Intensities of line features in vibration-rotational bands 2–0
to 6–0 of
14
N
16
O X
2
P
r
and experimental evaluation of a
radial function for electric dipolar moment
Yuan-Pern Lee
a,b, Swee-Lan Cheah
c, J.F. Ogilvie
d,*aDepartment of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001 Ta-Hsueh Road,
Hsinchu 30010, Taiwan
bInstitute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan
cDepartment of Chemistry, National Tsing Hua University, 101, Section 2, Kuang Fu Road, Hsinchu 30013, Taiwan dEscuela de Quimica, Universidad de Costa Rica, Ciudad Universitaria Rodrigo Facio, San Pedro de Montes de Oca,
San Jose 2060, Costa Rica Received 14 September 2004 Available online 25 March 2005
Abstract
We measured the strengths of individual line-like features, representing unresolved K doublets, in vibration-rota-tional bands 2–0 to 6–0 of14N16O within each substate of electronic ground state X2P1/2,3/2in mid and near infrared
regions. Analyses of these data to derive values of matrix elements for vibrational transitions enabled production of a radial function for electric dipolar moment, containing seven parameters, that satisfactorily reproduces the intensities of about 700 such features for vibrational states up to m = 6.
2005 Elsevier B.V. All rights reserved.
1. Introduction
We have measured the intensities of individual features in vibration-rotational bands of gaseous NO near 300 K from the first to the fifth overtone, or to vibrational states m0= 2, 3, 4, 5 and 6 from
m00= 0. Each such band within the electronic
ground state X2Pl/2,3/2has 24 branches, but 12 of
these occur in two somewhat separate satellite bands, which involve also a change of total elec-tronic angular momentum between elecelec-tronic sub-states, and of which their consequent weakness makes them generally difficult to detect. Because our spectral resolution was insufficient to separate the K doublets, we observed effectively six branches of line-like features, comprising two weak Q 1350-4495/$ - see front matter 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.infrared.2005.01.002
*
Corresponding author.
E-mail addresses: yplee@mail.nctu.edu.tw (Y.-P. Lee), ogilvie@cecm.sfu.ca(J.F. Ogilvie).
branches near the centre of each band and two intense P and R branches on each side of that centre. For each of five specified bands we mea-sured the strengths of these features and present here the results of our analysis of these intensity data; this analysis yields a radial function, i.e. of internuclear distance R, of electric dipolar moment p(R) that reproduces those data within experimen-tal uncertainties of measurement, which we com-pare with calculated data.
Previously reported measurements [1–3] of intensities of vibration-rotational bands of NO were based on gross features as a result of broad-ening through use of gaseous samples at large pres-sures, with these pertinent exceptions: there exist two independent measurements[4,5]of individual lines or features in the fundamental band, 1–0, two separate measurements of individual lines in the first overtone [6,7], 2–0, and single measure-ments of a band originating in m00= 1 in the
se-quence with Dm = 1, or band 2–1 [8], and a band originating in m00= 1 in the sequence with Dm = 2,
or band 3–1 [9], with our own previous work on bands 5–0 [10] and 6–0 [11]. We have reanalysed the latter spectra, and here combine those data with fresh measurements on bands 2–0, 3–0 and 4–0 in a consistent manner to derive p(R) accord-ing to an established method [12].
2. Experiments
We recorded all spectra with an interferomet-ric spectrometer (Bomem DA8, evacuated to 15 N m2) and a large vessel to contain gaseous samples. The latter cylindrical cell (Infrared Asso-ciates, model 100, of White type) has an optical path of length 1.375 m between mirrors for multi-ple internal reflections to produce an optical path variable between 8.25 m and 107.25 m in incre-ments 11 m. Spots in sequences on internal mirrors from the beam of a helium-neon laser enable a pre-cise count of a number of passes of the beam through a gaseous sample. The number of passes that we employ depends on the band being mea-sured and the pressure of NO required to yield maximum net absorbance, log10(I0/I), of the most
intense feature in each band less than 0.12, to
avoid distortion of signals from effects of satura-tion. With a capacitance manometer (MKS model 122A, precision 0.5% of reading), we measured the pressure of a gaseous sample before and after each collection of interferograms. We employed partial pressures (N m2) of NO in a range [68, 1.07· 105
] as appropriate for a particular band, within the limitations of properties of the cell; with NO at small partial pressures we added He gas to a total pressure at least 67,000 N m2to ensure broaden-ing of lines to widths substantially greater than the spectral resolution. We recorded interferograms for samples generally at three pressures, with at least two separate collections at each pressure. With two digital thermometers (Omega, thermo-couples of type K) that indicate maximum and minimum temperatures (in C) over a duration of spectral measurements, we monitored the tem-perature of the vessel at each end; for all spectra accepted for analysis here, the maximum variation was ±0.5 K about the mean.
To obtain the best conditions for each band, for spectral measurements we employed beam splitters made of CaF2 for band 2–0 and of quartz for
bands 3–0 to 6–0, a SiC or tungsten filament as source, and an InSb detector near 77 K or a Si photodiode at ambient temperature. To limit the spectral region of radiation reaching a detector, we selected optical filters (CVI, Optical Coatings Limited, Omega, and Spectrogon) to provide pass bands about the band of intended measurement. Optical resolution was set at 1.0 or 1.5, 1.3 or 1.5, 4.0, 6.0 and 6.0 m1 for bands 2, 3, 4, 5, 6–0, respectively, which was appreciably smaller than the apparent width of isolated individual spec-tral features in each case. We co-added interfero-grams numbering 300–750 and accumulated over periods up to 13.5 h. A ratio of a spectrum from each co-added interferogram with a reference spec-trum yielded an absorbance specspec-trum, of which a sample for band 4–0 appears inFig. 1. To reduce these spectra, and for further analysis of data, we employed software (Grams, Galactic Industries) designed to operate with the interferometric spectrometer; the characteristics (wave number, stature or apparent maximum net absorbance, width and area) of each feature were collected into a commercial spreadsheet, and further calculations
were conducted with a symbolic processor (Maple 9.5, Maplesoft, Waterloo Maple Inc.). Strengths of more intense lines in each band have uncertain-ties10%, but for weak lines up to 30%; statistical weights based on multiple spectra were applied in spectral reduction.
