Fine structure and hyperfine perturbations in the pure rotational
spectrum of the VCl radical in its X5r state
D. T. Halfen, L. M. Ziurys, and John M. Brown
Citation: J. Chem. Phys. 130, 164301 (2009); doi: 10.1063/1.3108538
View online: http://dx.doi.org/10.1063/1.3108538
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v130/i16
Published by the American Institute of Physics.
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THE JOURNAL OF CHEMICAL PHYSICS 130, 164301 共2009兲
Fine structure and hyperfine perturbations in the pure rotational spectrum
of the VCl radical in its X 5⌬r state
D. T. Halfen,1,a兲 L. M. Ziurys,1,b兲 and John M. Brown2,c兲
1
Department of Chemistry and Department of Astronomy, Arizona Radio Observatory and Steward
Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, Arizona 85721, USA
2
Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks
Road, Oxford OX1 3QZ, United Kingdom
共Received 27 August 2008; accepted 8 March 2009; published online 22 April 2009兲
The pure rotational spectrum of the VCl radical in its 5⌬r ground state has been recorded in the
range 236–417 GHz using millimeter/submillimeter direct absorption techniques. This species was
created in an ac discharge of VCl4 and argon. Ten rotational transitions of V 35Cl were measured in
all five ⍀ ladders; an additional nine transitions of the ⍀ = 1 spin state were recorded in order to
evaluate the 51V hyperfine structure. Hyperfine interactions associated with the 35Cl nucleus were
not resolved, consistent with the ionic structure of the molecule. Because of extensive perturbations
caused by the low-lying A 5⌸r excited state, the rotational spectrum of the ground state has been
found to be quite irregular. The four lowest ⍀ ladders exhibit unusually large lambda-doubling
interactions, with the ⍀ = 1 component showing the largest splitting, over 2 GHz in magnitude. The
⍀ = 1 transitions are also shifted to higher frequency relative to the other spin components. In
addition, the hyperfine structure varies widely between the ⍀ ladders, and an avoided crossing is
observed in two transitions of both the ⍀ = 1e and 2e components. The data have been analyzed with
a case 共c兲 Hamiltonian, and effective rotational, lambda-doubling, and hyperfine constants have
been determined for V 35Cl. Higher-order parity-dependent magnetic hyperfine terms d⌬2 and d⌬3
were required in the analysis, derived from perturbation theory, in addition to the usual d⌬
parameter. The local perturbations evident in these spectra indicate that the A 5⌸r excited state lies
within the spin-orbit manifold of the ground state, well below the predicted value of 517 cm−1.
Mixing of the A 5⌸r and X 5⌬r states apparently causes both local and global perturbations in the
ground state spectrum. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3108538兴
I. INTRODUCTION
Electronic transitions of various simple vanadium compounds have been measured at visible and infrared wavelengths, including VH, VN, VO, VF, and VS.1–5 The spectra
of these species are very complicated because of numerous
perturbations. The A-X transition of VH, in fact, has not been
analyzed to date because of such effects.1 Irregularities are
seen in the hyperfine structure of the electronic transitions of
VN, VO, VF, and VS.2–5 Pure rotational data of these species
in their ground electronic state, limited to VO and VN, also
reveal complexities.6,7 The hyperfine structure in VN, for
example, is perturbed by a nearby 1⌬ state,2 and VO exhibits
irregular hyperfine patterns due to an avoided crossing.3,7
One important aspect of the chemistry of vanadium is its
melting point of 1910 ° C, which is the highest among the 3d
transition metals.8 Therefore, working with vanadium in the
gas phase can be problematic. To generate gas-phase vanadium from the pure metal, carbon-tube furnaces and laser
ablation techniques have been employed.2,4–6 Another
a兲
NSF Astronomy and Astrophysics Postdoctoral Fellow. Electronic mail:
[email protected].
b兲
Electronic mail: [email protected].
c兲
Electronic mail: [email protected].
0021-9606/2009/130共16兲/164301/10/$25.00
method of producing vanadium vapor that has proven successful is the use of liquid vanadium-containing compounds,
such as VOCl3 or VCl4.3,7 The benefit of using these species
as precursors is that high temperature methods are not required. However, they are extremely toxic and air and moisture sensitive.
One vanadium-containing molecule of interest is VCl.
The visible spectrum of this radical was first observed by
Iacocca et al.9 in 1970, but these authors could not determine
the electronic ground state. More recently, Ram et al.10,11
measured the 0–2, 0–1, 0–0, and 1–0 bands of the E 5⌬r
− X 5⌬r transition using Fourier transform infrared emission
spectroscopy. In these studies, vanadium vapor was first produced in a carbon-tube furnace; VOCl3 was then used as the
precursor, which produced a cooler gas mixture that simplified the spectrum. The ⍀ = 1, 2, 3, and 4 subbands were
identified in the spectra, but the ⍀ = 0 component could not
be located due to heavy blending, even in the colder gas.
Spectroscopic constants were determined by these authors
for the v = 0, 1, and 2 levels of the electronic ground state
using a Hund’s case 共c兲 analysis, including effective lambdadoubling parameters. However, they found that in both the
E 5⌬r and X 5⌬r states, the rotational constants of the spin
components varied in an irregular manner with ⍀, suggestive
of perturbations. Furthermore, the lambda 共⌳兲-type doubling
130, 164301-1
© 2009 American Institute of Physics
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164301-2
J. Chem. Phys. 130, 164301 共2009兲
Halfen, Ziurys, and Brown
was determined to be unusually large for a ⌬ state. Theoretical calculations performed by Ram et al.11 indicated that the
A 5⌸r state lies very close in energy to the X 5⌬r state, which
could explain the unusual patterns.
