Table V. Reactions Used for Lead Isotope Ratio Measurements Threshold energy, MeV
Reaction
m4Pb(p,n)20'Bi 206Pb(p,2n)zo6Bi zo4Pb(d,2n),o4Bi 206Pb(d,2n)206Bi
Half-life
Principal y-rays, keV
Interfering reactions
376 703 376 803, 880
zosPb(p,3n)204Bi if E , 2 20 MeV zo7Pb(p,3n)205Bi if E , 2 18.5 MeV 206Pb(d,4n)20iBiif Ed 2 22 MeV 207Pb (d,3n) za6Biif E d 2 1 3 . 5 MeV
11.2 h 15.3 d
-10 -12 -8 -8
11.2h
6.24 d
Table VI. Results of zo4Pb/m6PbIsotope Ratio Measurements Sample
Lead isotope ratio standard SRM 982 Average Lead isotope ratio standard SRM 983 Average
Activating particle
1
deuteron deuteron proton
t
deuteron proton
tions are interference-free a t irradiation energies below 18.5 MeV. This method has been tested on samples with known isotopic ratios (NBS SRM's 982 and 983). The results obtained are given in Table VI; the certified values by NBS are also listed for comparison. In deuteron activation, the pertinent reactions used for measuring 204Pb/206Pbratios are 204Pb(d,2n)204Biand 2°6Pb(d,2n)206Bi,respectively. To avoid an interference due to 207Pb(d,3n)206Bi, the bombarding energy should be kept below 13.5 MeV. In practice, it appears that energies 2 to 3 MeV higher can be used without deteriorating the present accuracy of the 206Pb measurement. This technique has also been applied on samples with known isotopic abundances (Table VI), using the method of calculation described above. The data presented in Table VI establish the feasibility of this activation technique for the simultaneous measurement of zo4Pb/206Pbratios. The small number of determinations made precludes a refined assessment of the precision of this method. In this respect, it will clearly be lim(22) Certificates of Analysis, SRM's 981-983, Lead Isotope Ratio Standards, National Bureau of Standards, Washington, D C. (1969)
Value found
Certified value ( 2 2 )
0.0208
0.0217 0,0278 0.0234 i 0.0040 0.0004
0.027219 i 0.000027
0.0003 0.0003 f 0.0001
0.000371 i 0.000020
ited in comparison to mass spectrometry. The features of this approach are its inherent accuracy (freedom from reagent blanks) and its simplicity while still providing subppm level sensitivity.
CONCLUSIONS For total lead determination, techniques with sub-ppm detection limits can be devised with each of the four different activation modes examined. Deuteron activation appears to offer the most versatility with possibilities for very rapid part-per-thousand level analyses, nondestructive or destructive sub-part-per-million determinations, and for measuring 204Pb/206Pbratios under conditions providing maximum sensitivity.
ACKNOWLEDGMENT The assistance of the cyclotron operation personnel is gratefully acknowledged. Received for review July 13, 1973. Accepted October 5 , 1973. This work was supported by the Robert A. Welch Foundation, Grant A-339.
Theory of Heat-Flow Calorimeter Satohiro Tanaka and Kazuo Amaya National Chemical Laboratory for Industry, Hon-machi, Shibuya-ku, Tokyo, Japan
Boundary value problems for generalized, spherical, and cylindrical models of a heat-flow calorimeter are solved under some ideal conditions. The proportionality relation between the heat liberated and the time integral of the temperature deviation from steady state is deduced for the spherical and cylindrical models. Optimum conditions for maximum sensitivity of the calorimeter are evaluated for the two models. The method of transforming the temperature variation, as a function of time, to obtain the thermogenesis function is investigated through solution of an integral equation relating the temperature and thermogenesis function. 398
In previous papers (I, 2), idealized one- and three-dimensional models of a heat-flow calorimeter in which the heat transfer takes place only by conduction were presented. The proportionality relation between the quantity of heat evolved or absorbed in a reaction vessel of the calorimeter and the time integral of the temperature deviation from the steady state a t any point in the thermal conductor of the calorimeter was proved under some ideal conditions for both the models. Verhoff ( 3 ) has also de(1) M. Hattori, s. Tanaka. and K. Amaya, B u / / . SOC. Chem. Jap.. 43,
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, N O . 3, M A R C H 1974
1027 (1970). (2) S. Tanaka and K. Arnaya, ibid., 43, 1032 (1970). (3) F. H. Verhoff, Ana/. Chem., 43, 183 (1971).