To purify gaseous NO (AGA Specialty Gas, nominal chemical purity 99.5% in a cylinder as re-ceived), we passed the gas slowly through a trap charged with silica gel and cooled below 195 K
[13]; this treatment served to diminish greatly other oxides of nitrogen, but not N2. Tests of
pur-ity based on spectra in the mid infrared region indicated that remaining impurities occurred at a negligible level. Because the vessel has a large vol-ume and large area of internal surface, its evacua-tion through a narrow valve presented a problem for removal of residual water vapour, notorious for its retention on polar or metallic surfaces. In the regions of bands 2–0, 3–0, 4–0 and 6–0 of NO, vestigial water vapour either in the vessel or within the optical path inside the interferometer can absorb intensely, as transitions of H2O involve
fewer vibrational quanta than for NO in the same region. For this reason, to achieve minimal inter-ference from spectral lines of H2O at wave
num-bers of lines of NO, protracted pumping of both the vessel and the space inside the interferometer is essential. Comparison ofFig. 1with a published spectrum [14] for band 4–0 of NO indicates the extent of our success. Under our conditions of preparation for recording of spectra and our fitting
of all lines within each selected short segment of wave number scale, and in the absence of fortu-itous coincidences between intense lines of H2O
and of NO, interference from H2O with our
mea-surements of spectral features of NO is generally negligible.
3. Theoretical basis
For absorption by a diatomic molecular species in electronic state X1Rwithin a gaseous sample at temperature T, the strength Slof a line attributed
to a spectral transition from a state j0, J00i to
an-other state jm0, J0i in a vibration-rotational
spec-trum is represented according to [12]
Sl¼ ð8p3=3hcÞ½expðhcE0J=kBTÞ=4p0Q~m0
½1 expðhc~m0=kBTÞjijjhm0J0jpðxÞj0; J00ij2
ð1Þ in which appear fundamental physical constants h, c, kB and 0, total partition function Q, spectral
term E0J of the initial statej0, J00i of a transition,
and wave number ~m0 of the transition; i
1/2[J0(J0+ 1) J00(J00+ 1)] is a running number of
value J00+ 1 for a line in branch R orJ00for a line
in branch P. Injhm0J0p(x)j0, J00ij2
that is a square of an experimental matrix element for a transition be-tween specified states, p(x) is a radial function for electric dipolar moment of an absorbing molecular species in terms of reduced displacement x = (R Re)/Re with internuclear distances
instanta-neous R and equilibrium Re. Modification of this
formula to make it applicable to a molecule such as NO in an electronic state 2Pr with small
cou-pling between electronic spin and orbital angular momenta requires taking into account that the total partition function Q includes rotational states in the vibrational ground state of the electronic ground state 2P1/2 but also in the vibrational
ground state of electronic state 2P3/2 lying above 2
P1/2 by only 12313.361 m1 [15], and a
vibra-tional factor 1, with a factor 2 for K doubling. We calculated Qrotby direct summation over
rota-tional states [15] up to J = 40.5 at each tempera-ture. For application of formula (1) to NO in electronic state 2Pr, jij also becomes replaced by
wave number / 105 m-1 7.24 7.28 7.32 7.36 absorbance 0.00 0.02 0.04 0.06 0.08
Fig. 1. Spectrum of band 4–0 of gaseous14N16O at pressure
73,300 N m2and 298 K with length 8.25 m of absorbing path and spectral resolution 4.0 m1.
either (jij2 X2)/jij for transitions with DJ = ±1 or X2(2J + 1)/[J(J + 1)] for DJ = 0 [5]. The rota-tional dependence remains factorable according to[12], hm0J0jpðxÞj0; J00i2 ¼ hm0jpðxÞj0i2 ð1 þ Cm0 0iþ D m0 0i 2þ Þ ð2Þ in which jhm0jp(x)j0ij2
is the square of the pure vibrational matrix element of electric dipolar moment; the latter quantity is coefficient to a Herman-Wallis factor [12] containing coefficients Cm00 and Dm0
0. According to formula (1), strengths
of lines and bands are proportional to squares of matrix elements; as coefficients Cm00 and Dm0
0 are
linearly proportional to coefficients pj in an
expansion of p(x) [11], we apply this informa-tion to deduce the signs of hm0jp(x)j0i. A sum of
measured strengths Sl of all lines in a particular
band, Sb¼
X
Sl ð3Þ
provides an experimental estimate of a band strength Sb, whereas a theoretical measure depends
on the wave number ~mc characterising the origin of
a band and the square of the pure vibrational matrix element of electric dipolar moment, Sb¼ 8p2~mcjhm0jpðxÞj0ij
2
=ð12hc0Þ ð4Þ
A line strength is derived from integrated absor-bance of a particular spectral line, a known num-ber density N of molecules per unit volume in a gaseous sample at temperature T and an effective length ‘ of optical path:
Sl¼ ð1=N ‘Þ
Z
ln½I0ð~mÞ=Ið~mÞd~m ð5Þ
For NO under our conditions of measurement, spectral lines for vibration-rotational transitions occur in closely spaced couples due to K doubling, but each such couple appears as an unresolved line-like feature. We include factors two in pertinent formulae to reflect this condition, and assume equal probability of transition of each component of K doubling, according to precedent[5].
To represent the dependence of electric dipolar moment p as a function of internuclear distance R
or x we apply a simple polynomial [12] of order sufficient for data in a particular set:
pðxÞ ¼X
j¼0
pjxj ð6Þ
A Pade´ function[12]is a ratio of polynomials such that successive derivatives with respect to x, apart from factorial factors, agree exactly with as many coefficients pj in formula (6) as can be evaluated
from available data; tests on Pade´ functions of various orders indicate that, because of large mag-nitudes of coefficients pj, such a form is impractical
for the present data of NO. We employ SI units throughout this work and convert published val-ues accordingly.
