On the dissipation of the rotation energy of dust grains in interstellar magnetic fields

0.72 MB·

by  Renaud Papoular

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed8January2016 (MNLATEXstylefilev2.2)
On the dissipation of the rotation energy of dust grains in
interstellar magnetic fields
6
1 R. Papoular1⋆
0 1Service d’Astrophysique and Service de Chimie Moleculaire,
2 CEA Saclay, 91191 Gif-s-Yvette,France

n
a
J 8January2016

7

] ABSTRACT
A
A new mechanism is described, analyzed and visualized, for the dissipation of
G suprathermal rotation energy of molecules in magnetic fields, a necessary condition
. for their alignment. It relies upon the Lorentz force perturbing the motion of every
h
atom of the structure, as each is known to carry its own net electric charge because
p
of spatial fluctuations in electron density. If the molecule is large enough that the
-
o frequency of its lowest-frequency phonon lies near or below the rotation frequency,
r then the rotation couples with the molecular normal modes and energy flows from
t
s the former to the latter. The rate of this exchange is very fast, and the vibrational
a energyis radiatedawayinthe IRatastillfasterrate,whichcompletesthe removalof
[
rotationenergy.Theenergydecayratescaleslikethefieldintensity,theinitialangular

1 velocity, the number of atoms in the grain and the inverse of the moment of inertia.
v It does not depend on the susceptibility. Here, the focus is on carbon-rich molecules
9 which are diamagnetic. The same process must occur if the molecule is paramagnetic
7 or bathes in an electric field instead.
4 A semi-empirical method of chemical modeling was used extensively to illustrate
1 and quantify these concepts as applied to a hydrocarbon molecule. The motion of a
0
rotating molecule in a field was monitored in time so as to reveal the energy transfer
.
1 and visualize the evolution of its orientation towards the stable configuration.
0
6 Key words: astrochemistry—ISM:molecules—lines and bands—dust, extinction—
1 magnetic fields.
:
v
i
X
r 1 INTRODUCTION thegrain isparamagnetic, andtheimaginary componentof
a
its susceptibility induces a retarding torque. Its braking ef-
Theobservedpolarization oflightemittedorinterceptedby
fect is slower thehigherthemoment ofinertia, so thegrain
dustinmagneticfieldshasbeenattractingattentionforthe
ends up rotating about its highest moment of inertia, the
last six decades. A number of interpretations of this phe-
latter being parallel to the field, which is the orientation
nomenon were offered (see, for instance, reviews and bibli-
corresponding to theobserved polarization.
ography by Spitzer 1978, Hildebrand 1988, Roberge 2004,
This paradigm has been elaborated on (see Jones and
Lazarian and Cho 2005 and Lazarian 2007). In their pio-
Spitzer1967andSpitzer1978,Purcell1969and1979,forin-
neering work, Davies and Greenstein (1951) (to be referred
stance) and several variants were later proposed, especially
to below as DG), set the stage for later discussions. Their
in relation to the observed so-called anomalous microwave
scenarioconsidersarigid,solidstategrainwhich,somehow,
emission (AME),in particular those involvingrapidly spin-
has been given an angular momentum distributed over its
ning ultra small grains (see Draine and Lazarian 1998,
principal axes. As a result, it precesses and nutates about
Hoang and Lazarian 2014, Ysard and Verstraete 2010, Sils-
the angular momentum, which must be conserved absent
beeand Draine 2015).
any other perturbation, as illustrated by the Poinsot con-
struction (Goldstein 1980). Since the momentum is not, in Here,thesubjectisrestrictedtotherotationenergydis-
general,paralleltotheambientmagneticfield,B~,someway sipation. In this respect, the papers by Purcell (1979) and
must befound for at least those of its components that are Rouanet al. (1992) areremarkable bythebreadth of range
not parallel to B~ to decrease to zero. In the DG paradigm, of subjects discussed. In particular, Purcell suggested im-
perfect elasticity (internalfriction) asaveryefficient mech-
anism: internal heat is produced at the expense of rotation
⋆ E-mail:[email protected] energy, as a result of the periodic changes in internal me-
2 R. Papoular
chanical stresses due to centrifugal forces as the rotation reminiscent of a particle traveling with a high initial veloc-
axis wanders around the grain body. Rouan et al. studied ity through a fluid whose particles brake the initial linear
the quantum limit of this mechanism for small molecules, momentum. In the case of interest, both secular and ran-
whichtheycallInternalVibration-RotationEnergyTransfer dom componentsoftheangular momentumdecrease as the
(IVRET),based onthework ofNathansonand McClelland internal vibrations convert into IRradiation.
(1986).Purcell(1979)alsoconsideredconversionofrotation The molecule is then attracted into a state where its
energy into internal heat through the Barnett effect, which residual angular momentum becomes parallel to B~. For the
is the inverse of the Einstein-de Haas effect for para- and total energy to be a stable minimum, this orientation must
ferro-magnetic bodies. correspond to a maximum of induced magnetic moment
Now,iron’scosmicabundanceisrelativelylowandthis (maximumsusceptibility); thisgenerally coincides with one
elementisknowntoassociatepreferentiallywithsiliconand of the higher inertia principal axes. The whole process is
oxygen. On the other hand, carbon dust is one of the two very similar to the behavior of an axially symmetric top in
maincomponentsofcosmicdust,oftenmoreabundantthan theterrestrial gravitation field.
oxygen-rich dust, and it is responsible for a much richer IR The mechanism is governed by the same equations as
spectrum. It is therefore necessary to study its alignment in (9) and (10) of DG:
too. In the absence of para- or ferromagnetism, we are left
withdiamagnetismasthemainremainingplausiblecauseof
R˙ =L~ ·ω~ ;~˙J =L~ (1)
alignment. Long ago, Pascal (1910) showed that a diamag-
neticsusceptibilityisassociatedwiththeelectronpairsthat where R is the rotational kinetic energy (Iω2/2 in the
bond atoms in molecules. While this magnetism is much simplecaseofaspherical grain),J~theangularmomentum,
weaker than the paramagnetism associated with the elec- and L~, the retarding torque, except that, here, the expres-
tron spins of unpaired bonds or some heavy atoms, it is sion of Lmust beamended.Since,according tothepresent
more common and stronger in aromatic molecules. More- paradigm,thebrakingismediatedbytheLorentzforce,and
over, Pascal’s work and methods are still being discussed therateofenergytransferscalesliketheangularfrequency,
and updated with new data (see Bain and Berry, 2008). It Lmustincludeafactorω2B.ThiswillbedevelopedinSec.
