Crystalline and amorphous polymeric solid nitrogen
The structure and thermodynamic functions of solid high-density nonmolecular nitrogen in the crystalline and amorphous phases are studied by a Monte Carlo simulation technique on the basis of the potential model proposed earlier for the cubic gauche polymeric crystal. The solid amorphous state was c...
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irk-123456789-1289252018-01-15T03:03:03Z Crystalline and amorphous polymeric solid nitrogen Yakub, L.N. Low-Temperature Thermodynamics and Structure The structure and thermodynamic functions of solid high-density nonmolecular nitrogen in the crystalline and amorphous phases are studied by a Monte Carlo simulation technique on the basis of the potential model proposed earlier for the cubic gauche polymeric crystal. The solid amorphous state was created by fast melting followed by instant quenching of the crystalline structure. The computed atom-atom distribution functions in such solids are characteristic for amorphous structure. The simulation also reveals the negativity of thermal expansion coefficient of amorphous solid at high density and low temperatures, as was previously found to be the case in the cubic gauche nonmolecular nitrogen crystal. Analysis of the force model shows specific anharmonicity characteristics of the crystalline vibrations responsible for this effect. 2003 Article Crystalline and amorphous polymeric solid nitrogen / L.N. Yakub // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 1032-1035. — Бібліогр.: 10 назв. — англ. 0132-6414 PACS: 82.35.+t, 34.20.-b, 65.70.+y http://dspace.nbuv.gov.ua/handle/123456789/128925 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Low-Temperature Thermodynamics and Structure Low-Temperature Thermodynamics and Structure Yakub, L.N. Crystalline and amorphous polymeric solid nitrogen Физика низких температур |
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The structure and thermodynamic functions of solid high-density nonmolecular nitrogen in the crystalline and amorphous phases are studied by a Monte Carlo simulation technique on the basis of the potential model proposed earlier for the cubic gauche polymeric crystal. The solid amorphous state was created by fast melting followed by instant quenching of the crystalline structure. The computed atom-atom distribution functions in such solids are characteristic for amorphous structure. The simulation also reveals the negativity of thermal expansion coefficient of amorphous solid at high density and low temperatures, as was previously found to be the case in the cubic gauche nonmolecular nitrogen crystal. Analysis of the force model shows specific anharmonicity characteristics of the crystalline vibrations responsible for this effect. |
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Article |
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Yakub, L.N. |
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Yakub, L.N. |
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Yakub, L.N. |
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Crystalline and amorphous polymeric solid nitrogen |
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Crystalline and amorphous polymeric solid nitrogen |
| title_full |
Crystalline and amorphous polymeric solid nitrogen |
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Crystalline and amorphous polymeric solid nitrogen |
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Crystalline and amorphous polymeric solid nitrogen |
| title_sort |
crystalline and amorphous polymeric solid nitrogen |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2003 |
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Low-Temperature Thermodynamics and Structure |
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http://dspace.nbuv.gov.ua/handle/123456789/128925 |
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Crystalline and amorphous polymeric solid nitrogen / L.N. Yakub // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 1032-1035. — Бібліогр.: 10 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT yakubln crystallineandamorphouspolymericsolidnitrogen |
| first_indexed |
2025-07-09T10:14:32Z |
| last_indexed |
2025-07-09T10:14:32Z |
| _version_ |
1837163950093893632 |
| fulltext |
Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10, p. 1032–1035
Crystalline and amorphous polymeric solid nitrogen
L.N. Yakub
Department of Thermophysics, Odessa State Refrigeration Academy
1/3 Dvoryanskaya, Odessa 65026, Ukraine
E-mail: Lidiya@unive.odessa.ua
The structure and thermodynamic functions of solid high-density nonmolecular nitrogen in the
crystalline and amorphous phases are studied by a Monte Carlo simulation technique on the basis
of the potential model proposed earlier for the cubic gauche polymeric crystal. The solid amor-
phous state was created by fast melting followed by instant quenching of the crystalline structure.
