C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8

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Quantum excitation spectrum of hydrogen adsorbed innanoporous carbons observed by inelastic neutron scattering

Raina J. Olsen a,b,*, Matthew Beckner a, Matthew B. Stone c, Peter Pfeifer a,Carlos Wexler a, Haskell Taub a

a Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USAb Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USAc Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

A R T I C L E I N F O

Article history:

Received 30 October 2012

Accepted 11 February 2013

Available online 22 February 2013

0008-6223/$ - see front matter � 2013 Elsevihttp://dx.doi.org/10.1016/j.carbon.2013.02.026

* Corresponding author at: Materials ScienceE-mail address: olsenrj@ornl.gov (R.J. Ols

A B S T R A C T

Inelastic neutron scattering spectra have been collected over a wide range of momentum

transfer from H2 adsorbed in several high-porosity carbon substrates. We show theoretical

spectra which consider the relationship between rotational and translational transitions in

the highly anisotropic adsorption environment, proving that different rotational excita-

tions contain different amount of recoil broadening and motivating a new analysis method

which considers both types of transitions at once. Spectra for most of the samples,

including two activated carbons, are very similar to one another, supporting models of

nanoporous carbons which are quite similar on the sub-nanometer scale. The exception

is the low-energy side of the rotational peak, indicating important differences in the initial

distribution of motion. We also find more subtle differences in the spectra which may be

linked to differences in sample heterogeneity and surface rugosity. One sample does have

a very different spectrum, which is not explained by standard models of this system. We

also observe a significantly reduced effective mass in the spectrum of recoil transitions

and evidence of coupling of rotational and translational motion resulting from periodic

variations in orientation of the rotational states.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Because hydrogen is the lightest molecule in nature, its mo-

tion is the most strongly quantized. This is particularly true

in the confined environment of an adsorbed system. Incoher-

ent inelastic neutron scattering (IINS) provides a direct exper-

imental probe [1–18] of transitions between the quantum

states of adsorbed hydrogen. The adsorption potential per-

turbs the first rotational transition, J ¼ 0! 1 [19–23], which

occurs at 14.7 meV in the free gas. In the anisotropic adsorp-

tion environment, the three J ¼ 1 rotational states become

non-degenerate, with the parallel configuration (J1jj), in which

the molecular axis of the hydrogen molecule tends to be par-

allel to the adsorption plane, occurring at a slightly lower

er Ltd. All rights reserved

and Technology Divisionen).

energy than the perpendicular configuration (J1?) [19–23].

Analysis of this split in energy has been used to characterize

the adsorption potential [1–17], which is used in turn to infer

the composition and structure of the adsorbent. Transitions

in two types of translational motion can also be observed in

spectra, including the strongly quantized bound vibrational

motion of the hydrogen molecule within the adsorption po-

tential perpendicular to the plane of the substrate, and the

free motion in the periodic potential parallel to the plane.

Fig. 1(a) depicts the transitions which are predicted by theory

to be easily observed by IINS.

High porosity carbon materials have a wide variety of

applications, including adsorption and gas storage, separa-

tion of different molecules and different isotopes, and

.

, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA.

(a)

70

80M=0.64 amu

30

40

50

60

2M=2 amu

3.

2 4 6 8 10

10

20

30

1.

2.3.

Momentum Transfer (A-1)2 4 6 8 10

(b)

Ener

gy T

ranf

er (m

eV)

Fig. 1 – (a) Cartoon depicting different types of transitions of

the hydrogen molecule predicted to be observed by

incoherent inelastic neutron scattering (IINS). (b) IINS

intensity map of hydrogen adsorbed in multi-walled carbon

nanotubes after background subtraction, with intensity

plotted on a log scale. Incident neutron energy is 90 meV

and temperature is 23 K. The types of transitions are

identified, and a fit to the roto recoil tail is shown (green

curve). (For interpretation of the references to colour in this

figure legend, the reader is referred to the web version of

this article.)

C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8 47

purification of contaminants [24], all of which depend on

material properties, particularly at the sub-nanometer scale.

The utility of IINS as a characterization technique lies in its

sensitivity to sub-nanometer variations of the adsorption po-

tential and thus the local structure of the material [19–23] on

this scale of interest. In practice, IINS has been more com-

monly used with materials with stronger binding energies

than carbon [9–17], which results in a significant split in en-

ergy of the J1? and J1jj rotational peaks and makes interpreta-

tion of the results relatively simple. In contrast, the predicted

splitting for a carbon substrate is often on the order of the

instrument resolution, meaning interpretation is highly

dependent on the assumptions used during analysis. Previous

work has largely neglected transitions in translational motion

during analysis, despite the fact that they represent a signifi-

cant portion of the spectrum (Fig. 1(b)). In addition, there has

been no comparison between IINS spectra of multiple carbon

samples to verify the consistency of the spectral analysis

used. Only one activated carbon (AC) has been previously

measured with IINS [2], and that work concluded that their

AC contained a significant number sub-nanometer pores

without comparison of this result with independently deter-

mined pore size distributions or IINS from other samples.

Since sub-nanometer pores are the ones with the most vari-

able properties, it is highly desirable to address these issues.

In this paper, we present a set of IINS measurements col-

lected over a wide range of momentum transfer (Q) for hydro-

gen adsorbed on several carbon samples with distinctly

different pore structures (Fig. 3). To our knowledge, previous

published experiments [1–8] have probed a much narrower

range of the excitation spectrum. This is also the first data

set which compares Q-dependent spectra collected with the

same instrument for several different types of nanoporous

carbons. A representative spectral map, which is very similar

to most of our results as well as the previously published

spectrum for AC [2], is shown in Fig. 1(b).