scribed a similar theory based on the heat conduction in the calorimeter and has derived the proportionality relation. Recently, Churney and collaborators ( 4 ) have presented a cell model which considers that a calorimeter consists of a large number of volume elements or cells. Churney ( 4 , 5 ) has stated that the proportionality relation is true under more extended conditions. In this paper, we solve boundary value problems in a generalized and two specific models, give optimum conditions for the maximum sensitivity of the calorimeter, and investigate a method to derive the thermogenesis function from a given temperature function. Generalized Model. Figure 1 illustrates a generalized model of a heat-flow calorimeter composed of three domains, D1, D2, and D3. Domain D1, in which heat is evolved or absorbed, is surrounded by domain D2 of finite extension. Domain D2 is surrounded by domain D3 and the temperature a t DB is kept constant. Surfaces S1 and S2 are, respectively, the boundary surface between D1 and D2, and between D2 and D3. The three domains D1, D2, and D3 correspond, respectively, to the calorimeter reaction vessel, the space of the solid thermal conductor between the surfaces of the reaction vessel and the thermal bath, and the thermal bath. The following assumptions are adopted: 1) Calorific power w ( t ) is developed in the confined domain D1 a t time t. w ( t ) is also called the thermogenesis function. Let c be the heat capacity of D1. 2) Thermal properties such as heat capacity c, thermal diffusivity K , and thermal conductivity X of the calorimeter are independent of time and temperature. 3) Temperature B(r,t) a t a point represented by position vector r in D2 is given by Fourier's heat equation. 4) The temperature B and the temperature gradient a t a point of surface SI, (aO/an),,l,, are equal everywhere over the surface, where alan denotes differentiation along the outward-drawn normal to the surface. The Verhoff theory has a rather unnatural boundary condition that heat flux qn normal to the surface S1 can be written as a function of time g ( t ) multiplied by a function of position h(rn), q n = g ( t ) . h(rn).In our theory, the formula for the heat flux has a more reasonable form qn = Ai(r) T,(t) as will be described in the following section, where A l ( r ) and T , ( t )are a function of the position of the surface S1 and of time, respectively. 5 ) The thermal bath D3 is kept a t a constant temperature. 6) The initial temperature of the calorimeter is equal to the temperature a t D3 throughout D1, D2, and D3. 7 ) The temperature of the reaction vessel D1 is uniform. In our previous paper ( Z ) , the solution of the boundary value problem was somewhat complicated by the assumption of non-uniform temperature inside the reaction vessel. We, therefore, confine our model to the uniform temperature assumption to obtain a simple form of the solution of the problem. In actual experimental calorimetry, it is usual to assure uniform temperature and concentration distribution inside the vessel by adequate stirring of the contents. The temperature distribution in the model is given by the following differential equation and boundary conditions: The initial condition:
E,
B(r, 0) = 0
-
D3
@,
Figure' 1 . Generalized model of heat-flow calorimeter
( 3)
In D3 and on S2:
B(r,t) = 0
(4)
where 0 = temperature measured relative to the temperature of D3, t = time, r = position vector, w ( t ) = the thermogenesis function, heat given off per unit of time in D1, c = the heat capacity of D1, X = the thermal conductivity of D2, S1 = the area of the surface SI, K = the thermal diffusivity of D2, alan = differentiation in the direction of the outward normal to the surface SI, and A = the Laplacian. The Verhoff theory ( 3 ) has a boundary condition that B(r,t) = 0 when r is far away from the heat source. Our boundary condition 4 differs from that of Verhoff in being more realistic, and we can have an apparent estimate of optimum condition of calorimeter construction for the maximum sensitivity of the calorimeter. We transform these equations with respect to time using the Laplace transform: $s) = cs@r,s) - xs,zB(r,s) a -
(5 1
se(r,s> = KAe(r,s) (6) B(r,s) = o (7) where m ( s ) and B(r,s) are the Laplace transforms of the function w ( t ) and 8(r,t), respectively. Let Bo(r,t) be the temperature function when w ( t ) = w o = constant independent of time; then the Laplace transform of the temperature function, &(r,s) satisfies the following equations:
sgo(r,s) = KA&,(r,s)
-
B,(r,s) = 0 Now if we write &(r,s) as
(9) (10)
eCr, s) = s a s ) . &(r, s ) / w o (11) this satisfies all the conditions in Equations 5-7 as is easily seen from Equations 8-10. The inverse of the Laplace transform in Equation 11 can be written formally as the following.
