- Email: [email protected]
Positron two-photon annihilation coincidence technique: Difference mode
N U C L E A R I N S T R U M E N T S AND METHODS 151 (1978)
1 4 9 - 1 5 6 ; i~) N O R T H - H O L L A N D PUBLISHING CO.
POSITRON T W O - P H O T O N ANNIHILATION COINCIDENCE TECHNIQUE: DIFFERENCE M O D E * t PAUL J. KAROL and RICHARD L. KLOBUCHAR§
Department of Chemistry, Carnegie-Mellon Universty, Pittsburgh, PA 15213, U.S.A. Received 13 September 1977 A difference (or comparative) mode in the measurement of two-photon coincidences from positron or positronium annihilation has been developed. The method can be used to advantage, particularly in gases, in determining annihilation pa. rameters such as quenching cross sections for low concentration strongly quenching chemical species in a reference medium which is relatively non-quenching.
1. Introduction
Studies on the physical and chemical properties of complex systems using the positron as an informative probe have been developing for just over a quarter century. The major characteristic which makes this technique useful is that a metastable positron-electron system, positronium, is formed in two nearly degenerate states whose distinctly different half lives depend on the environment in which the system is found. Measuring annihilation characteristics is indirectly determining how positronium is interacting with neighboring species. If the mechanisms and physics of these quenching interactions are reasonably well understood, the annihilations quantitatively reflect microscopic structural properties of the quenching agents. If positronium is formed and lives long enough to thermalize, two possible states will result: a short-lived spin-singlet called parapositronium, pPs, decaying into two nearly collinear annihilation photons with a decay rate As = 8.0x 10 9 s -~ and a long-lived spin-triplet called ortho-positronium, oPs, with a three-photon annihilation decay rate AT = 7 . 2 × 1 0 6 S-~. The mechanisms by which Ps interacts with its environment are generally grouped into three broad classifications. In the pickoff process, Ps collides with an accessible atom or molecule and the positron in Ps annihilates rapidly with an electron in the collision partner. The noble gases are examples of species which shorten or quench the Ps lifetime by pickoff. The pickoff quenching rate, gp, * Work supported in part by the National Science Foundation. t Preliminary description of the method was presented at the Symposium on Nuclear Phenomena as Chemical Probes, ACS Meeting, Chicago 1975. § Center for Naval Analysis, Arlington, VA 22209.
is related to the concentration of quenching species, np (cm-3), and the thermalized Ps average velocity o(7.6 × 106 cm/s) by the pickoff quenching cross section or. =Ap/npO. At STP for the noble gases, crp ~ 10 -f~ cm 2 which is a very small fraction of the collision cross section with an atomic electron. If the collision partner has one or more unpaired electrons, the spin conversion process becomes likely. Here for example, the parallel intrinsic spin of the electron relative to the positron in ortho-positronium is "exchanged" with that of an unpaired electron with spin antiparallel to that of the positron at the time of collision. The overall result is conversion of long-lived o-Ps into shortlived p-Ps. This conversion rate K is related to the reverse para-to-ortho conversion rate because of the singlet-to-triplet spin statistics of the effectively degenerate system. Consequently, the para-toortho conversion rate is given by 3K. The conversion quenching rate Ok is related to the conversion cross section by ak =Ak/nkO. For paramagnetic species like OA(g) and NO(g), crk ~ 10 -19 cm 2. Finally, it is postulated that positronium can undergo several different types of chemical reactions including oxidation, addition and substitution. These are some of the more complex and also more intriguing quenching interactions in which Ps can be involved. The chemical quenching rate 2ch is related to the chemical quenching cross section a,h = Ach/ncn0 which is in general relatively large having values ranging up to ~ 10 -~5 cm 2 for NAO4(g). Several recent reviews l) discuss these features at length. Two major experimental methods have found wide use in studying positronium formation and decay. These are the angular correlation method, which is a measure ~of the divergence of annihilation photons from collinearity, and the lifetime
150
P. J. KAROL AND R. L. K L O B U C H A R
technique, which is a direct determination of annihilation quenching rates. Each of these proce, dures has been quite successful under a variety of circumstances. They are discussed in reasonable detail elsewhere2). Measurements of angular correlation functions or lifetime spectra do involve the necessary utilization of sophisticated nuclear electronics and either fairly intense positron sources or lengthy data accumulation times. We have developed a technique which has the advantage of requiting simpler equipment, weaker sources and short experimental acquisition times. Our method is based on the change in two-photon coincidence count rate relative to a reference count rate. The 23, count rate change is a function of the known composition of a quenching mixture containing the quenching species to be determined and a reference species of known quenching characteristics. 2. Method The expressions necessary for relating experimentally determined quantities to annihilation quenching cross sections or rates all derive from a series of coupled differential equations3). These differential equations describe the time-dependence of the number of positrons in each of its three possible conditions; as p-Ps (singlet), o-Ps (triplet) or as a free positron fl+. In terms of the various annihilation quenching rates we have:
dPF(t) = --)'F PF(t), dt
(1)
dPs(0 = - (3K+2s+2p+2~h) es(t) + KPT(O, dt dPT(0 dt = - (K+2T+2p+2ch) PT(t) + 3KPs(t),
(2) (3)
where PF(t), Ps(t) and PT(t) represent respectively the fraction of positrons which are free, or part of singlet and triplet positronium at time t. ;tF is the annihilation rate of free positrons and the other symbols have been defined in the introduction. The system of eqs (1-3) represents a series of linear homogeneous differential equations with constant coefficients. Such equations may be readily solved by standard means4). The solutions obtained are: PF(t) = (1 -- P) exp(-- 2F t), (:4)
Pr(t) = 3 P(2T+ RP + 2~h--Z') exp(--2't)+ 4(2'-Z')
"I- 3e()~I"It-~'P'Jr~'¢h--~")exp(--~.#t), where, following the notation of Ooldanskii~), ~., =
+ P(';Ls+2P+2ch--2')
4(2"--Z')
exp(-Z't),
(7)
g -- (g2__#2)I/2
and ,~,t = ~ ..[_ (~2__fl2)1/2 .
(8)
The quantities, o~ and //2 may be expressed in terms of the original rate constants as: = 2 K - F ~ p + ~ , c h -I (~'S't-'~T) 2
(9)
and f12 = (K.q_ ~p..[_~ch) ~S "~ 2T'~S+ (3K+;tp+2ch) Lr + + (4K+2,+2,h) (2,+;t~h).
