Visualization and measurement of quantum rotational dynamicsR. M. Dimeoa)

NIST Center for Neutron Research, National Institute of Standards and Technology, 100 Bureau Drive,Gaithersburg, Maryland 20899

~Received 30 July 2002; accepted 14 November 2002!

An introduction to quantum rotational tunneling and libration is presented with an emphasis onobtaining a qualitative understanding of this phenomenon through visualization of the dynamics,simple approximations, and measurements. The tunneling and librational dynamics of smallmolecular rotors are discussed using a very simple model of the rotational potential. Numericalcalculations of the evolution of probability packets are carried out for the low-lying states and theconnection is made between the quantum and classical librational dynamics. Finally, we presentmeasurements of these quantum rotations using inelastic neutron scattering and show in particularhow neutron scattering measurements of the ground state tunnel splitting and first librationaltransition can be used to characterize the magnitude and shape of the potential hindering the motionof the rotor. Some conceptual and computational problems are included that are suitable forundergraduate students. ©2003 American Association of Physics Teachers.

@DOI: 10.1119/1.1538575#

lab

lc

eor,o

de

p

m

la

ath

tast-oo

thth-in

eno

ulnsd

s-tion

ar-ceany

b-

theer-u-torglethend

ula

ex-

ne-

ingla-nd

t ofnalith

nicsthe

r-ge.

for

I. INTRODUCTION

Studying the quantum rotational dynamics of molecusystems can provide detailed information on the forcestween molecules~intermolecular interactions! and betweenthe molecular constituents within a single molecule~intramo-lecular interactions!. It is not surprising then that rotationaspectroscopy has been widely used as a means to charaize these interactions.1 Obtaining an understanding of throtational dynamics requires one to have a physically reasable model of the molecular system. In general, howevemicroscopic description of the rotational dynamics of a mlecular system can be quite complicated. For moleculeseven modest complexity there can be many sub-units ungoing simultaneous rotation1 translation thus requiringmultiple coordinate systems for a detailed analysis. A simthree-dimensional rotor such as solid hydrogen (H2) is es-sentially a free spherical rigid rotor and its rotational dynaics have been studied extensively.2 The rotational motion of atetrahedral molecule such as methane (CH4) is somewhatmore difficult to analyze but the dynamics of the molecusolid are still tractable.3

Arguably the simplest type of rigid rotors are those thhave a single rotational degree of freedom such as the mehalides containing methyl groups (CH3) or the ammoniacompounds~containing ammonia groups, NH3). In thesemolecular solids, the methyl or ammonia groups can roabout a fixed molecular axis, under the influence of a cryfield ~an orientational potential! hindering rotation, thus displaying a variety of different rotational motions. The typesmotions that can be observed are libration, stochastic reentation, and quantum rotational tunneling. Libration isquantum analog of classical torsional oscillation in whichCH3 group ~for instance! oscillates within the hindering potential. Stochastic reorientation is a rotational motionwhich thermal fluctuations in the crystal provide enoughergy to overcome the hindering barrier, thus enabling sptaneous reorientation of the CH3.

Quantum rotational tunneling is somewhat more difficto understand because it has no classical analog. Trational tunneling is better known and has been discusse

885 Am. J. Phys.71 ~9!, September 2003 http://ojps.aip.or

re-

ter-

n-a-ofr-

le

-

r

tyl

teal

fri-ee

-n-

tla-at

length.4–13 Pauling recognized in 1930 that molecular crytals such as hydrogen can exhibit quantized rotational moincluding rotational tunneling.14,15 Rotational tunneling is aphenomenon whereby a molecule~or a molecular sub-unit!whose rotational motion is hindered by an orientational brier, reorients itself via tunneling through the barrier. Sinthe 1930s, rotational tunneling has been observed in mmolecular systems from solids to gases.3,16 In rotational tun-neling the ‘‘particle’’ that tunnels can be a molecular sugroup such as a methylene group (CH2), a CH3 group, oreven an entire molecule such as H2 or CH4. The descriptionof the coherent motion of the constituent atoms is calledsingle particle model. The single particle motion is detmined by the molecular environment. If there is no moleclar field that hinders the single particle motion, then the roundergoes quantized free rotation. If the motion of the sinparticle is hindered, a situation that we will consider, thenquantum rotations can include rotational tunneling alibrations.3,16

The related problem of quantum rotations in pendwhere the potential,V(u)5V0(12cosu) for V05mgl, hasonly a single minimum rendering tunneling impossible.17–21

Nevertheless, the quantum pendulum provides a usefulample of torsional oscillations in the limit of largeV0 andfree quantum rotations in the limit of small values ofV0 .21

Recent experimental illustrations of free rotations in etha~containing CH3 groups! and torsional oscillations of potassium hexachloroplatinate~with a PtCl2 group! have beendiscussed in this context of quantum pendula.21

In the same way that the Josephson junction, scanntunneling microscope, and the tunnel diode exploit transtional tunneling, rotational tunneling spectroscopy has foua practical use, namely the identification and refinemenatom—atom potentials. Recently the method of rotatiotunneling spectroscopy has been used in conjunction wcrystallographic measurements and molecular mechamethods to provide a comprehensive set of techniques formicroscopic characterization of molecular solids.22 The sub-ject of quantum rotational tunneling, although straightfoward, has not appeared in this journal to our knowledHowever, the level of physics and mathematics necessary

885g/ajp/ © 2003 American Association of Physics Teachers

, au

nadala

wss

-anua

thonmrb

oero

on

inanulathondedul-

bewnen-thetheofo-tial

theof

re--

ofheds,

eeth

etateels

an understanding is at an advanced undergraduate levelwe illustrate the main features of rotational tunneling visally. In this paper we discuss the tunneling and libratiodynamics of CH3 groups within the single particle model anillustrate how measurements of tunneling and libration cbe used to extract detailed information on the local molecuenvironment.

