J . Phys. Chem. 1985,89, 774-776
114
triplet state tautomer can be calculated and is shown in Figure 7. As can be seen from Figure 7, the predicted spectrum does exhibit some of the spectral characteristics actually observed for 3-MeLc in glacial acetic, particularly with regard to the positions of the depletion and principal maxima. From the above considerations it is suggested that the transient spectra of both 3-MeLc
and lumichrome in glacial acetic acid represent the absorption of the isoalloxazine tautomer, although no definitive assignment can be made. Registry No. I (R,= CH3, R2 = CH,), 14684-48-1; I (R, = CH3, R2 = H), 18950-64-6; I (R, = H, R2 = CHg), 33174-44-6; I (Rl H, R2 = H), 1086-80-2.
Angular Momentum and Photocurrent Threshold Law for the Solvated Electron James K. Baird*+ and Claudio H. Moraled Department of Chemistry and Mathematics. University of Alabama in Huntsville, Huntsville, Alabama 35899 (Received: August 2, 1984; In Final Form: October 18, 1984)
We show that near the threshold energy, Eth,the solvated electron photocurrent, I , is given by I = BE(E - Eth)(2it1)/2,where B is a constant, 1 is an integer, and E is the photon energy. The value of 1 is related through selection rules to I’, the orbital angular momentum quantum number of the solvated electron in its ground state.
Introduction The optical absorption spectrum of the solvated electron is broad and featureless.’ Because of the lack of structure, it is difficult to assign designated portions of the spectrum to transitions involving specified quantum states. In particular, identifying the relative importance of bound-bound and bound-free transitions is always a problem. Some distinctions can be drawn, however, by comparing the optical absorption spectrum with the action spectrum of the photoconductivity.2 Whereas the absorption spectrum consists of a superposition of bound-bound and bound-free transitions, the photoconductivity spectrum depends exclusively on the bound-free transitions. Recently, Fueki and collaborators have carried out photoconductivity experiments on electrons solvated in glassy amines and alcohol^.^ They produced solvated electrons by @Coy radiolysis of the glasses at 77 K. The solvated electrons were subsequently detrapped by illumination with monochromatic light at wavelengths greater than 500 nm. They detected the resulting photocurrent using two different methods. In the first method, the photocurrent, I, was measured directly using a microvoltammeter. When the photon energy, E, was near the excitation threshold energy, &,, the current could be fit to the equation I = BE(E - &)’I2
(1.1)
Here B is a constant and Ethis the binding energy of the solvated electron with respect to the bottom of the conduction band of the glass. In the second method, the glass was doped with pyrene (Py) which rapidly scavenged the detrapped electrons to produce the negative ion, Py-. Py- absorbs light strongly at 492 nm. The change in optical density at 492 nm as a function of E was found to be proportional to the photocurrent given by eq 1.1. In this paper, we shall demonstrate that the three-halves power in eq 1.1 is determined by the angular momentum of the solvated electron in its ground state. Our theory follows the general line of argument introduced by Brodsky and collaborators to explain the threshold laws for external photoemission of electrons from liquids and solids.4 It is more closely related, however, to our own study of the cross section for negative ion photodetachment in ~ o l u t i o n . ~ Department of Chemistry. *Department of Mathematics.
Theory The cross section u for photoexcitation of an electron from a bound state described by the wave function +(r) to a free state in the conduction band having a wave function e“‘‘ is given by5
Here, h is Planck’s constant, h divided by 2a, c is the speed of light in vacuum, n is the refractive index of the medium, and v is the frequency of the light. The properties of the electron are e, the charge, k, the wavenumber, r, the position, and m*, the effective mass in the conduction band. The differential solid angle for the wavevector k is df&. The electric dipole matrix element for the transition is Jk = 1 d 3 r $(r)r exp(ik.r)
