Spectroscopic Study of Rotational Nonequilibrium in
Supersonic Free Molecular Flows
Hideo Mori, Ayumi Takasu, Kenji Niwa, Toshihiko Ishida and Tomohide Niimi
Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
Abstract. In highly rarefied gas flows, there appear nonequilibrium phenomena not only between translational and rotational
energy modes but also in rotational mode. To analyze these highly rarefied gas flows, we established the experimental system
for 2R+2 N2 -REMPI and applied it to the measurement of rotational temperature in a supersonic free molecular flow of
nitrogen. The data in the Boltzmann plot, obtained from the measured REMPI spectra, cannot be fitted by one line but
approximately by two lines, revealing the non-Boltzmann distribution of rotational energy in the ground state. In comparison
with the Boltzmann distribution at the source condition, it is clarified that the rotational energy distribution at relatively high
rotational level of J ≥ 13 is unchanged during the flow expansion, while there are rotational transitions from levels in J ≤ 12
to lower levels by the molecular collisions during the expansion.
INTRODUCTION
Equilibrium condition is established by a large number of intermolecular collisions. If there are enough intermolecular
collisions in a system consisting of a large number of gas molecules, distributions of translational, vibrational, and
rotational energy follow the Maxwell-Boltzmann distribution, and these distributions reveals the same temperature.
However, if the number of intermolecular collisions is very low, there appear nonequilibrium phenomena among
these energy modes, and these distributions reveals the different temperature. Moreover, deviation of rotational energy
distribution from Boltzmann distribution occurs if the molecular collision number is too low to maintain the Boltzmann
distribution.
In this study, the resonantly enhanced Multiphoton ionization (REMPI) technique is applied to an analysis of a
supersonic free molecular flow of nitrogen gas. In the 2R+2 N2 -REMPI technique[1, 2], nitrogen molecules are ionized
by two steps, i.e., the first step from the ground state to the resonance state (two photons) and the second step from
the resonance state to the ionization state (two photons). The nitrogen ions are detected as a signal and its spectrum
depending on the wavelength of an irradiated laser beam is analyzed to measure the rotational energy distribution, or
the rotational temperature.
The experimental setup for 2R+2 N2 -REMPI is established, and is applied to a measurement of the rotational
temperature in a supersonic free molecular nitrogen flow. From the Boltzmann plot of the REMPI spectra, the rotational
energy distribution of nitrogen in the free molecular flow is analyzed, and the deviation of the rotational energy
distribution from the Boltzmann distribution is discussed.
ROTATIONAL ENERGY RELAXATION
Now we consider the process of the rotational energy relaxation in supersonic free jets, by using the relaxation equation
by Gallagher and Fenn[3]. The relaxation time τr is defined as
DTrot
Ttr − Trot
=
,
Dt
τr
(1)
where Trot and Ttr are the rotational and translational temperature, respectively. For a one-dimension steady flow,
D/Dt is given as D/Dt = u · d/dx, using the flow velocity u and the distance x downstream from the nozzle exit. The
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
612
TABLE 1. Two-photon Hönl-London factors of nitrogen for the
a1 Πg ← X 1 Σ+
g transition using linearly polarized light
S(J )
Branch
O
P
Q
R
S
(∆J = −2)
(∆J = −1)
(∆J = 0)
(∆J = 1)
(∆J = 2)
M(O)J (J − 2)/15(2J − 1)
M(P)(J + 1)/30
M(Q)(2J + 1)/10(2J − 1)(2J + 3)
M(R)J/30
M(S)(J + 1)(J + 3)/15(2J + 3)
rotational collision number Zr is a number of collisions during the relaxation time τr and is defined as
C̄
Zr ≡ τr .
λ
(2)
C̄ and λ are the mean speed and the mean free path of molecules, respectively. If the isentropic flow is assumed, the
relaxation equation can be written with the Mach number Ma as
d(Trot /T0 ) −4Ddm 2 n0 π /γ (Trot /T0 ) 1 + (γ − 1)Ma 2 /2 − 1
=
(3)
γ /(γ −1)
d(x/D)
Zr Ma 1 + (γ − 1)Ma 2 /2
where D is the nozzle diameter. Eq. (3) can be solved numerically by the Runge-Kutta method with the empirical
formula of the dependence of Ma on x at the center line of supersonic free jets proposed by Ashkenas and Sherman[4].
