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On the Faraday effect in O2
JOURNAL
OF MOLECULAR
SPECTROSCOPY
9,
421-425 (1962)
On the Faraday
Effect
in O,*
J. FINKEL United States Naval Ordnance Laboratory,
White Oak, Silver Spring, Maryland
From Hamiltonians containing the interaction of the magnetic moment of the molecule with the magnetic field of the radiation, the imaginary parts of the magnetic susceptibilities for right and left circular polarization are calculated. The dispersion relations appropriate to tensor media is then applied yielding the difference between the real parts of the magnetic susceptibilities for the two states of circular polarization. Subject to the assumptions that the spin multiplet and adjacent rotational levels spacings are small compared to kT and that the frequency of the radiation is much greater than the frequencies of the absorption bands, an expression for the contribution of the magnetic susceptibility to the Verdet constant is obtained which is frequency independent. This contribution to the Verdet constant is calculated numerically for 02 , under experimental conditions, to be 1.96 pmin per gauss per cm. When the frequency-independent term is combined with a term of the earlier work of Serber, an expression for the total Verdet constant of 02 in the near infrared, visible, and ultraviolet is obtained which can be fit to the experimental data to 1%. The results obtained here are in complete agreement with those derived earlier by Hougen using a different method. INTRODUCTION
When a plane polarized electromagnetic wave traverses a medium along a dc magnetic field, it undergoes a rotation. This rotation is the Faraday effect. It was observed experimentally that the angle of rotation 0 is given by
e = VIB,
(1)
where V is Verdet’s constant which depends on the material of the medium and the frequency of the radiation, 1 is the path length in the medium, and B is the dc magnetic induction. From the solution of Maxwell’s equations and boundary conditions, one obtains for the general expression of the Verdet constant of a gas V = tw,‘4Bc) [(x:+ -
XL> +
(XL, -
xL)l,
(2)
where w is the angular frequency of the radiation, c is the velocity of light in are the real parts of the electric and magnetic susceptivacuum and xii, & * Thesis submitted to the faculty of the Graduate School of the University in partial fulfillment of the requirements for the degree of Master of Science. 421
of Maryland
422
FINKEL
bilities for right and left circularly polarized radiation with (+) corresponding to right and (-) to left. In their treatment of the Faraday effect, Rosenfeld (1) and Serber (a) considered the contribution of & to the Verdet constant but neglected XL* . Serber’s theory led to an expression for V which gives the proper frequency dependence for all gases tested with the exception of O2 (3). Hougen (4) showed that the anomalous behavior of 0, is due to the interaction of the spin magnetic moment of this molecule with the magnetic radiation field. This gives rise to xl+ and hence an additional contribution to the Verdet constant. This paper presents an alternative derivation of XL* based on the application of appropriat,e dispersion relations. THE
DISPERSIOS
RELATIONS
In isotropic media where the electric and magnetic susceptibilities are scalar quantities, the real and imaginary parts of the susceptibilities are related by the celebrated Kronig-Garners relations given by (5) (3a)
X
N
_--
m x’ dw 20 P . s0 ;2-_ n-
x represents the electric or magnetic susceptibility and P denotes the Cauchy principal value of the integral. With the introduction of a dc magnetic field into the medium, the susceptibiIities become tensor quantities and the above relations do not apply. The tensor analogues of the above relations which are particularly applicable to the calculation of the Faraday effect are (6) axtm22”p
s.a
AX”
T
~ a w2 -
dti w2
’
(-la)
where AX’
=
X+’ -
AX”
=
X+n -
X-‘,
X_N .
