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Mössbauer effect and solitary wave in one-dimensional molecular crystals at T ≠ 0
Physica 135A (1986) 446-454 North-Holland, Amsterdam
MiiSSBAUER
EFFECT
AND SOLITARY
MOLECULAR F. GASH1
AT T # 0
CRYSTALS and R. GASH1
Faculty of Natural Sciences.
B. STEPANtIC
WAVE IN ONE-DIMENSIONAL
Pri.Ctina, Yugoslavia
and R.B.
Institute of Nuclear Sciences “Boris
iAKULA
KidriC”, Beograd.
Yugoslaviu
Received 14 July 1983 In final form 25 July 1985
An attempt is made to analyse the influence of a solitary wave on the Mksbauer effect in a y-active atom situated in a one-dimensional crystal lattice mode. The expression for the y-radiative transition probability is derived, the y-quantum being exactly equal to the molecular excitation energy, i.e. in the case when the molecular chain does not change its state in the course of the emission. This state is a solitary wave solution of the nonlinear Schradinger equation, the crystal system being in contact with a thermostat at T#O. It is shown that, in general, the transition probability is lower than in the case when the crystal state is adequately described exclusively by thermal phonon modes. The analytical expression obtained opens the possibility to compute the transition probability as a function of temperature in the MGssbauer effect in our soliton case.
0. Introduction The object of this paper is to analyse and to derive the transition probability for the process in which the molecular chain remains in its initial quantum solitary
wave state after the y-quantum
emission
of the active nucleus
situated
at
the site no of the linear crystal lattice. This effect, is really a MGssbauer effect, and refers to purely phonon states’.2), but here we analyse a special form of the excitation localized in a region of the one-dimensonal lattice (in the neighbourhood of the site n,,), known as solitons. As is well known in the case of a linear molecular chain, solitons are rather stable quasiparticles appearing in the result of the coupling of an exciton with a phonon, due to an exciton-phonon interaction”-5). The domain of the crystal lattice where the soliton is present is subjected to a deformation traveling along the lattice with a velocity u < u,,, where I+ is the sound velocity. In the course of our analysis we find that the 0378-4371/86/$03.50 (Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
MbSSBAUER
EFFECT AND SOLITARY WAVE
447
crystal is in equilibrium with a thermal bath having T # 0. So that we use the solutions referring to solitons coupled with thermal phonons as described by Davydov in ref. 6.
1. The Hamiltonian
of the system
The linear molecular chain is described by Hamiltonian
H = He, + Hph+ Hi”,
(1.1)
7
where in the simplified model the exciton Hamiltonian
K, =
is
c[(go +-$)ml+g2 (W,+, + P:+,P.)] . n
(1.2)
0
0
go is the energy of the exciton zone. In the simplified two-level model, the excitons are described by creation and annihilation operators P,' , P, obeying the Pauli commutation rules [P,, P,‘,] = 6,,.(1-2P,+P,).
(1.3)
m is effective mass of the free electron defining the excitation levels; R, is the crystal lattice constant (1.4) in the longitudinal phonon Hamiltonian, where bi, b, are the creation and annihilation phonon operators having wave vectors q, 0, = uol q( is the phonon frequency, while u. = R06?%l is the sound velocity. M is the molecular mass in the node; K is the elasticity constant of the crystal. The Hamiltonian of the exciton-phonon interaction is described by Hint
=
&z
F(q)PzP,(b,
+ bzg)einRoq.
F(q) is the intensity of the exciton-phonon F(q) = i-($-$-)“2
fi
,
a=2R,x.
(1.5)
interaction with constant coupling x,
(1.6)
At T # 0 following the lines of ref. 6 we ask for the state of the system in the
F. GASH1
448
form of the Schrodinger
et al.
amplitude (1.7)
where v are phonon states given by the expression
(1.8) and where IO,,>, IO,,, > are the exciton and phonon vacuum states respectively. The unitary equilibrium mode displacement operator appearing in the result of the exciton-phonon coupling has the form u,(t) = exp{T
P,,
[p;,(t)b,
are chosen as modulated
P,,(t) = p,,,(t)
-
&,(r)b,‘l)
(1.9)
waves,
emi”“‘1y,
where the cp,(t) is normalised
(1.10) by the condition
In the formalism q,,(t) and P,,,(t) play the role of dynamical variables and are determined from the Hamiltonian equations (1.12a)
(1.12b) (H) is the functional averaged Hamiltonian function, i.e.
