J. Mod. Phys., 2010, 1, 9-16
doi:10.4236/jmp.2010.11002 Published Online April 2010 (http://www.scirp.org/journal/jmp)
Translational Motion of a Free Large Polaron and
Broadening of Absorption Spectra
Vladimir Mukhomorov
Agrophysical Institute, St.-Petersburg, Russia
Received February 16, 2010; revised March 17, 2010; accepted March 20, 2010
Abstract
The translational motion of a large polaron as whole is analyzed in the context of its effect on the broadening
of an absorption optical spectrum. It was open question how important the role of translational degrees of
freedom and the corresponding velocities are on the broadening. The Bogolyubov method of canonical
transformation of coordinates is formulated for a system of an electron and field, taking into account rigorous
fulfillment of the conservation laws. Separation of variables is carried out for the coordinates describing the
translational degrees of freedom and the electron oscillations in a polarization well. The equations obtained
for the electronic states explicitly depend on the velocity of the free polaron as a whole. An estimate is made
for free polaron in ammonia.
Keywords: Large Polaron, Translational Motion, Canonical Transformation, Broadening, Absorption Spectra,
Ammonia
1. Introduction
ITS is well known that the absorption spectra of free
large polaron consists of a broad featureless asymmetric
band with a long tail extended to the short-wave length
region. There is vast literature on the possible mecha-
nisms of broadening the optical spectrum of free polaron.
S. I. Pekar [1] have studied the broadening of the optical
absorption spectra of large polaron in crystal as a func-
tion of phonon dispersion. In the work [2] authors have
calculated the optical absorption coefficient for free po-
larons using the multiphonons mechanism. The main
idea in ref. [3] consist that broadening of the absorption
spectra depends on width of an electronic band. In this
case broadening of the absorption spectra to proportion-
ally electron effective mass at the bottom of a conductiv-
ity band [4]. However the question of a contribution to
the broadening from the translational degrees of freedom
and from the corresponding velocities is still unanswered.
In this work the thermal motion of free quasiparticle as a
whole is analyzed in the context of its effect on the
broadening of the optical absorption spectra.
2. Mathematical Method
To analyze the effect of translational motion of a free
large polaron on its absorption spectra, one must separate
in the Hamiltonian the translation-invariant degrees of
freedom from the coordinates describing the motion of
the polaron as whole and derive the velocity dependent
equations for the electron transitions. If the electron and
quantum field are strongly coupled then the collective
localized state of the field and particle is formed. In such
a formation the electron motion is rather intricate. On the
one hand the electron oscillated within a rather deep po-
larization potential well and undergoes the optical transi-
tions, and on the other, it moves together with the center
of inertia of the system and participates in the transla-
tional random walk. The problem is to separate these
motions correctly, rigorously taking into account the
conservation laws. This can be conveniently done using
Bogolyubov [5] method of canonical transformation to
the collective coordinates. The Bogolyubov method is
the most powerful analytic techniques available for dis-
cussing electron-phonon problems. This transformation
removes the translational degeneracy and allows one to
develop the successive approximation algorithm for the
energy and wave function while simultaneously fulfilling
the law of conservation of total momentum of the system.
Some of the transformed variables are generalized coor-
dinates whose canonically conjugated momenta are the
integrals of motion, which are defined by the symmetry
properties of the original Hamiltonian and hence ulti-
mately ensure fulfillment of the conservation laws. Fol-

V. MUKHOMOROV 10
lowing the Bogolyubov method, we reformulate the
adiabatic theory of the particle strongly interacting with
the quantum field. The resulting equations determine the
electron transitions and depend explicitly on the transla-
tional velocity of free polaron.
Within the effective mass one-electron continual ap-
proximation the Hamiltonian of the electron-phonon
system has the form
2
2(0) (0)*
*()
2
ii r
fff
f
p
HVebVe
m