The computed atom-atom distribution functions in such solids are characteristic for amorphous
structure. The simulation also reveals the negativity of thermal expansion coefficient of amorphous
solid at high density and low temperatures, as was previously found to be the case in the cubic
gauche nonmolecular nitrogen crystal. Analysis of the force model shows specific anharmonicity
characteristics of the crystalline vibrations responsible for this effect.
PACS: 82.35.+t, 34.20.–b, 65.70.+y
1. Introduction
McMahan and LeSar [1] and Martin and Needs [2]
first predicted the existence of a stable polymeric
(nonmolecular or atomic) phase of solid nitrogen at
zero temperature under pressure. They considered a
number of possible threefold-coordinated structures
and found the arseniclike A7 lattice to be the most sta-
ble one at T = 0. Later Mailhiot, Yang, and McMahan
reported new ab initio pseudopotential total-energy
calculations [3]. They indicate that a «cubic gauche»
(cg) distortion of simple cubic atomic structure, in
which all nitrogen atoms are threefold coordinated,
has an equilibrium total energy lower than that of
the arseniclike A7 structure mentioned above. Accord-
ing to Ref. 3, the diatomic form of nitrogen should
transform into this hypothetical polymeric cg phase at
50±15 GPa and remain the most stable nonmetallic
form of nitrogen at higher densities.
This polymeric phase of solid nitrogen, long only
discussed by theoreticians [1–3], was recently discov-
ered by experimentalists [4] at ambient temperatures
and has also been investigated at elevated tempera-
tures [5]. It was found that this phase very likely has
the cg structure and is not crystalline but amorphous.
In our earlier calculations [6–8], we predicted the
thermodynamic functions of polymeric nitrogen in the
A7 arseniclike structure at nonzero temperatures using
simple atom-atom potential model. Recently [9] we
proposed a new potential model representing the total
energy of polymeric nitrogen as a function of both in-
teratomic distances and angles between single chemi-
cal bonds attached to each atom. This model was used
in prediction of thermodynamic behavior of solid ni-
trogen in its polymeric cg crystalline phase at high
pressures and nonzero temperatures. The calibration of
the model was based on the ab initio quantum me-
chanical calculations of Mailhiot, Yang, and
McMahan [3] at T = 0. We applied this model in
Monte Carlo (MC) computer simulations of poly-
meric solid nitrogen at elevated temperatures.
Our simulations revealed that the polymeric cubic
gauche crystalline structure of nitrogen would proba-
bly exhibit quite unusual high-pressure behavior at
elevated temperatures. In particular, negative values
of the thermal expansion coefficient at high pressures
were predicted [9] over a wide range of temperatures.
This feature may play an important role in the estima-
tion of the relative stability of different crystalline
structures and in the prediction of polymorphous tran-
sitions at elevated temperatures. Here we report the
extension of the potential model proposed in Ref. 9 to
simulation of amorphous polymeric solid nitrogen.
2. Potential model
According to this model [9] the potential energy of
N interacting nitrogen atoms consists of two contribu-
tions:
U r U LN ij ij i j� ���( ) ( , , )
nonbonded atoms bonded atoms
θ θ� . (1)
© L.N. Yakub, 2003
The first sum in Eq. (1) is taken over all nonbonded
atoms, and the second contribution is a partial sum
over all nearest chemically bonded atoms, �( )R �
� �AR /13 2 is the nonvalence atomic repulsion [4]
(A/k = 3.86·105 K, and k is the Boltzmann’s con-
stant). Each atom is involved in three such chemical
(valence) interactions. In contrast with our older
model [4–6], valence potential U depends here not
only from interatomic distances but also from valence
angles between chemical bonds, attached to each in-
teracting atom.
Valence interaction of two single-bonded atoms U
is a function of the chemical bond length Lij as well as
the angular variables θ i i i i� { , , }( ) ( ) ( )� � �1 2 3 , and θ j �
� { , , }( ) ( ) ( )� � �j j j
1 2 3 , which represent angles between
bonds attached to each atom.