Here we present the first measurements of the pure rotational spectrum of VCl in its X 5⌬r state 共v = 0兲 using
millimeter-wave direct absorption techniques. VCl was produced in an ac discharge of gas-phase VCl4 and argon. Significant perturbations were observed in these spectra, as evidenced by unusual hyperfine patterns and large lambdadoubling splittings. In this paper, we describe our results and
analysis, and give an interpretation of the observed perturbations in the spectrum of this radical.
II. EXPERIMENT
The pure rotational spectrum of the VCl radical in its
X 5⌬r state 共v = 0兲 was measured using one of the millimeter/
submillimeter direct absorption spectrometers of the Ziurys
group. The instrument consists of a radiation source, freespace gas cell, and detector; it has been described in detail
elsewhere.12 The frequency source is a phase-locked Gunn
oscillator/Schottky-diode multiplier combination that can operate between 65 and 650 GHz. The source was modulated at
25 kHz and the signal detected at twice this frequency, producing a second-derivative line shape. The reaction vessel is
a 4 in. diameter glass cell containing two ring discharge electrodes and is surrounded by a cooling jacket, which is chilled
to −65 ° C with methanol. The detector is an InSb hotelectron bolometer. The spectral scanning is done under
computer control.
Vanadium chloride was produced in an ac discharge using gas-phase VCl4 and argon. 共VCl4 is an extremely toxic
and corrosive liquid, and caution should be taken when using
this compound.兲 The conditions that produced the strongest
signals were 1–2 mTorr of VCl4, 20 mTorr of Ar, and a
discharge power of 200 W at 600 ⍀ with a rate of 20 kHz.
The discharge plasma glowed a rich purple color while creating VCl. Within several minutes, this glow was obscured
by a dark purple residue that coated the inside of the reaction
cell.
A search for the spectrum of vanadium chloride was
conducted between 360 and 400 GHz, based on the rotational constants of the ⍀ = 1, 2, 3, and 4 components from
Ram et al.11 for the V 35Cl isotopologue 共v = 0 and 1兲. During
the spectral scanning process, numerous harmonically related
multiplets were found, and transitions for ⍀ = 1 – 4 were
quickly identified, including those of the 37Cl isotopologue.
The remaining multiplets, which were the strongest, were
attributed to the ⍀ = 0 spin components of V 35Cl 共v = 0 and
1兲 and V 37Cl 共v = 0兲. Hyperfine structure, arising from the
51
V nuclear spin of I = 7 / 2, was observed in all of the fine
structure components, although it was found to vary irregularly. 共The hyperfine structure associated with the Cl nucleus
was not observed.兲 Once the assignments were made, transition frequencies for V 35Cl 共v = 0兲 were determined by averaging pairs of 5 MHz wide scans taken in increasing and
decreasing frequency. Typically only one to two such pairs
were needed. Gaussian fits to the observed line profiles were
used to determine the center frequency and line width, which
ranged from 800 kHz at 236 GHz to 1300 kHz at 416 GHz.
The experimental error is estimated to be ⫾100 kHz.
III. RESULTS
Selected transition frequencies for VCl are listed in
Table I, and the complete data set is available from the
EPAPS.13 Ten rotational transitions were measured for the
⍀ = 0e / f, 2e / f, 3e / f and 4 ladders of V 35Cl, J = 33← 32 to
J = 42← 41, while 19 transitions of the ⍀ = 1e / f components,
J = 24← 23 to J = 42← 41, were recorded in order to unravel
the irregular hyperfine patterns observed. 共The e / f notation
refers to the lambda doublets.兲 The hyperfine structure of the
⍀ = 1e / f components is resolved into regular octets from the
J = 24← 23 to J = 34← 33 transition, after which the hyperfine components start to collapse and eventually blend into
two to three features at the J = 42← 41 line 共see the EPAPS
table兲.
As is evident from the data in Table I, the rotational
spectrum of the 5⌬r ground state is heavily perturbed, most
likely by the nearby A 5⌸r state. Manifestations of these perturbations are the unexpected ordering of the effective rotational constants for each spin component and the large
lambda-type doubling, both noted by Ram et al.11 The effects
are illustrated in Fig. 1, which shows the spin-orbit pattern of
the J = 38← 37 transition of V 35Cl 共v = 0兲 in the range 371–
378 GHz. A nonperturbed 5⌬ state would have evenly spaced
⍀ ladders from ⍀ = 0 to 4, and small lambda doubling14 since
this effect involves fourth-order perturbations when 兩⌳兩 = 2.15
In the case of VCl, by contrast, the centroid of the ⍀ = 1
component lies to higher frequency than that of the ⍀ = 2
features. The magnitude of the lambda doubling in this radical also changes considerably as a function of ⍀, and does
not follow the usual 关J共J + 1兲兴⍀ pattern.15 The separation of
the lambda doublets for the J = 38← 37 transition varies nonuniformly from ⬃1 GHz in ⍀ = 0, ⬃2.2 GHz for ⍀ = 1,
⬃1 GHz in ⍀ = 2, to 50 MHz for ⍀ = 3. It is unresolved for
⍀ = 4 共see Fig. 1 and Table I兲. There is a difference in intensity between the ⍀ = 1e and 1f components as well, as is
evident in the spectrum; this disparity amounts to a 20%–
40% decrease in the 1f lines relative to the 1e components,
and is seen in all measured transitions. This behavior can
also be attributed to mixing with the A excited state. This
pattern is exactly repeated in the V 37Cl isotopologue and the
V 35Cl v = 1 state, which are not shown in Fig. 1 for clarity.