(1)
(4) K. L. Churney. E. D. West, and G. T. Armstrong, N B S l R 73-184, National Bureau of Standards, Washington, D.C., 1973. ( 5 ) K. L. Churney, National Bureau of Standards, Washington, D.C., private communication, 1973.
This is the solution of the boundary problem, giving the temperature B(r,t) corresponding to a variable calorific power w ( t ) in terms of the temperature Oo(r,t) corresponding to a constant calorific power wo. ANALYTICAL CHEMISTRY, VOL. 46, NO. 3, M A R C H 1974
399
L
Figure 2. Spherical model and its geometry
Spherical Model. The geometry of the spherical model is indicated in Figure 2. Under the same assumptions made for the generalized model, equations for the temperature function 0 ( R ),'2 and the associated boundary conditions are: The initial condition:
8(R,0) = 0,
Figure 3. Cylindrical model and its geometry
( d8!-s))
1d sfI(r,s) = -r2 dr r*-
(13)
B(1,S) = 0
In D1 and on SI:
The solution of Equation 27 is of the form In Dz:
where A and B are functions of s. Substituting Equation 29 into Equations 26 and 28, and solving for A and B, we get finally
i15)
-
In D3 and on Sp:
B(r. s) =
0
=
8,
= constant
(16)
ui5 )
where: 0 = the temperature, T = time, R = distance from the center of the sphere, W = the heat given off per unit of time in D1, C1 = the heat capacity of D1, A = the thermal conductivity of Dz, SI = the surface area of the surface SI,K = the thermal diffusivity of Dz. Equations 13-16 can be made dimensionless by defining the following dimensionless reduced variables:
riz
(r,s
+ h ) sinh (1 - r , ) f i + r l h f i cosh(1 - r , ) f i sinh (1 - r ) f i r
X
(30)
B(r,s) is a single-valued function of s with a simple pole a t s = 0, and simple poles a t s = -on2 ( 6 ) , where fa, are the roots of Equation 31. - r p 2 ) sin (1 - r , ) a
+ hr,a cod1 - r , ) a = 0
(31) Then the inverse of the Laplace transform in Equation 30 can be found and the solution can be written formally as the following: (12
(21)
In dimensionless quantities, Equations 13-16 become
B(r,O)
0
B(1,t) = 0
(22)
Cylindrical Model. Figure 3 illustrates the cylindrical model, in which heat is conducted only through the curved surface of length L with both the end circular areas insulated. The boundary conditions are the same as those described in the preceding paragraph except that the Laplacian operator is that for cylindrical coordinates rather than spherical coordinates:
B(r.0) = 0
(34)
B(l,t) = 0
(37)
(25)
We apply the Laplace transform to Equations 23-25 with respect t o time, considering the initial condition in Equation 22.
(6) H. S. Carlslaw and J. C. Jager, "Conduction of Heat in Solids," 2nd ed., Oxford, 1959, p 325.
400
ANALYTICAL CHEMISTRY, VOL. 46, NO. 3, M A R C H 1974
The steady state temperature, Oo(r,m ) is obtained as:
where the notations and the dimensionless reduced variables are the same as described in the preceding paragraph, except that R denotes distance from the axis of the cylinder. We apply the Laplace transform to Equations 35-37 with respect to time, considering the initial condition, Equation 34.