Only one sample has a spectrum which is distinctly differ-

ent from the others, showing asymmetrical broadening of the

rotational peak on the low energy side. The sample does con-

tain a large number of pores �1 nm in width, but the theory

previously used to interpret these spectra predicts pore-

size-dependent changes to the energy of both J1jj and J1?,

which should result in more symmetric broadening. In addi-

tion, we show other general features of the spectral maps

which are unexpected based on previous theory, such as a re-

duced mass in the recoil. We discuss the relationship between

rotational and translational transitions, showing that even if

rotational motion and translational motion are independent

of one another, transitions in both of these types of motion

have a mutual dependence on the orientation of Q, resulting

in a very different shapes for the rotational peaks of J1jj and

J1?.

Both the unexpected experimental results and the theoret-

ically demonstrated relationship between rotational and re-

coil transitions show that there are sufficient issues with

the standard model of this system to justify development of

new analysis techniques. We present a provisional method,

which analyzes rotational and translational transitions to-

gether. Using this method, we find that for most of our car-

bons (including several ACs) the rotational splitting is equal

to that calculated for a sample with few sub-nanometer pores

[23]. Instead, we find more subtle differences in their spectra;

particularly in the Q-dependence, the width in energy trans-

fer (DE), and the relative proportions of the different peaks;

which are consistent with differences in the three-dimen-

sional structure of the samples. We also find differences in

the initial distribution of recoil motion in the samples which

are consistent with previous comparisons of diffusion in dif-

ferent carbon substrates [25]. Possible interpretations for the

IINS spectra of the unique sample are discussed. We also dis-

cuss preliminary numerical calculations which show rota-

tional and translational motion couple to one another,

meaning full five-dimensional solutions are likely needed

for a complete understanding of this system.

2. Background

Inelastic neutron scattering excites transitions in the quan-

tum states of the adsorbed hydrogen molecules. Experiments

are generally performed at cold enough temperatures that

molecules can be assumed to initially be in the ground state.

We expect to observe three different types of transitions in

spectra, all of which are illustrated in Fig. 1(a). The first is a

transition in the highly quantized rotational state of the

Fig. 2 – (a) Cartoon showing a transition to an excited

rotational state excited by two different orientations of the

momentum transfer, Q. (b) Theoretical neutron scattering

spectrum which includes rotational and roto-recoil

transitions. The total spectrum is composed of scattering

from the parallel and perpendicular orientations of the

excited rotational state (Sjj and S?), calculated using Eqs. 3

and 4. (c) Theoretical spectrum fit using a typical technique

by two Gaussian peaks which sit on top of a sloping

background.

48 C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8

diatomic molecule. Molecules start in the J ¼ 0 state which is

spherically symmetric, meaning the molecule is equally likely

to have any orientation. Spectra show a peak around

14.7 meV, representing the J ¼ 0! 1 transition. In an aniso-

tropic adsorption environment, the three J ¼ 1 rotational

states become non-degenerate, with the configuration in

which the molecular axis of the hydrogen molecule is more

likely to be parallel to the adsorption plane (J1jj) occurring at

a slightly lower energy than the perpendicular configuration.

(J1?) [19–23]. This occurs because the adsorption potential is

also dependent on the orientation of the molecule. The ex-

pected ratio of J1jj : J1? depends on the structure of the sub-

strate. For a planar surface, we expect a ratio of 2:1, but for

other structures, such as the inside of a nanotube, the ratio

might be 1:2.

Spectra also have a peak around 29 meV, representing a

roto-vibrational transition, where vibration refers to the mo-

tion of the molecular center of mass perpendicular to the sub-

strate within the adsorption potential. A pure vibrational

transition is not easily observable in spectra because transi-

tions are only proportional to the large incoherent cross sec-

tion when the neutron flips the spin of a proton. Since

protons are fermions, the H2 wave function must remain

anti-symmetric, requiring a spin flip to be accompanied by a

molecular rotational transition J! Jþ 2kþ 1, where k is any

integer. Pure translational transitions are proportional to the

coherent cross section, which is over 40 times smaller, and

thus are not easily observed.

In addition to the rotational and roto-vibrational peaks,

spectra also have a ‘‘tail’’ extending from the main rotational

peak on the high-energy side. This continuous spectrum cor-

responds to a transition in the translational motion parallel to

the adsorption plane that occurs along with a rotational tran-

sition; thus we refer to it as the roto-recoil tail.

Earlier work has analyzed the main rotational peak using

Gaussians or similar peak shapes after removal of the recoil

as a background [1–4,6,8], often by using a Gaussian with a

large standard deviation or some other gently sloping func-

tion to fit the recoil. The justification for this technique is

the assumption that the rotational motion of the hydrogen

atoms about their mutual center of mass can be treated sep-

arately from the translational motion of the hydrogen mole-

cule. This assumption is certainly true in free space.

However, we can show that rotational and recoil transitions

in an anisotropic environment are not separable due to their

mutual dependence on the orientation of Q.

In our previous calculations [23], we showed that neutrons

tend to excite rotational states in which the molecular axis is

parallel with Q. If h is the angle between Q and the normal to

the adsorption plane, the probability that the excited rota-

tional state will be oriented parallel to the adsorption plane

is sin2 h and the probability that the excited rotational state

will be oriented perpendicular to the adsorption plane is

cos2 h. Likewise, only the part of Q which is parallel to the

plane will be able to excite transitions in the motion parallel

to the plane and the part of Q which is perpendicular to the

plane will excite vibrational transitions. Thus the energy of

recoil along the plane should be proportional to Q2 sin2 h.