(H) =
cI’ ~,,Hvv
over phonon
states playing the role of the
3
p,, is the diagonal element of the phonon density matrix operator
(1.13)
MZjSSBAUER EFFECT AND SOLITARY
p
WAVE
449
(vJeeHph”lv)
=
(1.15)
Y”
7 (YJ ePHP”“lv) ’ From the Hamiltonian equation system (l.l2a,b) the nonlinear Schrodinger equation (referring to cp(x, t) and having a cubic nonlinearity) may be derived in the continuum approximation ifi % (x, t) -
@&+ -&
(1 - e-“)I&,
t) + --&
e-“R,f, ‘*l($’ l’ 0
+ G((P(*~J(x, t) = 0.
(1.16)
The particular solution of (1.16) is a solitary wave depending on the variable 5 = x - ut, where x is the position along the molecular chain. The explicit soliton solution has the form6)
’
exp k(x -x0)
112
+ &
‘(X,f)=[Roa2(eT’l
(a’(0
) - k*)t - 2
ch[a(e,)(x-x,Lut),
’
t -
(1.17) where
R,&*(O)
,+y,
a(&)= 3
G(8,) =
B=Z.
mG(%)exp
u2 ML$l - s2)
4
(Y(0) =
mRoa2 2Mv;fi*( 1 - s’) ’
(1.18)
Rima2
’ fm = 2 $ C4 i4lu + 2i,),
Ro 4MV31 - s2)h
with iq =
1 fr$lB7 _ 1 ’ e
,=v,
t$=k,T.
(1.18a)
UO
The expression e -’ IS ’ the Debye-Waller factor. The temperature effect in the case of solitons is such that the raising of temperature provokes a spreading of the soliton. The consequence of this fact is that the nonlinearity coefficient G tends to zero with the temperature, so that eq. (1.10) becomes linear and describes in a certain interval of the temperature a plane wave.
F. GASH1
450
2. The transition
probability
et al.
in the simplified
model
The probability of emission of a y-quantum from the nucleus situated in crystal lattice node n,, is calculated from the matrix element of the operator V,,, =
eiP.Rndfi . Qij, aj, ij) .
(2.1)
P is the momentum belonging to the emitted phonon, R,,, is the radius vector of the center of mass of crystal node, for which we assume that (2.la) where u “Dis the displacement of the emitting molecule from its node. The operator c(ij, cj, ej) depends on the coordinate, momentum and spin belonging to particle j constituting the y-active nucleus. It is not necessary to know explicitly the form of this operator. For the same reason the interaction affects only the movement of the center of mass of crystal nodes, so by the y-active molecule the number of internal degrees of freedom is fixed. The transition probability amplitude corresponding to the emission of the y-quantum is the following:
(2.2) (fl&j, pj, ej)li) is th e molecular matrix element depending on the internal structure. It will be treated as having a fixed value due to the reasons already mentioned. Since the matrix element (fl c(.G,, -i),, &,)li) is constant, it is not interesting any more for our analysis as we are interested only in relative transition probabilities determined by the matrix element
i.e. by the transition amplitude in our case when the same soliton is found in the crystal before and after the emission. The plane wave exp{iP *R,olfi} can be treated as an interaction operator initiating the transition. It can be written as
(2.4) umO expressed in terms of creation and annihilation phonon operators b T2and b2 has the form
MCiSSBAUER EFFECT AND SOLITARY WAVE
451
112 I.4
e,(b,
= “0
+ b_+,) eiqRono
(2.5)
when N is the total number of sites in the crystal chain and e, is the polarization vector of longitudinal phonons. Taking into account the definition of lFV) eq. (2.3) can be rewritten as (2.6)
where M,.(n)=(vlO,(r)exp[~~
x fiNI
(&)1’2(~*eq)(by+b~q)eiqR0no] 4
.
