J . Phys. Chem. 1988, 92, 1429-1431
1429
Dielectric Constants for Hydrogen, Helium, and One-Electron Ions S. A. Studniarz*+and J. P. Crawfordt Departments of Chemistry and Physics, The Pennsylvania State University, Fayette Campus, Uniontown, Pennsylvania 15401 (Received: March 31, 1987; In Final Form: October 7 , 1987)
The average dielectric constant, polarizability, and polarized radius for hydrogen, helium, and several one-electron ions, which were calculated from quantum mechanics, are reported in this paper. We show that the polarization charge can effectively be considered to lie on the surface of a sphere of uniform dielectric constant. The radius of the polarized charge ranged from 0.298 to 1.190 A while the dielectric constant ranged from 1.3 to 4.5. The trends observed for the variation of the dielectric constant and polarized radius with nuclear charge and also the electron4ectron repulsion effects were as expected. The results presented in this paper show that the polarizability reflects the size and dielectric constant of the atom while the dielectric constant reflects the ease with which the electronic distributions can be distorted.
Introduction A dielectric constant describes the response of a material to an applied electric field. The higher the dielectric constant, the greater the displacement of charge in the dielectric. It is usually applied to macroscopic matter, although there is no fundamental reason why it cannot be applied to systems with atomic and molecular dimensions. In fact, there has been at least one previous study of dielectric response on the atomic and molecular level over half a century ago,' and later H. L. Frisch and J. McKenna2 in 1965 described the dielectric response of isotropic molecules in ~ . W. ~ O ~ t o b yand ,~ a fluid. More recently, W. H. O r t t ~ n g , D. K. L. C. Hunt6 described with some success the dielectric constant (tensor) of atoms and molecules for a nonuniform dielectric model. The average dielectric constant is calculated here for several noninteracting atoms and ions. These atomic dielectric constants are of theoretical interest since they reflect how easily the electronic distributions can be distorted by an applied electric field,' and they appear in classical dielectric theory.* Atomic and molecular polarizabilities are readily available in the literature for many atoms and molecules, and the polarizability can even be calculated in certain cases. In fact, the polarizability of atomic hydrogen and helium was calculated over 50 years ago from quantum mechanic^.^-'' The atomic and molecular polarizability is a fundamental concept and is useful for interpretation of atomic and molecular responses to an electric field. However, since polarizability is a function of both volume and dielectric constant,'* the volume effects may obscure the dielectric constant effects, and vice versa. The polarized volumes, which are required to calculate the dielectric constant, are not known, and these volumes differ from those obtained from gas kinetics, crystallography, or the average quantum mechanical radius for many reasons, one of which is that the applied electric field distorts the charge distribution. A new feature presented in this paper is the calculation of the polarized radii from quantum mechanics which are then used to calculate the effective (or average) dielectric constants of the hydrogen atom, hydrogenic ions, and the helium atom. Polarized Radius and Effective Dielectric Constant The polarizability, a,of atomic hydrogen has been accurately calculated, and the wave function is readily available." Here the nuclear charge, 2,is included in the wave function since the functional form is the same for all one-electron species (21 1). The electric dipole moment, m, induced in a hydrogenic atom by an external field, Fo, is given by m = l J . * e x J .d V = l J . * e r ( c o s e)+ d V (1) The wave function, J., is given in ref 11 as 'Department of Chemistry. Present address: Westinghouse Electric Corp.,
M M T Lab, 469 Sharpsville Avenue, Sharon, PA 16146. *Department of Physics.
