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Analysis of quadrupole interaction at F- and FA-centres in KC1, KC1: Na and KC1: Li
Solid State Communications,
Vol. 8, pp. 167—169, 1970.
Pergamon Press.
Printed in Great Britain
ANALYSIS OF QUADRUPOLE INTERACTION AT F- AND FA-CENTRES IN Xci, KC1: Na and KC1: Li R. Kersten 2. Physikaiisches Institut der Universitat, Stuttgart (Received 16 December 1969 by G. Leibfried)
Calculated contributions to quadrupole interactions (envelope function, dipole fields and overlaps) are compared with experimental Endor results of F- and FA -centres in Xci. Sign and size of distortion are obtained for the F-centre.
QUADRUPOLE interactions (QI) for lattice nuclei in the neighbourhood of F-centres in KC1 have been measured previously by Endor out to 1, 1, 2 nuclei.’ Experimental data for PA-centres in KC1: Na and KC1: Li are reported at the end of this letter. A calculation to explain the electric field gradients at nuclear sites was made, and the results are reported. The following three contributions to QI were considered in this calculation: (i) the charge distribution of the center electron described by its orthogonalized envelope function 2,
case of the F-centre and extended out to 1, 1, 3 nuclei, allowing all electronic shell and core displacements consistent with symmetry, further nuclei being fixed. The unknown ‘spring constant’ between the F-centre and the 0, 0, 1 nucleus was left as a free parameter. Electrostatic effects due to changes in overlaps of free ion orbitals were calculated by a Löwdin orthogorialization procedure6 in a way similar to that used by Fukai,7 except that covalency contributions were neglected. The main contribution to the field gradient at ion o .
.
surrounded by the six nearest neighbours i 1.. 6 on the axes x, y, z is given by
(ii) electrostatic dipole fields due to displacement and polarization of lattice ions, (iii) changes in overlaps between neighbouring
V0 22
=
(S22 )
2
—
.2
~(S
)
22
2
.
(1)
2 =2+ 2 represents the main overlap contributions of the
ion wave-functions due to lattice distortion or replacement of lattice ions by impurities. QI due to envelope function were taken from
S
—
outermost ion orbitals and e ~
.
Gourary and Adrian or calculated using their type III wave-function.
—
Qzz
Lattice distortions around impurity ions in alkali halides and the resulting dipole fields and QI have been calculated by Dick and Das35
-
~
/
o 1 ‘~Pz ~
~\ Pz/
is the expectation value of the field gradient for the outermost p 2°-orbital, neglecting the small anti shielding factor for ion orbitals. Each of the squared overlap brackets represents an increased electron density in the corresponding orbital of
using electrostatic energies in an electronic shell model for ions, and Born—Mayer repulsive potentials. Their procedure was adapted to the 167
168
QUADRUPOLE INTERACTION AT F- AND 1~-CENTRES
Vol.8, No.3
Table 1. Experimental’ and calculated quadrupole interactions for the F-centre in KC1 (values in units of h. kc/s), e: QI due to the envelope function, d: QI due to dipole fields, o: QI due to overlap effects. q and q’ are the symmetric and asymmetric parts of the QI, E is the tilting angle between the QI principal axis and the [0011 direction (see reference 1). -
•
nucleus constant e d o e + d + o exp. nucleus constant e + d + o exp.
0, 0, 1 q —200 —18 +418 +200 +183
1, 0, q —13 —70 +37 —46 —24 1, 0, 2 q q’ —10.8 + 6.5 + 15.9° ±10.0 ~ 8.1 + 20.5°
1
1, 1, 1 0, 0, 2 q q —2 +4 +3 —103 +1 +45 +2 —54 ±5 ±1.7 1, 1, 2 q q’ e +5.2 + 5.3 + 38.2° ±5.7R 5.0 + 40°
q’ +40 +98 —102 +36 —8
Table 2. Experimental results of F- and FA-centres in KC1 (in units of h .Mc/s, T
F-centre Na-FA -centre
Li-FA-centre
1) 1) 1) 0) 0) 1)
a 20.72 6.90 53.4 19.8 21.5 10.61
constant b 0.92 0.52 1.94 0.88 0.91 j(a) 0.754
7Li (0, 0, 1) 39K (1, 0, 0) (0, 1,0) 35c1 (1, 0, 1)
7.91 23.62 23.31 9.75
0.63 0.983 (b) 0.989 0.654
39K 35Cl 23Na 39K
(0, (1, (0, (1, (0, 35C1 (1,
0, 0, 0, 0, 1, 0,
1
=
77K)
q 0.183 1—0.0241 0.215 0.213 ±0.545! 0.160 0.189 [±0.610!
