DIFFUSION COEFFICIENTS AND LAW OF MOLECULAR INTERACTION
By M . M A D A N
DKPARTMKNT ok PhYSICJS, t^NlVIiRSITV oF LUCKNOW.
{Received for publicikitni, September 27, u)S4)
ABSTRACT. The expeiiiueiital dat$ on thermal diffusion and self-diffusion in conjunction with viscosity of various isj^topic gases have been examined to study the law of molecular inlcraction on the basis an expniential potential energy function, the collision -ntegrals for which have l»|cn recently evaluated by Mason. The tempera
ture dependence of thermal diffusion and tSscositv has been used to obtain the intennolc- cular force constants The results have? been compared with those for a lycnnard-Jones 12:6 potential and also with those giveir’bv Mason from virial, viscosity and crystal properties for an exp : six potential. The agreement i.s reniarkabh good. Utilizing the force constants obtained from thermal diffusion, coeflficienl of self-diffusion has been calculated and compared with the experimental results. IToni the considerations based on thermal ddfusion datj, a reason has been suggested for the unsuccessful behaviour of the exp :six potential for non-spherical molecules. The effect of second approximation to the thermal diflusion latio on the mtermolecular force constants has also been studied, ll has been found that the conclusions arc in accord with the author’s statement made in an earlier paper.
I. I N T R O D U C T I O N
The best intennolecular potential wliich has been used to date for the study of tile transport properties and their correlation with other properties of gases is of the Leaiiard-Jones form,* with a repulsion term varying as the inverse twelfth power of the distance of separation between the centre of two molecules and an attraction term varying as the inverse sixth power of separation distance. It is well known that the repulsion energy is more suitably described by aii exponeiuial form and therefore a potential energy function containing an exponential term will decidedly be a more realistic potential torm thin what has been used previously. Recently Mason (1954) has reported the transport projierty collision integrals for the gases whose molecules obey an exponential ;six intermolccular potential of the form
E ( r ) = - ^ I ti)
1 - 6 / * a J
where E (r) is the potential energy of the molecules at separation distance r, is the value of r for which E (r) is minimum and * is the
K(r)=t |^(r„/r)'’ - 2 ( ^ ) l \^here r,„ ami > are defined in connection with cqaatiou (1) and « is the minium potential energy.
12 M. P. Madan
additional parameter which may be considered a measure of the steepness of repulsion energy. Therefore assuming the validity of such an exponential function, more accurate and reliable information can be obtained about
the law of molecular interaction.
Temperature variation cf transport and other pro|)erties of gases provides a very effective means of evaluating the force constants a, <?, and r,n of equa
tion (i). Mason and Rice (1954) have recently evaluated the potential parameters for a few gases using virial, viscosity and crystal properties of gabes. As is w'ell known, the coefficient of thermal diffusion is far more sensitive to the type of molecular interaction than the elementary gas coeffici
ents, a determination of the law of molecular interaction from thermal diffusion will be much more accurate and useful than their determination from other properties.
Further, the theory of transport properties of gases as given by Chapman and Cowling (1939) gives only the first approximation expression for the thermal diffusion coefficient. The second api>roximation involves very complicated algebra and no expression for it has been given so far. Mason (1954) has, however, recently reported the second approximation expression for the thermal diffusion ratio kr.
Keeping the foregoing facts in view, we have, in the present paper, utilized the observed temperature dependence of thermal diffusion to cor
rectly assess the law of force and investigate the effect of second approxima
tion to the thermal diffusion ratio on the inteimolecular force constants.
However, for the sake of comparison the force constants and have also been determined by a method, essentially different from that of Mason, using the temperature dependence of viscosity. The potential parameters thus obtained from thermal diffusion and viscosity have been compared with previous determinations by Mason for the ex p : six model, including those obtained by Srivastava and Madan (1953a, 1953b) and Hirschfelder ef a/
(1954) for the Lennard-Jones 12:6 model. To test the correctness cf the force constants obtained, the coefficient of self-diffusion, which is quite sensitive to the force law, has been used to compare the experimental results with those obtained from theory. The results are also compared for the case of Lennard-Jones model using the force constants obtained from thermal diffusion (Srivastava and Madan, 1953a and b) and from viscosity (Hirseb- felder et. al (1954).
