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Velocity distribution in spectral line shape
J. Qumr.
Specrrosc.
Rodiat.
Transfer.
VELOCITY
Vol.
7, pp. 505-515.
Pergamon Press Ltd., 1967. Printed in Great Britain
DISTRIBUTION
IN SPECTRAL
LINE
SHAPE
MASATAKA MIZUSHIMA Laboratory for Atmospheric and Space Physics, Department of Physics and Astrophysics, University of Colorado, Boulder, Colorado (Received 29 August 1966)
Abstract-In the derivation of the conventional Voigt profile formula, the width parameter of the Lorentz line shape formula is assumed to be independent of the velocity of radiating atoms. This assumption is acceptable when the pressure broadening is negligibly small in comparison with the natural broadening. When pressure broadening is more important than the natural broadening, however, the conventional Voigt profile formula is not valid since the width parameter depends appreciably on the velocity. The resulting profile is shown to yield about 30 per cent lower intensity than the Voigt profile in the tail region. Even when the Doppler effect is completely negligible (low-temperature limit), our profile deviates from the Lorentz profile, particularly in the tail region.
VOIGT IT IS generally
PROFILE
accepted”) that the Lorentz formula Z,(o ; o,,A) = (Z,/n)A/[(w- o,,)’ + A’]
(1)
represents the intensity distribution of a single spectral line quite well. Here w is the frequency, oO is the center frequency, A is the width parameter, and I, is the integrated intensity, or I, = j’? o3Z dw. If the radiating atom is moving at velocity v in the frame of the observer and if a is the angle between v and the vector from the observer to the atom, then,
In the non-relativistic
wg = w,,(l +(u/c) cos a’)/(1 - (V/C)2)“2,
(2)
cos a’ = (cos a - (u/c))/( 1 - (u/c) cos a).
(3)
approximation
of the Doppler effect, we see that
0, = u cos a = c(oo - ooo)/ooo. Assuming thermal equilibrium at temperature frequency between oO and oO + do,, is dN = N(m/2&T)“‘[exp(
(4)
T the number of atoms dN, with center - mut/2kT)] du,
= (N/7c1’26)[exp( -oO - oo,J2/S2)] dw,.
(5)
where N is the total number of atoms and 6 = (~,,,,/c)(2kT/m)“2 gives the width of the frequency distribution by the Doppler effect. 505
(6)
506
MASATAKA
MIZUSHIMA
If we assume that each atom with a given velocity Doppler
effect makes the envelope
of theintensity
produces line profile (l), then the distribution of _
00
A exp( - (oO -o~~)~/S~)/[(O - CO~)~ + A’] do,.
Z(w 7 . A 36 9woo) = (~,,N/Y?“c~)
s
(7)
-m If we further assume that A is independent
of o,,, equation
(7) reduces to
Z(w ; A, 6, woo) = (ION/n 1’26)H(Q, a),
(8)
where w,&/c!$ a = A/6 ;
!2 = (o-
(9)
the function m
H(R, a) = z is called
the Hjerting
function.
e-Y2 (10)
s (R-y)2+PdY --m
Some numerical
evaluations
of this function
have been
done.‘2’ The resulting line profile is called the Voigt profile. NATURAL
WIDTH
In obtaining conventional formula (8), the width parameter A is assumed to be constant, but this is not exactly true. When the width is due to the natural decay of the excited state of the atom, A cc l/t,,
(11)
where t, is the life time of the decay. When the atom is moving with respect to the observer,
the life time changes as t, = t&l
- (u/c)z)“z.
(12)
Thus, A E AOO[l++((v/c)~]. To the same order of approximation,
(13)
formula (2) gives
00 = oo()( 1 + (0,/c) -+(u/c,‘,. Using (13) and (14) and considering that I, is the transition which also changes with u in the same manner as A, we obtain,
I =
(14) probability
(zooN,7c%)(6,~)J~J (1 + (S/W,,)~ V2) e-“’
-cc
(C2-y+$(?l/000)1/2)2+a2(1
per unit time,
dV
+(S/W,,)~V~)
(15)
where V2 = y2+x2+z2,dV
= dxdydz
and
V = (r@T)‘12v.