(2.2)
where $(r) is the complex conjugate of +(r) and i = (-1)1/2. Evaluation ofJk. To evaluate the integral in eq 2.2, we let the z axis be parallel to k. We specify the spatial solid angle, Q = (O,+), in terms of the spherical polar angles, 0 I6 Ia and 0 I I$I2a, which are measured with respect to the three Cartesian unit vectors, t,, Cy, C,. The Rayleigh expansion6can be used to develop the plane wave m
exp(ik.r) = c ( 2 1 /=0
+ l)(i)’jl(kr)P/(cos 8)
(2.3)
where r = Irl,j,(kr) is the spherical Bessel function, and P, (cos (1) (a) M. D. Newton, J. Phys. Chem., 79,2795 (1975); (b) B. Webster, J . Phys. Chem., 79, 2809 (1975); (c) N. R. Kestner and J. Logan, J. Phys. Chem., 79, 2815 (1975); (d) B. S. Yakovlev, Russ. Chem. Reo. (Engl. Trawl.), 48, 615 (1979); (e) B. Webster, J . Phys. Chem., 84, 1070 (1980); (f) I. Carmichael, J . Phys. Chem., 84, 1076 (1980); (g) L. Kevan, J. Phys. Chem., 84, 1232 (1980); (h) D. F. Feng and L. Kevan, Chem. Reu. 80, 1 (1980); (i) A. Brodsky and A. V. Tsarevsky, Adu. Chem. Phys., 34, 483 ( 1981). (2) (a) S . A. Rice and L. Kevan, J . Phys. Chem., 81, 847 (1977); (b) S . A. Rice, G. Dolivo, and L. Kevan, J. Chem. Phys., 68, 4864 (1978); (c) S . A. Rice, G. Dolivo, and L. Kevan, J . Chem. Phys., 70, 18 (1979). (3) (a) N. Kato, S. Takagi, and K. Fueki, J . Phys. Chem., 85,2684 (1981). The amines were diisopropylamine and 1.2-propanediamine. The alcohols were 2-propanol, 1-butanol, 1-propanol, ethanol, and a solution of 5% water in methanol; (b) N. Kato, K. Akiyama, and K. Fueki, J . Phys. Chem., 85, 3087 (1981). (4) (a) A. I. Belkind, A. M. Brodsky, and V. V. Grechkov, Phys. Sratus Solidi E, 85,465 (1978); (b) A. M. Brodsky, J . Phys. Chem., 84, 1856 (1980). ( 5 ) J. K. Baird, J. Chem. Phys., 79, 316 (1983). (6) A. S. Davydov, “Quantum Mechanics”, Pergamon Press, New York, 1965, p 386.
0022-3654 I85 12089-0774%01.5010 0 1985 American Chemical Societv
The Journal of Physical Chemistry, Vol. 89, No. 5, 1985 775
Photocurrent Threshold Law for the Solvated Electron
e) is a Legendre polynomial. Equation 2.3 can be written in terms by use of the equation' of the spherical harmonic,
e(Q),
P/(COS e) = ( 4 a / ( 2 ~
+ i))l/2~(~)
(2.4)
v'(Q),
Since the spherical harmonics, form a complete orthonormal set over the spherical polar angles, $(r) can be expanded according to
(2.21) which shows to lowest order that both j/(kr)/k/ and K(1) are independent of k. Equation 2.1 depends upon IJkI2,which by virtue of eq 2.19 can be written
"
lJk12
where$'(r) is the complex conjugate of the radical wave function. The vector, r, can be expressed in spherical polar coordinates as
r = r(f/,(sin O)ei4e-
+ '/,(sin O)e-'+e+ + (cos O)e,)
(2.6)
In eq 2.6, C- and P+ are linear combinations of P, and E, 2- = E, - ie, (2.7)
E+ = P,
+ is,
(2.8)
and satisfy the relations
t+.L= 2
(2.9) (2.10)
= P-.E_ = 0
E+..?+
If we substitute eq 2.3-2.6 into eq 2.2, we obtain m
Jk = 5(47r(21 + l ) ) 1 / 2 ( i ) i i m dr3jl(kr)C r
+I'
/'=O "=-"
1-0
JdQ W(Q)[f/,(sin O)e"#&
$(r) X
=
+ +
r~
c(0)
'/,e+(sin
e)V(Q)= -40 G+lW)+ B ( 0 G-l(Q) e)V(Q)= A(I) yi:,(n) - B(1) Fjl(Q)
(COSe)u(n) = c ( i + i ) where A(1) =
I2[
B(1) = C(1) =
(2.12) (2.13)
Q+~(Q)+ C ( I ) ~ - ~ ( n )(2.14) (I
(21
+ 1)(1 + 2)
+ 1)(21+ 3) l(1- 1)
(21 - 1)(21
1'" 1
(2.17)
+ 1)
We substitute eq 2.12-2.14 into eq 2.11 and use the orthogonality relation JdQ
fl'(Q) v(Q) = 6i/t6mm,
(2.18)
where bllt and 6mmt are Kronecker deltas. The result is m
Jk = Ck'K(1)
(2.19)
/=0
where K(1) = (47r(21
+ l ) ) 1 / 2 ( i ) / I m dg(jl(kr)/k/)[(-A(l) r 0
J+l(r)
+
+ ( 4 0 AiIW - B ( 4 3-!1(r))G+ + (C(I+1) fl+,W + C(I)fl-I(r))Ezl (2.20) As k 0, eq 2.19 becomes a useful way of keeping track of the leading term in an expansion of Jk in powers of k, since9
4 4 J-1(4)2-
-
(7) J. D.Jackson, "Classical Electrodynamics", 1st ed.,Wiley, New York, 1962,p 66. (8) G. Arfken, "Mathematical Methods for Physicists", Academic Press, New York, 1970,p 587.