(10)
The above solutions were obtained by recognizing the following necessary boundary conditions: PF(0) = (l--P), (11) Ps(0) = P/4,
(12)
PT(0) = 3P/4,
(13)
assuming instantaneous thermalization and where P is t h e probability of positronium formation. Eqs. (11-13) reflect the statistical weights of the triplet state (7596) and the singlet state (25%). Inspection of eqs. (4-6) reveals that the annihilation of positronium can be conveniently described in terms of two decay constants, 2' and M', which contain the fundamental rate constants discussed above. Hence an experimental lifetime spectrum may yield values of the decay constants calculated from the logarithmic slopes by standard least squares techniques. Several approximations can be made which make the evaluation of K, ;tp and ;tea much simpler. In particular, if: As >>2r ~, ).p ~ K ~ 2oh,
(14)
then: f12 ,~ (K..l_J],p.jt. ~ch..l_,~,T) '~S
(15)
and a ~ 2s/2.
Ps(t) = P(2s-t'2P+2ch--2") exp(--,~'t) + 4(2'--2")
(6)
4(2" - Z )
(16)
It follows that: (5)
)]/ ~ ~-~-- [--~ -- (K q- ,~p-]-,~©h.Jr-,~r) q 1/2
(17)
POSITRON TWO-PHOTON ANNIHILATION COINCIDENCE TECHNIQUE and
2" ,~ ~2 "-I-I--~- (g+2~--b2eh-l-2T),~dl 1/2
P~ (18)
A Taylor series expansion retaining only the first terms leads to
2' ~ -b (Kq-2p-l-2ehq-2T)
(19)
and
2" ~ -t- 2s - (gq-2pq-2ehq-/~r).
(20)
Under these conditions, positronium possesses two decay constants, one of which (2') is determined primarily by the intrinsic decay constant for para-positronium and the other (2") by quenching rates and the intrinsic decay constant for orthopositronium. To test the validity of the above approximations, 2' and 2" were calculated by supplying decay constants 5) which correspond to 700 torr of oxygen ( K = l . 9 × 1 0 7 s -l) in 10 amagats of argon ( 2 p = 2 . 5 × 1 0 6 s - l ) . 2r and As were taken to be 7.2× 106 s -~, and 8.0× 109 s -l, respectively, and 2,h was set to equal zero. Such an experimental mixture would provide a rather severe test of the approximations for gas phase work as virtually all of the ortho-positronium would be quenched by the oxygen. The approximate value of 2' was calculated as 2.87× 107 s -~ compared to the exact value 2.86×107 s -~ (less than 1 percent error). A similar calculation for 2" also yields an error of less than 1 percent. However, one must exercise caution in analyzing lifetime spectra for K, 2p, and 2~h by eqs. (19) and (20) ascertaining that the inherent approximations have not broken down as may be the case for highly quenched positronium in the condensed phase. To calculate the instantaneous probability of two-gamma annihilation from positronium, one multiplies the probability of having either ortho or para positronium at any time by only those rate constants leading to two-gamma annihilation events. The total probability of two-gamma annihilation is consequently obtained by integration over time, from zero to infinity, of the previous relationships. For instance, triplet positronium can decay via two-gamma annihilation as a result of both pickoff (2p) and chemical reaction (2~h) mechanisms, but its intrinsic decay (2r) leads only to three-gamma annihilation. As a result, the total probability of two gamma annihilation from positronium, P1;2r, is given by:
3o
151
ka2, + a3&
x
o
(21,+2ch)Ps(t) dt +
\~2~+~3&, o (2p+2~h)
Pr(t) dr.
(21)
The factor, a2~/(a2r + a3~)= 371.3/372.3, in eq. (21) expresses the realization that during pickoff and chemical reaction there is a small probability that three-gamma annihilation can occur. This is similar in principle to the small probability for threegamma annihilation of free positrons. The exact relationship to intrinsic decay constants is~): a2~/(~2~+a3~) -- 2s/(2s+32r).
(22)
It should be recognized that the conversion process (K) does not lead to direct annihilation, but merely exchanges the ortho and para states. To calculate the fraction of positronium which decays by two-gamma annihilation, W2r one defines W27 -'~ (23)
P~[/P.
W2~ can be conveniently expressed in terms of dimensionless parameters or reduced decay rates defined as: (24)
x m K/2.r, y - 2p/&,
(25)
z - 2oh/&.
(26)
Performing the integration required by eq. (21) and using the substitution eq. (22) one obtains:
W2~,={(-~)(l+4x+y+z)+(~)(Y+Z) x
+ ( ~ r + 3) x + ( ~ - + 1) Y +
+(-~+ 1) z+(4x+y+z)(y+z)~.