(16)
Velocity
distribution
in spectral
line shape
507
The correction is of the order of 6/ooo, which is (2kT/m)“‘/c. For the hydrogen atom in the chromosphere of the sun (2 x lo6 “K), this ratio is about 10P3. Thus, the original Voigt profile (8) must be quite good when the pressure is low enough that pressure broadening is negligible. PRESSURE
BROADENING
When the width is due to pressure broadening, the Lorentz formula (1) is still applicable but A = n(q)
(17)
where n is the number of atoms in unit volume, u, is the relative velocity between the perturbing atom and the radiating atom, ( ) means the average over the velocity of the perturbing atoms, and0 = Re
do{Tr@/iT,y ipifTf)/Tr(pfipif)l}. (18) s Here Re means to the real part, ~1is the electric (or magnetic, depending on the radiation mechanism) dipole moment of the radiating atom, Tr means trace, and T is the time development operator due to a single collision, viz., T = exp{-i
i (H/h)dfi -03
(19)
with the parameter of collision in the range of da. Subscripts i and f refer to the initial and final states of the radiative transition, respectively. We assume each perturbing atom to be a classical particle and the collisions are essentially elastic so that the relative velocity does not change during each collision. Such assumptions are usually made in calculating the width parameter from interatomic fortes,(4) and they can be justified as a good approximation.* If there is no external field the average over the direction of the relative velocity can be taken care of by the Tr-operator, and da can be taken as do = 2nb db,
(20)
where b is the collision parameter illustrated in Fig. 1. The interaction Hamiltonian H which appears in formula (19) depends on the distance r between the atoms. This distance r depends on b, the shortest distance and x, the distance from the closest point: H = H(r) = H(b, x).
(21)
For our assumption of a point perturbing particle with constant velocity u,, we see that x = u,t
and
I = (b’ + @)1/Z
(22)
if time t is taken to be zero at the closest point. * From the uncertainty relation, AuAx = 0.7 x 10e9 in mks units for the argon atom. If we let Ax = 1 A then Au = 7 m/see. Since the average velocity is 50 m/set at room temperature for this atom, the inaccuracy of the Doppler effect in our calculation is about 14 per cent.
508
MASATAKA MIZUSHIMA
FIG. 1. Path of perturbing
atom.
Often H is given in the interaction representation. The extra time-dependent terms of the form exp( - iH,t) in that representation, however, do not contribute when there is no overlap of spectral lines.(3) Thus, 7 Hdt=
j
-a,
-U3
H(b, x) dx/u, = hh(b)/u,,
(23)
so that T(b ; v,) = exp( - ih(b)/u,) = 1 - i(h(b)/u,) - $(h(b)/v,)’ + . . .
(24)
When h/v,, or its matrix element, is less than one, we can have a good approximation by taking the first few terms only in the above expansion of the time development operator. Such approximation is good for large ur, at least. Let us assume that the approximation is good for any u,. Taking the first three terms only in the expansion (24), we find from formulas (18) and (19) that
+ 3(Cl/i~irh:)]/Tr~~i~i~)}/u~.(25) u(ur)= - 2nbWTr[0L,ibirh,) + i-@,J$pi,) Thus, from formula (17), A= n
i (constant). 0 *
(26)
Now u, is the relative velocity of the collision. If 5 and CCare velocities of the radiating and perturbing atoms, respectively, then u-l I
= (u,2+u~-2ucucose)-1’~
= u;’ +(u
0)+(u2,/u~)P2(cos t9)+ . . . , (27)
where u, and u< are larger and smaller than u, and u, respectively, and 0 is the angle between CCand I;; (u; ‘) in formula (26) means the average over the perturbing atom. Since the direction of CCis randomly distributed, (27) immediately gives
(t&-l) = (u;‘).
(28)
Velocity distribution in spectral line shape
509
Assuming the Boltzmann distribution of the velocity u,, we have
=
2a-“2(m/2kT)3’2{u-1
’ exp(-mu: s
*/2kT)uf
du,
0
cc
exp( -muz/2kT)u,
+
where a(x) is the error integral
du,} = u- ‘@((m/2kT)*‘2u),
(29)
X
O(x) = 2~ ‘j2 exp( - t2) dt. s
(30)
0
and m is the mass of the perturbing atom. Thus, (26) gives A = A,27c- “2D(V)/~
(31)
I/ = (m/2kT)‘12u.
(32)
where again
Formula (31) shows that the width parameter A is close to a constant A0 when Vis less than unity but proportional to v-’ when V is larger than unity. Figure 3 shows the relation between A and V:
FIG. 2. Relative velocity v,.
When the mass of the radiating atom is the same as that of the perturbing atom (selfbroadening), the line profile is given by Z =
(Z,N/X~‘~~)H’~‘(ZZ, ~1~)
(33)
where HCp’(Qa,) = 2aon312
{W)/V> _s o3(a-
exp( - V2) dV
Y)’+a~4/~{~~)/~}2
(34)
510
MASATAKA
MIZUSHIMA
O-2 t
I
O”O0.1
I 0.5 v FIG.
3.
I
I
I
I.0
I.5
2.0
I
2.5
(= “/JZTiG)
A/A,,as function of V[ = ~/(2&T/m)‘/~].
under relation (16). When the perturbing atoms are different from the radiating atom, formula (34) is to be modified by using Q(aV)/aVinstead of @(V)/V, where a = (mass of perturbing atom)/(mass of radiating atom).
