Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon
Infrared emission bands in the wavelength range of 6–12 μm observed in the ISO-SWS mission are assigned to rotational zero-phonon bands of solid parahydrogen by using Van Kranendonk’s approximate rigid-lattice method. This method is based on superposed electric quadrupole pair interactions and super...
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irk-123456789-1171272017-05-21T03:03:16Z Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon Schaefer, J. 7th International Conference on Cryocrystals and Quantum Crystals Infrared emission bands in the wavelength range of 6–12 μm observed in the ISO-SWS mission are assigned to rotational zero-phonon bands of solid parahydrogen by using Van Kranendonk’s approximate rigid-lattice method. This method is based on superposed electric quadrupole pair interactions and superposed quadrupole induced dipole moments of pairs in the hcp crystal. Accordingly, the approximate formalism uses zero-order H₂ pair wave functions. Symmetry effects of the hcp crystal require preference of rotational pair transitions. The interaction potential of the pairs is confined to the electric quadrupole–quadrupole interaction. Zero-phonon emission bands of H₂ pair transitions fitted to the spectrum contain at least one delocalized j = 2 state initially and/or finally because of their significantly enhanced emission rates. They also yield the characteristic band widths which fit nicely to the widths of the observed features. The frequency positions of the seven pure parahydrogen pair transitions used, obtained from experimentally determined rotational solid hydrogen energy levels, are in perfect agreement with the observed features, whereas the three mixed ortho–para pair transitions need a presently unknown frequency correction, caused by the migration of the ortho-H₂ molecules into the parahydrogen crystal prior to emission, the so-called initial excess binding energies. The astrophysical setup of the observed source is discussed in the end of the paper. Remove selected 2009 Article Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon / J. Schaefer // Физика низких температур. — 2009. — Т. 35, № 4. — С. 405-412. — Бібліогр.: 10 назв. — англ. 0132-6414 PACS: 95.30.Gv, 63.20.D– http://dspace.nbuv.gov.ua/handle/123456789/117127 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals |
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7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals Schaefer, J. Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon Физика низких температур |
| description |
Infrared emission bands in the wavelength range of 6–12 μm observed in the ISO-SWS mission are assigned to rotational zero-phonon bands of solid parahydrogen by using Van Kranendonk’s approximate rigid-lattice method. This method is based on superposed electric quadrupole pair interactions and superposed quadrupole induced dipole moments of pairs in the hcp crystal. Accordingly, the approximate formalism uses zero-order H₂ pair wave functions. Symmetry effects of the hcp crystal require preference of rotational pair transitions. The interaction potential of the pairs is confined to the electric quadrupole–quadrupole interaction. Zero-phonon emission bands of H₂ pair transitions fitted to the spectrum contain at least one delocalized j = 2 state initially and/or finally because of their significantly enhanced emission rates. They also yield the characteristic band widths which fit nicely to the widths of the observed features. The frequency positions of the seven pure parahydrogen pair transitions used, obtained from experimentally determined rotational solid hydrogen energy levels, are in perfect agreement with the observed features, whereas the three mixed ortho–para pair transitions need a presently unknown frequency correction, caused by the migration of the ortho-H₂ molecules into the parahydrogen crystal prior to emission, the so-called initial excess binding energies. The astrophysical setup of the observed source is discussed in the end of the paper.
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Schaefer, J. |
| title |
Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon |
| title_short |
Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon |
| title_full |
Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon |
| title_fullStr |
Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon |
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Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon |
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zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. an astrophysical phenomenon |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2009 |
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7th International Conference on Cryocrystals and Quantum Crystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117127 |
| citation_txt |
Zero-phonon emission bands of solid hydrogen at 6-12 μm wavelength. An astrophysical phenomenon / J. Schaefer // Физика низких температур. — 2009. — Т. 35, № 4. — С. 405-412. — Бібліогр.: 10 назв. — англ. |
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Физика низких температур |
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AT schaeferj zerophononemissionbandsofsolidhydrogenat612mmwavelengthanastrophysicalphenomenon |
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2025-07-08T11:41:36Z |
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2025-07-08T11:41:36Z |
| _version_ |
1837078831860547584 |
| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, No. 4, p. 405–412
Zero-phonon emission bands of solid hydrogen at
6–12 �m wavelength. An astrophysical phenomenon
J. Schaefer
Max-Planck-Institut für Astrophysik, 1 Karl-Schwarzschild-Strasse, 85741 Garching, Germany
E-mail: jas@mpa-garching.mpg.de
Received January 27, 2009
Infrared emission bands in the wavelength range of 6–12 �m observed in the ISO-SWS mission are as-
signed to rotational zero-phonon bands of solid parahydrogen by using Van Kranendonk’s approximate
rigid-lattice method. This method is based on superposed electric quadrupole pair interactions and superposed
quadrupole induced dipole moments of pairs in the hcp crystal. Accordingly, the approximate formalism uses
zero-order H2 pair wave functions. Symmetry effects of the hcp crystal require preference of rotational pair
transitions. The interaction potential of the pairs is confined to the electric quadrupole–quadrupole interaction.