Fig. 2(a) shows a likely transition for two different orienta-

tions of Q. We can calculate a theoretical spectrum to

demonstrate the expected result of these two separate orien-

tational dependencies, using the following equations to cal-

culate the scattering probabilities S as a function of energy

transfer,

Er ¼ ð�hQsinhÞ2=2M; ð1Þ

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kbTEr

p; ð2Þ

SjjðDEÞ ¼ 1ffiffiffiffiffiffiffiffi2rpp e�expððDE�Er�EJ1jj Þ2=ð2r2ÞÞFn¼1;J0!n;J1jj ðQ; hÞe�DQ2

dQdhsinh; ð3Þ

S?ðDEÞ ¼ 1ffiffiffiffiffiffiffiffi2rpp e�expððDE�Er�EJ1?Þ2=ð2r2ÞÞFn¼1;J0!n;J1? ðQ; hÞe�DQ2

dQdhsinh; ð4Þ

where M is the mass of the hydrogen molecules, T = 20 K is

the temperature, F are form factors defined in our previous

work [23], n is the vibrational state, and D ¼ 0:08 A2 is the

Debye–Waller factor. The first term inside the integrals is a

<10 10−20 20−30 30−40 >400

0.2

0.4

0.6

0.8

1

Pore Width (A)

Frac

tion

of S

urfa

ce A

rea

D

iffer

entia

l Por

e Vo

lum

e (c

m3 /(A

g))

5 10 15 20 25 30 35 400

0.1

Pore Width (A)

MSC−303KHS;0BMWCNT

Fig. 3 – (a) BET pore size distributions of samples calculated

from subcritical nitrogen adsorption isotherms, which give

the pore volume as a function of pore width. (b) Fraction of the

surface area concentrated in different ranges of pore size.

C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8 49

classical free recoil function [33], with the recoil energy (Eq. 1)

defined such that molecules only recoil along the plane. The

second term is a form factor which gives the probability for

a given rotational and vibrational transition, and takes into

account the orientation of Q. The final term takes into ac-

count the initial variance in the position of the molecule in

calculating the probability of a transition.

Fig. 2(b) shows both spectra, as well as the total, Sjj þ S?. Sjjrepresents transitions of molecules to J1jj; thus it peaks at a

lower energy. Because these rotational states tend to be ex-

cited when Q tends to be parallel to the plane and can easily

excite recoil, this curve is relatively broad. In contrast S?,

which represents transitions of molecules to J1?, peaks at a

higher energy and, because these rotational states tend to

be excited when Q is more perpendicular to the plane, it con-

tains much less recoil. We also note that the total theoretical

spectrum does not explicitly show two peaks, even though a

resolution function was not included in the calculation. The

reason for this is the different shape of the two peaks which

compose the total spectrum, in particular the broad shape of

the lower peak.

One can see that fitting techniques which assume that re-

coil occurs in equal amounts along with J1jj and J1? will not

correctly identify the peaks in this spectrum. In Fig. 2(c) we

have shown a fit to our theoretical spectrum which uses a

typical technique. First the recoil is removed as a gently slop-

ing background from the main rotational peak. Then the

remainder is fit with two or more peaks, generally Gaussians

or some other symmetric peak shape. We have demonstrated

the technique with two Gaussians, but often with experimen-

tal spectra it is found that three or more peaks are needed for

a good fit. One of these is generally quite broad, and is attrib-

uted to something like ‘‘a broader distribution of rotational

barriers resulting from different pore widths’’ [2]. However,

we attribute this broad peak to the recoil scattering which oc-

curs in greater amounts closer to the main rotational peak

and thus is not properly removed with the sloping back-

ground. The remaining peaks are significantly moved in en-

ergy by this process, and are generally attributed to some

combination of free hydrogen and highly split rotational

peaks in very narrow pores.

In this work, we address several deficiencies which exist in

the current literature. Firstly, we present the first Q dependent

measurements in the slow neutron limit which are able to

illuminate the nature of the recoil. It does not match the clas-

sical recoil function used in Eqs. 3–4, which works well for

higher incident energies (Fig. 5), but instead shows a reduced

mass in the Q-dependence. Secondly, we present an analysis

technique which better incorporates the relationship between

rotational and recoil transitions that we have demonstrated

in this section and takes into account the observed reduced

mass. Finally, we show spectra from multiple carbon samples

which have been characterized by other methods in order to

demonstrate the consistency of our results.

3. Experimental methods

Measurements were performed using the ARCS spectrometer

at the Spallation Neutron Source at Oak Ridge National

Laboratory. Its large detector coverage allowed data collection

over a wide range of Q and (DE). The high operating power of

the source provided enough flux to measure detailed spectro-

scopic features in multiple samples for several wavelengths

of incident neutrons.

Four carbon adsorbents were studied. Multi-wall carbon

nanotubes (MWCNTs) of outer diameter >8 nm and BET sur-

face area of 500 m2/g were purchased commercially. Because

the tubes are capped and have a large diameter, they offer a

nominally flat and relatively homogeneous surface to be used

as a reference. ‘‘AX-21 MSC-30’’ is a potassium hydroxide

(KOH) activated carbon (AC) produced by Kansai Coke and

Chemicals widely used as a standard sample because of its

high surface area of 2600 m2/g and large proportion of nanop-

ores. ‘‘3K’’ is a KOH AC with a surface area of 2700 m2/g pro-

duced at the University of Missouri from a corncob

precursor. ‘‘HS;0B’’ is a carbon produced at the University of

Missouri by the pyrolosis of poly (vinylidene chloride-co-vinyl

chloride) PVDC ((CH2CCl2)x(CH2CHCl)y) [26] with a surface area

of 700 m2/g.

