Hence in order to calculate the spectrum of the de- polarized ..... + £ y_1. Q'2. Q2[(l+e2T^2y-2. + Q2^'2]. + ^ y + y ' 1 ) 2 where y ==(joTF/a)j)=. (kl)"1 oc ///. (3.24a).
3)
For I ^ 2 the following trace formulas for triple products of ^-tensors hold: TR{7F T V H L } = 3 W ( / / 1 1 ; 2 / )
and
tr{T%
TÜ TjJz} = - 10 ^
XX
,
r ( / / 2 2 ; 2 /") Z l ^ y 4 a , / * ' 4 « . / a ' ,
(A7)
(A8)
where J F ( / / 1 1 ; 2 / ) and W(jj22;2j') are Racah coefficients. Formula ( A 8 ) together with the explicit expression for W { j j 2 2; 2 / + 2) has been used for deriving Eq. ( 4 . 1 9 b ) .
Spectrum of the Depolarized Rayleigh Light Scattered by Gases of Linear Molecules S I E G F R I E D HESS Institut für Theoretische Physik der Universität E r l a n g e n - N ü r n b e r g , Erlangen (Z. Naturforsch. 25 a, 350—362 [1970] ; received 24 October 1969) T h e spectrum of the depolarized Rayleigh light scattered b y a gas of linear molecules is calculated b y a kinetic theory approach based on the W a l d m a n - S n i d e r equation. Collisional and diffusional b r o a d e n i n g are studied. The line width is related to relaxation c o e f f i c i e n t s which are collision brackets obtained f r o m the linearized W a l d m a n n - S n i d e r collision term involving the binary m o l e c u l a r scattering amplitude and its adjoint. It is shown under which c o n d i t i o n s the relaxation c o e f f i c i e n t s characterizing the line width can be c o m p a r e d with data o b t a i n e d f r o m SentflebenB e e n a k k e r effect and nuclear magnetic relaxation measurements.
The "depolarized Rayleigh" component 1 of the light scattered by a gas of linear molecules is associated with fluctuations of the (2nd rank) tensor polarization of the rotational angular momentum of the molecules and, in particular, its spectrum is determined by the spectral function of the tensor polarization 2 ' 3 . The spectrum of the depolarized Rayleigh light has resently been measured 4 for some gases of linear molecules ( H 2 , N 2 , C 0 2 ) in the pressure region where the width of the line is primarily caused by collisional broadening (pressure broadening). In this paper, the spectrum of the depolarized Rayleigh ligth is calculated by a kinetic theory approach based on the WaldmannSnider equation 5 . Both collisional and diffusional broadening are studied. Collisional broadening of the depolarized Rayleigh scattering has been treated theoretically by G O R D O N 6 who developed a classical theory which is akin to Anderson's impact theory 7 for the pressure broadening of absorption and emission spectra. 1
2
By the kinetic equation approach used in this paper the calculation of the spectrum of the light scattered by a gas is based on the same generalized Boltzmann equation (Waldmann-Snider equation) as the calculation of transport properties of polyatomic gases. The line width is expressed in terms of collision brackets obtained from the linearized Waldmann-Snider collision term which involves the binary scattering amplitude operator and its adjoint. Thus a rigorous connection between the line width and the molecular (binary) collision processes is established. Furthermore it is possible to obtain relations between line widths and transport properties since transport coefficients can also be expressed in terms of collision brackets. This paper proceeds as follows: Firstly, after some preliminary remarks on the one-particle distribution function operator for a gas of linear molecules and the definition of the second rank tensor polarization, the connection between the spectrum of the depolarized Rayleigh light and the spectral
" D e p o l a r i z e d " refers to the c o m p o n e n t of the scattered light whose electric field vector is perpendicular to the electric field of the linearly polarized incident light. " R a y l e i g h " refers to the line of the spectrum of the scattered light which is centered at the f r e q u e n c y of the incident ( m o n o chromatic) light. S o m e of the results derived in this paper have b e e n reported earlier, cf. S. HESS, Z. Naturforsch. 2 4 a, 1852 [ 1 9 6 9 ] .
3 4
5
6 7
S. HESS, Z . Naturforsch. 2 4 a, 1675 [ 1 9 6 9 ] . V. G. COOPER, A . D . MAY, E. H. HARA, and H. F. P. KNAPP. Phys. Letters 2 7 A , 52 [ 1 9 6 8 ] . L. WALDMANN, Z. Naturforsch. 12 a, 6 6 0 [ 1 9 5 7 ] ; 13 a. 609 [ 1 9 5 8 ] ; R . F. SNIDER, J. Chem. Phys. 32, 1051 [ I 9 6 0 ] , see also S. HESS, Z. Naturforsch. 2 2 a, 1871 [ 1 9 6 7 ] . R . G. GORDON, J. C h e m . Phys. 44. 3 0 8 3 [ 1 9 6 6 ] . P. W . ANDERSON, P h y s . R e v . 76, 647 [ 1 9 4 9 ] .