Figure 1 shows the relaxation in the free molecular flow calculated by solving Eq. (3) using the conditions of the
study mentioned in the following chapters. The source pressure and the temperature are 1.2Torr (160Pa) and 293K,
respectively, and the nozzle diameter D is 0.5mm. The rotational collision number Zr is assumed as 4.2, proposed by
Miller and Andres[5]. In this case, at the ratio of the distance from the nozzle exit x to D is 10, translational temperature
is derived as 17K, and the rotational temperature is 210K. From the result, we can expect that the nonequilibrium, or
non-equipartition between the translational and the rotational energy, will occur in the condition, because of very low
number density.
THEORY OF REMPI SPECTRA
The theory of 2R+2 N2 -REMPI spectra [1, 2, 6] is used to analyze the possibility of rotational temperature measurement of nitrogen.
If the rotational energy distribution follows the Maxwell-Boltzmann distribution and the rotational temperature is
Trot , the number of molecules NJ in the rotational level J is given by[7]
NJ ∝ (2J + 1) exp(−Erot /kTrot ),
(4)
where k is Boltzmann’s constant and Erot is the rotational energy. Because each state with total angular momentum J has a (2J + 1)-fold degeneracy[7], the distribution is given by the product of the simple Boltzmann factor
exp(−Erot /kTrot ) and the deneracy 2J + 1. When the rotational energy distribution follows Eq. (4), the rotational line
intensity in 2R+2 N2 -REMPI spectra is given by
IJ ,J = Ag(J )S(J , J ) exp(−Erot /kTrot ),
(5)
where A is the constant independent of the rotational quantum number J of the ground state and J of the resonance
state, but including laser flux, number density, vibrational transition strength (Franck-Condon factor). g(J ) is the
nuclear spin degeneracy depending on the parity of J and the spin Is of the nuclei[7]. For N2 (Is = 1), g(J ) takes
3 and 6 for odd and even J , respectively. S(J , J ) is the two-photon Hönl-London factor for the a1 Πg ← X 1 Σ+
g
transition. In Table 1, the values of S(J , J ) for linearly polarized light[8, 9, 10] are listed (in this table, ∆J = J − J ).
The M(O)–M(S) in Table 1 are the relative transition intensity factors of the branches given by the products of the
electronic transition dipole moments. These factors depend not on J but on the kind of the branches.
613
Since the spectral line intensity I(J , J ) divided by g(J )S(J , J ) depends linearly on exp(Erot /kTrot ), as is easily
derived from Eq. (5), the rotational temperature can be deduced from the REMPI spectra using the Boltzmann plot.
Even the deviation of the rotational energy distribution from the Boltzmann distribution occurs, the rotational energy
distribution itself can be derived from REMPI spectra, by multiplying I/gS by the degeneracy (2J + 1).
Erot is a function of J and is given by[7]
(6)
Erot = hc Bv J(J + 1) − DvJ 2 (J + 1)2 + · · · ,
where h is Planck’s constant and c is the speed of light. Bv and Dv are the rotational constants in the vibrational level
v, and the values are Bv = 1.9895781cm−1 and Dv = 5.74118 × 10−6cm−1 for nitrogen molecules at X 1 Σ+
g electronic
state and v = 0[11].
EXPERIMENTAL APPARATUS
Figure 2 shows the experimental apparatus that we constructed for measurements of REMPI spectra of a free molecular
flow.
Nitrogen gas is issued via a sonic nozzle, whose diameter D is 0.50mm, into a vacuum chamber. The vacuum
chamber is evacuated by two turbo molecular pumps in parallel. For the stagnation pressure of 1.2Torr (160Pa) and
the temperature of 293K, the pressure in the chamber is kept at 3.3 × 10−5Torr (4.4 × 10−3 Pa). The background
temperature in the chamber is 293K, which is the room temperature of the laboratory.
We use Nd-YAG pumped dye laser (Lambda Physik, SCANMATE OG 2E C-400) with Rhodamine 6G dye as
the laser source, and the output is frequency-doubled by a BBO crystal with an auto-tracker. The energy, oscillation
frequency, and duration time of the laser is 7mJ/pulse, 10Hz, and 7ns, respectively. The beam is focused through a
quartz lens (f = 120mm) on the center line of the nitrogen free molecular flow.
