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Single-particle model for the frequency dependence of weak infrared absorption in crystals and molecules at T = 0 K
Volume
10, number
I
OPTICS
SINGLE-PARTICLE OF WEAK INFRARED
COMMUNICATIONS
January
MODEL FOR THE FREQUENCY
ABSORPTION
IN CRYSTALS
1974
DEPENDENCE
AND MOLECULES
AT
T =0K
Stanford P. YUKON * Parke Mathematical Laboratories, Carlisle, Massachusetts 01741, USA
and Bernard BENDOW Solid State Sciences Laboratory, Air Force Cambridge Research Laboratories, Bedford, Massachusetts 01730, USA Received
18 October
1973
We obtain a simple expression for the infrared absorption coefficient 01within the single-particle model at zero temperature. For potentials admitting harmonic approximation, a is found to decrease nearly exponentially with increasing frequency for small anharmonicity, with departures for larger anharmonicity. Potentials which do not admit harmonic approximation display a frequency dependence of the power-law variety. The present results in the quantum limit are similar to those obtained in the classical limit by Mills and Maradudin.
Recent measurements (see, e.g. ref. [ 11) have indicated that the infrared absorption coefficient U(W) in crystals decreases nearly exponentially with increasing frequency w, for values of w greater than several times wo, the fundamental lattice resonance frequency. To explain these observations, McGill et al. [2] (MH) introduced a non-interacting cell model of a crystal, with the lattice motion represented by a single particle assigned to each unit cell, and bound by a onedimensional potential V centered within the cell. Clearly, the same approach should serve equally well in describing weak vibrational absorption by molecules and by impurities in crystals. Although the model is admittedly rather artificial, MH demonstrated that with a Morse potential for V, a good fit to Deutsch’s experimental results [l] for (Yin ionic crystals could be obtained. The MH treatment was quantum mechanical; Mills and Maradudin [3] (MM) presented a classical treatment for the same model, valid only in the high-temperature limit, where the results of MH are * Research oratories,
supported by Air Force Cambridge Research LabAFSC,under Contract No. F19628-7 l-C-0142.
retrieved for the Morse case. MM also found that for a number of potentials not admitting a harmonic approximation, (Y- we2, rather than exponentially. The purpose of the present paper is: (a) to demonstrate that at T = 0 K (within the single particle model) a! versus w takes a simple form which is easily obtained without recourse to the lengthier and more complicated developments given previously; and (b) to explore the relation between the functional form of V and the frequency dependence of 01,in the quantum limit. This dependence on functional form is of special interest with respect to various impurities in crystals which are believed to “sit” in sharp-edged potential wells [3]. Following Ziman [4], the dielectric susceptibility at T = 0 K may be written as
x(w) =A
cn w2-,io+ie’ kOl
2%0
(1)
where x, o is the dipole matrix element (0 Ix In) connecting the ground state with the nth excited state, and %O the corresponding frequency difference; 53
Volume
10, number
1
OPTICS COMMUNICATIONS
A is a constant whose explicit form is not of concern here. The absorption coefficient follows directly as
January
I
I
1
I
4nw In1 x(w) Q(W) = -
a
KC
I
II
I
I
x
*
1974
n
I
*
-
.
-
A
-
. . . .
