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〈S0〉Cs and 〈S0〉Rb relaxation in the presence of buffer gas and the spin exchange between Cs and Rb
(SO),,
November 1974
OPTICS COMMUNICATIONS
Volume 12, number 3
AND (S”jR,
RELAXATION
IN THE PRESENCE
AND THE SPIN EXCHANGE
I. BANY-JACKOWSKA
BETWEEN
OF BUFFER
GAS
Cs AND Rb*
and K. ROS@SKI*
Instytut Fizyki PAN, Warszawa, Poland Received 20 July 1974 (S’)Rb and (S’)Cs relaxation is studied in the presence of buffer gas and spin exchange between Cs and Rb. Spin exchange collisions are found to equalize the relaxation rates of Rb and Cs polarization. This result is completely con firmed experimentally for the relaxation of the first diffusion mode.
The evolution of spin polarization Go(t)> under the action of relaxation processes is one of the more difficult problems for theoretical investigation, in the case of an alkali metal mixture (e.g. Rb + Cs in vapour) in the presence of a buffer gas. There are, in general, many contributing relaxation processes: collisions with cell walls (sometimes coated with e.g. paraffin) and buffer gas atoms as well as spin exchange collisions. When the vapour density (hence the temperature) is sufficiently high, the spin exchange collisions can be the dominant factor The experimental situation under investigation has thus far consisted in annihilating with a sufficiently strong radio frequency resonance field [ 1,2] the polarization transferred during spin exchange collisions from one polarized mixture component to the other. In this paper a different situation is considered: one mixture component is polarized by optical pumping and the polarization of the other one gained during spin exchange collisions is not annihilated by the rf field. The problem under consideration is how the spin exchange affects the polarization evolution (relaxation) of both mixture components, if a buffer gas is present and the spin exchange is the fastest of all processes.
* Work done under the Polish Academy of Sciences, Contract Nr PAN 3. * Address: Instytut Fizyki PAN, OD.2. Al. Lotnikhw 32146, 02-668 Warszawa, Poland.
For the investigation a mixture of 133Cs and 85Rb was used in the presence of 30 or 50 torr neon in an uncoated Cell.Relaxation ra03S Of (S')C~ and (S’)Rb were measured and the results compared as functions of the vapour density (temperature). For experimental details see ref. [2]. In this case the evolution equations of LS”)cs and (so),,, are coupled, nonlinear differential equations, which cannot be solved analytically. The question arises as to what extent, if at all possible, we are allowed to take no account of hyperfine structure in describing theoretically the situation in question. A two level description of the spin exchange between electrons and alkali atoms, which fitted rather well the experimental results has already been given by Dehmelt [3]. Hence a two-level model for Cs and Rb atoms is used here. In a static magnetic field HO a splitting takes place into 2 sublevels with mS = + $ and with populations Ni andNi (X=Cs or Rb). Let us putN_$ =N$ + N_$ and Px = (N$ - Nj)/Ng and look into the evolution of the polarization degree Px. The relaxat tion processes taken into account are: the relaxation on the cell walls after diffusion through the buffer gas and the relaxation during collisions with buffer gas atoms. Taking the exchange collisions into account, the evolution equations are (compare [3] ):
323
Volume
12, number
3
df’x (rJ> -------~ D,V2P,(r, dt
OPTICS COMMUNICATIONS
November
19 74
t) -- kXPX (r. t)
1 1 ~~P, (c t) + ----px, Txx ’ Txx ’
(r, t),
(1)
where X= Cs and X’ = Rb, Dx is the diffusion coefficient through the buffer gas, k, = relaxation rate on buffer gas atoms, Tiir =N$V,,, oex = exchange relaxation rate with bxx, relative velocity and oex exchange cross section. The equation for dPx(y, t)/dt is similar. The Cs polarization being proportional to the Rb one px (r) = OPX’ (r),
(2)
in the entire volume of the cell, the solution of eqs. (1) is possible. As the polarization Px (r) at t = 0 has been created by optical pumping and the polarization Px, (r) by the spin exchange, assumption (2) is valid for sufficiently fast transfer of the polarization from X to X’, subject to the conditions: T&,