(35)
In the remainder of this paper, we consider only self-broadening. LOW-TEMPERATURE
LIMIT
When the temperature is very low, the Doppler effect will become negligible. In formula (33), the parameter a0 becomes much larger than unity at low temperature. Since I/ 2 y, the integrant in (33) including exp( - I@) is negligibly small whenever y is comparable to $2 or ao. Thus we have HLp)(Q,a0 + co) = 2a,7c-3i2
m Q(V)/V exp( - Vz)V2 dV _m s Q2+a~4x-1{@(V)/V}2
(36)
in the limit of low temperature, or large ao. The limiting function H’p’(Q a0 + co) is shown in Fig. 4. From Fig. 4 we see that the resulting curve has the half-width at Q/a, = @65, or IW-WJha,f
= 065A(),
(37)
where A,, is the width parameter at o = 0 as formula (31) shows. The Lorentz formula I, = (0~86Z,N/~)0~65Ao/{(0 - oo)’ + (0.65Ao)2}
(38)
Velocity
distribution
in spectral
511
line shape
. . 0 4-
0 .3-
6 *
I N
g
2
b
0 .2 -
0 1.1 -
0.' 01 0
I
I
0.1
05
I
I
I.0
I.5
I 1.8
w-"oo AlI
FIG. 4. Spectral
Line Profiles at Lower Temperatures. (1) t1,, = CO, or d = 0; (2) 6, = 10, or A0 = 106. - - - is the Lorentz profile given by formula (38).
agrees with our result at the center and the half width. In Fig. 4 this Lorentz formula is compared with our result. It is seen that the Lorentz formula reproduces our resulting line profile fairly well. In the far-wing region, the Lorentz formula (38) gives I, + (0+36Z,N/K)065A,/(O = (ZeN/rr)O56A&~ Our formula,
- oo)*
- o~)~.
(39)
on the other hand, gives m Z -+ (Z,JV/~C)(~A~/~~“~)
{cD(V)/V} exp( - V2)V2 dV/(w-q,)2 I
0
=
(Z,N/n)O707A,/(o
-oo)‘.
Thus, the Lorentz formula with parameters intensity in the far-wing region to be about
(40) adjusted in the center region predicts the 25 per cent less than the “true” intensity.
MASATAKA MIZUSHIMA
512
If the integrated intensity is calculated from the same Lorentz formula, the result will be 0.86 times the “true” integrated intensity. One should also note that the average of the width parameter itself is 0707 AO, or A,/2 “’ , but the same parameter which appears in the equivalent Lorentz formula (38) is 0.65 A,. The difference has never been taken into account in the calculation of the width parameter from the interatomic forces. In Fig. 4, our spectral line profile for the case of A,, = 106 is also shown. If we take a typical atom such as Kr, this number corresponds to about i atmosphere for optical lines, or about 1 mm Hg pressure for infrared lines at room temperature.* Notice that an appreciable decrease in the peak intensity appears compared to the case 6 = 0, while the rest of the curve stays essentially the same as before. It is interesting to note that the present line profile agrees with a Lorentz profile better than the previous one for 6 = 0. The Lorentz formula to fit the present profile, however, should be I, = (0.86Zolv/n)0.66,A,/((o CORRECTIONS
TO
- WJ VOIGT
+(066,A,)2}
(41)
PROFILE
Numerical calculation of our profile function H (P’is made for cases of A,, = 26,6,26/3, and @. The results are shown in Figs. 4, 5 and Table 1. The Doppler effect is seen to determine the decrease of the peak intensity and broadening of the center region. It can easily be shown that formula (40) is valid in the tail region for any temperature. In Fig. 5, our resulting profiles are compared with the conventional Voigt profiles.
= 05 ,=05
O-I
= 213 ,=2/3 =I =2 0’1 0=2 0.01
FIG. 5. Spectral
line profiles when ao( = 6/A,) is around one, or A0 and 6 are comparable. Voigt profiles (~@‘~)ff(Q, a).
* For the Kr atom at room temperature,
6 = lo-’
ooO cps while A,/p is typically
about
- --
are
10’ cps/mm
Hg.
Velocity distribution in spectral line shape
513
TABLE1. (a,,/~~/~)If(~) a, = w
(v - Q,,)/A,
6=0 0.42 1 0.410 0.381 0.301 0227 0170 0.129 0.100 0.079 0.064 0.053 0.044
0.0 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
a0 = 0.1 A,, = 106
czg= 1.0 A,, = 6
0.412 0403 0.378 0.303 0.229 0.172 0.131 0.091 0.080 ox)64 0.053 0044
0.249 O-248 0246 0.235 0218 0.197 0.173 0.147 0122 O-099 0.079 O-064
As expected, the Voigt profile with a = a, agrees approximately with our profile in the center region, but an appreciable deviation appears in the tail region. As the matter of fact the conventional Voigt profile with a = a,-, gives Z, + (Z,NMAo/(o - 00)’
(42)
in the tail region as contrast to our formula (40) in the same region. PRESSURE
SHIFT
The atomic collisions shift the center frequency of a spectral line as they broaden it. The shift of the center frequency is proportional to the pressure and is called the pressure shift. With this shift, A’ in the Lorentz formula (1) should be written as Z(o ;coo,A, A’) =
(Zo/rr)A/{(o - o. - A’)’ + A2).