is therefore worthwhile exploring diamagnetism, especially 3.2 and 3.3.
in view of theabundanceof carbon-rich molecules. Chemical modeling helps illustrating and quantifying
The present work takesadvantage of theavailability of thisconceptasappliedtospecificmolecules(Sec.2).Inpar-
chemical modeling methods which allow for externally ap- ticular,itallows onetodefinetheessentialdependenceofL
pliedelectricandmagneticfields(e.g.semi-empiricalTNDO upon the various parameters. Unfortunately, present algo-
orTypedNeglectofDifferentialOverlap).Usingsuchmeth- rithms become very demanding in computation time when
ods,it becomespossible toexplorealarge numberofstruc- amagneticfieldisswitchedon.Fortunately,muchlesssofor
tures, rotating or otherwise, measure their diamagnetism, electricfields.Sincecarbon-richmoleculesusuallycarryafi-
and study their dynamicsin a magnetic field. niteelectricmoment(anexpressionofthenon-uniformityof
The main result of this study is a new scheme for the theelectronicchargedistribution),thedynamicsinanelec-
dissipation of dust grain rotation energy, a necessary con- tric field is qualitatively similar to that in a magnetic field,
dition for alignment. This starts with the observation that, differing only by the fact that the force on a given atom is
when a carbon molecule rotates or oscillates in a magnetic qE~ instead of qV~ ×B~. Using an electric field allows one to
field, its normal mode vibrations are excited to various ex- demonstrate the concept as applied to a larger molecule so
tents, all the more the stronger the field. This can be un- its lowest vibration frequency comes closer to the rotation
derstoodinseveralways.First,suchmoleculesbeingelectri- frequency (Section 3).
cally conducting to some extent, eddy currents are created In Sec. 4, we revert to magnetic fields applied to a
inthemastheymovethroughthefield.Theheatgenerated smaller molecule, first to study the magnetic susceptibility,
bythesecurrentstakestheformofinternalvibrations.More thentoillustratethedynamicsanddemonstratethebraking
basically,differentatomsofthestructurecarrydifferentnet mechanism,asinanelectricfield.Finally,asabyproductof
local electric charges although the molecule is neutral as a simulations, two types of drifts of the molecule as a whole
whole. The motion of an atom across the field, induces an are illustrated in theAppendix.
electromagnetic force (Lorentz force), qV~ ×B~, orthogonal
to both and acting upon the atom. As this force varies pe-
riodically,thesystemisreminiscentoftwodetunedcoupled
2 THE CHEMICAL MODELING APPARATUS
resonantcircuits(seeLouisell1960).Ifthemoleculeislarge
enough that its lowest frequency normal mode of vibration Thereareseveralwaysinwhichmodelingisessentialtothe
is lower than the rotation frequency, then energy begins to present purposes. First, while data relative to the param-
flow from the rotation motion into internal vibrations. Still agnetism of solids are quite abundant, this is not the case
anotherwaytounderstandthephenomenonistolookatthe for diamagnetism, if only because it is much weaker and,
periodic impulses given by the field to the atomic charges hence, difficult to measure. When it comes to particular IS
as random impacts of free flying ambient atoms, whose net molecules,suchknowledgebarelyexists.Itcanbedelivered
result is a friction force retarding theangular motion. by chemical modeling with reasonable accuracy. Modeling
As the internal vibrations grow stronger, the induced is also the only way to estimate other properties of the se-
molecular magnetic moment acquires an increasing random lected molecules, such as the moments of inertia, the net
component, giving rise to a random torque which tilts the electronic charges and dipole moments, the frequency and
angularmomentuminrandomsuccessivedirections.Thisis energyofrotation-inducingevents,etc.whichalsoaffectthe
Alignment of molecules 3
finaldegreeofmolecularalignment.Inthepresentinstance,
modeling provedtobeparticularly valuablein avoiding the
usualfar-reachingapproximationoftreatingthedustparti-
cleasasolidnon-deformablebodywithasimple,symmetri-
calshape.Thismadeitpossibletodemonstratethecoupling
between bulkrotation and internal vibrations.
The principles of chemical modeling in presence of a
magnetic field will now be briefly outlined (see Van Vleck
1932,Kittel1955).Thedynamicsofamoleculeiscontrolled
bytheelectric andmagneticfields,E~ andH~,which inturn Figure1.Thecarbon-richmoleculeusedinthisworktoillustrate
the argument: 59 atoms, 36 C, 20 H, 3 S (at the pentagons free
obey Maxwell’s equations
summits). Its 3 principal axes coincide with axes Ox, Oy, Oz of
thefigure.Theelectricdipolemomentrunsthroughthecenterof
δA mass,pointingdownwardsbecauseoftheexcesselectroniccharges
E~ =−∇V − ;B~ =∇×A~, (2) onthesulfuratoms.
δt
whereV andA~arethescalarandvectorpotentials,and
B~ the magnetic induction. Assuming the magnetic field to -Minimum normal mode wavenumber: 6.64 cm−1
beuniform andpointingtothezdirection,thecomponents (21011 Hz).
of A~ are found to be -Mean polarizability under electric field of 510−4 a.u.
A =−1yH,A = 1xH,A =0. (2.57105 V/cm): 65.4 ˚A 3.
x 2 y 2 z
-Partial atomic (Mulliken) charge, an expression of the
As a result, thedynamical relations for an atom of the
structure, at distance ~r from the origin, become (in the local fluctuations of the net electric charge and potential
Gauss system) energy:excesselectronicchargefrom+0.245eonthesulfur
p~=p~ +p~ =m~˙r+eA~/c atoms, to-0.025 eon carbon atoms.
kin pot
for the generalized momentum and, for the kinetic en- -Principal moments of inertia along Ox, Oy, Oz (axes
1, 2, 3): 1188.09, 27622.8, 28810.9 × 1.710−40 g.cm2 .
ergy,
T = 1mr˙2= 1 p2− e p~.A~+ e2 A2. The 3 principal axes of inertia are designated by 1, 2
2 2m mc 2mc2 and3(setherealongthefigureaxesOx,OyandOz,respec-
Sincethequantummechanicaloperatorforthemomen-
tumis−i~∇,thecorrespondingperturbationpotentialtobe tively),and runthrough theCM(centerof mass). TheDM
(electric dipole moment) also runs through the CM, point-
addedtotheHamiltonianH oftheatom(nottobeconfused
ingdown.Itsfinitenessindicatesthatthecentersofpositive
with themagnetic field) is
H = ie~(x δ −y δ )+ e2H2(x2 +y2). andnegativenetatomicchargesdonotcoincide;itsvalueis
2mc δy δx 8mc2 thesame as that of a dipole of length 1 ˚A and charge 0.65
Severalcommerciallyavailablechemicalmodelingpack-
electronic charge. This plays an essential role in molecular
ages provide for dynamics computations without magnetic
dynamicsunderelectric or magnetic fields.