Combining Eqs. (2) and (4a), the Faraday rotation in a frequency range where there are no absorptions can be calculated from a knowledge of the absorption as a function of frequency. Specifically, the ranges of interest in this paper are the near infrared, visible, and near ultraviolet, while the absorptions are in the microwave and far ultraviolet. The procedure pursued in this paper is to calculate the imaginary parts of the
ON THE FARADAY
EFFECT
IN 02
423
magnetic susceptibilities for right and left circularly polarized radiation and apply dispersion relation Eq. (4a) to obtain a~~,‘. CALCULATION
OF
Ax,,,’
The Hamiltonian of a molecule in a de magnetic field B (along 2 axis) and circularly polarized radiation field b is given by H* = Ho -
@~-‘bo(Srei”t
+ S*e+*),
(5)
where Ho is the Hamiltonian in the absence of the radiation but including the dc magnetic field; p is the Bohr magneton; Sk is the raising and lowering spin operator; w is the angular frequency of the radiation field. Applying first-order time-dependent perturbation theory to the solution of the Shrijdinger equation with the Hamiltonians given in (5)) we arrive at the expressions for the imaginary parts of the magnetic susceptibilities for right and left circularly polarized radiation given by
g xz*= 2B2~~~N w,i
1 (f
1 S,
j i)
!z(e-Ei’kT - e-Ef’kT)
“‘“f’, w’.
(6)
~0 is the permeability of free space (in M.K.S. units), Q is the partition function given by Q = xje-Ei’kT, N is the molecular concentration, wf; = (Ef - Bi)/fi, G(wfi - w) is the delta function in o. We now obtain the expression for AXE’ by substituting Axmn given by (6) in the dispersion relation (4a) with the result Ax,,,’ = -
41~0 ___ No@
fi3Q
e
i
Since wfi are in the microwave region of the spectrum while w is in the near infrared, visible, and near ultraviolet di < U” and for spin multiplets and adjacent rotational levels close compared to kT with only the ground electronic and vibrational levels occupied so that Ej << kT and eMEilkTE 1 - (Ej/kT). Furthermore, for moderate magnetic fields (less than 2000 gauss) the energies of the molecule are given, very nearly, by Ej = Ey’ + 2PBS, where Ej?’ is the energy in the absence of any applied fields and S, is the Z component of angular momentum. Taking into account the above considerations, Eq. (7) reduces to Ax,’
=
(16/3) (@3NBpo/wfikT) S (S + 1).
Combining (8) with (2), we obtain the frequency-independent the Verdet constant V,
= (S/3) (dNpo/hckT)
S (S + 1).
(8) contribution to
(9)
The result in (9) is in agreement with Hougen’s corresponding expression (4).
424
FINKEL TABLE
I
FARADAY DISPERSION OF 0~ x
4000 4360 4500 5000 5460 5500 5780 5893 6000 6500 7000 7500 8500 9000 9875 “X is a wavelength
V ohs
V talc
Error
11.69 9.72 8.49 8.05 6.87 6.04 5.98 5.59 5.45 5.32 4.78 4.35 4.03 3.79 3.58 3.41 3.14
11.64 9.82 8.49 8.07 6.85 6.02 5.96 5.57 5.42 5.30 4.79 4.38 4.06 3.80 3.58 3.40 3.15
0.47% 1.04 -
in Angstrom
units. V is the Verdet constant
APPLICATION
0.25 0.29 0.33 0.33 0.36 0.55 0.38 0.21 O.G9 0.75 0.26 0.29 0.32 in Fmin/gauss-cm.
TO OSYGES
CT, for Oe can be evaluated numerically under the experimental conditions of Ingersoll and Liebenberg (5’). These conditions consisted of a molecular concentration N of 2.69 X 10” molecules per cm’ and a temperature of 17°C. Making these substitutions in (9) together with a value of unity for S, since the O2 molecule has a spin of unity and devoid of orbital angular momentum in its normal state, we get I’, = 1.96 pmin per oersted per cm. This value of T/, is essentially in agreement with Hougen’s (4). The numerical value of T/, for O2 reported by Hougen is 1.95 which corresponds to a temperature of lY”C, two degrees higher than that reported by Ingersoll and Licbenberg (8). To obtain the total Verdet constant of the O2 molecule, account must be taken of the contribution from Ax,’ (Eq. (2)) calculated by Serber (2). The total Verdet constant is given by
v = [BX”/ (X” - X2)]+ v, ,
(10)
where I3 and 5 are determined from a least-square fit of the experimenhal dat,a. The best least-square fit of (10) with experiment yields the values: B = 104 pmin per oe per cm, x = lOSOA, V,
= 1.93 pmin per oe per rm.