(43)
According to Anderson’s theory, the pressure shift is given by A’ = n(~‘v,>,
(44)
where 6’ = Im
daTr(p/iT- l~~,-Tf)lTr(~~iQ. (45) f Here Im means to take the imaginary part, and the other notation has the same meaning as in formula (18). If we take the expansion (24) with only the first three terms, we obtain a’(v,) z
2zb db{Tr{@&Pif) - (~L/i~ish,)}/rrgLrii~)~/v,. (46) s We see from (44) and (46) that the pressure shift, in this approximation, is independent of the relative velocity of collisions. Thus, no corrections due to the velocity distribution appear in this approximation.
MASATAKA MIZUSHIMA
514
HEAD-ON
COLLISIONS
When the collision parameter b is less than the sum of the kinetic radii of colliding atoms, the assumption of a straight path, which is illustrated in Fig. 1, is not valid. For elastic collisions, which we are considering here, the magnitude of the relative velocity is still conserved, but the direction can be greatly changed. Before and after such “head-on collision”, the Doppler shift is usually quite different since the component of the velocity along a given direction changes greatly. The frequency shift determined by this mechanism, however, should not produce any net effect because, if the gas is in equilibrium, the number of atoms with a given velocity component must stay constant irrespective of atomic collisions, or the change in the Doppler shift must be zero when averaged over all perturbing atoms. The bending of the path is usually due to the repulsive interatomic force, which is of very short range. The relative velocity actually changes during the collision ; at least the radial component of the relative velocity becomes zero at the turning point. When the force which produces this bending is of short range, however, such change of the magnitude of the relative velocity can be neglected and we can still take the expression (23) provided that x is taken along the path. LOW
VELOCITY
CUT-OFF
Our theory is based on the approximation that inclusion of the first few terms in the expansion (24) for the time development operator is sufficient. This approximation is valid only when Ih(
4 v,.
(47)
Since JH( is zero at a large distance and increases as r, the interatomic distance decreases, is expected to increase as b decreases. As b becomes smaller than r,,, the sum of the kinetic radii of the colliding atoms, Ih(b is expected to stop increasing. This result is obtained because the atomic path bends, as is shown in Fig. 6. As a matter of fact, when IHI does not change greatly as a function of r in the region near ro, Ih( is expected to decrease as b decreases beyond r. because the effective length of the path decreases as b approaches zero. According to (29) or Fig. 3, (u; ‘) has the maximum value of (8kT/7rrr1)“~.When this maximum value of (u; ‘) is less than /I;:~, or when Ih(
(8kT/mn)“2
> h,,,.
(48)
our theory is valid.* When condition (48) is not satisfied, the V-dependence of width parameter A will deviate from formula (31) in the region of small K A plausible V-dependence is shown in Fig. 7, which is to be compared to Fig. 3. One way of approximating this complicated situation is to assume that A is constant for V smaller than some value, say hmx(mkT)1i2, * The repulsive potential can be as large at kT for a short distance of about 0.1 A during the head-on collision. Suppose a fraction l/r of the repulsive potential is effective for pressure broadening, then we obtain h,,, - k’T x is 300 m/set for the hydrogen (0.1 A)/th = 400/c m/set at room temperature; (IkT/nm) II2 at room temperature atom, while it is 50 m/set for the argon atom. If 5 is about 10, our condition (48) is satisfied for both atoms.
Velocity
distribution
in spectral
515
line shape
A
I
I h,,,m
vFIG. 6. Head-on
FIG. 7. Expected
collision.
dependence
of (h(h] on b.
while it is given by formula (3 1) for T/larger than this value. From formula (34) we can see that such modification makes our profile closer to the Voigt profile in the center region of the spectral line. Other approximations for the v-dependence of A in the region of small 1/ predict different line shapes in the center region. Since the tail region of the spectral line mostly depends on A for larger V, line shape in this region will be independent of the ambiguous I/-dependence of A in the region of small I/: Thus, our corrections on the conventional Lorentz and Voigt profiles in the tail region must be real even when the condition (48) is not satisfied. Acknowledgment-The author acknowledges Atmospheric and Space Physics.
support
under
grant
NSF GA-399
to
the
Laboratory
for
REFERENCES 1. W. Heitler, The Quantum Theory of Radiorion, Oxford University Press, London (1954). 2. H. C. VAN DE HULST and J. J. M. REESINK,Ascrophys. J. 106, 121 (1947); D. W. POSENER,Aust. J. Phys. 12, 184 (1959). 3. P. W. ANDERSON,Phys. Rev. 76,647 (1949). 4. H. M. FOLEY, Phys. Rev. 69,616 (1946); M. MIZUSHIMA,Phys. Rev. 83,94 (1951).
