530
VOl. 67
SOTES
bonding to the metal occurs through neutral binding sites (presumably the amino group and a peptide bond) rather than through the charged carboxyl group.* The decreased stability of triglycine complexes relative to diglycjne complexes5 is due to changes in both rate constants. The rates of dissociation are increased markedly for n = 2 and 3 in the case of triglyciiie. This enhancement probably is the result of increased electrostatic and steric interference among bulky ligand molecules forced into close proximity around a central ion. Steric interference also probably accounts for the observed decrease in lcs for triglycine relative to diglyciiie.
SIMPLIFICATION OF T H E BARTELLCHURCHILL REIATIOSS FOR T H E POLARIMETRIC DETERMINATION OF THE A4BSORPTIOK SPECTRA OF T H I N FILMS
Equating the two expressions for a in (4),eq. 5 is obtained [(@
+ z)(eN - H ) + (OX - p)(OH + LV)'I + (e2 + l)(il'ry + G z ) ( M H - G N ) X illy
+ Gx
There are two real roots of (5). Given the values of these roots, specific values of n and IC may be determined as follows: Using eq. 3 and the definitions of a and P,$ may be written
n
9 -
BY HILTONA. ROTHAND RICHARD D. SCHILE United Aircraft Corporation, Research Laboratories, East Hartford, Connectzcut
Solving (6) for n / k
Received Auoust SY, 1963
Bartell and Churchilll have demonstrated that the electronic absorption spectrum of a thin film supported by an optically absorbing substrate can be determined by an extension of the polarimetric method of Lucy.* The index of refraction and the absorption coefficient of the film are determined from the roots of two simultaneous third-order polynomials. The task of obtaining the real roots of these equations may be simplified to the point where a digital computer is no longer required for this calculation. The equations of Bartell and Churchill are reproduced as eq. 1 and 2 . P3y
+ B'(ax + G ) + P(a2y + H ) +
+ au) - yb(a2 - a ) + a2(G + N + X) = 0 + p y a y + 174) + P(-a2x + + y(a3 - a' + au) + xb(a2 - a ) + a2(M - H + y) = 0 - a'u
x(a3
-p3x
-
a2
(1)
14-1
Q2U
(2)
Let
n k
=
1 -[1 0
.\/e*
+ 11
NOW,solving (7) for n, substituting into a and solving for k 2
(7) =
n2 - k 2 ,
The procedure for finding n and k may now be summarized as follows: Given a value of $ from (;), tmo associated values of a are found from (4). Since n and k must be positive constaiits, only values of a and e of like sign are admissible. k may now be computed from (8). If a and 8 are positive, the plus sign in (8) is used; if a and 0 are negative, the minus sign is used. Finally, n is computed using (7), again taking account of the sign in order to obtain a positive result. It is thus seen that the problem has been reduced to that of finding the real roots of eq. 5 . If an electronic computer is not available, the evaluation of these roots may be accomplished with the aid of a desk calculator. An estimate of the location of the roots of (5) may sometimes be determined by inspection, depending Gx)(MH - GN). on the sign of the product ( M y Consider eq. 5 and let PI be the algebraically larger of
+
p
= a$
(3)
Substituting (3) into (1) and (a), it is found that a may be factored from both equations. Equations 1 and 2 are then quadratic in a and cubic in 0 and may therefore be solved for a. The result is 1
r
[G' -
4(BH
+ N)(By- ]+I h ]x) +1 $2
(4)
(1) L. S. Bartell and D. Churchill, J . Phys. Chem., 66, 2242 (1961). (2) F. -4.Lucy, J . Chem. Phgs., 16, 167 (1948).
and P2 the smaller. The first squared term in ( 5 ) is 1). always positive for real 8, as is the quantity ($2 Therefore, if the product (My G x ) ( M H - G N ) is positive the roots must be within the region where (0 - PI)($ - Pz) is negative. The real roots of ( 5 ) must thus be between PI and P 2 . However, if the G z ) ( M H - G N ) is negative then product ( M y (0 - P I ) ( @- P z ) must be positive and the real roots of (5) must lie outside of the region bounded by P I and Pz .
+
+
+
