejet expansion, transfers of energies among several degrees of freedom are fast in the region of higher pressures and enthalpies so that the various degrees of freedom are essentially in equilibrium in this region. This is consistent with the small relaxation time in comparison with the time required for the gas to pass through this region. In the region of lower pressures and enthalpies on the other hand, transfers of energies to and from certain degrees of freedom are so slow that negligible energy is transferred to or from these lagging degrees of freedom. Or in other words the relaxation time is large in comparison with the time required to pass through this region. It is convenient to divide these two regions by a “freezing point” and to use the terms “equilibrium flow” and “frozen flow” to identify the flows upstream and downstream, respectively, of this point.
molecular beams
A convenient freezing-point criterion is motivated by an examination of the hydrodynamic derivative D/Dt of the appropriate relaxation time, where D/ Dt is the rate of change observed in the reference frame of the moving gas. The appropriate relaxation time is identified by noting that, at any given point along the expansion path, the gas is relaxing toward the equilibrium state associated with the local macroscopic kinetic energy. Since constant macroscopic kinetic energy implies constant enthalpy and pressure, use of the relaxation time at constant enthalpy and pressure, r,,p, is motivated. If the flow begins from an equilibrium state, then, for equilibrium flow, rh,#<
c,
(1)
where C is a constant of the order of unity. On the basis of a recent examination of rotational, vibrational and chemical relaxations in free jets, Dao-Gibner and Knuth [ 221 suggest setting this constant equal to 1.6. If the dependence of rh,p on the thermodynamic state can be approximated by powers of thermodynamic variables, then a relatively simple alternative form of the above freezing-point criterion can be derived. Even greater simplicity is realized if one takes advantage of the finding [ 231 that the sudden-freeze model with constant collision cross section correlates the translational-freezing-point data well provided that one evaluates the cross section at the freezingpoint temperature. Define the effective collision cross section crcffvia the relaxation time at constant enthalpy and pressure: 7h.p
=bd~=
1 /fi
n&fffi
,
(2)
where & is the effective mean-free-path length and 8 the mean random speed: 8= (8kTlnm)‘l’.
(3)
Then one can write = 7h,pO 7hh,P
7h,p/7h,,O
=
7h.N
(&fd&
0 )GO
/4
(4)
where the index 0 denotes stagnation conditions in
G. Gundlach et al. /Relaxation in alkali dimer-monomer
the source. The effective cross section cancels in eq. (4) since as noted above it can be assumed that the cross section is constant at its sudden freeze value throughout the expansion: Z*,p=Zh,pO(noln)(TolT)1’2 *
(5)
If the expansion is isentropic and the gas behaves as a perfect gas, no/n= (TJT)I’Q--L)
)
(6)
one may write rh,P=r,,,
(T&-)(Y+1)‘2(y-‘).
(7)
molecular beams
133
on p,#, where p. is the source pressure, but also on TO. A more explicit expression for Mf results if an approximate relation between M and x/d* is introduced. Since the thermal energy added to the free jet as a result of dimer formation is typically of the order of only 5Ohof the terminal kinetic energy, both Na and Li expand as a monatomic gas and y= 513. For a monatomic gas and MB 1, the dependence of the centerline Mach number on axial distance may be written to good approximation [ 201 as M=3.22(~/d*)~‘).
(14)
For steady flow D/Dt=ud/dx, where u is the local hydrodynamic speed and x is the distance along the jet centerline. For adiabatic flow and a perfect gas,
Hence two convenient forms of the freezing-point criterion under these conditions result from eq. ( 13) with y= 5/3. Using eqs. (8) and ( 14) one gets with
T,JT=l+t(y-l)M*,
l+f(y-l)M*=f(y-l)M*
dTo/T -= dx
(8)
for MB 1:
(Y-w&
d*/7/,&-10= 1.85M:“.
where M is the Mach number ( =u/a) and a is the local speed of sound. The freezing-point criterion for steady flow, eq. ( 1) can be written
M,z [2To/T(y-1)]1’2,
Ma drh,P/dx= 1.6 .
d*/7h,poao=7.3(
(10)
In terms of a reduced distance x/d* with d* the nozzle throat diameter eq. ( 10) can be expressed as Mae
d7hp =1.6d*
(11) (Y+l)D(Y--L)--l
(12)
’
dTo/T
dx/P’
With eq. (9), eq. ( 12) can be rearranged to give finally (2--Y)/(Y-
1)
dx/P
d* f=Zh,poao’ (13)
where the subscript f refers to the sudden-freeze location. Since To/T, is a known function of Mf, this form of the freezing-point criterion relates (for given y) the freezing-point Mach number Mf to the dimensionless freezing-point parameter d*/rh,fla+ Note that according to eq. (2) this parameter depends not only
(15)
Substituting for Mf, using eq. (8) with
To/Tf)5’4.
(16)
Using the form which is more convenient, one can either, for given source conditions and known cross section, predict the location of the sudden-freeze point and terminal values of the relevant thermodynamic property or, from measured terminal values of the relevant thermodynamic property, extract the value of the effective cross section oeffvia eq. ( 2 ) . The latter application is of interest in the present work. The analysis to this point is applicable equally to translational and rotational relaxations. 2.1. Translational relaxation In this case, the terminal value traditionally measured is Mf. The effective cross section for translational relaxation is related frequently to the viscosity cross section gviscvia the collision number Ntrans, Ntran~= ovisc/“eff,trans
.
(17)
In a moment-method analysis of translational relaxations in monatomic free jets, Knuth and Fisher [ 241 find that Ntrans= 15/8. Hence from eq. (2) one gets
G. Gundlachet al. /Relaxation in alkalidimer-monomer molecularbeams
134
after multiplication with d* and using eq. ( 17 ) : d*/G,po = A Jz noaviscd*80 .
By definition of the mean random speed, eq. ( 3 ) , and the local speed of sound, a0 = (ykT/m)‘/2,
(18)
one arrives at -
d*
=
Th,pOaO
A (i +)I” Jz
no~visc d* .
(19)
If one eliminates d*/rhs oaofrom eqs. ( 15 ) and ( 19 ) , then one can extract values of the viscosity cross section using ovisc(q,,,)
= l.98M:‘2/nod*
,
(20)
where T,,,,, is the local temperature
at the translational freezing point. For convenient comparison with similar expressions found in the literature [ 231, eq. ( 15) is rearranged into the form
A&=0.78 (d*/T/,NUo)2’5.
(21)
Via eq. ( 19) the freezing-point parameter can then be related to the Knudsen number Kn = 1/Jz
no Oviscd*.
(22)
Then eq. (2 1) may be written alternatively M,=O.66 ( 1/Kn)2’5.
(23)
This relation correlates data for the rare gases well provided that the value of the (viscosity) cross section used in Kn is based on the local temperature at the translational freezing point as shown in fig. 3 of ref. [ 23 1. Note that frequently a correlation of terminal Mach number with Knudsen number is based on values of the cross section corresponding to the source temperature. This case will be discussed in the appendix. 2.2. Rotational relaxation Eqs. ( 15 ) and ( 16 ) are applicable also to rotational relaxations of dimers in a monomer/dimer free jet. If the dimers are present only in small mole fraction, then rh,px ~~~where, similar to eq. ( 2 ) , ~Ic-,~o =~OGT,,~
(~~TO/W)'/~,
(24)
where /I is the reduced mass for dimer-monomer collisions. If one eliminates rh,px rKPfrom eqs. ( 16) and (24)) then one can extract values of the effective rotational cross section using Gff,,rot- (4.83/nod*)
(25)
(To/T,,)"~,
where T,, is the local temperature at the rotational freezing point. Local rotational temperatures can be determined for dimers from measured LIF intensities for several rotational transitions of a selected v” +v’ transition. For a Boltzmann distribution, the emitted intensity in the non-saturated excitation regime is Ian P&,..g.QJ+
1) exp( -&IkT,,)
,
(26)
where n is the number density of particles, PL the laser power, S,.,.. the HGnl-London factor, gJ the degeneracy, E,,, the rotational energy and k the Boltzmann constant. Thus after normalization to the product of n, PL, SJp, g., and 2J+ 1 a semilogarithmic plot of measured transition intensity versus rotational energy or versus J(J+ 1) is expected to give a straight line, the slope corresponding to the inverse rotational temperature, T,,. In the measurements, which are made downstream from the rotational freezing point, however, one generally obtains a curved line with slope decreasing monotonically with increasing values of J. The slope of the curve at a given value of J is interpreted as the inverse of the local temperature in the expansion at which rotational states in the vicinity of this value of J have been frozen in. The monotonic decrease of the slope with increase in J therefore indicates that the states with larger values of J (i.e. those with smaller effective cross sections) have froze hearlier in the expansion at higher local temp, rutures. The most meaningful single cross section for such a distribution is a J-dependent cross section averaged and weighted according to the J-population for the given temperature. In a less precise, but more convenient, procedure one might approximate the cross section for a given temperature by the cross section for the most probable J= J* for that temperature. A relation between T,, and J* may be evaluated by determining the maximum of the distribution eq. (26) and using for a rigid rotor E,, =BJ( J+ 1) ,
(27)
G. Gundlach et al. /Relaxation in alkali dimer-monomer molecular beams
135
with the result TrO,= (2P+
l)*B/2k,
(28)
where B is the rotational constant of the electronic ground state of the molecule. For Na2: B= 0.154734 cm-’ [25] and for Li,: B=0.672592 cm-’ [26].
x
lmml
d lmml
2.3. Dimer formation
0 vor
P lmbarl
Dimer formation in the rare gases has been analyzed and correlated using the sudden-freeze model by Knuth [ 2 11. For terminal dimer mole fractions in the range from about 2 x 10m4to at least 0.03, the following simplified relation was found to give a good correlation of the data:
~0.5[n,,R~(~/kT,,)“~(d*/R~)*‘~]~’~,
(29)
where x, is the terminal dimer mole fraction, R0 is the zero-potential radius and 6 is the well depth. The coefficient 0.5 appearing here was identified as a function of the parameters in the rate coefficient for dimer formation. Hence, for the range of source conditions for which the alkali beams consist mainly of monomers, one might expect a relation of the general form of eq. (29) to be useful also for this case. However, since the reaction-rate coefficients for dimer formation in alkali gases and in rare gases are expected to differ significantly, one might expect that at least the coefficient 0.5 is modified in the case of alkali beams.