Specrrosc.
Rodiat.
Transfer.
VELOCITY
Vol.
7, pp. 505-515.
Pergamon Press Ltd., 1967. Printed in Great Britain
DISTRIBUTION
IN SPECTRAL
LINE
SHAPE
MASATAKA MIZUSHIMA Laboratory for Atmospheric and Space Physics, Department of Physics and Astrophysics, University of Colorado, Boulder, Colorado (Received 29 August 1966)
Abstract-In the derivation of the conventional Voigt profile formula, the width parameter of the Lorentz line shape formula is assumed to be independent of the velocity of radiating atoms. This assumption is acceptable when the pressure broadening is negligibly small in comparison with the natural broadening. When pressure broadening is more important than the natural broadening, however, the conventional Voigt profile formula is not valid since the width parameter depends appreciably on the velocity. The resulting profile is shown to yield about 30 per cent lower intensity than the Voigt profile in the tail region. Even when the Doppler effect is completely negligible (low-temperature limit), our profile deviates from the Lorentz profile, particularly in the tail region.
VOIGT IT IS generally
PROFILE
accepted”) that the Lorentz formula Z,(o ; o,,A) = (Z,/n)A/[(w- o,,)’ + A’]
(1)
represents the intensity distribution of a single spectral line quite well. Here w is the frequency, oO is the center frequency, A is the width parameter, and I, is the integrated intensity, or I, = j’? o3Z dw. If the radiating atom is moving at velocity v in the frame of the observer and if a is the angle between v and the vector from the observer to the atom, then,
In the non-relativistic
wg = w,,(l +(u/c) cos a’)/(1 - (V/C)2)“2,
(2)
cos a’ = (cos a - (u/c))/( 1 - (u/c) cos a).
(3)
approximation
of the Doppler effect, we see that
0, = u cos a = c(oo - ooo)/ooo. Assuming thermal equilibrium at temperature frequency between oO and oO + do,, is dN = N(m/2&T)“‘[exp(
(4)
T the number of atoms dN, with center - mut/2kT)] du,
= (N/7c1’26)[exp( -oO - oo,J2/S2)] dw,.
(5)
where N is the total number of atoms and 6 = (~,,,,/c)(2kT/m)“2 gives the width of the frequency distribution by the Doppler effect. 505
(6)
506
MASATAKA
MIZUSHIMA
If we assume that each atom with a given velocity Doppler
effect makes the envelope
of theintensity
produces line profile (l), then the distribution of _
00
A exp( - (oO -o~~)~/S~)/[(O - CO~)~ + A’] do,.
Z(w 7 . A 36 9woo) = (~,,N/Y?“c~)
s
(7)
-m If we further assume that A is independent
of o,,, equation
(7) reduces to
Z(w ; A, 6, woo) = (ION/n 1’26)H(Q, a),
(8)
where w,&/c!$ a = A/6 ;
!2 = (o-
(9)
the function m
H(R, a) = z is called
the Hjerting
function.
e-Y2 (10)
s (R-y)2+PdY --m
Some numerical
evaluations
of this function
have been
done.‘2’ The resulting line profile is called the Voigt profile. NATURAL
WIDTH
In obtaining conventional formula (8), the width parameter A is assumed to be constant, but this is not exactly true. When the width is due to the natural decay of the excited state of the atom, A cc l/t,,
(11)
where t, is the life time of the decay. When the atom is moving with respect to the observer,
the life time changes as t, = t&l
- (u/c)z)“z.
(12)
Thus, A E AOO[l++((v/c)~]. To the same order of approximation,
(13)
formula (2) gives
00 = oo()( 1 + (0,/c) -+(u/c,‘,. Using (13) and (14) and considering that I, is the transition which also changes with u in the same manner as A, we obtain,
I =
(14) probability
(zooN,7c%)(6,~)J~J (1 + (S/W,,)~ V2) e-“’
-cc
(C2-y+$(?l/000)1/2)2+a2(1
per unit time,
dV
+(S/W,,)~V~)
(15)
where V2 = y2+x2+z2,dV
= dxdydz
and
V = (r@T)‘12v.