J. = Ne'(1
+ Fx(1 + y2r))
and the normalization constant, N , is obtained from (3)
This wave function has been shown to give accurate values for both the energy and the polarizability for the case of the hydrogen atom. Here the dimensionless coordinate r and field strength F are in atomic units:
where ro and Fo are in ordinary units and a. is the Bohr radius (0.529 A). In conventional units, and neglecting terms nonlinear in F, we find
m =
(
z ) F o = aF0
(5)
which gives the polarizability, a, for the one-electron species. The distortion of the electronic distribution by the applied field produces a slightly higher charge density on one side of the atom and a corresponding lower charge density on the opposite side. It is natural then to define the polarization charge density as PP = e(lJ.# -
1J.Ol2)
(6)
where J.o is the wave function at zero field. The polarization field P can now be calculated from the polarization charge density yielding the dielectric tensor as a function of position. However, in order to simplify the situation, it is convenient to define an effective dielectric constant and an effective (1) Pauling, L. Proc. R. SOC.London 1927, 114, 181-211. (2) Frisch, H. L.; McKenna, J. Phys. Rev. 1965, 139, 568. (3) Orttung, W. H. Ann. N.Y.Acad. Sei. 1977,303,22; J. Am. Chem. Soc. 1978, 100,4769; J . Phys. Chem. 1985, 89, 3011. (4) Orttung, W. H.; St. Julien, D. J . Phys. Chem. 1983, 87, 1438. (5) Oxtoby, D. W. J. Chem. Phys. 1978, 69, 1184; 1980, 72, 5171. (6) Hunt, K. L. C. J . Chem. Phys. 1984, 80,393. (7) Studniarz, S. A,; Crawford, J. P. PA Acad. Sci. 1987. (8) Smyth, C. P. Dielectric Eehauior and Structure; McGraw-Hill: New York, 1955. (9) Epstein, P. S. Phys. Reu. 1926, 28, 695. (10) Wentzel, G. Z . Phys. 1926, 38, 512. (11) Hasse, H. R. Proc. Cambridge Philos. SOC.1930, 26, 542. (12) van Beek, L. K. In Progress in Dielectrics; Birks, J. B., Ed.; Iliffe Books: London, 1967; p 69.
0022-3654/88 /2092-1429%01.50/0 0 1988 American Chemical Society
1430 The Journal of Physical Chemistry, Vol. 92, No. 6, 1988
Studniarz and Crawford TABLE I: Calculated Atomic Dielectric Constant, Polarizability, and Polarized Radius for Several Substances
q:
species hydrogen atom helium ion (He+) lithium ion (Li2+) beryllium ion (Be3+)
dielectric constant
polarizability, A'
polarized radius, A
2.96 1.74 1.45 1.33 4.57
0.666 0.0416 0.00822 0.00260 0.2068
1.190 0.595 0.397 0.298 0.722
helium
respectively, and rI2is the distance between the electrons. These distances and the field strength F are in atomic units:
Figure 1. Spherical coordinate system with the second-order differential volume element used in the calculation of the polarized volumes of atoms.
polarized radius as follows. Consider the solid angle dQ = sin 0 d0 d 4 depicted in Figure 1. The total polarization charge contained in this volume is 3e d2q = Jmr2 d r pp dQ = -F cos 0 dQ (7) 4r and the average radius for this charge is given by
R = (d2q)-IJmr3 0 d r pp dQ =
(f
)ao
R2dQ
Fo cos 0
2 7 ~ 2
-(
= eJ$*$
dV2
P2F
=
dVl
PIP
=
P1F
- PI0
P2P
= P2F - P20
cos
(17)
(9)
P1P
= P2P
= PP
(18)
and for the dipole moment we have
m =
JP161
dV1 + JP2px2 dV2
= 2 S p , X d V = 1.38a03Fo =
aF0
e
(19)
which gives the well-known result for the polarizability. The polarized radius for each electron is obtained from d2ql = A m r l 2 dr, p l P dQ = d2q2 d2q
R , = (d2q1)-lJ2rl3 dr, plP dQ = R 2 0
= 3 -)Fo e-1 4r € + 2
(l6)
and the corresponding polarization charge densities
(8)
which is precisely the form for the induced charge density on the surface of a sphere of constant dielectric constant e: up
PIF
Since the (spacial) wave function is symmetric in 1 and 2, we find
where we have restored conventional units in eq 8 and all nonlinear terms in F have been dropped. Equation 8 is remarkable in two ways: first, the average radius is independent of the applied field; second, the average radius is also independent of the angles 0 and 4 and therefore defines a sphere. Furthermore, the surface charge density on this sphere is u p = - =d29 -
where roand Fo are in conventional units. Henceforth, only terms linear in F will be retained. To proceed, we define the single-electron charge distributions as follows