Zero-phonon emission bands of H2 pair transitions fitted to the spectrum contain at least one delocalized j = 2
state initially and/or finally because of their significantly enhanced emission rates. They also yield the charac-
teristic band widths which fit nicely to the widths of the observed features. The frequency positions of the
seven pure parahydrogen pair transitions used, obtained from experimentally determined rotational solid hy-
drogen energy levels, are in perfect agreement with the observed features, whereas the three mixed ortho–para
pair transitions need a presently unknown frequency correction, caused by the migration of the ortho-H2 mole-
cules into the parahydrogen crystal prior to emission, the so-called initial excess binding energies. The astro-
physical setup of the observed source is discussed in the end of the paper.
PACS: 95.30.Gv Radiation mechanisms; polarization;
63.20.D– Phonon states and bands, normal modes, and phonon dispersion.
Keywords: emission band, zero-phonon emission, quadrupole–quadrupole interaction, ortho–para transition.
1. Introduction
Zero-phonon emission bands of solid hydrogen have
not been measured yet in the laboratory, but an interesting
information emerged which could initiate new activities
in this particular field: the appearance of observed emis-
sion bands hypothetically related to dark baryonic matter
in the Milky Way and in other spiral galaxies. Therefore,
it seems to be an actual requirement to discuss a theoreti-
cal forecast of the simplest zero-phonon emission bands
of solid hydrogen, meaning rotational transitions of H2
molecules in the crystal which emit in the wavelength
range of interest. This subject has been discussed already
in a recently published paper of Schaefer (2007) [1].
There are three crucial aspects of the applied rigid-lat-
tice approximation of Van Kranendonk [2] to be briefly
introduced prior to the discussion of the characteristic
features in the wavelength range between 6 and 12 �m.
At first, a rigorous pair approximation is assumed,
based upon the validity of superposed pair interactions
and superposed pair dipole moments of the H2 molecules
in the lattice and approximately confined to electric quad-
rupole interactions.
At second, the symmetry properties of the solid para-
hydrogen hcp crystal cause a characteristic cancellation
effect valid in the rigid-lattice dipole moment function for
single rotational transitions, whereas rotational pair tran-
sitions are not affected. The rotational H2 pair transition
is a dipole transition of two rotating H2 molecules at near-
est neighbor (nn) distance, where either one or both mole-
cules undergo a rotational transition to emit or absorb one
photon. Consequently, prominent dipole transitions are
pair transitions, whereas single rotationally excited H2
states in the ground-state crystal — with the exception of
the j = 2 state — can be neglected in the discussion of
prominent dipole radiation.
At third, since H2, H2 pairs and the parahydrogen crys-
tal are Bose systems, the electric quadrupole–quadrupole
interaction makes the j = 2 state energy «hopping»
through the parahydrogen crystal; it is therefore called
© J. Schaefer, 2009
«delocalized j = 2 state». If contributed to a pair transition
initially and/or finally, it makes a great effect on the
squared dipole transition matrix element, with an en-
hancement factor called «lattice sum». Therefore, pair
transitions without a contributing delocalized j = 2 state
are less important and may be expected to contribute to
the background. Additionally, the typical band widths of
the zero-phonon pair emission bands used are mainly
caused by the energy spread of the delocalized j = 2
states in the lattice.
Details of Van Kranendonk’s rigid-lattice approxima-
tion will be discussed in the following sections, i.e., the
zero-order pair wave functions in Sec. 2, the dipole transi-
tions of H2 pairs in Sec. 3, widths and shifts of the emis-
sion bands in Sec. 4, and the effects caused by the depo-
larized j = 2 state in Sec. 5.
The actual zero-phonon emission spectroscopy is
started in Sec. 6 by presenting the spectrum observed in
the ISO-SWS mission and showing the discussed features
between 6 and 12 �m wavelength, of a source located
within the NGC7023 nebula, a so-called molecular cloud
and photo-dissociation region (PDR) in the Milky Way. I
should briefly mention the present state-of-the-art of the
physical interpretation. The astrophysical community de-
signates the source as mixture of polycyclic aromatic hy-
drocarbons (PAHs), more recently also as carbonaceous
composites, and it could be probably assigned to many
other molecules in the future. There is such a rich diver-
sity of molecules available on Earth or accessible by che-
mical synthesis, that more molecules should be found by
all means fitting somehow to the observed features. The
theoretical and experimental work on the subject summa-
rized under «PAHs» is quite impressive and highly pro-
fessional, nevertheless, this paper is aimed at showing
that the observed features are only emitted by visible hy-
drogen solids.
The prominent features in question have been fitted
by using all available rotational pair transitions in this
wavelength range with participating delocalized j = 2
states initially and/or finally. Seven of the ten prominent
zero-phonon emission bands used are pure parahydrogen
bands, the only ones available for testing the agreement
with the observed features, with regard to frequency posi-
tions and estimated band widths. The frequency position
is then determined by the difference of the experimentally
determined initial and final energy levels. This simple
procedure follows from the symmetry of the parahyd-
rogen pairs. This symmetry does not apply to mixed
ortho–para pairs because the substitutionally included
ortho-H2 impurities cause a distortion of the surrounding
lattice molecules and a so-called «excess binding energy»
of the initial JM pair states, with a frequency shift of the
band. Hence the fits of these pair emission bands are
rather artificial and not convincing.
