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Impurity-induced lattice absorption
J. Whys.Chem. Solids Pergamon Press 1965. Vol. 26, pp. 931-933.
TECHNICAL Impurity-induced
lattice absorption
Printed in Great 3ritain.
NOTES
may produce a first order dipole moment and thus absorption quite prominent compared to that of the perfect crystal.(*) In that case the dipole moment is proportional just to the square of us (if 0 is the site of the impurity) in contrast to the more complicated expression for ionic crystala. The main purpose of this letter is to show that a simple relations~p exists between these two apparently very different expressions. We begin, for simplicity, with the one-dimensional case, and a pure lattice with particles of mass M with nearest neighbor interaction constant CC.The equation of motion, after insertion of harmonic time dependence, for the lth particle is
(Received 26 October 1964)
ALTHouaH the study of the eigen frequencies of vibrating lattices began nearly 50 years ago,@) much attention has turned to the eigenvectors (atomic ~spla~~ents) only recently.(s*s) One motive for this recent interest is the realization that the absorption of light by crystals is closely related to the properties of these atomic displacements.{*JQJ) The detailed behavior of the displacements is required when the lattice absorption induced by an impurity is to be studied. The quantity that determines the absorption due to a given vibrational mode is the square of the dipole moment associated with it,
.M&+-
u(ur_1- 2ur+ l&l) = 0
{O = frequency). ~ultipl~ng summing over Zgives
where ur is the ~spla~ment amplitude of the tth atom and +a the charge on it. (We may use classical language, as the introduction of quantization changes the theory only ~sign~can~y~~s~ at least in the harmonic approximation which is here implied). For a perfect crystal this quantity is known to be zero for all frequencies except very few (usually, one, called the Reststrahl frequency). This is usually attributed to the symmetry of the normal modes (which causes the dipole moment of the whole lattice to vanish even though it may have a finite value in any individual cell) and that in turn is a consequence of the symmetry of the crystal; in an ~pe~ect crystal--one that contains impurities or one finite in size{*)--that symmetry is destroyed, each normal mode will be somewhat modified and will in general have a finite dipole moment. The same is true for impurity-induced ‘local modes’(st7) with a frequency entirely different from those of the lattice modes, if any. The preceding paragraph applies to ionic crystals. In non-polar crystals the atoms have no net charge and there is no first order dipole moment; absorption, if any, is due to second order effects such as anharmonicity@j or second order dipole moments depending on the displa~~ents~s) ; but a charged impurity in such a neutral lattice
(MS-4G9P
this by (-)z = 0
Cl.0 and (2)
where
(2) states that the dipole moment P vanishes for all frequencies other than (4oc/M)l/s and is thus just a statement of the ‘Restrahl’ theorem mentioned above (but derived in a simpler than usual way). Replacing the atom at site 0 by another of mass BP while leaving the force constants unchanged--(‘isotope impurity’) has the effect of putting a term (~-~)~~
(4)
on the right-hand side of {LO) and Q)I and the latter thus becomes AM p=:--“.--with
and
931
03
TECHNICAL
932
the Rest&l frequency. (5) shows that the lattice sum P defined by (3) is very simply related to the displacement of the impurity atom alone. The scheme for computing the impurity-induced absorption in non-polar crystals@) by computing ua is therefore of direct use in ionic crystals as well. (Of course computation of ua will be more involved for ionic crystals than for non-poIar ones). For the simple model above, the validity of (5) is known.(s) Our purpose is to show that it also holds in essence, for realistic models. Our first generalization is to a diatomic lattice, masses M_t and Ms. Then MP in (2) is replaced by Ml& - MsSs, where
2 w
Sl =
and
Ss = 2 ul.