field, in which case A~=0 and V reduces to the instanta-
neous electrostatic potential. The Hyperchem package used
here (available from Hypercube,Inc., USA)offers themag-
netic option only for some theoretical methods which ne- 3 DYNAMICS IN AN ELECTRIC FIELD
glectsomeoftheinteractionsbetweenelectronsofthestruc-
3.1 Some characteristic times
ture (NDO semi-empirical methods: Neglect of Differential
Overlap). Here, we use the TNDO (Typed NDO) method, As several processes are operating simultaneously in the IS
which is considered as the currently most elaborate semi- medium(ISM),theircharacteristictimesmustbecompared
empirical method. The general capabilities of the pack- in order to properly set the stage for the simulation exper-
age(drawing,energyoptimization,computationalgorithms, iments. Assume the flux of dissociating UV photons to be
molecular properties, molecular dynamics, atomic impacts, 107 photons.cm−2.s−1, typicalof thetenuous,high-latitude
etc.)havebeendescribedelsewhere(seePapoular2005,and ISM(G=1;seeDraine1978).TaketheUVabsorptioncross-
references therein). section of carbon atoms to be 10−17 cm2, and the average
WhileISdustismadeofaninfinitevarietyofmolecules number of such atoms per dust particle to be 1000. Then,
with a wide distribution of sizes, most of the proper- the average interval between UV impacts is about a year.
ties and interactions we wish to demonstrate here can Likewise, the average interval between H atom impacts is
be illustrated using the same, medium-size, representative about106 siftheambientdensityofsuchatomsis20cm−3,
“generic”moleculewhichdoesnotrequireexcessivecompu- theirspeed105 cm.s−1 andtheeffectivecross-section ofthe
tationaltimes.ThisissketchedinFig.1andwillsometimes target,100×100˚A 2.Sincetherelaxationtimeforthegrain
be referred to as the “3trioS”. A simple organic molecule tosettleintoastateofstatisticvibrationalequilibriumdoes
was chosen, as such molecules have long been known for not exceed 1 µs, and the IR radiative lifetime is less than
their strong diamagnetism (P. Pascal 1910, Pauling 1936). 1 s, it is clear that, most of the time, the grain’s internal
Any of the following modeling experiment can be repeated temperature is zero, although, on average, it may reach 20
on any molecule of reasonable size. K.
Some characteristics of the3trioS are as follows: However,uponan Hatom impact,theprobabilitythat
-Mass 548 a.u.; Binding (potential) energy: -7219.04 a H2 molecule is formed and ultimately ejected is not neg-
kcal/mole (1 kcal/mole=1/23 eV per molecule); Electric ligible (e.g. seePapoular 2005). Theattendantrocket effect
Dipole Moment: 3.76 Debye(1 Deb=10−18 escgs). then sends the grain into supra-thermal rotational motion.
4 R. Papoular
Ourproblem thensimplifies into studyingwhat happensto
sucharotatinggraininamagneticfield,inthetimeinterval
between such rocket effects.
Ananalysisoftheenergybudgetofthisprocess(opcit.
)showsthatmostofthereleasedenergygoesintothekinetic
energy of linear, rotational and vibrational motions of the
hydrogenmolecule.Theenergyleftinthedustgrainisusu-
ally less than 1 kcal/mole (0.05 eV). The resulting angular
velocity, ω, is generally taken to be as high as 109 s−1 (see
Spitzer 1978 ) for grain sizes of 10−5 cm. In the simulation
experiments below, with smaller molecules, this will be set
intheorderof1012 s−1,soastokeepthecomputationtime
within reasonable limits.
3.2 The braking process
Asmentionedabove,thesoftwarecannothandlesufficiently Figure 2. Timeevolution of the angle between a CH bond and
largemoleculesinamagneticfield.Thisisnotthecasewith an adjacent CC bond, from the moment the field is turned on
and the molecule starts rotating about Ox. Note the change in
electric fields. Since the proposed braking process relies on
characterofthemotion,fromtheresponseoftheHatomaloneto
theexcitation of internalvibrations bytheforce exertedby
the initial electric kick, during the first picosecond, to the later,
the field on individual atomic charges, it does not make a
much faster oscillations including in-plane and out-of-plane CH
fundamental difference whether this force is qE~ or qV~ ×
bendingmodes.
B~. Accordingly, the proposed braking process will first be
demonstratedonamoleculeinanelectricfield(absentany
external magnetic field.)
Themoleculesweareinterestedinusuallyhaveasignif-
icant electric dipole moment. As soon as the field is turned
on,themoleculegenerally startsrotatinginresponsetothe
couple formed by the field and the induced or permanent
dipolemoment.Onecaneither:a)starttheexperimentwith
themoleculeatrest,soitisfirstacceleratedthenbrakedby
thefield,orb)firstletthemoleculereachthedesiredangu-
larvelocityinapredeterminedmolecule/fieldconfiguration,
then set the field to the desired value and start monitoring
thebraking. Both yield similar results.
Theactionofrotationupontheatomsisbestevidenced
bystartingwiththemoleculeat restintheplaneofthefig-
ure (Fig. 1) and the field along Oz, orthogonal to it; the
torque on the dipole moment is then at its maximum and
the molecule begins to rotate as soon as the field is turned
on. Angular rotation and momentum are both parallel to Figure 3. Total molecular bonding (Epot) and kinetic (Ekin)
Ox.Thefieldintensityissetat0.01a.u.(1atomicunit=5.2 energies,andtheirsum(Etot),inunitsofkcal/moleasafunction
1011 V/m). This is a huge field by ISM standards, but it is of time, starting with application of the field. The variations of
the two quantities are exactly out of phase, as expected from
unavoidable if the effect is to be demonstrated with avail-
energy exchanges between different molecular degrees of liberty,
able modeling software and reasonable computation times.
excludingafieldcontribution.