Emission bands observed on both sides of the dis-
cussed wavelength range have not been included in the
calculations, but their frequency positions help identify
them as «extrapolations» of the prominent parahydrogen
emission bands inside the wavelength range.
Minor important intensities inside and outside the dis-
cussed wavelength range could also be seen as pair transi-
tion bands, with a delocalized j = 2 state sharing a rota-
tional single transition in the first excited vibrational
level. This is more or less speculative and needs experi-
mental proof.
Finally, in Sec. 7 the hypothetical astrophysical setup
of the chosen observed source is briefly explained with
the bright B-type star in the center of a sphere and observ-
able as well as dark hydrogen solids outside. It explains
infrared emission from the surroundings of young stars
wherever they are located, at solar Galactic radius in the
Milky Way or in spiral galaxies at remote regions of the
Universe as observed in the SPITZER mission.
2. Zero-order pair wave functions
The consequence of the pair approximation are
zero-order wave functions of H2 pairs introduced to de-
scribe the physics in the rigid lattice. The formalism is
known quite well from the physics of free pairs, where ab
initio calculated interaction potentials and ab initio in-
duced dipole moment functions have been used in colli-
sion induced radiation. I introduce zero-order pair wave
functions
� JM
jl
j j jl
JM
j j jl
JMj j R f R I( , , ; , ) ( ) ( �R r r r1 2 1 2
1
1
1 2 1 2
� �� , � , � ),r R2
with an effective radial function which is normalized at
the constant nn or rigid-lattice distance R0 of the H2 mol-
ecules at 3.75 �:
dR f R R R
j j jl
JM
jl
� � � �
1 2
2
0 1( ) ( )� ,
hence the pair wave function is also normalized. The an-
gular part shows the coupling of j1 and j2,
I C j j j m m m
j j jl
JM
m m mml
1 2
1 2
1 2 1 2 1 2( � , � , � ) ( ; )r r R � ��
C jlJ mm M Y Y Yl j m j m lml
( ; ) ( � ) ( � ) ( � ),
1 1 2 21 2r r R
where the C s are Clebsch–Gordan coefficients, and the
Y s are spherical harmonics. The angular momenta are
coupled in the order of j j j1 2
� , and j is coupled with
the orbital angular momentum l to the total J . The m s are
magnetic quantum numbers.
Now we must account for the fact the H 2, H 2 pairs and
parahydrogen crystals are Bose systems. I shall note a dif-
406 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4
J. Schaefer
ference: Mixed pairs of para- and ortho-H 2 contain dis-
tinguishable Bose particles, whereas pure parahydrogen
pairs and pairs in the parahydrogen crystal are pairs of in-
distinguishable Bose particles. Their wave functions
must be symmetrized with regard to the exchange of the
molecules, and their interaction potential is symmetric in
both directions.
Symmetrization of the pair wave functions is the basic
condition of describing pairs in the parahydrogen crystal.
Exchange of the molecules gives a new angular wave
function
I
j j jl
JM s
j j v v
1 2 1 2 1 2
2 1 1 2( ) /[ ( )]�
��� �
�
�
[ ( � , � , � ) ( ) ( �I I
j j jl
JM j j j l
j j jl
JM
1 2
1 2
1 2
1 2 11r r R r , � , � )]r R2 .
The two restricted symmetrized products, one for j j1 2�
and the other one for j j2 1� are wave functions of
parahydrogen free ( j j1 2, ) pairs as well as zero-order
wave functions of the pairs in the parahydrogen solids.
The parity is obtained from P
j j l
12 1 1 2� �
( ) which
shows, since j1 and j 2 are even in parahydrogen, that obvi-
ously a change of the orbital angular momentum l is re-
quired for the rotational dipole transitions of the pairs. This
is a curiosity of the rigid-lattice pair approximation. We
deal with that by assuming that any radiative dipole transi-
tion requires l � 0 initially, and the final orbiting energy is
immediately transferred to the lattice, thus contributing to
the phonon branches of the bands. It is a plausible assump-
tion because an anisotropic coupling to the lattice vibra-
tions exists. The final orbital angular momentum is conse-
quently determined by the dipole moment function.
3. Dipole transitions of pair states
Dipole radiation is almost entirely made by the qua-
drupole induction mechanism, where the quadrupole Q2
of molecule 1 polarizes molecule 2, the dipole moment is
located on molecule 2, but it depends upon the orientation
of molecule 1.
The rotationally invariant spherical expansion of the
induced dipole moment function of an H 2 pair can be
written in the form
�
� � �
� �
( , , )
( )
( )
/
R r rij i j L
L
ijB R� ��4
3
3 2
1 2
1 2
�
�
�
��C m m m m C L m m M
m m M
( ; , , ) ( ; , , )� � �1 2 1 2 1 2 1 2
1 2
1� �
�Y r Y r Y Rm i m j LM ij� �1 1 2 2
( � ) ( � ) ( � ),
where the symbols C and Y are the same as already ex-
plained in Sec. 2.