We note that the assignment of e and f parity was arbitrary—
the lower frequency lines are assigned to transitions between
the f levels and the higher frequency features to one between
the e levels for each spin component.
Not only is the fine structure affected by perturbations
but the hyperfine interactions are irregular as well. The vanadium spin should generate an octet pattern, which appears
as expected in the ⍀ = 0 components, displayed in Fig. 2 共top
panel兲. 共There is a frequency break in the spectrum to show
the lambda doublets.兲 Note that the hyperfine assignments
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164301-3
J. Chem. Phys. 130, 164301 共2009兲
Pure rotational spectrum of VCl
TABLE I. Selected rotational transitions of V 35Cl 共X 5⌬r : v = 0兲 共in megahertz兲.
J⬘ ← J⬙
F⬘ ← F⬙
⍀
␯obs
␯obs − ␯calc
⍀
␯obs
␯obs − ␯calc
33← 32
29.5← 28.5
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
0f
322 788.885
322 783.249
322 777.661
322 772.001
322 766.355
322 760.641
322 754.858
322 749.084
0.033
0.026
0.081
0.046
0.006
⫺0.122
⫺0.329
⫺0.422
0e
323 850.665
323 846.350
323 842.030
323 837.716
323 833.398
323 829.070
323 824.739
323 820.403
⫺0.271
⫺0.221
⫺0.138
⫺0.047
0.029
0.078
0.103
0.158
29.5← 28.5
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
1f
325 324.415
325 323.342
325 322.183
325 321.070
325 319.953
325 318.764
325 317.576
325 316.229
0.388
0.233
0.044
⫺0.051
⫺0.103
⫺0.179
⫺0.206
⫺0.351
1e
327 594.483
327 593.228
327 592.031
327 590.820
327 589.566
327 588.290
327 586.991
327 585.569
⫺0.630
⫺0.439
⫺0.209
⫺0.005
0.144
0.259
0.339
0.289
29.5← 28.5
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
2f
325 574.237
325 571.695
325 569.134
325 566.543
325 563.954
325 561.309
325 558.668
325 555.986
0.450
0.367
0.240
0.081
⫺0.070
⫺0.263
⫺0.429
⫺0.591
2e
326 662.066
326 660.437
326 658.839
326 657.139
326 655.838
326 654.306
326 652.861
326 651.379
⫺1.365
⫺0.791
⫺0.284
⫺0.003
0.543
0.714
0.821
0.715
29.5← 28.5
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
3f
327 314.780
327 312.948
327 311.123
327 309.132
327 307.164
327 305.042
327 302.899
327 300.700
⫺0.326
⫺0.248
⫺0.092
⫺0.033
0.118
0.185
0.301
0.429
3e
327 361.416
327 359.435
327 357.349
327 355.173
327 352.920
327 350.633
327 348.262
327 345.809
0.419
0.348
0.242
0.116
⫺0.017
⫺0.115
⫺0.227
⫺0.353
29.5← 28.5
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
4
327 829.708
327 826.230
327 822.602
327 818.785
327 814.775
327 810.613
327 806.250
327 801.746
0.015
⫺0.004
0.006
0.005
⫺0.011
⫺0.002
⫺0.019
0.000
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
37.5← 36.5
0f
332 563.013
332 557.547
332 551.960
332 546.488
332 540.921
332 535.175
332 529.476
332 523.807
0.072
0.151
0.136
0.220
0.190
⫺0.035
⫺0.221
⫺0.250
0e
333 649.011
333 644.752
333 640.478
333 636.225
333 631.950
333 627.665
333 623.386
333 619.100
⫺0.163
⫺0.147
⫺0.082
0.009
0.068
0.100
0.116
0.170
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
1f
335 164.062
335 163.360
335 161.902
335 160.811
335 159.735
335 158.603
0.464
0.624
0.081
⫺0.048
⫺0.114
⫺0.189
1e
337 470.042
337 469.348
337 467.848
337 466.721
337 465.581
337 464.373
⫺0.830
⫺0.120
⫺0.239
⫺0.002
0.207
0.330
34← 33
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164301-4
J. Chem. Phys. 130, 164301 共2009兲
Halfen, Ziurys, and Brown
TABLE I. 共Continued.兲
J⬘ ← J⬙
F⬘ ← F⬙
34← 33
36.5← 35.5
37.5← 36.5
⍀
␯obs
␯obs − ␯calc
335 157.461
335 156.142
⫺0.224
⫺0.394
⍀
␯obs
␯obs − ␯calc
337 463.132
337 461.734
0.404
0.309
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
37.5← 36.5
2f
335 407.608
335 405.020
335 402.432
335 399.806
335 397.167
335 394.523
335 391.843
335 389.150
0.398
0.314
0.212
0.077
⫺0.055
⫺0.170
⫺0.288
⫺0.361
2e
336 507.153
336 505.289
336 503.569
336 501.854
336 500.246
336 498.607
336 497.014
336 495.481
⫺1.009
⫺0.645
⫺0.231
0.065
0.336
0.434
0.425
0.300
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
37.5← 36.5
3f
337 198.838
337 197.059
337 195.267
337 193.413
337 191.476
337 189.506
337 187.472
337 185.321
⫺0.362
⫺0.289
⫺0.165
⫺0.039
0.066
0.202
0.337
0.418
3e
337 249.528
337 247.585
337 245.533
337 243.472
337 241.274
337 239.083
337 236.722
337 234.464
0.429
0.338
0.202
0.120
⫺0.035
⫺0.120
⫺0.312
⫺0.338
30.5← 29.5
31.5← 30.5
32.5← 31.5
33.5← 32.5
34.5← 33.5
35.5← 34.5
36.5← 35.5
37.5← 36.5
4
337 730.859
337 727.529
337 724.041
337 720.384
337 716.564
337 712.604
337 708.463
337 704.181
0.012
0.007
0.007
0.001
⫺0.005
0.009
0.004
0.018
were made from the observation that the higher F transitions
are slightly stronger than the lower ones. In contrast, for the
⍀ = 1e and 1f components, the octet hyperfine pattern is nonexistent; two to three single features are present instead 共Fig.