3. Experimental apparatus and procedures Fig. 1 is a schematic diagram of the experimental apparatus, which has been described in detail in several previous publications for Li/Li2 [ 14-181 and Na/Na2 [27] free jets. Note that the apparatus for measurements with Li/Li2 molecular beams is in principle similar to that for measurements with Na/ Na2 beams. Thus we have drawn the same diagram for both types of experiments and denote the different dimensions in the lower part of the figure. Briefly, an alkali beam is generated in a resistanceheated two-chamber stainless-steel oven and is expanded through a nozzle into a vacuum. After pass-
LO
BL
3
3
L95
1.3 lo-'
7L
x Imml
0
30
0.2
1.7 a4
13L0
1620 Li / Li2
L
10-6
d lmml P hlt.xl
207
lo-'O
395
L95
1.0 10-&
No/No2 1o‘7
Fig. I. Experimental setup for investigation of relaxation processes in alkali nozzle beams, including two-chamber alkali oven, heated skimmer, laser, photomultiplier and quadrupole mass spectrometer (QM ) . Characteristic apparatus dimensions such as diameters of nozzle and apertures, distances to the nozzle and typical background pressures in the different vacuum chambers are listed below the corresponding parts, used in the Li/L& (upper part) and Na/NaZ (lower part) experiments. For Li/L&, nozzle diameters of 0.2.0.3 and 0.4 mm have been used.
ing through a heated skimmer and an aperture the particles are excited state selectively by the light from an actively stabilized cw ring dye laser (Coherent CR 699-2 1) pumped by an Ar+ laser (Spectra 17 l-06). The fluorescence light is sampled at right angles to the molecular beam and laser beam with a photomultiplier. Further downstream a quadrupole mass spectrometer measures densities of atoms and molecules in the beam. The velocity distributions are measured using the Doppler-shift method [ 281. For this purpose, the laser beam is split into two components, one of them intersecting the beam perpendicularly, essentially not Doppler broadened or shifted, and the other at an angle (Ywith respect to the molecular beam axis. During all measurements the perpendicularly measured linewidth was less than 20 MHz for Li/Li2 and less than 30 MHz for Na/Na2. The velocity width Au of the particles in the beam leads to a Doppler broadening of the fluorescence line. Using (Y= 15” (Na) and (Y= 17.5 o (Li) and a sufficiently low laser power this broadening dominates other broadening effects such as lifetime, time-of-flight or power broadening. Under these conditions the flow velocity u and Au can
G. Gundlach et al. /Relaxation in alkali dimer-monomer
136
be determined from the measured line shift and linewidth. Several transitions X ‘Z + (v” = 0, J” ) + A ‘Z + (0 , J’ ), excited with an R6G dye laser, have been used for the characterization of the Li, and Naz dimer velocity distributions. With R6G as laser dye we have used the 3 ‘S, ,2 F= 2 + 3 2P3,2 F= 3 atomic hyperline transition for the Na atoms, whereas for Li atoms we have used DCM as laser dye for exciting the 2 2S,,2 F=2+2 2P3,2 F= 1,2, 3 atomic hyperflne transitions. The relative apparatus velocity resolution depends on both cy and the molecular beam divergence [ 28 ] and for a typical Li beam amounts to Au = 100 m/s; for a typical Na beam, Au x 320 m/s. Fig. 2 shows the reduced intensities versus J(J+ 1) for Na2(v”=0+17) and for Li2(v”=0412) [13]. Clearly, since the slopes of both curves fitted to the data decrease as J increases, both distributions are not Boltzmann distributions. This behaviour has been reported previously for Na2 [ 51 and Liz [ 141, and was anticipated in the preceding theoretical analysis. It has been shown for Na, [ 29 ] and Li2 [ 17 ] that the collisions of molecules with atoms during the nozzle expansion lead to an alignment of the angular momentum distribution of the molecules. The degree of this alignment (the number of molecules with J per-
pendicular to the beam axis) increases strongly with increasing rotational state J up to J= 10 and then levels out [ 291. This alignment affects the measured fluorescence intensity and leads to an additional uncertainty of about 5% in the determination of T,, for J< 10. This effect was neglected since uncertainty resulting from fluctuations of laser and molecular beam intensity is of the same order or higher.
4. Experimental results 4.1. Translational relaxation
Velocity distributions Au/u for the alkali atoms have been measured for various nozzle diameters using the Doppler-shift method described in section 3. The Mach numbers have been determined from the measured values of Au/u [ 301 and are tabulated in table 1 as function of nozzle temperature T,, for conTable 1 Measured Mach numbers and effective translational Knudsen numbers calculated from eq. (22) for Li atomic beams of various nozzle diameters d* and nozzle temperatures T0 for constant AT= 250 K d*
*\
I I0.8
0
100
200
300
I
,‘\t \ ,I
LOO
500
600
J lJ+ll Fig. 2. Measured and normalized fluorescence intensity for Na, X~A(v=0~17)transitionsatT0,=980KandT~=1285Kand Li,X-+A(~=0-r12)at7”,=1038K,T0=1138K[13]asfunction of rotational energy. The respective rotational states are given in parentheses. The solid lines are best-fit curves to the measured points.
molecular beams
TO W)
M
oviscCT,,“,) (A’)
Kn
(mm)
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2
1242 1287 1312 1352 1382 1412 1452 1222 1292 1322 1342 1362 1382 1412 1422 1432 1447 1462 1482 1500 1192 1222 1262
2.21 3.1 4.12 5.35 5.79 6.89 7.23 2.01 3.73 4.47 4.79 5.35 6.3 7.23 6.89 7.59 7.99 8.43 8.91 9.45 1.92 2.32 3.39
762 832 910 991 1015 1080 1094 747 882 943 954 990 1048 1100 1078 1116 1137 1156 1182 1201 743 774 859
3.33x 10-I 1.57x lo-* 1.02x lo-2 5.55 x low 3.56x 1O-3 2.40~ lO-3 1.36x lO-3 7.01 x 1o-2 1.95x 10-Z 1.15x10-* 8.62x 1O-3 6.78~ lo-’ 4.59x 10-3 3.12x 1O-3 2.69x 1O-3 2.41 x lo-’ 1.72~ IO-’ 1.49x 1o-3 1.33x lo-’ 1.02x IO-’ 1.84x10-’ 1.02x 10-I 4.85 x 10-Z
G. Gundlach et al. /Relaxation in alkali dimer-monomer
stant temperature difference AT=250 K between oven and nozzle. From the Mach numbers we use eq. (8) to calculate the corresponding local temperatures T,,. Then together with eq. (22) and the well known Landau-Lifshitz approximation for the purely elastic cross section: u= 8.083 ( C,/fig)2’5 [ 3 1 ] and we get the respective Knudsen g= (2kT,,,/m)“2 numbers. Values of the van der Waals force constant C, for Li-Li interactions have been taken from ref. [ 321. The observed dependence of A4ron Kn is presented in fig. 3. Also shown for comparison is the correlation (solid line) from eq. (23 ) found to Iit similar data for the rare gases. It is seen that the measurements for Li and different nozzle diameters fit well on the rare-gas curve. Note that this good correlation justifies a posteriori the use of eq. (20) for the determination of translational-relaxation cross sections Gvisc( T,,,,,) from the measured Mach numbers (also tabulated in table 1). Previous studies on KZ [ 1] and on Rb2 and CsZ [ 21 have shown that there does exist a velocity slip between the faster atoms and the slower molecules. Bergmann et al. on the other hand found in their stateselective investigation of Naz expansions [ 41 that for pod> 1 Torr cm only the v= 1 and v= 3 vibrationally excited molecules are slower than the atoms, whereas molecules in u= 0, J= 28 are significantly faster. We have observed in this investigation at TO= 1482 K and TO,= 1232 K that the Li2 ( ZI=1, J= 5) molecules are slower than the atoms. Moreover the Mach number for these molecules is 18% smaller than the corre-
1L lo-’
10-J 10-z Id Knudsen Number. Based on Freezing-Point
molecular beams
137
sponding Mach number for the atoms. Using vibrational states between v=O and v=4 and rotational states between J= 6 and J= 40 Niedner found that the Mach number of Li, decreases with increasing internal energy E of the molecules as Ma E -‘.’ [ 111. Thus also for Li2 molecules in the vibrational ground state we expect a smaller Mach number and velocity compared to the atoms, thus supporting the results found for Rb2, Csz and K2 in refs. [ 1,2]. ForNa2(0, 6), TO= 1100 K, andp&0.6 Torr cm we have measured a higher velocity than that of the corresponding atoms. The respective Mach number was 2OWhigher than the corresponding Mach number of the atoms. Note that this difference is of the same order of magnitude as that measured by Bergmann for Naz (0, 28) and p,d= 2 Torr cm. For pod= 0.5 Torr cm, however, Bergmann measured inside the error bars the same Mach number for atoms and molecules (0, 28 ), but suggested a steeper increase of Mach number of Naz(O, 28) than that of Mach number of the atoms with increasing pod. This is in agreement with our finding at p,d= 0.6 Torr cm. 4.2. Rotational relaxation Fig. 4 shows the LIF measured terminal rotational temperature as a function of oven temperature for J-c 12 for a Na/Na2 beam. In these measurements the nozzle was maintained at a 200 K higher temperature than the source chamber to prevent clogging. Under these conditions the dimer concentration at TO,= 962 K is about 4%, decreasing to about 2% at TO,= 864 K due to the decreasingp, #l. As expected, with decreasing T,,, the terminal rotational temperature increases since the number of collisions decreases with decreasing density in the source. For comparison, a value of T,, determined using the data for J> 12 is included, revealing the less efficient rotational cooling for high rotational quantum numbers (see fig. 2). Measured values of T,, as a function of nozzle temperature To at constant TO,of 900 K are shown in fig. 5. Again, the increase of T,,, with increasing TOis as expected since both (a) the number of collisions de-
Cross Section
Fig. 3. Terminal Mach number as function of Knudson number for Li, AT=250 K for various nozzle diameters of 0.2 (A ), 0.3 (0 ) and 0.4 mm (0 ). the solid line is the fitting curve for rare gases, eq. (23) from ref. [23 1.