On the other hand, the steady temperature Oo(r, m ) can be easily obtained from the steady state solution of the boundary value conditions, Equations 22-25, as:
(39)
From Equations 48 and 49, we obtain an r121- r
E-i.=hy
n=i a n
We now calculate the time integral of Equation 32 by a formal integration by parts:
A solution of Equation 39 is where JO is the Bessel function of order zero of the first kind and Yo is that of the second kind. Substituting Equation 41 in Equations 38 and 40, and solving for A and B, we get the final form of the solution as follows:
~~(ifir)~~(ifi r Y) o ( i f i ) Y o ( i f i r ) 8(r.s) = s(J,( i f i r i ) y,,(ifi) - ~ o ~ i Y"(ifir)l fi) i h f i { J , ( i f i r l ) Y ofi) ( i - ~ d i f iYl(ifirl)l ) (42) j ( r , s ) is a single-valued function of s with a simple pole a t s -
+
= 0, and simple poles a t s = -anz ( 6 ) , where an are the
roots of Equation 43
I t can be shown that the first term in the brace on the right-hand side of Equation 51 becomes zero if the total quantity of heat has a finite value
B(r,t)
=
i=ioit)di
(52)
A necessary and sufficient condition (7) for the convergence of the integral in Equation 52 is that, corresponding to any one positive number e, another positive number T should exist so that: (53)
(44)
cu,~Jl(a,)Yo(cu,rl) - Jo(a,,rJYl(an)l + cu,r,IJ,(a,)Y,(a,,r,) - JJa,rl)Yo(a,)l
whenever t > T. Then we have
The function e'nnZT is a positive increasing function and in the range [T,t]: has an upper bound,
+
hlJl(a,rl)Yl(an) - Jl(a,)Yi(a,,rl)l+ hr $ [Jo(a I Yo(a , r ) - Yd a , r1)I
- Yo(a,)lJo(a,rl)- J2(anrJl T h e Proportionality Relation and Sensitivity of the Calorimeter. In previous papers, the proportionality relation S = GQ (46) was proved for a one-dimensional model ( I ) and a generalized model ( 2 ) , where S = Jt Odt is the time integral of the temperature deviation from steady state a t any point in the thermal conductor Dz, Q = J; wdt is the total quantity of heat given off in the reaction vessel, and the constant G can be adopted as the definition of the sensitivity of the heat-flow calorimeter. We shall also establish the relation and evaluate the sensitivity G for spherical and cylindrical models by a method similar to that described in our previous paper ( I ) . Spherical Model. Let us a t first deduce a property of the series on the right hand of Equation 32, which is useful in establishing the relation in Equation 46. When w ( t ) is a constant equal to W O , the series Equation 32 becomes:
For the fixed number T , we can find number t large enough to make
Therefore, from Equations 54-56, we have
whenever t > T . From Equations 50-52 and 57, we have a final form
or (59) where S = G Q and 1 2 r 2 rl. (7) E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis,"
(47)
4th ed., Cambridge University Press, London, 1965, p 70. A N A L Y T I C A L C H E M I S T R Y , VOL. 46, N O . 3, M A R C H 1974
401
Cylindrical Model. By the same method as in the case of spherical model, we have 1 8&r,03) = ?In- r (60) (61) (62)
In the case of a thermokinetic study, it is desired to obtain the thermogenesis function w(t) from the measured temperature function @ ( t )(9). If we know the kernel, Equation 67, of Equation 66 of the calorimeter used in the study, the problem is how to solve integral Equation 66, and the solution can be formally found by the method written in the well known text book (IO). Here we investigate another method of solution of integral Equation 66 considering the series form of the kernel, Equation 67. It is found that Tian's equation
1 d8 a? d t ) = -+-I9 a dt a
and has the solution where an's are given by Equation 45, and 1 2 r 2 r l . Optimum Conditions for Maximum Sensitivity. We have now the sensitivity expressions for the two models, and the expressions and conditions for maximum sensitivity are easily found as follows: Spherical model 1 1 G,,, = when r = r l = 2 (64) Cylindrical model 1 1 when r = rl = 2 G,,,~,,= eh (65)
O(t) =
J'K ( t ,T)W(
T )dl
A:
w(t) = w,(t) K ( ~ , T=) K&t, T )
(71)
= a,e-"n'"-''.
-
(66 1
+ K,w, + K0u2)dT + ...
I 71
(67)
s(t) = J r0 K 0 u ~ , d r
2. Tian-Calvet's model (9) n=l
=O = h/c
n Z 2 n=l
=o
n 22
cy1
(8) T. Ozawa, Bull SOC.Chem. Jap., 39,2071 (1966). (9) E. Calvet and H. Prat. "Recent Progress in Microcalorimetry,"
H. A.
Skinner, Pergamon Press, New York, N . Y . ,
L1(
K,wo
+ Kowl)dT
+
0 = L r ( K 2 w o K,wl
+ KOW~MT
Rewriting these equations, we obtain the following iterated equations.