 
frfr
1()
2
f
ff ff
f
bb bb 


 (1)
where the interaction form-factor is defined as ()0
f
V

1/2 1/2
(/)(4/)
f
ifu V

 ,1/2*
(2/ )
f
um  and the di-
mensionless coupling constant is 2
c

 , (1 /
c




2
1/ )/2f
s
eu

 ; m* is the isotropic effective mass of
electron,
f

is the frequency of the long-wave length
longitudinal optical phonons of the polar medium;

and
s

are the high-frequency and low-frequency
dielectric constants of the isotropically polarizable di-
electric continuum, and r is the electron coordinate. The
quantum amplitudes bfand b
f of the polarization
field, respectively, annihilate and create the field quan-
tum

fand obey the Bose-Einstein commutation rule
[, ]bb


''
f
fff
.
In order to develop the iterative procedure for calculat-
ing the eigenfunctions and energy eigenvalues of Hamil-
tonian (1) we modify the canonical transformation of co-
ordinates. For this purpose the electron coordinate is
written as the vector sum of two variables
/

rλq, (2)
where q is independent of r and means the coordinate of
the center of gravity of the system, and λ describes the
electron motion relative to the center. Before developing
the perturbation theory, let us introduce in Hamiltonian
(1) instead of the phonon creation and annihilation op-
erators, the complex phonon coordinates qf and the cor-
responding operators of conjugated momentum pf :
()/2qbb 



fff , ()/2pibb




fff
,
(3)
which satisfy the commutation rule [, ]qp i



''
ffff
.
The 1/γ factor in (2) and (3) allows one to describe the
electron motion relative to the center of inertia even
within the lowest-order nonvanishing term of the expan-
sion of the Hamiltonian in γ powers. Using Equations (2)
and (3) and taking into account that λrλ/r



/)(/
λ/

one can transform Hamiltonian (1) as
22 2
2(0)
*22
2
i
HVqe qq
m






 


f(q λ/) λ
ff fff
ff
p
2
1
2pp



fff
f
. (4)
One can see that, after changing the variables, the en-
ergy of the electron - field interaction and the field po-
tential energy are indeed of the same order in the γ pa-
rameter. Hamiltonian (4) is translation-invariant. Ac-
cording to Equation (3) the operator of the total momen-
tum of the system can be written as /i

q
-i/-iqp




ff
f
r
f
, so that it is a strict integral of
motion. It then follows that the q vector indeed means
the coordinate of the center of gravity of the system. Be-
cause of a smallness of the last term in Equation (4), the
effect of interaction of the electron with the quantum
field reduces mainly to the appearance of a potential well
[second term in Equation (4)] whose depth depends on
the magnitude of the dimensionless coupling constant.
As a result of the strong interaction, the quasi-particle is
characterized by its own internal structure. The appropri-
ate internal states can be coupled to one another by the
electronic transitions.
The interaction of phonons with the charged particles
is known to shift the equilibrium positions of the field
oscillators relative to their unperturbed values. We thus
supplement transformation (2) by the transformation of
the field coordinates qf :
(/)exp(), quQi



fff
f
ruu

*
ff
, f QQ


f (5)
The translation-invariant Qf variables allow for the
quantum fluctuations of the field near its new self-con-
sistent classical value which is determined by the set of
c-numbers uf to be evaluated in the follows. Within the
new variables (5) the interaction potential between the
electron and the quantum polarization field retains its
order of magnitude in γ. Note that the introduction of
new coordinates (2) results in the appearance of three
extra degrees of freedom in comparison to the original
system. We therefore impose three additional conditions
on the Qf coordinates, which can be chosen in a linear
form without loss of generality:
0vQ 

*
ff
f
f (6)
This requirements, allow one to retain the number of
independent variables after introducing the new electron
and field coordinates. The vf values can be chosen in
such a way that the orthonormalization condition
*
ff
ffvu





f
, ,1,2,3

  (7)
is fulfilled together with the requirement that *
f
f
vv


.
The coordinate transformations (2) and (5) provide ful-
fillment of the conservation law for the total momentum.
Hamiltonian (4) can be further transformed after the op-
erator of momentum pf is expressed in the terms of the