The ionized nitrogen molecules are detected by a secondary electron multiplier (Murata, CERATRON®). The ion
signal is amplified by a current-input preamplifier and averaged by a boxcar integrator, and the intensity of the signal
is stored to a personal computer. To collect the ions by the detector effectively, an anode plate (repeller) and two
cathode plates with a hole (f = 10mm), which act as “lenses” for electric field, are placed in front of the detector. The
signal intensity is integrated 100 times for each wavelength. The wavelength range is 283–284.1nm and the step of the
scanning is 0.001nm.
RESULTS AND DISCUSSIONS
Rotational Energy Distribution Deduced from REMPI
Figure 3 is a REMPI spectrum measured using the experimental apparatus shown in Fig. 2. The horizontal axis of
Fig. 3 is wavelength of the laser and the vertical axis is the signal intensity normalized by the maximum. The scales
drawn in the upper side of the figure indicate the spectral lines of nitrogen molecules, and the numbers attached to the
scales indicate J of the spectral lines.
Figure 4 is a Boltzmann plot using the spectra. The horizontal axis indicates the rotational energy of the ground
level Erot divided by k. The vertical axis is logarithm of the signal intensity I divided by the statistical weight g and
the transition probability S. The ratio of M factors in S shown in Table 1 are determined by the relative intensity
of spectral lines with the same J but belonging to the different branch, and derived as M(P)/M(O) = 0.757 and
M(R)/M(O) = 0.596. The solid line in this figure is given by a least-square fitting using the points, and the rotational
temperature is deduced from the reciprocal of the slope as 306K. The translational temperature at the focal point is
estimated as 17K assuming isentropic flow (see Fig. 1), and there is a difference among these temperatures. From
the result, the nonequilibrium, or non-equipartition among the translational and the rotational energy exists. However,
the deduced rotational temperature is slightly higher than the source temperature T0 = 293K. There seems to exist no
(exothermic) chemical reaction in the plume of pure nitrogen gas, and no shockwave seems to be generated in such a
highly rarefied flows; there are no causes to raise the temperature in the plume.
In Fig. 4, the lower and the higher level look relatively highly populated, although the middle level looks less
populated. There seems to be nonequilibrium in the rotational energy distribution, i.e., deviation from the Boltzmann
distribution. If the Boltzmann plot is carried out with the rotational level of J ≤ 8, the slope is indicated by the solid
614
Temperature [K]
500
x/D = 10.0
Trot = 210K
Ttr = 17K
Ma = 9.0
13
3
n = 3.2×10 /cm
100
50
10
5
P0 = 1.2Torr
T0 = 293K
D = 0.5mm
Translational
Rotational (Zr=4.2)
T=T0
1
0
10
FIGURE 1.
20
30
40
Axial Distance [ x/D ]
50
Results of Relaxation Equation
Personal
Computer
Grating
Controller
YAG-pumped
Dye Laser
(Rhodamine 590)
Averaged
Sig.
Trigger
SHG Generator
(BBO with Auto-tracker)
Ref.
Beam
Sampler
N2
Sig.
Boxcar Integrator
Photodiode
Preamplifier
Chamber
Quartz
Lens
Secondary
Electron Multiplier
(CERATRON R )
Turbo molecular pump
Rotary pump
FIGURE 2.
Experimental apparatus
615
S
1
R
Q
P
10
O
10
4
19
20
20
0
16
P0 = 1.2Torr
T0 = 293K
x/D = 10
0.5
0
283
283.5
Wavelength[nm]
0
3
4
5
6
7
-1
-2
8
J''=2
9 10
O-Branch
P-Branch
R-Branch
11
12
13
-3
14
15
16
200
400
17 18
19
Trot =306K
-4
0
284
REMPI spectrum
FIGURE 3.
ln( I/gS) [a.u.]
Intensity[a.u.]
1.5
600 800
Erot / k [K]
FIGURE 4.
Boltzmann plot
616
1000 1200
line of Fig. 5, and the deduced “temperature” is 165K. On the other hand, if the Boltzmann plot is carried out with
the rotational level of J ≥ 8, the deduced “temperature” is 357K, as shown by the broken line. The temperature of
357K deduced from the Boltzmann plot at J ≥ 8 is higher than the background temperature of 293K. Therefore, the
population distribution cannot be assumed as a mixture of the distribution of the cooled gas flow and that of the
background gas. As a result, it should be thought that deviation from the Boltzmann distribution occurs in the plume
itself produced with the experimental condition.