” For Na the partial pressure p0 (Torr) is correlated with
TO,via 10g,~p~=7.51-5344.5/T,,(K) for 500
138 125
--/
G. Gundlach et al. /Relaxation in alkali dimer-m~~orner m~iecul~r beams 8,
1 N%
AT = 200K i
4 P P Oi Oven Tempemture, Tw [Kl
Fig. 4. Measured terminal rotationai temperature as function of oven temperature for a Na/Naz beam, constant difference AT= To- To,=200 K and f0rJ-z 12 (circles). The error bars are twice the standard deviation. For comparison, a value obtained from a distribution measured at J> 12 is included (triangle).
creases and (b) the initial population of molecules with high J increases. Values of the effective rotational cross sections for Na/Naz have been extracted from the deduced values of T,, ( 3 1;) using eq. (25). Note that one can extract in principle for each combination of To and T,, a value of the rotational cross section. In view of
fig. 2 we note, however, that the value of T,, is very sensitive to the J-values from which it is determined and gives a characteristic arot for the overall rotational freezing only if one uses an appropriate range of J-values which includes the most probable J derived from a Boltzmann distribution with temperature 1”,,. Hence we have omitted all other measurements of T,, presented in figs. 4 and 5 for the calculation of the most meaningful a,,,. The average deduced value of the cross section is then 478 A’ for Na/Naz with a most probable J* of 7. Analyzing previously reported values of T,, for Li/LiZ [ 111 with J*= 20 we get 148 A2. These values are compared in fig. 6 with values for Na/Naz obtained by Aerts et al. [8] from the relaxation of individual Jstates ( To,=850 K, AT=30 K, d*=O.S mm) by integrating the relaxation equations. Another value for Li/L& has been extracted using the present suddenfreeze model and the data of Fuchs and Toennies [ 13 ] (T,,=1138K,AT=100K,d*=0.3mm)withJ*=8. The solid line is the best-fit curve to the measured points, indicating essentially the same crmt(J) correlation for Na2 and Li2 for the investigated region of J-values. 4.3. Dimer formation Terminal Li2 mole fractions in LifL& free jets have been measured with a quadrupole mass spectrometer with a low ionizing electron energy of 20 eV. Under these conditions the Liz fragmentation induced by the ionizing electrons is expected to be negligible. In these -----
I,------
Solid Symbols No, Open Symbols Li,
zoi
$0
,
100 Temperature
200
t
300
Difference, AT [K1
Fig. 5. Terminal rotational temperature as a function of temperature difference AT= To - T,, for NaZ and constant T,,= 900 K. Low J-values have been used for the determination of T,*,. The error bars are twice the standard deviation. The line is the result of a calculation using the sudden-freeze model, fitted to the measured points at AT= 116 K.
Rotational
State.
_I
Fig. 6. Rotational cross section as a function of J for Naz expansion with AT=200 K, d*=O.Z mm (m), compared with values forNazfromref. [8] (m)andforLi,fromref. [II] (V)and ref. [13] (A).
G. Gundlach et al. /Relaxation in alkali dimer-monomer
experiments the nozzle diameter was 0.2 mm and the Li source vapor pressures ranged from 0.92 to 49.9 Torr corresponding to nozzle temperatures from 1170 to 1460 K. With the corresponding dimensionless parameters eq. (29) provides a good tit of the data after an adjustment of the prefactor:
=180[noR~i”(~/kT,)“’
(d*/R,in)2’5]5’3) (30)
where Rmin is the RKR minimum-potential radius, R,,=2.673 8, [26]. The’ coefficient 180 appearing here is more than two orders of magnitude larger than that used for rare gases, partly due to the much deeper potential wells of the alkali metals in the ‘X state compared to the weak van der Waals potentials of the rare gas dimers. Cf. fig. 2 of ref. [ 2 11.
molecular beams
139
of Z= 1.05 for high A4. Table 2 compares this value with values derived for Na and previously reported values for K, Rb and Cs [ 1,2 1. Those results are compared with values of Z,,,,, (atoms+ mol. ) , which is the expected value of Z if not only all the stagnation enthalpy but also all the binding energy released in the measured dimer formation were converted into kinetic energy of direct motion. The calculation of this value requires the value of the dimer mole fraction given by eq. (30) for Liz. The values for K2, Rb2 and Cs2 respectively can be found in refs. [ 1,2 1. Due to the relatively large and quantitatively unknown dimer fragmentations in the case of Na2 detection at an ionizing electron energy of 70 eV, no value of &,,(atoms+mol.) is given for Na. Finally, from Z( measured ) and Z,,,,, (atoms + mol. ) , the fraction 0 of the dimer binding energy which (from an energy balance with negligible dimer mole fraction in the source) went into excitation of internal degrees of freedom was calculated from
5. Discussion @(internal) 5. I. Translational relaxations
=
The ratio .Z= u/ (SkT,/m) ‘I2 provides additional information on the partitioning of energy between internal and translational degrees of freedom during the expansion. For negligible dimer formation with 1.0 [ 341. In fig. 7 we show the meaY=5/3,Zn,,= sured values of C for three nozzle diameters as function of T,, for Li. The curves level off at a maximum I
I
I
,
1
Li
I
1300
I200
Source
Temperature,
I
I
v.00
1500
1 I
I
To [Kl
Fig. 7. Atomic velocity ratio Z= u/ (SkT,Jm) ‘I2 as a function of source temperature To for Li and d* = 0.2 ( x ), d* = 0.3 ( V ) and d*=0.4 mm (0 ). The horizontal line shows Z,,,,, for a purely atomic beam.
C&,(atoms+mol.)-Z’(measured) Z&,(atoms+mol.)1
*
(31) The values calculated for Li, K, Rb and Cs are also listed in table 2. If the dimer mole fraction in the source is not negligible, then the realized value of @ is smaller than this calculated value. Since the data used in constructing table 2 are taken from three different references (using widely different experimental conditions), no effort is made here to extract trends from the listed values of 0. As noted earlier for Na the terminal velocity of the molecules (v= 0) is greater than that for the atoms, whereas in the case of Li and the heavier alkalis the velocity of the molecules is smaller than that of the atoms. In trying to explain this behaviour we note two counteracting effects. In going from the lighter to the heavier alkalis the binding energy decreases so that less energy is potentially available, but the vibrational energy spacing also decreases so that the VT transition probability increases. The unique behaviour of Na2 would be explained if as one went from Na2 to K2 the decrease in well depth dominated whereas as one goes from Li2 to Na2 the decrease in vibrational energy spacing dominates.
G. Gundlach et al. /Relaxation in alkali dimer-monomer
140
molecular beam
Table 2 Distribution of binding energy released in dimer formation between translational and internal degrees of freedom
Li Na K Rb
cs
z max
z
z
(atoms)
(measured )
(atoms+mol.)
CD (internal
1.oo 1.00 1.oo 1.oo 1.oo
1.05
1.12
0.6
this work this work
1.23 1.18 1.18
0.3 0.7 0.7
111
1.08 1.17 1.05 1.05
5.2. Rotational relaxations McGeoch and Schlier [ 12 ] have suggested that, for a given gas species and for a given nozzle geometry, a fixed value of the local free-jet density nf determines the location of the sudden-freeze point. If the expansion can be approximated by an isentropic expansion of a perfect monatomic gas, y= 513, this assumption corresponds to eq. (6): nolnf = ( TO/ Tf) ‘I2 . In an examination of this suggestion, we begin with the sudden-freeze criterion in the form of eq. ( 16), which may be written d*/zh,poao = 7.3 (no/nf)5/6 .
(32)
If the dimers are present only in small mole fraction, then one may write rh,poaOas a function of nOqot as before and write, using eq. (24), nf =6.6 ( 1/no) ‘I5 ( 1/d*qot)6/5 .
Inax
(33)
It is seen that, for a given gas species and for a given nozzle geometry, the value of the local free-jet density at the sudden-freeze point is only a weak function of the source density no and an implicit function of the temperature via the temperature dependence of a,,,. The effects of changes in no and a,,, on nf tend to cancel under certain variations in experimental conditions, thus explaining why McGeoch and S&her could conclude that nf has a constant value. Note, however, that especially in the case of a change in nozzle diameter d*, eq. (33) has to be used instead of a constant value of nf. Also, if dimer-formation effects cannot be neglected, then a more sophisticated analysis is required. The increase of the measured rotational temperature T,,, with increase of the temperature difference
Ref. )
121 [21
AT= To- TO,for constant d* and TO, shown in fig. 5 also can be clarified using the sudden-freeze model. Recall the sudden-freeze criterion in the form of eq. (25 ). For constant oven temperature TO,,the density no is inversely proportional to the nozzle temperature To. Also, for constant TO,, a,,, is approximately constant. Hence, for constant d* and TO, (implying also constant po), TfccT;‘5.
(34)
The curve drawn in fig. 5 which has been normalized to the measurement with the smallest error bars at AT= 116 K is based on this proportionality. It is seen that the increase in T,, with increase in AT (or To) is due to the shift of the sudden-freeze surface upstream with decreasing source density no.
