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VOLUME
61,
NUMBER
9
PHYSICAL
REVIEW
LETTERS
Water as a Free Electric Dipole Laser
Emilio Del Giudice
Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milano,
Italy
Giuliano
Preparata
Dipartimento di Fisica, Università di Mi/ano, Milano, Ita/y,
and
Sezione di Mi/ano,
Istituto Nazionale di Fisica Nucleare, Milano,
Italy
and
Giuseppe Vitiello
Dipartimento di Fisica, Università di Salerno, Salerno, Italy,
and
Sezione di Napoli,
Istituto
Nazionale di Fisica Nucleare, Napoli,
Italy
(Received 23
May
1988)
29
AUGUST
1988
We show
that
the usually neglected interaction between the electric dipole of the
water
molecule and
the quantized electromagnetic radiation field can be
treated
in the context of a recent
quantum
field
theoretical formulation of collective dynamics. We find the emergence of collective modes and the ap-
pearance of
permanent
electric polarization around any electrically polarized impurity.
PACS
numbers:
42.55.Tb,03.70.+k
I==2md; (d
g=0.82A)
(Ref. IO)
and
Ide I
==2ed
e(de
=0.2
A),
l2 •
H,>
2/Lj+A·dej,
where
the
sum
runs over
the
Nmolecules of
the
volume
V,
and
L, is
the
total
angular
momentum
of
the
single
molecule, / its (average)
momentum
of inertia,
and
de its
electric dipole vector. We set
where mis
the
proton mass
and
eits charge.
As we
are
interested in
the
collective aspects of
the
in-
teraction of
the
electric dipoles with
the
quantized
elec-
tromagnetic
field, we
concentrate
our
attention
only on
those electromagnetic modes whose wavelength is
either
larger
than
or
comparable
with
the
size Lof
the
system,
so
that
the
plane-wave
factor
eik 'xshall always be
dropped out.
Following
the
development of Ref. 9, we
can
cast
our
problem [Eqs. (1)
and
(2)] in
the
framework of a
quan-
tum
field
theory
of a complex
scalar
field
V/(u,t),
the
wave field, in interaction with
the
quantized
electromag-
netic field A.
By
standard
quantum-field-theory methods it is
straightforward
to write down
the
relevant
path
integrai
It is well known
that
liquid
water
is a very complicated
system,1,2 and
that
it
may
show significant
departures
from its average
bulk
behavior in
the
presence of macro-
molecules.:' colloidal particles,"
and
polarized impuri-
ties.i' In addition,
there
are
some experimental indica-
tions of its
important
role in
the
dynamics of macro-
molecules."
In recent times very
remarkable
progress has been
made
through
detailed
dynamical
calculations in
the
framework of a model which describes
water
as a net-
work of H-bonded molecules.Ì However, it seems legiti-
mate
to us to wonder
what
effect on
the
structure
of
liquid
water
could
the
quantized
electromagnetic field
have, which is usually neglected.
That
our
question
might
be totally nontrivial is suggested by
the
surprising-
ly close analogy
that
one can establish with
the
free-
electron laser. 8
There
the
undulator
field induces on a
bunch
of free electrons an oscillating electric dipole,
transverse to
their
motion, which gets coherently coupled
to
the
appropriate
modes of
the
electromagnetic
radia-
tion. On
the
other
hand, it is well known
that
the
water
molecules possess a considerable electric dipole.
The
aim of this
paper
is to investigate
whether
under
suitable conditions
the
electric dipoles of
water
molecules
can
interact
coherent1y with selected modes of the
radia-
tion field, as it
just
happens in
the
free-electron laser.
We shall employ atheoretical
approach
that
has
already
been applied to describe successfully some
important
as-
pects of free-electron-laser dynamics."
Th
us
our
model
for liquid
water
in a volume Vis defined by
the
following
Hamiltonian
(A
denotes
the
electromagnetic field in
the
radiation gauge):
N
H==
1:n;
j-l
(1)
©1988
The
American
Physical Society 1085
VOLUME
61,
NUMBER
9
PHYSICAL
REVIEW
LETTERS
29
AUGUST
1988
that
describes
the
dynamics or our system (we use the
natural
units h
==
c
==
1),
Z
=o
JId",d",*][da, da,*]exp(iW["" v"
,a,a*)),
where the action integraI (u is the unit vector in the dipole direction) is given by
W=
J
dnudt
Hu,d;
L(U,I)
==
-4
iI:a,*
(k.r
>a,
(k,d
+
",*
(u,di~(u,d
-
",*
(u,d~
",(u,t)
Jr
r,k
ut
4md
g
+
2e~~
iL[; ]1/2
(E,'
u)
",*
(u,d
",(u,tHa,
(k,de
rikt -a,*
(k,de
ikt},
V
r,k
L2is the square of the angular momentum operator In terms of the new functions ç
and
br,the Lagrangian
(7) scales as
l
[8
. 8 8
2]
L2
==
---
--sln8-+--
siné 80
ao
al/J2'
Liu.t)
==NI(u,t),
(11)
(15)
where l
(u.r
)has, in terms of çand br,precisely
the
same form as L
but
for the extremely
important
dif-
ference
that
the
coupling strength 2ed
e/V
1/2 now gets
multiplied by tbe very large
number
N1/2; furthermore,
Eq. (9) can now be rewritten as
J
dnuç*(u,dç(u,d
=1.