Pore size distributions, which give the pore volume as a

function of pore size, were calculated from nitrogen adsorp-

tion isotherms using quenched solid density functional the-

ory (QSDFT) [27,28] and are shown in Fig. 3(a). The MWCNTs

have few small pores, consistent with the presence of few

interstial sites and supporting our use of this sample as a ref-

erence. The ACs have a wide variety of pore sizes, resulting

from their origins in organic materials and the aggressive

activation process [24,29]. On the other hand HS;0B, which

is produced by a more controllable chemical process, has a

much narrower distribution of pore sizes, with the main peak

at 8 A and a smaller peak at approximately 30 A. Pore size

5 10 15 20 25

FWHM*

(a)MWCNTMSC−303KHS;0B

5 20 35 50 65 80

Energy Transfer (meV)

Inte

nsity

(arb

. uni

ts, l

og s

cale

)

FWHM*

(b)roto−vibrational peak

Fig. 4 – Inelastic neutron scattering intensity map at 60%

coverage summed over all Q collected with an incident

neutron energy of (a) 30 meV and (b) 90 meV at a

temperature of 23 K. Each spectrum has been normalized by

the amount of hydrogen in the sample cell. *The

instrumental resolution (FWHM) at DE = 14.7 meV for each

incident energy, which is 0.7 meV for an incident energy of

30 meV and 2.3 meV for an incident energy of 90 meV.

50 C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8

distributions were also calculated using non-local density

functional theory (NLDFT) [27,28] (not shown), and show false

minima for all samples due to the failure of this model to take

into consideration surface roughness and heterogeneity. In

addition, the small pore peak for all samples is shifted to

about 12 A, rather than 8 A as in the QSDFT pore size distribu-

tions. It seems nitrogen adsorption has enough uncertainty in

this region to be unable to distinguish robustly between pores

which are just large enough to be approximated as indepen-

dent pore walls or just small enough that there is significant

overlap between the potentials of both walls, creating higher

binding energies and significantly affecting the properties of

these samples. In contrast, the splitting of the rotational peak

measured with IINS is particularly sensitive to this distinc-

tion. We have found that peak locations vary significantly

only for pores under � 9 A in width [23]. Fig. 3(b) shows the

fraction of the surface area in different ranges of pore size

for each sample, using the results of QSDFT analysis. HS;0B

has nearly 90% of its surface area in pores in the sub-nanome-

ter region, whereas the ACs have a broad distribution of nano-

meter sized pores and the MWCNTs have a significant

proportion of very large pores.

Samples were prepared by outgassing at 125 �C in a vac-

uum oven at a pressure of 10�7 bar for 12 h. An aluminum

sample cell of cylindrical geometry with an inner diameter

of 0.6 cm and height of 6 cm was filled to capacity with

approximately 0.5 g of the powdered carbon sample under a

helium atmosphere. The lid of the cell was fitted with a cap-

illary tube to allow the addition of gas in situ. The sample cell

was loaded into a closed-cycle refrigerator (CCR) and flushed

with H2 three times, outgassing between each. After reaching

a temperature of 30 K, background spectra were taken at inci-

dent neutron energies of 30 and 90 meV.

Molecular hydrogen at room temperature to make 60% of a

monolayer was loaded, with the amount calculated based on

sample surface area, assuming a nominally flat surface and

an area per adsorption site of 10.7 A2 as estimated by neutron

diffraction of H2 on Grafoil [30]. Spectra were also collected at

90% and 120% coverage for some samples. The amount

loaded was measured by filling a known room temperature

volume to a given pressure, then opening a valve to the sam-

ple, which acted as a cryopump, causing a quick drop in pres-

sure of the room temperature volume. Experiments are

generally performed with the molecules initially in the

ground rotational state, J ¼ 0. After cooling to 15 K, successive

spectra were collected, and the J ¼ 0! 1; 1! 0 transitions at

DE � �14.7 meV compared to determine the population of

spin states over time. When the intensity of these peaks

was within error bars in subsequent spectra, it was assumed

that the majority of the molecules had reached J ¼ 0 and final

data collection began.

Due to the experimental arrangement, a temperature gra-

dient existed between the sample cell and CCR. During a sub-

sequent experiment with a sample cell containing sample 3K

and H2 at 85% coverage, the temperature was measured to be

15.0 � 0.3 K at the CCR, 23.1 � 0.1 K at the top of the cell, and

23.5 � 0.1 K at the bottom of the cell. Using the principle of

1 R.J. Olsen et al., unpublished.

detailed balance and the peak intensities summed over all Q

at DE � �14.7 meV, the sample temperature for sample 3K

and H2 at 90% coverage (from this data set) was calculated

to be 23.4 K. Thus we report the sample temperature of this

data set as 23 K.

Spectra were also collected1 for sample 3K and two vari-

ants of HS;0B in a subsequent experiment using a similar pro-

cedure. Spectra were taken at incident neutron energies of 30,

90 and 500 meV and H2 coverages of 25% and 85%.