ON THE FARADAY
EFFECT IN 02
425
Table I presents the observed values and those calculated from (10) with the values of B, X, and V, given by the best fit. The temperature dependence of the Verdet constant of O2 as given in (9) yields for the ratio V12.6~c/V~3.~0c at X = 44OlA the value of 1.032. The corresponding experimental value is 1.0275 f 0.005, in good agreement with theory. ACKNOWLEDGMENT The author is grateful to Professor Edward A. Stern of the University of Maryland for suggesting the approach to the problem and for guidance during the preparation of this research. RECEIVED: June 8, 1962 REFERENCES 1. L. ROSENFELD, Z. Physik 67,835 (1929). ,O. R. SERBER,Phys. Rev. 41,489 (1932). 5. L. R. INGERSOLL ANDD. H. LIEBENBERG, J. Opt. Sot. Am. 44,566 (1954); 46,538 (1956); 46,339 (1958). 4. J. T. HOUGEN,J. Chem. Phys. 32, 1122 (1960). b. R. DE L. KRONIG,Physica 3, 1009 (1936); H. A. KRAMERS,Atti Congr. Intern. Fisici, Como, 2,545 (1927); J. H. VAN VLECK,in “Propagation of Short Radio Waves,” D. E. Kerr, ed., Chapter 8. McGraw-Hill, New York, 1951. 6. B. S. GOURARY,J. Appl. Phys. 26,283 (1957); H. S. BENNETTANDE. A. STERN,Bull. Am. Phys. Sot. [II], 6,279 (1960).
OF MOLECULAR
SPECTROSCOPY
9,
421-425 (1962)
On the Faraday
Effect
in O,*
J. FINKEL United States Naval Ordnance Laboratory,
White Oak, Silver Spring, Maryland
From Hamiltonians containing the interaction of the magnetic moment of the molecule with the magnetic field of the radiation, the imaginary parts of the magnetic susceptibilities for right and left circular polarization are calculated. The dispersion relations appropriate to tensor media is then applied yielding the difference between the real parts of the magnetic susceptibilities for the two states of circular polarization. Subject to the assumptions that the spin multiplet and adjacent rotational levels spacings are small compared to kT and that the frequency of the radiation is much greater than the frequencies of the absorption bands, an expression for the contribution of the magnetic susceptibility to the Verdet constant is obtained which is frequency independent. This contribution to the Verdet constant is calculated numerically for 02 , under experimental conditions, to be 1.96 pmin per gauss per cm. When the frequency-independent term is combined with a term of the earlier work of Serber, an expression for the total Verdet constant of 02 in the near infrared, visible, and ultraviolet is obtained which can be fit to the experimental data to 1%. The results obtained here are in complete agreement with those derived earlier by Hougen using a different method. INTRODUCTION
When a plane polarized electromagnetic wave traverses a medium along a dc magnetic field, it undergoes a rotation. This rotation is the Faraday effect. It was observed experimentally that the angle of rotation 0 is given by
e = VIB,
(1)
where V is Verdet’s constant which depends on the material of the medium and the frequency of the radiation, 1 is the path length in the medium, and B is the dc magnetic induction. From the solution of Maxwell’s equations and boundary conditions, one obtains for the general expression of the Verdet constant of a gas V = tw,‘4Bc) [(x:+ -
XL> +
(XL, -
xL)l,
(2)
where w is the angular frequency of the radiation, c is the velocity of light in are the real parts of the electric and magnetic susceptivacuum and xii, & * Thesis submitted to the faculty of the Graduate School of the University in partial fulfillment of the requirements for the degree of Master of Science. 421
of Maryland
422
FINKEL