(27,
Making the substitution 2s/2r = 9 7r/[4 (rF-9)a¢] = 1114 where a¢ is the fine structure constant one obtains:
e2/~c,
152
P. J. KAROL AND R. L. KLOBUCHAR
W27 -- 11114 + 4456x + A.A.A.8(y+Z) +
The 1/+ source can be mounted on an ultrathin film which has a low stopping power for positrons. + 15.957 x(y+z) + 3.989 (y+z)2]/ To a good approximation then, the detectors are recording coincidences which correspond to events /I-4456 + 4468X + 4460 (y+z) + occurring only within the condensed phase sam+ 16 x ( y + z ) + 4(y+z)2"l. (28) ple. For gas phase work, however, the maximum positron (projected) path length is quite large (e.g. For small x , y and z, eq. (28) reduces to: 140cm/atm for argon) and pressure dependent. 1 + 4(x + y + z) (29) Furthermore, positrons annihilating on those reW2:, ,~, 4 + 4(x+ y+z)" gions of the sample chamber walls where the deInspection of eq. (29) reveals that in the absence tectors are efficient contribute two-gamma events of quenching phenomena (i.e. when x = y = z = 0) which add to unwanted background, decreasing 14/~r is 0.25 as expected. One may similarly derive sensitivity to changes in count rate. an expression for the fraction of positronium An often overlooked correction must also be which decays via three-gamma annihilation, W3r, made to account for a contribution from detection but it is more convenient to use: of three-gamma annihilations "disguised" as twoW3r -- 1 - W2~, (30) gamma events. Conceptually, this may arise when ortho-positronium annihilates into three photons when W2~ is known. of energies (for instance) 500, 500 and 22 keV. ExThe fraction of Ps undergoing two-gamma anni- perimentally, a 500keV and the usual 511 keV hilation W2~, and three-gamma annihilation W3r, quanta cannot be distinguished with a NaI(TI)detecare then used to calculate the two-gamma annihi- tor. Furthermore, the angle between the two lation rate detected experimentally by a collinear 500 keV quanta in the example is 177.5 °. With the (but not highly collimated) coincidence arrange- wide angle acceptance employed for gross twoment, shown in fig. 1. However, one cannot easily photon counting, it is clear from fig. 1 that a fracconfine detection to two-gamma (or three-gamma) tion of bona fide three-gamma events may be misannihilation of positronium alone. In practice, pos- takenly registered as two-photon coincidences. itronium is formed by only about half of the pos- Lastly, one is always confronted with some degree itrons. The remainder of the positrons annihilate of chance events and background caused primarily as free positrons or as positrons attached to a sub- by detecting Compton events in one or both destrate in the manner of a collision complex. Either tectors. way, the resulting annihilation is predominantly a Any expression for the total measured two-gamtwo-gamma process. Hence, free positrons intro- ma annihilation rate, A~, must contain correction duce a rather large and undesirable background terms for the contributions outlined above. Letting when performing gross two-gamma counting. In ca+ be the efficiency for detecting two-photon gas samples, the situation is further complicated events from free positrons and e~s be the efficienby the requirement that one stop a significant cy for detecting two-photon events from positronifraction of positrons in a spatial region where the urn, one may describe the observed two-gamma two collinear detectors are efficient. For condensed annihilation rate as: phase work, this presents little problem as the = 1,(l-P) 8B+f + PshfW2, + + range of 22Na positrons, for example, is approxi+ fw3,] + B, (31) mately 200 m g / c m 2 corresponding to a path length of only 2.0 m m (for a sample with unit density). where ¢ ~ + ) is the emission rate of the positron 2~ is the efficiency source, B is the background, 83~ for detecting three-gamma events as two-photon ms ] annihilations, 8, is essentially the combined efficiency for detecting annihilations in the walls and source, f ' is the fraction of positrons which stop in the walls and the source where the detectors Fig. 1. Schematic of the two-photon coincidence geometry; D: are viewing, and f is t h e fraction of positrons gamma detector, S: lead shielding, C: sample chamber, H: offwhich eventually annihilate in a region of the gas axis positron source holder, V: region viewed by collinear deviewed by the coUinear detectors. Briefly, the first tectors.
1
sI
1
POSITRON TWO-PHOTON
ANNIHILATION
term in eq. (31) is due to the two-photon annihilation of free positrons, while the second term arises from the 2~, annihilation of positronium. The third term is due to annihilations on the walls and source holder. The final term within the brackets reflects the registration of three-photon events as two-photon annihilations. As discussed by Goldanskiil), one cannot distinguish between twophoton annihilations from positronium or from free positrons by employing an apparatus with wide angle acceptance. Consequently, the efficiencies, ta+ and eps, are purposely equal in the wide angle approach and are relabelled s, the efficiency for detecting two-photon events irrespective of their origin. Note that the various efficiencies are more correctly re-expressed as averages over the geometric detection region. In general, one could calculate the efficiencies and fractions, f and f', and determine the background, B, experimentally. In practice, however, this approach is subject to large errors which are both systematic and random in nature. To avoid such uncertainties, fractional changes in the gross two-photon annihilation are measured. For a reference sample, one may express the reference two-photon rate, A°r as:
COINCIDENCE
TECHNIQUE
153
fective chamber constant" and is given by: l+c G = 1------R' (35) where
(ewf')
B
C ---- (sf-"--"'~ -I- ~b(jff+ ) ( s f )
(36)
and R is the ratio of three-gamma efficiency to the two-gamma efficiency: 27
R = .(sarf)
(37)
•
When dealing with a mixture of quenching sp ecies as is necessary here in the comparative technique, the various quenching rates associated with the different quenching processes are properly defined in the following manner: r
v,
(38)
-- (Z ni ap, t) v ,
(39)
= (Y i i
&h = (Y ni
o,
(40)
l
where n~ is the concentration of quenching species i; ak.~, ap,~ and 0rch.~ are respectively the conversion, pickoff and chemical quenching cross secA°~ = q~(jff+)[ ( l - P ° ) (e f ) + eO
154
P. J. KAROL AND R. L. KLOBUCHAR
tuation, has the form: AA27 =
A °,
P°(W27-W°r)
(41)
G - P ° ( 1 - W°,)"
An upper limit to the method's sensitivity can be obtained from eq. (41) by the boundary conditions: G = 1, p o = 1 and ~ = 0.25 as a result of which AA2r 3 nov
A°~
"~r
at low concentration. Factors which tend to reduce sensitivity are low Ps formation probabilities, high quenching rates for the reference system, and extraneous 27 contributions. The latter include "noise" effects such as annihilation in the source and walls and registration of 37 events from o-Ps decay as if they were 27 annihilations. It is also worth pointing out that if eq. (31) in which ep~ and e,+ are not yet set equal is used, one can enhance the sensitivity considerably by the introducing severe geometry requirements that make tp, greater than ~ + . That is, by discriminating against the "noise" from free positrons, the signal from positronium becomes more prominent. 3. Discussion As an illustration of the method, we show in fig. 2 a hypothetical example of fractional change in two-photon count rate as a function of relative I
I
I Illll
I
I
I
I lllll{
I
I
I lllll{
I
I
16-
,2-
p0=
Eq. (34) generates the broken curve in fig. 3 frc which an important observation may be draw That is, the shape of the data curve gives an i dication as to whether or not the positronium fi mation probability is changing with compositic
I IIlll
18
I0
pressure of a gas of interest in a reference ga The chamber constant G has the value 1.15. A t tal of 5.0 atm of reference gas (R2) with 25 and with a (pickoff) quenching rate 2.1 × 105 arm- 1 s- 1 plus a (conversion) quenchi] impurity (Q2) of known concentration is employ, in the illustration. On the initial assumption tl~ the Ps formation probability P does not chan with composition in this case, eq. (41) may used to fit the data with the best possible quenc ing cross section. Using W2~ from eq. (28) or its a proximation (29), a family of curves has been co structed with a k = 2 × l O - 1 9 c m 2, l × l O - 1 9 c r and 0 . 5 × 1 0 -19 c m 2 to demonstrate the accura with which cross sections can be obtained. Vari tion in values chosen for ak appear as horizon1 displacements from the fractional difference d~ as seen in fig. 2. With sufficiently precise data, it becomes pos: ble to determine whether or not P changes wi composition. This point is exemplified in fig. where data from the above example are compar with a variable P case. The variable P case w chosen with all the parameters of fig. 2 except tt. P varies linearly with composition from p 0 = 25 for pure R2 to P = 45 96 for pure Q2. Letting p i present the relative pressure of Q2 we have: P = P°(1 +0.8p). (z
//
t41|2
-J .=_
40Z ~ / I
tO-4
I
/
I Illlll
t
10-3
! I
Ill)Ill
,
*
|
tO-2
,111,1
I
,
,
till,
10-I
[o,1/([o,1 + [R,]) Fig. 2. Fractional change in two-photon coincidence rate, expressed as percent, as a function of relative 'Q21 pressure in R 2 at a total pressure of 5 atm. Curves from top to bottom correspond tO Gk -- 2 × 10-19 cm 2 ' 1 × 10-19 cm 2 and 0.5×10 -19 cm 2 , respectively in eq. (41). Positronium formation probability p = p O = 2 5 % , constant. S¢0 text for full discussion.