T$X
9
Ti’,
(T$V,
(T$“>%
(3)
where T-’ is the optical pumping rate; ( TTn))’ and 1p are the relaxation rates on the cell walls re( T2n)sulting from diffusion according to the mn mode, and on buffer gas atoms, respectively. (TF”)-’ and ( TFn)-l are to be determined by two independent equations obtained from (I) by rejecting the last two terms describing the spin exchange. For a cylindrical cell with radius a = 2.5 cm and length L =4 cm we get the solution in the form: I
X cos ‘y 2 exp (-t/Tmn), ( 1
(4)
where A,, are constants determined by initial conditions. h,, are given by boundary conditions, m=1,2,3 ,...., n= 1,3,5 ,.... Thusit turns out that for both polarizations P, and Pxl, the relaxation rates 7;; are equal: r,$
= 1/2[1/TFn
+ l/T,”
_~ l/2 [(l/TXm -l/TFn + 4/Txx’ Txrx] ‘12. 324
+ l/TX./’ + l/Tx,x] + l/Txsx
- l/Txx,)2 (5)
Fig. 1. The experimental ~ TX and Ti and theoretical -- 7,, longitudinal relaxation times versus temperature for Cs and Rb.
It should be kept in mind that r,$, is a function of the alkali vapour density (hence temperature of the saturated vapour), since Txxr and Txpx are functions of the alkali vapour density. The smooth lines in fig. 1 represent relaxation times for 6 initial diffusion modes, calculated for the cell used in the experiment. The necessary cross sections 0 as, oex and diffusion constants Dx have been taken Prom the appropriate literature. The experimental investigation of (SO),, and (so),, decay in the dark was carried out as in ref. [2]. The temperature of the cell was a parameter of the measurement. The decay curve could be decomposed into two exponential components for which the relaxation times TX and Ti were evaluated and have been plotted together with the theoretical results (fig. 1). The alkali metal being a component of the mixture, its vapour density is correspondingly lowered. By plotting in fig. 1 the theoretical results, the decrease in vapour density was taken into account as in ref. [2]. It can be stated directly from fig. 1 that at temperatures above - SO”C, within the error limits, Tkb = T&, in agreement with the theoretical prediction. At lower tempera-
Volume
12. number
3
OPTICS COMMUNICATIONS
tures the polarizations (so),,, and (so),, relax independently, condition (2) not being fulfilled, and T;+, < Tk,. The following qualitative interpretation can be given to the results just presented. The relaxation time for a cell filled with only 85 Rb or only 133Cs (with 30 torr neon), but otherwise identical to the one used in our experiment, should be 66 ms or 104 ms. Thus in the absence of spin exchange the decay of the 85Rb polarization can be expected to be about 2 times faster. Under conditions of sufficiently fast polarization exchange [see (3)] the rubidium polarization destroyed during collisions with the wall or with buffer gas atoms, is restored in the exchange collisions with cesium atoms which are less depolarized during collisions; thus the rubidium relaxation time Tib increases to rll according to (5). After performing the necessary calculations it is possible to see that for the cell under investigation the exchange time approaches rI1 if the temperature rises to about 45-50°C, which is in agreement with the experiment (fig. 1).
November
1974
The relations valid in the case of the longer relaxation time Ti (= 711) are in principle confirmed for the shorter one (Ti). However those relations are less pronounced, which is probably due to the relatively larger measurement errors and overlapping of effects, with closely spaced relaxation times (r2r , 713,~23, . .). We may conclude that the simple alkali atom twolevel model has once more provided a rather good approach. The authors would like to thank Professor T. Skalinski for his interest shown in the course of this work.
References [l] F. Grosset&e, J. Phys. (Paris), 25 (1964) 383. [2] I. Bany-Jackowska, Lett. N. Cim. 17 (1973) 430. [3] H.G. Dehmelt, Phys. Rev. 109 (1958) 381.