Solid state symmetry effects are important:
– The B L022 and the B L202 terms are responsible for sin-
g le ro ta t iona l t rans i t ions and neg lec ted in the
parahydrogen solids because their induced dipole mo-
ments are proportional to the isotropic polarizability times
the quadrupole moment, and the dipole moments on one
side of the rotating molecule and the diametrically oppo-
site one cancel, therefore, single �j � 2 transition matrix el-
ements are reduced by this so-called «cancellation effect».
The cancellation is total in a perfect fcc crystal.
– By contrast, double rotational transitions of a pair at
nn distance, determined by the B L22� terms with dipole
moments � �, the anisotropic polarizability, are not ef-
fected and are much stronger in the hcp crystal than single
transitions (after Van Kranendonk). I conclude from that.
– Pair transitions dominate the rotational emission
spectra of the H 2 molecules in solid hydrogen.
More specific details of solid hydrogen radiation can
be found by looking at the approximate matrix element of
dipole pair transitions obtained from the B L22� terms.
Integration over the angles gives
J M l JM�� � ��
B R L j j J j j JL
L
j
22 1 2 1 25 1�
�
�( ) [ ][ ][ ][ ][ ][ ][ ][ ]( ) 1 2
j l J �
�
��
�
�
��
�
�
��
�
�
��
�
�
�
�
�
� �
j j j j l L1 1 2 22
0 0 0
2
0 0 0
0
0 0 0
� �
�
�
�
�
�
�
�
�
�
�( ) [ ]
'
1
1
2
2
1 1
2 2
� �
�
L J
j
j
L
J J j
j j
j j
j J
�
�
�
�
�
�
�
�
�
�
�
�
�
��
( )
– –
'1
1J M J J
M M
�
�
,
where [ ]j j�
2 1, and the symbols in parentheses are 3- j,
6- j and 9- j Wigner symbols.
The Wigner symbols of this formula determine two ad-
ditional interesting conditions of the matrix elements:
– the 3- j symbol of the triple {l L' 0} in the second row
requires odd l and odd L �1or 3, but only L � 3(and �l 3!)
is significant, and that determines the final orbital angular
momentum �l 3;
– the triangle {�L1} in the third row yields three � s: 2,
3 and 4, but only B2233 is significant, with Tipping and
Poll [3]:
Zero-phonon emission bands of solid hydrogen at 6–12 �m wavelength. An astrophysical phenomenon
Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 407
B R Q Q R2233
1
2
2
2
1 2 42
15
( ) ( )( ) ( ) ( ) ( )� �
�� � ,
where the numbers in parentheses indicate the average
over the intramolecular motion. Hence, there is only one
term of the dipole moment expansion left in the rigid-lat-
tice approximation.
4. Widths and shifts of the emission bands
The interaction potentials in the rigid lattice are also
obtained from the interaction potential of pairs written in
the rotational invariant form
V V R r rij i j l l K
l l K
ij i j( , , ) ( ) ( , , )/
R r r � ��4 3 2
1 2
1 2
� � �C l l K m m m Y Y Y
m m m
l m i l m j Km ij( ; ) ( � ) ( � ) ( �
1 2 1 2
1 2
1 1 2 2
r r R ).
When considering only quadrupole interactions of the
anisotropic parts and the symmetry effects found for the
dipole moment transitions in the lattice, the only remain-
ing anisotropy is the electric quadrupole–quadrupole
term V224 , proportional to R �5 asymptotically, and ap-
proximately at nn distance. It determines the specific
widths of the rotational zero-phonon emission bands.
Normally several total J and J ' states are defined by a
rotationally excited ( j j1 2, ) pair, as, e.g., initially
| |j j J j j1 2 1 2� � �
.
The potential matrix element of the V224 term for a transi-
tion from JM states to J M states integrated over the
angles gives
J Mj j V JMj j V R j j J JMM1 2 224 1 2 224 1 215� �� ( )[ ][ ] [ ][ ]
�
�
�
��
�
�
��
�
�
��
�
�
��
�
�
�
�
j j j j
j j
j j
J J
1 1 2 2
1 1
2 2
2
0 0 0
2
0 0 0
2
2
4�
�
�
�
�
�
�
� � �
�
( ) ( ; )1 4 01 2j j J M
C JJ M M ,
with the results that M M� is required, whereas J is not
conserved by the EQQ term, therefore, the initial zero-or-
der JM pair states with l � 0 split into components of M
states which has been included in the calculated band
widths of the zero-phonon emission bands. A similar
splitting has been calculated for the final J M states ob-
tained from
| | , | | � � �
� � �
j j j j j j J j1 2 1 2 3 3.
The splitting result in an effective frequency shift of the
bands which is shown below. I skip the details. I may
note, all these splittings are local, bound to the local pair
at nn distance.