bilities for right and left circularly polarized radiation with (+) corresponding to right and (-) to left. In their treatment of the Faraday effect, Rosenfeld (1) and Serber (a) considered the contribution of & to the Verdet constant but neglected XL* . Serber’s theory led to an expression for V which gives the proper frequency dependence for all gases tested with the exception of O2 (3). Hougen (4) showed that the anomalous behavior of 0, is due to the interaction of the spin magnetic moment of this molecule with the magnetic radiation field. This gives rise to xl+ and hence an additional contribution to the Verdet constant. This paper presents an alternative derivation of XL* based on the application of appropriat,e dispersion relations. THE
DISPERSIOS
RELATIONS
In isotropic media where the electric and magnetic susceptibilities are scalar quantities, the real and imaginary parts of the susceptibilities are related by the celebrated Kronig-Garners relations given by (5) (3a)
X
N
_--
m x’ dw 20 P . s0 ;2-_ n-
x represents the electric or magnetic susceptibility and P denotes the Cauchy principal value of the integral. With the introduction of a dc magnetic field into the medium, the susceptibiIities become tensor quantities and the above relations do not apply. The tensor analogues of the above relations which are particularly applicable to the calculation of the Faraday effect are (6) axtm22”p
s.a
AX”
T
~ a w2 -
dti w2
’
(-la)
where AX’
=
X+’ -
AX”
=
X+n -
X-‘,
X_N .
Combining Eqs. (2) and (4a), the Faraday rotation in a frequency range where there are no absorptions can be calculated from a knowledge of the absorption as a function of frequency. Specifically, the ranges of interest in this paper are the near infrared, visible, and near ultraviolet, while the absorptions are in the microwave and far ultraviolet. The procedure pursued in this paper is to calculate the imaginary parts of the
ON THE FARADAY
EFFECT
IN 02
423
magnetic susceptibilities for right and left circularly polarized radiation and apply dispersion relation Eq. (4a) to obtain a~~,‘. CALCULATION
OF
Ax,,,’
The Hamiltonian of a molecule in a de magnetic field B (along 2 axis) and circularly polarized radiation field b is given by H* = Ho -
@~-‘bo(Srei”t
+ S*e+*),
(5)
where Ho is the Hamiltonian in the absence of the radiation but including the dc magnetic field; p is the Bohr magneton; Sk is the raising and lowering spin operator; w is the angular frequency of the radiation field. Applying first-order time-dependent perturbation theory to the solution of the Shrijdinger equation with the Hamiltonians given in (5)) we arrive at the expressions for the imaginary parts of the magnetic susceptibilities for right and left circularly polarized radiation given by
g xz*= 2B2~~~N w,i
1 (f
1 S,
j i)
!z(e-Ei’kT - e-Ef’kT)
“‘“f’, w’.
(6)
~0 is the permeability of free space (in M.K.S. units), Q is the partition function given by Q = xje-Ei’kT, N is the molecular concentration, wf; = (Ef - Bi)/fi, G(wfi - w) is the delta function in o. We now obtain the expression for AXE’ by substituting Axmn given by (6) in the dispersion relation (4a) with the result Ax,,,’ = -
41~0 ___ No@
fi3Q
e
i
Since wfi are in the microwave region of the spectrum while w is in the near infrared, visible, and near ultraviolet di < U” and for spin multiplets and adjacent rotational levels close compared to kT with only the ground electronic and vibrational levels occupied so that Ej << kT and eMEilkTE 1 - (Ej/kT). Furthermore, for moderate magnetic fields (less than 2000 gauss) the energies of the molecule are given, very nearly, by Ej = Ey’ + 2PBS, where Ej?’ is the energy in the absence of any applied fields and S, is the Z component of angular momentum. Taking into account the above considerations, Eq. (7) reduces to Ax,’
=
(16/3) (@3NBpo/wfikT) S (S + 1).