4. Raw data

Fig. 4 shows a spectrum for each sample summed over all Q

for both incident neutron energies. To account for the differ-

ent surface areas of each sample, spectra are normalized by

the amount of hydrogen in the sample cell whenever sam-

ple-dependent spectra are compared. The 30 meV incident

energy spectra (Fig. 4(a)) provide a detailed picture of the first

rotational peak, at �14.7 meV. As with similar spectra col-

lected from other carbon samples [1–7], the predicted splitting

of the rotational peak for a pore >1 nm (� 1:0 meV [19–23]) is

only slightly larger than the resolution of the instrument

around the main rotational peak (0.71 meV, shown in the fig-

ure), and it is not possible to distinguish the two or more sep-

arate peaks which are predicted by theory to compose the first

rotational peak. The peak is slightly broadened for the ACs

C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8 51

compared to the MWCNTs. On the other hand, this peak is

greatly broadened for HS;0B, with significantly more broaden-

ing on the low energy side. This broadening could be ex-

plained by a significant number of the sub-nanometer pores

in which the splitting of the rotational peak is greater [23],

but this hypothesis does not explain the asymmetric shape

of this peak, since theory predicts that the presence of a sig-

nificant number of sub-nanometer pores would result in

changes to all peak locations. The biggest difference between

the samples is on the low energy side of the rotational peak.

This part of the spectrum represents transitions in which

the hydrogen molecule gains rotational energy but loses re-

coil energy. Thus the differences between samples likely re-

flect the initial distributions of recoil motion in the samples,

with the MWCNT sample appearing to have the largest initial

distribution of motion. These results are consistent with pre-

vious comparisons of motion in different carbon substrates,

which find diffusion coefficients of hydrogen adsorbed in

ACs are much smaller than in substrates more similar to

MWCNTs, such as Grafoil and single-walled carbon nano-

tubes [25].

The 90 meV incident energy spectra (Fig. 4(b)) show more

of the recoil, as well as the roto-vibrational peak. For each

sample, the roto-vibrational peak occurs at DE � 29 meV,

which is within 1 meV of its calculated position [23] for a flat

graphene sheet. We previously found that [23], while the split-

ting of the rotational peak varies with pore size by only frac-

tions of a meV, the position of the roto-vibrational peak varies

by tens of meV, making analysis of the roto-vibrational peak a

more robust method for the detection of variations in pore

size. While the roto-vibrational peak is also broadened for

HS;0B, its location is not significantly different from the other

samples, a finding which is inconsistent with the presence of

a significant number of very narrow pores in this sample.

Free-recoil generally occurs with a quadratic dispersion,

DE ¼ ð�hQÞ2=2M, where M is the mass of the recoiling molecule.

Momentum Transfer (A−1)

Ener

gy T

rans

fer (

meV

)

2 6 10 140

100

200

300

400

Fig. 5 – Inelastic neutron scattering intensity map of

hydrogen adsorbed in 3 K, with intensity plotted on a log

scale. Incident neutron energy is 500 meV and the

temperature is 23 K. Free recoil lines with M ¼ 2 amu go

through the peaks, extending from the J ¼ 0! 1 transition

at DE = 14.7 meV, the J ¼ 0! 3 transition at DE = 88.2 meV,

and the J ¼ 0! 5 transition at DE = 220.5 meV.

Fig. 5 shows a spectrum collected with a 500 meV incident

neutron energy. The energy of each rotational level is given

by DEJ ¼ BJðJþ 1Þ, with B = 7.35 meV. Thus with the high en-

ergy of the incident neutrons, we expect to see the

J ¼ 0! 1;3;5 transitions at energy transfers of DE ¼ 14.7,

88.2, 220.5 meV. We have plotted a theoretical free recoil tail

with M ¼ 2 amu extending from each of these transitions in

Fig. 5 and these cover all of the areas of high intensity. The

high energy of the incident neutrons and the large values of

Q and DE mean that the impulse approximation can be used

and the molecule can largely be treated classically. This is

consistent with previous work with H2 on a carbon nanotube

substrate [31], which has also concluded that the molecule re-

coils freely when high energy incident neutrons are used.

In contrast, the Q of the ridge of maximum intensity as a

function of DE of the roto-recoil tail of the spectral map in

Fig. 1 is best fit with M ¼ 0:64� 0:07 amu. We find no signifi-

cant sample-dependent variation in this value. This data

was measured using neutrons with an incident energy of

90 meV and such small values of Q, that the slow neutron lim-

it is more appropriate. In this regime, the overlap of the initial

and final wavefunctions with the neutron momentum trans-

fer is used to calculate scattering probabilities. Previous theo-

retical work [32] in the slow neutron limit finds M ¼ 1:23 amu

for H2, still significantly larger than our value. A reasonable

hypothesis is that there is coupling between the roto-recoil

continuum and the roto-vibrational state, a possibility which

is consistent with the overlap of the roto-recoil tail and roto-

vibrational peak in both DE and Q in Fig. 1. This finding sug-

gests that the translational motion parallel and perpendicular

to the plane cannot be treated independently of one another.

