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r-centroids and Franck-Condon factors for the bands of the C2Σ+−X2Σ+ system of MgF
J.Quunr.Specwosc. Rrrdlur. Tmnsfer. Vol.IO,pp.945-948. PergamonPress 1970.Prknted inGreatBritain
NOTE r-CENTROIDS BANDS
AND OF
FRANCK-CONDON THE
C2C+--x2x+
FACTORS SYSTEM
OF
FOR
THE
MgF
T. V. RAMAKRISHNA RAO and S. V. J. LAKSHMAN SpectroscopicLaboratories,Department of Physics,S. V. University,Tirupati, India (Received21 Nocemher1969)
Abstract-The r-centroids and Franck-Condon factors for the bands of the C’E’-X*X+ system of the astrophysically important MgF molecule have been determined. The bands of this molecule are present both in the disk and spot spectra. The values of Franck-Condon factors as determined by the approximate analytical method of Fraser and Jarmain compare very well with those obtained by Bates’ method.
INTRODUCTION THE BAND
spectrum of the MgF molecule has been studied by DATTA,(‘) FOWLER,“’ and GRINFIELD and JEVONS.(~’ Three band systems designated as A’fl++ X2x+ (236863468 A), B2C + +-+X2x+ (12742-2649 A) and C2C+ +- X’C+ (23249-2387 A) are reported for this molecule. The detailed rotational analysis of the bands for this molecule has been carried out by BARROW and BEALE. ‘5’The molecular constants used in the present work are summarized in Table 1. Since the bands of the MgF molecule are astrophysically important (they are found in disk and spot spectra), the authors took up the present work to calculate the r-centroids and Franck-Condon factors for the bands of the C2C+-X*C+ system of MgF molecule. SINGH et ~1.‘~’found the state A and X of this molecule to obey Morse potential. To test how far the C state obeys the Morse potential, the potential energy curves for the C and X states have been calculated and presented in Table 4. JENKINS
TABLE ~.SPECTROSCOPICCONSTANTSOFTHE
Constant
co, (cm _ ‘) o&cm ‘) B,(cm-‘) re (‘Q ;(A-‘) P a,(cm-‘)
Upper
C'ZZ-X*Z'
state
823.2 5.04 0.55102 1.6988 1.780418 164.9741 (K,) 0.9799029 (pl) 0.00449
945
SYSTEMOFTHE
Lower state
721.6 4.94 0.51922 1.7500 1.762677 144.6130 (KZ) 1.0209388(p2) 0.00470
MgF MOLECULE
T. V. RAMAKRISHNA KAO
946
S. V. J. LAKSHMAY
and
r-CENTROID
(f,,.,?..)
The r-centroid V,,,,,..,representing the characteristic internuclear separation of a r’ + 1”’ transition in a diatomic molecular band system, has been defined (NICHOLLS and J.4RMAIN”)) as
where 1+4~, and Ic/,,,.are the vibrational wave functions and I’ is the internuclear distance. In the present work, both the graphical and quadratic-equation methods (Nicholls and Jarmain) have been employed for the evaluation of r-centroids for the bands of the C’C’ X2X+ system of the MgF molecule. The results are presented in Table 2. As has been observed by Nicholls and Jarmain for a system with rl,I < rt,r, a smooth curve has been obtained when a graph was drawn between r,,..,.,, and il,.,,.,, (the wavelength of the corresponding transition). The value of F,,, is slightly greater than the (r,.I + r, ?)i2 value, suggesting that the potentials are not very anharmonic. The difference A?-= ?,..,,.,. has been found to be a constant for a given sequence in accordance with r,. i I ,I ” + I the observations of Nicholls and Jarmain.
TABII 2. THE M’ENTKOIDS (A) AND WAV~LENGIHS
\
‘\
(A) OF IIANDS ot Ttib C“C
-____
SYSIEM OF Mgt-
I”
0
I”\‘i‘lL-__
2
3
obtained
4
i
I .729 1.729 7347.X
1.654 1.652 2381.9
I.330
1.499
1.57X
1.499
I.666 1.664 23x1 3
1.593 I .590
1.515 IS12
1.750
I .67X
I .750 2336.6
I .67X 2375.2
I.606 I .605
I.S.?l I 53
I.hlY 1.014
1.815
1.739
I.816 2303.6
1.740 2342. I
1.904 I.904 2262.0
I .x25 I .X26 229X.6
1.999 I.998
1.914 1.914 2757.7
2.104 2.088
2.009
I.923
2.006
1.924 2253.6
1 .x35
I.761
I .6X9
I .X36
1.760
I.689
2204. I
2.018 2.016
(a) r-centroids graphical method
’ X’Z’
by use of the m A.