~1~
6 4 2 0 i0-3
10-~'
I0-I
[ o 2 l / ( [ o z l + [R~] ) Fig. 3. Fractional change in two-photon coincidence rate. curves correspond to (rk = 1 × 10 -19 cmm as in fig. 2 fit data. Heavy curve has P = 25%, constant, broken curve ha,, varying with composition according to eq. (42), light cu{ shows effect of chamber calibration constant with G = 1 compared to G = 1.15 in heavy curve.
POSITRON T W O - P H O T O N A N N I H I L A T I O N C O I N C I D E N C E T E C H N I Q U E
Furthermore, if these data are sufficiently precise, the actual dependence of P on composition may be extracted. A caveat is required at this stage concerning evaluation of the P compositional dependence. In the hypothetical example illustrated in fig. 3, P was assumed to vary linearly with relative pressure. For this special case, the identical function may be achieved with a constant p0 and a chamber constant G = 1.24 rather than the value G = 1.15 given. To demonstrate the effect of the sample chamber constant, the thin curve in fig. 3 has been constructed. This curve corresponds to the heavy curve of constant po = 25% except that G has been revalued at 1.35. All of the above serve to emphasize the importance of maintaining c and R in eq. (35) as low as possible by design. Recalling eqs. (31-33), this is accomplished (1) by having a large enough chamber to virtually eliminate wall events, (2) by mounting the /~+ source off the detector-detector axis and "beaming" positrons into the region of high detection efficiency with the aid of a source backscatterer, and (3) by avoiding registration of chance coincidence events and three-gamma annihilation events using proper timing and energy discrimination in the electronics. In fact, G is readily kept at ~ 1.10. Also, other systems of known quenching characteristics can be utilized in calibrating the sample vessel, i.e. determining G, before an unknown is determined. In the example presented, there was no a priori reason for choosing the P dependence represented by eq. (42). Actually, the variation of P with gas composition is an interesting problem in itself and may, with the aid of the 23, difference method, help in elucidation of the mechanisms for Ps formation. The two-photon difference mode developed in this work was introduced in 1954 by Pond 6) who measured the percent change in 23, rate for various concentrations of diphenylpicrylhydrazine (DPPH) in benzene. The variation of zlA/A with concentration was interpreted as a Ps quenching cross section for DPPH of -~ 10 -17 c m 2. Although no discussion of the effect of the many contributing detection factors is given as in eq. (32) above, Pond's work did show that relative rate changes of less than 0.6% could be measured with ~0.1% uncertainty to provide quenching cross sections in condensed phases. In consideration of earlier discussions, Pond's experimental 25% source self-absorbtion factor in combination with source-on-de-
155
tector-axis geometry would account for an appreciable contribution to loss of "signal". As in all condensed phase studies, care must be exercised that the inequality (14) still justifies approximate eqs. (19) and (20). Otherwise, the equivalent to eq. (34) must be derived from the exact expressions (9) and (10). The method developed here is in many respects similar to the use of zllN, the percentage change in the narrow component of the annihilation angular correlation distribution. In angular correlation measurements, deviation of annihilation coincidences from collinearity are a direct reflection of the relative positron-electron momentum during annihilation. Annihilations of thermalized Ps result in mean stragglings of ~0.22 mrad, whereas annihilation of positrons with electrons at a few eV amounts to several miiliradians. The fraction of 23, events in the narrow region is proportional to the probability of positronium formation, P, and also to the probability that Ps decays by 23, annihilation, P ~ . From eq. (21), one may derive the expression for zIIN in terms of P, 2s, 2r, K, 2p and 2oh as was done by Goldanskiil). In measurements of dIN as a function of aqueous concentration of different manganese species, Goldanskii e t al. 7) were able to show that a 15% increase in the narrow component for concentrated MnSO4 relative to pure water was due to a large conversion rate for paramagnetic Mn ÷2 (K = 2.5 × 10 -12 [Mn÷÷]s-l). On the other hand, an observed 7% decrease in Ix for KMnO4 was attributable to a chemical (oxidation) quenching rate (2oh = 2.7× 10 -11 [MnO4+]s-l). Concentrations are in atoms/cm 3 so that these two rates are equivalent to quenching cross sections of 3.3× 10 -19 c m 2 and 3.6 × 10 -is cm 2 for manganous and permanganate respectively. A major difficulty with the technique based on percentage change in the narrow component has always been the deconvolution of the narrow component from the total distribution. Combined with the required intense /~+ source, stringent collimation requirements and long data acquisition times, this alternate method has not had wide acceptance. References 1) v. I. Goldanskii, danskii and V. G. 170; H. J. Ache, 179. 2) j. A. Merrigan, J. chemistry, voi. Ill
At. Energy Rev. 6 (1968) 1, V. I. (301Firsov, Ann. Rev. Phys. Chem. 22 (1971) Angew. Chem. internat. Edit. 11 (1972) H. Green and S. Tao, Physical methods of D (A. Weissberger and B. Rossiter, eds.
156
P. J. KAROL AND R. L. KLOBUCHAR
John Wiley and Sons, New York, 1972) p. 501. 5) R. L. Klobuchar and P. J. Karol, J. Phys. Chem., to be pub3) W. R. Dixon and L. E. H. Trainor, Phys. Rev. 97 (1955) lished. 733. 6) T. A, Pond, Phys. Rev. 93 (1954) 478. 4) D. Greenspan, Theory and solution of ordinary differential 7) V. I. Goldanskii, B. G. Egiazarov and V. P. Shantarovich, equations (MacMillan, New York, 1960). Fiz. Elem. Chastits (1966) 48, as reported in GoldanskiP).