Interatomic valence forces act between pairs of
neighboring single-bonded atoms only. The corre-
sponding interaction energy U (Lij, θ i , θ j ) was rep-
resented by the product of three factors:
U L q L q L u Lij i j ij i ij j M ij( , , ) ( , ) ( , ) ( )θ θ θ θ� � � , (2)
where uM(Lij) is the Morse potential and Lij is the
length of a single bond:
u L D L R L RM e e e( ) [exp { ( )} exp { ( )}].� � � � � �2 2� �
(3)
The factors q Lij i( , )θ in Eq. (2) depend on both the
bond lengths and the three plane valence angles
θ � { , , }( ) ( ) ( )� � �1 2 3 between three bonds attached to
atom (see Fig. 1 in Ref. 9):
q R
z R
( , )
exp { ( , )}
θ
θ
�
� �
2
1
. (4)
Here
z R R R c ci e i e( , ) ( ) ( )θ � � � � �
2 2
� � � � �� �( )( ) ( )R R c c ce i e i i1 4
, (5)
ci i i i� � �
1
3
2 1 2 2 2 3(cos cos cos )( ) ( ) ( )� � � , (6)
i i i i i� � � � �[cos cos ] [cos cos ]( ) ( ) ( ) ( )2 1 2 2 2 2 1 2 3 2� � � �
� �[cos cos ]( ) ( )2 2 2 3 2� �i i . (7)
If the atomic configuration is near the equilibrium
one, corresponding to minimum energy, all z R i( , )θ
tend to zero and all q Lij j( , )θ approach unity. The
equilibrium values of the valence angles ( )( )
� i
j
eq �
� 114 � in a free cg crystal at zero temperature [3] cor-
respond [9] to ce = 0.165.
The model has seven adjustable parameters: De,
Re, , �,
,�, and � (De is the depth of the potential
well or equilibrium bonding energy, and Re is the
equilibrium length of a Morse potential for single
bond). We used here the same values of parameters:
De/k = 35570 K, Re = 1.374 Å, = 12 Å
–2, � = 6.133,
= –14.25, and � = –118 Å
–1 as were found in Ref. 9.
This set of parameters reproduces i) the ab initio
total energy in the cubic gauche structure at T = 0 as a
function of volume, ii) the equilibrium bond angles,
and iii) the angular dependence of the total energy
within 103� < �( )i < 114�, reported as a result of the sen-
sitivity study by Mailhiot, Yang, and McMahan [3].
3. Monte Carlo simulation of amorphous solid
Computer simulations were carried out using the
same computer MC simulation program developed for
study of the cg crystal lattice [9] (512 nitrogen atoms
and periodic boundary conditions). Initially, all the
atoms were arranged within main cell near the sites of
a perfect cg lattice (see Ref. 9 for a more detailed de-
scription of the cg structure). The MC simulation was
initiated with a high initial temperature well above
the melting temperature. After some period the initial
long-range order disappears and the solid melts. Note
that the chemical bonds were fixed during the entirety
of the computer experiment and we kept a restriction
on allowable distances between bonded atoms (less
than 2.5 Å).
Then the system was quenched (the temperature is
suddenly reduced to room temperature) and the MC
simulation process continues. It was found that after
some period of relaxation (after 2000–3000 success-
ful steps per atom) a new quasi-equilibrium state re-
taining amorphous structure was reached. It keeps
the short-range order and threefold valence coordina-
tion in first coordination sphere but also has some
disorder in bond lengths and directions. This
quasi-equilibrium state was studied as a model for the
amorphous solid.
During the MC simulation process (namely after
each three steps per atom) the values of pressure, in-
ternal energy, isothermal compressibility, thermal ex-
pansion, and heat capacity were calculated. All of
the thermodynamic functions, along with the bond
lengths and angles, were averaged over the next 40000
steps per atom to compute the mean values as well as
their statistical errors. We also computed the radial
atomic distribution functions. Along with the mean
values of the angular variables we also monitored the
minimal and maximal angles between interatomic
bonds.