2. K V A h U A T I O N • O F T H B F O R C E C O N S T A N T S
A. Thermal diffusion coefficient and the effect of second approxiinaiion on froce constants
The only extensive data on thermal diffusion of gases at different temperatures are those of Slier (1942) for argon and neon. Unfortunately, the data on neon cannot be utilized due to the fact that the observed values
Diffusion Coefficients and law of Molecular Interaction 13
of thermal diffusion ratio for neon at high temperatures are much higher than those predicted by 12:6 tuodel or thi e x p : six model. The data on thermal diffusion of other gases are much less extensive. However, methane, out of spherical molecules and oxygen and nitrogen out of non-spherical molecules wer j selected for investigation, the data for which have been reported by Davenport and Winter ^1951).
The reduced thermal diffusion r|itio kr* is given to a first approxima tion for the case of binary mixtures <g heavy isotopes (Mason, 1954)
and to a second approximation by 5
[ fe v * ]3 = ( 5 /2 ) [A ’, - ? A ' 2 V , - X , y j l ... (3) vvliLM'e C* are ratio'* of ( the c'olli'^ion integrals given b y
and A’,, X3, Yj, Yj are coinplic|ted expressions in terms of the collision integrals. Tliese expressions have %ccn given by Mason.
The reduced thermal diffusion ratio ] is related to [A*/], the thermal diffusion ratio as
[/<,]= [hr M,) x>\.. (4)
where M2 are the inoleculai weights of species i and 2 and .Vj, X2 are the mo e fractions of the two components. The more frequently encoun
tered Rt, the thermal separation ratio is given by
/\’r = (118/105) k ’l^ ... (5) With the help of equations {2) and (5/ values of Rt for the values of tlic parameter a = 12 to a = i7 were calculated for both first and second approximations (see Table I). Using the experimental data on argon plots were made for the experimental Rt vs. T an 1 Iheoreticai Rr vs. k r h , which indicated that the curve for a = 12 is not in the least in accord with the experi
mental data and so is the case for the curve 2: = 13 but the plot for the value of a = 14 showed a very good paiallelisin and agreement with the experi
mental results. vSimilar procedure was applied for the case of methane, oxygen and nitrogen, but due to their scanty data it was difficult to arrive at any definite conclusion ; nevertheless, methane showed a clear tendency for « = I4 while the data on Oa and N3 showed a tendency for '-t>i5<Ci6 and a > 16 respectively. These values are in excellent agreement with those found by Mason and Rice (1954)-
Ft is interesting to note (see Table J) that the values of Rr for a > 14 do not show an inversion of sign at low temperatures, while there is suffici
ent theoretical and experimental evidence that Rr does change sign at low temperatures (Waldman, 1947; Troyer, van Itterbeek and Rietved, 19 5 1; Srivastava and Madan, 1953a), and thus the validity of ex p :six potential
14 M. P. Madan
for the values of parameter a > 14 is open to question. This might be one of 1 he important reasons that Mason and Rice (1954) find the exp : six potential unsuccessful in dealing with nou-spherical molecules, which have a tendency for > 14. Therefore, we restricted our detailed investiga
tion only to aigon and methane. An extensive data over long temperature range is desirable for a detailed investigation.
P'or the evaluation of the parameters e and the procedure adopted is analogous to that of Srivastva and Madan '1953a). Experimental thermal diffusion data of vSticr (1942) for argon were reduced to obtain the values of Rt at different temperatures Tr by using the relations J?7= .<4 ~ B /T +C/T*
and Rt^A-^BT at very low temperatures- The two curves Rt vs T (ex
perimental) and Rt v s kl'/e (theoretical) were plotted with arbitrarily different r and scales. Drawing the abscissa for different Rt values we get the corresponding and T values, from which e is easily deter
mined. These values of e were then utilized in conjunction with the experimentai viscosity data at different temperatures to obtain tlie value of r,n from equation (6). The values of « and r,u were found using both [Rt]i and [Rt]2» and are given in Table II. The same method was employed to get t' and for the case of methane after reducing the data with the help of equation RT^Ryi-^BiT (vSrivastava and Madan, 1953a!
the values being given in Table III.