FIG. 1. Spin-orbit pattern of the J = 38← 37 transition of V 35Cl 共X 5⌬r : v
= 0兲 in the range of 371–378 GHz. The spectrum consists of five spin-orbit
ladders labeled by quantum number ⍀. The ⍀ = 0, 1, 2, and 3 components
are split into e and f levels due to lambda doubling. 共The vanadium hyperfine splittings are not significant on this scale.兲 The ⍀ = 1 ladder has the
largest lambda doubling, and is shifted to higher frequency compared with
the rest of the spectrum, likely caused by perturbations by the nearby A 5⌸r
state.
2, second panel兲. The large difference in intensity between
⍀ = 1e and 1f lines is also visible in this spectrum. The third
and fourth panel of Fig. 2 show representative data for the
⍀ = 2 and 3 components, respectively. Again, there is a frequency break in the ⍀ = 2 spectrum. The hyperfine structure
is resolved into visible octets for both ⍀ ladders, although
the ⍀ = 2 lines show some uneven splittings. The lambda
doubling has collapsed in the ⍀ = 4 component, which has a
regular hyperfine structure, as shown in the bottom panel of
Fig. 2.
In addition, local hyperfine perturbations were observed
in the e-components of the lambda doublets of both the ⍀
= 1 and 2 levels; similar perturbations were not observed for
the f-components. Figure 3共a兲 displays the J = 39← 38 共top
panel兲 and J = 40← 39 共bottom panel兲 transitions of the ⍀
= 1e component of VCl. The spectra are centered on the harmonically related central frequency for the particular ⍀ component based on the nonperturbed transitions. Instead of the
unresolved hyperfine splittings seen in Fig. 2, irregularly
spaced multiplets are observed. One hyperfine component is
located near the predicted center frequency for both transitions. The rest of the hyperfine lines are widely spaced and
lie at higher frequencies for J = 39← 38 共top panel兲 and at
lower frequencies for J = 40← 39 共bottom panel兲. The components have a total spacing of 80–110 MHz for these transitions, as opposed to the spacing of 4–8 MHz found for
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164301-5
J. Chem. Phys. 130, 164301 共2009兲
Pure rotational spectrum of VCl
⍀ = 1e data. Here the components are widely spaced and lie
at lower frequencies for J = 40← 39 and at higher frequencies
for J = 41← 40, with a total spacing of 90–110 MHz. Again,
no unusual pattern was found for the ⍀ = 2f lines.
These sets of spectra most likely arise from a ⌬F = 0,
⌬J = ⫾ 1 avoided crossing between the X 5⌬1e and A 5⌸1e
levels at J = 39 for the ground state and J = 38 or 40 for the
excited state, and between the J = 40 level of the X 5⌬2e state
and J = 39 or 41 of the A 5⌸2e level. These results suggest
that the A 5⌸r state lies very close in energy to the X 5⌬r
state. The energies above the hypothetical J = 0 level for the
⍀ = 1e component at J = 39 and the J = 40 level of the ⍀
= 2e ladder calculated from the effective rotational constants
are 258.5 and 270.9 cm−1, respectively. This calculation implies that the A 5⌸r state must lie within the spin-orbit manifold of the X 5⌬r state.
IV. ANALYSIS
Because of the extensive perturbations, the data were
analyzed with a Hund’s case 共c兲 Hamiltonian. The highly
perturbed range of transitions, J = 35← 34 to J = 42← 41 of
⍀ = 1, J = 37← 36 to J = 38← 37 of ⍀ = 2f, and J = 39← 38 to
J = 42← 41 of ⍀ = 2e, were not included in the analysis. The
spin-orbit splitting is not known for VCl in its X 5⌬ state;11
we estimate that the spin-orbit intervals are about 78 cm−1
共see Sec. V兲. Each ⍀ component was fit individually using
an effective Hamiltonian that included rotational, lambdadoubling, and magnetic hyperfine interactions, including the
parity-dependent term,
Ĥeff = Ĥrot + Ĥld + Ĥmhf + Ĥmhf-ld .
FIG. 2. Laboratory spectrum of the J = 38← 37 transition of V 35Cl
共X 5⌬r : v = 0兲. All five ⍀ components are displayed in individual panels with
⍀ = 0 shown in the top panel and the successive ⍀ ladders from ⍀ = 1 to
⍀ = 4displayed below; each are on the same frequency scale. There are
frequency breaks in the spectra for the ⍀ = 0, 1, and 2 ladders, and the
lambda doublets for ⍀ = 0, 1, 2, and 3 are labeled as e and f. The hyperfine
structure for the five ⍀ components varies significantly as a function of ⍀.