(a) similar values were obtained by H. Ohkura, K. Miyoshi and Y. Mon.’2 (b) corrected values of reference.’3 ion o, accompanied by a reduction of electron density in the overlap region. V 2°2 vanishes only when there is cubic symmetry around ion o. One can get analogous expressions for V~°~ and V~°1, by permutating the summation indices i. In Table 1 the different theoretical contributions to the QI and their sum are compared to the experimental values. Experimental and calculated QI constants q were fitted
for 1, 0, 2 and 1, 1, 2 nuclei where d>> o>> e (see Table 1). For this fit one gets a radial displacement in units of the interionic distance d = 3.13 A of ci = + 0.010, + 0.0025, —0.0002 and + 0.0028 for nuclei 0, 0, 1; 1, 0, 1; 1, 1, 1 and 0, 0, 2. The sign of a could be determined by the much better agreement to experimental QI for the 1, 0, 1 nucleus. Sternheimer = 17.3, —56.6~and nuclear quadrupole moments Q = 0.062~and _O.079i0 barn were —
Vol.8, No.3
QUADRUPOLE INTERACTION AT F- AND
used for 39K4 and 38cr. The overlap effects were calculated with the tables of reference 11. For the overlap effects of the 0, 0, 1 and 1, 0, 1 nuclei the removal of the missing central ion is more important than the distortion relative to the nearest neighbours. The pronounced distortion along the [001] -axis results in large overlap and dipole effects of opposite sign for the 0, 0, 2 Cl-ion. Table 2 shows experimental results of Naand Li-I~-centres in KCI (0, 0, 1 ion replaced by the impurity cation). The 1, 0, 1 nearest neighbour to the impurity ion has a QI which exceeds that for the corresponding F-centre nucleus by more than an order of magnitude (circled numbers). For all other isotropic, anisotropic and quadrupole hyperfine constants, a, b, and q, the differences are comparatively small. The principle axis of the 1, 0, 1 QI is
~A -CENTRES
nearly in the [100] -direction, whereas for the F-centre it is in a [1011-direction. Changed overlaps with the impurity cation using equation (1) without distortion give the right order of magnitude: q/h —0.295 and —0.396MHz for the 1, 0, 1 nucleus in the Na- and Li-FA -centre respectively. Calculations of distortion have not been made for the FA-centre case. Additional data for F~-centresand details of the calculations will be published elsewhere. .
*
Note added: after finishing this letter the author got a recent publication of D. Ikenberry and T.P. Das (Phys. Rev. 184, 989 (1969)) where lattice distortions and overlaps were taken into account in a similar manner for some alkali-halide solid solutions.
Acknowledgment The author wishes to thank Dr. H. Seidel for helpful suggestions. —
REFERENCES 1.
KERSTEN R., Phys. Status. Solidi. 29, 575 (1968).
2.
GOURARY B.S. and ADRIAN F.J., Solid State Phys. 10, 127 (1960).
3.
DICK B.G. and DAS T.P., Phys. Rev. 127, 1053 (1962).
4.
DAS T.P. and DICK B.G., Phys. Rev. 127, 1063 (1962).
5.
DICK B.G., Phys. Rev. 145, 609 (1966).
6.
LOWDIN P.O., Adv. Phys. 5, 17 (1956).
7.
FUKAI Y., J. phys. Soc. Jap. 19, 175 (1964).
8.
STERNHEIMER R.M., Phys. Rev. 146, 140 (1966).
9.
NEY
J., Z. f. Phys.
169
223, 126 (1969).
10.
VARIAN NMR Table (1965).
11.
HAFEMEISTER D.W. and FLYGARE W.H., J. chem. Phys. 431, 795 (1965).
12.
OHKURA H., MIYOSHI K. and MORI Y., private communication.
13.
MIEHER R.L., Phys. Rev. Leu. 8, 362 (1962).
Beit~gezur Quadrupolwechselwirkung (Hüllfunktion, Dipolfelder und Uberlappeffekte) wurden benechnet und werden mit EndorMeBergebnissen an F- und FA-Zentren in KC1 verglichen. Für das F-Zentrum erh~ltman GröBe und Vorzeichen den Verzerrung.