(2.6a)
In (2.6a) the phonon matrix element can be rewritten in the form
M,,(n) =
nM,,W 4
(2.7) where y, = i (*)“2(p-eq)eiqRnno,
(2.8)
with the characteristic y_q = - y ,*. We are interested in the average value of the matrix element (2.6) over phonon states. It is given by
(2.9) where the soliton part of matrix element
(MsoAph = exp( -
( Mso,)ph is (2.10)
F. GASH1
452
where expressions
are obtained
(2.7)-(2.1(l)
et al
by the standard
procedure
applying
Weyl’s identity
e” e li
= ,“/*‘[A.
icl
eii +ri
(2.11)
for [A[A, 8]] = [B[A, 8]] =O. Since P,,,(t) is defined as a modulated
plane
wave, (2.12)
the solution
(1.12a)
is found
in the form’)
(2.13a)
(2.13b)
The expression for the transition probability amplitude reduces to a simplified form if the origin of the coordinate system is put at the site o,~ where the y-active atom is situated. Now the transition probability amplitude (2.6) is
It is clear that ~x&,(P* e,)qlp,,(t)l’ j&x Mv;?,hlq12(1 -sq 4 vanishes
since the function
cos qR,,n
of q:
d p * eq1~0sq&n is odd. In the continuum
approximation
we have C, --$ (1
lR,,) (Tzdx and in the
MijSSBAUER
EFFECT AND SOLITARY WAVE
453
long-wave approximation sin qR,n + qR,n + qx. In accordance with criteria in paper 6 we can put ((pn(*+ i aR, in the exponent of formula (2.14), so
(A%,,, =
exp{ -T (p%):(cq+ f)) 2MNno
+m
e -cc
iQx
(2.15)
ch’ C+ - x0 - u,) ’
where
(2.16) while the probability
of its amplitude is
1
?T’Q’
= a
sh2
The argument of the soliton part of the transition probability
rQ(v) -=-
244)
rr
xR;
2 M@(l
(2.17)
Qm exp1 2ff (4)
-.s2)
LC(p*eq). N 4
(2.17) is (2.18)
As one can see, the dependence of the probability IV+ on the coupling constant x has the same importance as for the existence of Davydov’s solitons, i.e., when x ---, 0 we get the term containing typically the pure phonon form factor only. From (2.17) it follows that in the absence of emission ( p = 0) one has Wi+ = 1 which proves that the expression for the probability has been adequately normalized.
3. Conclusion If we admit that the molecular chain is populated exclusively by solitons, the Mossbauer effect, generally speaking, is less probable then in the usual and well-known pure phonon modes that are simultaneously spread over the whole crystal. The latter case leads to the probability (2.17) having the factor
F. GASHI
454
et al
(3.1)
In the former
soliton
case the smaller
probability
is manifested
by the factor
(3.2)
which is to be multiplied
by the factor corresponding
to the pure phonon
states.
As one can see, the factor (3.2) has the typically localized character of soliton excitations. As to the effect of the temperature on the transition probability of the Mossbauer effect it is manifested through the factor I?, = [eh”q’f’ - 11-l in the case of pure phonon states, while in long-wave approximation (eq. (2.13)(2.18)) the soliton form factor (3.2) does not depend on the temperature. From the formula (2.17) it is also visible that static solitons (u = 0) have little impact on the transition sound velocity decrease
probability. The solitons having velocities close to the the Mossbauer effect transition probability significantly.
Acknowledgement The authors valuable
would
like to express
to Dr. M. Marinkovic
their gratitude
for
discussions.
References 1) R.P. Feynman, Statistical Mechanics (a set of Lectures), Reading, Massachusetts 1972), pp. 22-25. 2) H.J. Lipkin, Quantum Mechanics 3) A.S. Davydov and N.T. Kislukha,
Advanced
(North-Holland, Amsterdam, SW. Phys. JETP 71 (1976)
Book
Program
1973) chaps. 1090.
(Benjamin,
3 and 4.
4) V. Fedyanin and L.V. Yakushevich, Theor. Mat. Fiz. 37 (1978) 371. 5) AS. Davydov, Solitons in the Molecular Systems (Naukova Dumka, Kiev, 1984) (in Russian) 1, p. 18. 6) A.S.