Repeatingtheexperimentwithfieldsbetween0.01and0.001
a.u.suggeststhatthebrakingtimeisinverselyproportional
tothefieldstrength;thismakessenseifthecouplingbetween in the 13-µm band and i.p (in-plane) bending modes in the
rotation and internal vibrations is due to the electrostatic 7-µmrange.Thisis the core of the proposed braking mecha-
force acting on the atoms. This scaling rule will be used nism.
below for extrapolations to theISM. Other significant markers are the total kinetic energy,
The software provides several means by which to mon- E ,andtotalbondenergy,E ,(Fig.3).Bothincludethe
kin pot
itor the dynamics of the molecule and make the braking rotational and vibrational energies. Initially, the rotational
apparent. One of the most telling quantities that can be energy is dominant, so its variations in time roughly follow
monitoredistheanglebetweenaCHbondandanadjacent the variations of the electrostatic torque with the angular
CC bond. For thenearest CH bond to theright of the cen- position,andtheensuingmaximumsofpotentialenergy,co-
tral S atom, this angle is θ=120.5 at rest in the optimized incidingwithminimumsofkineticenergyandvice versa,at
configuration with no field. Its evolution starting from the intervalsdeterminedbytheacquiredangularvelocity.Thus,
moment when the field is turned on is illustrated in Fig. 2. thefirst peakofkineticenergy,∼15kcal/mole at ∼0.8 ps,
The starting kick has given theH atom a strong linear mo- coincides in time and amplitude with the first dip in po-
mentum,whichisconverted,within1ps,intonormalmode tential energy. At that point, the molecule has acquired its
vibrations, mainly theo.o.p. (out-of-plane) bendingmodes, maximum rotational speed, ∼3.31012 rad/s (17.5 cm−1).
Alignment of molecules 5
Figure 4. Black line: Fourier transform of the kinetic energy Figure 5. Angular frequency ω (open blue squares) and rota-
variations from 5 to 10 ps after staring the experiment. Nearly tionalenergy(reddots)asafunctionoftime.Thetimeconstants
allnormalmodes(reddotsatordinate100)areexcited,withthe of the superposed best fitting exponential functions are, respec-
energy flowing from low to high frequencies, as expected if the tively, 8.7 and 4.9 ps. Here, the molecule was initially given an
sourceistherotationatafrequencyof17.5cm−1. angular frequency 1.82 rad/ps parallel to its principal inertial
axis 1 and to coordinate axis Ox, with a rotational energy 5.65
kcal/mole.Theelectricfieldwasthensetat0.003a.u.parallelto
Afterafewsuchcycles,however,theamplitudeofthese Oz.
energyvariationsdecreasesvisiblyandgiveswaytoweaker,
faster and less regular variations as more and more rota-
tional energy is converted into internal vibrations. This is
confirmed by monitoring the molecule on the screen all
the while: one can visualize the undulations of the struc-
turecharacteristicofthoseofthenormalmodeswhichhave
the lowest frequencies (phonons), as described in Papoular
(2015).
Notethat,allalong,thetotalenergy(sumofthekinetic
andbondenergies)remainsessentially constant,confirming
that the energy exchanges are exclusively internal and not
with theelectricfield.Theonlycontributionof thelatteris
intheform ofthepotentialenergywhichwasdepositedthe
moment the field was turnedon.
Details oftheenergytransfer canbequantifiedbytak-
ingtheFouriertransform(FT)ofeitherE orE .Figure
kin pot
4displays theFT of thekineticenergy variations from 5to
10 ps after launching the experiment. It all but coincides
Figure 6. Trajectory of the positive end of the electric dipole
with the FT of the bond energy variations, except beyond
2000cm−1,wheretheydifferonlyinbackgroundcontinuum. momentin3D.Here,themoleculewasinitiallygivenanangular
frequency1.82rad/psparalleltoitsprincipalinertialaxis1andto
Comparisonwiththenormalmodespectrumofthemolecule
coordinateaxisOx,witharotationalenergy5.65kcal/mole.The
(red dots at ordinate 100) indicates that most modes have electric field is then set at 0.003 a.u. parallel to Oz. For clarity,
been excited. The general slope of the spectrum suggests onlyaninitialleg,1-2(0to5.22ps)andafinalleg,3-4(20to30.57
that energy has been flowing from low to high frequencies, ps)ofthetrajectoryareshown.Themotionisinitiallypendulum-
asexpectedifthesourceistherotationenergyvariationsat likeaboutOxintheyzplane;asrotationenergyisconvertedinto
a frequency of 17.5 cm−1. internalvibrationenergy,itbecomesmoreandmoreirregular,but
Thatthevibrationalenergyincreaseisattheexpenseof withacleartrendtowardsthebottomofthestablepotentialwell
createdbythefield(withitsaxisofmaximummomentofinertia
the angular velocity and rotational energy is illustrated by
paralleltothefield).However,becausethefinalradiativeremoval
another experiment. Here, the molecule was initially given
of its energy cannot be simulated here, it ends up as pendulum
anangularfrequency1.82rad/psparalleltoitsprincipalin-
oscillationsaboutOz,inthexyplane
ertialaxis1andtocoordinateaxisOx,witharotationalen-
ergy5.65 kcal/mole. Theelectric fieldwasthensetat 0.003
a.u.parallel toOz.Fig. 5displaysthosetwoquantitiesasa only on the angular velocity, but also on the varying com-
function of time during the following 37 ps. The time con- binationoftheinstantaneouspositionsandvelocities ofthe
stants of the superposed best fitting exponential functions vibrating atoms.
are,respectively,8.7 and4.9ps.Theformer isherereferred ThisismadeclearerinFig.6,whichdisplaysthetrajec-
to as the braking time (or retarding, or decay time). The tory,in3D,ofthepositivetipoftheelectricdipolemoment.