; (c)wavelengths
quadratic-equation
method;
1.x4.5 I .X46 22x9.5 I.933 I.934 224Y.7
(h) r-centrmds
I.771 1.772 1.x55 1.856 1285.0
obtained
I.701 I .700 I .7x2
I .7x4
by use of the
r-Centroids
and Franck-Condon
FRANCK-CONDON
The relative
vibrational
transition
947
factors
FACTORS
probability
is written
as
P l9.0” = R,2(?“~,“&,~,,~~. Here R, is the electronic transition moment, FL,,,,,.is the r-centroid and qvr,uf.denotes the Franck-Condon factor. This parameter qvs,usrcontrols the intensity distribution in the emission of molecular bands. The Franck-Condon factors calculated by the approximate analytical method of FRASER and JARMAIN@)are given in Table 3 along with the values obtained by BATES’(~) method. The two values are in excellent agreement. TABLE 3. FRANCK--CONDON FACTORSFOR BANDS OF THE C’E’-X*C’
B
SYSTEMOF MgF
crs
0
1
2
3
4
0
0.123 (0.721)
0.227 (0.229)
0.043 (0.045)
0.006 (0.006)
0.00
1
0.239 (0.240)
0.326 (0.319)
0.310 (0.314)
0.100
0.023
2
0.035 (0.037)
0.348 (0.349)
0.110
0.280
3
0.003 (0.003)
0.089
0.328
4
0.000
0.002
L.’
Values given in parentheses
were derived
True
r max (A)
Morse
42938.3 43751.5 44554.5 45347.5 46130.3 46903.1 41665.9 48418.5
1.639 1.575 1.555 1.539 1.525 1.513 1.499
1.639 1.607 1.575 1.555 1.539 1.525 1.514 1.500
1.687 1.645 1.618 1.597 1.580 1.567 1.554 1.543
1.687 1.646 1.619 1.599 1.581 1.567 1.555 1.544
1.607
z = 0 359.6 1067.4 1766.3 2457.3 3140.7 3815.3 4481.0 5140.0
True
Morse
1.764 1.811 1.856 I.890 1.921 1.949 1.976 2.000
1.764 1.811 I .856 I .890 1.922 1.950 1.977 2.000
1.820 1.877 1.920 1.957 1.987 2.02 1 2.050 2.078
1.820 I .879 1.920
C’C+ state
T, = 42528.0 0
0.000
SYSTEMOF THE MgF MOLECULE
r min (A) (I+ z(cm-‘)
I
by using Bates’ method
TABLE 4. POTENTIAL ENERGY CURVESFOR THE C’Z’-X*X’
L’
5
X’C+ state
1.957 1.988 2.021 2.050 2.080
94x
7. V. RAMAKKISHNA RAO and S. V. .I. LAKSIIMAN
From the magnitude of the FranckkCondon factors, we conclude that it is very unlikely that the bands 3.0 and beyond in the r’-progression with 1”’ = 0 and the bands 0.3 and beyond in the c”-progression with I.’ = 0 will be observed.
POTENTIAL
ENERGY
CURVES
Singh et ul. found the A and X states of the MgF molecule to obey closely the Morse potential. In order to test how far the state C of this molecule follows the Morse-potential function, the potential energy curves have been calculated for the bands of the CX system between the two using both RKRV”” and MORSE(“) methods. The close agreement values (Table 4) proves that the state C also obeys the Morse potential function closely.
A(,kno~,/c~~rmm/s The authors wish to express their grateful thanks tu the authorities of this unlversit) lor providing all the necessary facilities and to PROF. J. BHIMASENA~HARfor his encouragement and Interest in this work.
REFERENCES
I S. DATTA, Pro<. R. Sot. 99, 436 ( I92 I ). 2. c‘. A. FOWLEK, Phys. Rev. 59,645 (194 I ). 3. 4. 5. 6. 7. 8. 9. IO. I I.
F. A. JENKINSand R. GRINFELU. Phw Ru. 45, 229 ( I Y34). W. JEVONS, Proc. R. Sot. 122,211 (1929). R. F. BARROW and J. R. BEALE, Proc~. PII~J. Sot. 91, 483 (1967). I. D. SINGH, M. M. SHUKLA and R. C. MAHESHWAKI, Jo.SRT 9, 533 ( lY6Y). K. W. NKHOLLS and W. R. JARMAIN, PIW. phy.v. SK. 69A. 253 (1956). P. A. FRASER and W. R. JARMAIN. Proc. ph,s. SW. 66A, 1145 (1953). D. R. BATES, Mm Not. R. ustr. Sot. 112, 614 (1952). J. T. VANDERSLICE. E. A. MASON, W. G. MAIS~II and E. R. LIPPINCOTT. J. .!4olr,c,. Sprc.tro.si~. 5, X3 (1960) P. M. MORSE, Phm Rrr. 34, 57 (1929).