In particular, the ⍀ = 1e / f components have only partially resolved hyperfine splittings at this J, while the other ⍀ components are resolved into
octets, as expected for I = 7 / 2 for 51V. Each spectrum is an average of two
scans 100 MHz wide each with a duration of 70 s.
adjacent transitions 共see Fig. 2兲. Several components of the
⍀ = 1e, J = 40← 39 transition actually blend into the ⍀ = 3e
multiplet, also labeled in the spectra.
As illustrated in Fig. 3共b兲, a similar pattern is observed
for the ⍀ = 2e component of the J = 40← 39 共top panel兲
and J = 41← 40 共bottom panel兲 transitions, but the hyperfine
lines spread out in opposite directions compared with the
共1兲
The lambda doubling was fit using an empirical expression
similar to that of Ram et al.,11
Ĥld = ⫾ 1/2兵qJ共J + 1兲 + qD关J共J + 1兲兴2 + qH关J共J + 1兲兴3其.
共2兲
In this equation, the upper and lower sign choices correspond
to the e and f parity levels. The ordering of the lambda
doubling is not yet established for VCl in its 5⌬ state. We
have arbitrarily chosen the parameter q to be positive
throughout so the e levels lie above the f levels for all five
spin components. This empirical modeling was adopted because when an appropriate Hund’s case 共a兲 lambda-doubling
expression15 was used, the rms values for all the ⍀ components deteriorate, in one case by several orders of magnitude.
This behavior could imply that the coupling at moderately
high J is intermediate between cases 共b兲 and 共a兲. It could
TABLE II. Matrix elements for the parity-dependent lambda-doubling-type magnetic hyperfine parameters for a 5⌬ state 共upper and lower signs refer to the
+ and ⫺ parity levels, respectively, with x = J共J + 1兲 and C = 关F共F + 1兲 − J共J + 1兲 − I共I + 1兲兴兲.
具⌺⬘ , ⍀⬘兩Ĥmhf-ld兩⌺ , ⍀典
具−2 , 0兩
具−1 , 1兩
具0 , 2兩
具1 , 3兩
具2 , 4兩
兩−2 , 0典
兩−1 , 1典
兩0 , 2典
兩1 , 3典
兩2 , 4典
0
⫿共−1兲Jd⌬3共3C / 2x兲 x1/2
⫿共−1兲Jd⌬23C / 4
⫿共−1兲Jd⌬261/2 共C / 4x兲x1/2共x − 2兲1/2
⫿共−1兲Jd⌬161/2C / 4 共x − 2兲1/2
0
⫿共−1兲Jd⌬1共C / 2x兲 x1/2共x − 2兲1/2共x − 6兲1/2
0
0
0
0
0
0
0
0
Symmetric
Downloaded 31 Oct 2011 to 150.135.211.246. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
35
5
V Cl (X 'r): : = 1e
J = 39
: = 3e
386.66
386.70
386.74
38
386.78
386.82
: = 3e
(a)
396.51
386.86
J = 40
396.55
396.59
396.63
396.67
39
396.71
Frequency (GHz)
386.90
396.75
395.34
395.38
395.42
395.46
395.50
395.54
395.58
395.62
405.15
405.19
405.23
405.27
405.31
405.35
405.39
405.43
also be caused by the perturbation with the A state, or perhaps a combination of the two. We note that Eq. 共2兲 does not
include the J-independent term that dominates the lambda
doubling in the 5⌬0 spin component.15 This term is not determinable from pure rotational, ⌬J = 1 transitions.
Two new terms were added to the Hamiltonian to account for unusually large differences in the magnetic hyperfine interactions between the lambda doublets. There are
three possible forms of the parity-dependent lambdadoubling-type magnetic hyperfine parameter, d, in the effective Hamiltonian for a molecule in a 2S+1⌬ electronic state,
Ĥmhf-ld = − 21 d⌬1共J+2I+S+ + J−2I−S−兲
− 21 d⌬2共J+I+S+2 + J−I−S−2兲
− 21 d⌬3共I+S+3 + I−S−3兲.