Vol. 8, pp. 167—169, 1970.
Pergamon Press.
Printed in Great Britain
ANALYSIS OF QUADRUPOLE INTERACTION AT F- AND FA-CENTRES IN Xci, KC1: Na and KC1: Li R. Kersten 2. Physikaiisches Institut der Universitat, Stuttgart (Received 16 December 1969 by G. Leibfried)
Calculated contributions to quadrupole interactions (envelope function, dipole fields and overlaps) are compared with experimental Endor results of F- and FA -centres in Xci. Sign and size of distortion are obtained for the F-centre.
QUADRUPOLE interactions (QI) for lattice nuclei in the neighbourhood of F-centres in KC1 have been measured previously by Endor out to 1, 1, 2 nuclei.’ Experimental data for PA-centres in KC1: Na and KC1: Li are reported at the end of this letter. A calculation to explain the electric field gradients at nuclear sites was made, and the results are reported. The following three contributions to QI were considered in this calculation: (i) the charge distribution of the center electron described by its orthogonalized envelope function 2,
case of the F-centre and extended out to 1, 1, 3 nuclei, allowing all electronic shell and core displacements consistent with symmetry, further nuclei being fixed. The unknown ‘spring constant’ between the F-centre and the 0, 0, 1 nucleus was left as a free parameter. Electrostatic effects due to changes in overlaps of free ion orbitals were calculated by a Löwdin orthogorialization procedure6 in a way similar to that used by Fukai,7 except that covalency contributions were neglected. The main contribution to the field gradient at ion o .
.
surrounded by the six nearest neighbours i 1.. 6 on the axes x, y, z is given by
(ii) electrostatic dipole fields due to displacement and polarization of lattice ions, (iii) changes in overlaps between neighbouring
V0 22
=
(S22 )
2
—
.2
~(S
)
22
2
.
(1)
2 =
ion wave-functions due to lattice distortion or replacement of lattice ions by impurities. QI due to envelope function were taken from
S
—
outermost ion orbitals and
.
Gourary and Adrian or calculated using their type III wave-function.
—
Qzz
Lattice distortions around impurity ions in alkali halides and the resulting dipole fields and QI have been calculated by Dick and Das35
-
~
/
o 1 ‘~Pz ~
~\ Pz/
is the expectation value of the field gradient for the outermost p 2°-orbital, neglecting the small anti shielding factor for ion orbitals. Each of the squared overlap brackets represents an increased electron density in the corresponding orbital of
using electrostatic energies in an electronic shell model for ions, and Born—Mayer repulsive potentials. Their procedure was adapted to the 167
168
QUADRUPOLE INTERACTION AT F- AND 1~-CENTRES
Vol.8, No.3
Table 1. Experimental’ and calculated quadrupole interactions for the F-centre in KC1 (values in units of h. kc/s), e: QI due to the envelope function, d: QI due to dipole fields, o: QI due to overlap effects. q and q’ are the symmetric and asymmetric parts of the QI, E is the tilting angle between the QI principal axis and the [0011 direction (see reference 1). -
•
nucleus constant e d o e + d + o exp. nucleus constant e + d + o exp.
0, 0, 1 q —200 —18 +418 +200 +183
1, 0, q —13 —70 +37 —46 —24 1, 0, 2 q q’ —10.8 + 6.5 + 15.9° ±10.0 ~ 8.1 + 20.5°
1
1, 1, 1 0, 0, 2 q q —2 +4 +3 —103 +1 +45 +2 —54 ±5 ±1.7 1, 1, 2 q q’ e +5.2 + 5.3 + 38.2° ±5.7R 5.0 + 40°
q’ +40 +98 —102 +36 —8
Table 2. Experimental results of F- and FA-centres in KC1 (in units of h .Mc/s, T
F-centre Na-FA -centre
Li-FA-centre
1) 1) 1) 0) 0) 1)
a 20.72 6.90 53.4 19.8 21.5 10.61
constant b 0.92 0.52 1.94 0.88 0.91 j(a) 0.754
7Li (0, 0, 1) 39K (1, 0, 0) (0, 1,0) 35c1 (1, 0, 1)
7.91 23.62 23.31 9.75
0.63 0.983 (b) 0.989 0.654
39K 35Cl 23Na 39K
(0, (1, (0, (1, (0, 35C1 (1,
0, 0, 0, 0, 1, 0,
1
=
77K)
q 0.183 1—0.0241 0.215 0.213 ±0.545! 0.160 0.189 [±0.610!