(12)
By changing variables in
the
path
integrai (5), we clearly
see
that
the solution of
our
quantum
field theory consists
simply of the "classical" solution of the variational prob-
lem
(lO)
oJldnudt=O,
(13)
modulo
"quantum
fluctuations" whose size is
0(1/
NI/2) o9
The
variational problem (13) yields the follow-
ing Euler-Lagrange equations:
/Jç~u,d
=
~ç(u,d
-i
led,
[N
v]1/2
L(E,.
ul ( k
2]1/2
{b,(k,t)e
rikt -b,*
(k,t
)eikt}
ç(u,t),
l4mdg k,r
oBb,
(k,l)
_o
ikt
[
Nk
]1/2 ( ) J*( ) ( )
l
al
-le
2V
Led;e, ko
dnuuç
u,t
ç
u,t
.
.,,(U,/)
==
N1/2ç(u, / ),
and
ar(k,/)
is the amplitude of
the
mode k with polariza-
tion r[k·
€r(k)
==0,
from the transversality conditionl.
The
Noether
theorem leads to tbe following conserved
quantity:
where Nis
just
the
number of dipoles contained in the
volume V.
As emphasized in Ref. 9, Eq, (9), implying
tbat
the
paths contributing to Z must have the "macroscopic"
size N1/2, suggests the relevance of the folIowing rescal-
ings:
In order to proceed any
further
we must get back to
the
meaning of
the
complex functions
ç(u,t)
and
br(k,t)o
It is clear
that
N1
ç(u,t)
12represents the aver-
age number of dipoles in the volume V
that
are polarized
in
the
udirection, while NIbr(k, t)I2is the average
number of pbotons in
the
mode k,r, By expansion of
ç(u,/)
in spherical harmonics, i.e., writing
(16)
NI
c;
(t)
12denotes the number of molecules
that
popu-
late the rotational
state
Il
.m)
of
our
rigid rotator,
which, neglecting alI interactions, at
thermal
equilibrium
should follow a Boltzmann distribution with El
==
l (l
+
1)/
4mdjo
Under
normal conditions (unpolarized bulk
waterlthe levels l
==0
and l
==
l have comparable popula-
tions. To simplify
our
problem, II from now on we shall
work in the subspace spanned by the four states 10,0)
and Il,
m),
and restrict
the
electromagnetic modes to the
resonating ones, i.e., those for which k
==wo
==
1/2mdj.
12
Setting
ro
(t )
==
C00
(/
),
Ym(t )
==
Clm(t )e-
iooot,
and calling b
(t)
the
amplitude of
the
electromagnetic
mode coupled to transition Il,m)+-+ 10,0), one can cast
the system (14) and (15) in
the
form
ro(t)
==
nI:mb:
(t)
Ym
(t),
rm
(l
)
==
-nbm
(l
)
Yo
(t
), ( l7)
bm(1)
==
2n
Yd
(t )Ym(1),
1086
VOLUME
61,
NUMBER
9
PHYSICAL
REVIEW
LETTERS
29
AUGUST
1988
where
Aed, [N]
1/2
n=
J3
2lùOV
lùO-Glù().
(}8)
Note
that
in
pure
water
(H20),
G---17.
Taking
advan-
tage
of
the
rotational
symmetry
of
the
problem,
our
sys-
tem
can
be
further
simplified by
setting
Ym
(t)
==
rl
(t)
and
bm
(t
)
==
b
(t
).
One
thus
obtains
ro(t)
==
3nb*
(t
)rl
(t
),
rl(t)
==
-
nb(t)ro(t),
6
(t)
==
2
il
r6
(t)
rl
(t),
which
admits
the
following
constants
of
motion:
IroI2+31
rl
I2
==
l,
21
rl
12+1b12
==
tsin
290,
(20)
(21 )
and
the
small
oscillations
around
it
are
controlled
by
the
pulsation
Wl
==
2.J2l1
-tsin
22Bo]
1/4
il.