6. Conclusions Translational freezing in Li free jets is found to be well correlated by the sudden-freeze model with viscosity cross section evaluated at the freezing-point temperature and with a translational-relaxation collision number of 15/8 in agreement with an earlier analysis [ 24 1. The local slope of a non-linear semilogarithmic plot of measured fluorescence intensity of a rotational transition versus J(J+ 1) is interpreted in the present study as the inverse of the temperature in the expansion at which the rotational states in the vicinity of the local J froze. The suddenfreeze model with constant rotational cross section yields a very convenient relationship for extracting values of the J-dependent rotational cross section from values of this J-dependent rotational freezingpoint temperature. The good agreement between the J-dependent cross sections for Li/Li2 and Na/Na2
G. Gundlach et al. /Relaxation in alkali dimer-monomer
obtained here using the sudden-freeze model and those for Na/Na* obtained previously for individual J-states from a direct integration of a relaxation equation provides confidence in the use of the more convenient sudden-freeze model. The dependence of the terminal dimer mole fraction in Li/LiZ free jets on source pressure and temperature is found to be described satisfactorily by the dimensionless relationship developed earlier for the rare gases with only the value of the multiplying constant changed. It is suggested that the measurements be extended to additional values of the source-orifice diameter d* in order to either confirm or modify for alkali free jets the dependence on d* used for the rare gases as described in eq. ( 30). Finally, the simplicity of the sudden-freeze model is found to facilitate physical insights into observations and empirical relations. For example, use of the model clarifies (a) the limitations of using the sourcetemperature cross section in correlations of free-jet terminal values, (b) the limitations of associating the location of the sudden-freeze point with a fixed value of the free-jet density and (c) the observation that the terminal rotational temperature in a Na/Na;! free jet increases as To- To, increases for constant d* and To,. In summary, the present measurements and analyses of terminal states in free jets demonstrate the convenience and effectiveness of the suddenfreeze model in either (a) extracting the relaxationrate cross sections from measurements of terminal states or (b) predicting terminal states given values of relevant relaxation-rate cross sections.
molecular beams
141
Appendix
Recall that the values of the Knudsen number in eq. (23) were based on values of the translational cross section corresponding to the local translational temperature. If the correlation of terminal Mach number with Knudsen number is based on values of the cross section corresponding to the source temperature, then A&= 1.2 &i-0.4
(35)
is typical. The two correlations are equivalent if &( Tf) = Mf( TO 1 or G,,,
( T,)
l~trans
(To)=
( 1.2/0.66)2.5=4.5 .
A case for which this requirement is met exactly is one in which o,,,, is proportional to T - ‘I3 and To/ T+ 90. For a monatomic gas, this temperature ratio is obtained for M,= 16. Although a,,,, proportional to T-1/3 and To/T+ 90 might be representative of many experimental situations, they certainly are not universal conditions. Hence the use of eq. (35) and the cross section evaluated at the source temperature is recommended for convenience, e.g., in system-design calculations. The use of eq. (23) and the cross section evaluated at the translational-freezing-point temperature is recommended for analyses where a higher level of precision is desired, e.g., in extractions of cross sections from measured values of terminal Mach numbers.
References Acknowledgement
The authors are indebted to G. Niedner for permission to use his previously unpublished results on Li/Li, terminal rotational temperatures and thank Professor Dr. C. Nyeland and A. Slenczka for carefully reading the manuscript. Financial support by the SFB 93 “Photochemistry with Lasers” of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
[ 1] P.B. Foreman, G.M. Kendall and R. Grice, Mol. Phys. 23 (1972) 117.
[ 21 R.J. Gordon, Y.T. Lee and D.R. Herschhach, J. Chem. Phys. 54 (1971) 2393. [3] M.P. Sinha, A. Schultz and R.N. Zare, J. Chem. Phys. (1973) 549. [4] K. Bergmann, U. Hefter and P. Hering, Chem. Phys. (1978) 329; 44 (1978) 23. [5] K. Bergmann, U. Hefter and J. Witt, J. Chem. Phys. (1980) 4777. [ 61 M. Kompitsas, K. Kolwas and H.G. Weber, Chem. Phys. (1981) 221. [ 71 F. Aerts, M. Hulsman and P. Willems, Chem. Phys. (1982) 233. [8] F. Aerts, M. Hulsman and P. Willems, Chem. Phys. (1984) 319.
58 32 72 55 68 83
142
G. Gundlach et al. /Relaxation in alkali dimer-monomer
[9] C.-Y.R. Wu, J.B. Crooks, S.-C. Yang, K.R. Way and W.C.
Stwalley, Rev. Sci. Instr. 49 (1978) 380. [ lo] M.J. Alguard, J. Clendenin, R.D. Ehrlich, V.W. Hughes, J.S. Ladish, M.S. Lubell, K.P. Schiller, G. Baum, W. Raith, R.H. Miller and W. Lysenko, Nucl. Instr. Methods 163 ( 1979) 29. [ 1 I ] G. Niedner, MPI fur Striimungsforschung Giittingen, Bericht 104/1985. [ 121 M.W. McGeoch and R.E. Schlier, Phys. Rev. A 33 ( 1986) 1708. [ 131 M. Fuchs and J.P. Toennies, J. Phys. B 20 ( 1987) 605. [ 141 M. Fuchs, Ph.D. Thesis, Bericht 8/ 1985, MPI fur Stromungsforschung Gottingen, ( 1986). [ 151 M. Fuchs and J.P. Toennies, J. Chem. Phys. 85 ( 1986) 7063. [ 161 H.-G. Rubahn, N. Sathyamurthy and J.P. Toennies, Faraday Discussions 84 ( 1987). [ 171 H.-G. Rubahn and J.P. Toennies, J. Chem. Phys. (1988) to be published. [ 181 H.-G. Rubahn, Ph. D. Thesis, University of Gottingen, FRG (1988). [ 191 E.L. Knuth, Proceedings of the 1972 Heat Transfer and Fluid Mechanics Institute, eds. R.B. Landis and G.H. Hordemann, Stanford (1972) p. 89. [20] E.L. Knuth, in: Engine emissions: pollutant formation and measurements, eds. G.S. Springer and D.J. Patterson (PlenumPress, New York, 1973 ) p. 3 19. [21] E.L. Knuth, J. Chem. Phys. 66 (1977) 3515. [ 221 L.T. Dao-Gibner and E.L. Knuth, in: Rarefied gas dynamics, Vol. 2, eds V. Boff~ and C. Cercignani, (Teubner, Stuttgart, 1986) p. 514.
molecular beams
[ 231 S.S. Fisher and E.L. Knuth, AIAA J. 7 ( 1969) 1174. [24] E.L. Knuth and S.S. Fisher, J. Chem. Phys. 48 (1968) 1674. [25] P. Kusch andM.M. Hessel, J. Chem. Phys. 68 (1978) 2591. [26] P. Kusch and M.M. Hessel, J. Chem. Phys. 67 ( 1977) 586. [ 271 G. Gundlach, Diplomthesis, University of Giittingen, FRG (1987). [ 281 K. Bergmann, W. Demtriider and P. Hering, Appl. Phys. 8 (1975) 65. [ 291 U. Hefter, G. Ziegler, A. Mattheus, A. Fischer and K. Bergmann, J. Chem. Phys. 85 ( 1986) 286, and references therein. [30] J.B. Anderson and F.S. Sherman, Rarefied Gas Dyn. 2 (1966) 84; U. Buck, Ph.D. Thesis, University of Bonn, FRG ( 1969); M. Gllser, MPI fllr Strijmungsforschung Gdttingen, Bericht 112 (1972); H. Haberland, U. Buck and M. Tolle, Rev. Sci. Instr. 56 (1985) 1712; R. Engelhardt, Th. Lorenz, K. Bergmann, Th. Mietzner and A. Palczewski, Chem. Phys. 95 (1985) 417. [ 3 1 ] H.G. Bennewitz, K.H. Kramer, W. Paul and J.P. Toennies, 2. Physik 177 (1964) 84. [ 321 K.T. Tang, J.M. Norbeck and P.R. Certain, J. Chem. Phys. 64 (1976) 3063. [ 331 D’Ans Lax, Taschenbuch fur Chemiker and Physiker, III (Springer, Berlin, 1970). [ 341 R.J. Gordon, R.R. Herman and D.R. Herschbach, J. Chem. Phys. 49 (1968) 2684.
CROSS SECTIONS FOR TRANSLATIONAL AND ROTATIONAL IN Li/L& AND Na/Na, MOLECULAR BEAMS
RELAXATION
G. GUNDLACH, E.L. KNUTH ‘, H.-G. RUBAHN and J.P. TOENNIES Max-Planck-Institutfdr Striimungsforschung, Bunsenstrasse IO, D-3400 Gijttingen, FRG Received 3 March 1988
Translational and rotational relaxation processes in alkali monomer-dimer free-jet expansions are investigated both theoretically and experimentally. Based on a sudden-freeze model, simple formulas are derived for the extraction of effective translational relaxation cross sections from terminal Mach numbers and effective rotational relaxation cross sections from measured terminal rotational populations. These formulas are verified in a state-resolved laser-molecular beam study. The magnitude of the rotational cross section is found to be strongly state dependent and to be essentially the same for Na, and Liz. The terminal dimer velocity exceeds that for the monomers in Na/NaZ but not in Li/L&. An earlier sudden-freeze analysis for rare gases is used to correlate terminal dimer mole fractions measured for Li/L& beams with dimensionless parameters. From this correlation and applying energy conservation to the measured terminal velocities we found that at least 60% of the binding energy released in dimer formation in Li/L& beams resides terminally in internal degrees of freedom.