1 4 9 - 1 5 6 ; i~) N O R T H - H O L L A N D PUBLISHING CO.
POSITRON T W O - P H O T O N ANNIHILATION COINCIDENCE TECHNIQUE: DIFFERENCE M O D E * t PAUL J. KAROL and RICHARD L. KLOBUCHAR§
Department of Chemistry, Carnegie-Mellon Universty, Pittsburgh, PA 15213, U.S.A. Received 13 September 1977 A difference (or comparative) mode in the measurement of two-photon coincidences from positron or positronium annihilation has been developed. The method can be used to advantage, particularly in gases, in determining annihilation pa. rameters such as quenching cross sections for low concentration strongly quenching chemical species in a reference medium which is relatively non-quenching.
1. Introduction
Studies on the physical and chemical properties of complex systems using the positron as an informative probe have been developing for just over a quarter century. The major characteristic which makes this technique useful is that a metastable positron-electron system, positronium, is formed in two nearly degenerate states whose distinctly different half lives depend on the environment in which the system is found. Measuring annihilation characteristics is indirectly determining how positronium is interacting with neighboring species. If the mechanisms and physics of these quenching interactions are reasonably well understood, the annihilations quantitatively reflect microscopic structural properties of the quenching agents. If positronium is formed and lives long enough to thermalize, two possible states will result: a short-lived spin-singlet called parapositronium, pPs, decaying into two nearly collinear annihilation photons with a decay rate As = 8.0x 10 9 s -~ and a long-lived spin-triplet called ortho-positronium, oPs, with a three-photon annihilation decay rate AT = 7 . 2 × 1 0 6 S-~. The mechanisms by which Ps interacts with its environment are generally grouped into three broad classifications. In the pickoff process, Ps collides with an accessible atom or molecule and the positron in Ps annihilates rapidly with an electron in the collision partner. The noble gases are examples of species which shorten or quench the Ps lifetime by pickoff. The pickoff quenching rate, gp, * Work supported in part by the National Science Foundation. t Preliminary description of the method was presented at the Symposium on Nuclear Phenomena as Chemical Probes, ACS Meeting, Chicago 1975. § Center for Naval Analysis, Arlington, VA 22209.
is related to the concentration of quenching species, np (cm-3), and the thermalized Ps average velocity o(7.6 × 106 cm/s) by the pickoff quenching cross section or. =Ap/npO. At STP for the noble gases, crp ~ 10 -f~ cm 2 which is a very small fraction of the collision cross section with an atomic electron. If the collision partner has one or more unpaired electrons, the spin conversion process becomes likely. Here for example, the parallel intrinsic spin of the electron relative to the positron in ortho-positronium is "exchanged" with that of an unpaired electron with spin antiparallel to that of the positron at the time of collision. The overall result is conversion of long-lived o-Ps into shortlived p-Ps. This conversion rate K is related to the reverse para-to-ortho conversion rate because of the singlet-to-triplet spin statistics of the effectively degenerate system. Consequently, the para-toortho conversion rate is given by 3K. The conversion quenching rate Ok is related to the conversion cross section by ak =Ak/nkO. For paramagnetic species like OA(g) and NO(g), crk ~ 10 -19 cm 2. Finally, it is postulated that positronium can undergo several different types of chemical reactions including oxidation, addition and substitution. These are some of the more complex and also more intriguing quenching interactions in which Ps can be involved. The chemical quenching rate 2ch is related to the chemical quenching cross section a,h = Ach/ncn0 which is in general relatively large having values ranging up to ~ 10 -~5 cm 2 for NAO4(g). Several recent reviews l) discuss these features at length. Two major experimental methods have found wide use in studying positronium formation and decay. These are the angular correlation method, which is a measure ~of the divergence of annihilation photons from collinearity, and the lifetime
150
P. J. KAROL AND R. L. K L O B U C H A R
technique, which is a direct determination of annihilation quenching rates. Each of these proce, dures has been quite successful under a variety of circumstances. They are discussed in reasonable detail elsewhere2). Measurements of angular correlation functions or lifetime spectra do involve the necessary utilization of sophisticated nuclear electronics and either fairly intense positron sources or lengthy data accumulation times. We have developed a technique which has the advantage of requiting simpler equipment, weaker sources and short experimental acquisition times. Our method is based on the change in two-photon coincidence count rate relative to a reference count rate. The 23, count rate change is a function of the known composition of a quenching mixture containing the quenching species to be determined and a reference species of known quenching characteristics. 2. Method The expressions necessary for relating experimentally determined quantities to annihilation quenching cross sections or rates all derive from a series of coupled differential equations3). These differential equations describe the time-dependence of the number of positrons in each of its three possible conditions; as p-Ps (singlet), o-Ps (triplet) or as a free positron fl+. In terms of the various annihilation quenching rates we have:
dPF(t) = --)'F PF(t), dt
(1)
dPs(0 = - (3K+2s+2p+2~h) es(t) + KPT(O, dt dPT(0 dt = - (K+2T+2p+2ch) PT(t) + 3KPs(t),
(2) (3)
where PF(t), Ps(t) and PT(t) represent respectively the fraction of positrons which are free, or part of singlet and triplet positronium at time t. ;tF is the annihilation rate of free positrons and the other symbols have been defined in the introduction. The system of eqs (1-3) represents a series of linear homogeneous differential equations with constant coefficients. Such equations may be readily solved by standard means4). The solutions obtained are: PF(t) = (1 -- P) exp(-- 2F t), (:4)
Pr(t) = 3 P(2T+ RP + 2~h--Z') exp(--2't)+ 4(2'-Z')
"I- 3e()~I"It-~'P'Jr~'¢h--~")exp(--~.#t), where, following the notation of Ooldanskii~), ~., =
+ P(';Ls+2P+2ch--2')
4(2"--Z')
exp(-Z't),
(7)
g -- (g2__#2)I/2
and ,~,t = ~ ..[_ (~2__fl2)1/2 .
(8)
The quantities, o~ and //2 may be expressed in terms of the original rate constants as: = 2 K - F ~ p + ~ , c h -I (~'S't-'~T) 2
(9)
and f12 = (K.q_ ~p..[_~ch) ~S "~ 2T'~S+ (3K+;tp+2ch) Lr + + (4K+2,+2,h) (2,+;t~h).