3. One-dimensional model ( I )
where an's are the roots of equation a sin a - h cos a = 0 4. Spherical model (see Equations 31-33) 5 . Cylindrical model (see Equations 43-45)
(73)
Comparing coefficients of powers of X in Equation 73, we get
0= a,, = l / c
(72)
Substituting Equations 70 and 71 into Equation 66, we get
has the following form for each model: 1. Generalized model (see Equation 12)
9
+ X K , ( ~ . T+) X ' K , ( ~ , T )+ ...
K,(t, T )
"-1
402
(70)
X2Jt(K,uo
~ ( Tt ) , =
transtated by 1963, Chap. 4.
+ Xw,(t) + X2w2(t)+ ...
where X is a parameter and
where the kernel ir
(69)
The temperature function B ( t ) and thermogenesis function w ( t ) have properties of a reciprocal nature as may be seen by comparing Equations 68 and 69. If B ( t ) is known and w(t) is unknown, Equation 68 is the solution. We notice that the kernel, Equation 67, of Equation 66 is a series of terms which are identical in form with that of Equation 69 and the value of the term in the series becomes small as n increases. Remembering the above relations, we attempt to solve Equation 66 by means of a power series in
T. Ozawa (8) also obtained similar but somewhat different results, rl = l/d; on the optimum condition for the sensitivity of the cylindrical model of his quantitative DTA sample holder. Transform of Temperature Function to Thermogenesis Function. We now have the solutions of our problems; they are identical in their integral form known as Volterra's equation of the first kind, and will be shown once more for easy reference: 19(t ) =
W(T)dT
sr0 ae-U2't-T,
8 ( t ) = J 0' K ~ ~ j d T .
...
(74)
...
When B ( t ) is known, the unknown w o ( t ) is given by Equation 68. Then the left hand side of Equation 75 can be
(10) For example, see H. Margenau and G . M . Murphy, "The Mathematics of Physics and Chemistry," Van Nostrand, New York, N.Y., 1956, Chap. 14.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 3, MARCH 1974
evaluated and w l ( t ) is also given by Equation 68. Thus these successive procedures give the terms u nas
w, =
2%
{-wn-t
+ (a,? - ~
ACKNOWLEDGMENT The authors thank M. Hattori of Tokyo Institute of Technology for his helpful discussions.
~ ~ ) ~ f ~ - ~ ~ , ~(77) , ~ - Received ~ ~ w " -for, dreview ~ } June 25, 1973. Accepted September
a0
24, 1973.
Precipitation in Gels under Conditions of Double Diffusion: Critical Concentrations of the Precipitating Components Zvonirnir PuEar, Biserka Pokric, and Ante Graovac lnstitute "Ruder BoSkovic," Zagreb. Yugoslavia
A new method is proposed for determining critical concentrations at which the precipitation of two components begins in gels under conditions of double diffusion in Ushaped tubes. The method can be applied to the evaluation of both inorganic and organic precipitating systems, as well as of antigen-antibody immunoprecipitates. The exact solution of Fick's second-order differential equation was applied to one-dimensional diffusion in U-tubes. To solve the double diffusion problem for practical purposes, the algebra of the Gauss error function was processed in a computer and the data were summarized in a table. In this way, it was possible to perform calculations of "precipitating titres" of solutions and critical concentrations of precipitating components in gels and to obtain exact results. The precipitating system, Pb( NO3)*K2Cr04 in agar gel, at different concentrations at 25 "C was treated as an example. The critical molar concentrations in agar gel of both of the components were 5.6 X and were independent of their initial molar concentration ratios. The concentration of and/or the aging of agar gel had no influence on the critical concentration of the components. It has been proved both theoretically and experimentally that precipitations under conditions of double diffusion obey the "equivalency rule" CA = CB = ccr,t rather than the ionic solubility law CACB = Ks.
