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A frequency spectrum moment method for the study of the vibrational properties of amorphous substances
VOIume 14, number 2
15 May 1972
CCIEMICAL PHYSICS LEEERS
A FREQtJENCV SPECTRUM MOMENT METHOD FOR THE STUDY OF THE VIBRATIONAL PROPERTIES OF AMORPHOUS SUBSTANCES 0. WERES* Deparrmcnt of Chemistry and The Jurnes Franck insfitute. Chimgo, Ilitiois Received
Uliversiry
of Chicago,
60637, US.4
6 hlarch 1972
A previously dcveloprd method for generating the moments of the frequency spectra df cdbichtliozs is extended to amorphoos substimes. The extension invokes little approximation and allows one to calculate the moments in terms of the interatomic potent&4 function and low-order atomic distribution functions. A frequency spectrum cluster ehansion technique with well defined Lirnits of error may be derived from it.
Our method isbased on the meehod of generating the even moments of the frequency spectra of simple cubic lattices developed by Litzman and Klvava [l] . Their fundamental result was that the equations of motion for such lattices may be written in the form
cy spectrum
pti =(l/M”)Tr(D”)=
Q = (x~‘~2’....~‘~.YI,Y~,...:_v~~t,“~.....z_~) The (VP) are t_bequadratic coupling atoms whose relative lattice positions terms of the primitive vectors are (p) null vector corresponding to V*, the field) x52P2
(Ijni”)
c
Tr(WW’2...)
X Tr(@‘* -@‘2-__) ,,
where Q is thp vector of displacements from equilibrium positicns
@P=flPl
as a function of w to be ex-
PI.Pz-..-*Pn
~~~~(V~X~)--14v~~~~zJ.B,
0=
represented
pressed as
-
tensors linking expressed in (inchiding the Einstein
where D is the dynamical matrix. It is easy to show that the second factor of each term is non-zero onIy if 5p.4. i=] ’ The equations of motion may be rearranged to yield
mean
XQP3,
where ~1, j2, ~3 ae
where the
componenk
of the givenp,
and ti= 16ii+l! , and G *is the set of vectors p associated with non-neg--
ligible W’s_ This form allows the
even moments
.* Fan&e and John Hertz Foundation
Fellow.
ofthe
frequen-
Q’= (~,,~~,z~.-..,~~,~~~,z~~.
’
The expressions for the even moments do not change. The property of the dynamical matrix in the latter representation which leads to the above expressions for the even moments is its block structures not.the constancy of VP,for all pairs of atoms with a given relative displacement p as assumed in &e.ori@ial derivahon. Indeed; @’ may be different fcr e.ach such pair. 155’
Volume
i4, number 2
CHEMICAL
PHYSICS
The p~~sibiii~ of variabfe M may be tribally dealt with by suitabtc averaging of the factors I/&p. The above result allows one to treat an arbitrary mixed lattice. The transition to an amorphous substance is niade 5y assuming an arbitrarily Gne-grained lattice over which the atoms of the given substance are distributed, fXing a very smaIf. fraction of the lattice Fositions. The remaining lattice positions are filled with a hypothetical second ccmponeni’, the atoms of which are assumed io interact very weakly with those of the substance and each other. If these interactions are further weakened so as to become infinitesimal, the second corn~l~Ii~nt becomes an interpenetrating ideal gas and all contributions to the moments arising from sequences of vectors which include one or more vectors which connect a lattice position occupied by a second component atom go to zero. The result is a set of expressions for the even moments of the frequency spectrum of a glass which are exact and which contain no assumptions of structural regularity. The accu:acy with which various quantities may be calculated from the-m may be estimated from the number of moments empioyed. Their values are obviously convergent as the number of moments employed increases. in what follows we restrict the discussion to the case of an amorphous, monoisotopic element; the exiension to arbitrary glasses is simple. It is obvious that a closed sequence of n fixed vectors connecting atoms can be described in terms of the positions of n atoms {or fewer, since 3 vector sequence may contain nulf vectors or it may cross partialIy or fully retrace itself). Of course, any such cluster of atoms corresponds to several vector sequences due to the possibility of connecting the atoms in several different ways and starting the tracing from any vector in the sequence and in either direction. It is easy to demonstrate that the appropriate linear combinations of the frequency spectra of the appropriate clusters of up to and including tt atoms calculated wiih the remaining atoms fixed in their equilibrium positions gives the same expressions for the even moments of up to and inchading the Znth. if attention is restricted to thus quantities whic?l may be expressed as inte. yrals over the sycctrum, the appPOpiiatC integraIs over the component cluster spectra may be replaced by phase integrals, thus yielding a general cluster expansion. A simpfe app:o:timation fe3ds to a W!iy convenient IS6