Deviation of Rotational Energy Distribution from the Boltzmann Distribution
In the experimental conditions in this study, the molecular collision number in the expanding flow seems to be
very low because of low source pressure of P0 = 1.2Torr. As a result, the equilibrium cannot be maintained when the
rotational energy is transfered to the translational energy along with the cooling of expanding flows, and deviation
from the Boltzmann distribution occurs.
Figure 6 is the deduced rotational energy distribution of molecules in the plume. The horizontal axis is J, and
the vertical one indicates the population ratio, that is, the number of molecules NJ divided by ∑J NJ . The closed
circles display the distribution that is deduced experimentally. The solid line indicates the Boltzmann distribution of
Trot = 210K, which is the theoretical rotational temperature of nitrogen at x/D = 10 (see Fig. 1), and the broken line
is that of the source temperature T0 = 293K. To calculate ∑J NJ and NJ / ∑J NJ of the experimetnal results, the NJ
values at J < 2 and J > 19 are deduced by extrapolating the experimental results at 2 ≤ J ≤ 19, because they cannot be
determined by the REMPI spectra, The NJ=0 and NJ=1 can be extrapolated by the Boltzmann plot by J ≤ 8 as shown in
Fig. 5 and multiplying the deduced I/gS by (2J + 1). Using the Boltzmann plot for J ≥ 8, NJ at J > 19 can be deduced
in the same manner.
If the experimental distribution is compared with the Boltzmann distribution of T0 = 210K, the experimental
population at x < 11 is lower than the theoretical one, while the experimental one at x > 11 is higher than the theoretical
one. It is also worth noting that the population ratio at J ≥ 13 is very close to that of the source condition at Trot = 293K
(the broken line of Fig. 6). This result reveals that there are rotational transitions from levels in J ≤ 12 to lower levels
by the molecular collisions during the expansion, but no transitions from levels in J ≥ 13. It can be expected that the
probability of rotational energy exchange at high J is lower than that at low J.
CONCLUSION
In this study, the rotational energy distribution in supersonic free molecular flow with very low number density is
measured using 2R+2 N2 -REMPI method, and the nonequilibrium phenomena of a highly rarefied gas flows are
analyzed. Following concluding remarks are obtained.
1. The nonequilibrium of rotational energy distribution, i.e., deviation from the Boltzmann distribution, is detected
experimentally in supersonic free molecular flow. If the Boltzmann plot is carried out with the rotational level of
J ≤ 8, the “temperature” is deduced as 165K, while the temperature deduced with the rotational level of J ≥ 8
is 357K, which is higher than the source temperature T0 = 293K. The rotational energy distribution cannot be
assumed as a mixture of the energy distribution of the cooled gas flow and that of the background gas.
2. In our experimental condition, the population ratio at J < 11 is lower than that of the Boltzmann distribution
at Trot = 210K, which is deduced using a rotational relaxation equation, while higher rotational energy level of
J > 11 are overpopulated. Moreover, the population ratio at J ≥ 13 is very close to that of the source condition.
From the result, it can be expected that the probability of rotational energy exchange at high J is lower than that
at low J.
ACKNOWLEDGMENTS
The present work was supported by "Molecular Sensors for Aero-Thermodynamic Research (MOSAIC)", the Special
Coordination Funds, and a grant-in-aid for Scientific Research (B) of Ministry of Education, Culture, Sports, Science
and Technology. Hideo Mori also wishes to thank Japan Society for the Promotion of Science for a research fellowship.
617
3
4
ln( I/gS) [a.u.]
0
-1
5
6
7
8
J''=2
9
10
-2
-3
O-Branch
P-Branch
R-Branch
11
12
13
14
15
16
J ≤ 8 Trot =165K
J ≥ 8 Trot =357K
-4
0
200
400
600 800
Erot / k [K]
17
18
19
1000 1200
FIGURE 5. Boltzmann plot using two lines
Experimental
Theoretical
(Trot=210K)
Trot=293K
NJ /ΣNJ
0.1
0.05
0
0
5
FIGURE 6.
10
J
15
Rotational energy distribution of molecules
618
20
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