where B = 2rA/~, with c the speed of light, and K the refractive index, which is nearly a constant for frequencies of interest hem. For the most part, one is concerned with discrete energy spectra which characterize strongly binding potentials, and which result in line spectra for the absorption; i.e., (Yconsists of a series of delta-function peaks at w = o,,,” of strength wBlx,,012. In a real crystal the line spectrum is broadened through interactions between cells [5]. Thus one views the envelope of the peaks in the single-particle model as providing a semi-quantitative indication of the frequency dependence of o for more realistic situations, a procedure which has, in fact, found justification in recent studies of the tail absorption applicable to real crystals [5]. From eq. (2) it is evident that all that is required to calculate (Yversus w are the matrix elements xnO. We here consider the results for some simple potentials : (a) 1-D square-well. For a well extending from 0 to a and with infinitely high walls, one has 1H) 1. Then~,~-n/(1-,1~)~. sin (mrx/a), w, O -n-so that for n 9 1, a(w) - we 2. (b) Harmonic oscillator. The matrix element x,, vanishes except for n = m t 1; thus the harmonic oscillator (in any number of dimensions) is characterized by just a single absorption peak at its natural frequency o,,, a well-known result. (c) I-D Morse potential. For V - e- 2bu _ Ie-bu, where u is the dimensionless displacement, one obtains an energy spectrum -(n +$ ++b2(n+$)2, and the results for x nO quoted in ref. [2] for example. In the limit of small anharmonicity (b + 1) one finds lxnt,12 -n.(,b1 1 2) +I, so that LYis very near to exponential for small b and intermediate values of n. Departures from exponential behavior result for larger values of b, as demonstrated graphically in fig. 1, with 01then falling off more slowly than exponential. (d) The potential V = -~ V, coshk2(x/a). Here -(h ~ n)* (where h = +(J8Voa2+1 11, in % 0 54
. l
!G Y g
15 -
E 6 i
lo-
A
. A A A
2
A
I a - 0,892 0 B - 0.282 A a - 0.089 0
1 0
1
I
I
I
I
2
4
6
8
10
I 12
DIMENSIONLESS FREQUENCY
Fig. 1. Logto of the absorption coefficient versus dimensionless frequency for Morse potential, for different values of the anharmonicity parameter b (frequency measured in units of the harmonic approximation frequency wo).
. a
-9.m
A
a -7.w
x
;L -3.728
I .. X .
.
X
x
. .
.
. .
0
I
1
50
im
1
150 DIRNSIWLESS
1
im
1
29J
,
.
XII
FREOUENCY
Fig. 2. Loglo of the absorption coefficient versus dimensionless frequency for cash-‘(u) potential. The dimensionless frequency is defined as 2ma2 w/h.
Volume
10, number
AOJ 0
1
20
OPTICS
40
COMMUNICATIONS
January
1974
60
DIMENSIONLESS FREOUENCY
0
n Fig. 3. Logro of the absorption coefficient versus dimensionless frequency for Coulomb potential, with dotted line indicating power law fit (frequency in units of the Coulomb ionization frequency).
units of fi = m = 1). The xnO are computed numerically using the wave functions displayed in ref. [6]. Results for a(w), which display an exponential-like behavior, are illustrated in fig. 2. (e) (3-D) Coulomb potential. In this case we investigate just the continuous absorption spectrum; numerical results computed employing the wave functions given in ref. [7] are illustrated in fig. 3. The absorption is seen to be close to a power law tin with n--2. (f)Spherical well. The expression for the matrix elements for this case are computed using the wave functions given in ref. [6, p. 1551, and the results for a(w) illustrated in fig. 4. The behavior is close to a power law wn with n - - 2. The above results suggest that exponential-like decreases in absorption at high frequencies above the fundamental are a general feature of potentials which admit harmonic approximations. With increasing anharmonicity o begins to deviate from exponential behavior. For potentials which do not admit the harmonic approximation at all, such as the square well or Coulomb, say, the absorption is of the power law variety. MM obtained essentially the same results for the 1-D potentials they investigated in the classical
I
I
10
20
I
30
I
I
40
50
1 60
DIMENSIONLESS FREQUENCY
Fig. 4. Logre of the absorption coefficient versus dimensionless frequency for the spherical well, with dotted line indicating power law fit (frequency measured in units of wto).
limit. It is thus probable that the above conclusions hold (within the single-particle model) at all temperatures ranging from the quantum to the classical limit.
References [II F.A. Horrigan
and T.F. Deutsch, Quarterly Tech. Report No. 2 (Raytheon, Waltham, Mass., 1972); A.J. Barker, J. Phys. CS (1972) 2276. 121 T.C. McGill, R.W. Hellwarth, M. Mangir and H.V. Winston, J. Phys. Chem. Sol. (in press). 131 D.L. Mills and A.A. Maradudin, Phys. Rev. B8 (1973) 1617. [41 J.M. Ziman, Principles of the theory of solids (Cambridge University Press, England, 1969). 151 M. Sparks and L.J. Sham, Sol. State Comm. 11 (1972) 1451 and Phys. Rev. B (in press); B. Bendow, S.C. Ying and S.P. Yukon, Phys. Rev. B8 (1973) 1689; B. Bendow, Phys. Rev. B (in press); K.V. Namjoshi and S.S. Mitra (unpublished communication). I61 S. Flugge, Practical quantum mechanics I (Springer Verlag, Berlin, 1971) p. 94. I71 L.Y.C. Chiu, Phys. Rev. 154 (1967) 56.
