PHYSICAL REVIEW A VOLUME 22, NUMBER 6 DECEMBER 1980

Shift and broadening of hyperflne components of the first doublet of cesium perturbed by
foreign gases

E. Bernabeu and J. M. Alvarez
Departamento de F(sica Fundamental, Catedra de Optica, Facultad de Ciencias, Universidad de Zaragoza, Spain

(Received 18 December 1979)

The collision broadening and shift of the hyperfine structure components of the ground-state splitting for the D,
(8943 A) and D, (8521 A) cesium lines by noble gases (He, Ne, Ar, Kr, and Xe) and by light molecular gases

(H, and Ng under pressures not greater than 150 Torr and at a temperature of 295 K are investigated. The

Lennard-Jones (12-6) interatomic potential constants are calculated for these systems, using the semiclassical

theory of Lindholm-Foley (for Cs with Ne, Ar, Kr, Xe, and N, systems) and the quantum theory of Baranger

(for Cs with He and H, systems), by comparing with our experimental results in fine structure. These potential

constants are employed for the determination of the shift and broadening in hyperfine structure, using the

hyperfine theory of collision broadening.

I. INTRODUCTION II. EXPERIMENTAL METHOD

The spectral lines of an atom experience mod-
ifications in their profiles when the atom is per-
turbed by the presence of other atoms or mole-
cules. ' A great many experimental studies of the
shift, broadening, and asymmetry of the first reso-
nance doublets of cesium perturbed by foreign
gases have been made. ' But there are few experi-
mental results of the hyperfine structure of ces-
ium: The broadening of the 4555 A line by argon'
and of the second doublet (X =4555 and 4593 A) by

helium, argon, and xenon4 have been investigated.
The broadening and shift of this doublet has been
measured under pressure of neon, argon, and

xenon. ' For the first doublet we have measured
the shift of the hfs (hyperfine-structure} compon-
ents of the ground state for the D~ (8943 A) and D2

(8521 A) lines in the presence of noble gases and

light molecular gases (hydrogen, nitrogen) at pres-
sures not greater than 150 Torr. The results for
the light noble gases helium, neon, and argon have
been reported in a previous paper. '

In this paper we present our measurements for
krypton, xenon, hydrogen, and nitrogen as foreign
gases as well as new values for helium, neon, and

argon. These latter values correspond to the ones
previously reported by us' corrected with a more
accurate deconvolution method of the instrumental
function from the experimental profiles. The
method is described in this paper. We also present
here the values of the interatomic potential con-
stants for cesium-noble gas and cesium-light
molecular gas, calculated for a Lennard-Jones
(12-6) potential using the shift and broadening in
fine structure, derived from our experimental re-

. sults in hfs. Moreover, we have calculated with
these interatomic potential the shift and broadening
of the hfs components of the D, line.

n(c) = W(o) eX(o); (2)

in our case X&-0.3, so that the condition is ful-
filled.

The true absorption coefficient is the convolution
of the D(o) inhomogeneous process function (Dop-
pler broadening) and the v(g) homogeneous process
function (pressure effects, resonant broadening,
and natural width). Therefore, if the Doppler
broadening is a Gaussian function and v(o) is a
Lorentzian function, the experimental absorption
coefficients may be considered as Voigt profiles.
Moreover, the instrumental function W(c) of our
Fabry-Perot spectrometer may be approximated

The absorption coefficient of the cesium vapor
with and without foreign gas has been measured in
order to make a comparative study on the influence
of the foreign gases. The profile of the absorption
coefficient has been obtained by the double mea-
surement method, which allows the obtaining of
comparable emission gnd transmission recordings
with a common origin. One can obtain the n.'(a)
absorption coefficient profiles by the expression

a(c) = (1/l) in[I, (o)/1(o) j,
where l is the length of the absorption cell, I(o)
and I,(o}are the spectral distributions of the trans-
mitted and emitted light, respectively. These
spectral distribution profiles were measured with

a piezoelectrically scanned Fabry-Perot spectro-
meter with synchronous detection, described in

Ref. 6.
For small absorption, such that &g ~0.4, the ab-

sorption coefficient can be expressed by a convolu-
tion of the instrumental function of the spectro-
meter W(o), and the true absorption coefficient
X(o) (Ref. 7).