dispersionofthedatapointsaboutthefitsisduetotheen- Hereagain,themoleculeisinitiallygivenarotationalveloc-
ergy transfer being an irregular process which depends not ityω=1.81012 rad/s, corresponding toarotational energy
6 R. Papoular
of 5.65 kcal/mole (0.25 eV), parallel to Ox, which initially cycle. Then V =r ω for atom i, and the sum over the grain
i i
coincides with inertial axis 1. The electric field is then set is Nωr , where N is the total number of atoms and r ,
av av
at 0.003 a.u. (1.56107 V/cm) parallel to Oz, and the elec- theiraveragedistancefromtherotationaxis.Finally,define
tricdipolemomentismonitoredforthefollowing30ps.For
clarity,onlyaninitialleg(0to5.22ps)andafinalleg(20to
I
30.57ps)areshown.Theformershowstheinitialpendulum τ0 = . (5)
λNBravω0
oscillations about Ox, in the yz plane. In the later phase,
much of the rotation energy has gone into vibrations and Then, from eq.(1),
the dipole moment tends towards the configuration where
ω2
energy is minimum. This is the case when the dipole mo- ω˙ = , (6)
ment (like theaxis of symmetry) is parallel to Ox,which is ω0τ0
thebottom of thestable potential well of thisstructure. whose solution is
However, it is clear that the dipole moment is not at
restinthisconfigurationbutoscillates aboutOz,inaplane ω0
nearlynormaltothataxis.Thisisbecausethevibrationen- ω= . (7)
1+t/τ0
ergyhasgrowntoaconsiderablelevel.Inspace,thisenergy
would have been radiated away as soon as it was gained. Here,therefore,thedecayisinitiallyexponentialwitha
Unfortunately, the available software does not allow for a characteristic time τ0, then progressively turns hyperbolic.
simulation ofthisremoval, soall theenergy convertedfrom Note the dependence of τ0 on B−1, as opposed to B−2 for
rotation to vibration is available for pendulum oscillations, paramagnetic braking.
at least for a very long time. That τ0 scales like I does not mean that the molecule
Thedemonstration ofthebrakingprocessiscompleted will necessarily settle in the configuration where the prin-
by checking that, if the field is null or parallel to both the cipal axis of highest inertia is parallel to the field. Rather,
momentumandrotationvectors,theinitialangularmomen- itwill settleinthemoststablemagnetic configuration:axis
tum is unaffected, and therotation goes on indefinitely. of symmetry parallel to the field. For an axially symmetric
Returningtoeq.(1)withournewunderstandingofthe ellipsoid, thetwo configurations coincide.
braking mechanism, we think of a larger grain made of the If the frequency of successive rocket events, ν, is not
samestructureasintheexampleabove,andexpresstherea- muchless than 1/τ0, then,the variation of ω with time has
sonableassumptionthattherateofrotationenergytransfer, no analytical expression; however, it is fairly well described
R˙, is proportional to the electric force, to the angular fre- asfollows.AsrecalledinSec.3.1,previousexperimentsshow
quencyand to thegrain volume(as the field permeates the that the rotation energy deposited in the grain by a rocket
hole grain). Hence, effect does not exceed 1 kcal/mole (∼ 0.05 eV) whatever
the grain size. To remain on the safe side, take this as the
common value, R0. If the relevant moment of inertia is I,
L=λENω, (3) thenR0 = 12Iω02andatermνR0mustbeaddedtotheright-
hand side of the first eq. (1). The rotational kinetic energy
where N is the number of atoms in the molecule: for the
will stop decreasing (R˙ ∼0) roughly around thetime when
sake of clarity, we make no difference between atoms; λ ac- thistermis equalto 1Iω2,whereω is givenbyeq.(7).This
counts for the distribution of charge among atoms and of 2
thestructuraldetails,whichareassumedtobeindependent time is τ = τ0/(ντ0)1/3. Thereafter, the angular velocity
of size. The solution of eq. (1) then becomes does not fall below ∼ω0(ντ0)1/3.
We have still to specify the scaling law for the grain
ω=ω0exp(−t/τ); R= 1Iω02exp(−2t/τ) ; τ = I . size. This is done by writing I ∝ N(r2)av and N ∝ (r¯)3 ,
2 λEN so, from (5),
(4)
In the experiment reported above, τ = 8.710−12 s, E =
0a.t0o0m3saa.un.d, Ior=5.11.1120410s3taat.Vu/.,comr,2o1r01−.3571g0.7cmV2/.cmHe,nNce,=λ5∼9 τ0 ∝ (r(2r¯))3a3/v2rr¯ r¯3.5. (8)
av
8.510−33. It is of interest to note that τ does not depend
Toa form factor, thisscales approximately likethenumber
explicitly on the angular velocity nor on the susceptibility.
of atoms, N.
We are finally in a position to deduce, from the above,
specificnumericalvaluesofτ forspecificcasesofinterestto
3.3 The braking time in interstellar space
astronomy.Take,forinstance,themagneticfieldintensityin
In spite of theobvious difference of force directions,there is theISMtobe510−6G,andthemolecularangularvelocity2
no essential difference between acting on the atoms of the rad/ps,soatypicalatomvelocityis400m/s(forourgeneric
moleculethroughtheelectricorthemagneticforce; wealso molecule,3trioS);thentheequivalentelectricfieldintensity
assume that a magnetic field acting alone, is as effective is roughly 210−7 V/m, which is about 1016 times weaker
as an electric field acting alone, provided |ΣqE~ |=|ΣqV~ × than the 0.003 a.u. electric intensity used for Fig. 5. If the
B~ |.Consideracomponentofangularvelocityorthogonalto extrapolation is valid, the braking time in the ISM, for the
thefield (as that which is parallel will not beaffected). For moleculeconsideredabove(59atoms),wouldbeintheorder
simplicity, assume all net atomic charges to be equal, and of 105 s.
reduce this relation to the corresponding scalar quantities: Depending on the rate of formation of hydrogen
ΣE=BΣV.AlsoconsiderV assomeaverageoverarotation molecules on the grain surface, a more or less significant
Alignment of molecules 7
Figure7.Bondingenergyincreaseasafunctionofappliedmag- Figure 8. Profiles of the magnetic potential well along two or-
netic field, H. Thefield isapplied parallelto the symmetryaxis thogonal planes:a)(full)theplaneofprincipalaxes 1and2(xy
ofthemolecule(axis2). plane);b)(dashed)theplaneofaxes1and3(xzplane).θ isthe
angle between the symmetry axis and the particular field orien-
tation.
fractionofthelargergrainswillbeexcludedfromthealign-
mentprocess.However,theverysteepdistributionofgrains
modeling molecules. In that case, χ must be deduced from
in linear size should greatly mitigate this effect. But this
the change in bonding (potential) energy upon turning the
must be left for furtherscrutiny.
magnetic field on; this is given by ∆E = −χH2V, where
V = 1/61023 cm3. The modeling code gives the field in
unitsof 1.7107 G(a.u.: atomic units)and themagnetic en-
4 SIMULATING A MOLECULE IN A
ergy in kcal/mole (1 kcal/mole is equivalent to 1/23 eV, or
MAGNETIC FIELD
710−14 erg, per molecule).