Davidov,
Sov. Phys.
JETP
78 (1980)
789.
chap.
MiiSSBAUER
EFFECT
AND SOLITARY
MOLECULAR F. GASH1
AT T # 0
CRYSTALS and R. GASH1
Faculty of Natural Sciences.
B. STEPANtIC
WAVE IN ONE-DIMENSIONAL
Pri.Ctina, Yugoslavia
and R.B.
Institute of Nuclear Sciences “Boris
iAKULA
KidriC”, Beograd.
Yugoslaviu
Received 14 July 1983 In final form 25 July 1985
An attempt is made to analyse the influence of a solitary wave on the Mksbauer effect in a y-active atom situated in a one-dimensional crystal lattice mode. The expression for the y-radiative transition probability is derived, the y-quantum being exactly equal to the molecular excitation energy, i.e. in the case when the molecular chain does not change its state in the course of the emission. This state is a solitary wave solution of the nonlinear Schradinger equation, the crystal system being in contact with a thermostat at T#O. It is shown that, in general, the transition probability is lower than in the case when the crystal state is adequately described exclusively by thermal phonon modes. The analytical expression obtained opens the possibility to compute the transition probability as a function of temperature in the MGssbauer effect in our soliton case.
0. Introduction The object of this paper is to analyse and to derive the transition probability for the process in which the molecular chain remains in its initial quantum solitary
wave state after the y-quantum
emission
of the active nucleus
situated
at
the site no of the linear crystal lattice. This effect, is really a MGssbauer effect, and refers to purely phonon states’.2), but here we analyse a special form of the excitation localized in a region of the one-dimensonal lattice (in the neighbourhood of the site n,,), known as solitons. As is well known in the case of a linear molecular chain, solitons are rather stable quasiparticles appearing in the result of the coupling of an exciton with a phonon, due to an exciton-phonon interaction”-5). The domain of the crystal lattice where the soliton is present is subjected to a deformation traveling along the lattice with a velocity u < u,,, where I+ is the sound velocity. In the course of our analysis we find that the 0378-4371/86/$03.50 (Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
MbSSBAUER
EFFECT AND SOLITARY WAVE
447
crystal is in equilibrium with a thermal bath having T # 0. So that we use the solutions referring to solitons coupled with thermal phonons as described by Davydov in ref. 6.
1. The Hamiltonian
of the system
The linear molecular chain is described by Hamiltonian
H = He, + Hph+ Hi”,
(1.1)
7
where in the simplified model the exciton Hamiltonian
K, =
is
c[(go +-$)ml+g2 (W,+, + P:+,P.)] . n
(1.2)
0
0
go is the energy of the exciton zone. In the simplified two-level model, the excitons are described by creation and annihilation operators P,' , P, obeying the Pauli commutation rules [P,, P,‘,] = 6,,.(1-2P,+P,).
(1.3)
m is effective mass of the free electron defining the excitation levels; R, is the crystal lattice constant (1.4) in the longitudinal phonon Hamiltonian, where bi, b, are the creation and annihilation phonon operators having wave vectors q, 0, = uol q( is the phonon frequency, while u. = R06?%l is the sound velocity. M is the molecular mass in the node; K is the elasticity constant of the crystal. The Hamiltonian of the exciton-phonon interaction is described by Hint
=
&z
F(q)PzP,(b,
+ bzg)einRoq.
F(q) is the intensity of the exciton-phonon F(q) = i-($-$-)“2
fi
,
a=2R,x.
(1.5)
interaction with constant coupling x,
(1.6)
At T # 0 following the lines of ref. 6 we ask for the state of the system in the
F. GASH1
448
form of the Schrodinger
et al.
amplitude (1.7)
where v are phonon states given by the expression
(1.8) and where IO,,>, IO,,, > are the exciton and phonon vacuum states respectively. The unitary equilibrium mode displacement operator appearing in the result of the exciton-phonon coupling has the form u,(t) = exp{T
P,,
[p;,(t)b,
are chosen as modulated
P,,(t) = p,,,(t)
-
&,(r)b,‘l)
(1.9)
waves,
emi”“‘1y,
where the cp,(t) is normalised
(1.10) by the condition
In the formalism q,,(t) and P,,,(t) play the role of dynamical variables and are determined from the Hamiltonian equations (1.12a)
(1.12b) (H) is the functional averaged Hamiltonian function, i.e.