5. Effects caused by the delocalized j � 2 state
The largest contribution to the band widths, by far
more significant than all local JM and �J M l 3
splittings, is made by the energy spread of the j � 2 state
over the parahydrogen crystal, therefore called «deloca-
lized j � 2 state», an effect of the quadrupole–quadrupole
interaction term applied to the symmetric pair wave func-
tion of a single j � 2 state in the ground-state crystal
yielding
� � ��02
2 2,m mEQQ �
� �EQQ R d d d I EQQ Im m( ) � � � ; ( � , � , � )r r R R r r1 2 02
2
1 2 20
2 .
The result is an exchange of j � 2 in the angular part of the
matrix element from I 20 to I 02. This matrix element is
called «hopping matrix element» because the energy of
the j � 2 state is transferred to the other pair molecule,
similar to the next one and so on, to the parahydrogen
molecules in the crystal.
The potential terms responsible for single rotational
transitions do not contribute to the «hopping». This is an-
other reason for neglecting them.
Evaluation of the matrix element for the transitions of
the j � 2 state energy between the positions R i and R j
gives the general expression
2 2 70
5
12 2
5
m EQQ n
Q Q
R
i j
ij
m, , ( )R R � � �
� � � �C m n m n Y n m ij( ; ( )) ( � )( )224
4
3
4
R ,
where a transition of the z components from n to m is in-
cluded because of different orientations of the H2 mole-
cules in the lattice.
Solid state theory tells us: The hopping motion of the
j � 2 state through the lattice is described by the Bloch
wave, with the amplitude determined by the hopping ma-
trix elements. But this formalism has not been applied yet
to the parahydrogen crystal. Calculated estimates of the
line widths (provided by Van Kranendonk) and an avail-
able data of an absorption band measured by Balasub-
ramanian et al. (1982) [4] can be used for a roughly esti-
mated general typical emission bandwidth.
It will be generally assumed that each initial and/or fi-
nal j � 2 state at the vibrational H2 ground-state level, oc-
curring in a rotational pair transition �J M JM in the
parahydrogen lattice, contributes with � 20 cm �1 to the
width of this band.
408 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4
J. Schaefer
There is a second important effect of the delocalized
j � 2 state in dipole moment pair transitions, a significant
enhancement factor of the squared dipole moment matrix
elements, called «lattice sum S». It applies to all transi-
tions of pair states with a delocalized j � 2 state initially
and/or finally.
Van Kranendonk used the summation method of Nijboer
and Wette [5] and calculated a lattice sum of 12.8 enhanc-
ing the (local) squared dipole moment matrix elements of
the S S( ) ( )0 0
absorption coefficient by including the in-
duction of the delocalized j � 2 states in about two to three
hundred neighbors of the local pair. Proof has been pro-
vided in a measurement by Kiss [6]. I will not repeat this
procedure, but use Van Kranendonk’s results for a general
estimated enhancement factor: S = 6.4 will be applied to
the rotational squared dipole moment matrix element
J M j j JMj j1 2 1 2
2
�� for each j � 2 state involved ini-
tially and/or finally.
Consequently, rotational pair transitions not including
a j � 2 state molecule are smaller by this factor and are ex-
pected to occur normally in the background. Now I have
all the tools for an approximate calculation of solid hy-
drogen zero-phonon emission bands.
The computational procedure of calculating band
emission profiles starts with the calculation of the local
EQQ splitting and shifts. Then the squared dipole mo-
ment matrix elements of the JM J M� transitions are
calculated, J M j j JMj j1 2 1 2
2
�� with initial l � 0 and
final �l 3, as required by the B2233 term. The correct
zero-phonon approximation would then continue with the
calculation of the effective band width and the effective
«lattice sum S» of each JM transition caused by the
delocalized j = 2 state. Programs for this are not avail-
able, therefore, the provisional estimates of 20 (40) wave-
numbers for the effective width and an enhancement fac-
tor of 6.4 (12.8) are used to obtain the contribution of
each squared matrix element to the band profile of the
pair transition with an assumed Gaussian profile placed at
the experimental band transition frequency and corrected
by the resulting local EQQ shift.
With an intermediate expression for the sum over the
magnetic quantum numbers, over the polarizations, and
over the final total angular momentum [7],
�j j j j J1 2 1 2
2
�
�
��1
2 1
1 2 1 2
2
( )J
J M j j JMj j
M MJ
��
�
,
the emission rate of the zero-phonon band is obtained in
the form
A j j j j
E
h c
e
( ) � � �1 2 1 2
3 3
3 3
232
3
�
�
�
�a
j j
j j j j J
J
0
2
1 2
1 2 1 2
2
2 1 2 1( )( )
� .
The theory is now applied to the infrared emission
spectrum observed in the ISO-SWS mission.
6. Spectroscopy of rotational pair transitions
The source has been the prominent photo-dissociation
region (PDR) of the NGC7023 (Iris) nebula. I will de-
scribe the astrophysical conditions of this radiation at the
end of my paper. The observed spectrum is shown in
Fig. 1.