This paper is organized as follows. In Sec. II we reviethe problem of hindered rotation for CH3 groups and discusthe different types of motion possible. Section III containbrief review of neutron scattering, particularly from CH3

groups, and measurements from CH3I are presented and discussed. Section IV contains a number of conceptualcomputational problems suitable for advanced undergradstudents.

II. METHYL GROUP DYNAMICS

The single particle model often can be used to describedynamics of solids composed of molecules with at leastmethyl group. A methyl group is best described as a pyracomposed of three hydrogen atoms at the base and a caatom at the apex with an H–H distancedHH.1.78 Å. Oneparticular small molecule, methyl iodide (CH3I!, contains asingle methyl group and is shown in Fig. 1. The momentinertia about the C atom perpendicular to the plane of thatoms is 5.3310247kg m2, which sets the energy scale fothe quantum rotational dynamics. Unhindered rotationsthe CH3 group can be described in terms of the Schro¨dingerequation,

2\2

2I

d2

du2cn~u!5Encn~u!, ~1!

whereu is the coordinate specifying the angular orientatiof the CH3 group. The free rotor wave functions~see Appen-dix A! form a complete basis set for Eq.~1! and can be usedto obtain the energy levels,Ej5B j2, where B5\2/2I5654meV andj is the rotational quantum number.

The motion of the CH3 group sub-unit can be describedterms of a potential composed of an intramolecular terman intermolecular term. In the gas phase the intramolecterm dominates whereas the intermolecular term is dominin the dense solid phase. We restrict our discussion inpaper to the low temperature solid phase and henceconsider the intermolecular potential. This term can becomposed into an attractive van der Waal’s term, a termto steric~short-range! repulsion, and other electrostatic mu

Fig. 1. Stick diagram of methyl iodide, CH3I. Rotations considered in thispaper are shown with the arrow.

886 Am. J. Phys., Vol. 71, No. 9, September 2003

nd-l

nr

a

dte

ee

idon

fH

f

darntisly-e

tipole forces.23 A calculation of the overall contribution tothe intermolecular potential of each of these terms canperformed for a molecular solid whose structure is knoand can often result in a simplification where the total pottial is replaced by a term encapsulating the symmetry ofrotating group and a term encapsulating the symmetry oflocal environment. In this view the threefold symmetrythe CH3 group constrains the symmetry of the total intermlecular potential to be at least threefold. Thus the potencan be expanded in a Fourier series in multiples of 3u,

V~u!5 (m50

`

v3m cos~3mu1am!. ~2!

If the local molecular environment has no symmetry thenfirst two terms (m50,1) provide an adequate descriptionthe dynamics and the phase factors,a0 anda1 , can be set tozero without loss of generality.16 Then the full Hamiltoniancan be expressed as

H52\2

2I

d2

du21

V3

2~12cos 3u!. ~3!

This is the Mathieu equation whose solution has been psented in this journal.21 This Hamiltonian can be solved numerically for a particular value of the barrier height,V3 , asdiscussed in Appendix A. The solution over a broad rangebarrier heights is shown in Fig. 2. As a reference, the daslines in Fig. 2 indicate the barrier heights for two systemsodium acetate trihydrate (CH3COONa•3H2O) and CH3I.There are two limits for the motion shown in Fig. 2. In thlimit of high barrier, small deviations from one of the threequilibrium positions results in nearly harmonic motion withe usual quantized energy levels,

En5~nlib1 12!\v0 , ~4!

where the oscillation frequency is given by

v053AV3

2I, ~5!

Fig. 2. Eigenvalues for a CH3 group as a function of the barrier height,V3 .The dashed line labeled a corresponds to the barrier for sodium actrihydrate and b corresponds to methyl iodide. The first two librational levare labeled by the quantum numbernlib .

886R. M. Dimeo

reb

toicIndt

elb

lcangthglid

ellu

d

mecil

i.a

thn

arieunh

la

ingistictun-lag-

n-hesetheee-s.

tionthethe

tesn

atearly

and I is the moment of inertia.21 The librational transitions~the energy difference between the librational levels! in thislimit are approximately constant, given by\v0 . In the limitof low barrier, the energy levels are approximately the frotation levels discussed previously with transitions givenDEj5Ej2Ej 215(2 j 21)B for j 51,2,3, . . . .

In the intermediate regime where the barrier is nothigh or too low, we can understand the rotational dynamfrom the simplified energy-level diagram shown in Fig. 3.this figure, which emphasizes the eigenvalues intersecteone of the dashed lines in Fig. 2, we have exaggeratedspacing between the levels. The lowest two librational levnlib50 and 1, are shown as well as their tunnel split sulevels. The transition from the ground state (nlib50) to thefirst excited librational state (nlib51), denoted by\v01, candiffer from the ground state tunnel splitting (\v tunnel

0 ) byseveral orders of magnitude. This large difference in thebrational and tunneling transitions is an important practipoint to consider in measurements of rotational tunneliEffectively this means that, at low temperature wherethermal population of the excited librational states is negible, the system can be approximated by a two-state mo

We note also that the invariance of the Hamiltonian~3!under rotations of 2p/3 causes a degeneracy in the tunnsplit librational states. In the language of group theory, sotions of H in Eq. ~3! belong to theC3 point group whichpossesses two symmetry species.24 These species, designateA ~a singlet!, which is symmetric and E~a doublet!, which isantisymmetric, are labeled accordingly in Fig. 3. The symetry of the rotational wave function has a profound effon the overall molecular wave function and its influence wbe discussed in Sec. III.