4. Results
In Tables 1–5 we present strengths of line-like
features in vibration-rotational bands m0–0, with
m0= 2, 3, 4, 5 and 6, respectively; each feature is
characterised by both a value of quantum num-ber J00 for total angular momentum of the state
of lesser energy and a wave number calculated as the difference of published spectral terms[15]. From a direct sum of these strengths Sl of lines
according to formula (3) for a particular band we estimated a strength of each band Sb. In the
case of band 2–0, the resulting value was about 6 per cent less than one published value[6], but the same, within experimental error, as another published value [7], either of which might be ex-pected to be accurate; although problems with sample handling and adsorption of NO on sur-faces within the large gas cell might slightly affect our measurements of that band, our inde-pendent coincidence with the later published result supports the correctness of that value [7]. Because we used the same samples of NO for corresponding measurements of band 3–0, for which only the length of optical path varied through an increased number of passes of the light through the cell, the strength of band 3–0 is as accurate as that of band 2–0; for the same reason ratios of bands 4, 5, 6–0 are likewise accurate. With large pressures of nominally pure NO used
Table 1
Values of quantum number J00for total angular momentum of the lower state of a vibration-rotational transition in band 2–0, wave
number/m1 of each apparent line in branches P, Q and R, and line strength Sl/1023m observed and calculated for14N16O in
substates of electronic ground state X2P
rat 298 K
J P(J) Sl,ob Sl,cal Q(J) Sl,ob Sl,cal R(J) Sl,ob Sl,cal
Sub-band 2–0 for substate2P 1/2 0.5 372406.7 0.22 0.247 372898.0 0.50 0.500 1.5 371905.1 0.48 0.476 372396.4 0.07 0.096 373215.2 0.91 0.886 2.5 371560.4 0.84 0.816 372379.3 0.04 0.060 373525.6 1.23 1.226 3.5 371208.9 1.14 1.093 373829.2 1.55 1.515 4.5 370850.5 1.34 1.306 374125.9 1.66 1.744 5.5 370485.3 1.49 1.455 374415.6 1.89 1.906 6.5 370113.2 1.60 1.539 374698.6 1.97 2.000 7.5 369734.2 1.61 1.563 374974.6 1.90 2.028 8.5 369348.4 1.61 1.533 375243.7 1.73 1.994 9.5 368955.8 1.52 1.459 375505.9 1.81 1.906 10.5 368556.3 1.42 1.350 375761.1 1.66 1.780 11.5 368149.9 1.28 1.219 376009.5 1.49 1.621 12.5 367736.7 1.16 1.074 376250.8 1.37 1.443 13.5 367316.6 0.99 0.926 376485.2 1.16 1.258 14.5 366889.6 0.85 0.781 376712.5 1.00 1.073 15.5 366455.8 0.71 0.645 376932.9 0.82 0.898 16.5 366015.1 0.57 0.522 377146.2 0.68 0.736 17.5 365567.6 0.45 0.414 377352.5 0.54 0.592 18.5 365113.3 0.34 0.322 377551.6 0.44 0.467 19.5 364652.1 0.27 0.246 377743.7 0.33 0.361 20.5 364184.1 0.21 0.184 377928.6 0.25 0.275 21.5 363709.3 0.15 0.136 378106.4 0.19 0.205 22.5 363227.6 0.11 0.098 378277.1 0.12 0.150 23.5 362739.1 0.07 0.069 378440.5 0.10 0.109 24.5 362243.9 0.09 0.048 378596.8 0.07 0.077 25.5 361741.8 0.03 0.033 378745.8 0.05 0.053 26.5 361233.0 0.02 0.022 378887.5 0.04 0.036 27.5 360717.4 0.01 0.015 379022.0 0.03 0.024 28.5 360195.0 0.01 0.010 379149.1 0.02 0.016 29.5 379269.0 0.01 0.010 30.5 379381.4 0.01 0.007
Sub-band 2–0 for substate2P 3/2 1.5 372353.0 0.54 0.486 373195.0 0.29 0.326 2.5 371493.0 0.30 0.305 372335.0 0.32 0.300 373513.7 0.52 0.558 3.5 371131.0 0.48 0.509 372309.7 0.23 0.210 373825.1 0.74 0.738 4.5 370761.9 0.66 0.656 372277.2 0.15 0.155 374129.1 1.02 0.872 5.5 370385.7 0.81 0.755 372237.6 0.12 0.117 374425.8 0.93 0.962 6.5 370002.4 0.86 0.813 372190.7 0.10 0.090 374715.2 1.02 1.011 7.5 369612.1 0.94 0.833 372136.6 0.08 0.070 374997.2 1.01 1.022 8.5 369214.8 0.89 0.821 372075.3 0.06 0.054 375271.8 0.90 0.999 9.5 368810.5 0.81 0.782 375538.9 0.95 0.948 10.5 368399.2 0.79 0.724 375798.7 0.89 0.876 11.5 367980.9 0.71 0.652 376051.0 0.79 0.789 12.5 367555.8 0.61 0.573 376295.9 0.73 0.695 13.5 367123.7 0.54 0.492 376533.3 0.59 0.598 14.5 366684.8 0.47 0.413 376763.2 0.50 0.504 15.5 366239.1 0.39 0.339 376985.7 0.54 0.416 16.5 365786.5 0.32 0.273 377200.7 0.34 0.336 17.5 365327.1 0.25 0.215 377408.2 0.27 0.266
for the latter measurements, problems with adsorption of NO or measurement of pressures are negligible.
By fitting all data for each band, including both sub-bands for each electronic substate, according to formulae (1) and (2), we derived values of squares of pure vibrational matrix elements and Herman-Wallis coefficient Cm00; for no band is a value of second coefficient Dm0
0 evaluated
statisti-cally significantly.Table 6presents for each band a summary of results for pure vibrational matrix element and its statistical uncertainty, a band strength Sb derived therefrom at 298 K, Cm
0
0, and
corresponding theoretical quantities to be ex-plained separately. To convert those values of pure vibrational matrix element into coefficient pjin
for-mula(6) through the following relation: hm0jpðxÞj0i ¼X
6
j¼0
pjhm0jxjj0i ð7Þ
we require further experimental information of two types: an accurate magnitude of h0jp(x)j0i is ob-tained from application of the Stark effect on a molecular beam of14N16O[16], which we associate with a polarityNO+and consequently a positive sign for p0 as a sign convention, consistent with
previous usage for CO[17]; as a value ofh1jp(x)j0i we accepted the more recent value of Spencer et al.