22 2690 1980 The American Physical Society



SHIFT AND BROADENING OF HYPERFINE COMPONENTS OF. . . 2691

26 P3

6 P,
1/2

L+

Q~ by Q b

Qeb

C+

Q„
b-

C-

62S
1/2

100
Qy

71.5
b+

100
b

w 3

47.7

33I
Q 159

Dl G

100
C+

D2

47. 7
b

45.5
Q-

L+

FIG. 1. Hyperfine pattern of 6 Si/~-6 Pi/2& 6 ~i/2-
6 P3/~ transition. The theoretical intensity is given for
each of their hyperfine components.

width (10 ' cm ')
Lorentzian component
Gaussian component

49.1 51.1
10.3 10.3
43.3 45.3

Finally, with these considerations the experimental
absorption coefficient can be expressed by means
of a Voigt function.

To obtain the homogeneous process contribution
v(o}, one must perform deconvolution of v(o),
W(o}, and D(o). We have used a deconvolution
method consisting in applying an analytic approxi-
mation to the Voigt function. ' Pyrex glass absorp-
tion cells of 5-cm width with cesium and foreign
gases at pressures of 10, 20, 50, 100, and 150
Torr were used. The measurements were made
for a temperature of 295+1 K.

III. RESULTS

We have determined the shift and broadening of
the hfs components (i„i ) corresponding to the
ground-state splitting of the Dy Dg lines of cesium
(Fig. 1), starting from the experimental absorption

by a Voigt function, and an experimental determin-
ation of this function has been made by us. The
instrumental function for the first optical doublet
of cesium is characterized by the following widths:

Instrumental Voigt function

D~ D~

coefficients for the test cells of cesium-buffer
gas. The absorption coefficient of a gasless cesi-
um cell has been taken as a reference.

As the hfs components (i„i ) of the D„D, lines
are complex spectral lines, it has been necessary
to carry out an analysis of the absorption coef-
ficient profiles. ' The absorption coefficient of the
gasless cesium cell (reference coefficient) for each
hfs component (i„i ) of the lines D„D, is obtained
by the addition of the absorption coefficient of hfs
components corresponding to the excited state. For
the hfs components i„i of the line D„ the width
of the absorption coefficient of each hfs component
a+, b„c, is -52.8&10 cm ', much higher than the
hfs splitting of the centers of gravity of the a„b„
c, components (-7.6&&10 ' cm ~). In this case the
absorption coefficient of the components i„i of
the line D, can be considered as a Voigt profile. "
For the line D„ the width of the absorption coef-
ficient of the hfs components a„b, is -50.7&10 '
cm ', it is of the same order as the hfs splitting
of the centers of gravity (-40.6 &&10 ' cm '}. Each
one of the components i„i of the line D, is then
obtained by addition of two shifted Voigt profiles.

The Voigt profile corresponding to the hfs com-
ponents (a„b„c)of the absorption coefficient has
a Gaussian component, which is the sum of the
Doppler broadening and the Gaussian component of
the instrumental function. For our experimental
conditions, the Lorentzian component is the sum
of the natural width" (-10 ' cm '), the resonant
broadening I -10 ' cm ', for an atomic density of
3 &10" atom cm ' at T =295 K (Ref. 12)], and the
Lorentzian component of the instrumental function.

The analysis of the hfs components i+, i of the
absorption coefficient of cesium with gas cells has
been made by fitting to each component i, , i a,

Voigt profile for the line D„and an addition of two
Voigt profiles for the line D, . The Lorentzian con-
tribution, due only to broadening by pressure, is
obtained by deconvolution of the absorption coef-
ficient of the cesium-gas system from the one of
the gasless cesium. The shifts have been evalu-
ated by considering the gravity center of the i„i
components of the absorption coefficient of cesium
with gas respective to the gasless cesium.

The values obtained for the shift and broadening
for each pressure value of the perturbing gases
have been fitted by a least-squares method to a
straight line, obtaining a standard deviation of
the slope between 1.0 to 3.(Pp. Each straight line
crosses the origin. The values of the shifts and
broadenings are given in Table I and Table II, re-
spectively.