TAm.E I
IR r]. LRr].,
feT/f _ __ __________
0 - 1.^
1
a - 16 a - 12 0 - 1 3 )
• a - 14 a = i5
0..1 -0.0305 0.1765 0.0519 0.0817
0-5 - 0 0932 -6,0437 — 0.0042 0.0257
0.6 — 0.0:287 D.0382 — 0.1166 - ■ -0777 — 0.0288 0 0023
0.7 -0.0302 0.0004 0,0360 - 0 .1158 — 0.0700 -■ 0.0304 o.ooo6
0.8 -0.0183 o.oi 17 0.0472 -0.1014 -0.0572 - 0.0182 0.0118
O.C)1.0 — 0.00260.0280 0.03260 o5^’>8 0.067.^o.()9io -0.0775-0.0513 -0.0356— O.OO99 0.0*'»260.0274 0 03230.0560
1 . 2 0.0836 0 1120 0.1461 -*.-r8o 0.0471 i 0.0824 0.1103
14 0.1397 0.1665 0.2012 0.0652 0. JO38 0.1378 0.1647
1.6 0.190g 0.2165 n.2506 o.i j8o ''6155^ O.I88S 02 1 1 ;
1.8 0.2355 0.2607 0.2956 0.1659 *'..2019 0.3341 0.2506
2.0 0.2746 o.29c;9 0.3360 0.2068 0.2429 0.2741 0 2995
2.5 0.3514 0.3775 ^>•4135 0.2888 0.3167 0.3535 0.3798
3.0 0.4054 0.4297 0.4686 <'•3487 6.3^^5 0.41 I I 0.4371
3*5 0.4437 0.4686 0.5091 0.3898 <^.4255 0.4S34 0.4798
4.0 0.4720 0.4967 0.5371 0.42-.-8 0.4562 0.4848 0.5107
5 0,5084 "•5.W7 0.5765 0.461Q 0.4987 o.5-\S8 0.5542
6 0.5314 ‘’ •■ S543 0.596*8 0.4846 0.5218 0.5520 0.57^
7Q 0.5439 0.5682 0.6113 o..4g6i 0.5.369 0.5667 0 5936
0 0.5520 '•5762 0.6181 0.5051 0.5459 0.5766 0.6020
9 0.5555 0.5820 0.6261 O.5093 0.5404 0.5822 0.6 99
l u 1 0.5589 0.5830 0.6294 0.5098 0.5520 0.5853 0.6120
Di0usion Coefficients and law of Molecular Interaction 15 Table 11—ArRon
Temp (“K)
J.SO
I S o
25f) 300 35f>
480
*//c ("K)
117.7 1^0.1
1 .:o.o
1 ?7 3 T42.O 147-1 147-7
I i 8 . g
[Rrl2
Table III—Methnne
'IVnip. r.'itige T,
1
1 R r 1 t f k C K ) t n, (A)
*0 5“ 155 >Si 0.25 *55 5 4-211
2 0 5 . 8 445 0,41 ' 147 7 4..>0-1
Tables I and II show that the difference between \Rr]i and \Rt]2 values is about 7%, which produces a difference of about r% in the values of e.
npto 400 'K, while for sliRhtly higher temperatures this difference is within 5%, taking also into account the most probable graphical and coniputional errors. Though this difference is more pronouticed at higher temperatures, we cannot determine its magnitude: as the theortical values practically show no variation in this region ; nevertheless it clearly indicates that the error involved in taking J^7’= f KrlI i.e., assuming the theoretical value of Rr to be given by is considerably less and lies well within the experimental errors in the measurement of Rr and its effect on the values of the inter- molecular force constants is almost negligible. This is in agreement with our previous statement bSrivastava and Madan 1953a).
B. Viscosity
The coefficient of viscosity, for a single gas is given by
t] X io^ = 266.93(Mr,)^ ... (6)
where M is the molecular weight, T is the absolute temperature tm is the
/6 M. P. Madan
separation distance for which E(r) is iijinimum, is the reduced collision integral and is just the collision integral of Chapman and Cowling divided by its value for rigid spheres of diameter r^. while /n represents the infinite series and is a complicated function of the collision integral. This term is nearly unity and is a slowly varying function of fe7'/e. Mason has tabulated the collision integrals in terms of and for values of a = i2 to a= 7. is related to as
Z(W== [fe7V«(i ~6/a)]^ (7)
Utilizing his tables were first converted into and then the ratio vvhicJi is used for a detailed analysis of viscosity data was calculated for various values of /e77«.