LETTERS
1.5 May 1972
form for the moment expressions. TMs approx~ma~on is defined by the assumption that V0 is spherically symmetrical, which is certainly reasonable in the case of tetrahedrally bound atoms. Then the contributions corresponding to any given cluster depend only upon the relative positions of the atoms within it, the assumed ~ntermolecu1~ potential, and the assumed mean field. The effect of a null vector is now to simply introduce a scalar factor into the given term. The contribution of a given cluster to the 32th moment, now takes the form
where the subscript i ranges over the various vector sequences corresponding to the cluster which make distinct numericai contributions, wi is the degeneracy factor, rrzi is the number of null vectors in the ith sequence, and s is the force constant of the mean field. Summing this over the singlets, doublets, .. . tl-tupiets and integrating over the probability distributions of the internal coordinates of the clusters gives the full value of the 21th moment. A particularly simple way of carrying out the integration step would be to partition the ir-tupfet phase space into regions conesponding to distinct arrangements of covaIent bonds among the tz atoms and then approximate the configurational distributions of ?he various subtypes using gaussian distributions of interatomic distances and bond angles fitted to the experimentaf radial distribution function. With the exception of integrating over the subtype’s configurational distributions, we have carried out an analogous calculation for liquid water with tt = 5 with. out any great difficulty. The positions of the two distinct peaks which appear in the appropriate portion of the neutron diffraction; and IR spectra were accurately reproduced [ 2j . I wish to thank the Hertz Foundation, the Air Force Office of Scientific Research and t!ze Advanced Research Projects Agency for supporting this work and Dr. Stuart A. Rice for helpful discussions. References Litzman and F. Klwva, Phys. Stat. Sol. 2 (1962) 121 0. Weres and S.A. Rice, to be published.
111 0.
42.
15 May 1972
CCIEMICAL PHYSICS LEEERS
A FREQtJENCV SPECTRUM MOMENT METHOD FOR THE STUDY OF THE VIBRATIONAL PROPERTIES OF AMORPHOUS SUBSTANCES 0. WERES* Deparrmcnt of Chemistry and The Jurnes Franck insfitute. Chimgo, Ilitiois Received
Uliversiry
of Chicago,
60637, US.4
6 hlarch 1972
A previously dcveloprd method for generating the moments of the frequency spectra df cdbichtliozs is extended to amorphoos substimes. The extension invokes little approximation and allows one to calculate the moments in terms of the interatomic potent&4 function and low-order atomic distribution functions. A frequency spectrum cluster ehansion technique with well defined Lirnits of error may be derived from it.
Our method isbased on the meehod of generating the even moments of the frequency spectra of simple cubic lattices developed by Litzman and Klvava [l] . Their fundamental result was that the equations of motion for such lattices may be written in the form
cy spectrum
pti =(l/M”)Tr(D”)=
Q = (x~‘~2’....~‘~.YI,Y~,...:_v~~t,“~.....z_~) The (VP) are t_bequadratic coupling atoms whose relative lattice positions terms of the primitive vectors are (p) null vector corresponding to V*, the field) x52P2
(Ijni”)
c
Tr(WW’2...)
X Tr(@‘* -@‘2-__) ,,
where Q is thp vector of displacements from equilibrium positicns
@P=flPl
as a function of w to be ex-
PI.Pz-..-*Pn
~~~~(V~X~)--14v~~~~zJ.B,
0=
represented
pressed as
-
tensors linking expressed in (inchiding the Einstein
where D is the dynamical matrix. It is easy to show that the second factor of each term is non-zero onIy if 5p.4. i=] ’ The equations of motion may be rearranged to yield
mean
XQP3,
where ~1, j2, ~3 ae
where the
componenk
of the givenp,
and ti= 16ii+l! , and G *is the set of vectors p associated with non-neg--
ligible W’s_ This form allows the
even moments
.* Fan&e and John Hertz Foundation
Fellow.
ofthe
frequen-
Q’= (~,,~~,z~.-..,~~,~~~,z~~.