As our results are for hyperfine structure and
those of other authors for fine structure, we
thought it was useful to calculate the shift and



2692 E. BERNABEU AND J. M. ALVAREZ 22

TABLE I. Summary of results for the shifts of all hfs components of the D&, D2 lines of C,'s
in He, Ne, Ar, Kr, Xe, H2, and N2.

hfs components of the
ground-state splitting

shift (10 cm /Torr; T =295 K)
D (8943 A) . D2(8521 A)

He

Ne

Ar

Kr

Xe

H2

N2 l+
z

0.17+0.03
0.25+ 0.04

-0.09+ 0.02
-0.12+ 0.04
-0.26 + 0.03
-0.32+ 0.03
-0.09+0.05
-0.29+ 0.01
-0.24 + 0.05
-0.30+0.03
0.001+0.008
0.148+ 0.004
-0.21 + 0.02
-0.28+ 0.03

0.08 + 0.01
0.18+ 0.04

-0.10 + 0.01
-0.21 + 0.01
-0.23 + 0.02
-0.28 + 0.03
-0.28+ 0.02
-0.24 + 0.05
-0.33 + 0.02
-0.26 + 0.03
-0.35+ 0.01

0.46 + 0.01
-0.18+0.07
-0.30 + 0.06

broadening in fine structure from our values in
hfs. We have hence calculated the shifts of the
centers of gravity of the D„D, lines, obtained by
Jackson's method. " We obtained the broadenings
by the weighted mean of the broadenings of the hy-
perfine components i+, z of each one of the lines.
These values are shown in Table III together with
those of other authors. These experimental and
theoretical values correspond to a linear extrapo-
lation to our experimental conditions from the val-
ues given by these authors for higher pressures,
except for the values of Qranier and collaborators"
given for low pressure (~3 relative density). The
extrapolation to the temperature range of interest
in our measurements has been obtained using the
expression given by Kielkopf. '

P =2NV sing p pdp,
0

(3)

2y = 4mNv 1 —cosy p p dp,
0

where p is the impact parameter of the collision,
v is the mean relative velocity, N is the number

IV. INTERATOMIC POTENTIALS

We present here the calculation of the interatom-
ic potential constants for Cs-gas systems with a
Lennard-Jones (C»r '2-CBr 6) potential using our
experimental results in fine structure. 'The Lind-
holm-Foley theory" has been used to relate the
shift (43) and broadening (2y) to the interatomic po-
tential constants by the expressions"

TABLE II. Summary of results for the broadening of the hfs components of the ground-
state splitting of the D&, D2 lines of Cs in He, Ne, Ar, Kr, Xe, H2, and N2.

System hfs components
Broadening (10 3 cm ~/Torr; T = 295 K)

D, (8943 A) Dp(8521 A.}

Cs-He

Cs-Ne

Cs-Ar

Cs-Kr

Cs-Xe

Cs-H2

Cs-N2

+

+

0.66 + 0.03
0.62+ 0.06
0.10+ 0.01
0.57+ 0.04
0.69+ 0.03
0.62+ 0.05
0.70+ 0.10
0.61+ 0.06
0.74+ 0.11
0.69 + 0.06
1.44 + 0.28
1.25+ 0.13
1.01+ 0.33
1.02 + 0.13

0.92+ 0.05
0.86 + 0.06
0.39+0.04
0.30+0.03
0.73 + 0.03
0.79 + 0.02
0.40+ 0.04
0.33+0.02
1.89 + 0.10
1.95+ 0.30
2.00 + 0.32
1.98+0.21
1.19+ 0.22
1.38+0.31



22 SHIFT AND BROADENING OF HYPERFIIVE COMPONENTS OF. . . 2693

TABLE III. Shift (p/N) and broadening (y/N) constants for D&, D2 lines of the Cs in He, Ne,
Ar, Kr, Xe, H2, and N2. (p/N and y/N are in units for 10 + cm (atom/cm) ). (ist de-
notes isotropic potential; ans, anisotropic potential).