S. HESS, Phys. Letters 2 9 A , 108 [ 1 9 6 9 ] .
Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Creative Commons Namensnennung-Keine Bearbeitung 3.0 Deutschland Lizenz.
This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution-NoDerivs 3.0 Germany License.
Zum 01.01.2015 ist eine Anpassung der Lizenzbedingungen (Entfall der Creative Commons Lizenzbedingung „Keine Bearbeitung“) beabsichtigt, um eine Nachnutzung auch im Rahmen zukünftiger wissenschaftlicher Nutzungsformen zu ermöglichen.
On 01.01.2015 it is planned to change the License Conditions (the removal of the Creative Commons License condition “no derivative works”). This is to allow reuse in the area of future scientific usage.
function of the spatial Fourier transform of the tensor polarization is stated ( § 1 ) . Then, by using the moment method, transport-relaxation equations for the (time- and space-dependent) tensor polarization and "tensor polarization flow" are derived from the linearized Waldmann-Snider equation. The relaxation coefficients involved are collision brackets which are proportional to the number density of the gas ( § 2 ) . Next, the spectral function of the spatial Fourier transform of the tensor polarization is calculated from the transport-relaxation equations (§ 3.) For high pressures where the mean free path Z of a molecule is very short compared with the wave length 1 of the light (/ ^ X) the resulting spectrum of the depolarized Rayleigh light has a Lorentzian line shape with a half-width determined by the relaxation coefficient for the tensor polarization. At lower pressures where I £ / the line shape is no longer Lorentzian. Diffusional broadening characterized by a diffusion coefficient which is different from the self-diffusion coefficient contributes to the half-width. Furthermore it is indicated that propagating tensor polarizaton waves may exist (under suitable conditions) which cause a splitting of the line into two components or lead to an effective broadening when the splitting cannot be resolved. Then the relaxation coefficients are studied in more detail (§ 4 ) . The relaxation coefficient for the tensor polarization is expressed by a collision bracket where only the nonspherical part of the scattering amplitude gives a nonvanishing contribution. Finally the interrelation between the half-width of the depolarized Rayleigh line and certain other transport and relaxation phenomena is considered (§ 5 ) . It is discussed under which conditions the relaxation coefficient for the tensor polarization can also be extracted from measurements of the (magnetic) Senftleben-Beenakker e f f e c t 8 ' 9 on the viscosity and of nuclear spin relaxation times and when the diffusion coefficient for the "tensor polarization" can be compared with data obtained from the SenftlebenBeenaker effect on the heat conductivity.
For a gas at room temperature the translational motion of the molecules can be treated classically, i. e. position and linear momentum of a molecule are specified by the classical variables X and p . The rotational motion of a molecule, treated quantum mechanically, is characterized by the rotational angular momentum operator J (in units of h ) . The Cartesian components of J obey the commutation relations 10 JH JV ~ j
j
H
i
=
£ nvl
j
X
(1-1)
where eßVx is the 3rd rank isotropic tensor with £123 = 1. It is convenient to define the projection operator Pj=
I\jm)(jm\, m
(1.2)
where j rn) is a normalized internal wave function for a linear molecule in the y'-th rotational state and m is the magnetic quantum number with respect to an arbitrary axis of quantization. Then the magnitude of the rotational angular momentum (in units of h ) is given by j = 0,1,2,....
PjP = PPj = j(j + l)Pi,
(1.3)
Obviously J is treated as an operator with respect to both magnetic and rotational quantum numbers; it is diagonal with respect to the latter. Now a brief discussion of the one-particle distribution function operator of a gas of linear molecules can be given. The part of the distribution function which is diagonal with respect to the rotational quantum numbers is written as (1.4)
f(t,x,p,J).