5. Data analysis

In materials which bind hydrogen strongly [9–17], IINS has

found the J ¼ 0! 1 rotational peak splits into two distinct

peaks, representing transitions to J1jj and J1? in which the ex-

cited rotational state describes a molecule whose axis tends

to align either parallel or perpendicular to the plane, with

J1jj lower in energy. We do not see two distinct rotational

peaks in our spectra, even though calculations with carbon

adsorbents predicts they should be there [19–23]. Previous

work with carbon substrates often does not see distinct split-

ting [1,2,5,7,8], or sees two distinct rotational peaks only at

low coverage [3,4,6]. Thus explicit splitting is likely not seen

in our spectra because the predicted splitting, �1 meV, is only

slightly larger than the instrument resolution, and because

we have taken spectra at relatively high coverage. However,

because of the high interest in the structure of nanoporous

carbon materials, previous work has fit the rotational peak

with two or more peaks [2–4,6–8], just as is done with strongly

binding materials which show distinct peaks. The fact that

one of our samples is significantly different from the others

is an additional motivation to analyze the rotational peak,

in order to understand the origin of this difference.

In Fig. 6(a) we compare an experimental spectrum with the

theoretical spectrum (same as Fig. 2(b)). Comparison of this

theoretical spectrum with the experimental data reveals sev-

eral differences. The location of the main rotational peak is

1 2 3 4 5 6

Momentum Transfer (A−1)

Aver

age

Inte

nsity

(arb

. uni

ts)

peak of Ib

fit with F⊥

peak of Im

fit with F||

Fig. 7 – Q-dependence of the average intensity in a

DE ¼ 0:8 meV region around the peak location for Ib and Im

for the MWCNT sample at a temperature of 23 K. Fits to

these functions using our previously calculated [23] form

factors are also shown, where Ib has been fit with F?ðQÞ and

Im has been fit with FjjðQÞ.

(a) experimenttheoryS||

S⊥

10 12 14 16 18 20 22

Energy Tranfer (meV)

Inte

nsity

(arb

. uni

ts)

(b) I totalImIb

Fig. 6 – (a) A comparison of theoretical and experimental

spectra. (b) Experimental intensity I (error bars are smaller

than the symbols) decomposed into Ib and Im summed over

Q ¼ 2� 4 A�1 for the MWCNT sample. Incident energy is

30 meV, temperature is 23 K.

52 C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8

shifted downward by 0.2 meV in the experimental spectrum,

consistent with previous work [1–6,30] which has found the

first rotational peak of hydrogen adsorbed in carbons at

14.5–14.6 meV rather than the 14.7 meV of the free gas. The

theoretical spectrum also has significantly more recoil scat-

tering on the high-energy side. This difference could be re-

lated to our approximation of the recoil as classical; when

instead in the slow-neutron limit the full five-dimensional

wavefunctions which include both rotational and transla-

tional motion should be used to calculate form factors.

To fit our data, we have tried various methods to fit both

the rotational and recoil parts of the spectrum together, uti-

lizing the full two-dimensional data set. In this process, we

found an effective method which separates the experimental

intensity into a broad peak which includes all of the recoil,

and a Gaussian which contains no recoil. This procedure

(which uses the full two dimensional spectral maps and is de-

scribed in more detail in the Appendix), fits the roto-recoil tail

as a function of both Q and DE, extends that fit into the main

rotational peak and subtracts it from the experimental inten-

sity, and finds the remaining intensity is well-described by a

Gaussian. The results of the analysis are shown in Fig. 6(b),

which shows a one-dimensional projection of the experimen-

tal intensity I, decomposed into the two peaks, which we call

Im and Ib (where the latter is the Gaussian). We will discuss

possible physical interpretations for these peaks in the next

section.

The analysis technique we have used, which splits spectra

into a broad recoil peak and a narrow Gaussian, does not ex-

actly match the theoretical spectrum in Fig. 2(b), which pre-

dicts a broad peak containing significant recoil and a

narrow peak containing much less recoil. This could be

because the current theory needs to be modified, and may ex-

plain why our theoretical spectrum contains more recoil over-

all than the experimental spectrum. This could also be

because the experiment is not sensitive enough to distinguish

between a Gaussian and a peak containing little recoil. How-

ever, we can offer several justifications that the technique re-

sults in a reasonable separation of the rotational peak.

Fig. 7 shows the average Q-dependence of Ib and Im at their

respective peak locations, DEb ¼ 14:7 meV and DEm ¼13:8 meV. Clearly, the two types of transitions have a signifi-

cantly different Q-dependence, both in shape and peak Q, a

justification of our technique. (In fact, because the peak Q of

Im shifts upwards as DE increases, the difference between

the peak Q locations increases when both are plotted at the

same energy transfer DE ¼ 14:7 meV.) The difference in the

Q-dependence of Ib and Im, as well as their different shapes

and peak locations in DE (Fig. 6(b)) creates a distinctive al-

mond shape to the composite rotational peak visible in exper-

imental spectra (see Appendix, Fig. A1(a)).

Depending on the resolution of the instrument and the

peak splitting of the material, some previous experimental

work with carbon materials has been able to see two explicit

peaks at low coverage [3,4,6]. Our analysis finds a larger split-

ting for HS;0B, and in the second experiment we were able to

distinguish the two peaks in the full spectra at low coverage.

Fig. 8 shows the raw spectrum for a variant [26] of HS;0B at

25% coverage, with the peak locations found by our analysis

for the original spectrum of HS;0B at 60% coverage marked.

The new spectrum clearly shows two separate peaks, with

the same approximate locations as the values found with

our analysis technique, a further justification of the method-

ology we have presented in this work.