Perturbing
gas

Di (8943 A)
P/N y/N

D2(8521 A.)
P/N y/N References

He

Ne

Ar

Kr

Xe

6.72 + 1.02
7.18
4.76
0.616

12.95

-2.93+ 0.09
-1.86
-4.40

-8.87+ 0.41
-8.85
-6.28

-7.44

-2.69+ 0.1
-7.44
—7.44
-6.28
-4.48

-8.42 + 1.11
-9.30

-11.90
-8.55
-6.96
-5.21

19.49+ 1.37
24.18
26.01
15.78
20.47

10.29+ 0.87
8.78
8.41

20.13+0.23
21.22
19.16

29.75

20.16+ 2.59
10.41
18.60
19.02
17.85

21.84 + 2.64
18.23
31.99
18.60
20.95
22.98

3.94+ 1.71
1.14
0.56
0.616

-4.33+ 0.33
3 ~ 72

-4.40

-7.54+ 0.33
-7.99
-6.28
-7.88

-4.91
-6.40

-8.10+ 1.13
-7.81
-7.44
-6.28
-4.07
-5.02

-9.07+ 0.84
-9.29

-8.56
-6.96
-5.32
-5.09

27.20 + 1.72
24.18
24.05
15.78

10.51+1.10
8.78
8.41

23.21+ 0.11
21.22
19.16
21.20

19.32
22.69

11.18+ 0.98
10.42
18.60
19.02
17.48
18.97

58.62 + 6.26
8.55

18.60
20.95
22.95
20.35

Present work
S. Y. Ch'en '
Ref. 2(b)'
Ref. 16
Ref. 23(b)

Present work
Ref. 2(d)
Ref. 16

Present work
Ref. 2(a)~
Ref. 16
S.Y. Ch'en

ef 23(b) b, c

b (ist)
(ans)

Present work
Ref. 2(c)~
J.Duperier'
Ref. 16

b(ist)
(ans)

Present work
Ref. 2(e) and 2(f)
Ref. 23(a)
Ref. 2(c)
Ref. 16

b(ist)
(ans)

, H2 2.29+0.19 41.08+ 6.30 1.81+ 0.26 47.44+ 8.10 Present work

N2 -7.38+ 0.11 30.93+ 5.71 -7.25+ 0.24 39.38+ 9.73
-6.42

Present work
E. Bernabeu '

~ Experimental value.
Theoretical value.
S. Y. Ch'en, in Proceedings of the International Conference on Optical Jumping and

Atomics Lines Shape, edited by T. Skalinski (Panstwowe Wydawnictwo Naukowe, Warsaw,
1969), p. 403.

J.Duperier, Diplom. Et. Sup. Paris (1966).
'E. Bernabeu, Rev. Acad. Cienc. Exactas Fis. Quim. Nat. Zaragoza.

63m C~ ~j ~s C~
256 )fV 6 ae (5)

The validity range of the impact approximation for
the Lennard-Jones (12-6) potential is determined
by19

y (((v/2@v'wc )(C,/C„) ". (6)

density of the buffer gas, and g(p) is the phase
change caused by a collision.

For the impact approximation, and assuming that
the atoms move in classical straight paths one ob-
tains

For neutral atom pairs t"6-10 "ergcme, Q~
-10 '~ erg cm', and v -10' cm sec '. With these
values, we find that the condition (6) is y «6.45
cm '. This condition is largely fulfilled by all our
experimental results.

The path described by the colliding atoms can in-
fluence the results for shift and- broadening. In
particular, if the interaction potential between
these atoms is much smaller than the kinetic en-
ergy, the path can be considered as straight. This
condition is verified in the applicability range of
semiclassical theory.

A qualitative calculation can be made in order to



2694 E. BERNABEU AND J. M. ALVAREZ 22

TABLE IV. Summary of results for Ca and C&2 forD&, D2 lines of Cs in He, Ne, Ar, Kr,
Xe, H2, and N2. (C6 and C&2 are in units of 10 ergcm and 10 ergcm ).

System
Di(8943 A)

C&2 c,
Di(8521 A)