Combining (8) with (2), we obtain the frequency-independent the Verdet constant V,
= (S/3) (dNpo/hckT)
S (S + 1).
(8) contribution to
(9)
The result in (9) is in agreement with Hougen’s corresponding expression (4).
424
FINKEL TABLE
I
FARADAY DISPERSION OF 0~ x
4000 4360 4500 5000 5460 5500 5780 5893 6000 6500 7000 7500 8500 9000 9875 “X is a wavelength
V ohs
V talc
Error
11.69 9.72 8.49 8.05 6.87 6.04 5.98 5.59 5.45 5.32 4.78 4.35 4.03 3.79 3.58 3.41 3.14
11.64 9.82 8.49 8.07 6.85 6.02 5.96 5.57 5.42 5.30 4.79 4.38 4.06 3.80 3.58 3.40 3.15
0.47% 1.04 -
in Angstrom
units. V is the Verdet constant
APPLICATION
0.25 0.29 0.33 0.33 0.36 0.55 0.38 0.21 O.G9 0.75 0.26 0.29 0.32 in Fmin/gauss-cm.
TO OSYGES
CT, for Oe can be evaluated numerically under the experimental conditions of Ingersoll and Liebenberg (5’). These conditions consisted of a molecular concentration N of 2.69 X 10” molecules per cm’ and a temperature of 17°C. Making these substitutions in (9) together with a value of unity for S, since the O2 molecule has a spin of unity and devoid of orbital angular momentum in its normal state, we get I’, = 1.96 pmin per oersted per cm. This value of T/, is essentially in agreement with Hougen’s (4). The numerical value of T/, for O2 reported by Hougen is 1.95 which corresponds to a temperature of lY”C, two degrees higher than that reported by Ingersoll and Licbenberg (8). To obtain the total Verdet constant of the O2 molecule, account must be taken of the contribution from Ax,’ (Eq. (2)) calculated by Serber (2). The total Verdet constant is given by
v = [BX”/ (X” - X2)]+ v, ,
(10)
where I3 and 5 are determined from a least-square fit of the experimenhal dat,a. The best least-square fit of (10) with experiment yields the values: B = 104 pmin per oe per cm, x = lOSOA, V,
= 1.93 pmin per oe per rm.
ON THE FARADAY
EFFECT IN 02
425
Table I presents the observed values and those calculated from (10) with the values of B, X, and V, given by the best fit. The temperature dependence of the Verdet constant of O2 as given in (9) yields for the ratio V12.6~c/V~3.~0c at X = 44OlA the value of 1.032. The corresponding experimental value is 1.0275 f 0.005, in good agreement with theory. ACKNOWLEDGMENT The author is grateful to Professor Edward A. Stern of the University of Maryland for suggesting the approach to the problem and for guidance during the preparation of this research. RECEIVED: June 8, 1962 REFERENCES 1. L. ROSENFELD, Z. Physik 67,835 (1929). ,O. R. SERBER,Phys. Rev. 41,489 (1932). 5. L. R. INGERSOLL ANDD. H. LIEBENBERG, J. Opt. Sot. Am. 44,566 (1954); 46,538 (1956); 46,339 (1958). 4. J. T. HOUGEN,J. Chem. Phys. 32, 1122 (1960). b. R. DE L. KRONIG,Physica 3, 1009 (1936); H. A. KRAMERS,Atti Congr. Intern. Fisici, Como, 2,545 (1927); J. H. VAN VLECK,in “Propagation of Short Radio Waves,” D. E. Kerr, ed., Chapter 8. McGraw-Hill, New York, 1951. 6. B. S. GOURARY,J. Appl. Phys. 26,283 (1957); H. S. BENNETTANDE. A. STERN,Bull. Am. Phys. Sot. [II], 6,279 (1960).