R
(20) (21)
and is found to be
Equations 9 and 10 then yield the effective dielectric constants for hydrogenic atoms (ions): 8 1 2 64 e = (11) 8 1 2 - 32
+
Finally, the total dipole moment for the sphere is given by
R = 1.365~0 For the effective surface charge density we have up
d29, +-=d292 =RI2dR Rz2dQ
2-
d29 R2dQ
= O.1297Fo COS 0
(23)
Now making use of eq 10 we obtain the effective dielectric constant for helium: which agrees with eq 5 . Equations 8 and 11 constitute the main results for the hydrogenic case. The effective dielectric constant for the helium atom can be calculated in a similar fashion with only slight complications arising from the fact that helium has two electrons. The wave function should give accurate spacial distributions for the two electrons in order to accurately calculate the polarized radius. The helium wave function given by Hassell gives both the energy and the polarizability to better than 0.3% and can be written as
+= Ne-"('I+'2)(1
+ Cr12)(1+ AFxl(l + r , ) ) ( l + A F X ~ ( I+ r2)) (13)
where N , n, C, and A are constants: N = 1.391, n = 1.849, C = 0.364, A = 0.3845 (14) Here rl and r2 are the radial distances for electrons 1 and 2,
e = 4.57
(24)
Equations 22 and 24 are the main results for helium.
Results and Discussion The dielectric constants and polarized radii for hydrogen and several hydrogenic ions (calculated from eq 8 and 11) and for helium (from eq 22 and 24), along with the corresponding polarizabilities, are presented in Table I. We now make several qualitative observations concerning the numbers appearing in Table 1. The polarized radius for helium is smaller than that of hydrogen due to the increased nuclear charge but larger than that of the helium ion due to the electron repulsion. The polarized radius for hydrogen is considerably greater than the point at which the radial electron distribution function is maximum and is even beyond the average radius, which for hydrogen is 3a0/2 = 0.194 A. This is generally true since the electron distribution farther
J. Phys. Chem. 1988,92, 1431-1436 from the nucleus is more easily distorted due to the reduced field of the nucleus; this results in a larger polarized radius. The dielectric constant of helium is larger than that of hydrogen because there are two electrons to be polarized, but not a factor of 2 larger because each electron is more tightly bound to the nucleus. However, since the polarizability strongly depends on the radius (see eq 12) the polarizability of helium is significantly less than that of hydrogen even though the dielectric constant is larger. The dielectric constants for the one-electron species decrease with increasing nuclear charge since the electrons become more tightly bound and are therefore less easily distorted. The polarized radius also decreases with increasing nuclear charge since the stronger field of the nucleus “draws in” the electron distribution. Therefore, the polarizability of the oneelectron species decreases more rapidly than the dielectric constant. In fact, the polarizability decreases as the fourth power of the nuclear charge (eq 5 ) for the one-electron isoelectronic series, in agreement with the results
1431
obtained for heavier atoms.’ In contrast, the dielectric constant has a much weaker dependence on nuclear charge (seeeq 12) since it does not depend on polarized volume.
Conclusions W e have shown that for the ground state of one- and twoelectron atoms and ions the polarized radius is independent of angle and field strength (at low fields). This forms the basis for a dielectric model of atoms as spheres with uniform dielectric constant. The complementary properties of polarizability and dielectric constant can be understood as follows: the polarizability reflects more the size of the atom while the dielectric constant reflects the ease in which the electronic charge distribution is distorted. The trends observed for the variation of dielectric constant and polarized radius with nuclear charge were as expected. Registry No. H, 12385-13-6; He, 7440-59-7; He+, 14234-48-1; Li’, 14336-95-9; Be3+, 15721-53-6.