V. MUKHOMOROV 11
new variables q, λ and Qf :
Q
pii i
qqQqq
 
 

k
f
f
ffkff
qλ
qλ. (8)
Putting uf independent of qk and differentiating (5)
with respect to qk we get
ii
e
q Qiqe
qq q

 
 
 
ffq fq
f
kkk
fq
f
1
[() ]
ii
eiuQe
q
 

 
fq fq
fkf f
k
q
f. (9)
The derivative /q
k
qis found by inserting (5) in the
additional condition (6) and differentiating the identity
obtained. The result is
()0
ii
veiqe
q




*fq fq
fkf f
ff
q
ff . (10)
The equation for the partial derivative of q with re-
spect to qk is obtained from Equation (10) taking into
account transformation (5) and the condition (7)
1()
i
ivevQ
qq 




*kq *
kff
f
kk
qq
kff (11)
This equation can be solved by iteration, with 1/γ as a
small parameter. The following solution is then obtained
within an accuracy of the terms on the order of 1/γ2:
*
**
2
1
[()
1()()...]
i
fl
ievv vQ
q
vv vQQ


 




kq **
kkff
f
k
*
kfl
f,l
qkffk
kflfl
(12)
Using the transformation of variables (2), one deter-
mines the partial derivative //qq



ff
λq. Now,
using this equality and Equation (12), one finally obtains
the following expansion series for the /qf operator
in terms of the λ, q, and Qf variables
{()
i
k
k
ei ivvQ
qQ


 


qf **
ff f
fk
ffk
λ
P
() ()vvQ iuv
Q 


 
 

** *
fk klf
kl
l
kfklf q
*
1[()() ()vvvQQ iuQ





**
fklk ll
k,ll l
fkl kll
()() ]vvQ QvvQ
Q




** **
fk kfk k
klk
l
l
kkflkkf q
*
2
1[()()vvvQQQ Q





**
fklk lm
mm k,l
fkl klm
*
()() ]..}vvvQQ 

**
fklk l
k,l
fkl klq (13)
where Pf stands for the field generalized momentum;
the latter is expressed as a linear combination of the
/iQ


f
mome n t u m:
//iQvuQ

 

P*
fffkk
k
f
k.
As the q coordinate is a cyclic variable, the corre-
sponding canonically conjugated operator of momentum
/ i

q (which coincides with the total momentum
of the system) commutes with Hamiltonian (4).
Correspondingly, the /

q operator will be further
replaced throughout by the total momentum


r
P
p
bb



f
ff
f. In order to allow for the momentum even in
the first approximation, we introduce the I vector such
that P = γ2I. As a result, translational effects appear even
in the first order. The total eigenfunction of the system
can then be written as
2
(,,)exp(/)(,)Qi Q   
ff
λqIqλ. (14)
This function realizes a certain representation of the
translation group and corresponds to the state with a
fixed total momentum P of the system. It is convenient to
perform, according to [5], one more unitary transforma-
tion of the wave function with respect to the Qf variable
and rewrite the total wave function as
2
(,)exp() exp(/)(,)
f
QisQi Q   


fff f
λ,qIq λ,
(15)
The complex numbers *
s
s 
ff
and can be chosen in
a way to satisfy the condition
0us 

ff
f
f. (16)
Transformation (15) can be used to expand the collec-
tive coordinates Hamiltonian in descending powers of
the γ parameter:
....
01
2
2HHHH
 (17)
where the following notations are used:
2
(0)
2*22
2
i
HVueuu
m






f
f/ λ
ff ff
ff
p
1()()
2
ii
s
vs v 




**
fffff
f
IfIf , (18)
1()( )
i
Hsvv 






**
fff -f
f
-f
If fλP
(0) /
[2 i
f
Veu





fλ
ff
f
()()()]
ii
s
vsvvQ 




***
ffmm mmf
m
fIIm fm , (19)

V. MUKHOMOROV 12
0
11
[2( )
22
i
HQQ sv 





*
ffff ff
ff
If
-
(() )vvQuvQ
QQ




 
**
fk kmfk
km k
mk
kfkmf k
-
(())vvsQivvQ


 
 
** **
f-ffkk fkk
kk
I
ffkkfkP
(vvsQ



**
fffkk
k
ffkP
() )]ivvQ

**
fk k
k
Ikfk . (20)
We also require that
fC-If/ f
*
fff iuivs  )( 
 . (21)
The physical meaning of the C vector will be given
below. Let us expand the total wave function Φ and en-
ergy E in powers of γ:
....
012
2 EEEE
 ,
01 2
2
11...