(16)
Velocity
distribution
in spectral
line shape
507
The correction is of the order of 6/ooo, which is (2kT/m)“‘/c. For the hydrogen atom in the chromosphere of the sun (2 x lo6 “K), this ratio is about 10P3. Thus, the original Voigt profile (8) must be quite good when the pressure is low enough that pressure broadening is negligible. PRESSURE
BROADENING
When the width is due to pressure broadening, the Lorentz formula (1) is still applicable but A = n(q)
(17)
where n is the number of atoms in unit volume, u, is the relative velocity between the perturbing atom and the radiating atom, ( ) means the average over the velocity of the perturbing atoms, and0 = Re
do{Tr@/iT,y ipifTf)/Tr(pfipif)l}. (18) s Here Re means to the real part, ~1is the electric (or magnetic, depending on the radiation mechanism) dipole moment of the radiating atom, Tr means trace, and T is the time development operator due to a single collision, viz., T = exp{-i
i (H/h)dfi -03
(19)
with the parameter of collision in the range of da. Subscripts i and f refer to the initial and final states of the radiative transition, respectively. We assume each perturbing atom to be a classical particle and the collisions are essentially elastic so that the relative velocity does not change during each collision. Such assumptions are usually made in calculating the width parameter from interatomic fortes,(4) and they can be justified as a good approximation.* If there is no external field the average over the direction of the relative velocity can be taken care of by the Tr-operator, and da can be taken as do = 2nb db,
(20)
where b is the collision parameter illustrated in Fig. 1. The interaction Hamiltonian H which appears in formula (19) depends on the distance r between the atoms. This distance r depends on b, the shortest distance and x, the distance from the closest point: H = H(r) = H(b, x).
(21)
For our assumption of a point perturbing particle with constant velocity u,, we see that x = u,t
and
I = (b’ + @)1/Z
(22)
if time t is taken to be zero at the closest point. * From the uncertainty relation, AuAx = 0.7 x 10e9 in mks units for the argon atom. If we let Ax = 1 A then Au = 7 m/see. Since the average velocity is 50 m/set at room temperature for this atom, the inaccuracy of the Doppler effect in our calculation is about 14 per cent.
508
MASATAKA MIZUSHIMA
FIG. 1. Path of perturbing
atom.
Often H is given in the interaction representation. The extra time-dependent terms of the form exp( - iH,t) in that representation, however, do not contribute when there is no overlap of spectral lines.(3) Thus, 7 Hdt=
j
-a,
-U3
H(b, x) dx/u, = hh(b)/u,,
(23)
so that T(b ; v,) = exp( - ih(b)/u,) = 1 - i(h(b)/u,) - $(h(b)/v,)’ + . . .
(24)
When h/v,, or its matrix element, is less than one, we can have a good approximation by taking the first few terms only in the above expansion of the time development operator. Such approximation is good for large ur, at least. Let us assume that the approximation is good for any u,. Taking the first three terms only in the expansion (24), we find from formulas (18) and (19) that
+ 3(Cl/i~irh:)]/Tr~~i~i~)}/u~.(25) u(ur)= - 2nbWTr[0L,ibirh,) + i-@,J$pi,) Thus, from formula (17), A= n
i (constant). 0 *
(26)
Now u, is the relative velocity of the collision. If 5 and CCare velocities of the radiating and perturbing atoms, respectively, then u-l I
= (u,2+u~-2ucucose)-1’~
= u;’ +(u
0)+(u2,/u~)P2(cos t9)+ . . . , (27)
where u, and u< are larger and smaller than u, and u, respectively, and 0 is the angle between CCand I;; (u; ‘) in formula (26) means the average over the perturbing atom. Since the direction of CCis randomly distributed, (27) immediately gives
(t&-l) = (u;‘).
(28)
Velocity distribution in spectral line shape
509
Assuming the Boltzmann distribution of the velocity u,, we have
=
2a-“2(m/2kT)3’2{u-1
’ exp(-mu: s
*/2kT)uf
du,
0
cc
exp( -muz/2kT)u,
+
where a(x) is the error integral
du,} = u- ‘@((m/2kT)*‘2u),
(29)
X
O(x) = 2~ ‘j2 exp( - t2) dt. s
(30)
0
and m is the mass of the perturbing atom. Thus, (26) gives A = A,27c- “2D(V)/~
(31)
I/ = (m/2kT)‘12u.
(32)
where again
Formula (31) shows that the width parameter A is close to a constant A0 when Vis less than unity but proportional to v-’ when V is larger than unity. Figure 3 shows the relation between A and V:
FIG. 2. Relative velocity v,.
When the mass of the radiating atom is the same as that of the perturbing atom (selfbroadening), the line profile is given by Z =
(Z,N/X~‘~~)H’~‘(ZZ, ~1~)
(33)
where HCp’(Qa,) = 2aon312
{W)/V> _s o3(a-
exp( - V2) dV
Y)’+a~4/~{~~)/~}2
(34)
510
MASATAKA
MIZUSHIMA
O-2 t
I
O”O0.1
I 0.5 v FIG.
3.
I
I
I
I.0
I.5
2.0
I
2.5
(= “/JZTiG)
A/A,,as function of V[ = ~/(2&T/m)‘/~].
under relation (16). When the perturbing atoms are different from the radiating atom, formula (34) is to be modified by using Q(aV)/aVinstead of @(V)/V, where a = (mass of perturbing atom)/(mass of radiating atom).