[5]: both these data are reported inTable 6. Vibra-tional matrix elements of reduced displacement x to various powers require for their evaluation a knowledge of coefficients cj of z 2(R Re)/
(R Re) in a function for potential energy,
VðzÞ ¼ hcc0z2 1þ X j¼1 cjzj ! ð8Þ and c 2Be/xe in terms of conventional spectral
parameters. For this purpose we fitted values of vibrational terms G(m) and rotational parameters Bmlisted by Amiot[14]to polynomials in (m + l/2)
up to m = 22; from the resulting coefficients Yk,0
and Yk,1, respectively, we applied expressions
[18,19] of these quantities in terms of coefficients
cj to derive directly the required values, involving
no fitting. The results are c = 0.001790727 and this formula for adiabatic potential energy,
VðzÞ=hc m1¼ 53166650:6z2ð1 1:91599565z
þ 1:4963702z2 0:3822858z3
0:5913666z4 1:6270814z5
þ 25:86692z6 95:37679z7Þ ð9Þ
with equilibrium internuclear distance Re=
1.1507825· 1010m of 14N16O; for vibrational terms within the range of states, m 6 6, for which we measured intensities, this formula is essentially exact, in that uncertainties of measured intensities dominate the eventual values of parameters for dipolar moment, but this formula might become unreliable for vibration-rotational terms at large val-ues of m and J. Using efficient procedures[12,19]to generate algebraic expressions for accurate vibra-tional matrix elementshm0jp(x)j0i, we evaluated these
quantities and solved seven simultaneous linear equations implied in formula(7)to evaluate seven coefficients pj, according to the following formula:
Table 1 (continued)
J P(J) Sl,ob Sl,cal Q(J) Sl,ob Sl,cal R(J) Sl,ob Sl,cal
18.5 364860.9 0.17 0.166 377608.1 0.21 0.207 19.5 364387.9 0.14 0.126 377800.6 0.18 0.168 20.5 363908.2 0.11 0.094 377985.5 0.13 0.118 21.5 363421.7 0.09 0.068 378163.0 0.09 0.087 22.5 362928.5 0.07 0.049 378332.8 0.07 0.062 23.5 362428.7 0.04 0.034 378495.2 0.05 0.044 24.5 361922.1 0.03 0.024 378649.9 0.03 0.031 25.5 361408.8 0.02 0.016 378797.2 0.02 0.021 26.5 360888.8 0.02 0.011 378936.8 0.02 0.014 27.5 360362.3 0.01 0.007 379068.9 0.01 0.009 28.5 379193.4 0.01 0.006
Table 2
Values of quantum number J00for total angular momentum of the lower state of a vibration-rotational transition in band 3–0, wave
number/m1 of each apparent line in branches P, Q and R, and line strength Sl/1025m observed and calculated for14N16O in
substates of electronic ground state X2P
rat 296.6 K
J P(J) Sl,ob Sl,cal Q(J) Sl,ob Sl,cal R(J) Sl,ob Sl,cal
Sub-band 3–0 for substate2P 1/2 0.5 554402.1 0.58 0.513 554888.3 1.09 1.048 1.5 553900.5 1.12 0.984 554386.7 0.09 0.200 555196.9 1.95 1.868 2.5 553550.7 1.71 1.681 555495.3 2.47 2.598 3.5 553190.6 2.38 2.241 555783.4 3.37 3.227 4.5 552820.2 2.81 2.669 556061.1 3.76 3.733 5.5 552439.4 3.13 2.960 556328.5 4.05 4.101 6.5 552048.4 3.21 3.118 556585.6 4.04 4.324 7.5 551647.1 3.40 3.152 556832.4 4.33 4.404 8.5 551235.5 3.33 3.078 557068.8 4.36 4.350 9.5 550813.6 3.14 2.915 557294.8 4.13 4.180 10.5 550381.4 2.82 2.686 557510.5 3.59 3.916 11.5 549938.8 2.60 2.413 557715.7 3.16 3.581 12.5 549486.0 2.43 2.116 557910.5 2.83 3.202 13.5 549022.8 2.01 1.815 558094.8 2.48 2.801 14.5 548549.3 1.73 1.522 558268.6 2.01 2.399 15.5 548065.4 1.36 1.250 558432.0 1.59 1.803 16.5 547571.2 1.26 1.006 558584.8 1.51 1.656 17.5 547066.8 0.90 0.794 558727.1 1.38 1.336 18.5 546551.9 0.89 0.614 558858.9 0.99 1.057 19.5 546026.8 0.58 0.466 558980.0 0.80 0.821 20.5 545491.3 0.43 0.347 559090.5 0.57 0.626 21.5 544945.6 0.17 0.254 559190.3 0.31 0.468 22.5 544389.5 0.09 0.182 559279.5 0.23 0.344 23.5 559358.0 0.16 0.248