even
Odd
Sl and Ss can be eliminated by the use of Sl + S’s = P and Mu& + MpQ = AMua, the latter being the center-of-mass theorem obtainable formally by summing the analog of (1.2) over I without first multiplying by (-)J. It follows that (5) may be retained as it stands providing only M is replaced by Ml in (5) and (6) and by ZMlMz~(~l+Ms) in (7). To generalize to 3 dimensions, we again work out a special, fairly realistic, example before presenting general conclusions. Write
as the analog of (1 .Z) for a three dimensional crystal with short range interaction between first and second neighbors and coulomb interaction between ail ions. Here a is the distance between atoms, +e the charge on each, cc and /3 force constants and the F’s are
FiAr =
F%=
ul+l,m,n
-I- UCI,~,~,
-
2ulmn
~U1+l,mfl,nS.~~+l,m-l,n+~ltl,m,n+l +UZ+l,m,n-lf(+
+-+ -)-SUlmfl]
+Evl+l,m+l,n-Vl+l,m-I,n+Vl+l,m,nfl --z+l,m,.+l1-($_t+-_)]+(v~w]
NOTES
with Al = 1’-I, etc., Au = u~J~‘~’--ulmlz etc., and u, v, w the displacements in the x, y, z directions, respectively, of the ions designated by the subscript. Multiplying by (-)t+m+n and summing over E,m, n we find that the contributions from the v’s and w’s vanish, and we obtain: from the first two terms, the same contribution as from equation rep, no contribution at (1) ; from the third term Ft2)$ all; from the coulomb term a contribution -ZPV, where P now means z
( -)z+m+nWmn,
lmn
and V, defined as the sum of &s--q2 - rs) x (pz+ps+ra)-s/saver all points forwhichp+q+r is odd, has the value@O) 3.754. Thus, for the perfect crystal one again sees that P,vanishes unless & is equal to CJJ~= [4~-7.508es/us]l2~, and for a lattice with an isotopic impuri~ of mass IM’, quite analogous to (5) P=
AM ~2 --uooo 2/L “s-w;
(9)
Readers will be able to derive analogs of this for more involved force schemes or lattice symmetries. It is worth noting that forces between all like atoms (and not only between second neighbors as in the above example) will contribute nothing-a fact related to a similar absence of contributions from such forces to the sum rule for lattice vibrationsfll) For impurities not isotopic in nature but incurring a change in the short range force constants, relations similar to (9) are easily derivable; the righthand sides will then contain not only tisaa, but the u’s of all the atoms to which the impurity is coupled by a changed force constant. Finally, the present method may lend itself to the study of the effects of boundaries on lattice absorption, as boundaries can formally be treated as impurities of a special sort in an infinite lattice. (Observe that even in the derivation of (3), the assumption that all equations are alike-no boundaries-was
TECHNICAL
implicit.) Indeed, the main feature of (5) (the peak caused by the denominator) is mathematically similar to what has been found elsewhere(495) from boundary effects. Acknozuledgement-I thank Dr. R. F. WALLISfor a critical reading of the above. U.S.
Naval Research
H. B. ROSENSTOCK
Laboratory
Washington D.C. References 1. MARADUDINA. A., MONTROLLE. W. and WEISS G. H., Lattice Dynamics in the Harmonic Approximation, Academic Press, New York (1963). 2. ROSENSTOCK H. B. and KLICK C. C., Phys. Rev. 119,
1198(1960). 3. WALLIS R. F. and MARADUDINA. A,, Progr. Theor.
Phys. 24, 1055 (1960). 4. ROSENSTOCK H. B., J. Chem. Phys. 23 2415 (1955). 5. ROSENSTOCK H. B.; J. Chem. Phys. 27, 1194.(1957). 6. DAWBERP. G. and ELLIOTTR. 5.. Proc. Rov. Sot. A273, 222 (1963); Proc. Phy;: Sot., Lo& 81, 453 (1963). 7. MONTROLLE. W. and POTTSR. B., Phys. Rev. 100, 525 (1955). 8. MARADUDINA. and WALLIS R. F., Phys. Rev. 120, 442 (1960). 9. LAX M. and BURSTEIN E., Phys. Rev. 97, 39 (1955). 10. ROSENSTOCK H. B., J. Phys. Chem. Solids 4, 201 (1958). 11. ROSENSTOCK H. B., Phys. Rev. 129, 1959 (1963). BORN M. and MAYERM. G., Handbuch de-r Physik Vol. 24/2, p. 632, Springer, Berlin (1933).
J. Phys.
Chem. Solids
Vol. 26, pp. 933-934.
Energy of the formation of the anion Frenkel pair in calcium fluoride* (Received
12 October
1964; in revised form 1964)
12 November
A CALCULATION haa been made of approximate energies required to form an isolated anion vacancy in CaFa by removing a lattice F- ion to infinity, and to form an interstitial by inserting an F- ion from infinity into the body-center position in an empty cube of F- ions. The Born model of ionic solids was used, including Coulombic, repulsive (BORN-MAYER(~) potential), and polarization (monopole-dipole and dipole-dipole energies). The * Work performed while temporarily attached to the Theoretical Physics Division, United Kingdom Atomic Energy Research Establishment, Harwell, Didcot, England.