The rotational excitations observed in measurementsvolve transitions between the eigenvalues shown in FigMeasurement techniques such as nuclear magnetic reson~NMR! and neutron scattering~discussed in Sec. III! probethese transitions directly by the transfer of preciselyamount of energy by which the eigenvalues differ. The tuneling transition, also called the tunnel splitting of the librtional ground state, is shown in Fig. 4 as a function of barheight. Clearly, the Bohr frequency corresponding to the tneling energy decreases as the barrier increases. This beior is due to the fact that the wave functions tend to over

Fig. 3. Schematic illustration of the first two librational levels as intercepby one of the dashed lines in Fig. 2. The symmetries of the eigenstatedenoted by labels A and E and the degeneracy of the states is giveparentheses.

887 Am. J. Phys., Vol. 71, No. 9, September 2003

ey

os

byhes,-

i-l.

e-el.

--

-tl

n-2.nce

e-

-r-av-p

less for larger barriers. Thus the frequency of the tunnelevents diminishes. Perhaps the most important characterof Fig. 4 is the apparent exponential dependence of thenel energy on barrier height,V3 . The sensitivity of the tunneenergy on barrier height can be exploited to probe the mnitude of the barrier.

The barrier height dependence of the first librational trasition calculated from a numerical diagonalization of tHamiltonian in Eq.~3! using the harmonic approximation ishown in Fig. 5. The first librational transition is simply thdifference between the average of the lowest two andfollowing two eigenvalues in the limit of high barriers. Wexpect the harmonic approximation for the librational frquency, Eq.~5!, to improve as the barrier height increaseHowever, Fig. 5 clearly shows that whenV3 /B@1, theslopes of the curves agree, but the harmonic approximasystematically overestimates the actual value. In factagreement improves with increasing barrier height butconvergence is quite slow. It can be shown~see Appendix B!

dareinFig. 4. Tunnel splitting of the ground librational state. The approximexponential dependence of the tunnel splitting on the barrier height is cleseen.

Fig. 5. Barrier height dependence of the first librational transition~solidline!. The dashed line is the harmonic approximation,\v0.

887R. M. Dimeo

er

raluA

ingl t

d

hemob-

tialhe

x-undhelityialalnrem,ano-

alr

the

ss,

odeli-

inTh

fots

ed

po-

ale

that the classical turning point decreases only asV321/4, and

thus the small angle approximation is approached vgradually.

We can calculate the dynamics of the tunneling and libtion processes using the eigenfunctions and eigenvafound through the numerical diagonalization procedure.will be described in Sec. III, a neutron is capable of induca transition between the rotational eigenstates. In generaprocess can be described by

P~u,t !5uc~u,t !u2, ~6!

which is the probability of finding the methyl group orienteat an angleu ~with respect to some arbitrary direction! attime t ~with respect to some arbitrary origin in time!. In Eq.~6! c is represented as a sum over the eigenstates,^uun&, ofthe Hamiltonian, Eq.~3!, as

c~u,t !5 (n51

`

an exp~2 iEnt/\!^uun&, ~7!

where

an5E0

2p

duc~u,0!^uun&. ~8!

Note that in Eqs.~6! and ~7! n denotes the~possibly degen-erate! eigenstates. For simplicity, we will assume that tneutron couples the lowest two eigenstates of the Hatonian, Eq.~3!. The results of such a coupling are shown fV/B563 in Fig. 6. It is clear that the region of largest pro

Fig. 6. Time evolution of the probability density when transitions areduced between the two levels in the tunnel split librational ground state.upper right-hand panel shows the potential barrier forV3542 meV and theprobability densities for the lowest three stationary librational statesreference. The bottom panels show the probability densities at two pointime, t150.3 ns andt252.1 ns.

888 Am. J. Phys., Vol. 71, No. 9, September 2003

y

-ess

he

il-r

ability density oscillates in time among the three potenminima, whereas the region of least probability is at tmaxima of the potential.

The librational dynamics can be illustrated similarly ecept that now we assume that the neutron couples the grolibrational state and the first excited librational state. Tresult is shown in Fig. 7. We can see that the probabidensity ‘‘sloshes’’ back and forth within each of the potentminima. This type of motion is consistent with the classicnotion of libration in which the librating particle oscillates ispace. This behavior, a consequence of Ehrenfest’s theois similar to the behavior of a Gaussian wave packet inharmonic potential where the expectation values of the mmentum and position operators follow classictrajectories.25,26 In fact the classical probability density fothe librating system is given by

Pcl@u~ t !#5d@u~ t !2ucl~ t !#, ~9!

whereucl(t) can be written as

ucl~ t !5u tp sin~vt1a!. ~10!

The classical turning point,u tp , is given by

u tp51

3cos21S 12

\v

V3D , ~11!

anda is the phase shift andv5v01 is the frequency differ-encev between the librational levels. Equation~10! is su-perposed on Fig. 7 to show the correspondence betweenquantum and classical librational motion.

The evolution of the excited state tunneling proce\v tunnel

1 , is shown in Fig. 8. The similarity with Fig. 6 isobvious and the main difference is the appearance of a nin the center of each well. The time-development of thebrational transition,nlib51→2 or \v12, is shown in Fig. 9.

-e

rin

Fig. 7. Evolution of the probability density when transitions are inducbetween the ground librational state and first excited librational state,\v01 .The probability density sloshes back and forth periodically within eachtential minimum. The white solid lines are given by Eq.~10! indicating theclassical oscillatory equation of motion. Note the difference in time sccompared with Fig. 6.