For the evaluation of the force constants from the temperature dependence of viscosity, tlie method adopted has already been described in detail by Srivastava and Madan (iQS2a and h), for determining the force constants from self-diffusion and viscosity for the case of 12:6 model. The values of « and rm are given in Table IV for the gases investigated together with the temperature ranges for which they have been calculated, along with their average mean values.
3. C O E F F I C I E N T O F S E L F - D T F F U S T O N
Mason and Rice (1954) give for the diffusion coefficient the e.xpression n V , , , 4 ^ 2 6.2 8o( T7M , ^ -M, ) !2M,M,V ( yn
in which D12 the coefficient of inter-diffusion in cn r. sec"*’ and M,, represent here the molecular weights and and fn are slowly varying
functi* ns of kTh- as already defined in section 2,
Kxcept thermal diffusion coefficient, the coefficient of self-diffusion or inter-'J ffusion of a gas is much more sensitive to the law of molecular inteiaction than any other gas coefficient. Utilizing the self-diffusion data of Winn ^1950), a comparison of the experitnental values has been made with tlK5se obtained from theory, using the force constants derived ftom thermal diffusion both for the ex p :six potential, as well as for the Lennaid Jones 12: 6 potential. Since the experimental data on self- diffusion usually refer to low temperatures values of e and rm have been selected for the temperature range 100'' to 300""K. These values along with values of « and obtained by Srivastava and Madan (1953a and 6), Hirschfelder et al (1954) for the 1 2: 6 potential including those obtained by Mason and Rice (1954) tfi® exp :six potential are given in Tables V and VI.
The data of Winn (1950) give which is calculated from the relation
••• (9)
^ut the quantity actually measured by him is the inter-diffusion of one
Diffusion Coefficients and law of Molecular Interaction 17
isotope into the other. So the value of was calculated and converted to Rive Dll with the help of equations (8) and (9) and are given in Table Vll- For the sake of comparison values obtained by Hirschfelder et al (1954) using tlie force constants obtained from viscosity data have also been lasted in this table.
Ta b i,r I V
Al gon Methane
T e m p r a n g e
C K ) • / A - ( - K ) ( A )
j
Tem p range |
( ’ K ) ; / / / . ' ( ° K ) I
1, , ( A )
150-300 12.’ .9 3-*^65 iof»-3 0 122.0 A351
122 g 3 865 150.300 I3t^ 3 4 308
■ 200-/100 128.2 200-400 160. * ■ 4.122
200-600 T i g . S 3 S90 2i)0-6(;0 l6g 4 4.104
250.500 J1S.5 3.S« 3 250-750 168 g 4.133
Mean 127.^16 3. S68 M e a n FS2 1 ; 4 a o3
T a b l e V IC xp. : s i x p o le s itia l.
From thermal diff.
(present work)
I"rom virial viscf sitv and crystal propcMties (Mas' n
ami Rice, 1951)
From the tem pera
ture dependence of vi'-cosity.
a j e / k i D t 1 k 1 « / 1 r«.
Argon : 122 9 3 11 J
1 1 123.2 I
!
3 S o6 1 1224^ \1 3 86S
M e t h a n e 1^1 150 ^ 1
1 4 222 1 14 ! 132.K
' . 1
4.206 1 152 20 ^ 4 203
T a b i.u V I 12 : ( S P o t e n t i a l
1 From thermal diff l?ti-
vastJiva and Madnn 1951a, 6)
From visc'ositv (Ilir-
•schfcldt r c't al, o;54)
j hroni self-diff, (Sii- I va'-fava and >radan,
! I9S26»
f//c i j
1 1!
1
f / k 1 j . I k 1 r»v
A rgon 124.9 3.^42 124 3.837 125 5 3 801
Methane .5 6 7 4.150 136 5
1
4 290 154.1 4.096
IvSq iP—i
/6 M. P. Madan
separation distance for which E(r) is imuimuni, is the reduced collision integral and is just the collision integral of Chapman and Cowling divided hy its value for rife’id spheres of diameter r„,. while fn represents the infinite series and is a complicated function of the collision integral. This term is nearly unity and is a slowly varying function of kTje, Mason has tabulated the collision integrals in terms o f a n d for values of a = i2 to
7 is related to as
[ kT/ t i i -~6/a)]' ... (7) Utilizing his tab’cs were first converted into and then the r a t i o w h i c h is used for a detailed analysis of viscosity data was calculated for various values of kT/*s,
For the evaluation of the force constants from the temperature dependence of viscosity, the method adopted has already been described in detail by Srivastava and Madan ^ 1952a and b), for determining the force constants from self-diffusion and viscosity for the case of 12-6 model. The values of « and tm aie given in Table IV for the gases investigated together with the temperature ranges for which they have been calculated, along with their average mean values.