Sub-band 3–0 for substate2P 3/2 1.5 554320.0 0.88 1.008 555153.0 0.62 0.696 2.5 553459.9 0.49 0.622 554292.9 0.47 0.622 555459.0 1.10 1.209 3.5 553088.9 1.08 1.033 554255.0 0.31 0.434 555754.1 1.53 1.619 4.5 552707.2 1.46 1.323 554206.3 0.41 0.320 556038.3 1.98 1.938 5.5 552314.7 1.80 1.514 554146.7 0.36 0.243 556311.5 2.14 2.187 6.5 551911.5 1.78 1.619 554076.3 0.26 0.186 556573.7 2.31 2.307 7.5 551497.7 1.69 1.647 553995.1 0.15 0.144 556825.0 2.26 2.361 8.5 551073.3 1.82 1.612 557065.3 1.82 2.336 9.5 550638.3 1.58 1.525 557294.5 2.06 2.245 10.5 550192.7 1.36 1.401 557512.7 1.91 2.099 11.5 549736.5 1.27 1.253 557719.9 1.81 1.905 12.5 549269.8 1.02 1.093 557916.1 1.67 1.706 13.5 548792.7 1.04 0.931 558101.2 1.26 1.485 14.5 548305.1 0.76 0.775 558275.3 1.27 1.266 15.5 547807.0 0.42 0.632 558438.3 1.00 1.056 16.5 547298.5 0.62 0.504 558590.2 0.94 0.864 17.5 546779.6 0.45 0.394 558731.0 0.51 0.692 18.5 546250.4 0.26 0.302 558860.8 0.50 0.544 19.5 545710.8 0.35 0.227 558979.5 0.40 0.419 20.5 545160.9 0.11 0.167 559087.0 0.23 0.317 21.5 544600.6 0.05 0.121 559183.5 0.35 0.235 22.5 559268.9 0.08 0.172 23.5 559343.2 0.06 0.123
Table 3
Values of quantum number J00for total angular momentum of the lower state of a vibration-rotational transition in band 4–0, wave
number/m1of each apparent line in branches P, Q and R, and line strength Sl/1026m observed and calculated for14N16O in
substates of electronic ground state X2P
rat 298.8 K
J P(J) Sl,ob Sl,cal Q(J) Sl,ob Sl,cal R(J) Sl,ob Sl,cal
Sub-band 4–0 for substate2P 1/2 0.5 733598.3 0.16 0.232 734079.3 0.36 0.472 1.5 733096.7 0.49 0.445 733577.7 0.10 0.091 734379.3 0.92 0.839 2.5 732741.7 0.75 0.760 733543.3 0.03 0.056 734665.6 1.14 1.164 3.5 732373.0 1.03 1.015 734938.2 1.40 1.442 4.5 731990.5 1.31 1.210 735197.0 1.56 1.665 5.5 731594.3 1.47 1.343 735442.0 1.74 1.825 6.5 731184.3 1.48 1.415 735673.2 1.73 1.920 7.5 730760.6 1.55 1.432 735890.7 1.70 1.952 8.5 730323.1 1.51 1.400 736094.3 1.59 1.925 9.5 729871.9 1.44 1.327 736284.0 1.53 1.846 10.5 729406.8 1.49 1.224 736459.9 1.46 1.727 11.5 728928.0 1.21 1.101 736622.0 1.31 1.577 12.5 728435.5 1.00 0.967 736770.1 1.26 1.408 13.5 727929.1 0.90 0.830 736904.3 1.06 1.230 14.5 727408.9 0.76 0.697 737024.5 0.89 1.052 15.5 726874.9 0.64 0.573 737130.7 0.74 0.882 16.5 726327.1 0.51 0.462 737223.0 0.56 0.725 17.5 725765.5 0.38 0.365 737301.2 0.58 0.584 18.5 725190.1 0.33 0.283 737365.3 0.25 0.462 19.5 724600.8 0.24 0.215 737415.3 0.29 0.358 20.5 723997.8 0.20 0.160 737451.2 0.27 0.273 21.5 723380.9 0.14 0.118 737472.9 0.24 0.200 22.5 722750.2 0.09 0.085 737480.5 0.03 0.150 23.5 722105.6 0.08 0.060 737473.8 0.08 0.108 24.5 721447.3 0.02 0.041 737452.9 0.11 0.077 25.5 720775.1 0.04 0.028 26.5 720089.1 0.02 0.019
Sub-band 4–0 for substate2P 3/2 1.5 733486.5 0.48 0.458 734310.4 0.34 0.317 2.5 732626.4 0.23 0.284 733450.3 0.35 0.282 734603.7 0.55 0.554 3.5 732246.3 0.33 0.474 733399.7 0.18 0.197 734882.5 0.79 0.750 4.5 731851.9 0.54 0.608 733334.7 0.15 0.146 735146.8 0.92 0.909 5.5 731443.1 0.54 0.698 733255.2 0.12 0.110 735396.5 1.06 1.032 6.5 731020.1 0.64 0.749 733161.3 0.13 0.085 735631.6 1.23 1.119 7.5 730582.7 0.69 0.765 733053.0 0.09 0.065 735852.1 1.22 1.169 8.5 730131.1 0.70 0.751 736058.0 1.11 1.184 9.5 729665.3 0.66 0.713 736249.2 1.17 1.167 10.5 729185.4 0.63 0.657 736425.9 1.07 1.122 11.5 728691.2 0.59 0.590 736587.9 1.06 1.055 12.5 728183.0 0.53 0.516 736735.2 0.82 0.970 13.5 727660.6 0.44 0.441 736868.0 0.65 0.873 14.5 727124.2 0.38 0.369 736986.0 0.53 0.771 15.5 726573.7 0.30 0.302 737089.4 0.39 0.667 16.5 726009.3 0.25 0.242 737178.2 0.21 0.566 17.5 725430.8 0.24 0.190 737252.3 0.18 0.472 18.5 724838.4 0.18 0.146 737311.7 0.13 0.386 19.5 724232.0 0.15 0.110 737356.4 0.16 0.310 20.5 723611.7 0.09 0.089 737386.5 0.11 0.245
pðxÞ=1030C m¼ 0:558274 8:649479x þ 5:048787x2
0:866241x3þ 28:3255x4
81:2569x5þ 270:714x6 ð10Þ
As presented inTable 6, the signs ofhm0jp(x)j0i are
selected to produce best agreement between exper-imental values of Herman-Wallis coefficient Cm00 and values calculated on the basis of formula
(10); these calculated values of Cm00 and also Dm00 are included inTable 6. With points resulting from calculations of molecular electronic structure[20]
for comparison, formula (10)is plotted in Fig. 2; the range/1010m of validity of this formula, cor-responding to the amplitude of vibration in state m= 6, is [1.0, 1.38]. With these values ofhm0p(x)j0i,
Cm00, Dm0
0 for each band and the wave number of
each spectral feature, as reported in Tables 1–5, we calculated with formula(1)the strength of each feature for comparison with observed strengths, also presented inTables 1–5.