888R. M. Dimeo

ion

rethin-t

illlielhe

Iing

ros-therrboutlarotHthechbe

u-

then–

d inonry

cat-he

inga--H

ledlarsthe

theythe

he

Intheis,

the

c,s,e. At

er-hevems.

s.

-ve

ve

t a

Again we can immediately see the similarity to the transit\v01 in Fig. 7. The classical trajectory, Eq.~10!, based onthe excited state classical turning point and librational fquency,v5v12, has also been superposed to illustratesemi-classical nature of the motion. In both cases involvtransitions between~or within! the higher librational eigenstates, the increase in the kinetic energy is made manifesthe increased ‘‘wiggliness’’ of the probability density. In sthigher excited states, differences in the amount of ‘‘wiggness’’ can be seen in different regions of the potential wthus illustrating the ‘‘wiggliness’’ as a local measure of tkinetic energy.27,28

Fig. 8. Same as Fig. 6 but for the transition induced between the two leof the tunnel split first excited librational state,\v tunnel

1 .

Fig. 9. Same as Fig. 7 except for the transition induced between the firssecond excited librational states,\v12 .

889 Am. J. Phys., Vol. 71, No. 9, September 2003

-eg

by

-l,

III. MEASUREMENTS OF ROTATIONALTUNNELING

The hindered motions of a CH3 group discussed in Sec. Ican be measured using a number of techniques includNMR, neutron scattering, heat capacity, microwave spectcopy, optical spectroscopy, infrared spectroscopy, and otechniques.16 Microwave spectroscopy is typically used fosystems in the gas phase where one seeks information athe intramolecular interactions rather than intermolecuinteractions.1 In general, light scattering techniques cannbe used to measure quantum rotational tunneling of C3groups because such rotations involve a change innuclear spin state of the molecular wave function. Suchanges, which require coupling to the nuclear spin, caninduced using NMR, via dipole–dipole coupling, and netron scattering. The neutron, possessing nuclear spin\/2,couples directly to the nucleus of the scatterer vianucleon–nucleon interaction. The short-range neutronucleon interaction for low-energy neutrons can be treatethe first Born approximation, thus making the interpretatiof the scattering data relatively straightforward. In a vesimple view, rotational spectra measured with neutron stering provide direct information on the energy levels of tscattering system.

To appreciate the utility of neutron scattering in measurquantum rotations, we must first clarify the notion of rottional tunneling from the point of view of the rotor’s constituents. For our purposes the hydrogen atoms in the C3

group can be viewed as protons. If we imagine a CH3 groupwith a well-defined orientation such that we have labeprotons on labeled sites, then we can identify a particuorientation of the CH3 group by the proton location such au123&. The successive state of a 120° reorientation ofmolecule via tunneling is then represented asu312&. Thesestates are often referred to as ‘‘pocket states,’’ becausecorrespond to wave functions that are highly localized inminima of the potential well, that is, in the ‘‘pockets.’’3,16

The tunneling dynamics evolve according tou123&→u312&→u231& with the Bohr frequency given byEtunnel/h.

Section II alluded to the fact that the symmetry of trotational wave functions, the solutions to~3!, has a pro-found influence on the overall molecular wave function.fact, the nature of a quantum rotation is directly related tosymmetry of the wave function. In order to understand thwe must examine the components ofCmol , the molecularwave function. In the Born–Oppenheimer approximationmolecular wave function can be decomposed asCmol

5c transcelcvibc rotcns, which are the translational, electronivibrational, rotational, and nuclear spin functionrespectively.29 In order to induce a quantum rotation of thCH3 group, a change in the nuclear spin state must occurlow temperature we can assume thatCmol is in the groundelectronic state. A quantum rotation of a CH3 group is asymmetry operation and thus it is equivalent to a cyclic pmutation of the nuclei. Permutation of the nuclei leave ttranslational and vibrational parts of the molecular wafunction unchanged so that we may ignore these terTherefore the only portions ofCmol relevant to rotationalmotion are the rotational and nuclear spin wave function

Because the CH3 group is composed of three~indistin-guishable! fermions,Cmol must be antisymmetric under exchange of any two protons. Therefore, the rotational wa

ls

nd

889R. M. Dimeo

th

ar

r-iIncttaseicserr-uc

i

ol-gh

etsngdi

eathspgeurentradr

ittioth

e

e

ooriththinob

ig

ntth

iver

en-

Aspa-

nel-e

lowtateedund

toere

inghe

thator

n.

rom

function must be of the symmetric A species whennuclear spin wave function is antisymmetric.3,16,30Thereforethe allowed combinations of the molecular wave function

Cmol5c rotE cns

A c transcelcvib , ~12!

and

Cmol5c rotA cns

E c transcelcvib . ~13!

Equations~12! and~13! state that, in order to keep the oveall antisymmetry of the molecular wave function, a changethe rotational state requires a change in the nuclear spin.neutron scattering experiment, the neutron couples direto the nuclear spin of the target and can change the spin s

As mentioned in Sec. II, the two-state model can be uas an approximate description of the methyl dynamTherefore one might consider using heat capacity to obsthe Schottky anomaly31 associated with the energy diffeence,\v tunnel

0 , of the two states. This method has been scessful in measurements of one particular tunnel splittingCH4 where phonons induce the transition,32 a phenomenonknown as spin conversion. However, the heat capacitynitromethane (CH3NO2) showed no evidence for the tunneing transition.33 Neutron scattering measurements, thourevealed a tunneling excitation at 34meV.34,35 The disagree-ment between the measurements was appreciated by thsearchers who performed the calorimetric measurementhey pointed out ‘‘it is thus possible for neutron scatteriand calorimetric measurements to yield apparently contratory views of an array of tunneling states.’’33 The origin ofthe different results comes from the nature of the two msurement processes. In the calorimetric measurements,is no mechanism that can induce a change in the nuclearstate that can subsequently induce the necessary chanthe rotational state. Contrary to the heat capacity measment of CH3NO2 where phonons induce a transition betwestates of the same symmetry, phonons cannot induce asition between states of different symmetry. As discussethe previous paragraph, the tunnel split states for methyltors have different symmetry~see Fig. 3!, so they require achange in the nuclear spin state which is not possible wphonons. Thus neutron scattering allows direct observaof the tunneling excitations regardless of the symmetry ofstates.