3. C ( > K F F T C T K N T O F S K L F - T) T F F U S T O N
Mason and Rice (195/1) Jiive for the diffusion coefficient the expression
/ ) , I X J O4^ 26.28o ( r ( M ,
(8)
in which P12 is the coefficient of inter-diffusion in cm*’. sec“^ and il/,, Mg re])reseiit here the molecular weights and and /n are slowly varying
functif ns of A:T/t as already defined in section 2.
Kxcept thermal diffusion coefficient, the coefficient of self-diffusion or inter-d (fusion or a gas is much more sensitive to the law^ of molecular inteiaction than any other gas coefficient. Utilizing the selUdiffusioii data of Winn (1950), a comparison of the ex[>erimental values has been made with tlr)^e obtained from theory, using the force constants derived fiom thermal diffusion both for the exp :six potential, as well as for the Lennaid Jones 12:6 potential. Since the experimenlal data on self
diffusion Usually refer to low tein]>eralures values of e and tm have been selected for the temperature lange 100° to 3 0 0 These values along with values of « and r,„ obtained by Srivastava and Madan (1953a and M, Hirschfelder et al {1954) for the 1 2: 6 potential including those obtained by Mason and Rice (1954) for the exp :six potential are given in Tables V and VT.
The data of Winn (1950) give /In w’hich is calculated from the relation P n -[ 2 M g /( M ,+ M 2 ) ] ^ ^ ^ [ P ,g ] ... (9) hut the quantity actually measured by him is the inter-diffusion of one
Diffusion Coefficients and law of Molecular Interaction 17
isotope into the other. So the value of /),a was calculated and converted to give Dll with the help of equations (8) and (9) and are given in Table VIL For the sake of comparison values obtained by Hirschfelder ct al (1954) using the force constants obtained from viscosity data have also been lasted in this table.
TAB U t I V
Argon M e th a n e
Tem p, range
("K) C K ) 1 1... (A)
1 ^
1
"J'einp r.'inq:e I
('^K) ^ (“K)
1
1
(X)
150-300 100-3' 0 122 0 435^
155-450 122 y 150-300 3 4
■ 200-400 128.2
1 3 : 200-400 166.0 4.122
2 0 0 - 6 0 0 ji 1:9.8 3 <'^90 200-6tJ0 i6g 4 4.104
250-500 { JIS5 3.^^«3 25(^-750 168 Q 4.133
INJcan 1 122.46 3.868 " j i
1 Mean
j
4 203Tabi.e V Kxj). : six ])otential.
Troni thermal diff.
(present work)
From virial visorsilv and crystal pnjperlies (Mas'ii
niul Rice. iQ5|)
From the tempera
ture dependence of vis('ositv.
a !
i r,„
1
. 1 ' y,„ e//,
1 j
Argon 11 122 Q 3 !
1 It 123.2 3 866 ! 122.46 1 3 86S
Methane 14 150 6 4 222
1
11 152.8 4.206 j 152 20 ; 4 2<^3
Ta w.e .V I i 2 : f i P o t e n t ia l
From thcrm.'il diff. (Sti-
vastava and Madaii From viscosity iTIir- schfclder et al, 1954).
From s( If-diff (Sri- va'.tava and Madan,
19526'
r f k f m
1
f / k 1
1 ! ./fr
1
j r,„
1 Argon
Methane
124.9 15^ 7
3.842 't-15‘5
124 1365 1
3 S37 4 390
125 5
154.1
3 801 4.096 3 - i8<;ir’- I
18 M. P. Madan
Ta7u,e VII n,t in cin\ sec“‘
1 2 : 6
?sent work)
i 12 : 6 i (Hirschfeldcr et.al.