  (22)
Upon substituting (22) in the equation HΦ = EΦ with
Hamiltonian (17) and collecting the terms with the same
γ powers, we obtain the following set of equations:
20 20
HE,
12011201  EEHH,
021120021120 

 EEEHHH , ...(23)
Because the H2 operator acts only on the field vari-
ables Qf the zero-order wave function can be written in
a multiplicative form )()(),( 00 ff
λ
λ
QQ


 , where
)(f Q

is an arbitrary function of the Qf coordinates.
Taking into account that the functions Φ0 and Φ1 are or-
tonormal, one has from the second equation in (23):
0221
()|| 0HE

 
so that the ()Q

f
function
obeys the following equation: 010
()|| () ()
f
H
Q


1()
f
EQ

. This equation has a regular solution χ(Qf)
only if 010
()||()H




is equal to zero, because the
H1 operator is linear in the Qf variables. Taking into ac-
count the form of Hamiltonian (19) and the obvious re-
quirement that E1 = 0, one obtains from (23) the follow-
ing relation for determining the uf values for an arbitrar-
ily chosen χ(Qf):
(0) /
00
2()||()( /)
i
Ve usiv



 
fλ*
fffff
λλ If
()()0siv v  

**
mm mm
m
mI /fm. (24)
The Equation (24) is derived under the assumptions
that the operators P
f
satisfy the condition u

ff
f
fP
0 that directly follows from the P
f
definition. Sub-
stituting the additional requirement (21) and the condi-
tion (7) in (24) and assuming that the ground electronic
state is described by the wave function )(
0λ
 , one finds
from Equation (24) the self-consistent classical field
components
(0)
00
22
2()|exp()|()
(())
Vi
u
 


 
*
ff
f
f
λf/ λ
fC . (25)
In the strong coupling limit, the H2 term in the Hamil-
tonian expansion (17) dominates and bears nontrivial
information about the system. Using transformations (21)
and taking into account that Hamiltonian H2 depends
only on the λ variable, one can write the energy eigen-
value of the state as 20 20
(, )(, )
H
QE Q


ff
λλ, where
the notation
2
22
22 || ||
22
EW uu 


 




ff ff
ff f
fC (26)
is introduced. The first two terms in Equation (26) define
the internal energy of large polaron.
The equation for determining the lowest energy state
wave function )(
0λ
 has the form
2
(0)
020
2()()
2*
i
Vue W
m








f/
ff
f
p. (27)
Using Equation (25), it can be recast as
2
200
0
2
00
||()|| ()
2()
(()) 2
()
i
i
*
Vee
m
W

 














(0)
2

fλ/
ff f/
ff
p
fC
(28)
which parametrically depends on the C vector. The inte-
gro-differential Equation (28) must be generally solved
using the self-consistent method because the classical
component of the field is influenced by the electronic
state to the same extent as uf influence the electronic
state.
Let us now clarify the physical meaning of the C vec-
tor. For this purpose, we differentiate E2 in (26) with
respect to C
22
2C C
uu
EW uu
CC
  








 

*
ff f
*
ff
f
22
2()
1|uf











f
f
ff
fC fC
|, α = 1, 2, 3, (29)
The C /
2
Wderivative can be found using Equation
(27). This can be done by differentiating (27) with re-
spect to C,
2
(0) (0) 0
0
22
2
ii
*
u
VueV e
CC m
  



 







f
f/ f/
ff f
ff
p

V. MUKHOMOROV 13
0 2
02 0
WW
CC



 


(30)
The average of Equation (30) for the state with the
wave function )(
0λ
 is
(0)
2
00
2()| |()
i
u
WVe
CC







ff/
f
f
(31)
Using the value obtained of the classical field compo-
nent f u(25), Equation (31) can be transformed to
(0)*2 2
00
2()||() (())
iu
Ve