’
The expressions for the even moments do not change. The property of the dynamical matrix in the latter representation which leads to the above expressions for the even moments is its block structures not.the constancy of VP,for all pairs of atoms with a given relative displacement p as assumed in &e.ori@ial derivahon. Indeed; @’ may be different fcr e.ach such pair. 155’
Volume
i4, number 2
CHEMICAL
PHYSICS
The p~~sibiii~ of variabfe M may be tribally dealt with by suitabtc averaging of the factors I/&p. The above result allows one to treat an arbitrary mixed lattice. The transition to an amorphous substance is niade 5y assuming an arbitrarily Gne-grained lattice over which the atoms of the given substance are distributed, fXing a very smaIf. fraction of the lattice Fositions. The remaining lattice positions are filled with a hypothetical second ccmponeni’, the atoms of which are assumed io interact very weakly with those of the substance and each other. If these interactions are further weakened so as to become infinitesimal, the second corn~l~Ii~nt becomes an interpenetrating ideal gas and all contributions to the moments arising from sequences of vectors which include one or more vectors which connect a lattice position occupied by a second component atom go to zero. The result is a set of expressions for the even moments of the frequency spectrum of a glass which are exact and which contain no assumptions of structural regularity. The accu:acy with which various quantities may be calculated from the-m may be estimated from the number of moments empioyed. Their values are obviously convergent as the number of moments employed increases. in what follows we restrict the discussion to the case of an amorphous, monoisotopic element; the exiension to arbitrary glasses is simple. It is obvious that a closed sequence of n fixed vectors connecting atoms can be described in terms of the positions of n atoms {or fewer, since 3 vector sequence may contain nulf vectors or it may cross partialIy or fully retrace itself). Of course, any such cluster of atoms corresponds to several vector sequences due to the possibility of connecting the atoms in several different ways and starting the tracing from any vector in the sequence and in either direction. It is easy to demonstrate that the appropriate linear combinations of the frequency spectra of the appropriate clusters of up to and including tt atoms calculated wiih the remaining atoms fixed in their equilibrium positions gives the same expressions for the even moments of up to and inchading the Znth. if attention is restricted to thus quantities whic?l may be expressed as inte. yrals over the sycctrum, the appPOpiiatC integraIs over the component cluster spectra may be replaced by phase integrals, thus yielding a general cluster expansion. A simpfe app:o:timation fe3ds to a W!iy convenient IS6
LETTERS
1.5 May 1972
form for the moment expressions. TMs approx~ma~on is defined by the assumption that V0 is spherically symmetrical, which is certainly reasonable in the case of tetrahedrally bound atoms. Then the contributions corresponding to any given cluster depend only upon the relative positions of the atoms within it, the assumed ~ntermolecu1~ potential, and the assumed mean field. The effect of a null vector is now to simply introduce a scalar factor into the given term. The contribution of a given cluster to the 32th moment, now takes the form
where the subscript i ranges over the various vector sequences corresponding to the cluster which make distinct numericai contributions, wi is the degeneracy factor, rrzi is the number of null vectors in the ith sequence, and s is the force constant of the mean field. Summing this over the singlets, doublets, .. . tl-tupiets and integrating over the probability distributions of the internal coordinates of the clusters gives the full value of the 21th moment. A particularly simple way of carrying out the integration step would be to partition the ir-tupfet phase space into regions conesponding to distinct arrangements of covaIent bonds among the tz atoms and then approximate the configurational distributions of ?he various subtypes using gaussian distributions of interatomic distances and bond angles fitted to the experimentaf radial distribution function. With the exception of integrating over the subtype’s configurational distributions, we have carried out an analogous calculation for liquid water with tt = 5 with. out any great difficulty. The positions of the two distinct peaks which appear in the appropriate portion of the neutron diffraction; and IR spectra were accurately reproduced [ 2j . I wish to thank the Hertz Foundation, the Air Force Office of Scientific Research and t!ze Advanced Research Projects Agency for supporting this work and Dr. Stuart A. Rice for helpful discussions. References Litzman and F. Klwva, Phys. Stat. Sol. 2 (1962) 121 0. Weres and S.A. Rice, to be published.
111 0.
42.