In an inelastic neutron scattering measurement the incidneutron can impart both energy~E! and momentum (p) tothe scattering system.36 Cold neutrons refer to those whosde Broglie wavelength (l5h/p) is on the order of 4–10 Å,and energies in the range of 0.1–5 meV. The variationscattering intensity with momentum transfer provides infmation on the geometry of the motion and the variation wenergy transfer provides information on the eigenstates ofsystem. The combination of geometrical and dynamicalformation make neutron scattering uniquely suited as a prof molecular dynamics.

The geometry of the scattering process is illustrated in F10. The incident neutron has an energyEi and a momentumpi5\Qi , whereQi denotes the wave vector of the incideneutron. The magnitude of the wave vector is related towavelength viaQi52p/l i , so that the dimension ofQ isinverse length and the popular units used are Å21. Similarly,the energy and momentum of the scattered neutron are gby Ef and pf5\Qf , respectively. Thus the energy transfand magnitude of the momentum transfer are given byE

890 Am. J. Phys., Vol. 71, No. 9, September 2003

e

e

na

lyte.d.

ve

-n

f

,

re-as

c-

-ereinin

e-

n-ino-

hne

nt

f-

e-e

.

e

en

5Ei2Ef and p25pi21pf

222pipf cosu. Neutron spectrom-eters typically can measure molecular dynamics in theergy ~time! range from less thanmeV ~ns! to meV ~ps!.

Usually one measures rotational tunneling below 50 K.the temperature increases, phonons with an energy comrable to the librational transition energy,\v01, modulate thepotential barrier thus destroying the coherence of the tuning motion of the rotor.37,38Because the librational levels arso far apart (;O ~10 meV!!, thermal population of thehigher librational states does not become appreciable attemperature. Because the tunnel-split librational ground senergy is much lower in magnitude than the first excitstate, the effects of the higher excited states on the grostate can safely be ignored at low temperatures.

The scattered intensity of CH3I for very low energy trans-fers collected over a wave vector transfer spanning 0.621.68 Å21 is displayed in Fig. 11. These measurements wperformed at the NIST Center for Neutron Research~NCNR!using a technique known as neutron backscatterspectroscopy.39 There are three prominent features in tspectrum. The large scattering intensity centered onE50 isthe elastic peak and is due to scattering from moleculeseither do not move on the time scale of the instrumentwhose original orientation is identical to its final orientatioThe two satellite peaks located around62.4meV are peaks

Fig. 10. Geometry of an inelastic neutron scattering experiment.

Fig. 11. Neutron scattering data for mi at 10 K. Satellite peaks arise ftransitions within the tunnel-split ground librational state.

890R. M. Dimeo

herdon

atgeet

e-yztcn

rdnta

as

unsi-th

iahotto

stht,onn

n-f

tosi-eV

n-the

g isnalsi-nrep-un-

entsi-n.be-o-

edrtedrect

o-ys-

nd

e-und

neeri-

ct

is--. A

tond

ld

-

ate

lidsed

re

arising from the tunnel split librational ground state. Tinterpretation of this inelastic spectrum is straightforwaThe incident neutron has approximately equal probabilitygaining or losing 2.4meV, thus causing a transition betweethe E and A states (\v tunnel

0 ) in the ground librational state(nlib50). Note that the measurements here were takentemperature low enough that the excited states were larunpopulated. Thus the shape of each of the peaks is dmined by the resolution of the instrument.

An inelastic neutron scattering measurement of the lowlibrational transition,\v01, is shown in Fig. 12. The measurements shown here were carried out using a filter-analneutron spectrometer.40 In this technique, only neutrons thalose energy are detected. This means that the neutroncreate a librational excitation. Unfortunately, measuremeat these larger energy transfers are not as straightforwainterpret as the tunneling measurements. The main reasothe difficulty is that there are many different types of excitions that the incident neutron can create whose energiesof the same order as the librational transitions. The dshown in Fig. 12 clearly show at least two peaks whocenters are very close together. Disentangling the originthe peaks can be aided by deuterating the methyl gro(CH3 becomes CD3). The deuteration increases the momeof inertia by a factor of 2 and reduces the librational trantion by a factor ofA2 @see Eq.~5!#. Indeed such measurements were performed resulting in a clear separation oftwo peaks with the lower energy peak decreased byA2.41

Thus the lower energy peak is assigned as the\v01 transi-tion. The value of the transition based on a fit of a Gaussto the data is 13.4 meV. In a similar type of analysis, thigher energy component in the figure is assigned to rtions (Rxy) of the molecule about an axis perpendicularthe main molecular axis.

The values for the tunneling energy and librational trantion energy can be used to determine the validity ofmodel Hamiltonian, Eq.~3!, and to extract the barrier heighV3 . As discussed in Appendix A, numerical diagonalizatiof the Hamiltonian is used to determine the tunneling a

Fig. 12. Neutron scattering data for mi at 10 K. The two dominant featuare the 0→1 librational transition (\v01) and a rotation of the moleculeabout an axis perpendicular to the main molecular axis (Rxy).