1. Introduction
During the early 1970’s nozzle beams of the heavier alkalis were used extensively since they provided, together with Langmuir-Taylor detectors, a relatively simple tool for the investigation of chemical reactions under single-collision conditions. These beams have been characterized in terms of velocity distributions [ 1 ] and polymer concentrations [ 2 1. First insight into the internal state distributions of the expanding alkali dimers was made possible by analysis of the fluorescence following excitation by the light from high-pressure lamps or coherent light from an Ar+ laser [ 31. The advent of tunable visible cw dye lasers in the late 1970’s and the accessibility of the first electronic excited states of most of the alkalis by these lasers renewed interest in a more detailed analysis of the properties of these beams. For Na/Naz beams, several relaxation processes during the beam expansion have been investigated state selectively [ 4-8 1. Due to the significantly greater difficulties in generating stable Li/LiZ beams, the in’ Permanent address: Chemical Engineering Department, University of California at Los Angeles, Los Angeles, CA 90024, USA.
vestigation of properties of these beams has not reached the degree of accuracy of the Naz studies [ 913 1. For (v, J, m,) state-selective investigations of elementary chemical interactions, however, Liz has two advantages compared to Naz. For one, it is possible to prepare a small number of high vibrational states of the electronic ground state (up to 75O%1 of the dissociation energy) via optical pumping [ 14-18 1. Secondly, the ab initio calculation of Liz interactions is much more feasible due to the simpler electronic structure compared with Na,. In section 2 we use the sudden-freeze model [ 1922] to develop a procedure for extracting from terminal values of free-jet properties (e.g., the Mach number of the monomers and the rotational population of the dimers) the magnitudes of the associated effective cross sections. Relative to methods which require integration of a relaxation equation along the free-jet axis from the source to some far-downstream location [ 81, the present method has two obvious advantages. First, the extracted cross section is related clearly to the local temperature at the sudden-freeze location in agreement with the fact that the values of the cross section at upstream locations where thermodynamic equilibrium exists have no direct bearing on the terminal values. Second, the simplicity of
0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
132
G. Gundlach et al. /Relaxation in alkali dimer-monomer
the calculation procedure allows one to identify the physically important parameters. In the analysis of rotational relaxations, a procedure for handling the important case in which the terminal rotational population is non-Boltzmann is suggested. In a closely related analysis, the terminal dimer concentrations in Li free jets are correlated using the relation developed previously for rare gases [ 2 1 ] and changing only the value of the multiplying coefficient. In section 3 the experimental setup and procedures used for timeof-flight (TOF), laser-induced-fluorescence (LIF) and mass-spectrometer measurements on Li/L& and Na/Na2 beams are described. In section 4, the theoretical procedures developed previously are applied to the experimental results to extract values of effective translational and rotational cross sections and to adapt the dimer-mole-fraction relation to Li free jets. Section 5 includes a relatively simple scaling rule for the free-jet density at the sudden-freeze point as a function of source conditions, an analysis of the terminal distribution of the binding energy released in dimer formation and a discussion of some of the differences between Li/L& and Na/Na* expansions.
2. Sudden-freeze analysis The simplest useful method for extracting cross sections from terminal values of free-jet properties uses the sudden-freeze model [ 19-221. This model is motivated by the observation that, in a typical freejet expansion, transfers of energies among several degrees of freedom are fast in the region of higher pressures and enthalpies so that the various degrees of freedom are essentially in equilibrium in this region. This is consistent with the small relaxation time in comparison with the time required for the gas to pass through this region. In the region of lower pressures and enthalpies on the other hand, transfers of energies to and from certain degrees of freedom are so slow that negligible energy is transferred to or from these lagging degrees of freedom. Or in other words the relaxation time is large in comparison with the time required to pass through this region. It is convenient to divide these two regions by a “freezing point” and to use the terms “equilibrium flow” and “frozen flow” to identify the flows upstream and downstream, respectively, of this point.
molecular beams
A convenient freezing-point criterion is motivated by an examination of the hydrodynamic derivative D/Dt of the appropriate relaxation time, where D/ Dt is the rate of change observed in the reference frame of the moving gas. The appropriate relaxation time is identified by noting that, at any given point along the expansion path, the gas is relaxing toward the equilibrium state associated with the local macroscopic kinetic energy. Since constant macroscopic kinetic energy implies constant enthalpy and pressure, use of the relaxation time at constant enthalpy and pressure, r,,p, is motivated. If the flow begins from an equilibrium state, then, for equilibrium flow, rh,#<
c,
(1)
where C is a constant of the order of unity. On the basis of a recent examination of rotational, vibrational and chemical relaxations in free jets, Dao-Gibner and Knuth [ 221 suggest setting this constant equal to 1.6. If the dependence of rh,p on the thermodynamic state can be approximated by powers of thermodynamic variables, then a relatively simple alternative form of the above freezing-point criterion can be derived. Even greater simplicity is realized if one takes advantage of the finding [ 231 that the sudden-freeze model with constant collision cross section correlates the translational-freezing-point data well provided that one evaluates the cross section at the freezingpoint temperature. Define the effective collision cross section crcffvia the relaxation time at constant enthalpy and pressure: 7h.p
=bd~=
1 /fi
n&fffi
,
(2)
where & is the effective mean-free-path length and 8 the mean random speed: 8= (8kTlnm)‘l’.
(3)
Then one can write = 7h,pO 7hh,P
7h,p/7h,,O
=
7h.N
(&fd&
0 )GO
/4
(4)
where the index 0 denotes stagnation conditions in
G. Gundlach et al. /Relaxation in alkali dimer-monomer
the source. The effective cross section cancels in eq. (4) since as noted above it can be assumed that the cross section is constant at its sudden freeze value throughout the expansion: Z*,p=Zh,pO(noln)(TolT)1’2 *
(5)
If the expansion is isentropic and the gas behaves as a perfect gas, no/n= (TJT)I’Q--L)
)
(6)
one may write rh,P=r,,,
(T&-)(Y+1)‘2(y-‘).
(7)
molecular beams
133
on p,#, where p. is the source pressure, but also on TO. A more explicit expression for Mf results if an approximate relation between M and x/d* is introduced. Since the thermal energy added to the free jet as a result of dimer formation is typically of the order of only 5Ohof the terminal kinetic energy, both Na and Li expand as a monatomic gas and y= 513. For a monatomic gas and MB 1, the dependence of the centerline Mach number on axial distance may be written to good approximation [ 201 as M=3.22(~/d*)~‘).
(14)
For steady flow D/Dt=ud/dx, where u is the local hydrodynamic speed and x is the distance along the jet centerline. For adiabatic flow and a perfect gas,
Hence two convenient forms of the freezing-point criterion under these conditions result from eq. ( 13) with y= 5/3. Using eqs. (8) and ( 14) one gets with
T,JT=l+t(y-l)M*,
l+f(y-l)M*=f(y-l)M*
dTo/T -= dx
(8)
for MB 1:
(Y-w&
d*/7/,&-10= 1.85M:“.
where M is the Mach number ( =u/a) and a is the local speed of sound. The freezing-point criterion for steady flow, eq. ( 1) can be written
M,z [2To/T(y-1)]1’2,
Ma drh,P/dx= 1.6 .
d*/7h,poao=7.3(
(10)
In terms of a reduced distance x/d* with d* the nozzle throat diameter eq. ( 10) can be expressed as Mae
d7hp =1.6d*
(11) (Y+l)D(Y--L)--l
(12)
’
dTo/T
dx/P’
With eq. (9), eq. ( 12) can be rearranged to give finally (2--Y)/(Y-
1)
dx/P
d* f=Zh,poao’ (13)
where the subscript f refers to the sudden-freeze location. Since To/T, is a known function of Mf, this form of the freezing-point criterion relates (for given y) the freezing-point Mach number Mf to the dimensionless freezing-point parameter d*/rh,fla+ Note that according to eq. (2) this parameter depends not only
(15)
Substituting for Mf, using eq. (8) with
To/Tf)5’4.
(16)
Using the form which is more convenient, one can either, for given source conditions and known cross section, predict the location of the sudden-freeze point and terminal values of the relevant thermodynamic property or, from measured terminal values of the relevant thermodynamic property, extract the value of the effective cross section oeffvia eq. ( 2 ) . The latter application is of interest in the present work. The analysis to this point is applicable equally to translational and rotational relaxations. 2.1. Translational relaxation In this case, the terminal value traditionally measured is Mf. The effective cross section for translational relaxation is related frequently to the viscosity cross section gviscvia the collision number Ntrans, Ntran~= ovisc/“eff,trans
.
(17)
In a moment-method analysis of translational relaxations in monatomic free jets, Knuth and Fisher [ 241 find that Ntrans= 15/8. Hence from eq. (2) one gets
G. Gundlachet al. /Relaxation in alkalidimer-monomer molecularbeams
134
after multiplication with d* and using eq. ( 17 ) : d*/G,po = A Jz noaviscd*80 .
By definition of the mean random speed, eq. ( 3 ) , and the local speed of sound, a0 = (ykT/m)‘/2,
(18)
one arrives at -
d*
=
Th,pOaO
A (i +)I” Jz
no~visc d* .
(19)
If one eliminates d*/rhs oaofrom eqs. ( 15 ) and ( 19 ) , then one can extract values of the viscosity cross section using ovisc(q,,,)
= l.98M:‘2/nod*
,
(20)
where T,,,,, is the local temperature
at the translational freezing point. For convenient comparison with similar expressions found in the literature [ 231, eq. ( 15) is rearranged into the form
A&=0.78 (d*/T/,NUo)2’5.
(21)
Via eq. ( 19) the freezing-point parameter can then be related to the Knudsen number Kn = 1/Jz
no Oviscd*.
(22)
Then eq. (2 1) may be written alternatively M,=O.66 ( 1/Kn)2’5.