(a) similar values were obtained by H. Ohkura, K. Miyoshi and Y. Mon.’2 (b) corrected values of reference.’3 ion o, accompanied by a reduction of electron density in the overlap region. V 2°2 vanishes only when there is cubic symmetry around ion o. One can get analogous expressions for V~°~ and V~°1, by permutating the summation indices i. In Table 1 the different theoretical contributions to the QI and their sum are compared to the experimental values. Experimental and calculated QI constants q were fitted
for 1, 0, 2 and 1, 1, 2 nuclei where d>> o>> e (see Table 1). For this fit one gets a radial displacement in units of the interionic distance d = 3.13 A of ci = + 0.010, + 0.0025, —0.0002 and + 0.0028 for nuclei 0, 0, 1; 1, 0, 1; 1, 1, 1 and 0, 0, 2. The sign of a could be determined by the much better agreement to experimental QI for the 1, 0, 1 nucleus. Sternheimer = 17.3, —56.6~and nuclear quadrupole moments Q = 0.062~and _O.079i0 barn were —
Vol.8, No.3
QUADRUPOLE INTERACTION AT F- AND
used for 39K4 and 38cr. The overlap effects were calculated with the tables of reference 11. For the overlap effects of the 0, 0, 1 and 1, 0, 1 nuclei the removal of the missing central ion is more important than the distortion relative to the nearest neighbours. The pronounced distortion along the [001] -axis results in large overlap and dipole effects of opposite sign for the 0, 0, 2 Cl-ion. Table 2 shows experimental results of Naand Li-I~-centres in KCI (0, 0, 1 ion replaced by the impurity cation). The 1, 0, 1 nearest neighbour to the impurity ion has a QI which exceeds that for the corresponding F-centre nucleus by more than an order of magnitude (circled numbers). For all other isotropic, anisotropic and quadrupole hyperfine constants, a, b, and q, the differences are comparatively small. The principle axis of the 1, 0, 1 QI is
~A -CENTRES
nearly in the [100] -direction, whereas for the F-centre it is in a [1011-direction. Changed overlaps with the impurity cation using equation (1) without distortion give the right order of magnitude: q/h —0.295 and —0.396MHz for the 1, 0, 1 nucleus in the Na- and Li-FA -centre respectively. Calculations of distortion have not been made for the FA-centre case. Additional data for F~-centresand details of the calculations will be published elsewhere. .
*
Note added: after finishing this letter the author got a recent publication of D. Ikenberry and T.P. Das (Phys. Rev. 184, 989 (1969)) where lattice distortions and overlaps were taken into account in a similar manner for some alkali-halide solid solutions.
Acknowledgment The author wishes to thank Dr. H. Seidel for helpful suggestions. —
REFERENCES 1.
KERSTEN R., Phys. Status. Solidi. 29, 575 (1968).
2.
GOURARY B.S. and ADRIAN F.J., Solid State Phys. 10, 127 (1960).
3.
DICK B.G. and DAS T.P., Phys. Rev. 127, 1053 (1962).
4.
DAS T.P. and DICK B.G., Phys. Rev. 127, 1063 (1962).
5.
DICK B.G., Phys. Rev. 145, 609 (1966).
6.
LOWDIN P.O., Adv. Phys. 5, 17 (1956).
7.
FUKAI Y., J. phys. Soc. Jap. 19, 175 (1964).
8.
STERNHEIMER R.M., Phys. Rev. 146, 140 (1966).
9.
NEY
J., Z. f. Phys.
169
223, 126 (1969).
10.
VARIAN NMR Table (1965).
11.
HAFEMEISTER D.W. and FLYGARE W.H., J. chem. Phys. 431, 795 (1965).
12.
OHKURA H., MIYOSHI K. and MORI Y., private communication.
13.
MIEHER R.L., Phys. Rev. Leu. 8, 362 (1962).
Beit~gezur Quadrupolwechselwirkung (Hüllfunktion, Dipolfelder und Uberlappeffekte) wurden benechnet und werden mit EndorMeBergebnissen an F- und FA-Zentren in KC1 verglichen. Für das F-Zentrum erh~ltman GröBe und Vorzeichen den Verzerrung.