(35)
In the remainder of this paper, we consider only self-broadening. LOW-TEMPERATURE
LIMIT
When the temperature is very low, the Doppler effect will become negligible. In formula (33), the parameter a0 becomes much larger than unity at low temperature. Since I/ 2 y, the integrant in (33) including exp( - I@) is negligibly small whenever y is comparable to $2 or ao. Thus we have HLp)(Q,a0 + co) = 2a,7c-3i2
m Q(V)/V exp( - Vz)V2 dV _m s Q2+a~4x-1{@(V)/V}2
(36)
in the limit of low temperature, or large ao. The limiting function H’p’(Q a0 + co) is shown in Fig. 4. From Fig. 4 we see that the resulting curve has the half-width at Q/a, = @65, or IW-WJha,f
= 065A(),
(37)
where A,, is the width parameter at o = 0 as formula (31) shows. The Lorentz formula I, = (0~86Z,N/~)0~65Ao/{(0 - oo)’ + (0.65Ao)2}
(38)
Velocity
distribution
in spectral
511
line shape
. . 0 4-
0 .3-
6 *
I N
g
2
b
0 .2 -
0 1.1 -
0.' 01 0
I
I
0.1
05
I
I
I.0
I.5
I 1.8
w-"oo AlI
FIG. 4. Spectral
Line Profiles at Lower Temperatures. (1) t1,, = CO, or d = 0; (2) 6, = 10, or A0 = 106. - - - is the Lorentz profile given by formula (38).
agrees with our result at the center and the half width. In Fig. 4 this Lorentz formula is compared with our result. It is seen that the Lorentz formula reproduces our resulting line profile fairly well. In the far-wing region, the Lorentz formula (38) gives I, + (0+36Z,N/K)065A,/(O = (ZeN/rr)O56A&~ Our formula,
- oo)*
- o~)~.
(39)
on the other hand, gives m Z -+ (Z,JV/~C)(~A~/~~“~)
{cD(V)/V} exp( - V2)V2 dV/(w-q,)2 I
0
=
(Z,N/n)O707A,/(o
-oo)‘.
Thus, the Lorentz formula with parameters intensity in the far-wing region to be about
(40) adjusted in the center region predicts the 25 per cent less than the “true” intensity.
MASATAKA MIZUSHIMA
512
If the integrated intensity is calculated from the same Lorentz formula, the result will be 0.86 times the “true” integrated intensity. One should also note that the average of the width parameter itself is 0707 AO, or A,/2 “’ , but the same parameter which appears in the equivalent Lorentz formula (38) is 0.65 A,. The difference has never been taken into account in the calculation of the width parameter from the interatomic forces. In Fig. 4, our spectral line profile for the case of A,, = 106 is also shown. If we take a typical atom such as Kr, this number corresponds to about i atmosphere for optical lines, or about 1 mm Hg pressure for infrared lines at room temperature.* Notice that an appreciable decrease in the peak intensity appears compared to the case 6 = 0, while the rest of the curve stays essentially the same as before. It is interesting to note that the present line profile agrees with a Lorentz profile better than the previous one for 6 = 0. The Lorentz formula to fit the present profile, however, should be I, = (0.86Zolv/n)0.66,A,/((o CORRECTIONS
TO
- WJ VOIGT
+(066,A,)2}
(41)
PROFILE
Numerical calculation of our profile function H (P’is made for cases of A,, = 26,6,26/3, and @. The results are shown in Figs. 4, 5 and Table 1. The Doppler effect is seen to determine the decrease of the peak intensity and broadening of the center region. It can easily be shown that formula (40) is valid in the tail region for any temperature. In Fig. 5, our resulting profiles are compared with the conventional Voigt profiles.
= 05 ,=05
O-I
= 213 ,=2/3 =I =2 0’1 0=2 0.01
FIG. 5. Spectral
line profiles when ao( = 6/A,) is around one, or A0 and 6 are comparable. Voigt profiles (~@‘~)ff(Q, a).
* For the Kr atom at room temperature,
6 = lo-’
ooO cps while A,/p is typically
about
- --
are
10’ cps/mm
Hg.
Velocity distribution in spectral line shape
513
TABLE1. (a,,/~~/~)If(~) a, = w
(v - Q,,)/A,
6=0 0.42 1 0.410 0.381 0.301 0227 0170 0.129 0.100 0.079 0.064 0.053 0.044
0.0 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
a0 = 0.1 A,, = 106
czg= 1.0 A,, = 6
0.412 0403 0.378 0.303 0.229 0.172 0.131 0.091 0.080 ox)64 0.053 0044
0.249 O-248 0246 0.235 0218 0.197 0.173 0.147 0122 O-099 0.079 O-064
As expected, the Voigt profile with a = a, agrees approximately with our profile in the center region, but an appreciable deviation appears in the tail region. As the matter of fact the conventional Voigt profile with a = a,-, gives Z, + (Z,NMAo/(o - 00)’
(42)
in the tail region as contrast to our formula (40) in the same region. PRESSURE
SHIFT
The atomic collisions shift the center frequency of a spectral line as they broaden it. The shift of the center frequency is proportional to the pressure and is called the pressure shift. With this shift, A’ in the Lorentz formula (1) should be written as Z(o ;coo,A, A’) =
(Zo/rr)A/{(o - o. - A’)’ + A2).