C&2

Cs-He

Cs-Ne

Cs-Ar

Cs-Kr

Cs-Xe

Cs-H2

Cs-N2

0.87+ 0.26
0.37
0.76

-1.04

1.57+ 0.28
0.71
1.66
0.91

5.40 + 0.38
2.94
7.49

20.0

13.16+ 0.53
4.44

12.19

9.18+ 0.31
7.19

20.90
9.2

-2.16+0.71

26.91+0.43

-0.245+ 0.31

40.0
-0.43

34.0 + 5.6

340.0
11.0

84.22 + 15.1

143.76
8400

16 360

243.86

308.22

423.348
11.8

29.52+ 82

26 760

0.66+ 0.28
0.42

-0.32
-0.43

0.64 + 0.13
0.81

-0.52
1.2

4.73+ 0.65
3.35

-0.41

3.85

7.29+ 0.42
7.14
0.55

249.27
11.59
3.74

3.35+ 0.64

15.10 + 0.51

—3.38+ 0.61

-2.12

0.76 + 0.36

-19.67
11.0

55.20 + 8.60

-3.41

7.69

461.8 + 20.5

3.279

9.89x 106

-2.74

56.9 + 10.1

17460

Ref. 22
Ref. 21
Ref. 23b

Ref. 22
Ref. 21
S.P. Ch'en~

Ref. 22~

Ref. 21
S.Y. Ch en~
Ref. 23

Ref. 22
Ref. 21

Ref. 22
Ref. 21
Ref. 23b

Theoretical values.
Experimental value.
S. Y. Ch'en, in Proceedings of the International Conference on Optical Pumping and

Atomics I ines Shape, edited by T. Skalinski (Panstwowe Wydawnictwo Naukowe, Warsaw,
1969), p. 403.

estimate the range of applicability of the straight-
path approximation. The orbit will be significantly
curved if the impulse of the force exerted between
the colliding atoms becomes of the order of the
momentum of either of them. The relevant ratio
p is therefore

P (6/g)(R/v'C, )' ', (7)

if p-1 the path is significantly curved and it is
necessary to use the quantum theory of Baranger. '
If p «1 the path is practically unaltered and the
semiclassical theory can be used. Substituting in
expression (7) the typical values of C, for the sys-
tems of our interest leads to the following values:

Gas He Ne Ar Kr Xe H, N,
P 0.2 0.06 0.01 0.009 0.006 0.3 0.02 .
From these considerations, we analyze our ex-

perimental results using the semiclassical theory
of Lindholm-Foley for all the systems studied by
us except for Cs-He, and Cs-H, where we use the
quantum theory of Baranger. In 'Table IV we show
the calculated values of the interatomic potential

constants for the Dy D2 lines of Cs interacting
with the different buffer gases. For Cs-noble gas
systems our values of the C, constant for the Dy
line are in agreement with the theoretical values
obtained by Baylis ', for the D2 line such values
are in agreement with the theoretical values of
Mahan and the empirical values pf Jacpbspn.
The values of the constant Qy2 are in disagreement
ampng all the authprs, ' " but the values pf this
constant have very little influence on the shifts and
broadenings of the lines in the theories employed
by us.

V. SHIFT AND BROADENING OF THE HYPERFINE
COMPONENTS OF Di LINE

The general theory of collision broadening and
shift of the hfs components of atomic spectral lines
has been developed by Omont" and Rebane. " The
shift and broadening of the hyperfine components
are determined by the eigenvalues of the collision-
al relaxation matrix of the electronic-nuclear
multipole moments. If we consider the transition
j=2 j'= —,', (D, line) we find for the impact ap-



22 SHIFT AND BROADENING OF HYPERFI1VE COMPONENTS OF ~ ~ ~ 2695

TABLE V. Theoretical and experimental values of shift and broadening of hfs components
of D& line of Cs in Cs-gas system (in units of 10 cm-'/Torr).

System
Broaden&no

Theoretical Experimental Theoretical
Shift

Experimental

Cs-Ar
Cs-Kr
Cs-Xe

0.5
0.51
0.57

0.66 + 0.04
0.66 + 0.08
0.72 + 0.09

-0.18
-0.18
-0.21

-0.29+ 0.03
-0.29+ 0.01
-0.27 + 0.04

~ Reference 25.

proximation two purely electronic constant relaxa-
tions, and if very weak magnetic interactions are
neglected one obtains'4 an equal value for the shift
and broadening in both hfs components.

In Table V we show the theoretical values of
shift and broadening of the hfs components of the

Qj line of cesium -heavy noble gas system s calcu-
lated by Rebane. " These values are somewhat
lower than the ones calculated by us by a weighted
average from our experimental results for the

(i+, i ) components of the D, line. 26 The experi-
mental shift shown for the Cs-Kr system corres-
ponds only to the i component, because the ex-
perimental values for the two hfs components are
in disagreement. For other systems measured by
us (Cs-He, Cs-Ne, Cs-H„and Cs-N, ) the values
for the shift and broadening of hfs (i„i ) compon-
ents, are also in disagreement, but they are not
shown in the table.