325
November 1974
OPTICS COMMUNICATIONS
Volume 12, number 3
AND (S”jR,
RELAXATION
IN THE PRESENCE
AND THE SPIN EXCHANGE
I. BANY-JACKOWSKA
BETWEEN
OF BUFFER
GAS
Cs AND Rb*
and K. ROS@SKI*
Instytut Fizyki PAN, Warszawa, Poland Received 20 July 1974 (S’)Rb and (S’)Cs relaxation is studied in the presence of buffer gas and spin exchange between Cs and Rb. Spin exchange collisions are found to equalize the relaxation rates of Rb and Cs polarization. This result is completely con firmed experimentally for the relaxation of the first diffusion mode.
The evolution of spin polarization Go(t)> under the action of relaxation processes is one of the more difficult problems for theoretical investigation, in the case of an alkali metal mixture (e.g. Rb + Cs in vapour) in the presence of a buffer gas. There are, in general, many contributing relaxation processes: collisions with cell walls (sometimes coated with e.g. paraffin) and buffer gas atoms as well as spin exchange collisions. When the vapour density (hence the temperature) is sufficiently high, the spin exchange collisions can be the dominant factor The experimental situation under investigation has thus far consisted in annihilating with a sufficiently strong radio frequency resonance field [ 1,2] the polarization transferred during spin exchange collisions from one polarized mixture component to the other. In this paper a different situation is considered: one mixture component is polarized by optical pumping and the polarization of the other one gained during spin exchange collisions is not annihilated by the rf field. The problem under consideration is how the spin exchange affects the polarization evolution (relaxation) of both mixture components, if a buffer gas is present and the spin exchange is the fastest of all processes.
* Work done under the Polish Academy of Sciences, Contract Nr PAN 3. * Address: Instytut Fizyki PAN, OD.2. Al. Lotnikhw 32146, 02-668 Warszawa, Poland.
For the investigation a mixture of 133Cs and 85Rb was used in the presence of 30 or 50 torr neon in an uncoated Cell.Relaxation ra03S Of (S')C~ and (S’)Rb were measured and the results compared as functions of the vapour density (temperature). For experimental details see ref. [2]. In this case the evolution equations of LS”)cs and (so),,, are coupled, nonlinear differential equations, which cannot be solved analytically. The question arises as to what extent, if at all possible, we are allowed to take no account of hyperfine structure in describing theoretically the situation in question. A two level description of the spin exchange between electrons and alkali atoms, which fitted rather well the experimental results has already been given by Dehmelt [3]. Hence a two-level model for Cs and Rb atoms is used here. In a static magnetic field HO a splitting takes place into 2 sublevels with mS = + $ and with populations Ni andNi (X=Cs or Rb). Let us putN_$ =N$ + N_$ and Px = (N$ - Nj)/Ng and look into the evolution of the polarization degree Px. The relaxat tion processes taken into account are: the relaxation on the cell walls after diffusion through the buffer gas and the relaxation during collisions with buffer gas atoms. Taking the exchange collisions into account, the evolution equations are (compare [3] ):
323
Volume
12, number
3
df’x (rJ> -------~ D,V2P,(r, dt
OPTICS COMMUNICATIONS
November
19 74
t) -- kXPX (r. t)
1 1 ~~P, (c t) + ----px, Txx ’ Txx ’
(r, t),
(1)
where X= Cs and X’ = Rb, Dx is the diffusion coefficient through the buffer gas, k, = relaxation rate on buffer gas atoms, Tiir =N$V,,, oex = exchange relaxation rate with bxx, relative velocity and oex exchange cross section. The equation for dPx(y, t)/dt is similar. The Cs polarization being proportional to the Rb one px (r) = OPX’ (r),
(2)
in the entire volume of the cell, the solution of eqs. (1) is possible. As the polarization Px (r) at t = 0 has been created by optical pumping and the polarization Px, (r) by the spin exchange, assumption (2) is valid for sufficiently fast transfer of the polarization from X to X’, subject to the conditions: T&,