共3兲
The general expressions for the matrix elements of these operators in a case 共a␤兲 basis set, using standard spherical tensor notation, are as follows:
具␩,⌳⬘,S,⌺⬘,J⬘,⍀⬘兩Ĥ共d⌬1兲兩␩,⌳,S,⌺,J,⍀典 = d⌬1共− 1兲J+I+F
再
冎
I J⬘ F
关I共I + 1兲共2I + 1兲兴1/2
J I 1
⫻关共2J⬘ + 1兲共2J + 1兲兴1/2
1
兺 ␦⌳ ,⌳⫿4共− 1兲S−⌺⬘
2冑6 q=⫾1 ⬘
再
⫻关S共S + 1兲共2S + 1兲兴1/2 共− 1兲J⬘−⍀⬘
⫻共− 1兲J−⍀⬙
冉
冉
J
2
J
− ⍀⬙ − 2q ⍀
+ 共− 1兲J⬘−⍀⬘
J⬘
冊
J⬘
1
J
− ⍀⬘ − q ⍀⬙
S
1 S
− ⌺⬘ q ⌺
冊
冊
关共2J − 1兲共2J兲共2J + 1兲共2J + 2兲共2J + 3兲兴1/2
J⬘
2
冉
冉
− ⍀⬘ − 2q ⍀⵮
冊
共− 1兲J⬘−⍀⵮
冉
J⬘
1
J
− ⍀⵮ − q ⍀
冎
冊
⫻关共2J⬘ − 1兲共2J⬘兲共2J⬘ + 1兲共2J⬘ + 2兲共2J⬘ + 3兲兴1/2 ,
具␩,⌳⬘,S,⌺⬘,J⬘,⍀⬘兩Ĥ共d⌬2兲兩␩,⌳,S,⌺,J,⍀典 = d⌬2共− 1兲J+I+F
⫻
再
冎
共4兲
I J⬘ F
关I共I + 1兲共2I + 1兲兴1/2关共2J⬘ + 1兲共2J + 1兲兴1/2
J I 1
冉
冊
冊
冉
冉
冊
S 1 S
S 1 S
1
兺 ␦⌳ ,⌳⫿4共− 1兲S−⌺⬘ − ⌺⬘ q ⌺⬙ 共− 1兲S−⌺⬙ − ⌺⬙ q ⌺ 关S共S + 1兲
2 q=⫾1 ⬘
再
冉
J⬘
1
J
− ⍀⬘ − q ⍀⬙
共− 1兲J−⍀⬙
2j/4069.78Tm(J)F1f53-Dcq+,andS
⫻共2S + 1兲兴 共− 1兲J⬘−⍀⬘
⫻共2J + 1兲兴1/2 + 共− 1兲J2⬘
J
1
J
− ⍀⬙ − q ⍀
冊
关J共J + 1兲
164301-8
J. Chem. Phys. 130, 164301 共2009兲
Halfen, Ziurys, and Brown
TABLE III. Spectroscopic constants for V 35Cl 共X 5⌬r : v = 0兲. 共In megahertz; errors are 3␴ in the last quoted
ⴱ
, etc. are given in text.兲
digits. Definitions of parameters bⴱ, bDⴱ , d⌬1
Millimeter wave
Parameter
B
D
H
q
qD
qH
h
hD
hH
bⴱ
bDⴱ
ⴱ
d⌬1
ⴱ
d⌬2
ⴱ
d⌬3
rms
⍀=0
⍀=1
⍀=2
⍀=3
⍀=4
4900.495共19兲
0.000 659共14兲
−1.507共32兲 ⫻ 10−7
18.413共39兲
⫺0.001 166共28兲
−7.98共64兲 ⫻ 10−8
⫺545a
0.628a
0.000 443共64兲
1.530共29兲
−8.75共83兲 ⫻ 10−5
4955.012共26兲
0.004 186共30兲
1.11共11兲 ⫻ 10−7
43.054共50兲
⫺0.004 353共58兲
−2.30共22兲 ⫻ 10−7
489共149兲
⫺1.24共31兲
⫺0.000 168共81兲
4951.562共97兲
0.004 873共72兲
3.7共1.8兲 ⫻ 10−8
21.818共37兲
⫺0.002 463共14兲
−2.6⫻ 10−8 a
⫺508共188兲
⫺1.52共10兲
−5.4⫻ 10−5 a
4967.375 9共73兲
0.003 591 7共24兲
4974.544 93共48兲
0.003 510 18共14兲
0.000 349 4共52兲
1.84共23兲 ⫻ 10−8
443共93兲
⫺0.341共64兲
−1.2⫻ 10−5 a
835.9共8.6兲
⫺0.259共12兲
−5.8共1.6兲 ⫻ 10−6
0.317
0.012
4965.40共51兲
0.002 906共67兲
−4.72共24兲 ⫻ 10−8
4974.54共35兲
0.003 495共25兲
−1.66共24兲 ⫻ 10−5
0.114共17兲
⫺0.051 9共12兲
0.119
0.243
0.407
Opticalb
B
D
H
q
qD
qH
4953.61共65兲
0.003 451共99兲
4951.3共1.4兲
0.004 36共60兲
39.2共1.3兲
⫺0.002 72共19兲
23.47共33兲
⫺0.002 42共24兲
−5.26共31兲 ⫻ 10−5
−1.850共99兲 ⫻ 10−9
a
Held fixed.
Ram et al., 2003; originally given in cm−1.
b
⍀ = 0,
⌬E⍀ = − 2C共Bⴱb/⌬E01兲,
共11兲
where the other constants are defined above. These paramⴱ
ⴱ
, d⌬2
,
eters will be discussed in the rest of the paper as d⌬1
ⴱ
ⴱ
ⴱ
ⴱ
ⴱ
ⴱ
d⌬3, and b , where d⌬1 = B d⌬1 / ⌬E21, d⌬2 = d⌬2, d⌬3
= Bⴱd⌬3 / ⌬E01, and bⴱ = Bⴱb / ⌬E01.
Using these new terms in the analysis for the ⍀ = 0, 1,
and 2 states led to improvements in the fits, especially for
⍀ = 0, where the rms changed from ⬃2 MHz to 119 kHz. In
this case, h and hD were not well determined when allowed
to float in the analysis, and therefore were fixed to the fitted
value in the final iteration.
The effective rotational, lambda-doubling, and magnetic
hyperfine parameters for each ⍀ ladder obtained from this
analysis are listed in Table III. For several of the spin components, a number of higher-order centrifugal distortion constants 共H, qH, and hH兲 were necessary for a good fit. Also, the
constant q for ⍀ = 3 was fixed to zero, and qH for the ⍀ = 2
ladder and hH for the ⍀ = 2 and 3 ladders had to be fixed to
values obtained from the initial analysis, as they were not
defined. The fits to the individual spin components have rms
values of less than 320 kHz, except for the ⍀ = 2 ladder; here
the rms was as high as 407 kHz.