1 1054)
lixptl.
n.245 0 245 0.249
0.177 — 0.178
154 0.15.1 0.156
0,0815 — 0.0830
0.0178 -- o.oiSo
0 0132 0.0133 0 0134
0 309 0.293 0.318
0.227 — (. 240
0,192
1 0 183 0 206
1 o.ioo '
! - ^.0992
0 0214 1 0 0187 0.0266
Argon
M e t h a n e
'JVmp "K I Kxp. : six
353«2 2‘)5 2
273.2 191 7 ()) 2 777
33.3 2
273.2
191 7 go 2
O 2/[J
o 178
0.0S24 0 di8j o 0 1 3 4
0.226 0.192 0 100 o 0216
4. D I S C U S S T O N O F R K S U I v T S
A comparison of the force constants obtained by usiiiR different methods (see Table V) shows that the values of »•: and from thermal diffusion and viscosity by the present author are in excellent agreement with those reported by Mason and Rice (1954).
A comparison of the calculated and observed value.s of coefficient of self-diffusion (sec Table VII) reflects that the agreement is very good for both argon and methane and is definitely soinewliat better for the case of argon based on an ex]) :six potential though the experimental eirors in the measure
ment of self-diffusion coefficient foibid arriving at a definite conclusion.
Ncveitheless there is a clear indication of the superiority of exp :six potential over the 12 T) potential. For the case of methane, lack of experimental data on thermal diffusion over a wide temperature range makes it impossible to establish the supremacy of one potential over the other. However, it ipay be remarked that Mason and Rice (1954) hi^d a very poor agreement for the case of methane usmg the force constants from virial, viscosity and crystal properties. Kven for the case of 12:6 potential, it is clear from
Table V ll that the values calculated by us are more in accord with the experi
mental data than those calculated by Hirchfelder et al (1954) using force constants from viscosity.
It I S interesting to note from Table II. that and r,„ show a slight varia- ion with temperature. In the case of viscosity, as the calculations are made
for various small ranges, and further because the viscosity is not so sensitive to the force law as thermal diffusion, thJs variation is not so marked.
Similar variation in the values of the force constants have been noted by Srivastava and Madan (1953a;, Keyes (1951) and Whalley and Schneider (1952). Srivastava and Madan have also given uasoiis for this variation based on the I^ennard-Joiies 12:6 potential. No conclusions can be arrived at the present stage regarding the variation of these force constants with temperature. This requires a more thorough investigation by piopcrly pooling together all the transport properties. Whether this variation in the values of <•: and tm mean that the force field is temperature dependent or is simply a rcfleclion on the inadequacy of the potential form chosen, also, awaits a confirmation of Stier’s thermal diffusion data along with an additional accimite and extensive experimental data over a wide temperature
range for othei gases.
A C K N O'W h li 1) G M Iv N T S
It is great [>leasiire to tliank Prof. 13. N. vSrivastava, of the Indian Association for the CultkVaiiou of vScience Calcutta, for useful advice and encouragement and to Pi of. T. N.vSharma 'Lucknow) and Prof. J. 0 . Hirscli- felder (W'is:oiisiii) lor much help and continued interest in the work presen
ted here.
Dij^usion Coefficients and law of Molecular Interaction 19
R n V K K N C K S
Chapnum,vS ami Cowling, T.Ct , K)39, ‘ TIk.' IMatlicniatK'al theory of non-uiiifcnn gases’ (Cambridge Uiiiv Press).
Davenport, A.N., and Winter, K.R S , igsi, Trans, tauui. Soc., 47, iiir).
Hiischfekler,J.()., Curtiss, C P , Ihrd, R . ami Sp .1^, p l y , iQ5P Molecular theory of Gases and lyiipiids' (John Wilev and Sons, New \o rk).
Keyes, P G., 1951, Turns. I m a . Soc. IMcch. 73. sSg Ma.son, P^ A., 1954, ]. i 'hem. Thy.^.22, 169
Mason, R A., and Rice, W K , 1954, /■ 22, 813.
Srivastava, n.\'., ami Madan, M,P , 1953 a, /^T’/ M'lii , 4j, 918.
,, igs-’ h, Tioc. Nat. .icad. Sci, 21,254.
I95.PO /• (- ticni. Phys. 21, 807
,, iQ53h, Ptoc. Phys, Soc (London}^ A 66, 277.
Slier, h . 195^’, Thy.^. Rci'. 62, 518
Trover, A , van lUcrbeek, \ , ami Rietvcid A (j , 1951, Phy.dca, 17, 938.
Waldrnan, ^ , Naturfor.‘>th A, 1917» 2, 35S Whalley, K . and Si'lincides W G , ig.sui- Winn, K.R., 1950, Phys. Rev.80, loj^.