(26)
It is
easy
to see
that
access to
the
limit
cycle
can
be
obtained
only for
Bo:>
Te14.
Note
that
the
"thermal
start"
lBoltzmann
distribution,
in
normal
conditions
ro(O)
---r 1(O)]
corresponds
to
Bo
==
Te13.
On
the
other
hand,
for
Bo
<
Tel4
the
system
(19)
goes
through
a
quasi-
periodic
motion
with
average
pulsation
iii
==
(2 cos2Bo)
1/2
n. (27)
From
our
definitions it is
straightforward
to
calculate
the
polarization
along
any
direction
n; one finds
Pn==(çln-ulç>
==
(21
J3
)ro(t)r
1
(t)
COS(W2
-wo)t.
where
the
angle
Bo
(O
<:
Bo
-<
Te/2)
specifies
the
initial
conditions as
1ro(O) 12==cos
2BO,
1
rl
(O) I2
==
tsin
290,
Ib(O)12==0.
If
we call
ro,rl,B
and
l/Jo,l/Jl,1fI
the
modulus
and
the
phase
of
re-
rl,b, respectively,
the
system
(19)
admits
a
limit
cycle:
Let
us now suppose
that
within
the
volume
Vwe have
an
impurity
with
asizable
electric
dipole,
which
gen-
erates
an
electric
field Ed
oriented
in
the
zdirection.
The
static
part
of
the
Hamiltonian
H,
[Eqs.
(1)
and
(2)]
will
acquire
a new
term
Vd
==
-de}·
Ed,
(29)
which will mix
the
states
10,0>
and
Il,0>,
to
produce
the
new
eigenstates
lO)
==cosa
1O,O>+sina
Il,0>,
(30)
1i>
==
-sina
10,0>+
cosa
Il,0>,
ro(t)
==fo
==
(1I
J3
)[1+cos 290+(1 - tsin
22(
0)
1/2]
1/2,
r,
(t)
==fl
==
(1/J3)(1
-
fJ)
1/2,
(24)
B(t)
==
li
==
(t )
1/2(fJ
- cos 2(0)
1/2,
with eigenvalues
ÀO,1
==(wo/2)[1
+=
(1
+4V}l(6)
1/2],
and
tga
==
[WO -
(w6+4V})
II2]/2Vd.
(31)
larization
around
an
impurity
that
carries
asizeable
electric dipole
(organic
macromolecules
are
known to
have
this
property).
The
amount
of
such
polarization
turns
out
to be
(P3(t»
==
(1/J3)
sin2a
(f6
-
fr),
(34)
which in
the
region
where
2Vd/
Wo
is
--1
attains
very
respectable
values.
Note
that
in
the
absence
of
the
col-
lective
interaction
analyzed
in
this
paper,
one
must
have
f6=fr,
as
prescribed
by
thermal
equilibrium,
and
no
significant
electric
polarization
can
emerge
even for
rath-
and
For
Vd
not
too
strong
(a <
TeIS)
the
dynamics
of
our
sys-
tem
is
practically
unchanged,
except
for
the
rotation
(30),
induced
by
the
electrostatic
mixing,
which
modifies
the
form (2S) of
the
polarization
P3 as
P3(t)
==
(I/J3)
fsin2a
(ra
-
rr)
+cos2a
2ro(t
)rl
(t
)COS([W2
-
(w6+4V})
1/2]t
>I.
Equations
(24)
to (2S)
and
(31)
to
(33)
display
the
essence of
this
work.
Let
us see
what
their
physical
meaning
is.
For
different initial conditions (90),
the
sys-
tem
will
exhibit
different
dynamical
behaviors; however,
for 90<
Te/4
the
frequencies involved will fall in
the
in-
terval
lsee
(27)]
O<v<
500
cm
-l;
while for Te/3> 90
>
Te/4
the
frequencies
populate
three
distinct
bands
around
1600, 750,
and
400
cm - l. It
should
be recalled
that
in
our
frequency
range
the
main
absorption
bands
of
pure
water
are
located
at 1640, 5S0,
and
180 cm
-1.13
But
the
most
intriguing
result
of this work is
the
pre-
diction
(33)
of
the
emergence
of a
permanent
electric
po-
1087
VOLUME
61,
NUMBER
9
PHYSICAL
REVIEW
LETTERS
29
AUGUST
1988
er strong electric fields.
So far no account has been made for the radiative en-
ergy losses from the volume V, which clearly is not an
electromagnetic cavity. By taking such losses into ac-
count one can easily see
that
the lifetimes of our collec-
tive modes are of the order of
27C/
lOo, and thus consider-
abIy larger than the periods
27C/
lO1,
27C/
102, and
27C/
iii of
the collective dynamics.