First to be mentioned are the nicely resolved gas phase
quadrupole transition lines of H2 marked on top of the fig-
ure, ranging from S(1) up to S(5), emitted in locally sepa-
rated gas regions of this prominent PDR, where a mixture
of H and excited H2 gas can be observed. We know they
are emitted from low-density gas sources because col-
lisional relaxation of excited H2 gas interacting with H2
or H prevails already at reasonable gas densities much
smaller then those above the solids.
There are four features marked with roman numbers to
be explained by using ten bands of rotational pair transi-
tions of solid H2. I may note again: these are rotational
pair transitions which include at least one delocalized
j � 2 state initially or finally.
Three different types of pair transitions have been
used:
– both molecules relax in 6 bands,
– the j � 2 state is excited in 3 bands,
– and one band is a single transition S(4) with a j � 2
partner at nn distance.
The transition wavenumbers are shown in Table 1.
Zero-phonon emission bands of solid hydrogen at 6–12 �m wavelength. An astrophysical phenomenon
Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 409
20
30
25
15
10
5
0
600 800 1000 1200 1400 1600 1800
Wavenumber, cm
–1
I
II III IV
0
–
0
S
(1
)
0
–
0
S
(2
)
0
–
0
S
(3
)
0
–
0
S
(4
)
0
–
0
S
(5
)
In
te
n
si
ty
,
1
0
W
/(
cm
H
z)
–
3
0
2
Fig. 1. The emission spectrum of the NGC7023 (Iris) nebula.
Table 1. Wavenumbers of transition
No. Transition �E, cm
–1
I
S(1) + S(0)
S(4) – S(0)
941.47 – x
887.84 (exp)
II
S(2) + S(0)
S(4) with a j = 2 host
1167.1 (exp)
887.84 (exp)
II
S(6) – S(0)
S(3) + S(0)
S(1) + S(2)
1278.18 (exp)
1389.14 – x
1396.9 – x
IV
S(4) + S(0)
S(2) + S(2)
S(8) – S(0)
1599. (exp)
1623.0 (exp)
1625.
Experimentally determined energy levels of H2 in
solid hydrogen up to j � 8 can be used to determine the
frequency positions of the pure parahydrogen emission
bands, i.e., the pair transition frequencies are simply de-
termined by the sum of the experimental S j( ) (�j � 2) en-
ergies marked with «exp». The energy levels of the
ortho-H 2 species observed in absorption as single impuri-
ties or mixed ortho–para pairs cannot be used this way.
The ortho-H2 impurity normally migrates slowly into the
parahydrogen solid, where «slowly» means: the ortho–para
pair emission rates in solid hydrogen may be generally
expected small enough to normally assume a completed
migration process prior to emission. I shall note that the
migration process of the final state starts up after radia-
tion, therefore, only the excess binding energies of the
initial pair states produce a shift of the transition fre-
quency. In more detail, the 2 1j
possible orientations of
an ortho-H2 j state in the lattice give rise to generally
2 1j
different local distortions with Rij distances ! R0 ,
as determined by the orientation-dependent interaction.
Consequently, all the 2 1j
excess binding energies must
be evaluated and averaged.
Significant band widths as well as large excess binding
energies can be expected for mixed ortho–para pairs con-
taining initially an excited ortho-H2 j state and a de-
localized j � 2 state, where contributions to the effective
band widths as well as to the averaged excess binding en-
ergies exceed by far the range of the next shell around the
ortho-H2 impurity. Efforts of doing this computationally
have not yet been started.
The calculated emission profiles are finally expressed
in Jansky units (10 30� W/cm2 Hz) for the comparison
with the measurement and multiplied with an appropriate
scaling factor fitted to the observed band intensity, keeping
in mind that some phonon radiation should be included in
the fits, normally expected on the blue side of the bands.
The Fig. 2 contains the result of this procedure.
I will discuss, first of all, the 7 pure parahydrogen
bands by comparing their frequency positions with the
positions of the observed features and by comparing their
estimated band widths with the required band widths of
the observed features. I want to note: all observed band
widths have been the same at all observed sources!
– The first parahydrogen band in feature I, S(4)–S(0) at
888 wavenumbers, shares the intensity with the mixed
ortho–para S S( ) ( )1 0
band. This is an assumption based
on the expected large excess binding energy of the
S S( ) ( )1 0
band. The width of the S(4)–S(0) band of � 20
wavenumbers is reasonable, but the portions of flux dis-
tributed by the two bands are uncertain.
– Feature II is in perfect agreement with the double
de-excitation band S S( ) ( )0 2
at 1167 wavenumbers,
considering the position and band widths, and an ex-
pected phonon intensity on the blue side.
– The next parahydrogen pair transition band S(6)–S(0)
at 1278 wavenumbers does not show up with a separate
feature, however, it explains the measured flux at the right
position. (The fit neglects a small shift to the red side and
some minor extra splitting due to spin–lattice coupling
called «self-energy shift» of the j � 8 state.) There is no
pair transition known explaining the intensity on its red side,
410 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4
J. Schaefer
20
15
10
5
0
In
te
n
si
ty
,
1
0
W
/(
cm
H
z)
–
3
0
2
15
10
5
0
15
10
5
0
1100
860 880 900 920 940
1400
Wavenumber, cm
–1
I
II
III
IV
1150 1200 1250 1300 1350
1400 17001450 1500 1550 1600 1650
Wavenumber, cm
–1
Wavenumber, cm
–1
Fig. 2. The same as in Fig. 1. but with the calculated emission
profile.
but a single S( )4 transition at 1243 wavenumbers with an
unchanged j � 2 partner at nn distance fits in at the right po-
sition with reasonable band width.