Paramagnetic Relaxation of Alkyl Radicals In Solid Alkanes As Studied by Electron Spln Echo Method: Posslblllty of Hlgh-Resolution Solld-State ESR Tsuneki Ichikawa Faculty of Engineering, Hokkaido University, Sapporo, 060 Japan (Received: May 6, 1987; I n Final Form: October 5, 1987)
Electron spin echo detected ESR spectra of alkyl radicals in y-irradiated solid alkane (n-hexane, n-octane, n-decane, n-dodecane, polyethylene, and cyclohexane) have been measured at 77 K as a function of extemal magnetic fields and transverseor longitudinal relaxation rate in order to investigate the possibility of the selective detection and the resolution enhancement of the ESR spectra of radical species in solids by the electron spin echo method. Among the two types of alkyl radicals stably trapped at 77 K in the solids, the longitudinal and transverse relaxation rates of CH3eHCH2Rradicals are much faster than those of RCH2CHCH2R’radicals, so that the RCH2eHCH2R’radicals are selectively detected by the electron spin echo method. The relaxation rates of the RCH2eHCH2R’radicals decrease with increasing length of the C-C chain and aye lowest for cyclohexyl radicals, which indicates that relaxation is mainly caused by the irregular twisting motion of the C-C bond. The ESR spectra of the magnetically saturated RCHzCHCH2R’radicals become sha,qxr with increasing time during saturation recovery. This effect is interpreted as due to the spectral diffusion of the RCH2CHCH2R’radicals caused by the slow time fluctuation of hyperfine fields. It is concluded that the resolution of a solid-state ESR spectrum is improved by measuring the saturated component of the spectrum when the line width of the ESR spectrum is comparable to the width of the spectral broadening caused by the time fluctuation of the hyperfine fields.
Introduction The ESR spectrum of radical species in a solid is inhomogeneously broadened due to anisotropic hyperfine interactions and the distribution of radical conformations. Although magic angle spinning or a sequential radio frequency pulse technique is used for obtaining high-resolution solid-state N M R spectra,’ such a technique is not applicable to ESR spectroscopy because of the width of ESR spectra. Suppose a radical species with a spectral width of 30 mT and g = 2 is measured at X-band, then the bandwidth of the entire spectrum is about 1 GHz. However, the spinning frequency of 1 G H z or a microwave pulse with the bandwidth of 1 GHz (pulse width of about 1 ns) is not currently available. A new approach to obtain a high-resolution solid-state ESR spectrum is to measure the ESR spectrum of a magnetically unrelaxed component as a function of longitudinal or transverse relaxation rate. When the relaxation rates are not the same for all the spin packets, the spin packets that relax faster disappear faster than the slower ones. The sharpness of the ESR spectrum
is therefore expected to increase as the time interval between the excitation and the observation increases. Such an ESR measurement is easily performed by measuring an electron spin echo (ESE) signal while sweeping the external magnetic field. This type of the ESR measurement is called an ESE detected ESR or a two-dimensional ESE measurement. In the previous works on the ESE detected ESR measurement the attention has been focused on the relation between the relaxation time and the molecular motion of radical species, so that no attempt has been made on the resolution enhancement of the ESR spectra of radical species in solids.* In the present work the ESE detected ESR spectra of alkyl radicals generated by the y-irradiation of polycrystalline alkanes (n-hexane, n-octane, ndecane, n-dodecane, polyethylene, and cyclohexane) have been measured at 77 K as a function of the time interval between excitation and observation in order to investigate the possibility of high-resolution solid-state ESR spectroscopy and the mechanism of paramagnetic relaxation. In these alkanes only two types of alkyl radicals, CH3CHCH2R and RCH2CHCH2R’ (R and R’
( 1 ) Mehring, M. High Resolution NMR Spectroscopy in Solids; Springer-Verlag: N e w York, 1976.
37.
0022-3654/88/2092- 1431$01.50/0
(2) See, for example: Millhauser, J.; Freed, J. J . Chem. Phys. 1984, 81,
0 1988 American Chemical Society