Early investigations of the formation of precipitates in gels are related to immunology. H. Berchhold ( 1 ) has shown that immunoprecipitates are formed if two reactants, antigen and antibody, instead of being mixed in a test tube, are allowed to meet each other by diffusion across a gel. The description of Liesegang's phenomenon (2-9) has initiated considerable interest in precipitation processes and diffusion in gels. The method became known through immunological studies (10-26), although (1) H.Berchhold, Z.Phys. Chem. (Leipzig), 52,185 (1905). (2) R. E. Ltesegang, Z. Phys. Chem. (Leipzig), 88,1 (1914). (3) S . C. Bradford, Biochem. J., 10,169 (1916). (4) D . N. Ghosh, J. Indian Chem. Soc.. 7,509 (1930). (5) R. Fricke. Koliofd-Z.. 64,62 (1933). (6) R . Fricke, Z. Elektrochem., 39,629 (1933). (7) H. Erbing. K o l l o d Z . . 56,194 (1931). (8) B. Tetak, Koiioid-Z., 68, 60 (1934). (9) C . J. van Ossand P. Hirsch-Ayalon, Science, 129,1365 (1959). (10) C. G.Pope and M . Healey, Brit. J. E x p . Pathol., 19,397 (1938). ( 1 1 ) J. Oudin, C. R . Acad. Sci.. 222,115 (1946); 228,1890 (1949). (12)J. Oudin. Ann. inst. Pasfeur, Paris. 75,30 and 109 (1948). (13) S.D. Eiek, J . Ciin. Pathol., 2,259 (1949). (14)0.Ouchterlony, Ark. K e m i , 1 , 4 3 a n d 55 (1950). (15) C. L . Oakiey and J. Fulthorpe, J. Pathoi. Bacterio:., 65,49 (1953)
considerable attempts to study the nucleation and growth of crystals in gel matrices were also made (27-34). Various processes concerning precipitations in gels have also been treated theoretically (16-18, 20-23, 25, 26). However, the problem of precipitation under conditions of double diffusion has not yet been treated in the way that would make it possible to determine parameters of analytical interest by applying exact solutions of the diffusion law. Our studies on tissue calcification have shown that the double diffusion method in gels may give valuable results concerning critical concentrations of soluble mineral components a t the place of gel mineralization. In this paper, we propose a simple and rapid method for determining critical concentrations a t which the precipitation of two components starts in gels under conditions of one-dimensional double diffusion in U-shaped tubes. This method will also meet requirements of various studies of precipitation processes in gels.
THEORETICAL General Experimental Behavior. One-dimensional double diffusion experiments in a certain gel may be performed in U-shaped tubes, which are schematically shown in Figure 1A. The two vertical branches of the U-tube contain solutions of the precipitating components A and B, their concentrations being coA and c o ~ respectively. , The gel is situated in the horizontal part of the tube (16)J. R. Preer. Jr., J. lmmunol., 77,52 (1956) (17) A. Polson. Biochem. Biophys. Acta, 29,426 (1958). (18) A. Poison, sci. roois, 5, 18 (1958). (19)0. Ouchterlony, "Progress in Allergy," P. Kallos, Ed., Karger. New York, N.Y., 1958,Vol. 5, p 1. (20) J. A. Spiers and R. Augustin, Trans. Faraday Soc., 54,287 (1958) (21) M. H. V. van Regenmortel, Biochem. Biophys. Acta, 34, 553 (1959). (22) J. Engelberg, J. lmmunol., 82,467 (1959) (23) J. van Oss and Y. S. L. Heck, Z. lmmunitaetsforsch.-Allergieforsch., 122,44,(1961). (24) G. Mancini, J. P. Vearman, A. 0. Carbonara, and J. F. Heremans, "Protides of Biological Fluids, Proceedings of the 1 l t h Colloquium, Bruges, 1963,"H. Peeters. Ed., Eisevier, Amsterdam, 1964,p 370. (25) F. Aladjem, J. lmmunol., 93,682 (1964). (26) R. Trautman. Biophys. J., 12, 1474 (1972);13, 409 (1973). (27) H. K. Henisch, J. Dennis, and J. I. Hanoka, J. Phys. Chem. Solids,
26,493 (1965). (28)J. Dennis and H. K. Henisch, J. Electrochem. Soc.. 114, 263 (1967). (29) S. E. Halberstadt, H. K. Henisch, J . Nickl, and E. White, J . Coiloid lnterface Sci., 29,469 (1969). (30) B. Rubin and A. Saffir, Nature, ( L o n d o n ) , 225,78 (1970). (31) H. M. Liaw and W. Faust. Jr., J. Cryst. Growth, 13/14,471 (1972). (32) 2 . LeGeros and J. P. LeGeros, J. Cryst. Growth, 13/14, 476 (1 972). (33) E. Banks, R. Chianelli, and F. Pintchovsky, J. Cryst. Growth, 18, 185 (1973). (34) A. R. Patel and S. K. Arora, J . Cryst. Growth, 18,199 (1973) A N A L Y T I C A L C H E M I S T R Y , VOL. 46, N O . 3, M A R C H 1974
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