– The next two bands are mixed ortho–para bands
available to fit the rest of the feature lebeled III.
– The feature IV is fitted with three emission bands,
starting with the S S( ) ( )4 0
band on the red side at 1599
wavenumbers. Additional intensity on the red side could be
explained again by a single rotational transition S( )6 in the
first vibrational level, at �1553 wavenumbers with a j � 2
partner and a width of 40 wavenumbers. It is not included in
the fit. The S(8)–S(0) band at �1612 wavenumbers with � 20
wavenumbers width and the S S( ) ( )2 2
band at 1623
wavenumbers with � 40 wavenumbers width fit properly to
the frequency position and to the width of the feature IV.
Despite of expected changes of the calculated band
profiles to be provided in the future, I am able to state per-
fect agreement of the zero-phonon band positions and
useful band widths of the seven pure parahydrogen bands
presented as convincing facts in favor of assignable inter-
stellar solid hydrogen sources.
In contrast to this result, the transition frequencies of
the three bands of mixed ortho–para pairs cannot be
shown in agreement with measurements because of their
unknown initial excess binding energies and their un-
known profiles, certainly not Gaussian. And even their
portions of intensity contributing to the measured fea-
tures are unclear, therefore, their fits shown in the figure
are largely artificial. More work is needed to do better.
The fitted flux and the calculated emission rates of the
zero-phonon bands are used to determine pair column
densities of the source. The SWS aperture has been
14�20 arcsec2 � 6.5812 10–9 rad. Assuming a source dis-
tance of � 440 pc (1.3574 1019 m), it spans an area of
1.2 1030 m 2. The flux density corrected for the aperture
and the emission rate A if [sec �1] determine the (aver-
aged) column density of the observed pair transition,
N � 4
� Flux/(hcA if ) [cm �2]. Filling factors of the
sources are unknown. Estimates of column densities are
shown in Table 2, last column. They are as much uncer-
tain as the estimated emission rate coefficients.
It needs a cloud of several parsec depth and a huge
amount of hydrogen solids to explain these column densi-
ties summed over many different sources.
Also shown are the three significant still uncertain ex-
cess binding energies of the mixed ortho–para bands and
the small calculated effective shifts of the pure parahyd-
rogen bands caused by the local splitting at fifth column
and estimated emission rates at sixth column.
Less significant single transitions with participating
j � 2 states at 6–12 �m wavelengths have not been calcu-
lated but should be mentioned: The shoulder above 1600
wavenumbers could contain a small contribution of the
S( )6 band with a width of � 40 wavenumbers. And some
S j( ) transitions in the first vibrational excited level could
also contribute to shoulders, as, e.g.
1 1 4� S( ) at �1180 cm �1,
1 1 5� S( ) at �1370 cm �1,
1 1 6� S( ) at �1553 cm �1.
Of course, solid hydrogen emission bands can also be
found outside the 6–12 �m wavelength interval, again
with participating j � 2 states and � 40 cm �1 widths:
below 800 cm �1: the 0–0S( )2 and the 1–1S( )2 band,
below 610 cm �1: the 0–0S( )1 and the 1–1S( )1 band.
The series of de-excitation plus excitation pairs has
been started with the S( )4 –S( )0 band. It should be contin-
ued to include also the S( )2 –S( )0 pair transition at � 22 �m
(450 cm �1). And indeed, it has been observed twice in the
ISO mission at the Carina nebula, together with a single
S( )0 band at � 28 �m (356 cm �1), both with at least
40 cm–1 widths. The complete width of this double band
is not yet clear because of different beams used below and
above 27.5 �m, therefore, I did not apply calculations.
Zero-phonon emission bands of solid hydrogen at 6–12 �m wavelength. An astrophysical phenomenon
Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 411
Table 2. The parameters of the features I–IV of Fig. 2.