5. Discussion
With only seven parameters in a radial function for electric dipolar moment—coefficients of x to various powers in formula (10)—and with c and seven parameters cj, j > 0, in a radial function for
potential energy in formula(9), we reproduce the strengths of more than 500 spectral features, each having two components of supposed equal contri-bution, in our measurements of bands 2–0 up to 6–0, almost within the uncertainties of their measurement. With the same parameters we can likewise reproduce more than 100 separate mea-surements on band 1–0[5]and another 79 measure-ments on bands 3–1[9]and 2–1[8]; the latter results are more compatible with the results [5]on band 1–0 that we used than with earlier measurements
[4]. Measurements of strengths of lines in some R branches are complicated by overlapping of fea-tures due to not only the presence of band heads but also branches of separate substates. Except for band 2–0, our experimental magnitudes of Herman-Wallis coefficient Cm00 are consistent with calculated values; as integrated areas of lines are small and subject to uncertainties from interfer-ences between overlapping lines, evaluation of a quadratic term to evaluate Dm0
0 is impracticable in
this work. Both reported values C2
0¼ ð3:57
0:14Þ 103 [6] and (3.66 ± 0.08)· 103 [7] are consistent with our calculated value 3.75· 103 indicated in Table 6, but a reported value D2
0¼
ð4:74 0:50Þ 105 [7] differs appreciably from our calculated value 1.64· 105; even with great spectral resolution and full separation of K dou-blets, an accurate evaluation of coefficient Dm0
0 is
evidently difficult. Because the total strength of each band, estimated as the sum of measured strengths according to formula (3), is consistent with the band strength, presented in Table 6, that is derived from the deduced pure vibrational matrix element based on strengths of individual features through formula(4), likely the calculated strengths of features, which are double the strengths of each component of a feature, are more reliable than the observed strengths. The ratios of strengths of successive vibration-rotational bands—65.2, 48.1, 22.1, 13.4, 8.6—of NO decrease regularly with increasing vibrational quantum number m, whereas the corresponding ratios— 128, 177, 289, 250—for CO tend to increase in the same order[20]; the radial functions for electric dipolar moment of NO and CO share common features of two extrema, as exhibited in Fig. 2, with a consequent reversal of polarity between these two extrema and small magnitudes of per-manent electric moment p(Re). In each case the
polarities, NO+ and CO+, at that condition Table 3 (continued)
J P(J) Sl,ob Sl,cal Q(J) Sl,ob Sl,cal R(J) Sl,ob Sl,cal
21.5 722977.5 0.07 0.059 737401.8 0.10 0.190
22.5 722329.5 0.04 0.042 737402.5 0.08 0.145
23.5 721667.5 0.04 0.030 737388.5 0.05 0.108
Table 4
Values of quantum number J00for total angular momentum of the lower state of a vibration-rotational transition in band 5–0, wave
number/m1of each apparent line in branches P, Q and R, and line strength Sl/1028m observed and calculated for14N16O in
substates of electronic ground state X2P
rat 297.2 K
J P(J) Sl,ob Sl,cal Q(J) Sl,ob Sl,cal R(J) Sl,ob Sl,cal
Sub-band 5–0 for substate2P 1/2 0.5 909998.7 1.67 1.748 910474.5 3.50 3.523 1.5 909497.1 3.43 3.385 909972.9 0.71 0.682 910765.9 6.16 6.220 2.5 909136.9 6.56 5.820 909929.9 0.42 0.421 911040.1 8.26 8.575 3.5 908759.6 7.99 7.851 909869.8 0.10 0.295 911297.2 10.7 10.56 4.5 908365.0 9.41 9.367 911536.9 12.7 12.10 5.5 907953.2 10.3 10.46 911759.5 13.9 13.18 6.5 907524.3 11.6 10.09 911964.8 12.7 13.77 7.5 907078.1 11.2 11.28 912152.8 14.1 13.90 8.5 906614.7 10.4 11.09 912323.5 14.5 13.61 9.5 906134.0 10.4 10.57 912477.0 12.6 12.96 10.5 905636.1 9.68 9.807 912613.0 13.2 12.04 11.5 905121.0 8.55 8.866 912731.8 11.7 10.91 12.5 904588.6 7.43 7.826 912833.1 7.72 9.668 13.5 904038.9 6.67 6.752 912917.0 8.33 8.383 14.5 903471.9 5.89 5.701 912983.4 6.30 7.117 15.5 902887.6 4.77 4.712 913032.4 4.48 5.920 16.5 902286.1 4.01 3.816 913063.9 4.01 4.827 17.5 901667.2 2.56 3.030 913077.8 3.29 3.860 18.5 901031.0 2.37 2.359 913074.2 3.25 3.028 19.5 900377.5 1.84 1.801 913052.9 2.98 2.331 20.5 899706.6 1.40 1.350 913014.0 0.79 1.762 21.5 899018.5 1.18 0.993 912957.4 1.58 1.307 22.5 898313.0 0.45 0.707 912883.1 0.82 0.952 23.5 897590.1 0.46 0.509 912791.1 0.97 0.681 24.5 912681.3 0.44 0.479
Sub-band 5–0 for substate2P 3/2 1.5 909855.9 3.10 3.438 910670.7 2.60 2.321 2.5 908995.8 1.93 2.170 909810.7 2.31 2.120 910951.4 3.49 3.996 3.5 908606.7 3.86 3.635 909747.4 1.28 1.481 911213.8 6.15 5.303 4.5 908199.5 4.33 4.696 909666.0 0.90 1.093 911458.1 6.32 6.292 5.5 907774.4 4.79 5.423 909566.6 0.99 0.829 911684.2 7.85 6.974 6.5 907331.4 5.73 5.850 909449.1 0.63 0.636 911892.1 6.32 7.357 7.5 906870.5 7.16 6.007 909313.5 0.48 0.491 912081.8 7.80 7.461 8.5 906391.7 7.06 5.930 912253.2 8.79 7.319 9.5 905895.1 6.15 5.661 912406.4 7.05 6.970 10.5 905380.6 5.44 5.247 912541.4 7.11 6.461 11.5 904848.4 4.64 4.735 912658.1 6.36 5.841 12.5 904298.5 4.59 4.166 912756.6 5.89 5.157 13.5 903730.8 3.36 3.580 912836.8 6.35 4.452 14.5 903145.5 2.87 3.008 912898.7 3.98 3.761 15.5 902542.6 2.44 2.473 912942.4 3.04 3.111 16.5 901922.0 2.37 1.991 912967.9 2.16 2.521 17.5 901283.8 1.90 1.570 912975.0 1.64 2.003 18.5 900628.1 1.33 1.214 912963.9 1.08 1.560 19.5 899954.8 0.91 0.921 912934.6 0.85 1.192 20.5 899264.0 0.77 0.685 912886.9 1.15 0.894 21.5 898555.7 0.47 0.500 912821.0 0.64 0.658
contradict expectations from crude consideration of conventional electronegativities.