10, number
I
OPTICS
SINGLE-PARTICLE OF WEAK INFRARED
COMMUNICATIONS
January
MODEL FOR THE FREQUENCY
ABSORPTION
IN CRYSTALS
1974
DEPENDENCE
AND MOLECULES
AT
T =0K
Stanford P. YUKON * Parke Mathematical Laboratories, Carlisle, Massachusetts 01741, USA
and Bernard BENDOW Solid State Sciences Laboratory, Air Force Cambridge Research Laboratories, Bedford, Massachusetts 01730, USA Received
18 October
1973
We obtain a simple expression for the infrared absorption coefficient 01within the single-particle model at zero temperature. For potentials admitting harmonic approximation, a is found to decrease nearly exponentially with increasing frequency for small anharmonicity, with departures for larger anharmonicity. Potentials which do not admit harmonic approximation display a frequency dependence of the power-law variety. The present results in the quantum limit are similar to those obtained in the classical limit by Mills and Maradudin.
Recent measurements (see, e.g. ref. [ 11) have indicated that the infrared absorption coefficient U(W) in crystals decreases nearly exponentially with increasing frequency w, for values of w greater than several times wo, the fundamental lattice resonance frequency. To explain these observations, McGill et al. [2] (MH) introduced a non-interacting cell model of a crystal, with the lattice motion represented by a single particle assigned to each unit cell, and bound by a onedimensional potential V centered within the cell. Clearly, the same approach should serve equally well in describing weak vibrational absorption by molecules and by impurities in crystals. Although the model is admittedly rather artificial, MH demonstrated that with a Morse potential for V, a good fit to Deutsch’s experimental results [l] for (Yin ionic crystals could be obtained. The MH treatment was quantum mechanical; Mills and Maradudin [3] (MM) presented a classical treatment for the same model, valid only in the high-temperature limit, where the results of MH are * Research oratories,
supported by Air Force Cambridge Research LabAFSC,under Contract No. F19628-7 l-C-0142.
retrieved for the Morse case. MM also found that for a number of potentials not admitting a harmonic approximation, (Y- we2, rather than exponentially. The purpose of the present paper is: (a) to demonstrate that at T = 0 K (within the single particle model) a! versus w takes a simple form which is easily obtained without recourse to the lengthier and more complicated developments given previously; and (b) to explore the relation between the functional form of V and the frequency dependence of 01,in the quantum limit. This dependence on functional form is of special interest with respect to various impurities in crystals which are believed to “sit” in sharp-edged potential wells [3]. Following Ziman [4], the dielectric susceptibility at T = 0 K may be written as
x(w) =A
cn w2-,io+ie’ kOl
2%0
(1)
where x, o is the dipole matrix element (0 Ix In) connecting the ground state with the nth excited state, and %O the corresponding frequency difference; 53
Volume
10, number
1
OPTICS COMMUNICATIONS
A is a constant whose explicit form is not of concern here. The absorption coefficient follows directly as
January
I
I
1
I
4nw In1 x(w) Q(W) = -
a
KC
I
II
I
I
x
*
1974
n
I
*
-
.
-
A
-
. . . .
where B = 2rA/~, with c the speed of light, and K the refractive index, which is nearly a constant for frequencies of interest hem. For the most part, one is concerned with discrete energy spectra which characterize strongly binding potentials, and which result in line spectra for the absorption; i.e., (Yconsists of a series of delta-function peaks at w = o,,,” of strength wBlx,,012. In a real crystal the line spectrum is broadened through interactions between cells [5]. Thus one views the envelope of the peaks in the single-particle model as providing a semi-quantitative indication of the frequency dependence of o for more realistic situations, a procedure which has, in fact, found justification in recent studies of the tail absorption applicable to real crystals [5]. From eq. (2) it is evident that all that is required to calculate (Yversus w are the matrix elements xnO. We here consider the results for some simple potentials : (a) 1-D square-well. For a well extending from 0 to a and with infinitely high walls, one has 1H) 1. Then~,~-n/(1-,1~)~. sin (mrx/a), w, O -n-so that for n 9 1, a(w) - we 2. (b) Harmonic oscillator. The matrix element x,, vanishes except for n = m t 1; thus the harmonic oscillator (in any number of dimensions) is characterized by just a single absorption peak at its natural frequency o,,, a well-known result. (c) I-D Morse potential. For V - e- 2bu _ Ie-bu, where u is the dimensionless displacement, one obtains an energy spectrum -(n +$ ++b2(n+$)2, and the results for x nO quoted in ref. [2] for example. In the limit of small anharmonicity (b + 1) one finds lxnt,12 -n.(,b1 1 2) +I, so that LYis very near to exponential for small b and intermediate values of n. Departures from exponential behavior result for larger values of b, as demonstrated graphically in fig. 1, with 01then falling off more slowly than exponential. (d) The potential V = -~ V, coshk2(x/a). Here -(h ~ n)* (where h = +(J8Voa2+1 11, in % 0 54
. l
!G Y g
15 -
E 6 i
lo-
A
. A A A
2
A
I a - 0,892 0 B - 0.282 A a - 0.089 0
1 0
1
I
I
I
I
2
4
6
8
10
I 12
DIMENSIONLESS FREQUENCY
Fig. 1. Logto of the absorption coefficient versus dimensionless frequency for Morse potential, for different values of the anharmonicity parameter b (frequency measured in units of the harmonic approximation frequency wo).