T$X
9
Ti’,
(T$V,
(T$“>%
(3)
where T-’ is the optical pumping rate; ( TTn))’ and 1p are the relaxation rates on the cell walls re( T2n)sulting from diffusion according to the mn mode, and on buffer gas atoms, respectively. (TF”)-’ and ( TFn)-l are to be determined by two independent equations obtained from (I) by rejecting the last two terms describing the spin exchange. For a cylindrical cell with radius a = 2.5 cm and length L =4 cm we get the solution in the form: I
X cos ‘y 2 exp (-t/Tmn), ( 1
(4)
where A,, are constants determined by initial conditions. h,, are given by boundary conditions, m=1,2,3 ,...., n= 1,3,5 ,.... Thusit turns out that for both polarizations P, and Pxl, the relaxation rates 7;; are equal: r,$
= 1/2[1/TFn
+ l/T,”
_~ l/2 [(l/TXm -l/TFn + 4/Txx’ Txrx] ‘12. 324
+ l/TX./’ + l/Tx,x] + l/Txsx
- l/Txx,)2 (5)
Fig. 1. The experimental ~ TX and Ti and theoretical -- 7,, longitudinal relaxation times versus temperature for Cs and Rb.
It should be kept in mind that r,$, is a function of the alkali vapour density (hence temperature of the saturated vapour), since Txxr and Txpx are functions of the alkali vapour density. The smooth lines in fig. 1 represent relaxation times for 6 initial diffusion modes, calculated for the cell used in the experiment. The necessary cross sections 0 as, oex and diffusion constants Dx have been taken Prom the appropriate literature. The experimental investigation of (SO),, and (so),, decay in the dark was carried out as in ref. [2]. The temperature of the cell was a parameter of the measurement. The decay curve could be decomposed into two exponential components for which the relaxation times TX and Ti were evaluated and have been plotted together with the theoretical results (fig. 1). The alkali metal being a component of the mixture, its vapour density is correspondingly lowered. By plotting in fig. 1 the theoretical results, the decrease in vapour density was taken into account as in ref. [2]. It can be stated directly from fig. 1 that at temperatures above - SO”C, within the error limits, Tkb = T&, in agreement with the theoretical prediction. At lower tempera-
Volume
12. number
3
OPTICS COMMUNICATIONS
tures the polarizations (so),,, and (so),, relax independently, condition (2) not being fulfilled, and T;+, < Tk,. The following qualitative interpretation can be given to the results just presented. The relaxation time for a cell filled with only 85 Rb or only 133Cs (with 30 torr neon), but otherwise identical to the one used in our experiment, should be 66 ms or 104 ms. Thus in the absence of spin exchange the decay of the 85Rb polarization can be expected to be about 2 times faster. Under conditions of sufficiently fast polarization exchange [see (3)] the rubidium polarization destroyed during collisions with the wall or with buffer gas atoms, is restored in the exchange collisions with cesium atoms which are less depolarized during collisions; thus the rubidium relaxation time Tib increases to rll according to (5). After performing the necessary calculations it is possible to see that for the cell under investigation the exchange time approaches rI1 if the temperature rises to about 45-50°C, which is in agreement with the experiment (fig. 1).
November
1974
The relations valid in the case of the longer relaxation time Ti (= 711) are in principle confirmed for the shorter one (Ti). However those relations are less pronounced, which is probably due to the relatively larger measurement errors and overlapping of effects, with closely spaced relaxation times (r2r , 713,~23, . .). We may conclude that the simple alkali atom twolevel model has once more provided a rather good approach. The authors would like to thank Professor T. Skalinski for his interest shown in the course of this work.
References [l] F. Grosset&e, J. Phys. (Paris), 25 (1964) 383. [2] I. Bany-Jackowska, Lett. N. Cim. 17 (1973) 430. [3] H.G. Dehmelt, Phys. Rev. 109 (1958) 381.
325