The local instantaneous mean value of an operator # = @(p,J) is given by ( )
=
Tr
f
d3p 0 / ,
(1.5)
where " T r " denotes the trace over magnetic and rotational quantum numbers and n(t,X)=Trfd*pf(t,X,p,J)
8
§ 1. General Remarks In this section the relation between the spectrum of the depolarized Rayleigh light and the spectral function of the tensor polarization is stated. Firstly, however, a number of preliminary remarks are necessary.
v
9
10
(1.6)
H. SENFTLEBEN, Phys. Z . 3 1 , 822, 961 [ 1 9 3 0 ] . - J. J. M . BEENAKKER, G. SCOLES, H. F . P. KNAPP, and R . M . JONKMAN, Phys. Letters 2, 5 [ 1 9 6 2 ] . J. J. M . BEENAKKER, T h e Influence of E l e c t r i c and M a g n e tic Fields on the Transport Properties of P o l y a t o m i c Dilute Gases, in ed. O. MADELUNG, F e s t k ö r p e r p r o b l e m e V I I I , V i e w e g , Braunschweig 1968. — J. J. M . BEENAKKER and F. R . MCCOURT, A n n . Rev. Phys. C h e m . 1 9 7 0 . Cartesian components of vectors and tensors are denoted be Greek subscripts. T h e summation convention is used.
is the local number density. Averages over an equilibrium distribution / 0 specified by the equilibrium density n 0 and temperature T0 will also be needed and are denoted by ( . . . ) 0 .
macroscopic deviations from the equilibrium state of the gas the function A(t\k) can be obtained from "transport-relaxation" equations which are derived in the next section.
The 2nd rank tensor polarization (which in the following will always be referred to as the "tensor polarization") is defined by
§ 2. Transport-Relaxation Equations
(1.7)
afn.(t,x) ={ ( . . . ) is defined by (Yo>(0))o.
(2.6)
If XF and are dimensionless quantities the collision bracket (2.6) has the dimension of an inverse time (effective "collision time"). It is of importance to notice that the collision term OJ(. . . ) and consequently the collision brackets are proportional to the equilibrium number density n 0 . Furthermore o>(. . .) is positive semi-definite, i. e. o ^ O .
(Wco(W))
By expanding the quantity 0 occurring in Eq. (2.5) with respect to a (complete) set of orthogonal "expansion tensors" depending on p and J and taking moments of the Waldmann-Snider equation (2.5) the (infinite) set of transport-relaxation equations for the time- and space-dependent "expansion coefficients" can be d e r i v e d 1 2 . To obtain the transport-relaxation equations that are required for the calculation of the spectral function of the tensor polarization at "high and medium" pressures the following ansatz is made for 0 :
with
+
0
1
m
ax, nv = ( p v )
(2.10)
„ V ) 0 = öxx' 4 , , „ V .
(2.11)
Due to the isotropy of the collision operator c o ( . . . ) the collision bracket pertaining to the expansion tensor 0 f l v can be written as (& t l v a> ( 0 f l V ) ) o = a>r 4 , , . „ V ,
12
(2.13)
Assuming that the relaxation coefficients for the 3 irreducible parts of the tensor ax, ^ are approximately equal to each other ("spherical approximation" 13 ) one has ) 0 = a> T F d l x'A M V , M v
(2.14)
where tt>TF= J5
(2.15)
is the relaxation coefficient of the tensor polarization flux. Using the ansatz ( 2 . 8 ) , the Eqs. ( 1 . 7 ) , (2.10) to ( 2 . 1 5 ) , and
(1.10),
1 / — ( 2 . 1 6 ) \
m
the following transport-relaxation equations are obtained from the Waldmann-Snider equation (2.5) :
+
j/^F
"
=
'
(2 18)
The spectrum of the depolarized Rayleigh line will be calculated from Eqs. (2.17) and (2.18) in the next section.
§ 3. Spectrum, Line Width
(2.9)
is the "tensor polarization flux". The normalization occurring in (2.9) has been chosen such that
11
= £ (&HV M ( ^HV) )o •
(2.8)
The 3rd rank tensor
( „ V
( °T
(2.7)
In (2.7) the equality sign occurs only if W is a "conserved quantity". The "isotropy" of the collision operator implies that the collision bracket vanishes for two irreducible tensors of different rank. For further (symmetry) properties of the WaldmannSnider collision brackets which, however, are not needed in this paper, see Ref. n ' 1 2 .