(23)
This relation correlates data for the rare gases well provided that the value of the (viscosity) cross section used in Kn is based on the local temperature at the translational freezing point as shown in fig. 3 of ref. [ 23 1. Note that frequently a correlation of terminal Mach number with Knudsen number is based on values of the cross section corresponding to the source temperature. This case will be discussed in the appendix. 2.2. Rotational relaxation Eqs. ( 15 ) and ( 16 ) are applicable also to rotational relaxations of dimers in a monomer/dimer free jet. If the dimers are present only in small mole fraction, then rh,px ~~~where, similar to eq. ( 2 ) , ~Ic-,~o =~OGT,,~
(~~TO/W)'/~,
(24)
where /I is the reduced mass for dimer-monomer collisions. If one eliminates rh,px rKPfrom eqs. ( 16) and (24)) then one can extract values of the effective rotational cross section using Gff,,rot- (4.83/nod*)
(25)
(To/T,,)"~,
where T,, is the local temperature at the rotational freezing point. Local rotational temperatures can be determined for dimers from measured LIF intensities for several rotational transitions of a selected v” +v’ transition. For a Boltzmann distribution, the emitted intensity in the non-saturated excitation regime is Ian P&,..g.QJ+
1) exp( -&IkT,,)
,
(26)
where n is the number density of particles, PL the laser power, S,.,.. the HGnl-London factor, gJ the degeneracy, E,,, the rotational energy and k the Boltzmann constant. Thus after normalization to the product of n, PL, SJp, g., and 2J+ 1 a semilogarithmic plot of measured transition intensity versus rotational energy or versus J(J+ 1) is expected to give a straight line, the slope corresponding to the inverse rotational temperature, T,,. In the measurements, which are made downstream from the rotational freezing point, however, one generally obtains a curved line with slope decreasing monotonically with increasing values of J. The slope of the curve at a given value of J is interpreted as the inverse of the local temperature in the expansion at which rotational states in the vicinity of this value of J have been frozen in. The monotonic decrease of the slope with increase in J therefore indicates that the states with larger values of J (i.e. those with smaller effective cross sections) have froze hearlier in the expansion at higher local temp, rutures. The most meaningful single cross section for such a distribution is a J-dependent cross section averaged and weighted according to the J-population for the given temperature. In a less precise, but more convenient, procedure one might approximate the cross section for a given temperature by the cross section for the most probable J= J* for that temperature. A relation between T,, and J* may be evaluated by determining the maximum of the distribution eq. (26) and using for a rigid rotor E,, =BJ( J+ 1) ,
(27)
G. Gundlach et al. /Relaxation in alkali dimer-monomer molecular beams
135
with the result TrO,= (2P+
l)*B/2k,
(28)
where B is the rotational constant of the electronic ground state of the molecule. For Na2: B= 0.154734 cm-’ [25] and for Li,: B=0.672592 cm-’ [26].
x
lmml
d lmml
2.3. Dimer formation
0 vor
P lmbarl
Dimer formation in the rare gases has been analyzed and correlated using the sudden-freeze model by Knuth [ 2 11. For terminal dimer mole fractions in the range from about 2 x 10m4to at least 0.03, the following simplified relation was found to give a good correlation of the data:
~0.5[n,,R~(~/kT,,)“~(d*/R~)*‘~]~’~,
(29)
where x, is the terminal dimer mole fraction, R0 is the zero-potential radius and 6 is the well depth. The coefficient 0.5 appearing here was identified as a function of the parameters in the rate coefficient for dimer formation. Hence, for the range of source conditions for which the alkali beams consist mainly of monomers, one might expect a relation of the general form of eq. (29) to be useful also for this case. However, since the reaction-rate coefficients for dimer formation in alkali gases and in rare gases are expected to differ significantly, one might expect that at least the coefficient 0.5 is modified in the case of alkali beams.
3. Experimental apparatus and procedures Fig. 1 is a schematic diagram of the experimental apparatus, which has been described in detail in several previous publications for Li/Li2 [ 14-181 and Na/Na2 [27] free jets. Note that the apparatus for measurements with Li/Li2 molecular beams is in principle similar to that for measurements with Na/ Na2 beams. Thus we have drawn the same diagram for both types of experiments and denote the different dimensions in the lower part of the figure. Briefly, an alkali beam is generated in a resistanceheated two-chamber stainless-steel oven and is expanded through a nozzle into a vacuum. After pass-
LO
BL
3
3
L95
1.3 lo-'
7L
x Imml
0
30
0.2
1.7 a4
13L0
1620 Li / Li2
L
10-6
d lmml P hlt.xl
207
lo-'O
395
L95
1.0 10-&
No/No2 1o‘7
Fig. I. Experimental setup for investigation of relaxation processes in alkali nozzle beams, including two-chamber alkali oven, heated skimmer, laser, photomultiplier and quadrupole mass spectrometer (QM ) . Characteristic apparatus dimensions such as diameters of nozzle and apertures, distances to the nozzle and typical background pressures in the different vacuum chambers are listed below the corresponding parts, used in the Li/L& (upper part) and Na/NaZ (lower part) experiments. For Li/L&, nozzle diameters of 0.2.0.3 and 0.4 mm have been used.
ing through a heated skimmer and an aperture the particles are excited state selectively by the light from an actively stabilized cw ring dye laser (Coherent CR 699-2 1) pumped by an Ar+ laser (Spectra 17 l-06). The fluorescence light is sampled at right angles to the molecular beam and laser beam with a photomultiplier. Further downstream a quadrupole mass spectrometer measures densities of atoms and molecules in the beam. The velocity distributions are measured using the Doppler-shift method [ 281. For this purpose, the laser beam is split into two components, one of them intersecting the beam perpendicularly, essentially not Doppler broadened or shifted, and the other at an angle (Ywith respect to the molecular beam axis. During all measurements the perpendicularly measured linewidth was less than 20 MHz for Li/Li2 and less than 30 MHz for Na/Na2. The velocity width Au of the particles in the beam leads to a Doppler broadening of the fluorescence line. Using (Y= 15” (Na) and (Y= 17.5 o (Li) and a sufficiently low laser power this broadening dominates other broadening effects such as lifetime, time-of-flight or power broadening. Under these conditions the flow velocity u and Au can
G. Gundlach et al. /Relaxation in alkali dimer-monomer
136
be determined from the measured line shift and linewidth. Several transitions X ‘Z + (v” = 0, J” ) + A ‘Z + (0 , J’ ), excited with an R6G dye laser, have been used for the characterization of the Li, and Naz dimer velocity distributions. With R6G as laser dye we have used the 3 ‘S, ,2 F= 2 + 3 2P3,2 F= 3 atomic hyperline transition for the Na atoms, whereas for Li atoms we have used DCM as laser dye for exciting the 2 2S,,2 F=2+2 2P3,2 F= 1,2, 3 atomic hyperflne transitions. The relative apparatus velocity resolution depends on both cy and the molecular beam divergence [ 28 ] and for a typical Li beam amounts to Au = 100 m/s; for a typical Na beam, Au x 320 m/s. Fig. 2 shows the reduced intensities versus J(J+ 1) for Na2(v”=0+17) and for Li2(v”=0412) [13]. Clearly, since the slopes of both curves fitted to the data decrease as J increases, both distributions are not Boltzmann distributions. This behaviour has been reported previously for Na2 [ 51 and Liz [ 141, and was anticipated in the preceding theoretical analysis. It has been shown for Na, [ 29 ] and Li2 [ 17 ] that the collisions of molecules with atoms during the nozzle expansion lead to an alignment of the angular momentum distribution of the molecules. The degree of this alignment (the number of molecules with J per-
pendicular to the beam axis) increases strongly with increasing rotational state J up to J= 10 and then levels out [ 291. This alignment affects the measured fluorescence intensity and leads to an additional uncertainty of about 5% in the determination of T,, for J< 10. This effect was neglected since uncertainty resulting from fluctuations of laser and molecular beam intensity is of the same order or higher.
4. Experimental results 4.1. Translational relaxation
Velocity distributions Au/u for the alkali atoms have been measured for various nozzle diameters using the Doppler-shift method described in section 3. The Mach numbers have been determined from the measured values of Au/u [ 301 and are tabulated in table 1 as function of nozzle temperature T,, for conTable 1 Measured Mach numbers and effective translational Knudsen numbers calculated from eq. (22) for Li atomic beams of various nozzle diameters d* and nozzle temperatures T0 for constant AT= 250 K d*
*\
I I0.8
0
100
200
300
I
,‘\t \ ,I
LOO
500
600
J lJ+ll Fig. 2. Measured and normalized fluorescence intensity for Na, X~A(v=0~17)transitionsatT0,=980KandT~=1285Kand Li,X-+A(~=0-r12)at7”,=1038K,T0=1138K[13]asfunction of rotational energy. The respective rotational states are given in parentheses. The solid lines are best-fit curves to the measured points.