(H) =
cI’ ~,,Hvv
over phonon
states playing the role of the
3
p,, is the diagonal element of the phonon density matrix operator
(1.13)
MZjSSBAUER EFFECT AND SOLITARY
p
WAVE
449
(vJeeHph”lv)
=
(1.15)
Y”
7 (YJ ePHP”“lv) ’ From the Hamiltonian equation system (l.l2a,b) the nonlinear Schrodinger equation (referring to cp(x, t) and having a cubic nonlinearity) may be derived in the continuum approximation ifi % (x, t) -
@&+ -&
(1 - e-“)I&,
t) + --&
e-“R,f, ‘*l($’ l’ 0
+ G((P(*~J(x, t) = 0.
(1.16)
The particular solution of (1.16) is a solitary wave depending on the variable 5 = x - ut, where x is the position along the molecular chain. The explicit soliton solution has the form6)
’
exp k(x -x0)
112
+ &
‘(X,f)=[Roa2(eT’l
(a’(0
) - k*)t - 2
ch[a(e,)(x-x,Lut),
’
t -
(1.17) where
R,&*(O)
,+y,
a(&)= 3
G(8,) =
B=Z.
mG(%)exp
u2 ML$l - s2)
4
(Y(0) =
mRoa2 2Mv;fi*( 1 - s’) ’
(1.18)
Rima2
’ fm = 2 $ C4 i4lu + 2i,),
Ro 4MV31 - s2)h
with iq =
1 fr$lB7 _ 1 ’ e
,=v,
t$=k,T.
(1.18a)
UO
The expression e -’ IS ’ the Debye-Waller factor. The temperature effect in the case of solitons is such that the raising of temperature provokes a spreading of the soliton. The consequence of this fact is that the nonlinearity coefficient G tends to zero with the temperature, so that eq. (1.10) becomes linear and describes in a certain interval of the temperature a plane wave.
F. GASH1
450
2. The transition
probability
et al.
in the simplified
model
The probability of emission of a y-quantum from the nucleus situated in crystal lattice node n,, is calculated from the matrix element of the operator V,,, =
eiP.Rndfi . Qij, aj, ij) .
(2.1)
P is the momentum belonging to the emitted phonon, R,,, is the radius vector of the center of mass of crystal node, for which we assume that (2.la) where u “Dis the displacement of the emitting molecule from its node. The operator c(ij, cj, ej) depends on the coordinate, momentum and spin belonging to particle j constituting the y-active nucleus. It is not necessary to know explicitly the form of this operator. For the same reason the interaction affects only the movement of the center of mass of crystal nodes, so by the y-active molecule the number of internal degrees of freedom is fixed. The transition probability amplitude corresponding to the emission of the y-quantum is the following:
(2.2) (fl&j, pj, ej)li) is th e molecular matrix element depending on the internal structure. It will be treated as having a fixed value due to the reasons already mentioned. Since the matrix element (fl c(.G,, -i),, &,)li) is constant, it is not interesting any more for our analysis as we are interested only in relative transition probabilities determined by the matrix element
i.e. by the transition amplitude in our case when the same soliton is found in the crystal before and after the emission. The plane wave exp{iP *R,olfi} can be treated as an interaction operator initiating the transition. It can be written as
(2.4) umO expressed in terms of creation and annihilation phonon operators b T2and b2 has the form
MCiSSBAUER EFFECT AND SOLITARY WAVE
451
112 I.4
e,(b,
= “0
+ b_+,) eiqRono
(2.5)
when N is the total number of sites in the crystal chain and e, is the polarization vector of longitudinal phonons. Taking into account the definition of lFV) eq. (2.3) can be rewritten as (2.6)
where M,.(n)=(vlO,(r)exp[~~
x fiNI
(&)1’2(~*eq)(by+b~q)eiqR0no] 4
.