(10)
The above solutions were obtained by recognizing the following necessary boundary conditions: PF(0) = (l--P), (11) Ps(0) = P/4,
(12)
PT(0) = 3P/4,
(13)
assuming instantaneous thermalization and where P is t h e probability of positronium formation. Eqs. (11-13) reflect the statistical weights of the triplet state (7596) and the singlet state (25%). Inspection of eqs. (4-6) reveals that the annihilation of positronium can be conveniently described in terms of two decay constants, 2' and M', which contain the fundamental rate constants discussed above. Hence an experimental lifetime spectrum may yield values of the decay constants calculated from the logarithmic slopes by standard least squares techniques. Several approximations can be made which make the evaluation of K, ;tp and ;tea much simpler. In particular, if: As >>2r ~, ).p ~ K ~ 2oh,
(14)
then: f12 ,~ (K..l_J],p.jt. ~ch..l_,~,T) '~S
(15)
and a ~ 2s/2.
Ps(t) = P(2s-t'2P+2ch--2") exp(--,~'t) + 4(2'--2")
(6)
4(2" - Z )
(16)
It follows that: (5)
)]/ ~ ~-~-- [--~ -- (K q- ,~p-]-,~©h.Jr-,~r) q 1/2
(17)
POSITRON TWO-PHOTON ANNIHILATION COINCIDENCE TECHNIQUE and
2" ,~ ~2 "-I-I--~- (g+2~--b2eh-l-2T),~dl 1/2
P~ (18)
A Taylor series expansion retaining only the first terms leads to
2' ~ -b (Kq-2p-l-2ehq-2T)
(19)
and
2" ~ -t- 2s - (gq-2pq-2ehq-/~r).
(20)
Under these conditions, positronium possesses two decay constants, one of which (2') is determined primarily by the intrinsic decay constant for para-positronium and the other (2") by quenching rates and the intrinsic decay constant for orthopositronium. To test the validity of the above approximations, 2' and 2" were calculated by supplying decay constants 5) which correspond to 700 torr of oxygen ( K = l . 9 × 1 0 7 s -l) in 10 amagats of argon ( 2 p = 2 . 5 × 1 0 6 s - l ) . 2r and As were taken to be 7.2× 106 s -~, and 8.0× 109 s -l, respectively, and 2,h was set to equal zero. Such an experimental mixture would provide a rather severe test of the approximations for gas phase work as virtually all of the ortho-positronium would be quenched by the oxygen. The approximate value of 2' was calculated as 2.87× 107 s -~ compared to the exact value 2.86×107 s -~ (less than 1 percent error). A similar calculation for 2" also yields an error of less than 1 percent. However, one must exercise caution in analyzing lifetime spectra for K, 2p, and 2~h by eqs. (19) and (20) ascertaining that the inherent approximations have not broken down as may be the case for highly quenched positronium in the condensed phase. To calculate the instantaneous probability of two-gamma annihilation from positronium, one multiplies the probability of having either ortho or para positronium at any time by only those rate constants leading to two-gamma annihilation events. The total probability of two-gamma annihilation is consequently obtained by integration over time, from zero to infinity, of the previous relationships. For instance, triplet positronium can decay via two-gamma annihilation as a result of both pickoff (2p) and chemical reaction (2~h) mechanisms, but its intrinsic decay (2r) leads only to three-gamma annihilation. As a result, the total probability of two gamma annihilation from positronium, P1;2r, is given by:
3o
151
ka2, + a3&
x
o
(21,+2ch)Ps(t) dt +
\~2~+~3&, o (2p+2~h)
Pr(t) dr.
(21)
The factor, a2~/(a2r + a3~)= 371.3/372.3, in eq. (21) expresses the realization that during pickoff and chemical reaction there is a small probability that three-gamma annihilation can occur. This is similar in principle to the small probability for threegamma annihilation of free positrons. The exact relationship to intrinsic decay constants is~): a2~/(~2~+a3~) -- 2s/(2s+32r).
(22)
It should be recognized that the conversion process (K) does not lead to direct annihilation, but merely exchanges the ortho and para states. To calculate the fraction of positronium which decays by two-gamma annihilation, W2r one defines W27 -'~ (23)
P~[/P.
W2~ can be conveniently expressed in terms of dimensionless parameters or reduced decay rates defined as: (24)
x m K/2.r, y - 2p/&,
(25)
z - 2oh/&.
(26)
Performing the integration required by eq. (21) and using the substitution eq. (22) one obtains:
W2~,={(-~)(l+4x+y+z)+(~)(Y+Z) x
+ ( ~ r + 3) x + ( ~ - + 1) Y +
+(-~+ 1) z+(4x+y+z)(y+z)~.
(27,
Making the substitution 2s/2r = 9 7r/[4 (rF-9)a¢] = 1114 where a¢ is the fine structure constant one obtains:
e2/~c,
152
P. J. KAROL AND R. L. KLOBUCHAR
W27 -- 11114 + 4456x + A.A.A.8(y+Z) +
The 1/+ source can be mounted on an ultrathin film which has a low stopping power for positrons. + 15.957 x(y+z) + 3.989 (y+z)2]/ To a good approximation then, the detectors are recording coincidences which correspond to events /I-4456 + 4468X + 4460 (y+z) + occurring only within the condensed phase sam+ 16 x ( y + z ) + 4(y+z)2"l. (28) ple. For gas phase work, however, the maximum positron (projected) path length is quite large (e.g. For small x , y and z, eq. (28) reduces to: 140cm/atm for argon) and pressure dependent. 1 + 4(x + y + z) (29) Furthermore, positrons annihilating on those reW2:, ,~, 4 + 4(x+ y+z)" gions of the sample chamber walls where the deInspection of eq. (29) reveals that in the absence tectors are efficient contribute two-gamma events of quenching phenomena (i.e. when x = y = z = 0) which add to unwanted background, decreasing 14/~r is 0.25 as expected. One may similarly derive sensitivity to changes in count rate. an expression for the fraction of positronium An often overlooked correction must also be which decays via three-gamma annihilation, W3r, made to account for a contribution from detection but it is more convenient to use: of three-gamma annihilations "disguised" as twoW3r -- 1 - W2~, (30) gamma events. Conceptually, this may arise when ortho-positronium annihilates into three photons when W2~ is known. of energies (for instance) 500, 500 and 22 keV. ExThe fraction of Ps undergoing two-gamma anni- perimentally, a 500keV and the usual 511 keV hilation W2~, and three-gamma annihilation W3r, quanta cannot be distinguished with a NaI(TI)detecare then used to calculate the two-gamma annihi- tor. Furthermore, the angle between the two lation rate detected experimentally by a collinear 500 keV quanta in the example is 177.5 °. With the (but not highly collimated) coincidence arrange- wide angle acceptance employed for gross twoment, shown in fig. 1. However, one cannot easily photon counting, it is clear from fig. 1 that a fracconfine detection to two-gamma (or three-gamma) tion of bona fide three-gamma events may be misannihilation of positronium alone. In practice, pos- takenly registered as two-photon coincidences. itronium is formed by only about half of the pos- Lastly, one is always confronted with some degree itrons. The remainder of the positrons annihilate of chance events and background caused primarily as free positrons or as positrons attached to a sub- by detecting Compton events in