& =
where the relaxation coefficient (inverse relaxation time) (Of for the tensor polarization is given by
(2.12)
L . WALDMANN, Z . N a t u r f o r s c h . 1 5 a , 19 [ 1 9 6 0 ] ; 1 8 a , 1 0 3 3 [1963], S. HESS a n d L . WALDMANN, Z . N a t u r f o r s c h . 2 1 a , 1 5 2 9 [1966],
To calculate the spectrum of the depolarized Rayleigh light it is firstly assumed that the time derivative in Eq. (2.18) can be neglected compared with the relaxation term. This approximation leads to a Lorentzian line shape and is refered to as the " L o rentzian approximation" (part a). Secondly, in part b of this section, the spectral function is calculated from the full Eqs. (2.17) and ( 2 . 1 8 ) . Then (parte) the posible existence of tensor polarization waves leading to a splitting of the depolarized Rayleigh line into two components is discussed. 13
A . C. LEVI and F . R . M C C O U R T , Physica 3 8 , 4 1 5 [ 1 9 6 8 ] , F . R . M C C O U R T , H . F . P. K N A A P , and H . M O R A A L , Physica 43, 4 8 5 [ 1 9 6 9 ] .
a) Lorentzian
Approximation
where / = VkB Tjm
Firstly, Eq. (2.18) is approximated by kß Tri _ i 3a„ r & Wtf ^ - , / -
ax, ßV = -
\
m
/o i \ (3.1)
öxx
so that, using ( 3 . 1 ) , Eqs. (2.17) reduces to the following closed equation for the tensor polarization: 3a - Z)T AaMV + coT a,v = 0.
(3.2)
Here the "tensor polarization diffusion coefficient" h To DT = (3.3) m COyp
has been introduced and A = d2/dxQ dx0 is the Laplacian. A spatial Fourier transformation of Eq. (3.2) leads to [cf. Eq. ( 1 . 1 2 ) ] dt
a„v(f, k) + (k2 Dt + OJT) a„v{t, k) = 0,
(3.4)
and hence äßV{t,
with
k)
= A (t\k)
r
+
k)
k2 DT)
tj
(3.5)
for t ^ 0 . From this result, the following Lorentzian line is obtained for the spectrum of the depolarized Rayleigh line [cf. Eq. ( 1 . 1 3 ) ] o /
Sl(OJ
11_\
1
1
ti
k)
=
0jr + k2DT (o>p + A; Z ) t )
= — co2 +
.
3.6)
The half of the width at half height is (AOJ)
i/j = oj-p -+- k2 Dj .
(3.7)
Clearly the line width is determined by two additive contributions: "collisional" or "pressure" broadening characterized by the relaxation coefficient OJt oc n0 for the tensor polarization and diffusional broadening determined by k2 D^ oc The latter contribution depends on the scattering angle 7 between the wave vectors k0, kt of the incident and the scattered light, for k ^ ^ k ^ 2 one has k2^k02
2(1 - c o s * ) .
(3.8)
The relative importance of diffusional and collisional broadening is characterized by the ratio f = wT 1 k 2 D T = E ' 1 k 2 l 2 14
R . N. DICKE. Phys. Rev. 89. 472 [ 1 9 5 3 ] .
(3.9)
(3.10)
cotf1
is the mean free path of the tensor polarization flux and £ = coT/C>TF (3.11) is the ratio of the relaxation coefficients for the tensor polarization and the tensor polarization flux. For most gases of linear molecules — except for the hydrogen isotopes where fi^lO-1 — £ is of the order of 1. Since the wave number k is of the order of for scattering under 9 0 ° , collisional broadening dominates if (l/l)2i x , ßy (viv x have been neglected in ( 2 . 8 ) . Inclusion of (3.12) leads to a set of three coupled transport-relaxation equations instead of (2.17), ( 2 . 1 8 ) , and these yield a line width of the form (Aco)1/l
= a>Tp[e+(lß)2
+
...(llW]
[cf. ( 3 . 9 ) ] instead of ( 3 . 7 ) . Thus Eq. (3.7) is only valid for small values of ( l / l ) 2 such that terms of order (//A) 4 can be disregarded. Furthermore, the fact that Eq. (3.7) yields an infinite line width for n 0 - ^ - 0 (i. e. IjX 00) whereas the line width for small pressures is finite and determined by the 15
V. G. COOPER, Thesis, T o r o n t o 1968. - V. G. COOPER, A. D. MAY, E. H. HARA, and H. F. P. KNAAP. Can. J. Phys. 46. 2 0 1 9 [ 1 9 6 8 ] .
Doppler broadening clearly indicates that Eq. ( 3 . 7 ) is not valid if the mean free path I is larger than the wave length A of the light. So far, the spectrum of the depolarized Rayleigh light has been investigated experimentally only in the pressure broadening r e g i o n 4 (l/X ^ 1 ) . The validity of the approximation (3.1) is discussed in the followig two parts of this section. b) Non-Lorentzian
Line Shape
Now the spectral function 5 ( t o I fc) is calculated from Eqs. ( 2 . 1 7 ) , (2.18) without the approximation (3.1). Elimination of a x , ^ from Eqs. ( 2 . 1 7 ) , ( 2 . 1 8 ) yields the 2nd order differential equation «TF "V, (*> *) + (l+e) -D^
AaßV{t,X)
=0,
(3.13)
where the dot denotes differentiation with respect to time. As far as the calculation of the spectral function is concerned, the initial conditions imposed on the tensor polarization and the tensor polarization flux are a^„(0,X) =0.