molecular beams
TO W)
M
oviscCT,,“,) (A’)
Kn
(mm)
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2
1242 1287 1312 1352 1382 1412 1452 1222 1292 1322 1342 1362 1382 1412 1422 1432 1447 1462 1482 1500 1192 1222 1262
2.21 3.1 4.12 5.35 5.79 6.89 7.23 2.01 3.73 4.47 4.79 5.35 6.3 7.23 6.89 7.59 7.99 8.43 8.91 9.45 1.92 2.32 3.39
762 832 910 991 1015 1080 1094 747 882 943 954 990 1048 1100 1078 1116 1137 1156 1182 1201 743 774 859
3.33x 10-I 1.57x lo-* 1.02x lo-2 5.55 x low 3.56x 1O-3 2.40~ lO-3 1.36x lO-3 7.01 x 1o-2 1.95x 10-Z 1.15x10-* 8.62x 1O-3 6.78~ lo-’ 4.59x 10-3 3.12x 1O-3 2.69x 1O-3 2.41 x lo-’ 1.72~ IO-’ 1.49x 1o-3 1.33x lo-’ 1.02x IO-’ 1.84x10-’ 1.02x 10-I 4.85 x 10-Z
G. Gundlach et al. /Relaxation in alkali dimer-monomer
stant temperature difference AT=250 K between oven and nozzle. From the Mach numbers we use eq. (8) to calculate the corresponding local temperatures T,,. Then together with eq. (22) and the well known Landau-Lifshitz approximation for the purely elastic cross section: u= 8.083 ( C,/fig)2’5 [ 3 1 ] and we get the respective Knudsen g= (2kT,,,/m)“2 numbers. Values of the van der Waals force constant C, for Li-Li interactions have been taken from ref. [ 321. The observed dependence of A4ron Kn is presented in fig. 3. Also shown for comparison is the correlation (solid line) from eq. (23 ) found to Iit similar data for the rare gases. It is seen that the measurements for Li and different nozzle diameters fit well on the rare-gas curve. Note that this good correlation justifies a posteriori the use of eq. (20) for the determination of translational-relaxation cross sections Gvisc( T,,,,,) from the measured Mach numbers (also tabulated in table 1). Previous studies on KZ [ 1] and on Rb2 and CsZ [ 21 have shown that there does exist a velocity slip between the faster atoms and the slower molecules. Bergmann et al. on the other hand found in their stateselective investigation of Naz expansions [ 41 that for pod> 1 Torr cm only the v= 1 and v= 3 vibrationally excited molecules are slower than the atoms, whereas molecules in u= 0, J= 28 are significantly faster. We have observed in this investigation at TO= 1482 K and TO,= 1232 K that the Li2 ( ZI=1, J= 5) molecules are slower than the atoms. Moreover the Mach number for these molecules is 18% smaller than the corre-
1L lo-’
10-J 10-z Id Knudsen Number. Based on Freezing-Point
molecular beams
137
sponding Mach number for the atoms. Using vibrational states between v=O and v=4 and rotational states between J= 6 and J= 40 Niedner found that the Mach number of Li, decreases with increasing internal energy E of the molecules as Ma E -‘.’ [ 111. Thus also for Li2 molecules in the vibrational ground state we expect a smaller Mach number and velocity compared to the atoms, thus supporting the results found for Rb2, Csz and K2 in refs. [ 1,2]. ForNa2(0, 6), TO= 1100 K, andp&0.6 Torr cm we have measured a higher velocity than that of the corresponding atoms. The respective Mach number was 2OWhigher than the corresponding Mach number of the atoms. Note that this difference is of the same order of magnitude as that measured by Bergmann for Naz (0, 28) and p,d= 2 Torr cm. For pod= 0.5 Torr cm, however, Bergmann measured inside the error bars the same Mach number for atoms and molecules (0, 28 ), but suggested a steeper increase of Mach number of Naz(O, 28) than that of Mach number of the atoms with increasing pod. This is in agreement with our finding at p,d= 0.6 Torr cm. 4.2. Rotational relaxation Fig. 4 shows the LIF measured terminal rotational temperature as a function of oven temperature for J-c 12 for a Na/Na2 beam. In these measurements the nozzle was maintained at a 200 K higher temperature than the source chamber to prevent clogging. Under these conditions the dimer concentration at TO,= 962 K is about 4%, decreasing to about 2% at TO,= 864 K due to the decreasingp, #l. As expected, with decreasing T,,, the terminal rotational temperature increases since the number of collisions decreases with decreasing density in the source. For comparison, a value of T,, determined using the data for J> 12 is included, revealing the less efficient rotational cooling for high rotational quantum numbers (see fig. 2). Measured values of T,, as a function of nozzle temperature To at constant TO,of 900 K are shown in fig. 5. Again, the increase of T,,, with increasing TOis as expected since both (a) the number of collisions de-
Cross Section
Fig. 3. Terminal Mach number as function of Knudson number for Li, AT=250 K for various nozzle diameters of 0.2 (A ), 0.3 (0 ) and 0.4 mm (0 ). the solid line is the fitting curve for rare gases, eq. (23) from ref. [23 1.
” For Na the partial pressure p0 (Torr) is correlated with
TO,via 10g,~p~=7.51-5344.5/T,,(K) for 500
138 125
--/
G. Gundlach et al. /Relaxation in alkali dimer-m~~orner m~iecul~r beams 8,
1 N%
AT = 200K i
4 P P Oi Oven Tempemture, Tw [Kl
Fig. 4. Measured terminal rotationai temperature as function of oven temperature for a Na/Naz beam, constant difference AT= To- To,=200 K and f0rJ-z 12 (circles). The error bars are twice the standard deviation. For comparison, a value obtained from a distribution measured at J> 12 is included (triangle).
creases and (b) the initial population of molecules with high J increases. Values of the effective rotational cross sections for Na/Naz have been extracted from the deduced values of T,, ( 3 1;) using eq. (25). Note that one can extract in principle for each combination of To and T,, a value of the rotational cross section. In view of
fig. 2 we note, however, that the value of T,, is very sensitive to the J-values from which it is determined and gives a characteristic arot for the overall rotational freezing only if one uses an appropriate range of J-values which includes the most probable J derived from a Boltzmann distribution with temperature 1”,,. Hence we have omitted all other measurements of T,, presented in figs. 4 and 5 for the calculation of the most meaningful a,,,. The average deduced value of the cross section is then 478 A’ for Na/Naz with a most probable J* of 7. Analyzing previously reported values of T,, for Li/LiZ [ 111 with J*= 20 we get 148 A2. These values are compared in fig. 6 with values for Na/Naz obtained by Aerts et al. [8] from the relaxation of individual Jstates ( To,=850 K, AT=30 K, d*=O.S mm) by integrating the relaxation equations. Another value for Li/L& has been extracted using the present suddenfreeze model and the data of Fuchs and Toennies [ 13 ] (T,,=1138K,AT=100K,d*=0.3mm)withJ*=8. The solid line is the best-fit curve to the measured points, indicating essentially the same crmt(J) correlation for Na2 and Li2 for the investigated region of J-values. 4.3. Dimer formation Terminal Li2 mole fractions in LifL& free jets have been measured with a quadrupole mass spectrometer with a low ionizing electron energy of 20 eV. Under these conditions the Liz fragmentation induced by the ionizing electrons is expected to be negligible. In these -----
I,------
Solid Symbols No, Open Symbols Li,
zoi
$0
,
100 Temperature
200
t
300
Difference, AT [K1
Fig. 5. Terminal rotational temperature as a function of temperature difference AT= To - T,, for NaZ and constant T,,= 900 K. Low J-values have been used for the determination of T,*,. The error bars are twice the standard deviation. The line is the result of a calculation using the sudden-freeze model, fitted to the measured points at AT= 116 K.
Rotational
State.
_I
Fig. 6. Rotational cross section as a function of J for Naz expansion with AT=200 K, d*=O.Z mm (m), compared with values forNazfromref. [8] (m)andforLi,fromref. [II] (V)and ref. [13] (A).
G. Gundlach et al. /Relaxation in alkali dimer-monomer
experiments the nozzle diameter was 0.2 mm and the Li source vapor pressures ranged from 0.92 to 49.9 Torr corresponding to nozzle temperatures from 1170 to 1460 K. With the corresponding dimensionless parameters eq. (29) provides a good tit of the data after an adjustment of the prefactor:
=180[noR~i”(~/kT,)“’
(d*/R,in)2’5]5’3) (30)
where Rmin is the RKR minimum-potential radius, R,,=2.673 8, [26]. The’ coefficient 180 appearing here is more than two orders of magnitude larger than that used for rare gases, partly due to the much deeper potential wells of the alkali metals in the ‘X state compared to the weak van der Waals potentials of the rare gas dimers. Cf. fig. 2 of ref. [ 2 11.
molecular beams
139
of Z= 1.05 for high A4. Table 2 compares this value with values derived for Na and previously reported values for K, Rb and Cs [ 1,2 1. Those results are compared with values of Z,,,,, (atoms+ mol. ) , which is the expected value of Z if not only all the stagnation enthalpy but also all the binding energy released in the measured dimer formation were converted into kinetic energy of direct motion. The calculation of this value requires the value of the dimer mole fraction given by eq. (30) for Liz. The values for K2, Rb2 and Cs2 respectively can be found in refs. [ 1,2 1. Due to the relatively large and quantitatively unknown dimer fragmentations in the case of Na2 detection at an ionizing electron energy of 70 eV, no value of &,,(atoms+mol.) is given for Na. Finally, from Z( measured ) and Z,,,,, (atoms + mol. ) , the fraction 0 of the dimer binding energy which (from an energy balance with negligible dimer mole fraction in the source) went into excitation of internal degrees of freedom was calculated from
5. Discussion @(internal) 5. I. Translational relaxations
=
The ratio .Z= u/ (SkT,/m) ‘I2 provides additional information on the partitioning of energy between internal and translational degrees of freedom during the expansion. For negligible dimer formation with 1.0 [ 341. In fig. 7 we show the meaY=5/3,Zn,,= sured values of C for three nozzle diameters as function of T,, for Li. The curves level off at a maximum I
I
I
,
1
Li
I
1300
I200
Source
Temperature,
I
I
v.00
1500
1 I
I
To [Kl
Fig. 7. Atomic velocity ratio Z= u/ (SkT,Jm) ‘I2 as a function of source temperature To for Li and d* = 0.2 ( x ), d* = 0.3 ( V ) and d*=0.4 mm (0 ). The horizontal line shows Z,,,,, for a purely atomic beam.
C&,(atoms+mol.)-Z’(measured) Z&,(atoms+mol.)1
*
(31) The values calculated for Li, K, Rb and Cs are also listed in table 2. If the dimer mole fraction in the source is not negligible, then the realized value of @ is smaller than this calculated value. Since the data used in constructing table 2 are taken from three different references (using widely different experimental conditions), no effort is made here to extract trends from the listed values of 0. As noted earlier for Na the terminal velocity of the molecules (v= 0) is greater than that for the atoms, whereas in the case of Li and the heavier alkalis the velocity of the molecules is smaller than that of the atoms. In trying to explain this behaviour we note two counteracting effects. In going from the lighter to the heavier alkalis the binding energy decreases so that less energy is potentially available, but the vibrational energy spacing also decreases so that the VT transition probability increases. The unique behaviour of Na2 would be explained if as one went from Na2 to K2 the decrease in well depth dominated whereas as one goes from Li2 to Na2 the decrease in vibrational energy spacing dominates.
G. Gundlach et al. /Relaxation in alkali dimer-monomer
140
molecular beam
Table 2 Distribution of binding energy released in dimer formation between translational and internal degrees of freedom
Li Na K Rb
cs
z max
z
z
(atoms)
(measured )
(atoms+mol.)
CD (internal
1.oo 1.00 1.oo 1.oo 1.oo
1.05
1.12
0.6
this work this work
1.23 1.18 1.18
0.3 0.7 0.7
111
1.08 1.17 1.05 1.05
5.2. Rotational relaxations McGeoch and Schlier [ 12 ] have suggested that, for a given gas species and for a given nozzle geometry, a fixed value of the local free-jet density nf determines the location of the sudden-freeze point. If the expansion can be approximated by an isentropic expansion of a perfect monatomic gas, y= 513, this assumption corresponds to eq. (6): nolnf = ( TO/ Tf) ‘I2 . In an examination of this suggestion, we begin with the sudden-freeze criterion in the form of eq. ( 16), which may be written d*/zh,poao = 7.3 (no/nf)5/6 .