one or both destrate in the manner of a collision complex. Either tectors. way, the resulting annihilation is predominantly a Any expression for the total measured two-gamtwo-gamma process. Hence, free positrons intro- ma annihilation rate, A~, must contain correction duce a rather large and undesirable background terms for the contributions outlined above. Letting when performing gross two-gamma counting. In ca+ be the efficiency for detecting two-photon gas samples, the situation is further complicated events from free positrons and e~s be the efficienby the requirement that one stop a significant cy for detecting two-photon events from positronifraction of positrons in a spatial region where the urn, one may describe the observed two-gamma two collinear detectors are efficient. For condensed annihilation rate as: phase work, this presents little problem as the = 1,(l-P) 8B+f + PshfW2, + + range of 22Na positrons, for example, is approxi+ fw3,] + B, (31) mately 200 m g / c m 2 corresponding to a path length of only 2.0 m m (for a sample with unit density). where ¢ ~ + ) is the emission rate of the positron 2~ is the efficiency source, B is the background, 83~ for detecting three-gamma events as two-photon ms ] annihilations, 8, is essentially the combined efficiency for detecting annihilations in the walls and source, f ' is the fraction of positrons which stop in the walls and the source where the detectors Fig. 1. Schematic of the two-photon coincidence geometry; D: are viewing, and f is t h e fraction of positrons gamma detector, S: lead shielding, C: sample chamber, H: offwhich eventually annihilate in a region of the gas axis positron source holder, V: region viewed by collinear deviewed by the coUinear detectors. Briefly, the first tectors.
1
sI
1
POSITRON TWO-PHOTON
ANNIHILATION
term in eq. (31) is due to the two-photon annihilation of free positrons, while the second term arises from the 2~, annihilation of positronium. The third term is due to annihilations on the walls and source holder. The final term within the brackets reflects the registration of three-photon events as two-photon annihilations. As discussed by Goldanskiil), one cannot distinguish between twophoton annihilations from positronium or from free positrons by employing an apparatus with wide angle acceptance. Consequently, the efficiencies, ta+ and eps, are purposely equal in the wide angle approach and are relabelled s, the efficiency for detecting two-photon events irrespective of their origin. Note that the various efficiencies are more correctly re-expressed as averages over the geometric detection region. In general, one could calculate the efficiencies and fractions, f and f', and determine the background, B, experimentally. In practice, however, this approach is subject to large errors which are both systematic and random in nature. To avoid such uncertainties, fractional changes in the gross two-photon annihilation are measured. For a reference sample, one may express the reference two-photon rate, A°r as:
COINCIDENCE
TECHNIQUE
153
fective chamber constant" and is given by: l+c G = 1------R' (35) where
(ewf')
B
C ---- (sf-"--"'~ -I- ~b(jff+ ) ( s f )
(36)
and R is the ratio of three-gamma efficiency to the two-gamma efficiency: 27
R = .(sarf)
(37)
•
When dealing with a mixture of quenching sp ecies as is necessary here in the comparative technique, the various quenching rates associated with the different quenching processes are properly defined in the following manner: r
v,
(38)
-- (Z ni ap, t) v ,
(39)
= (Y i i
&h = (Y ni
o,
(40)
l
where n~ is the concentration of quenching species i; ak.~, ap,~ and 0rch.~ are respectively the conversion, pickoff and chemical quenching cross secA°~ = q~(jff+)[ ( l - P ° ) (e f ) + eO
154
P. J. KAROL AND R. L. KLOBUCHAR
tuation, has the form: AA27 =
A °,
P°(W27-W°r)
(41)
G - P ° ( 1 - W°,)"
An upper limit to the method's sensitivity can be obtained from eq. (41) by the boundary conditions: G = 1, p o = 1 and ~ = 0.25 as a result of which AA2r 3 nov
A°~
"~r
at low concentration. Factors which tend to reduce sensitivity are low Ps formation probabilities, high quenching rates for the reference system, and extraneous 27 contributions. The latter include "noise" effects such as annihilation in the source and walls and registration of 37 events from o-Ps decay as if they were 27 annihilations. It is also worth pointing out that if eq. (31) in which ep~ and e,+ are not yet set equal is used, one can enhance the sensitivity considerably by the introducing severe geometry requirements that make tp, greater than ~ + . That is, by discriminating against the "noise" from free positrons, the signal from positronium becomes more prominent. 3. Discussion As an illustration of the method, we show in fig. 2 a hypothetical example of fractional change in two-photon count rate as a function of relative I
I
I Illll
I
I
I
I lllll{
I
I
I lllll{
I
I
16-
,2-
p0=
Eq. (34) generates the broken curve in fig. 3 frc which an important observation may be draw That is, the shape of the data curve gives an i dication as to whether or not the positronium fi mation probability is changing with compositic
I IIlll
18
I0
pressure of a gas of interest in a reference ga The chamber constant G has the value 1.15. A t tal of 5.0 atm of reference gas (R2) with 25 and with a (pickoff) quenching rate 2.1 × 105 arm- 1 s- 1 plus a (conversion) quenchi] impurity (Q2) of known concentration is employ, in the illustration. On the initial assumption tl~ the Ps formation probability P does not chan with composition in this case, eq. (41) may used to fit the data with the best possible quenc ing cross section. Using W2~ from eq. (28) or its a proximation (29), a family of curves has been co structed with a k = 2 × l O - 1 9 c m 2, l × l O - 1 9 c r and 0 . 5 × 1 0 -19 c m 2 to demonstrate the accura with which cross sections can be obtained. Vari tion in values chosen for ak appear as horizon1 displacements from the fractional difference d~ as seen in fig. 2. With sufficiently precise data, it becomes pos: ble to determine whether or not P changes wi composition. This point is exemplified in fig. where data from the above example are compar with a variable P case. The variable P case w chosen with all the parameters of fig. 2 except tt. P varies linearly with composition from p 0 = 25 for pure R2 to P = 45 96 for pure Q2. Letting p i present the relative pressure of Q2 we have: P = P°(1 +0.8p). (z
//
t41|2
-J .=_
40Z ~ / I
tO-4
I
/
I Illlll
t
10-3
! I
Ill)Ill
,
*
|
tO-2
,111,1
I
,
,
till,
10-I
[o,1/([o,1 + [R,]) Fig. 2. Fractional change in two-photon coincidence rate, expressed as percent, as a function of relative 'Q21 pressure in R 2 at a total pressure of 5 atm. Curves from top to bottom correspond tO Gk -- 2 × 10-19 cm 2 ' 1 × 10-19 cm 2 and 0.5×10 -19 cm 2 , respectively in eq. (41). Positronium formation probability p = p O = 2 5 % , constant. S¢0 text for full discussion.