933
NOTES
method
of
calculation
was
that
described by the region I ions (interactions treated explicitly) included only the nearest neighbor ions. For the vacancy, region I contained the next nearest neighbors as well. The displacements of the latter were approximated with the MOTT-LITTLETON@) dielectric continuum theory, while the displacements of the nearest neighbor ions were found by minimizing the energy to create the vacancy. The repulsive parameters used were those of REITZ et aZ.@) With these, the Born model is capable of reproducing reasonably well the experimental cohesive energy,(d) inter-ionic spacing,c4) elastic constants,@) and ion-displacement contribution to the dielectric constant.@ They are also consistent, within the Born-Mayer model, with an ionic radius for the Caa+ ion of the same size or slightly larger than that for the F- ion, as in the set given by HUGGINS,@) and in conformity with the electron density maps of WITTE and WOLFE(~) for CaFa. This implies a radius for the F- ion in the range of 1.1 to 1.2 A, as found also in the alkali halides by TOSI and FuMI.@) For the interstitial, the body-center position is not necessarily that of least energy. A complete calculation including displacements of the surrounding ions has not been made for other positions, but some information can be obtained from the Madelung and repulsive energies for insertion of the interstitial into a rigid crystal. The sum of these energies exhibits a single minimum, to within 0.02 eV, centered on the body-centered position. This minimum is very broad, particularly in the (100) direction, toward the cube facecenter. Even if the isolated interstitial occupies the body-center position, it could easily be displaced in this direction, so that association between the interstitial and an immediately adjacent impurity cation of valence higher than 2 should be favored. THARMALINGHAM.(~) For the interstitial,
UREcg) finds from conductivity of association
between
data that the energy
the interstitial
and sub-
stitutional Y3+ ion is about 1 a4 eV. The
results of the calculation,
for the body-
center position, are, Defect Interstitial Vacancy
61
0.080 0.085
82
- 0.023
-%
-1.56 eV 4.27 2.7 eV
TECHNICAL Impurity-induced
lattice absorption
Printed in Great 3ritain.
NOTES
may produce a first order dipole moment and thus absorption quite prominent compared to that of the perfect crystal.(*) In that case the dipole moment is proportional just to the square of us (if 0 is the site of the impurity) in contrast to the more complicated expression for ionic crystala. The main purpose of this letter is to show that a simple relations~p exists between these two apparently very different expressions. We begin, for simplicity, with the one-dimensional case, and a pure lattice with particles of mass M with nearest neighbor interaction constant CC.The equation of motion, after insertion of harmonic time dependence, for the lth particle is
(Received 26 October 1964)
ALTHouaH the study of the eigen frequencies of vibrating lattices began nearly 50 years ago,@) much attention has turned to the eigenvectors (atomic ~spla~~ents) only recently.(s*s) One motive for this recent interest is the realization that the absorption of light by crystals is closely related to the properties of these atomic displacements.{*JQJ) The detailed behavior of the displacements is required when the lattice absorption induced by an impurity is to be studied. The quantity that determines the absorption due to a given vibrational mode is the square of the dipole moment associated with it,
.M&+-
u(ur_1- 2ur+ l&l) = 0
{O = frequency). ~ultipl~ng summing over Zgives
where ur is the ~spla~ment amplitude of the tth atom and +a the charge on it. (We may use classical language, as the introduction of quantization changes the theory only ~sign~can~y~~s~ at least in the harmonic approximation which is here implied). For a perfect crystal this quantity is known to be zero for all frequencies except very few (usually, one, called the Reststrahl frequency). This is usually attributed to the symmetry of the normal modes (which causes the dipole moment of the whole lattice to vanish even though it may have a finite value in any individual cell) and that in turn is a consequence of the symmetry of the crystal; in an ~pe~ect crystal--one that contains impurities or one finite in size{*)--that symmetry is destroyed, each normal mode will be somewhat modified and will in general have a finite dipole moment. The same is true for impurity-induced ‘local modes’(st7) with a frequency entirely different from those of the lattice modes, if any. The preceding paragraph applies to ionic crystals. In non-polar crystals the atoms have no net charge and there is no first order dipole moment; absorption, if any, is due to second order effects such as anharmonicity@j or second order dipole moments depending on the displa~~ents~s) ; but a charged impurity in such a neutral lattice