(32)
If the dimers are present only in small mole fraction, then one may write rh,poaOas a function of nOqot as before and write, using eq. (24), nf =6.6 ( 1/no) ‘I5 ( 1/d*qot)6/5 .
Inax
(33)
It is seen that, for a given gas species and for a given nozzle geometry, the value of the local free-jet density at the sudden-freeze point is only a weak function of the source density no and an implicit function of the temperature via the temperature dependence of a,,,. The effects of changes in no and a,,, on nf tend to cancel under certain variations in experimental conditions, thus explaining why McGeoch and S&her could conclude that nf has a constant value. Note, however, that especially in the case of a change in nozzle diameter d*, eq. (33) has to be used instead of a constant value of nf. Also, if dimer-formation effects cannot be neglected, then a more sophisticated analysis is required. The increase of the measured rotational temperature T,,, with increase of the temperature difference
Ref. )
121 [21
AT= To- TO,for constant d* and TO, shown in fig. 5 also can be clarified using the sudden-freeze model. Recall the sudden-freeze criterion in the form of eq. (25 ). For constant oven temperature TO,,the density no is inversely proportional to the nozzle temperature To. Also, for constant TO,, a,,, is approximately constant. Hence, for constant d* and TO, (implying also constant po), TfccT;‘5.
(34)
The curve drawn in fig. 5 which has been normalized to the measurement with the smallest error bars at AT= 116 K is based on this proportionality. It is seen that the increase in T,, with increase in AT (or To) is due to the shift of the sudden-freeze surface upstream with decreasing source density no.
6. Conclusions Translational freezing in Li free jets is found to be well correlated by the sudden-freeze model with viscosity cross section evaluated at the freezing-point temperature and with a translational-relaxation collision number of 15/8 in agreement with an earlier analysis [ 24 1. The local slope of a non-linear semilogarithmic plot of measured fluorescence intensity of a rotational transition versus J(J+ 1) is interpreted in the present study as the inverse of the temperature in the expansion at which the rotational states in the vicinity of the local J froze. The suddenfreeze model with constant rotational cross section yields a very convenient relationship for extracting values of the J-dependent rotational cross section from values of this J-dependent rotational freezingpoint temperature. The good agreement between the J-dependent cross sections for Li/Li2 and Na/Na2
G. Gundlach et al. /Relaxation in alkali dimer-monomer
obtained here using the sudden-freeze model and those for Na/Na* obtained previously for individual J-states from a direct integration of a relaxation equation provides confidence in the use of the more convenient sudden-freeze model. The dependence of the terminal dimer mole fraction in Li/LiZ free jets on source pressure and temperature is found to be described satisfactorily by the dimensionless relationship developed earlier for the rare gases with only the value of the multiplying constant changed. It is suggested that the measurements be extended to additional values of the source-orifice diameter d* in order to either confirm or modify for alkali free jets the dependence on d* used for the rare gases as described in eq. ( 30). Finally, the simplicity of the sudden-freeze model is found to facilitate physical insights into observations and empirical relations. For example, use of the model clarifies (a) the limitations of using the sourcetemperature cross section in correlations of free-jet terminal values, (b) the limitations of associating the location of the sudden-freeze point with a fixed value of the free-jet density and (c) the observation that the terminal rotational temperature in a Na/Na;! free jet increases as To- To, increases for constant d* and To,. In summary, the present measurements and analyses of terminal states in free jets demonstrate the convenience and effectiveness of the suddenfreeze model in either (a) extracting the relaxationrate cross sections from measurements of terminal states or (b) predicting terminal states given values of relevant relaxation-rate cross sections.
molecular beams
141
Appendix
Recall that the values of the Knudsen number in eq. (23) were based on values of the translational cross section corresponding to the local translational temperature. If the correlation of terminal Mach number with Knudsen number is based on values of the cross section corresponding to the source temperature, then A&= 1.2 &i-0.4
(35)
is typical. The two correlations are equivalent if &( Tf) = Mf( TO 1 or G,,,
( T,)
l~trans
(To)=
( 1.2/0.66)2.5=4.5 .
A case for which this requirement is met exactly is one in which o,,,, is proportional to T - ‘I3 and To/ T+ 90. For a monatomic gas, this temperature ratio is obtained for M,= 16. Although a,,,, proportional to T-1/3 and To/T+ 90 might be representative of many experimental situations, they certainly are not universal conditions. Hence the use of eq. (35) and the cross section evaluated at the source temperature is recommended for convenience, e.g., in system-design calculations. The use of eq. (23) and the cross section evaluated at the translational-freezing-point temperature is recommended for analyses where a higher level of precision is desired, e.g., in extractions of cross sections from measured values of terminal Mach numbers.
References Acknowledgement
The authors are indebted to G. Niedner for permission to use his previously unpublished results on Li/Li, terminal rotational temperatures and thank Professor Dr. C. Nyeland and A. Slenczka for carefully reading the manuscript. Financial support by the SFB 93 “Photochemistry with Lasers” of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
[ 1] P.B. Foreman, G.M. Kendall and R. Grice, Mol. Phys. 23 (1972) 117.
[ 21 R.J. Gordon, Y.T. Lee and D.R. Herschhach, J. Chem. Phys. 54 (1971) 2393. [3] M.P. Sinha, A. Schultz and R.N. Zare, J. Chem. Phys. (1973) 549. [4] K. Bergmann, U. Hefter and P. Hering, Chem. Phys. (1978) 329; 44 (1978) 23. [5] K. Bergmann, U. Hefter and J. Witt, J. Chem. Phys. (1980) 4777. [ 61 M. Kompitsas, K. Kolwas and H.G. Weber, Chem. Phys. (1981) 221. [ 71 F. Aerts, M. Hulsman and P. Willems, Chem. Phys. (1982) 233. [8] F. Aerts, M. Hulsman and P. Willems, Chem. Phys. (1984) 319.
58 32 72 55 68 83
142
G. Gundlach et al. /Relaxation in alkali dimer-monomer
[9] C.-Y.R. Wu, J.B. Crooks, S.-C. Yang, K.R. Way and W.C.
Stwalley, Rev. Sci. Instr. 49 (1978) 380. [ lo] M.J. Alguard, J. Clendenin, R.D. Ehrlich, V.W. Hughes, J.S. Ladish, M.S. Lubell, K.P. Schiller, G. Baum, W. Raith, R.H. Miller and W. Lysenko, Nucl. Instr. Methods 163 ( 1979) 29. [ 1 I ] G. Niedner, MPI fur Striimungsforschung Giittingen, Bericht 104/1985. [ 121 M.W. McGeoch and R.E. Schlier, Phys. Rev. A 33 ( 1986) 1708. [ 131 M. Fuchs and J.P. Toennies, J. Phys. B 20 ( 1987) 605. [ 141 M. Fuchs, Ph.D. Thesis, Bericht 8/ 1985, MPI fur Stromungsforschung Gottingen, ( 1986). [ 151 M. Fuchs and J.P. Toennies, J. Chem. Phys. 85 ( 1986) 7063. [ 161 H.-G. Rubahn, N. Sathyamurthy and J.P. Toennies, Faraday Discussions 84 ( 1987). [ 171 H.-G. Rubahn and J.P. Toennies, J. Chem. Phys. (1988) to be published. [ 181 H.-G. Rubahn, Ph. D. Thesis, University of Gottingen, FRG (1988). [ 191 E.L. Knuth, Proceedings of the 1972 Heat Transfer and Fluid Mechanics Institute, eds. R.B. Landis and G.H. Hordemann, Stanford (1972) p. 89. [20] E.L. Knuth, in: Engine emissions: pollutant formation and measurements, eds. G.S. Springer and D.J. Patterson (PlenumPress, New York, 1973 ) p. 3 19. [21] E.L. Knuth, J. Chem. Phys. 66 (1977) 3515. [ 221 L.T. Dao-Gibner and E.L. Knuth, in: Rarefied gas dynamics, Vol. 2, eds V. Boff~ and C. Cercignani, (Teubner, Stuttgart, 1986) p. 514.
molecular beams
[ 231 S.S. Fisher and E.L. Knuth, AIAA J. 7 ( 1969) 1174. [24] E.L. Knuth and S.S. Fisher, J. Chem. Phys. 48 (1968) 1674. [25] P. Kusch andM.M. Hessel, J. Chem. Phys. 68 (1978) 2591. [26] P. Kusch and M.M. Hessel, J. Chem. Phys. 67 ( 1977) 586. [ 271 G. Gundlach, Diplomthesis, University of Giittingen, FRG (1987). [ 281 K. Bergmann, W. Demtriider and P. Hering, Appl. Phys. 8 (1975) 65. [ 291 U. Hefter, G. Ziegler, A. Mattheus, A. Fischer and K. Bergmann, J. Chem. Phys. 85 ( 1986) 286, and references therein. [30] J.B. Anderson and F.S. Sherman, Rarefied Gas Dyn. 2 (1966) 84; U. Buck, Ph.D. Thesis, University of Bonn, FRG ( 1969); M. Gllser, MPI fllr Strijmungsforschung Gdttingen, Bericht 112 (1972); H. Haberland, U. Buck and M. Tolle, Rev. Sci. Instr. 56 (1985) 1712; R. Engelhardt, Th. Lorenz, K. Bergmann, Th. Mietzner and A. Palczewski, Chem. Phys. 95 (1985) 417. [ 3 1 ] H.G. Bennewitz, K.H. Kramer, W. Paul and J.P. Toennies, 2. Physik 177 (1964) 84. [ 321 K.T. Tang, J.M. Norbeck and P.R. Certain, J. Chem. Phys. 64 (1976) 3063. [ 331 D’Ans Lax, Taschenbuch fur Chemiker and Physiker, III (Springer, Berlin, 1970). [ 341 R.J. Gordon, R.R. Herman and D.R. Herschbach, J. Chem. Phys. 49 (1968) 2684.