Far
eo>
n/4 the energy losses
will bring the Fo and Tr to their limit-cycle values, and
the system will resume "lasering" in such a regime onIy
when the thermai processes will have brought 80 above
n/4.
Even though, it must be admitted, our analysis is in
many ways at a preliminary,
rather
rudimentary stage,
nevertheless, we believe
that
we can draw anumber of
relevant eonelusions: (i) In the study of the dynamics of
water, the neglect of the eoherent interaetion of water
molecules with the quantized radiation field is complete-
ly unjustified, for we have shown
that
its coupling to col-
lective
quantum
states of size
Zn]
lOo (a few hundreds of
microns) is indeed very large (G =17); (ii) because of
the
latter
fact, the time scale associated with the
eoherent interaetion is much shorter
(=
lO- 14 s) than
those connected with short-range interactions. In such
coherent dynamics, our analysis has
a1so
recognized the
relevanee of frequeney bands which can be related to the
observed absorption bands of pure water; (iii) amacro-
scopic, permanent polarization can easily arise in water
in the presence of a small eleetric disturbance such as the
Ioeai field produeed by a macromolecule, or the field at
the surface of a colloid grain or within clays.
As a result, one can envisage the possibility
that
the
eoherent interaction between the water electric dipoles
and the radiation field fulfills the very important task of
generating ordered structures in macroscopic domains
(Le., within a few hundred microns) which could then
have a fundamental role in the organization of inanimate
as well as living
matter
14 in the wonderful ways
that
physical analysis is incessantly revealing.
Of
course
mueh more work is needed in this direetion.
1088
We wish to express our thanks to Professar Silvia Do-
glia far her encouragement and for giving us useful in-
formation on the present experimental knowledge of wa-
ter.
IWater, A Comprehensive Treatise,
edited
by F.
Franks
(Plenum,
New
York,
1972-1982),7
Vois.
2C. A. AngelI,
Annu.
Rev. Phys.
Chem.
34, 593 (1983);
S. H.
Chen
and
J.
Teixeira,
Adv.
Chem.
Phys. 64, l (1986).
3Biophysics of Water,
edited
by F.
Franks
and S.
Mathias
(Wiley,
Chichester,
1982).
4D.
Eagland,
in Ref. 1, VoI. 4,
Chap.
1, p. 305.
5J. B.
Hasted,
H. M.
Millany,
and
D. Rosen, J.
Chem.
Soc.
Faraday
Trans.
77, 2289 (1981).
6V.
Dahlborg,
V. Dimic,
and
A.
Rupprecht,
Phys. Scr. 22,
179 (1980); G. Albanese, A.
Deriu,
F. Ugozzoli,
and
C. Vig-
nali,
Nuovo
Cimento
9D, 319 (1987).
7H. E.
Stanley,
J.
Teixeira,
A.
Geiger,
and
R. L.
Blumbery,
Physica
(Amsterdam)
l06A,
260 (1981); E.
Clementi,
in
Structure
and
Dynamics
oJ
Nucleic
Acids, Proteins
and
Mem-
branes,
edited
by E.
Clementi
and
S.
Chin
(Plenum,
New
York, 1986).
8G.
Dattoli
and A.
Renieri,
in
Laser
Handbook
Volume
4,
edited
by M. L.
Stitch
and
M.
Bass
(North-Holland,
New
York, 1985).
9G.
Preparata,
Phys. Rev. A
38,233
(1988).
10This value refers to
the
highest
moment
of
inertia
of
the
water
molecule.
See
Ref. 1.
110ne
can
see
that
in
thermal
equilibrium
the
ratios R[ be-
tween
the
populations of
the
state
I
Lm)
and
the
state
10,0>
are
R
1=0.887,
R2
=0.698,
R3
=0.487,
R4
=0.301,
Rs
=0.165,
etc.
For
heavy
water
020,
the
same
ratios
are
0.942, 0.835,
0.698, 0.549, 0.406,
...
,respectively.
12This obviously implies
that
the
size Lof
our
Vis now con-
strained
to be less
than
2Jr/mD
==
4.3xlO - 2cm.
13G. E.
Walrafen,
in Ref. 1, VoI. 1, p. 151.
14E. Del Giudice, S. Doglia, M.
Milani,
and
G. Vitiel1o, in
Modern
Bioelectrochemistry,
edited
by F.
Guttmann
and
H. Keyzer
(Plenum,
New
York,
1986),
and
Nucl. Phys.
8275
[FSl7],
185 (1986).