 

 
f
f/
ff
f
fC ,
(32)
Then, instead of (31), the required derivative can be
represented as
22
2(())
uu
W
CC





 

*
ff
f
ff
fC . (33)
Substituting Equation (33) into (30), we get
2
2() uu
Euu
CCC









*
ff
*
ff
ff
fC
2
2
||uf






ff
ff
f
C (34)
We now determine the I vector. For this purpose, let
us multiply condition (21) by fuf and sum over the wave
vector f. After applying the requirement (7) and condi-
tion (16), we obtain the following expression for the I
vector:
2
()| |u

 f
ff
ffC
I. (35)
Let us differentiate (35) with respect to the C vector,
2
()( )
|| f
u
uu


 



ff
f
f
ff
Iff fC
f
CC
() u
u





*
f
f
ff
fC
fC (36)
One can easily see from a comparison of Equation (34)
with (36) that
2
I
EC
CC





 

, α, β=1, 2, 3. (37)
We finally obtain
22
CIC
EE C
I
CI CI
I
CC
I


 












 






. (38)
Consequently, the following result is obtained after
using the definition for the total momentum P = γ2I:
PC/
2
2E
 . However, by definition, P /
2
E is
merely the velocity v. Therefore, the C vector is related
to the translational velocity of the polaron by expression:
22
2/E

  vСP, (39)
and determines to the γ2 factor, the mean velocity of the
center of inertia of system. Hence the energy eigenvalue
(28) of the self-consistent ground electronic state W2
explicitly depends on the translational velocity of the
quasiparticle.
Let us now determine the translational effective mass
of the polaron. Using (26) and assuming that the velocity
of the center of inertia is small, which ordinary holds for
thermal motion, we expand the energy eigenvalue of the
system in series
2
(0)(0)(0) 2(0) 2
22
()
|| ||...
22
EWuu



 


f
ff
ff
f
fC
(40)
The quantities that correspond to the zero translational
velocity of the polaron are labeled by superscripts in
Formula (40). After substituting relation (39) in (40), the
following expression is finally obtained for the ground
state energy of the system:
42
(0)(0)(0) 2(0) 2
22
()
|||| ...
22
EW uu



 


f
ff
ff f
fv
(41)
The last term in (41) can be regarded as the kinetic
energy of the translational motion of the free particle as a
whole: ** /2
kin
Em2 v, where the notation m** stands
for the ground state translational mass of the large pola-
ron:
(0) 2
4
** ()| |
3
u
m




f
ff
ff
(0) 22
4
00
3
2| |()|exp()|()|()
3
|V i 



f
ff
λλf/λff (42)
If the electron is trapped by the polarization field, the
interaction of the particle with the field fully “consumes”
the rest mass of an electron. Indeed, it follows from the
order-of-magnitude analysis of the variables in Equation
(42) that the effective mass m** ≈ γ8m* >> m* is domi-
nated by the field inertia.
The translational mass m** can be calculated using (42)
if the wave function 0()

 is known. It can be found
by solving the nonlinear integro-differential Equation
(27). However, in practice, it is convenient to determine
the ground state wave function using a direct variational
method and varying the total energy functional
2
2(0)2
000
*
1
[()]()|| ()||
2 2
Fu
m
 
ff
f
λλλ
(43)

V. MUKHOMOROV 14
with
(0)*
00
(0) 2()|exp( )|()Vi
u



 
f
f
f
λf/ λ
.
The approximate analytic form of a trial variational
wave function
0
()


of a nondegenerate ground state
can be established by expanding the exponential in
Equation (27). Upon restricting ourselves to the quad-
ratic terms in the resultant series, we obtain the oscillator
equation
2
2
(0) (0)
0
*
1
21 ...()
2 2Vu i m
 





 


 