(2.6a)
In (2.6a) the phonon matrix element can be rewritten in the form
M,,(n) =
nM,,W 4
(2.7) where y, = i (*)“2(p-eq)eiqRnno,
(2.8)
with the characteristic y_q = - y ,*. We are interested in the average value of the matrix element (2.6) over phonon states. It is given by
(2.9) where the soliton part of matrix element
(MsoAph = exp( -
( Mso,)ph is (2.10)
F. GASH1
452
where expressions
are obtained
(2.7)-(2.1(l)
et al
by the standard
procedure
applying
Weyl’s identity
e” e li
= ,“/*‘[A.
icl
eii +ri
(2.11)
for [A[A, 8]] = [B[A, 8]] =O. Since P,,,(t) is defined as a modulated
plane
wave, (2.12)
the solution
(1.12a)
is found
in the form’)
(2.13a)
(2.13b)
The expression for the transition probability amplitude reduces to a simplified form if the origin of the coordinate system is put at the site o,~ where the y-active atom is situated. Now the transition probability amplitude (2.6) is
It is clear that ~x&,(P* e,)qlp,,(t)l’ j&x Mv;?,hlq12(1 -sq 4 vanishes
since the function
cos qR,,n
of q:
d p * eq1~0sq&n is odd. In the continuum
approximation
we have C, --$ (1
lR,,) (Tzdx and in the
MijSSBAUER
EFFECT AND SOLITARY WAVE
453
long-wave approximation sin qR,n + qR,n + qx. In accordance with criteria in paper 6 we can put ((pn(*+ i aR, in the exponent of formula (2.14), so
(A%,,, =
exp{ -T (p%):(cq+ f)) 2MNno
+m
e -cc
iQx
(2.15)
ch’ C+ - x0 - u,) ’
where
(2.16) while the probability
of its amplitude is
1
?T’Q’
= a
sh2
The argument of the soliton part of the transition probability
rQ(v) -=-
244)
rr
xR;
2 M@(l
(2.17)
Qm exp1 2ff (4)
-.s2)
LC(p*eq). N 4
(2.17) is (2.18)
As one can see, the dependence of the probability IV+ on the coupling constant x has the same importance as for the existence of Davydov’s solitons, i.e., when x ---, 0 we get the term containing typically the pure phonon form factor only. From (2.17) it follows that in the absence of emission ( p = 0) one has Wi+ = 1 which proves that the expression for the probability has been adequately normalized.
3. Conclusion If we admit that the molecular chain is populated exclusively by solitons, the Mossbauer effect, generally speaking, is less probable then in the usual and well-known pure phonon modes that are simultaneously spread over the whole crystal. The latter case leads to the probability (2.17) having the factor
F. GASHI
454
et al
(3.1)
In the former
soliton
case the smaller
probability
is manifested
by the factor
(3.2)
which is to be multiplied
by the factor corresponding
to the pure phonon
states.
As one can see, the factor (3.2) has the typically localized character of soliton excitations. As to the effect of the temperature on the transition probability of the Mossbauer effect it is manifested through the factor I?, = [eh”q’f’ - 11-l in the case of pure phonon states, while in long-wave approximation (eq. (2.13)(2.18)) the soliton form factor (3.2) does not depend on the temperature. From the formula (2.17) it is also visible that static solitons (u = 0) have little impact on the transition sound velocity decrease
probability. The solitons having velocities close to the the Mossbauer effect transition probability significantly.
Acknowledgement The authors valuable
would
like to express
to Dr. M. Marinkovic
their gratitude
for
discussions.
References 1) R.P. Feynman, Statistical Mechanics (a set of Lectures), Reading, Massachusetts 1972), pp. 22-25. 2) H.J. Lipkin, Quantum Mechanics 3) A.S. Davydov and N.T. Kislukha,
Advanced
(North-Holland, Amsterdam, SW. Phys. JETP 71 (1976)
Book
Program
1973) chaps. 1090.
(Benjamin,
3 and 4.
4) V. Fedyanin and L.V. Yakushevich, Theor. Mat. Fiz. 37 (1978) 371. 5) AS. Davydov, Solitons in the Molecular Systems (Naukova Dumka, Kiev, 1984) (in Russian) 1, p. 18. 6) A.S.
Davidov,
Sov. Phys.
JETP
78 (1980)
789.
chap.