6. Peak identification

Because of the shape of the peaks, we tentatively identify the

Gaussian (Ib) as transitions to excited rotational states which

are translationally bound with respect to motion parallel to

the adsorption plane and the broad curve (Im) as transitions

to excited rotational states which are mobile with respect to

10 12 14 16 18

Energy Transfer (meV)

Inte

nsity

(arb

. uni

ts)

Fig. 8 – Experimental neutron scattering spectrum summed

over all Q for a variant of HS;0B. Incident neutron energy is

30 meV, the sample contains H2 at 25% coverage, and the

temperature is 23 K. The peak locations of HS;0B in Table 1

are also marked.

C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8 53

motion along the adsorption plane. Because the peaks are sig-

nificantly split in energy, with Ib at 14.7 meV and Im at

13.8 meV, it is also natural to associate Ib with transitions to

J1? and Im with transitions to J1jj. If this peak identification is

correct, it represents strong coupling between rotational and

translational degrees of freedom, with recoil occurring only

with a parallel rotational excitation. Rotational motion has

been treated separately from translational motion along the

adsorption plane by all previous theory [19–23].

To investigate the possibility of roto-translational cou-

pling, we have numerically calculated wavefunctions [34] for

a four-dimensional adsorption potential (a slice of a five

dimensional graphene adsorption potential [35,23]) using a

limited rotational basis (J ¼ 0;1). These preliminary calcula-

tions do indeed show significant coupling of rotational mo-

tion and translational motion along the plane. Specifically,

we find mixing between the different orientations of J1, which

result in mu (the magnetic rotational quantum number which

gives the projection of the angular momentum in the direc-

tion u) no longer being a constant of the motion, resulting

in the angular momentum varying periodically by as much

as 0.3 �h (depending on the state).

This result is caused by the corrugation of the adsorption

potential, another feature of the problem which has not been

included in previous calculations of quantum states [19–23].

With our four-dimensional potential (as well as with the full

five-dimensional potential), we find that not only does the po-

tential minimum change by �45 K as the lateral position var-

ies between the center of a hexagonal cell and a position

directly above a carbon atom [23], but the distance of the min-

ima from the plane varies by �0.1 A. We also find that while

the rotational states tend to align parallel or perpendicular

Fig. 9 – Excited J = 1 rotational state performing a rocking and bo

to the plane at symmetry points [23], as the lateral position

varies between these points, the orientation of the diagonal-

ized J ¼ 1 states varies by more than 10�. Fig. 9 depicts the

simultaneous variation in the equilibrium orientation and

perpendicular position as a function of lateral position (x). Be-

cause the binding energy, equilibrium orientation, and equi-

librium distance from the plane all have a dependence on x,

states calculated by treating rotation and vibration separately

from recoil interfere through the @2=@x2 term of the

Hamiltonian.

Clearly, improved theory which solves the full five-dimen-

sional problem is required to explore these effects. We also

propose that an experiment with an oriented adsorbent,

which collects spectra as a function of the direction of Q rel-

ative to the adsorption plane, will be quite useful in fully

understanding the different types of transitions which are

contained in the powder spectra.

While there is still uncertainty in identifying these peaks,

we nevertheless associate them with different types of excita-

tions which exist simultaneously at any location on the sur-

face. This represents a significantly different interpretation

than those which would identify the peaks as two different

populations of H2 at different locations.

One such alternative peak identification put forth by previ-

ous work [2,6,7], is that Ib represents free (or very weakly

bound) hydrogen and Im represents hydrogen strongly bound

to the surface. This hypothesis is based on the peak location

of Ib near 14.7 meV, which is also the unperturbed energy of

transitions of free hydrogen to an excited rotational state.

However, binding energies of hydrogen on carbon substrates

are on the order of 500 K [35], meaning as long as molecules

have access to a surface, they have <1% probability of being

free at typical temperatures for these experiments (5–40 K).

Conversely, in our spectra Ib represents a significant propor-

tion of intensity. Additionally Georgiev et al. [2] also found

that the intensity of the peak near 14.7 meV as a fraction of

the total J ¼ 0! 1 peak decreases as the temperature in-

creases. We also observed Ib to decrease in intensity with

temperature. In contrast, an increase in temperature lowers

the probability of adsorption, and should result in more free

molecules.

Another alternative, based on the shape of the peaks, is

that Ib represents solid hydrogen which is not free to recoil

and Im represents liquid or gaseous hydrogen which is free

to recoil. A similar hypothesis has been used to explain a peak

at 14.7 meV in spectra of H2 adsorbed in Xerogel at 11 K [36].

This could not be merely a temperature or pressure depen-

dent effect, as the temperature (23 K) is well above the melt-

ing point (14 K) and the pressures (measured on the order of

uncing motion as it moves across the corrugated potential.

54 C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8

millibars) are too small to create a solid phase. Instead, we

must imagine that there is a high enough binding energy in

some pores to immobilize molecules adsorbed in these sites.

Fig. 10(a) shows coverage dependent spectra for sample

3K. We have applied our analysis to each spectrum, and show

the ratio of Ib=Im in Fig. 10(b). The fraction of scattering from

bound states increases slightly as the coverage increases. This

is consistent with our interpretation of Ib as bound states, be-

cause we expect that states with no (or little) recoil should be-

come more highly populated as the coverage increases and

interactions restrict the motion of neighboring molecules.

For comparison, the ratio of Ib=Im is also plotted for a scenario

in which Ib represents a solid fraction of molecules which are

immobilized by a 10 K increase in binding energy in 15% of

the sites. The high binding energy sites are occupied first,

and the ratio of Ib=Im drops quickly as a function of coverage

as these sites fill up. Changing these parameters changes

the slope and y-intercept of the resultant curve, but the ratio

of Ib=Im always drops as a function of coverage in this sce-

nario. This is precisely the opposite of the observed experi-

mental trend, in which Ib=Im rises as a function of coverage.