λ
ff
f
pff λ
(0)
20
()W  λ, (44)
whose solutions are the Hermite polynomials. These
functions can be regarded as good approximations to the
wave functions of the ground and low excited states of a
system with large γ. Therefore, for slow translational
motion of particle we choose the trial ground state wave
function in the following analytic form: 3
0()(



66 1/42
)exp(/2)



, where α is the variational
parameter. Such an approximation for the trial function is
consistent with the results of the shifted – 1/N – expan-
sion numerical technique that was applied in [6] and
technique solving of nonlinear integral Equation [7] to
the analysis of Equation (43).
It was established earlier that the optical transition
from the ground to the lowest lying electronic p-state is
most probable (oscillator strength of 0.77 [1]). Because
the transition time 115
0(/) 10Es


  is much
shorter than the relaxation time 13 10
s


 of the polar
lattice, the optical transition can be considered as vertical,
i.e., proceeding at a fixed value (0)
ufof the classical
component of the polarization field in the lowest elec-
tronic state. According to this premise and based on
Equation (28), the initial electronic state is described by
the equation:
(0) 2
000
*
2|
2()||()()
2
ii
Vee
m
 











ff/ f/
ff
|
pλλλ
2
(0) 2
00 0
|
2()||()()
ii
Vee
 
 







ff/ f/
fff
|fC
λλ λ
)()( 0
)1(
0
)0(
0λ
 WW , (45)
whereas the final state (k) of electronic optical transition
obeys the equation
(0) 2
2
00
*
|
2()||()()
2
ii
k
|V ee
m
 










ff/ f/
λ
ff
pλλλ
2
(0) 2
00
|
2()||() ()
ii
k
|V ee
 
 







ff/ f/
fff
fC
λλ λ
(0) (1)
()()
kkk
WW  λ. (46)
In Equations (45) and (46), the translational velocity
of large polaron is assumed to be small, i.e.,
22
()

f fvand only quadratic terms are retained in the
expansion of the potential. The wave function of the ex-
cited electronic p-state is chosen in the form
2
1/2
(/2 )
3/2 55
2
() cos
ke










λ (47)
where β is the variational parameter.
It is convenient to transfer from Equation (45) to the
equations
(0) 2(0)(0)*
2
(0) (1)2
00 00
|
()|| ()2
2*
V
WW m


 

 

fff
ff
|
p
λλ
2
(0) 2(0)(0)*
|
2ff
V 






2f
fff
|fv,
(0) 2(0)(k)*
2
(0) (1)2|
()|| ()2
2
kk kk
*
V
WW m


 

 

fff
ff
|
p
λλ
2
(0) 2(0)(k)*
2|
2ff
V 






f
fff
|fv (48)
Here (0)

fand (k)

f are the Fourier transforms of elec-
tron densities in the ground and excited electronic states.
In the adopted approximation, the frequency of the most
active electronic dipole transition can be written as
(0)(0)(1)(1)(0) (1)
00 000
()()
kkkkk
WW WW

, (49)
where the second term depends on the polaron velocity,
whereas the first term determines the optical transition
frequency at the band maximum in the state with the zero
center-of-mass velocity. The frequency )1(
0k
 can be
found from Equation (48)
(1) 2
00 kk
Sv. (50)
The following notation is used in Equation (50)
(0)
4(0)(0)(k)
03
()|V
2()
k
S 



f
fff
ff
ff . (51)
Such an approach in the phototransition calculation is
justified if the impurity absorption spectrum lies between
the IR-absorption region of the polar lattice oscillating
and the absorption region of the strongly bound electrons
of base material. We assume that the polarons are in
thermal equilibrium and that the quasiparticle distribu-
tion over the velocities v is Maxwellian: 3/2 ()F



v
2
(/ )
30
0e

vv
v, where 2**
02/kTm

v. Then, the full width
at half maximum of the optical absorption spectra is re-
lated to the standard deviation D as 1/222ln2WD. In

V. MUKHOMOROV 15
this approximation, the intensity is symmetrically dis-
tributed relative to the (0)
0
frequency. The band be-
comes asymmetric in the presence of photo transitions to
high-lying electron excited states. With the Maxwellian
velocity distribution, the variance is 22
0k
D