891 Am. J. Phys., Vol. 71, No. 9, September 2003

.f

aly

er-

st

er

antstofor-aretaeofpst-

e

nea-

i-e

d

librational energies for a particular barrier height. The tuneling energy, 2.43meV, corresponds to a barrier height oV3542.2 meV. The librational transition correspondingV3542.2 meV is 14.0 meV. Conversely, a librational trantion of 13.4 meV corresponds to a barrier height of 39.0 mand a tunneling energy of 3.5meV. These differences in theresulting tunnel splitting illustrate the sensitivity of the tuneling energy to changes in the barrier height. We selectfirst value, 42.2 meV, for the barrier because the tunnelinso much more sensitive to the barrier than the libratiotransition. Thus our measurement of the librational trantion, \v01513.4 meV is close but too low by a margilarger than the uncertainty in the measurement. The discancy could come from a number of sources. First, the trcation of the Fourier series for the potential, Eq.~2!, at thefirst two terms may be premature. In fact, the measuremby Prager and co-workers41 were analyzed using an addtional small sixfold term in the potential energy functioAnother source of the disagreement could be a couplingtween the rotations of the methyl group about the main mlecular axis and the perpendicular axis (Rxy) not accountedfor in the potential. Such discrepancies could be explainvia a calculation of thetrue potential barrier via a moleculamechanics calculation in which the potential is calculabased on the low temperature solid structure and the corinteractions between the atoms.

IV. PROBLEMS FOR FURTHER DISCUSSION

~1! For a molecular gas, why does the intramolecular ptential term generally dominate the total potential of the stem?

~2! For an n-fold symmetric barrier,V(u)5 (Vn/2) (12cosnu), calculate the approximate frequency of the groulibrational state. Make sure that you recover Eq.~5! for n53.

~3! For then-fold symmetric barrier considered in the prvious problem, how many sub-states comprise the grolibrational state?

~4! Describe a suitable set of basis functions in which ocould expand the Hamiltonian~3! so that convergence to thexact eigenvalues would be rapid in the limit of high barers.

~5! In the limit of small barrier height, what do you expethe ‘‘tunneling’’ plot shown in Fig. 6~upper left panel! tolook like?

~6! What will be the measurable effects, if any, of a dtribution of barrier heights,g(V3)? Note that we have considered a single barrier height exclusively in this paperdistribution of barrier heights has been used extensivelymodel the effects of disorder on tunneling in polymers aconfined molecular solids.42–45

~7! What effect on the ground state tunnel splitting woudeuterating the CH3 group have~that is, replace CH3 groupswith CD3 groups!? Hint: consider how the librational frequency of the ground state changes upon deuteration@via Eq.~5!# and how this affects the overlap of the new ground stwave function through the barrier.

V. CONCLUSION

The quantum rotational dynamics for a molecular socomposed of methyl group sub-units have been discus

s

891R. M. Dimeo

enr-ic

hed

thr-

mnicllaioa

scfint

set aoam

tn

r tn-an

dbe

nge

c

-r

ot

s,

nro

ln

ei--

torly

as aix.

inof

l-

using a simple parametrization of the intermolecular pottial. Using this simple model we obtained a wealth of infomation on the dynamics of the rotor in a variety of dynamregimes ranging from quantum tunneling to libration. Tdynamics can be easily computed using standard matrixagonalization routines, thus allowing one to visualizetime-evolution of the rotational dynamics. A simple hamonic oscillator approximation of the Hamiltonian, Eq.~3!,can be used to estimate the librational transitions in the liof infinite barrier height. Furthermore, the simple harmooscillator approximation can be used to illustrate the oscitory nature of the quantum motion, appealing to our notof the classical picture of libration. Finally, we showed ththe dynamics can be measured using inelastic neutrontering which probes the eigenvalue spectrum. The valuesthe tunneling and librational transitions as measured usneutron scattering are in reasonable agreement withsimple model of the rotational potential discussed here.

The simplicity of rotational tunneling, especially its eaof observation using inelastic neutron scattering, makes iideal example of quantum dynamics. There are numerexamples of systems that display quantum rotational dynics as discussed here. In fact, there are over 300 systemshave been studied,16 many of them using inelastic neutroscattering. The system presented in this paper, CH3I, hasserved as one of the featured hands-on experiments foNCNR biennial summer school on Methods and Applicatioof Neutron Spectroscopy.46 Those interested in more information about the experiment can download the student hout containing instrumental and experimental details.47 Fi-nally, animations for rotational tunneling in the grounlibrational state and the first librational transition canfound on-line.48

ACKNOWLEDGMENTS

We are grateful to Dr. Terry Udovic for assistance usithe FANS spectrometer at the NIST Center for Neutron Rsearch~NCNR!, Dr. Dan Neumann~NCNR!, Dr. SamuelTrevino ~U.S. Army Research Laboratory!, Dr. ZemaChowdhuri~NCNR, U.MD.!, Dr. Sankar Nair~NCNR!, andDr. Ronald Cappelletti~NCNR! for useful discussions. Thiswork is based upon activities supported by the National Sence Foundation under Agreement No. DMR-0086210.

APPENDIX A: MATRIX ELEMENTS OF THEMETHYL ROTOR

The Hamiltonian for the rigid rotor in the threefold symmetric potential is given by Eq.~3! and reproduced here foconvenience,

H52\2

2I

d2

du21

V3

2~12cos 3u!. ~A1!

The matrix elements can be calculated using the free-rbasis eigenfunctions,

^uun&FR51

A2pexp~ inu! ~n50,61,62,63, . . . !.

~A2!