Repeating,inamagneticfield,theexperimentsdemonstrat- Thus, the static susceptibility in a given direction can
ing the braking and alignment mechanisms as in electric be obtained from the increase in binding energy (algebraic
fieldswouldbevaluable.However,asmentionedintheIntro- decreaseofthemolecular potentialenergy)asanincreasing
duction, using the magnetic field algorithm with the 3trioS magnetic field is applied parallel to this direction. This is
molecule requires impossibly long computational times. We illustratedinFig.7fordirectionOy,whichisalsotheaxisof
are therefore compelled to turn to a smaller molecule: the symmetry.Agoodfittothepointsisgivenby10−5H2V,so
structurewillbereducedtoasingleoneofthe3components χ=10−5. By comparison, the susceptibilities of the highly
of3trioS,thecentralone,calledtrioS.Organicmoleculesare diamagnetic materials, graphite and diamond, are of order
generally diamagnetic to some extent, as will now be illus- 210−5 (see Young1992).
trated in this particular case. Some characteristics of the Thesametrendwasfoundforthe2otherprincipalaxes
trioS are as follows: albeitwithsomewhatdifferentsusceptibilities.These3par-
-Mass184a.u.;Bindingenergy:-2476.5kcal/mole;Elec- ticular directions of H~ define three different local extrema
tric Dipole Moment: 2.72 Debye(1 Deb=10−18 escgs). ofthepotentialenergy,i.e.threeequilibriumconfigurations.
-Netatomic chargesinthesame rangeas3trioS above. Theonlystableoneoccurswhenthemagneticfieldliesalong
-Principal moments of inertia along Ox, Oy, Oz (axes the(in-plane)axisof symmetryofthemolecule, perpendic-
1, 2 3): 318.5, 866.6, 1185.1 × 1.710−40 erg. ulartoitslongdimension,asinFig.1.Figure8outlinesthe
profilesofthepotentialwellalongtwoorthogonaldirections
relative to the stable one. Obviously the anisotropy of the
4.1 Diamagnetic susceptibility
wellespouses theasymmetryofthemolecule. Thesuscepti-
A given material, subjected to a magnetic field, H~ will de- bilitymatrixisapproximatelydiagonalandsymmetricwith
velop a magnetic moment, M~ per unit volume ,such that respect to thecenterof mass.
M~ =χH,with χ=χ′+iχ′′.
whereχisnon-dimensional,andH andM areextensive
4.2 Molecular dynamics in a magnetic field
′′ ′
quantities. As χ is small, we shall only be interested in χ
andrefertoitasχ.Now,considerafinitevolume,V,ofthis In the absence of braking torques, the initial angular mo-
material,possiblyamicroscopicvolume.Itwillbeassumed, mentumimparted tothemolecule bypreviouscollision and
for convenience, that the total magnetic moment, M (note rocketprocessesmustbeconserved,sothemoleculecanonly
the change of meaning), of this chunk of material is χHV. andindefinitelyexperiencechangesinthedistributionofthe
Itmustbeemphasizedthatχissometimesalsogiveninthe rotation components but never align with the field. This is
chemical physics literature in units of g−1, or mole−1 or by illustrated by thefollowing experiment.
only specifying thedensity of thematerial (e.g. mole/cm3). To begin with, the molecule must be set in rotation as
In each case, the expression for M should be modified cor- in the ISM. For this purpose, in the absence of a magnetic
respondingly. Still another case must be confronted when field,with themoleculelyinginthexyplanesymmetrically
8 R. Papoular
Figure 9. Motionof the trioS, initiallyspinningabout its prin- Figure 10. Trajectory of the positive tip of the dipole moment
cipal axis 3, in a magnetic field pointing in direction Oz, with ofatrioSinamagneticfieldof50a.u.alongOy.Redcurvewith
intensity 20 a.u., as symbolized by the trajectory of the tip of dots:from0to1.5ps(firstpendulumoscillationaboutOx);blue
its electric dipole moment. Open blue squares: from 0 (point B) line:13.5to15ps(finalrandommotionaroundthebottomofthe
to 10 ps; filled red circles: from 10 to 15 ps (point E). Upper stablepotential well).SeeSec.2forcomments.
blacksquare:centerofthecircularinitialtrajectory.Lowerblack
square:stabletargetpositioninthisfield,foramoleculespinning
nowaboutitsprincipalaxis2,paralleltothefield.Althoughthe
rotationvectorcontinuouslywandersoverthemolecule,fromone attracted back upwards with increasing radius, only to re-
principal axis to the other, there is no apparent rotation energy turnlater,alwaysridingonthesame3Dsurface.Atconstant
dissipation and the dipole moment tip indefinitely pursues the totalenergy,adescenttowardslowerpotentialenergiesisac-
explorationofthesameaxiallysymmetricsurface. companiedbyanincreaseinkineticenergywhichimpliesan
increase in rotation velocity and a decrease of the moment
ofinertia. Figure9shows just this.Thefinalalignment can
aboutOy,weapplyanelectricfieldE~ =0.03a.u(1a.u.=107 only be reached if a braking mechanism enters the picture,
V/cm)paralleltoOx,i.e.orthogonaltothemoleculardipole to absorb enough of the kinetic energy so the molecule can
moment. The resulting torque launches the rotation about settle in the most stable magnetic configuration. Here, ap-
axis Oz (parallel to principal molecular axis 3; moment of parently, the anelestacity invoked by Purcell (1979) is not
inertia:1185a.u.).After1/4ofarevolution,theangularve- sufficient to do the job, in spite of the very high intensity
locity,ω,reachesamaximumof3.36rad/ps,withanenergy ofthefield.Nonetheless,observation ofthemoleculeon the
of 13.36 kcal/mol. This rotation frequency is much higher screen reveals indeed its deformations (phonon modes) un-
than expected in the ISM, but is necessary to enhance all dertheperiodic centrifugal stresses it experiences.
mechanical effects and keep the computation times within As suggested above, the braking mechanism invoked
reasonable limits. here relies on the mediation of the Lorentz force qV~ ×B~,
The electric field is then turned off and the magnetic actingontheindividualnetatomicchargesinthesameway
field turned on, with an intensity of 20 a.u., parallel to Oz as the qE~ force did above. As the molecule rotates, each
(again, this is much higher than in the ISM, but is also atomicchargefeelsafieldvaryingattherotationfrequency,
necessary for the same reasons). This configuration is not whichperiodicallyperturbsitsrotationalmotion.Ifthisfre-
a stable one, as shown in the previous section; while spin- quency falls within, or not far from, the internal vibration
ning, the molecule is therefore attracted towards the sta- spectrumof themolecule, thenthelatteris excitedand en-
ble configuration where its symmetry axis (axis 2; moment ergy can flow from rotation to vibrations and soon be radi-
of inertia:866.6 a.u.) is parallel (or anti-parallel, because of ated away. With the available simulation software, we can
its diamagnetic character) to the magnetic field. The mo- illustratetheconversionofrotationtovibrationenergy(but
tion is governed by the couple between the field and the not its evacuation), which we proceed to do.