For band 3–0 of 14N16O our strength Sb=
(1.48 ± 0.01)· 1023 m, reported in Table 6, is consistent with the most recent result, Sb=
(1.52 ± 0.08)· 1023 m, from measurements on
samples with He or Ar at pressures large enough to obliterate rotational structure [21]. Like our independent reproduction of the strength of band 2–0 [7], this agreement confirms the efficacy of our procedure of measurement involving fitting, with a mixed gaussian and lorentzian profile, of Table 5
Values of quantum number J00for total angular momentum of the lower state of a vibration-rotational transition in band 6–0, wave
number/m1of each apparent line in branches P, Q and R, and line strength Sl/1028m observed and calculated for14N16O in substates
of electronic ground state X2P
rat 297.4 K
J P(J) Sl,ob Sl,cal Q(J) Sl,ob Sl,cal R(J) Sl,ob Sl,cal
Sub-band 6–0 for substate2P 1/2 0.5 1083605.8 0.22 0.202 1084076.4 0.44 0.406 1.5 1083104.2 0.47 0.392 1083574.8 0.12 0.079 1084359.1 0.79 0.717 2.5 1082738.8 0.68 0.674 1083523.1 0.04 0.049 1084621.2 1.17 0.988 3.5 1082352.8 0.96 0.906 1084862.6 1.38 1.214 4.5 1081946.1 1.15 1.087 1085083.3 1.38 1.391 5.5 1081518.7 1.14 1.215 1085283.4 1.52 1.513 6.5 1081070.7 1.27 1.290 1085462.6 1.53 1.580 7.5 1080602.0 1.24 1.314 1085621.2 1.55 1.593 8.5 1080112.5 1.35 1.293 1085758.9 1.53 1.558 9.5 1079602.4 1.29 1.234 1085875.9 1.47 1.483 10.5 1079071.5 1.04 1.145 1085972.1 1.32 1.375 11.5 1078519.9 1.08 1.037 1086047.4 1.29 1.246 12.5 1077947.6 0.94 0.916 1086101.8 1.16 1.103 13.5 1077354.5 0.64 0.791 1086135.3 1.30 0.955 14.5 1076740.6 0.79 0.669 1086147.8 0.65 0.810 15.5 1076105.9 0.60 0.553 1086139.3 0.51 0.673 16.5 1075450.4 0.54 0.449 1086109.9 0.77 0.549 17.5 1074774.1 0.30 0.357 1086059.4 0.37 0.438 18.5 1074077.0 0.22 0.278 1085987.8 0.33 0.343 19.5 1073359.0 0.24 0.212 1085895.0 0.28 0.264 20.5 1085781.1 0.18 0.199
Sub-band 6–0 for substate2P 3/2 1.5 1083430.4 0.34 0.397 1084236.2 0.25 0.268 2.5 1082570.4 0.28 0.251 1083376.1 0.24 0.245 1084504.1 0.50 0.460 3.5 1082172.1 0.47 0.421 1083300.1 0.15 0.171 1084750.2 0.68 0.610 4.5 1081752.3 0.41 0.545 1083202.3 0.15 0.126 1084974.5 0.73 0.723 5.5 1081310.8 0.66 0.630 1085176.9 0.68 0.801 6.5 1080847.7 0.74 0.680 1085357.5 0.72 0.844 7.5 1080363.2 0.79 0.699 1085516.3 0.69 0.855 8.5 1079857.1 0.66 0.691 1085653.2 0.93 0.838 9.5 1079329.6 0.59 0.660 1085768.3 0.86 0.797 10.5 1078780.6 0.64 0.613 1085861.4 0.68 0.738 11.5 1078210.3 0.65 0.553 1085932.8 0.66 0.667 12.5 1077618.5 0.41 0.488 1085982.2 0.49 0.588 13.5 1077005.5 0.35 0.419 1086009.8 0.52 0.507 14.5 1076371.2 0.38 0.353 1086015.5 0.45 0.428 15.5 1075715.6 0.25 0.290 1085999.3 0.43 0.354 16.5 1075038.7 0.23 0.234 1085961.3 0.29 0.286 17.5 1074340.7 0.25 0.185 1085901.4 0.28 0.227 18.5 1073621.5 0.14 0.143 1085819.6 0.13 0.177 19.5 1085716.0 0.08 0.135
each perceived feature that comprises two lines in close proximity, each of which has hyperfine struc-ture due to coupling of rotational and angular momenta and the nuclear quadrupolar moment of 14N. As already explained, our value of the strength of band 3–0 results from an accurate ratio with band 2–0. Our strength of band 2–0 is also consistent with a measurement of matrix element for band 3–1 [9], which implies a corresponding matrix element (2.284 ± 0.029)· 1032C m for
band 2–0. For band 4–0, extensive quantitative measurements of intensity are unavailable. For bands 5–0 and 6–0 our results reported here super-sede those reported in previous publications
[10,11].
From calculations of molecular electronic struc-ture [20] that indicate the polarity at Re to be
NO+ as we applied, points for electric dipolar moment of NO, included inFig. 2, at internuclear distances in the range of experimental obser-vations lie consistently slightly above a curve cor-responding to formula (10). These differences reflect a systematic deficiency of those calculations, previously explained[20]to be related to an under-estimate of the electron affinity of oxygen, with re-spect to the permanent dipolar moment, p(Re); as
the latter value that is based essentially on an accu-rate measurement through the Stark effect [16] is included in the basis of our generation of formula
(10), the differences persist between our experimen-tally derived function and the calculated points
[20]. Apart from that slight systematic shift, our results are entirely compatible with those compu-tational results. Some estimates [22] of band strengths from results of other, unpublished calcu-lations of molecular electronic structure are great-er than eithgreat-er reported value for band 2–0[6,7]and thus also our present value, but smaller [22] for band 5–0 than our experimental result; compari-son with graphical depictions [22] indicates that those estimates of line strengths in bands 3–0 Table 6
Origins of vibration-rotational bands m0–0, experimental matrix elements of electric dipolar moment, band strengths at 298 K and
experimental and calculated values of Herman-Wallis coefficients for14N16O
m0 ~m c/m1 hv0jp(x)j0i/C m Sb/ma C0m0/·103b C0m0/·103c D0m0/·105c 0 0 (5.29433 ± 0.00066)· 1031d – – 0 10.5 1 187598.9277 (2.5778 ± 0.0017) · 1031e 4.63· 1020e 0.51 ± 0.13e 0.414 0.178 2 372388.7584 (2.2579 ± 0.0035)· 1032f 7.10· 1022 2.52 ± 0.22 3.751 1.638 3 554374.4427 (2.6605 ± 0.0077) · 1033f 1.48· 1023 4.99 ± 0.57 5.675 1.606 4 733560.2566 (4.9386 ± 0.0150)· 1034f 6.69· 1025 5.86 ± 0.48 5.527 0.074 5 909949.6772 (1.2135 ± 0.0058) · 1034f 5.01· 1026 3.44 ± 0.73 4.822 2.011 6 1083544.980 (3.791 ± 0.019)· 1035f 5.83· 1027 3.18 ± 0.89 3.892 1.917
a calculated from matrix element in preceding column bfitted from spectra with formulae(1) and (2) ccalculated on the basis of formulae(9) and (10) d
jh0jp(x)j0ij from Ref.[15]
e
jh1jp(x)j0ij, Sband C10from Ref.[5]
funcertainties from fits of line strengths; relative error of absolute magnitude is likely ±2%.