NOTE r-CENTROIDS BANDS
AND OF
FRANCK-CONDON THE
C2C+--x2x+
FACTORS SYSTEM
OF
FOR
THE
MgF
T. V. RAMAKRISHNA RAO and S. V. J. LAKSHMAN SpectroscopicLaboratories,Department of Physics,S. V. University,Tirupati, India (Received21 Nocemher1969)
Abstract-The r-centroids and Franck-Condon factors for the bands of the C’E’-X*X+ system of the astrophysically important MgF molecule have been determined. The bands of this molecule are present both in the disk and spot spectra. The values of Franck-Condon factors as determined by the approximate analytical method of Fraser and Jarmain compare very well with those obtained by Bates’ method.
INTRODUCTION THE BAND
spectrum of the MgF molecule has been studied by DATTA,(‘) FOWLER,“’ and GRINFIELD and JEVONS.(~’ Three band systems designated as A’fl++ X2x+ (236863468 A), B2C + +-+X2x+ (12742-2649 A) and C2C+ +- X’C+ (23249-2387 A) are reported for this molecule. The detailed rotational analysis of the bands for this molecule has been carried out by BARROW and BEALE. ‘5’The molecular constants used in the present work are summarized in Table 1. Since the bands of the MgF molecule are astrophysically important (they are found in disk and spot spectra), the authors took up the present work to calculate the r-centroids and Franck-Condon factors for the bands of the C2C+-X*C+ system of MgF molecule. SINGH et ~1.‘~’found the state A and X of this molecule to obey Morse potential. To test how far the C state obeys the Morse potential, the potential energy curves for the C and X states have been calculated and presented in Table 4. JENKINS
TABLE ~.SPECTROSCOPICCONSTANTSOFTHE
Constant
co, (cm _ ‘) o&cm ‘) B,(cm-‘) re (‘Q ;(A-‘) P a,(cm-‘)
Upper
C'ZZ-X*Z'
state
823.2 5.04 0.55102 1.6988 1.780418 164.9741 (K,) 0.9799029 (pl) 0.00449
945
SYSTEMOFTHE
Lower state
721.6 4.94 0.51922 1.7500 1.762677 144.6130 (KZ) 1.0209388(p2) 0.00470
MgF MOLECULE
T. V. RAMAKRISHNA KAO
946
S. V. J. LAKSHMAY
and
r-CENTROID
(f,,.,?..)
The r-centroid V,,,,,..,representing the characteristic internuclear separation of a r’ + 1”’ transition in a diatomic molecular band system, has been defined (NICHOLLS and J.4RMAIN”)) as
where 1+4~, and Ic/,,,.are the vibrational wave functions and I’ is the internuclear distance. In the present work, both the graphical and quadratic-equation methods (Nicholls and Jarmain) have been employed for the evaluation of r-centroids for the bands of the C’C’ X2X+ system of the MgF molecule. The results are presented in Table 2. As has been observed by Nicholls and Jarmain for a system with rl,I < rt,r, a smooth curve has been obtained when a graph was drawn between r,,..,.,, and il,.,,.,, (the wavelength of the corresponding transition). The value of F,,, is slightly greater than the (r,.I + r, ?)i2 value, suggesting that the potentials are not very anharmonic. The difference A?-= ?,..,,.,. has been found to be a constant for a given sequence in accordance with r,. i I ,I ” + I the observations of Nicholls and Jarmain.
TABII 2. THE M’ENTKOIDS (A) AND WAV~LENGIHS
\
‘\
(A) OF IIANDS ot Ttib C“C
-____
SYSIEM OF Mgt-
I”
0
I”\‘i‘lL-__
2
3
obtained
4
i
I .729 1.729 7347.X
1.654 1.652 2381.9
I.330
1.499
1.57X
1.499
I.666 1.664 23x1 3
1.593 I .590
1.515 IS12
1.750
I .67X
I .750 2336.6
I .67X 2375.2
I.606 I .605
I.S.?l I 53
I.hlY 1.014
1.815
1.739
I.816 2303.6
1.740 2342. I
1.904 I.904 2262.0
I .x25 I .X26 229X.6
1.999 I.998
1.914 1.914 2757.7
2.104 2.088
2.009
I.923
2.006
1.924 2253.6
1 .x35
I.761
I .6X9
I .X36
1.760
I.689
2204. I
2.018 2.016
(a) r-centroids graphical method
’ X’Z’
by use of the m A.