22**2
0
06(/)
k
kSkTm . Then, the band width W1/2 is
equal
**
1/2 0
4||(/) 3ln2
k
WSkTm. (52)
The approach presented to estimating the broadening
of the absorption spectra is valid if the inequality
(0)
0
/k tc v is fulfilled, where t is the mean free path
time of the quasi-particle, and c is the light velocity. This
inequality is fulfilled for the transition frequencies and
temperatures of interest.
3. Discussions and Conclusions
The theory is applied to free polaron in ammonia. The
electron is self-trapped owing to strong interaction with
the quantum polarization field, which is generated by the
dipole ammonia molecules librating around their equilib-
rium positions. Various investigations [8-12] have shown
that many properties of electrons in ammonia may be
described using the model of continual polarons. Within
the framework of this model the possibility of existence
coupled of two-electronic bipolaron formations in singlet
state [13,14] has been established and magnetic and opti-
cal properties of metal-ammonia systems are explained
[8,9]. The criteria for validity of the theory reduce to the
following inequality: mef



  . For an elec-
tron in ammonia, 0.885
eeV

is the energy of the
most active optical transition of a self-trapped electron
[11], e
V
f4.0

is the energy of the longitudinal po-
larization oscillations of the medium [15], and m


6eVis the excitation energy of electrons of the main
substance [11]. The orientational oscillations of mole-
cules about their equilibrium position in a polar liquid
form elastic waves that may be treated as in crystal. As a
result of the directionality and saturation of the intermo-
lecular hydrogen bonds for ammonia, the “quasicrystal-
linity” of the structure is comparatively well defined. Far
from the critical point, the thermal vibrations of the
molecules may be reduced to a set of Debye waves, as in
a polar crystal, where the spectrum of collective oscilla-
tions in the liquid has a cutoff at longer wavelengths than
in crystals [16] on account of the translational motion of
the particles. The elastic continuum approximation does
not generally allow for anisotropy and is far better appli-
cable to a liquid than to a crystal [17].
The width W1/2 of the optical spectrum of free polaron
in ammonia can be numerically estimated if the numeri-
cal parameters of the theory are given. At low concentra-
tions of the polarons, the dielectric constants ε∞ and εs
can be set equal to their values in pure ammonia; i.e., ε∞
= 1.756 and εs = 22.7. The electron effective mass m* is
usually determined from a comparison of the experi-
mental and theoretical positions of the absorption band
maximum. At sufficiently low temperatures, the transi-
tion frequency is dominated by the first term in (49).
Indeed, for the experimental measurements at tempera-
ture T = 225 K [11], we have the ratio
(1) 2
00 0
(0)(0)** (0)4
00 0
311
kk k
kk k
SSkT
m 



 


v (53)
In this estimate, it is taken into account that, according
to Formula (42), the effective mass of the solvated elec-
tron is m** = 0.02γ8m*. Therefore, the translational veloc-
ity contributes only insignificantly to the optical
transition. It is mainly determined by the (0)
0k


(0) (0)
0k
WW

term. A comparison of the theoretical posi-
tion of the band maximum with its experimental value
0.88 eV [11,18] yields the value of m* = 1.73m for the
electron effective mass, where m is the mass of a free
electron.
Let us estimate numerically the contribution from the
translations of quasiparticle as whole to the full width at
half maximum of the absorption spectra. For definiteness,
we use the following parameter values: 08.5



113
10

s
[8], γ2 = 13.5. Then, Formulas (51) and (52)
yield the value of W1/2 = 0.23 eV for the contribution
from the thermal motion of the quasiparticle, which
represents an appreciable part of the experimentally ob-
served value 0.46 eV [11,18]. The remaining part in the
broadening of the absorption spectra of the polaron is
likely to be due to the fluctuations of the polarization
field [4] or other mechanisms which short discussed in
Introduction. Equations (51) and (52) can be also used to
calculate the temperature band-width coefficient; it
occurred to be equal to 3
1/2 /1.0310/
K
dW dTeV