(3.14)
= -a>ra„v(0, X).
(3.15)
d(x), Thus Eq. (2.17) implies
By taking the spatial Fourier transform of Eq. (3.13) and using the definition ( 1 . 1 4 ) f o r the function A(t\k) the following differential equations f o r A is obtained: cofp1
Ä + (1 + — ) \
rv
0)
A +
J
(k2
A(0\k)
(o
=
\ (l+£)
and e ~co r t
(3.21) „juck
reduces to the exponential function (3.5) does for high pressures.
as
Next the spectral function S(co \ k) is obtained from ( 3 . 2 1 ) according to ( 1 . 1 3 ) . Using the dimensionless frequency variable Q = co/coDOp
di2 = S(eo|fc) dco,
S(Q\k) 7iS(Q\k)
(3.24) £ y + y_1
(3.19)
. ,
Q'2
+ Q2^'2]
+
^ y + y '
1
)
2
where y
==(joTF/a)j)=
(kl)"1
oc / / /
(3.24a)
is a dimensionless collision frequency which is proportional to the number density of the gas. The theory presented here can be considered to be applicable f o r only. The Lorentzian function (3.6) is equivalent to £ y + y 7iSL{&
fc)
=
Q - 2
+ ( £ y + y
- i
)
2
•
(3.25)
TI S 1.5 // / / / / /
/ / / /
//
(3.18)
COTF,
+ £ y_1
Q2[(l+e2T^2y-2
(3.17)
= -CDJ.
oj -o>r
by
(3.23)
=
/ / / /
,
„
(3.22)
and introducing the spectral function S(£?|fc)
!
cosh co t +
£ ) 2
a ) " « \ (1 — £) 0>rF
are
/
/ ' / / '/
/y/7 /
/
\\\\ A i A\ \ \ \ 1.0 \ \ \
\
\
\
/
y
05
\ v
(3.20)
the solution of Eq. ( 3 . 1 6 ) subject to the initial conditions (3.17) is given by k)
_
one has
/
o / ' = io>TF[(l-£)2_4o/i)0p/Co|F]1/i,
A(t
< ( 1
COJF
With the abbreviations &>DoP = k2-kB Tjm
C O | O P
D t + o>r) A = 0. ( 3 . 1 6 )
The initial conditions for A(t\k) ^ (0 i fc) = 1 ;
4
one finds
dßV(t,X)
+ o*ra/xv(t,x)
where
,,
77— sinh co t
a>
(3.21) Clearly the function A(t\k) as given by ( 3 . 2 1 ) is different from ( 3 . 5 ) . For £ =+= 1 and high pressures
Q Fig. 1. T h e spectrum of the depolarized Rayleigh light for £=&>T/COTF = 0.1. Curves 1 and 2 are the graphical representation of the functions n S{Q) as given by Eq. (3.24) for y = 2 and ?/ = 10. For the definition of Q and y see Eqs. (3.22) and ( 3 . 2 4 a ) . T h e dashed curve is the Lorentzian approximation ti SL ( ß ) [as given by Eq. (3.25] to curve 1. The Lorentzian curve for ?/ = 10 cannot be distinguished from curve 2 on the scale of the figure.
In Fig. 1, 7 i S { ü )
and n 5 L ( ß )
are plotted
for
For £ =)= 1 these tensor polarization waves do not
e = 0.1, y = 2 and £ = 0.1, y = 10. In the latter case
show up unless the pressure of the gas is sufficiently
both spectral functions cannot be distinguished on
low such that the inequality ( 3 . 2 7 ) is fulfilled. For
the scale of the figure. For y = 2, the spectral func-
e = l , i. e.,ft>R= COTF , however, ( 3 . 2 7 ) always holds
tion ( 3 . 2 4 ) is broader than the Lorentzian function
and ( 3 . 2 1 ) takes the simple form
(3.25)
f o r small Q and falls off faster for large
values of Q .
0)
rather than [ S ( ß ) -SL(Q)]/S{Q)
(3.26)
cal analysis f o r £ = 0.1, 0.5, 1.0, 1.5 showes that 14%,
< ; 6%
and
for
d ,( +*