The Hamiltonian~A1! can be decomposed into two partHmn5Hmn

0 1Vmn , where the elementsHmn0 are related to the

free-rotor energy levels as

892 Am. J. Phys., Vol. 71, No. 9, September 2003

-

i-e

it

-ntat-org

he

nus

-hat

hes

d-

-

i-

or

Hmn0 5

n2\2

2Idmn , ~A3!

wheredmn is the Kronecker delta. This matrix element cabe found via the eigenvalue equation for the free rotor Sch¨-dinger equation,

2\2

2I

d2

du2^uun&FR5En^uun&FR, ~A4!

where the eigenfunctions are given by Eq.~A2! and theeigenfunctions are given byEn5Bn2 with B5\2/2I5654meV. The elements of the potential matrix,Vmn , arefound by calculating the integral,

Vmn5E0

2p

du^muu&FR

V3

2~12cos 3u!^uun&FR. ~A5!

Explicit calculation of~A5! and including the zero-potentiaenergy matrix elements~A3! results in the final expressiofor the matrix elements of the total Hamiltonian,

Hmn5Hmn0 1Vmn , ~A6!

where

Vmn5V3

4~2dmn2dn2m13,02dn1m13,0!, ~A7!

andHmn0 is given by Eq.~A3! for n,m50,61,62,63, . . . .

Standard numerical routines can be used to compute thegenvalues and eigenvectors of Eq.~A6! to any desired accuracy.

APPENDIX B: CONVERGENCE OF THEHARMONIC APPROXIMATION TO THELIBRATIONAL TRANSITION

The rate of convergence of the harmonic approximationthe exact first librational transition can be seen quite cleain Fig. 5. We obtain an estimate of the convergence ratefunction of barrier height for large barriers in this append

We can obtain the classical turning point about theu0

50 equilibrium position for the ground librational statethe harmonic approximation via Taylor series expansionthe potential energy term in Eq.~3!,

\v0

2'

V3

2 F12S 121

2~3u tp!

2D G . ~B1!

Next we can substitute the expression forv0 , Eq. ~5!, intoEq. ~B1! and arrive at the result,

u tp5S 2

9

\2

V3I D1/4

. ~B2!

Table I. Evaluation of Eq.~B2! for a number of barrier heights~in units ofB whereB50.654 meV!, illustrating the slow improvement of the smalangle approximation with increasing barrier height.

V3 /B u tp (°) u tp/60°

1 62 1.010 35 0.58

100 20 0.331 000 11 0.18

10 000 6 0.1

892R. M. Dimeo

n

i.

v/

ve

s

ys

s,’

’

n-

ls,

ng

J.

,’’

arre

v.

a

g’’ Im

ro

ys

cs

ys

m

m

ive

eu-

ues

ms,

o-

yl

se’’ J.

lid

un-m:

g

tter-ion

Z.

the

is-

tronJ.

m

hn,ore

p

n-

The evaluation ofu tp for a few barrier heights is shown iTable I. Because the classical turning point~that is, the maxi-mum deviation from the equilibrium position! decreases soslowly, it is clear that the small angle approximation usedthe harmonic approximation will converge slowly as well

a!Electronic mail: robert.dimeo@nist.gov; URL: http://www.ncnr.nist.gostaff/dimeo

1Chun C. Lin and Jerome D. Swalen, ‘‘Internal rotation and microwaspectroscopy,’’ Rev. Mod. Phys.31, 841–891~1959!.

2I. F. Silvera, ‘‘The solid molecular hydrogens in the condensed phaFundamentals and static properties,’’ Rev. Mod. Phys.52, 393–452~1980!.

3W. Press,Single-Particle Rotations in Molecular Crystals~Springer Ver-lag, Berlin, 1981!.

4Tung-Lin Tsai and G. Thomas, ‘‘Analog between optical waveguide stem and quantum-mechanical tunneling,’’ Am. J. Phys.44, 636–638~1976!.

5E. Condon, ‘‘Tunneling—how it all started,’’ Am. J. Phys.46, 319–323~1978!.

6Jeff. D. Chalk, ‘‘A study of barrier penetration in quantum mechanicAm. J. Phys.56, 29–32~1988!.

7D. Kiang, T. Ochiai, and S. Date´, ‘‘Tunneling through Fibonacci barriers,’Am. J. Phys.58, 1200–1201~1990!.

8Charles W. Leming and Andrew Van Smith, ‘‘A numerical study of quatum barrier penetration in one dimension,’’ Am. J. Phys.59, 441–443~1991!.

9Oscar J. Riveros, ‘‘Tunneling between two inequivalent potential welAm. J. Phys.60, 88–89~1992!.

10C. Julian Chen Author and Walter F. Smith, ‘‘Introduction to scannitunneling microscopy,’’ Am. J. Phys.62, 573–574~1994!.

11H. J. Reittu, ‘‘Fermi’s golden rule and Bardeen’s tunneling theory,’’ Am.Phys.63, 940–944~1995!.

12A. Garg, ‘‘Tunnel splittings for one-dimensional potential wells revisitedAm. J. Phys.68, 430–437~2000!.

13W. M. Robertson, J. Ash, and J. M. McGaugh, ‘‘Breaking the sound brier: Tunneling of acoustic waves through the forbidden transmissiongion of a one-dimensional acoustic band gap array,’’ Am. J. Phys.70,689–693~2002!.

14L. Pauling, ‘‘The rotational motion of molecules in crystals,’’ Phys. Re36, 430–443~1930!.

15It is interesting to note that Pauling did not refer to the tunneling motionrotational tunneling. Rather he described the torsional motion~for a di-atomic molecule! as one of ‘‘oscillational states, the molecule oscillatinabout an equilibrium orientation and changing end for end only rarely.is the rare ‘‘end for end’’ motion that is now referred to as quanturotational tunneling. See Ref. 14, p. 434.