(2.23)
If the relations
E = hv
(2.24)
and
h 2k2 E - Eth = 2m * are used in eq 2.23, we obtain finally
(2.15) (2.16)
+
=327re2 -m*vk(21+1)1K(1)12 (n2 + 2)2 27h2c n
The products of trigonometric functions with appearing within the solid angle integral in eq 2.1 1 can be written as8 1/2e i4(sin
+ + +
Given that one or more of the functions,F(r), in eq 2.20 may be identically zero, we are interested in finding in eq 2.22 the lowest power of k which has a nonzero coefficient. This suffices to determine the threshold behavior of the photocurrent cross section. Selection Rules and the Threshold Behavior of the Cross Section. If K(0) # 0, then the leading term in eq 2.22 is IK(0)12. If K(0) = 0 and K( 1) # 0, then the leading term is k21K(1)12. If K(0) = K(l) = 0 and K(2) # 0, then the leading term is k4(K(2)I2. In general, if the first nonzero coefficient is K(I), then lJk12 = k2'lK(1)I2for small values of k. We put this result into eq 2.1. Because Jk depends only upon the magnitude of k, the solid angle integral in eq 2.1 is just equal to 47r. We thus obtain from eq 2.1 the result
+ y2(sin B)e-'+P+ + (COSe ) ~ , l q y ~(2.1 ) 1)
+ + +
IK(0)l2 + k[K(O)*K(l) K(l)*K(O)] + k2[K(0)-K(2) K(l)*K(l) K(2).K(0)] k3[K(0).K(3) + K(l)*K(2) K(2)*K(l) K(3)*K(O)] k4[K(0).K(4) + K(l)-K(3) K(2)-K(2) K(3)*K(l) + K(4)*K(O)] + ... (2.22)
(2.25)
(2.26) which is an absolute expression for the photocurrent cross section in the threshold region. We now investigate the selection rules which insure that K(1) is the first nonzero coefficient. If K(0) # 0, then according to eq 2.15-2.17 and eq 2.20, one of the radial wave functions,f;?'(r), with I' = 1 and m' = 0 or f l is nonzero. The conditions on m' follow because of the definitions given by eq 2.7 and 2.8 and the orthogonality relations expressed by eq 2.9 and 2.10. Note also that eq 2.26 involves lK(1)I2. If K(0) = 0 and K(l) # 0, there must exist radial wave functions,fl'(r), with quantum numbers I ' = 0, m' = 0 and/or I f = 2, m' = 0 or f l . Radial wave functions with I' = 1, m' = 0 or f 1 cannot exist; otherwise, K(0) f 1 which is contrary to the assumption we started with at the beginning of this paragraph. The possibility that I' = 2 leads to a further point. Since -I' 5 m' I+If, azimuthal substates with m' = f 2 can occur. However, eq 2.20 indicates that states other than m' = 0 or f 1 play no role in determining the cross section; hence, no conclusions one way or another can be drawn concerning the existence of substates other than these. If K(0) = K ( l ) = 0 and K(2) # 0, there exist radial wave functions,f;?'(r), with I f = 3, m' = 0 or f l . Radial wave functions with I ' = 0, m' = 0 and I f = 1, m' = 0 or f l and I f = 2, m' = 0 or *1 are forbidden; otherwise, K(0) and K(l) would not be zero, which is contrary to assumption. Beyond this point, our conclusions can be generalized. Our selection rule results are summarized in Table I. We look at these (9) Reference 6,p 390.
J. Phys. Chem. 1985, 89, 776-779
776
TABLE I: Catalog of Possible and Forbidden Solvated Electron States As Determined by Threshold Photoconductivity Spectra”
I’ I 0
1 2 3
I
possible angular momenta
forbidden angular momenta
4
none 1 0 , 1, 2 0, 1, 2, 3
I+ 1
0 , 1 , 2 ,..., I
1 0, 2 3
“The integer I appears in eq 2.26. The ground-state solvated electron orbital angular momentum quantum number is I’, and the quantum number for the z component of the orbital angular momentum is m‘. For I = 0, the selection rule requires the existence of a state with I t = 1, m’= 0 or f l . For I = 1, the existence of a state with I’ = 1, m’ = 0 and/or I‘ = 2, m’ = 0 or f l is required. The state with I‘ = 1, m‘ = 0 or f l is forbidden. For I 2 2, the existence of a state with I’ = I + l , m ’ = O o r & l isrcquired. S t a t e s w i t h I ’ I I a n d m ’ = O o r f l are forbidden. results from the experimental point of view. Consider the case where the photocurrent cross section has been experimentally determined to obey eq 2.26 with a value of I I2. We may conclude that K(0) = K(l) = ... = K(1-1) = 0 and K(f) # 0. Find the relevant value of I in the first column of the table. From the second column, one may conclude that solvated electron wave functions with I’ = I 1, m’ = 0 or f 1 exist. Wave functions with 1’5 I, m‘ = 0 or f l are forbidden; otherwise, the cross section in the threshold region would depend upon a smaller value of I than the one found. The forbidden values of I are given in the third column of the table. On the basis of the threshold behavior of the cross section alone, however, nothing can be concluded concerning the existence of states with 1’1(I 2). These angular momentum states contribute powers of k in eq 2.19 which are higher than the one which determines the threshold behavior.