For a review see S. Y. Ch'en and M. Takeo, Rev. Mod.
Phys. 29, 20 (1957); and R. G. Breene, Sr. , The Shift
and Shape of Spectral Lines (Pergamon, New York,
1963), among others.

(a) S. Y. Ch'en and R. O. Garret, Phys. Rev. 144, 59
(1966); (b) R. O. Garret and S. Y. Ch'en, ibid. 144, 66
(1966); (c) S. Y. Ch'en, E. C. Looi, and R. Garret,
ibid. 155, 38 (1967); (d) R. O. Garret, S. Y. Ch'en,
and E. C. Looi, ibid. 156, 48 (1967); (e) S. Y. Ch'en,
D. E. Gilbert, and D. K. L. Tan, ibid. 148, 51 (1960);
(f) D. E. Gilbert and S. T. Ch'en, ibid. 188, 40 (1969);
(g) F. Rostas and J. L. Lemaire, J. Phys. B 4, 555
(1971).

S. Y. Ch'en, E. L. Lewis, and D. N. Stacey, J. Phys.
B 2, 275 (1969).

4Y. V. Evdokimov, Opt. Spektrosk. 24, 832 (1968) tOpt.
Spectrosc. (USSR) 24, 448 (1968)].

56. Smith, J. Phys. B 8, 2273 (1975).
SE. Bernabeu, F. Garcia Peralta, and J. M. Alvarez,
J. Opt. Soc. Am. 67, 241 (1977).

VJ. M. Alvarez, Ph. D. thesis, University of Zaragoza,
Spain, 1977 (unpublished); E. Bernabeu and J. M.
Alvarez, in Proceedings of the Eleventh Congress
of the International Commision for Optics, edited by
J. Bescos, A. Hidalgo, L. Plaza, and J. Santamaria
(Sociedad Espanola de Optica, Madrid, 1978), p. 476.
E. Bernabeu and J. M. Alvarez, Opt. Pur. Apl. 12, 113
(1979).

9J. M. Alvarez and E. Bernabeu, Opt. Pur. Apl. 11, 99
(1978).
Zhe experimental fits of these absorption coefficients
to a Voigt profile give a standard deviation of -6 x 10+.
O. S. Heavens, J.Opt. Soc. Am. 10, 1058 (1961); J. K.

Link, ibad. 9, 1195 (1966).
C. L. Chen and A. V. Phelps, Phys. Rev. 173, 62
(1968).
For the pressure range employed, the theoretical pre-
dictions give a linear dependence of the shift and
broadening with the pressure.
D. A. Jackson, Proc. R. Soc. London 147, 2095 (1969);
H. Kleiman, J. Opt. Soc. Am. 52, 441 (1961).

SR. Granier, J. Granier, and F. Schuller, J. Quant.
Spectrosc. Radiat. Transfer 16, 143 (1976).
J. F. Kielkopf, J. Phys. B 9, L547 (1976).
E. Lindholm, Ark. Mat. Astron. Fys. 32A, 17 (1945);
H. M. Foley, Phys. Rev. 69, 616 (1946).

fSETT-w'. Behmenburg, J. Quant. Spectrosc. Radiat. Transfer
4, 177 (1964); W. R. Hindmarsh, A. D. Petford, and
G. Smith, Proc. R. Soc. London 247, 296 (1967).
W. R. Hindmarsh and J. M. Farr, Progress in Quantum
Electronics (Pergamon, New York, 1973).
M. Baranger, Phys. Rev. 111, 494 (1958); 11]., 481
(1958).
W. E. Baylis, J. Chem. Phys. 51, 2665 (1969); JILA,
Report No. 100, University of Colorado, 1969 (unpub-
lished).
G. D. Mahan, Chem. Phys. 50, 2735 (1969).
(a) H. C. Jacobson, Phys. Rev. A 4, 1363 (1971); (b)
4, 1368 (1971).
A. Omont, J. Phys. (Paris) 34, 179 (1973).
V. N. Rabane, Opt. Spektrosk. 41, 372 (1976) [Opt.
Spectrosc. (USSR) 41, 214 (1976)l; Opt. Spektrosk. 41,
894 (1976) IOpt. Spectrosc. (USSR) 41, 526 (1976)l.
This can be due, in part, to the fact that the interatom-
ic potential considered has only a van der Waals term.