The present constants can be compared with those from
Ram et al.,11 which are also listed in Table III. As seen in the
table, the rotational and lambda-doubling constants are in
excellent agreement. Note that the lambda-doubling parameter q has a maximum value at ⍀ = 1 of q
= 43.054共50兲 MHz. This parameter is about twice as large as
the values for ⍀ = 0 and 2, q⍀=0 = 18.413共39兲 and q⍀=2
= 21.818共37兲 MHz. Both constants are empirical modeling
terms.
The large variation in the hyperfine structure in VCl can
be seen in the values of the diagonal h hyperfine parameter,
where
h = a⌳ + 共b + c兲⌺.
共12兲
A plot of the effective h for each ⍀ component is displayed
in Fig. 4. As shown in the graph, the magnitude of h oscillates from negative to positive from ⍀ = 0 through ⍀ = 4. For
a regular case 共a␤兲 progression, the values of h would fall on
a nearly straight line, following Eq. 共12兲.
V. DISCUSSION
A. Fine structure perturbations
The dominating effect in the rotational spectrum of VCl
is the perturbation of the X 5⌬r state by the excited but very
low-lying A 5⌸r state. A 5⌺− state is predicted to lie
1587 cm−1 above the ground state;11 the proximity of this
state and the 5⌸ state could account for the large lambda
doubling observed in VCl. However, the 5⌸r state is the
main perturbing state. The ⌬-⌸ interaction is most likely due
to homogeneous 共⌬⍀ = 0兲 spin-orbit perturbations, which affect the ⍀ = 0, 1, 2, and 3 components. The local hyperfine
structure perturbations in the ⍀ = 1e and 2e components also
suggest that this state lies very low in energy, almost cer-
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164301-9
J. Chem. Phys. 130, 164301 共2009兲
Pure rotational spectrum of VCl
FIG. 4. Graph of the effective diagonal magnetic hyperfine constant h
plotted against ⍀ derived in this study.
共Note that perturbed lines were not
used in determining these h parameters, hence the term “effective.”兲 To
first order, h should be equal to a⌳
+ 共b + c兲⌺, such that the points on this
plot fall on a straight line, as indicated.
The interaction of the X 5⌬r and A 5⌸r
states causes the hyperfine parameters
for ⍀ = 1 and 2 to deviate significantly
from this relationship, even when the
most perturbed lines are excluded
from the fit.
tainly within the spin-orbit manifold of the ground state. Unfortunately, the electronic spectra of Ram et al.11 showed
only ⌬⍀ = 0 transitions. Consequently, there is no information on the spin-orbit energies of the ⍀ ladders for either
electronic state of VCl and spin-orbit constants have not yet
been determined. The ⍀ = 1 component seems to be most
affected by the interaction with the A 5⌸r state. In addition,
the large lambda doubling observed in the spectrum is rather
unexpected since this effect is a fourth-order interaction in ⌬
electronic states.15 Lambda doubling has been seen in the
spectra of several other molecules with ⌬ electronic states,
such as CoO and FeF.17,18 However, the lambda doubling in
the lowest ⍀ ladders is at most several hundreds of megahertz, an order of magnitude less than the splitting seen in
the lower ⍀ ladders of VCl.
The interaction between the X 5⌬r and A 5⌸r states
causes the wave function for each ⍀ component of the 5⌬r
state, 兩␩ ; ⌳ , ⌺典, to contain smaller contributions from the 5⌸r
state, i.e.,
兩⍀⫾0典 = a0
1
冑2 兵兩
+ b0
1
5
⌬; + 2,− 2典 ⫾ 兩 5⌬;− 2, + 2典其
冑2 兵兩
5
⌸; + 1,− 1典 ⫾ 兩 5⌸;− 1, + 1典其,
兩⍀⫾1典 = a⫾1兩 5⌬; ⫾ 2, ⫿ 1典 + b⫾1兩 5⌸; ⫾ 1,0典
+ c⫾1兩 5⌸; ⫿ 1, ⫾ 2典,
兩⍀⫾2典 = a⫾2兩 5⌬; ⫾ 2,0典 + b⫾2兩 5⌸; ⫾ 1, ⫾ 1典,
共13兲
兩⍀⫾3典 = a⫾3兩 5⌬; ⫾ 2, ⫾ 1典 + b⫾3兩 5⌸; ⫾ 1, ⫾ 2典,
兩⍀⫾4典 = a⫾4兩 5⌬; ⫾ 2, ⫾ 2典 with a⫾4 = 1.0,
where ␩ specifies the electronic state, and ai, bi, and ci coefficients give the relative contribution from each state 共ai
⬎ bi ⬃ ci兲, and are labeled with the sign of ⌳ for each ket and
the value of ⍀ for that component. 共This form of mixed
wave functions was described by Hougen19 for 4⌺ states.兲
The magnitude of the coefficients depends on the energy
separation of the 5⌬r and 5⌸r states. The stronger the local
perturbation by the 5⌸ state, the more the rotational intensity
is transferred from the transitions in the 5⌬ state, which
would explain the intensity variation between the ⍀ = 1e and
1f lines. The contributions to each lambda-doubling component from the 5⌸ state are likely to be different, i.e., c+1
⫽ c−1 because of the large lambda doubling in this state,
leading to dissimilar wave functions and variations in transition probabilities.