No. Band in v = 0 �E, cm
–1
Width, cm
–1
Shift, cm
–1
Aif , 10
–6
sec
–1
Flux, 10
–18
W/cm
2
N, 10
16
cm
–2
I
S(1) + S(0)
S(4) – S(0)
941.47 (exp)
887.84 (exp) 20.0
� –54.6
– 0.06
4.62
2.47
3.69
2.59
8.63
11.4
II S(2) + S(0) 1167.1 (exp) 40.0 – 1.05 39.4 9.25 1.94
III
S(4)
S(6) – S(0)
S(3) + S(0)
S(1) + S(2)
1243.44 (exp)
1278.18 (exp)
1389.14 (exp)
1396.9 (exp)
40.0
20.0
32.0
32.0
+ 1.31
+ 0.08
�– 73.8
� –46.8
7.92
7.36
13.0
21.3
5.98
5.08
12.6
5.73
5.84
5.19
7.06
1.91
IV
S(4) + S(0)
S(8) – S(0)
S(2) + S(2)
1599.04 exp
1612.5
1623.00 (exp)
20.0
20.0
40.0
– 0.01
– 0.09
42.9
14.5
50.2
4.26
4.22
4.67
0.597
1.74
0.551
The general series of rotational double de-excitation
emission bands is expected to continue above 1600
wavenumbers, up to about 3000 wavenumbers, where
the emission bands of nrs IX and X overlap, containing
transitions from up to j = 13 and 12, respectively. Above
3000 wavenumbers starts a new pair transition series, the
1-0O j S( ) ( )
0 series. In front of those we have a gener-
ally observed feature at � 3040 wavenumbers probably
built up from some 1-0O j S( ) ( )� 0 pair transitions. Ex-
pecting still a dominant role of the delocalized j � 2 state
in pair transitions at this frequency range seems to be an
appropriate assumption.
7. The astrophysical phenomenon
Hydrogen solids should be formed already in the halos
of a spiral galactic, which is not discussed here. It means
baryonic dark matter is available to form stars either at the
edges of the optical galactic disks or in socalled molecu-
lar clouds.
The distance between a star and the position of the
source is important for observing solid hydrogen, because
in surroundings close to the star solid hydrogen either
evaporates completely shortly after the star starts burn-
ing, or heavy planets gained sufficient gravitational at-
traction to retain their hydrogen at surfaces. The situation
is different at larger distances, where the selfshielding at-
mospheres of hydrogen solids are opaque for the disso-
ciative radiation from the star, and the supersonic hydro-
gen atoms produce a shock front in front of the solids.
This situation has been found in the ISO mission at the
prominent PDR of the NGC7023 (Iris) nebula, where we
have a scheme published by Fuente et al. [8], see Fig. 3,
showing H column density maps in contours, 13CO abun-
dances in grey scales observed by Fuente et al. [9], and
filaments in H2 fluorescent emission observed by
Lemaire et al. [10]. The SWS beam of 14�20 arcsecs
squared has been included in the figure. H2 fluorescent
emission has been observed there and shown in the SWS
spectra, I refer to the gas phase emission lines in Fig. 1.
The circle shows the beam of the ISO long wavelength
spectrometer.
Solid hydrogen is expected to be observable behind
the fluorescent emission at about 60 arcsecs distance from
the star. Assuming a source distance of � 440 pc (parsec),
the distance from the star is about 0.128 pc (� 26000 A.U.;
Pluto’s maximum distance from the Sun is 49 A.U.) and
the SWS source observed in the middle of the rectangular
beam is located approximately 0.114 pc outside the pro-
jection plane which contains the star, on one or on both
sides of the sphere with radius 0.128 pc around the star.
I assume that part of the supersonic H penetrates the
shock front and recombines in the solids at temperatures
below the triple point of 13.8 K. It is plausible to further
assume a great part of the H 2 binding energy being
collisionally transferred to surrounding lattice molecules
prior to restoration of the crystal structure. The result is a
much smaller ratio of excited ortho/para H 2 abundances
after recombination than the statistical ratio of 3:1. The
last steps of radiative relaxation are expected to be rota-
tional pair transitions at zero vibrational level to be ob-
served in the frequency range of the ISO-SWS, where
solid hydrogen as a relatively weak source, meaning a
huge amount of hydrogen solids of unknown sizes, be-
came observable in the interstellar space.
1. J. Schaefer, Chem. Phys. 332, 211 (2007).
2. J. Van Kranendonk, Solid Hydrogen, Plenum Press, N.Y.
(1983).
3. R.H. Tipping and J.D. Poll, in: Molecular Spectroscopy:
Modern Research, Vol. III, K. Narahari Rao and C. Weldon
Mathews (eds.), Academic Press Inc., London (1985), p. 421.
4. T.K. Balasubramanian, C.H. Lien, J.R. Gaines, K. Nara-
hari, and E.K. Damon, J. Mol. Spectr. 92, 77 (1982).
5. B.R.A. Nijboer and F.W. De Wette, Physica 23, 309 (1957).
6. Z.J. Kiss, Ph.D. Thesis, University of Toronto, Toronto,
Ontario (1959).
7. E.U. Condon and G.H. Shortley, The Theory of Atomic
Spectra, Cambridge University Press (1967).
8. A. Fuente, J. Martin-Pintado, N.J. Rodriguez-Fernández, J.
Cernicharo, and M. Gerin, A&A 354, 1053 (2000).
9. A. Fuente, J. Martin-Pintado, N.J. Rodriguez-Franco, and
G.D. Moriarty-Schieven, A&A 339, 575 (1998).
10. J.L. Lemaire, D. Field, M. Gerin, S. Leach, G. Pineau des
For�ts, F. Rostas, and D. Rouan, A&A 308, 895 (1996).
412 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4
J. Schaefer
50
0
–50
50 0 –50
�", arcsec
�
#,
ar
cs
ec
Fig. 3. The emission spectrum of the NGC7023 (Iris) nebula.
|