induced magnetic moment, as well as by the resultant of SincethelowestphononfrequencyoftrioSisstillmuch
theLorentzforces qV~ ×B~ appliedtothemovingindividual higherthanareasonably highrotation frequency,oneisled
electric charges. This is illustrated in Fig. 9, which displays tousethemaximumpossiblemagneticintensitysoastoin-
the trajectory in space of the positive tip of the dipole mo- crease the coupling between them. The molecule is initially
ment (assumed permanently coincident with axis 2 of the at rest in the xz plane and centered at the origin with its
molecule). It begins at B and ends at E: open blue squares principal axis 1 coincident with the Ox coordinate axis. A
from 0 to 10 ps, filled red circles from 10 to 15 ps. The in- magnetic field of 50 a.u. is then applied along the Oy axis.
tervalsbetweensymbolscorrespondto0.03ps.Initially,the Thisisanunstableconfigurationandamagnetictorqueacts
tipdescribesacircleofradius∼2˚A abouttheorigin ofco- to rotate the molecule about the Ox axis, starting an oscil-
ordinates (upper black square). As it descends towards one latorymotion.Thefirstperiodofthis(1.5ps)isrepresented
ofitstwostablepositionsinthisfield(0,0,-2.7:lowerblack in Fig. 10 as the trajectory of the positive tip of the dipole
square),theradiusofthecirclesdecreases.However,itonly moment (red curve and dots). During this time, the latter
reachesanaltitudeof-2.6,andbeforeitcangofurther,itis remains closely parallel to the yz plane, but already shows
Alignment of molecules 9
Figure11.ThepotentialenergyvariationscorrespondingtoFig. Figure12.FTofthepotentialenergyvariationsduringthelast
10.Inthe absence of amagnetic field, the bindingenergy ofthe 5 ps of Fig. 11 (black line). The red dots at ordinate 10000 rep-
optimizedmoleculeis-7667.1kcal/mole.Notethebreakbetween resenttheIRabsorptionlinesofthemolecule.Clearly,thewhole
2and13ps.Duringthefirstperiod,thecurveisregularbecause vibrationalspectrumofthemoleculehasbeenexcited.Thepeak
the motion is a nearly pure pendulum oscillation. In the end, near50cm−1 correspondstothepseudo-periodofthetrajectory
the curve is dominated by the much faster internal vibrations, aroundthebottomofthestablepotentialwell.Exceptbelow100
whichalsoinducetherandomfluctuations ofthedipolemoment cm−1 the spectrum is quite close to that of the same molecule
apparentinthesamefigure. heated up to 200 K and left to vibrate in vacuum (no magnetic
field).Itsexcessenergycontent(kineticandpotential)isthen24
kcal/mole,orabout1eVThisistheclearestevidenceofconversion
signs of tilting. For the sake of clarity, we skip the next 12 ofrotationtointernalenergy.Bycomparison,thecorresponding
ps and show only thelast 1.5 ps. This differs in two impor- spectrum of the same molecule at rest with an internal kinetic
tant respects: a) the tip now describes a curve around the temperatureof20Kismuchlessrichinmodesanditspeaks do
notexceed 100a.u.
Oyaxis,i.e.thefielddirection,whichisastableorientation
for the molecule, if it could be reached; b) the trajectory
is obviously much more erratic, a first indication that the
5 CONCLUSION
dipolemomentnolongerexactlycoincideswiththesymme-
try axis of the molecule due to the increased amplitude of The present study differs from others in three respects: a)
the internal atomic vibrations, which suggests a conversion it is restricted to the consideration of a single rotation en-
of rotation into vibrational energy. ergy dissipation mechanism; b)it appliestoboth para- and
AnotherexpressionofthisconversionisapparentinFig. dia-magnetic grains; c) it is illustrated by semi-empirical
11 , which displays the variations of the molecular bonding chemical modeling experiments.
energy during the first and last 2 ps of the experiment (cf. The said mechanism can be mediated byLorentz mag-
Fig.3). During the first 0.5 ps, the molecule “hesitates” to neticorelectricforces,actingonpartialelectricchargescar-
leave its unstable position, after which it executes one and ried by the individual constitutive atoms of the grains. It
a half period of relatively regular pendular motion, with converts rotation energy into internal (vibrational) energy:
exchangesbetweenpotentialandkineticenergy.Afterwards, it is an “internal dissipation” mechanism in the terms of
this regular oscillation is more and more drowned into the Purcell(1979).Theconversionisefficientonlyiftherotation
much faster and erratic internal vibrational motions. Since vectorisnotparalleltoasymmetryaxisofthegrainandto
thisenergy cannot findan oulet, themotion takestheform thefield,andprovidedthegrain islarge enoughand/orthe
ofarandomwalkaroundthebottomofthestablepotential angular frequency high enough that the latter falls within
well (0, -2.3, 0), the farther from it, the higher the residual thespectral range of thephonons.The smallest grains can-
kinetic energy. not meet thelatter condition in theISM.
A more quantitativeassessment of theaccumulated vi- The Lorentz mechanism differs from anelasticity and
brational energy is provided by a Fourier analysis of the IVRET in that it does not consider the grain as an elec-
total potential or kinetic energy, as in Fig. 12. Clearly, the trically uniformly neutral body (even if it is, as a whole,
whole vibrational spectrum of the molecule (red dots) has electricallyneutral).Thecomputationoftheelectronicwave
been excited. The peak near 50 cm−1 corresponds to the function gives the distribution of electron density over the
pseudo-period of the trajectory around the bottom of the molecule. The Lorentz force can then be computed on each
stable potential well. Except below 100 cm−1 the spectrum atom. In this way, the modeling accounts for both anelas-
is quite similar to that of the same molecule heated up to ticity and electromagnetic forces. Asa result, it isobserved
200 K and left to vibrate in vacuum (no magnetic field). that, during rotation, energy is continuously fed into the
Its excess energy content (kinetic and potential) is then 24 wholespectrumofinternalvibrations,whichisnotthecase
kcal/mole, or about 1eV. This is the clearest evidence of with anelasticity alone. At the same time, our experiments
conversion of rotation to internal energy. indeed exhibit the molecule deformations (phonon modes)
According to eq. (5), if this experiment were repeated due to the purely mechanical centrifugal forces invoked by
in a field of 510−6 G, it would last about 3000 s. Purcell (1979). It must be stressed that the vibrational en-
10 R. Papoular
ergy is provided by the rotation energy, not the magnetic ChiarJ.andPendleton Y.2008,IAUSymposium251,26
(or electric) field, although, at thesame time, thefield also DaviesL.andGreensteinJ.1951,ApJ.114,206
periodicallyexchangesenergywiththemoleculethroughits DraineB.andLazarianA.1998, ApJ.508,157
couple with themagnetic (or electric) moment. DraineB.1978ApJSup.36,600
GoldsteinH.1980,ClassicalMechanics,AWInc.