4. Results and discussion
The atom–atom distribution functions of polymeric
crystalline and amorphous nitrogen solid at high den-
Crystalline and amorphous polymeric solid nitrogen
Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 1033
sity (V = 6 cm3/mol) and two temperatures are com-
pared in Fig. 1. As one can see, the long-range order
in the amorphous solid is completely lost, but the
short-range first coordination peak formed by chemi-
cally bonded atoms is clearly expressed. The second
coordination sphere, formed by closest nonbonded at-
oms, is also present but much less noticeable compared
to the crystalline solid. The position of this peak is
closely related (see below) to the mean valence angle
between single bonds attached to each atom.
In Table we compare calculated values of the ther-
mal expansion T and isothermal compressibility �T
coefficients, as well as the isochoric heat capacity Cv,
and the mean valence angles � �� and bond lengths � �L
of crystalline and amorphous polymeric nitrogen at
T = 300 and 500 K. As one can see, according to our
model, the negativity of thermal expansion should
persist in the amorphous state of nonmolecular nitro-
gen solid at high densities and lower temperatures. It
is obviously related to the short-range pyramidal
structure formed by neighboring nitrogen atoms. As
the mean distance between closest nonbonded atoms
increases (see Fig. 1, T = 500 K), thermal expansion
became less positive.
Table
Some properties of solid nitrogen in the crystalline and
the amorphous polymeric phases at V = 6 cm3/mol
Property 300 K 500 K
*
Expansivity T ,
kK–1
Crystalline
Amorphous
–0.018
–0.011
–0.014
–0.003
0.002
Compressibility
�T , GPa–1
Crystalline
Amorphous
0.0017
0.0015
0.0018
0.0012
0.0001
Heat capacity
C
v
/R
Crystalline
Amorphous
3.07
3.11
3.10
3.08
0.003
Mean valence
angle � �� , degrees
Crystalline
Amorphous
106.3
103.6
106.2
105.4
0.2
Mean bond
length � �L , Å
Crystalline
Amorphous
1.37
1.52
1.38
1.42
0.02
* Estimated statistical error.
It is well known that the anharmonicity of crystal-
line vibrations is responsible for thermal expansion.
The rigorous analysis of all anharmonic contributions
to the energy of vibration modes is a complicated task.
It requires much effort and is well beyond the scope of
this work. However, we have performed an examina-
tion of symmetric longitudinal vibrations as the most
important kind of crystal vibration in this regard. Our
simplified analysis has been performed for the adopted
force model within the cg lattice. We took into ac-
count that the chemical bonds between atoms are
much stiffer than the nonvalence interatomic repul-
sion. Therefore a symmetric squeezing of the crystal
decreases the valence angles between single bonds and
correspondingly decreases the distance between near-
est nonbonded atoms (see Fig. 1 for the positions of
the second peaks near R = 2.3 Å).
In Fig. 2 we compare the harmonic and anharmonic
contributions to the variation of interaction energy
with this distance. It is clearly seen that the anhar-
monic contribution to the interaction energy within
our model behaves just opposite to the usual depend-
ence of the potential energy in diatomic molecules,
where anharmonicity leads to elongation of the chemi-
cal bonds with increase of temperature. In Fig. 2 the
interaction energy is more negative at short inter-
atomic distances than its harmonic counterpart. The-
refore, as temperature increases, the mean distance be-
1034 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10
L.N. Yakub
T = 300 K
Distance, Å
R
a
d
ia
l d
is
tr
ib
u
tio
n
f
u
n
ct
io
n
1.0 2.0 3.0 4.0 5.0
0
4
8
12
16
20
a
T = 500 K
Distance,Å
1.0 2.0 3.0 4.0 5.0
R
a
d
ia
l d
is
tr
ib
u
tio
n
f
u
n
ct
io
n
0
4
8
12
16
20
b
Fig. 1. Radial distribution functions of the amorphous
(solid line) and crystalline (dashed line) polymeric nitro-
gen at 6.0 cm3/mol and two temperatures 300 and 500 K.
tween nonbonded atoms should decrease. We believe
that the negative thermal expansion predicted is due
to this specific anharmonicity in our potential model.