 .
The experimentally measured [19,20] range (0.6 – 1.6) ×
10-3 eV/K of the temperature coefficient is in satisfactory
agreement with the calculated value.
4. References
[1] S. I. Pekar, “Research in Electron Theory of Crystals,”
US AEC Report, 1960.
[2] M. J. Goovaerts, J. M. De Sitter and J. T. Devreese, “Nu-
merical Study of Two-Phonon Sidebands in the Optical Ab-
sorption of Free Polarons in the Strong-Coupling Limit,”
Physical Review B, Vol. 7, No. 6, 1973, pp. 2639-2644.
[3] D. Emin, “Optical Properties of Large and Small Pola-
rons,” Physical Review B, Vol. 48, 1993, pp. 13691-
13696.
[4] Y. T. Mazurenko and V. K. Mukhomorov, “On the
Mechanism of Optical Spectra Broadening of Polarons,”
Optics and Spectroscopy (USSR), Vol. 41, 1976, pp.
51-56.

V. MUKHOMOROV 16
[5] N. N. Bogolyubov, “About One New Form of an Adia-
batic Perturbation Theory in a Problem about Interaction
Particles with a Quantum Field,” Ukrainskii Matemati-
cheskii Zhurnal (Russian Edtion), Vol. 2, 1950, pp. 3-24.
[6] V. K. Mukhomorov, “On a Spectra of the Coupled States
of Polaron in a Limit of Adiabatic and Strong Coupling,”
Optics and Spectroscopy (USSR), Vol. 71, 1991, pp.
958-965.
[7] V. K. Mukhomorov, “Singlet and Triplet States of a Con-
tinuum Bipolaron for Adiabatic and Strong Coupling
Limit,” Optics and Spectroscopy (USSR), Vol. 74, 1993,
pp. 544-655.
[8] V. K. Mukhomorov, “Bipolaron States of Electrons and
Magnetic Properties of Metal-Ammonia Systems,” Phy-
sica Status Solidi B, Vol. 219, 2000, pp. 71-89.
[9] V. K. Mukhomorov, “Ground and Excited States of a
Three-Dimensional Continual Bipolaron,” Physica Status
Solidi B, Vol. 231, 2002, pp. 462-476.
[10] J. Jortner, “Energy Levels of Bound Electrons in Liquid
Ammonia,” Journal of Chemical Physics, Vol. 30, 1959,
pp. 839-846.
[11] J. C. Thompson, “Electrons in Liquid Ammonia,” Clare-
don, Oxford, 1976.
[12] N. F. Mott, “Metal-Insulator Transitions,” Taylor and
Francis, London, 1974.
[13] V. K. Mukhomorov, “Stability of Bipolarons, Elec-
tron-Electron Correlations, the Variational Principle, and
the Virial Theorem,” Physics of the Solid State, Vol. 48,
2006, pp. 864-870.
[14] V. K. Mukhomorov, “About the Problem of the Existence
of a Three-Dimensional Bipolaron,” Physica Scripta, Vol.
69, 2004, pp. 139-145.
[15] V. K. Mukhomorov, “Bipolaron Formations and Interpo-
laron Interactions in Dielectric Layers,” Physica Scripta,
Vol. 79, No. 6, 2009, pp. 065704-065709.
[16] I. Z. Fisher, “Statistical Theory of Liquids,” University of
Chicago Press, Chicago, 1965.
[17] A. I. Gubanov, “Quantum Electron Theory of Amorphous
Conductors,” Consultants Bureau, New York, 1965.
[18] E. J. Hart and M. Anbar, “The Hydrated Electron,” Wiley,
New York, 1970.
[19] R. C. Dauthit and J. L. Dye, “Absorption Spectra of So-
dium and Potassium in Liquid Ammonia,” Journal of the
American Chemical Society, Vol. 82, 1960, pp. 4472-
4478.
[20] S. Arai and M. G. Sayer, “Absorption Spectra of the Sol-
vated Electron in Polar Liquids: Dependence on Tem-
perature and Composition of Mixture,” Journal of Chemi-
cal Physics, Vol. 44, 1966, pp. 2297-2301.