Thedissipation timeis foundtobeproportional tothe
HildebrandR.1988, Q.Jl.R.astr.Soc.39,327
moment of inertia about the relevant principal axis and in-
HoangT.,LazarianA.andMartinP.G.2013,ApJ779,152
versely proportional to the magnetic field, the initial angu-
HoangT.andLazarianA.2014,MNRAS,438,680
lar velocity and the number of atoms in the molecule. For
JonesR.andSpitzer L.1967,ApJ.147943
an aromatic molecule of 59 carbon atoms, the computation
KittelCh.1955,Introductiontosolidstatephysics,J.Wiley,N.Y.
gives 105 s in a field of 510−6 G, which shows that the LazarianA.andDraineB.2000, ApJLett.536,L15
Lorentz mechanism can contribute significantly to rotation LazarianA.andChoJ.2005, ASPConf.Ser.343
braking. Using this data, it is possible to extrapolate these LazarianA.2007,J.Quant. Spec.Rad.Trans.106,225
results to other molecules havingsimilar structures. Louisell W. 1960, Coupled-mode and parametric electronics, J.
With the Lorentz mechanism, it is not possible to dis- Wiley,N.Y.
tinguish external from internal alignment for both rely on Nathanson G.M and McClelland G. 1986, J. Chem. Phys. 84,
3170
the field and occur simultaneously, contrary to the anelas-
PapoularR.2005,MNRAS359,683
ticity or Barnett mechanisms, for instance (Purcell 1979).
PapoularR.2012,arXiv1207.7217
Themechanismisactiveforbothfullrotationsandpen-
PapoularR.2015,MNRAS450,2539
dulumoscillations. Itdoesnotdependonthemagneticsus-
PascalP.1910,Ann.Chim.Phys.19,5
ceptibility, except that the latter determines the ultimate PlanckCollaboration,A&ASpecialFeatures2011,536and2014,
orientation of the molecule when the rotation energy has 571
been removed. PaulingL.1936,J.Chem.Phys.4,673
Modelingallowsthedeterminationofthemagneticsus- PurcellE.1969,Physica41,100
ceptibilities along theprincipal axes.Onlyoneofthesecor- PurcellE.1979,ApJ.231,404
responds to a stable static configuration when it is parallel RobergeW.2004,APSConf.Ser.309,edrs.WittA.etal.
RouanD.etal.1992, A&A253,498
tothefield;ithasoneofthetwohighestmomentsofinertia.
SilsbeeK.andDraineB.2015,arXiv1508.00646
Modeling of the motion in a field shows that, with enough
SpitzerL.andTukeyJ.1951, ApJ.114,187
braking, the grain orients itself so it rotates or oscillates
SpitzerL.1978,Physicalprocessesintheinterstellarmedium,J.
about thisaxis.
Wiley,N.Y.
Finally,modelingunveilstwotypesofdriftsofthegrain Van Vleck J. H. 1932, The theory of electric and magnetic sus-
mass center, also due to the Lorentz force: one accelerated ceptibility,ClarendonPr.,Oxford
andonewithconstantspeed,dependingonthedirectionsof vanDishoekE.2008,IAUSymposium251,3
thefield and rotation vectors.. YoungH.1992,UniversityPhysics,7thed.,AddisonWesley
As the modeling package provides capabilities for YsardN.andVerstraeteL.2010,A&A509,12.
Langevin dynamics at constant temperature, it is possible
to study the grain motion in more complex situations, e.g.
where the photon influx and/or the ambient H atoms fric- 6 APPENDIX: DRIFTS IN A MAGNETIC
tion cannot beignored. But this cannot bedeveloped here. FIELD
As recalled above, the frequency of the lowest-energy
phononof thesmallest molecules is toohigh tocouplewith As shown above, the braking mechanism relies upon the
therotationfrequency,sothesemoleculescannotaligninthe coupling between rotation and vibration motions provided
fieldandcontributetopolarization.However,thisshouldnot bytheLorentzmagneticforce.Anothermanifestationofthis
be a major handicap for the proposed mechanism, nor, for force is the drift of the molecule as a whole. There are two
that matter, for other candidate mechanisms, such as spin- kinds of drifts of rotating molecules, according to whether
lattice relaxation (Lazarian and Draine 2000, Hoang et al. the rotation axis is perpendicular or parallel to molecular
2013).For,largemoleculesarecertainlyamajorcomponent axis3(orthogonaltothemolecularplane).Genericexamples
ofdust,aswitnessedbytheextensionofobserveddustemis- are given here.
siondowntoafewGHz(seePlanckCollaboration2011-2014
and analysis by Papoular 2015). Also, many PAH models
6.1 Accelerated drift parallel to the rotation axis
now include quite large PAHs (van Dishoeck 2000, Chiar
and Pendleton 2000), and I have shown that the Unidenti- Consider a trioS rotating about its principal axis 1 (coinci-
fiedInfraredBandscanbeemittedbylargemolecules,PAH dent with Ox), in a magnetic field parallel to Oz, H = 10
andkerogentypes(Papoular2012).Nonetheless,thesequal- a.u.TheatomsexperienceaLorentzforceparallelto+Oxor
itative considerations need to be backed by an exhaustive -Oxaccordingtothesignoftheirnetelectriccharge(higher
quantitative study of the polarizing efficiency of grains as orlowerlocalelectrondensity).Ifthemoleculehasaperma-
a function of size and shape, a task that remains to be ad- nentdipolemoment,thenthecentersofmassofthetwosets
dressed. do not coincide so the resultant force is finite and propor-
tional to both the field and the rotation velocity. Because
of the torque between the field and the induced magnetic
moment, the rotation velocity periodically increases or de-
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