5. Conclusions
Anomalous structural and thermodynamic behavior
of polymeric fourfold-coordinated solids like ice, sili-
con, carbon etc. has been known for a long time. Nega-
tive thermal expansion has also been predicted re-
cently in molecular-dynamic simulations of silicon
both in the crystalline and amorphous states [10]. Ac-
cording to our simulation this peculiarity appears in
threefold-coordinated structures like polymeric nitro-
gen only in the helical cg structure and not in the lay-
ered A7 structure.
The radius of the second coordination sphere in the
amorphous cg solid at T = 300 K is close to that for the
crystalline (see Fig. 1). On the contrary, at T = 500 K
this radius is markedly shifted to larger distances,
which leads to larger mean valence angles and the
thermal expansion at 500 K is less negative.
It should be noted that the amorphous state exam-
ined in this work as well as the real amorphous nitro-
gen solid studied experimentally [4,5] is actually not
the real thermodynamic equilibrium state. Its proper-
ties depend on the frozen structure inherited from the
high-temperature melt. Fast quenching of a high-tem-
perature polymeric melt can produce different amor-
phous states having slightly different properties.
We have chosen several samples at V = 8.04 and
6 cm3/mol and selected two of them having no bro-
ken bonds, i.e. having lengths close to limiting 2.5 Å,
see above. We have also continuously monitored the
arithmetic mean as well as the maximum and mini-
mum values of the bond lengths and angles between
bonds to avoid such bond breaking after quenching.
At T = 300 K and V = 6 cm3/mol all the thermody-
namic characteristics stabilize quickly and remain con-
stant during the continuance of the computer simula-
tion run. The constancy of the heat capacity computed
is another strong piece of evidence of stability of the
amorphous state under investigation.
This is not the case at higher temperatures and vol-
umes. We performed several runs at T = 1000 K which
demonstrate the essential instability of the amorphous
solid and gradual recovery of the crystalline structure.
Probably much longer MC runs could reach the com-
plete or partial annealing of a stable crystalline struc-
ture. But this process is very slow, and such a study is
far beyond the limit of our facilities.
1. A.K. McMahan and R. LeSar, Phys. Rev. Lett. 54,
1929 (1985).
2. R.M. Martin and R.J. Needs, Phys. Rev. B34, 5082
(1986).
3. C. Mailhiot, L.H. Yang, and A.K. McMahan, Phys.
Rev. B46, 14419 (1992).
4. A.F. Goncharov, E. Grigorianz, H.-K. Mao, Z. Liu,
and R. Hemley, Phys. Rev. Lett. 85, 1262 (2000).
5. E. Grigorianz, A.F. Goncharov, R. Hemley, and H.-K.
Mao, Phys. Rev. B64, 052103 (2001).
6. L.N. Yakub, Fiz. Nizk. Temp. 19, 531 (1993) [Low
Temp. Phys. 19, 377 (1993)].
7. L.N. Yakub, Fiz. Nizk. Temp. 21, 1174 (1995) [Low
Temp. Phys. 21, 905 (1995)].
8. L.N. Yakub, Fiz. Nizk. Temp. 22, 222 (1996) [Low
Temp. Phys. 22, 174 (1996)].
9. L.N. Yakub, J. Low Temp. Phys. 122, 501 (2001).
10. P. Keblinski, M.Z. Bazant, R.K. Dash, and M.M.
Treacy, Phys. Rev. B66, 064104 (2002).
Crystalline and amorphous polymeric solid nitrogen
Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 1035
E
n
e
rg
y,
a
rb
. u
n
its
2.34 2.35 2.36 2.37
R
R, Å
Fig. 2. Potential energy of cg crystalline nitrogen as a
function of the interatomic distance between two non-
bonded atoms at volume V = 6.0 cm3/mol (solid line).
The harmonic approximation is presented by the dashed
line.
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