(43)
According to Anderson’s theory, the pressure shift is given by A’ = n(~‘v,>,
(44)
where 6’ = Im
daTr(p/iT- l~~,-Tf)lTr(~~iQ. (45) f Here Im means to take the imaginary part, and the other notation has the same meaning as in formula (18). If we take the expansion (24) with only the first three terms, we obtain a’(v,) z
2zb db{Tr{@&Pif) - (~L/i~ish,)}/rrgLrii~)~/v,. (46) s We see from (44) and (46) that the pressure shift, in this approximation, is independent of the relative velocity of collisions. Thus, no corrections due to the velocity distribution appear in this approximation.
MASATAKA MIZUSHIMA
514
HEAD-ON
COLLISIONS
When the collision parameter b is less than the sum of the kinetic radii of colliding atoms, the assumption of a straight path, which is illustrated in Fig. 1, is not valid. For elastic collisions, which we are considering here, the magnitude of the relative velocity is still conserved, but the direction can be greatly changed. Before and after such “head-on collision”, the Doppler shift is usually quite different since the component of the velocity along a given direction changes greatly. The frequency shift determined by this mechanism, however, should not produce any net effect because, if the gas is in equilibrium, the number of atoms with a given velocity component must stay constant irrespective of atomic collisions, or the change in the Doppler shift must be zero when averaged over all perturbing atoms. The bending of the path is usually due to the repulsive interatomic force, which is of very short range. The relative velocity actually changes during the collision ; at least the radial component of the relative velocity becomes zero at the turning point. When the force which produces this bending is of short range, however, such change of the magnitude of the relative velocity can be neglected and we can still take the expression (23) provided that x is taken along the path. LOW
VELOCITY
CUT-OFF
Our theory is based on the approximation that inclusion of the first few terms in the expansion (24) for the time development operator is sufficient. This approximation is valid only when Ih(
4 v,.
(47)
Since JH( is zero at a large distance and increases as r, the interatomic distance decreases, is expected to increase as b decreases. As b becomes smaller than r,,, the sum of the kinetic radii of the colliding atoms, Ih(b is expected to stop increasing. This result is obtained because the atomic path bends, as is shown in Fig. 6. As a matter of fact, when IHI does not change greatly as a function of r in the region near ro, Ih( is expected to decrease as b decreases beyond r. because the effective length of the path decreases as b approaches zero. According to (29) or Fig. 3, (u; ‘) has the maximum value of (8kT/7rrr1)“~.When this maximum value of (u; ‘) is less than /I;:~, or when Ih(
(8kT/mn)“2
> h,,,.
(48)
our theory is valid.* When condition (48) is not satisfied, the V-dependence of width parameter A will deviate from formula (31) in the region of small K A plausible V-dependence is shown in Fig. 7, which is to be compared to Fig. 3. One way of approximating this complicated situation is to assume that A is constant for V smaller than some value, say hmx(mkT)1i2, * The repulsive potential can be as large at kT for a short distance of about 0.1 A during the head-on collision. Suppose a fraction l/r of the repulsive potential is effective for pressure broadening, then we obtain h,,, - k’T x is 300 m/set for the hydrogen (0.1 A)/th = 400/c m/set at room temperature; (IkT/nm) II2 at room temperature atom, while it is 50 m/set for the argon atom. If 5 is about 10, our condition (48) is satisfied for both atoms.
Velocity
distribution
in spectral
515
line shape
A
I
I h,,,m
vFIG. 6. Head-on
FIG. 7. Expected
collision.
dependence
of (h(h] on b.
while it is given by formula (3 1) for T/larger than this value. From formula (34) we can see that such modification makes our profile closer to the Voigt profile in the center region of the spectral line. Other approximations for the v-dependence of A in the region of small 1/ predict different line shapes in the center region. Since the tail region of the spectral line mostly depends on A for larger V, line shape in this region will be independent of the ambiguous I/-dependence of A in the region of small I/: Thus, our corrections on the conventional Lorentz and Voigt profiles in the tail region must be real even when the condition (48) is not satisfied. Acknowledgment-The author acknowledges Atmospheric and Space Physics.
support
under
grant
NSF GA-399
to
the
Laboratory
for
REFERENCES 1. W. Heitler, The Quantum Theory of Radiorion, Oxford University Press, London (1954). 2. H. C. VAN DE HULST and J. J. M. REESINK,Ascrophys. J. 106, 121 (1947); D. W. POSENER,Aust. J. Phys. 12, 184 (1959). 3. P. W. ANDERSON,Phys. Rev. 76,647 (1949). 4. H. M. FOLEY, Phys. Rev. 69,616 (1946); M. MIZUSHIMA,Phys. Rev. 83,94 (1951).