. a
-9.m
A
a -7.w
x
;L -3.728
I .. X .
.
X
x
. .
.
. .
0
I
1
50
im
1
150 DIRNSIWLESS
1
im
1
29J
,
.
XII
FREOUENCY
Fig. 2. Loglo of the absorption coefficient versus dimensionless frequency for cash-‘(u) potential. The dimensionless frequency is defined as 2ma2 w/h.
Volume
10, number
AOJ 0
1
20
OPTICS
40
COMMUNICATIONS
January
1974
60
DIMENSIONLESS FREOUENCY
0
n Fig. 3. Logro of the absorption coefficient versus dimensionless frequency for Coulomb potential, with dotted line indicating power law fit (frequency in units of the Coulomb ionization frequency).
units of fi = m = 1). The xnO are computed numerically using the wave functions displayed in ref. [6]. Results for a(w), which display an exponential-like behavior, are illustrated in fig. 2. (e) (3-D) Coulomb potential. In this case we investigate just the continuous absorption spectrum; numerical results computed employing the wave functions given in ref. [7] are illustrated in fig. 3. The absorption is seen to be close to a power law tin with n--2. (f)Spherical well. The expression for the matrix elements for this case are computed using the wave functions given in ref. [6, p. 1551, and the results for a(w) illustrated in fig. 4. The behavior is close to a power law wn with n - - 2. The above results suggest that exponential-like decreases in absorption at high frequencies above the fundamental are a general feature of potentials which admit harmonic approximations. With increasing anharmonicity o begins to deviate from exponential behavior. For potentials which do not admit the harmonic approximation at all, such as the square well or Coulomb, say, the absorption is of the power law variety. MM obtained essentially the same results for the 1-D potentials they investigated in the classical
I
I
10
20
I
30
I
I
40
50
1 60
DIMENSIONLESS FREQUENCY
Fig. 4. Logre of the absorption coefficient versus dimensionless frequency for the spherical well, with dotted line indicating power law fit (frequency measured in units of wto).
limit. It is thus probable that the above conclusions hold (within the single-particle model) at all temperatures ranging from the quantum to the classical limit.
References [II F.A. Horrigan
and T.F. Deutsch, Quarterly Tech. Report No. 2 (Raytheon, Waltham, Mass., 1972); A.J. Barker, J. Phys. CS (1972) 2276. 121 T.C. McGill, R.W. Hellwarth, M. Mangir and H.V. Winston, J. Phys. Chem. Sol. (in press). 131 D.L. Mills and A.A. Maradudin, Phys. Rev. B8 (1973) 1617. [41 J.M. Ziman, Principles of the theory of solids (Cambridge University Press, England, 1969). 151 M. Sparks and L.J. Sham, Sol. State Comm. 11 (1972) 1451 and Phys. Rev. B (in press); B. Bendow, S.C. Ying and S.P. Yukon, Phys. Rev. B8 (1973) 1689; B. Bendow, Phys. Rev. B (in press); K.V. Namjoshi and S.S. Mitra (unpublished communication). I61 S. Flugge, Practical quantum mechanics I (Springer Verlag, Berlin, 1971) p. 94. I71 L.Y.C. Chiu, Phys. Rev. 154 (1967) 56.