+
+
Discussion and Conclusions The photocurrent I is proportional to the cross section u. If eq 1.1 and 2.26 are to be proportional, we see that I must have the value unity. Because B was not determined in the experiments,3 it is not possible to compare it with the factors multiplying E(E - Eth)(21+1)/2 in eq 2.26. Nevertheless, because I = 1, we conclude on the basis of Table I that the ground state of the solvated electron in the amine and alcohol glasses must have s character (I’ = 0) or d character (I’ = 2) or a linear combination of both. Due to Table I, moreover, if d character is involved, m’ = 0 or f l . In the case of d character, we cannot conclude anything one way or the other concerning the existence of substates m’ = f2. The existence of p character (1’ = 1 ) is forbidden, as indicated in the Table. Our conclusion that l’can be zero is consistent with the frequently made assumption that the ground state of the solvated electron is a pure s state.’O Inasmuch as glasses are isotropic, it may be that the Hamiltonian of the solvated electron is spherically symmetric. In that case, the orbital angular momentum operator would commute with the Hamiltonian, and the angular momentum and energy would have simultaneous eigenfunctions. A pure s-energy eigenstate would then be possible. Our derivation of the selection rules is similar to that of Geltman, who treated the cross section for electron photodetachment from negative ions of diatomic molecules.” Geltman’s result was subsequently extended to cover molecular negative ions of any symmetry by Reed et al.12 Finally, it is to be noted that eq 2.26 is an absolute formula for the photocurrent cross section. It is accurate to the extent that an electron in the conduction band of the glass can be represented by the function exp(ik-r). (10) T. Shida, S. Iwata, and T. Watanake, J . Phys. Chem., 76, 3683 (1972). (11) S. Geltman. Phvs. Rev.. 112. 176 11958). (12) K. J. Reed, A. H. Zimmerman, H. C. Anderson, and J. I. Brauman, J . Cheni. Phys., 64,1368 (1976).
Picosecond and Nanosecond Laser Photolyses of p-( Dimethy1amfno)phenyl Azide in Solution Takayoshi Kobayashi,* Hiroyuki Ohtani,? Department of Physics, Faculty of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan
Kaoru Suzuki, Department of Chemistry, Faculty of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan
and Tsuguo Yamaoka Department of Image Science and Technology, Faculty of Engineering, Chiba University, Yayoi-cho, Chiba-shi, Chiba 260, Japan (Received: August 6, 1984)
Photodissociation of p(dimethy1amino)phenyl azide (DMAPA) in toluene at rmm temperature takes a place from the excited singlet state of azide within 6 ps. A transient absorption spectrum of a precursor of the relevant nitrene, p-(dimethylamino)phen lnitrene (DMAPN), was measured, and the formation time of DMAPN in the ground triplet state was determined to be 120-,, +sr ps.
introduction Photochemical or thermal decomposition of azides produces highly reactive intermediates, nitrenes’ the ground state of which is triplet. Nitrenes are reactive because of electron deficiency of the nitrogen atom. Reactions are, for example, dimerization, insertion, and hydrogen abstraction.’ The chemistry of azides and Hamamatsu Photonics, Ichino-cho, Hamamatsu-shi, Shizuoka 435, Japan.
0022-3654/85/2089-0776$01.50/0
nitrenes attracts attention because of the interest both in application and in fundamental research. Reactions of nitrenes are being applied to materials for Photoima€$ng and microlithograPhic Processes.* From the viewpoint of fundamental research, azides (1) (a) Lwowski, W. “Nitrenes”; Interscience: New York, 1970; pp 99-162. (b) Reiser, A.; Wagner, H. M. In “Photochemistry of the Azido Group”; Patai, S., Ed.; Interscience: New York, 1971; pp 441-501.
0 1985 American Chemical Society