(MS-4G9P
this by (-)z = 0
Cl.0 and (2)
where
(2) states that the dipole moment P vanishes for all frequencies other than (4oc/M)l/s and is thus just a statement of the ‘Restrahl’ theorem mentioned above (but derived in a simpler than usual way). Replacing the atom at site 0 by another of mass BP while leaving the force constants unchanged--(‘isotope impurity’) has the effect of putting a term (~-~)~~
(4)
on the right-hand side of {LO) and Q)I and the latter thus becomes AM p=:--“.--with
and
931
03
TECHNICAL
932
the Rest&l frequency. (5) shows that the lattice sum P defined by (3) is very simply related to the displacement of the impurity atom alone. The scheme for computing the impurity-induced absorption in non-polar crystals@) by computing ua is therefore of direct use in ionic crystals as well. (Of course computation of ua will be more involved for ionic crystals than for non-poIar ones). For the simple model above, the validity of (5) is known.(s) Our purpose is to show that it also holds in essence, for realistic models. Our first generalization is to a diatomic lattice, masses M_t and Ms. Then MP in (2) is replaced by Ml& - MsSs, where
2 w
Sl =
and
Ss = 2 ul.
even
Odd
Sl and Ss can be eliminated by the use of Sl + S’s = P and Mu& + MpQ = AMua, the latter being the center-of-mass theorem obtainable formally by summing the analog of (1.2) over I without first multiplying by (-)J. It follows that (5) may be retained as it stands providing only M is replaced by Ml in (5) and (6) and by ZMlMz~(~l+Ms) in (7). To generalize to 3 dimensions, we again work out a special, fairly realistic, example before presenting general conclusions. Write
as the analog of (1 .Z) for a three dimensional crystal with short range interaction between first and second neighbors and coulomb interaction between ail ions. Here a is the distance between atoms, +e the charge on each, cc and /3 force constants and the F’s are
FiAr =
F%=
ul+l,m,n
-I- UCI,~,~,
-
2ulmn
~U1+l,mfl,nS.~~+l,m-l,n+~ltl,m,n+l +UZ+l,m,n-lf(+
+-+ -)-SUlmfl]
+Evl+l,m+l,n-Vl+l,m-I,n+Vl+l,m,nfl --z+l,m,.+l1-($_t+-_)]+(v~w]
NOTES
with Al = 1’-I, etc., Au = u~J~‘~’--ulmlz etc., and u, v, w the displacements in the x, y, z directions, respectively, of the ions designated by the subscript. Multiplying by (-)t+m+n and summing over E,m, n we find that the contributions from the v’s and w’s vanish, and we obtain: from the first two terms, the same contribution as from equation rep, no contribution at (1) ; from the third term Ft2)$ all; from the coulomb term a contribution -ZPV, where P now means z
( -)z+m+nWmn,
lmn
and V, defined as the sum of &s--q2 - rs) x (pz+ps+ra)-s/saver all points forwhichp+q+r is odd, has the value@O) 3.754. Thus, for the perfect crystal one again sees that P,vanishes unless & is equal to CJJ~= [4~-7.508es/us]l2~, and for a lattice with an isotopic impuri~ of mass IM’, quite analogous to (5) P=
AM ~2 --uooo 2/L “s-w;
(9)
Readers will be able to derive analogs of this for more involved force schemes or lattice symmetries. It is worth noting that forces between all like atoms (and not only between second neighbors as in the above example) will contribute nothing-a fact related to a similar absence of contributions from such forces to the sum rule for lattice vibrationsfll) For impurities not isotopic in nature but incurring a change in the short range force constants, relations similar to (9) are easily derivable; the righthand sides will then contain not only tisaa, but the u’s of all the atoms to which the impurity is coupled by a changed force constant. Finally, the present method may lend itself to the study of the effects of boundaries on lattice absorption, as boundaries can formally be treated as impurities of a special sort in an infinite lattice. (Observe that even in the derivation of (3), the assumption that all equations are alike-no boundaries-was
TECHNICAL
implicit.) Indeed, the main feature of (5) (the peak caused by the denominator) is mathematically similar to what has been found elsewhere(495) from boundary effects. Acknozuledgement-I thank Dr. R. F. WALLISfor a critical reading of the above. U.S.