Although the spin-orbit intervals for VCl in its X 5⌬
state have not been measured experimentally, they can be
calculated quite reliably from a knowledge of the electronic
wave function.11 Using values for ␨3p共Cl−兲 and ␨3d共V+兲 of
326 and 152 cm−1, respectively,20 the spin-orbit splittings in
the X 5⌬ and A 5⌸ states are estimated to be 78 and
48 cm−1. The two levels in the X 5⌬ state that show the
strongest local perturbations are J = 39 in the 5⌬1e spin component and J = 40 in the 5⌬2e component. Using the B-values
in Table III with the estimated spin-orbit splitting of
78 cm−1, these levels are separated by 91.0 cm−1. The significantly smaller spin-orbit splitting in the A 5⌸ state suggests that the perturbing levels are 5⌸1e, J = 38 and 5⌸2e, J
= 41. The ab initio calculation by Ram et al.11 shows that the
B-value of VCl in the A 5⌸ state is very similar to that in the
X 5⌬ state. With this assumption, we obtain a value of
51.3 cm−1 for the separation of the 5⌸1 and 5⌸2 components. This is in good agreement with the theoretical estimate, particularly so since the 共probably兲 large lambdadoubling effects in the 5⌸1 component have been ignored. It
can be seen from this analysis that the A 5⌸ state lies even
closer to the X 5⌬ state than the separation of 517 cm−1 predicted by the ab initio calculation.11
If the levels perturbing J = 39 in the 5⌬1e spin component
and J = 40 in the 5⌬2e component are the 5⌸1e, J = 38 and
5
⌸2e, J = 41 levels, respectively, as stated above, then one
hyperfine transition should be unperturbed. As clearly visible
in Figs. 3共a兲 and 3共b兲, one hyperfine feature lies close to the
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164301-10
J. Chem. Phys. 130, 164301 共2009兲
Halfen, Ziurys, and Brown
center of each spectrum near the predicted unperturbed frequency. The unperturbed transitions for the 5⌬1e spin component are J = 39← 38, F = 42.5← 41.5 and J = 40← 39, F
= 43.5← 42.5. For the 5⌬2e component, the J = 40← 39, F
= 36.5← 35.5 and J = 41← 40, F = 37.5← 36.5 transitions are
unperturbed.
In addition to the spectral features assigned to V 35Cl in
the v = 0 and v = 1 levels and V 37Cl in the v = 0 level of the
X 5⌬r state, three other monotonic progressions of harmonically related lines have been identified in the rotational spectrum. The corresponding B-values are 4858.175, 4876.645,
and 4878.298 MHz. The analysis of the local perturbations
described in the previous paragraph strongly suggests that
these harmonically related features correspond to rotational
transitions of V 35Cl in the A 5⌸ state. Further work is
needed to confirm this assignment.
B. Hyperfine structure perturbations
The chaotic nature of the hyperfine constants for the ⍀
ladders, demonstrated in Fig. 4, renders a determination of
the Frosch and Foley21 hyperfine parameters 共a, b, and c兲
impossible. The mixing of the X 5⌬r and A 5⌸r states quite
possibly involves a cross term between the spin-orbit and
nuclear-spin orbit operators, 共L · S兲共I · L兲, which would generate a term of the form 共I · S兲 in the effective Hamiltonian.
Following the observed nature of the perturbations, the magnitude of this effective term would be J-dependent.
The ⍀ = 4 component is not affected by the spin-orbit
perturbation with the A 5⌸r state, and therefore h in this case
reflects the unperturbed hyperfine constants. The ⍀ = 3 component also shows a regular hyperfine pattern similar to that
for ⍀ = 4, and therefore it is not likely to be highly perturbed.
The hyperfine parameters of these two spin components 共h4
= 836 MHz and h3 = 443 MHz兲 are in reasonable agreement
with that of the V atomic spin-orbital hyperfine constant P
= 437.6 MHz,22 as well as the a constant for VN of 338.8
MHz,7 if 共b + c兲 ⬃ 0 关see Eq. 共12兲兴.
VI. CONCLUSION
The measurement of the pure rotational spectrum of VCl
has demonstrated that this species is highly perturbed in its
ground electronic state. A number of transitions in the ⍀
= 1 and 2 ladders could not be analyzed, and the data that
were fit produced unusually large lambda-doubling terms
and erratic hyperfine parameters. Conventional case 共a兲
lambda-doubling expressions could not be effectively employed, necessitating the use of an empirical formula for this
interaction. Two new parity-dependent magnetic hyperfine
parameters had to be incorporated into the analysis to obtain
reasonable fits, as well, derived by perturbation theory. These
terms account for additional interactions between the spinorbit components. The irregular pattern for the magnetic hy-
perfine parameter h is likely a result of 共L · S兲共I · L兲 interactions between the ground and excited state. If the analysis of
the perturbations is correct, it implies that the A 5⌸r state lies
only slightly above the ground state 共about 50 cm−1兲, lower
than the theoretical predictions. Indeed, it is quite possible
that the 5⌸ state is the true ground state of VCl, as measured
by the relative positions of the central spin component with
⌺ = 0. In this case, Ram et al.10,11 actually observed transitions to a very low-lying but excited state of VCl in their
work. Their assignment of the 5⌬ state as the ground state
was based on an ab initio calculation, which may not be as
reliable as required. Further study of the perturbations in VCl
is clearly desirable.
ACKNOWLEDGMENTS
We are very grateful to the referee of this paper for
pointing out the implications of the local perturbations and
for several other helpful comments. This research was supported by the NSF under Grant No. CHE-0718699 and
NASA under Grant No. NNX06AB64G. D.T.H. was supported by the NSF Astronomy and Astrophysics Postdoctoral
Fellowship under Award No. AST-0602282.
R. S. Ram and P. F. Bernath, private communication 共August 26, 2008兲.
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