; (c)wavelengths
quadratic-equation
method;
1.x4.5 I .X46 22x9.5 I.933 I.934 224Y.7
(h) r-centrmds
I.771 1.772 1.x55 1.856 1285.0
obtained
I.701 I .700 I .7x2
I .7x4
by use of the
r-Centroids
and Franck-Condon
FRANCK-CONDON
The relative
vibrational
transition
947
factors
FACTORS
probability
is written
as
P l9.0” = R,2(?“~,“&,~,,~~. Here R, is the electronic transition moment, FL,,,,,.is the r-centroid and qvr,uf.denotes the Franck-Condon factor. This parameter qvs,usrcontrols the intensity distribution in the emission of molecular bands. The Franck-Condon factors calculated by the approximate analytical method of FRASER and JARMAIN@)are given in Table 3 along with the values obtained by BATES’(~) method. The two values are in excellent agreement. TABLE 3. FRANCK--CONDON FACTORSFOR BANDS OF THE C’E’-X*C’
B
SYSTEMOF MgF
crs
0
1
2
3
4
0
0.123 (0.721)
0.227 (0.229)
0.043 (0.045)
0.006 (0.006)
0.00
1
0.239 (0.240)
0.326 (0.319)
0.310 (0.314)
0.100
0.023
2
0.035 (0.037)
0.348 (0.349)
0.110
0.280
3
0.003 (0.003)
0.089
0.328
4
0.000
0.002
L.’
Values given in parentheses
were derived
True
r max (A)
Morse
42938.3 43751.5 44554.5 45347.5 46130.3 46903.1 41665.9 48418.5
1.639 1.575 1.555 1.539 1.525 1.513 1.499
1.639 1.607 1.575 1.555 1.539 1.525 1.514 1.500
1.687 1.645 1.618 1.597 1.580 1.567 1.554 1.543
1.687 1.646 1.619 1.599 1.581 1.567 1.555 1.544
1.607
z = 0 359.6 1067.4 1766.3 2457.3 3140.7 3815.3 4481.0 5140.0
True
Morse
1.764 1.811 1.856 I.890 1.921 1.949 1.976 2.000
1.764 1.811 I .856 I .890 1.922 1.950 1.977 2.000
1.820 1.877 1.920 1.957 1.987 2.02 1 2.050 2.078
1.820 I .879 1.920
C’C+ state
T, = 42528.0 0
0.000
SYSTEMOF THE MgF MOLECULE
r min (A) (I+ z(cm-‘)
I
by using Bates’ method
TABLE 4. POTENTIAL ENERGY CURVESFOR THE C’Z’-X*X’
L’
5
X’C+ state
1.957 1.988 2.021 2.050 2.080
94x
7. V. RAMAKKISHNA RAO and S. V. .I. LAKSIIMAN
From the magnitude of the FranckkCondon factors, we conclude that it is very unlikely that the bands 3.0 and beyond in the r’-progression with 1”’ = 0 and the bands 0.3 and beyond in the c”-progression with I.’ = 0 will be observed.
POTENTIAL
ENERGY
CURVES
Singh et ul. found the A and X states of the MgF molecule to obey closely the Morse potential. In order to test how far the state C of this molecule follows the Morse-potential function, the potential energy curves have been calculated for the bands of the CX system between the two using both RKRV”” and MORSE(“) methods. The close agreement values (Table 4) proves that the state C also obeys the Morse potential function closely.
A(,kno~,/c~~rmm/s The authors wish to express their grateful thanks tu the authorities of this unlversit) lor providing all the necessary facilities and to PROF. J. BHIMASENA~HARfor his encouragement and Interest in this work.
REFERENCES
I S. DATTA, Pro<. R. Sot. 99, 436 ( I92 I ). 2. c‘. A. FOWLEK, Phys. Rev. 59,645 (194 I ). 3. 4. 5. 6. 7. 8. 9. IO. I I.
F. A. JENKINSand R. GRINFELU. Phw Ru. 45, 229 ( I Y34). W. JEVONS, Proc. R. Sot. 122,211 (1929). R. F. BARROW and J. R. BEALE, Proc~. PII~J. Sot. 91, 483 (1967). I. D. SINGH, M. M. SHUKLA and R. C. MAHESHWAKI, Jo.SRT 9, 533 ( lY6Y). K. W. NKHOLLS and W. R. JARMAIN, PIW. phy.v. SK. 69A. 253 (1956). P. A. FRASER and W. R. JARMAIN. Proc. ph,s. SW. 66A, 1145 (1953). D. R. BATES, Mm Not. R. ustr. Sot. 112, 614 (1952). J. T. VANDERSLICE. E. A. MASON, W. G. MAIS~II and E. R. LIPPINCOTT. J. .!4olr,c,. Sprc.tro.si~. 5, X3 (1960) P. M. MORSE, Phm Rrr. 34, 57 (1929).