Fig. 2. Radial function p(R) of14N16O from spectral data (solid
curve) with points from calculations of molecular electronic structure from reference 19 for comparison.
and 4–0 are also only roughly consistent with our direct measurements. Further, accurate calcula-tions are desirable.
6. Conclusion
Based on experimental measurements of line strengths in five bands of the infrared spectrum of14N16O in vibration-rotational transitions with-in each substate of its electronic ground state X
2
Pr, we have derived a formula for electric dipolar
moment as a function of internuclear distance that reproduces satisfactorily the observed strengths of about 700 features in vibration-rotational spectra up to m = 6. Our data will facilitate monitoring the concentration of NO in the terrestrial atmo-sphere.
Acknowledgments
TaiwanÕs National Science Council (grant NSC92-2113-M007-034) and Ministry of Educa-tion Program for Promoting Academic Excellence of Universities (grant 89-FA04-AA) provided sup-port for this project.
References
[1] L.A. Pugh, K.N. Rao, Intensities from infrared spectra, in: K.N. Rao (Ed.), Molecular spectroscopy: modern research, Academic Press, New York, 1976, pp. 165–181.
[2] M.A.H. Smith, C.P. Rinsland, B. Fridovich, K.N. Rao, Intensities and collisional broadening parameters from infrared spectra, in: K.N. Rao (Ed.), Molecular Spectros-copy: Modern Research, Academic Press, New York, 1985, pp. 111–248.
[3] M.A.H. Smith, C.P. Rinsland, V.M. Devi, L.S. Rothman, K.N. Rao, Intensities and collisional-broadening parame-ters from infrared spectra—an update, in: K.N. Rao, A. Weber (Eds.), Spectroscopy of the EarthÕs Atmosphere and Interstellar Media, Academic Press, Boston, 1992, pp. 153– 260.
[4] J. Ballard, W.B. Johnston, B.J. Kerridge, J.J. Remedies, Experimental spectral line parameters in the band 1 0 of NO, J. Mol. Spectrosc. 127 (1988) 70–82.
[5] M.N. Spencer, C. Chackerian, L.P. Giver, L.R. Brown, The fundamental band of NO: frequency and shape parameters for rovibrational lines, J. Mol. Spectrosc. 165 (1994) 506–524.
[6] A.S. Pine, A.G. Maki, N.-Y. Chou, Pressure broadening, line shapes and intensity measurements in the band 2 0 of NO, J. Mol. Spectrosc. 114 (1985) 132–147.
[7] J.-Y. Mandin, V. Dana, L. Regalia, A. Barbe, X. Thomas, Ksplittings and line intensities in the first overtone of NO, J. Mol. Spectrosc. 185 (1997) 347–355.
[8] V. Dana, J.-Y. Mandin, L.H. Coudert, M. Badaoui, F. LeRoy, G. Guelachvili, L.S. Rothman, K splittings and line intensities in the hot band 2 1 of NO, J. Mol. Spectrosc. 165 (1994) 525–540.
[9] J.Y. Mandin, V. Dana, L. Regalia, A. Barbe, P. von der Heyden, K splittings and line intensities in the hot band 3 1 of14
N16O; the spectrum of nitric oxide in the first-overtone region, J. Mol. Spectrosc. 187 (1998) 200–205. [10] Y.-P. Lee, J.F. Ogilvie, Strengths of absorption lines in
vibration-rotational band m = 5 m = 0 of 14N16O X2P r,
Infrared Phys. 28 (1988) 321–324.
[11] Y.-P. Lee, S.-L. Cheah, J.F. Ogilvie, Strengths of absorp-tion features in vibraabsorp-tion-rotaabsorp-tional band m = 6 m = 0 of
14NI6O X2P
r in the near infrared region, Infrared Phys.
Technol. 44 (2003) 199–205.
[12] J.F. Ogilvie, The Vibrational and Rotational Spectrometry of Diatomic Molecules, Academic Press, London, UK, 1998, and references therein.
[13] J.D. Ray, R.A. Ogg, A new method of preparing nitric oxide, J. Amer. Chem. Soc. 78 (1956) 5893.
[14] C. Meyer, C. Haeusler, Spectre de vibration-rotation de lÕoxyde nitrique NO; etude de la bande 4 0 a 7336 cm1,
C. R. Acad. Sci. Paris 280 (1965) 4182–4185.
[15] C. Amiot, The infrared emission spectrum of NO; analysis of the sequence Dm = 3 up to m = 22, J. Mol. Spectrosc. 94 (1982) 150–172.
[16] R.M. Neumann, Highly precise radio-frequency spectrum of14N16O, Astrophys. J. 161 (1970) 779–784.
[17] J.F. Ogilvie, S.-L. Cheah, Y.-P. Lee, S.P.A. Sauer, Infrared spectra of CO in absorption and evaluation of radial functions for potential energy and electric dipolar moment, Theor. Chem. Ace. 108 (2002) 85–97, and unpublished results.
[18] J.F. Ogilvie, Spectral energy coefficients for vibration-rotational states of diatomic molecules, Comput. Phys. Commun. 30 (1983) 101–105.
[19] F.M. Fernandez, J.F. Ogilvie, Symbolic programmes for analysis of diatomic vibration-rotational spectra, Maple-Tech. 5 (1998) 42–46.
[20] S.R. Langhoff, C.W. Bauschlicher, J. Partridge, Theoret-ical dipole moment for the X2Pstate of NO, Chem. Phys. Lett. 223 (1994) 416–422.
[21] G. Chandraiah, P. Gillard, Absorption intensity measure-ments of the second overtone band of NO, Can. J. Phys. 69 (1991) 597–602.
[22] A. Goldman, L.R. Brown, W.G. Schoenfeld, M.N. Spencer, C. Chackerian, L.P. Giver, H. Dothe, C.P. Rinsland, L.H. Coudert, V. Dana, J.-Y. Mandin, NO line parameters: review of 1996 HITRAN update and new results, J. Quant. Spectrosc. Radiat. Transfer 60 (1998) 825–838.