~1~
6 4 2 0 i0-3
10-~'
I0-I
[ o 2 l / ( [ o z l + [R~] ) Fig. 3. Fractional change in two-photon coincidence rate. curves correspond to (rk = 1 × 10 -19 cmm as in fig. 2 fit data. Heavy curve has P = 25%, constant, broken curve ha,, varying with composition according to eq. (42), light cu{ shows effect of chamber calibration constant with G = 1 compared to G = 1.15 in heavy curve.
POSITRON T W O - P H O T O N A N N I H I L A T I O N C O I N C I D E N C E T E C H N I Q U E
Furthermore, if these data are sufficiently precise, the actual dependence of P on composition may be extracted. A caveat is required at this stage concerning evaluation of the P compositional dependence. In the hypothetical example illustrated in fig. 3, P was assumed to vary linearly with relative pressure. For this special case, the identical function may be achieved with a constant p0 and a chamber constant G = 1.24 rather than the value G = 1.15 given. To demonstrate the effect of the sample chamber constant, the thin curve in fig. 3 has been constructed. This curve corresponds to the heavy curve of constant po = 25% except that G has been revalued at 1.35. All of the above serve to emphasize the importance of maintaining c and R in eq. (35) as low as possible by design. Recalling eqs. (31-33), this is accomplished (1) by having a large enough chamber to virtually eliminate wall events, (2) by mounting the /~+ source off the detector-detector axis and "beaming" positrons into the region of high detection efficiency with the aid of a source backscatterer, and (3) by avoiding registration of chance coincidence events and three-gamma annihilation events using proper timing and energy discrimination in the electronics. In fact, G is readily kept at ~ 1.10. Also, other systems of known quenching characteristics can be utilized in calibrating the sample vessel, i.e. determining G, before an unknown is determined. In the example presented, there was no a priori reason for choosing the P dependence represented by eq. (42). Actually, the variation of P with gas composition is an interesting problem in itself and may, with the aid of the 23, difference method, help in elucidation of the mechanisms for Ps formation. The two-photon difference mode developed in this work was introduced in 1954 by Pond 6) who measured the percent change in 23, rate for various concentrations of diphenylpicrylhydrazine (DPPH) in benzene. The variation of zlA/A with concentration was interpreted as a Ps quenching cross section for DPPH of -~ 10 -17 c m 2. Although no discussion of the effect of the many contributing detection factors is given as in eq. (32) above, Pond's work did show that relative rate changes of less than 0.6% could be measured with ~0.1% uncertainty to provide quenching cross sections in condensed phases. In consideration of earlier discussions, Pond's experimental 25% source self-absorbtion factor in combination with source-on-de-
155
tector-axis geometry would account for an appreciable contribution to loss of "signal". As in all condensed phase studies, care must be exercised that the inequality (14) still justifies approximate eqs. (19) and (20). Otherwise, the equivalent to eq. (34) must be derived from the exact expressions (9) and (10). The method developed here is in many respects similar to the use of zllN, the percentage change in the narrow component of the annihilation angular correlation distribution. In angular correlation measurements, deviation of annihilation coincidences from collinearity are a direct reflection of the relative positron-electron momentum during annihilation. Annihilations of thermalized Ps result in mean stragglings of ~0.22 mrad, whereas annihilation of positrons with electrons at a few eV amounts to several miiliradians. The fraction of 23, events in the narrow region is proportional to the probability of positronium formation, P, and also to the probability that Ps decays by 23, annihilation, P ~ . From eq. (21), one may derive the expression for zIIN in terms of P, 2s, 2r, K, 2p and 2oh as was done by Goldanskiil). In measurements of dIN as a function of aqueous concentration of different manganese species, Goldanskii e t al. 7) were able to show that a 15% increase in the narrow component for concentrated MnSO4 relative to pure water was due to a large conversion rate for paramagnetic Mn ÷2 (K = 2.5 × 10 -12 [Mn÷÷]s-l). On the other hand, an observed 7% decrease in Ix for KMnO4 was attributable to a chemical (oxidation) quenching rate (2oh = 2.7× 10 -11 [MnO4+]s-l). Concentrations are in atoms/cm 3 so that these two rates are equivalent to quenching cross sections of 3.3× 10 -19 c m 2 and 3.6 × 10 -is cm 2 for manganous and permanganate respectively. A major difficulty with the technique based on percentage change in the narrow component has always been the deconvolution of the narrow component from the total distribution. Combined with the required intense /~+ source, stringent collimation requirements and long data acquisition times, this alternate method has not had wide acceptance. References 1) v. I. Goldanskii, danskii and V. G. 170; H. J. Ache, 179. 2) j. A. Merrigan, J. chemistry, voi. Ill
At. Energy Rev. 6 (1968) 1, V. I. (301Firsov, Ann. Rev. Phys. Chem. 22 (1971) Angew. Chem. internat. Edit. 11 (1972) H. Green and S. Tao, Physical methods of D (A. Weissberger and B. Rossiter, eds.
156
P. J. KAROL AND R. L. KLOBUCHAR
John Wiley and Sons, New York, 1972) p. 501. 5) R. L. Klobuchar and P. J. Karol, J. Phys. Chem., to be pub3) W. R. Dixon and L. E. H. Trainor, Phys. Rev. 97 (1955) lished. 733. 6) T. A, Pond, Phys. Rev. 93 (1954) 478. 4) D. Greenspan, Theory and solution of ordinary differential 7) V. I. Goldanskii, B. G. Egiazarov and V. P. Shantarovich, equations (MacMillan, New York, 1960). Fiz. Elem. Chastits (1966) 48, as reported in GoldanskiP).