Naval Research
H. B. ROSENSTOCK
Laboratory
Washington D.C. References 1. MARADUDINA. A., MONTROLLE. W. and WEISS G. H., Lattice Dynamics in the Harmonic Approximation, Academic Press, New York (1963). 2. ROSENSTOCK H. B. and KLICK C. C., Phys. Rev. 119,
1198(1960). 3. WALLIS R. F. and MARADUDINA. A,, Progr. Theor.
Phys. 24, 1055 (1960). 4. ROSENSTOCK H. B., J. Chem. Phys. 23 2415 (1955). 5. ROSENSTOCK H. B.; J. Chem. Phys. 27, 1194.(1957). 6. DAWBERP. G. and ELLIOTTR. 5.. Proc. Rov. Sot. A273, 222 (1963); Proc. Phy;: Sot., Lo& 81, 453 (1963). 7. MONTROLLE. W. and POTTSR. B., Phys. Rev. 100, 525 (1955). 8. MARADUDINA. and WALLIS R. F., Phys. Rev. 120, 442 (1960). 9. LAX M. and BURSTEIN E., Phys. Rev. 97, 39 (1955). 10. ROSENSTOCK H. B., J. Phys. Chem. Solids 4, 201 (1958). 11. ROSENSTOCK H. B., Phys. Rev. 129, 1959 (1963). BORN M. and MAYERM. G., Handbuch de-r Physik Vol. 24/2, p. 632, Springer, Berlin (1933).
J. Phys.
Chem. Solids
Vol. 26, pp. 933-934.
Energy of the formation of the anion Frenkel pair in calcium fluoride* (Received
12 October
1964; in revised form 1964)
12 November
A CALCULATION haa been made of approximate energies required to form an isolated anion vacancy in CaFa by removing a lattice F- ion to infinity, and to form an interstitial by inserting an F- ion from infinity into the body-center position in an empty cube of F- ions. The Born model of ionic solids was used, including Coulombic, repulsive (BORN-MAYER(~) potential), and polarization (monopole-dipole and dipole-dipole energies). The * Work performed while temporarily attached to the Theoretical Physics Division, United Kingdom Atomic Energy Research Establishment, Harwell, Didcot, England.
933
NOTES
method
of
calculation
was
that
described by the region I ions (interactions treated explicitly) included only the nearest neighbor ions. For the vacancy, region I contained the next nearest neighbors as well. The displacements of the latter were approximated with the MOTT-LITTLETON@) dielectric continuum theory, while the displacements of the nearest neighbor ions were found by minimizing the energy to create the vacancy. The repulsive parameters used were those of REITZ et aZ.@) With these, the Born model is capable of reproducing reasonably well the experimental cohesive energy,(d) inter-ionic spacing,c4) elastic constants,@) and ion-displacement contribution to the dielectric constant.@ They are also consistent, within the Born-Mayer model, with an ionic radius for the Caa+ ion of the same size or slightly larger than that for the F- ion, as in the set given by HUGGINS,@) and in conformity with the electron density maps of WITTE and WOLFE(~) for CaFa. This implies a radius for the F- ion in the range of 1.1 to 1.2 A, as found also in the alkali halides by TOSI and FuMI.@) For the interstitial, the body-center position is not necessarily that of least energy. A complete calculation including displacements of the surrounding ions has not been made for other positions, but some information can be obtained from the Madelung and repulsive energies for insertion of the interstitial into a rigid crystal. The sum of these energies exhibits a single minimum, to within 0.02 eV, centered on the body-centered position. This minimum is very broad, particularly in the (100) direction, toward the cube facecenter. Even if the isolated interstitial occupies the body-center position, it could easily be displaced in this direction, so that association between the interstitial and an immediately adjacent impurity cation of valence higher than 2 should be favored. THARMALINGHAM.(~) For the interstitial,
UREcg) finds from conductivity of association
between
data that the energy
the interstitial
and sub-
stitutional Y3+ ion is about 1 a4 eV. The
results of the calculation,
for the body-
center position, are, Defect Interstitial Vacancy
61
0.080 0.085
82
- 0.023
-%
-1.56 eV 4.27 2.7 eV