16M. Prager and A. Heidemann, ‘‘Rotational tunneling and neutron spectcopy: A compilation,’’ Chem. Rev.97, 2933–2966~1997!.

17E. U. Condon, ‘‘The physical pendulum in quantum mechanics,’’ PhRev.31, 891–894~1928!.

18T. Pradhan and A. V. Khare, ‘‘Plane pendulum in quantum mechaniAm. J. Phys.41, 59–66~1973!.

19R. Aldrovandi and P. Leal Ferreira, ‘‘Quantum pendulum,’’ Am. J. Ph48, 660–664~1980!.

20G. P. Cook and Clyde S. Zaidins, ‘‘The quantum point-mass penduluAm. J. Phys.54, 259–261~1986!.

21G. L. Baker, J. A. Blackburn, and H. J. T. Smith, ‘‘The quantum penduluSmall and large,’’ Am. J. Phys.70, 525–531~2002!.

22M. Neumann and M. R. Johnson, ‘‘Methyl group tunneling-a quantitat

893 Am. J. Phys., Vol. 71, No. 9, September 2003

n

e:

-

’

’’

--

s

t

s-

.

,’’

.

,’’

:

probe of atom-atom potentials,’’ J. Chem. Phys.107, 1725–1731~1997!.23C. J. Carlile and M. Prager, ‘‘Rotational tunneling spectroscopy with n

trons,’’ Int. J. Mod. Phys. A7, 3113–3151~1993!.24H. W. Kroto,Molecular Rotation Spectra~Dover, New York, 1992!, p. 56.25D. F. Styer, ‘‘The motion of wave packets through their expectation val

and uncertainties,’’ Am. J. Phys.58, 742–744~1990!.26R. W. Robinett,Quantum Mechanics: Classical Results, Modern Syste

and Visualized Examples~Oxford U. P., New York, 1997!, pp. 206–208.27Reference 26, p. 59.28R. W. Robinett, ‘‘Quantum and classical probability distributions for p

sition and momentum,’’ Am. J. Phys.63, 823–832~1995!.29Charles E. Hecht,Statistical Thermodynamics and Kinetic Theory~Free-

man, New York, 1990!, p. 115.30J. Peternelj, T. Kranjc, and M. M. Pintar, ‘‘Rotational motion of meth

groups in solids,’’ Phys. Rev. B54, 955–961~1996!.31R. K. Pathria,Statistical Mechanics~Butterworth-Heinemann, Oxford,

1996!, 2nd ed., pp. 77–79.32M. A. White and J. A. Morrison, ‘‘Deductions about the structure of pha

III from thermodynamic measurements on solid isotopic methanes,Chem. Phys.70, 5384–5390~1979!.

33K. J. Lushington and J. A. Morrison, ‘‘Search for tunnel states in sonitromethane,’’ J. Chem. Phys.73, 2015–2016~1980!.

34S. F. Trevino, ‘‘An observation of one-dimensional reorientation and tnel splitting of the ground and first excited state in a low barrier systeSolid nitromethane,’’ J. Chem. Phys.71, 1973–1974~1979!.

35S. F. Trevino, ‘‘The measurement of tunnel states in solid CH3NO2 andCD3NO2 ,’’ J. Chem. Phys.76, 2578–2579~1982!.

36G. L. Squires,Introduction to the Theory of Thermal Neutron Scatterin~Dover, New York, 1997!.

37A. C. Hewson, ‘‘The temperature dependence of inelastic neutron scaing in rotational tunneling systems. I. Formulation and perturbattheory,’’ J. Phys. C15, 3841–3853~1982!.

38A. Wurger, ‘‘The temperature dependence of rotational tunneling,’’Phys. B: Condens. Matter76, 65–76~1989!.

39P. M. Gehring and D. A. Neumann, ‘‘Backscattering spectroscopy atNIST Center for Neutron Research,’’ Physica B241–243, 64–70~1998!.

40J. R. D. Copley, D. A. Neumann, and W. A. Kamitakahara, ‘‘Energy dtributions of neutrons scattered from solid C60 by the beryllium detectormethod,’’ Can. J. Phys.73, 763–771~1995!.

41M. Prager, J. Stanislawski, and W. Hausler, ‘‘Inelastic incoherent neuscattering study of the methyl rotation in various methyl halides,’’Chem. Phys.86, 2563–2575~1987!.

42J. Colmenero, R. Mukhopadhyay, A. Alegria, and B. Frick, ‘‘Quanturotational tunneling of methyl groups in polymers,’’ Phys. Rev.80, 2350–2353 ~1998!.

43C. Gutt, B. Asmussen, I. Krasnov, W. Press, W. Langel, and R. Ka‘‘Dynamics of methane molecules in the mesopores of controlled-pglass at low temperatures,’’ Phys. Rev. B59, 8607–8614~1999!.

44A. J. Moreno, A. Alegria, J. Colmenero, and B. Frick, ‘‘Methyl-groudynamics from tunneling to hopping in CH3COONa•3H2O: Comparisonbetween a crystal and its glassy counterpart,’’ Phys. Rev. B65, 134202~2002!.

45R. M. Dimeo and D. A. Neumann, ‘‘Low-temperature dynamics of cofined methyl iodide,’’ Phys. Rev. B63, 014301-1–014301-9~2001!.

46For more details see<http://www.ncnr.nist.gov/programs/CHRNS/outreach.html> .

47Available via ftp at <ftp://ftp.ncnr.nist.gov/pub/staff/dimeo/hfbss_01.pdf> .

48<http://www.ncnr.nist.gov/staff/dimeo/rotations.html> .

893R. M. Dimeo