Fig. 10 shows the total scattering from Ib as a function of cov-

erage. It grows slightly faster than the coverage, but then sat-

10 15 20 250

5

10

15

Energy Transfer (meV)

Inte

nsity

(arb

. uni

ts) 25 %

60 %

85 %

90%

120 %

0 20 40 60 80 100 120

0.2

0.30.4

0.5

0.6

I b/I m

experimentaltheory,solid fraction

0 20 40 60 80 100 1200

50

100

Coverage (%)

I b (arb

. uni

ts)

Fig. 10 – (a) Experimental neutron scattering spectrum

summed over all Q as a function of coverage for sample 3 K.

Incident neutron energy is 30 meV and temperature is 23 K.

(b) Ratio of Ib=Im, summed over all Q and

5 meV < DE < 25 meV, as a function of coverage. Data points

from the second data set are circled. The experimental trend

is compared with a theory in which Ib represents a solid

fraction of molecules which are immobilized by a 10 K

increase in binding energy in 15% of the sites. (c) Ib, summed

over all Q and 5 meV < DE < 25 meV, as a function of

coverage.

urates near monolayer coverage. This behavior is consistent

with the interpretation of Ib as bound surface excitations in

sites with no significant difference in binding energy from

the rest of the surface, which saturate when the first adsorbed

layer saturates.

We can also use these coverage dependent results to ex-

plain why two distinct peaks have been observed in the rota-

tional peak only at low coverage ([3,4,6] and Fig. 8). At low

coverage (and depending on the resolution), Ib is a small part

of the spectrum and appears as a distinct peak on the sloping

high-energy side of the broad Im. But as the proportion of Ib

grows, it merges into Im at the location of its peak, obscuring

its distinct presence.

7. Results

Separation of spectra into Ib and Im and interpretation of these

peaks as bound and mobile excited states results in conclu-

sions which are consistent with other methods of structural

characterization.

Fig. 11 shows spectra at 60% coverage for each sample

decomposed into mobile and bound states and Table 1 gives

the peak locations, widths, and areas. For the MWCNTs and

the ACs, both Im and Ib peak at approximately the same loca-

tion for every sample. Coupled with the similar location of the

roto-vibrational peak (Fig. 4(b)) in all samples, this suggests

that there are relatively few sub-nanometer pores in the

ACs and the average binding energy is the same in all of these

samples. The area of Im for the ACs is � 95% that of the

MWCNT sample, and is slightly wider. The area of Ib for the

ACs is � 105% that of the MWCNT sample, and is significantly

wider, consistent with a greater surface heterogeneity.

The small decrease in scattering to mobile states and

small increase in scattering to bound states of the ACs is con-

sistent with slightly decreased planarity of the adsorption

surfaces in AC. This explanation is also consistent with the

significantly lower scattering on the low energy side of the

rotational peak for the ACs (Fig. 4(a)), meaning the decreased

planarity of the ACs results in a smaller amount of transla-

tional motion in the initial distribution. Low-resolution

10 12 14 16 18

Rel

ativ

e In

tens

ity

Energy Transfer (meV)

MWNCTMSC−303KHS;0B

(b) Ib

(a) Im

Fig. 11 – Experimental neutron scattering spectrum

summed over all Q separated into (a) Im and (b) Ib for

hydrogen in several carbon adsorbents. Spectra are

normalized to the amount of hydrogen in the sample.

Table 1 – Peak location (meV), width (meV), and area (arb. units) of the mobile and bound states, calculated from the 30 meVspectra as for each sample. The step size in DE is 0.05 meV, and the resolution at 14.7 meV is 0.7 meV.

Im peak Im HWHMa Im areab Ib peak Ib FWHM Ib area b Ib=Im

MWCNT 13.8 1.0 4.0 14.6 1.1 1.0 0.25MSC-30 13.8 1.2 3.8 14.7 1.5 1.0 0.283K 13.9 1.2 3.8 14.7 1.3 1.1 0.28HS;0B 13.5 1.9 5.5 14.7 1.6 1.22 0.22

a The half width at half maximum (HWHM) is given for Im and gives the distance between the peak and the location of half intensity on the low

energy side.b The area is equal to the intensity summed over all Q and DE = 5–22 meV and is normalized by the unity area of Ib for the MWCNT.

C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8 55

transmission electron microscopy (TEM) images of 3K previ-

ously collected [26] are similar to TEM images of many other

ACs [24,29]. Higher resolution images of ACs [37] are able to

detect individual atoms, and observe graphene-like sheets

with mostly hexagonal rings, with some pentagonal rings

which cause curvature of the sheets. This result is consistent

with our IINS results, as well as previously proposed models

of AC composed of graphene-like plaquettes [24] which have

only a small amount of curvature along the plane on the

nanometer length scale.

The fact that the MWCNT and AC samples are so similar,

despite significant differences in their pore size distributions

(Fig. 2), is further evidence against an interpretation which

would identify Ib with H2 molecules immobilized in the small-

est pores. Likewise, an interpretation which identifies Ib as

free molecules does not explain why there should be more

free molecules in the ACs or why the peak broadens.

Table 2 also gives the Debye–Waller factors for all samples

as a function of coverage, and they are also quite consistent

between the MWCNT and AC samples. The finding that these

values decrease as a function of coverage for each sample is

consistent with the interpretation of the Debye–Waller factor

as the average variance in position of the center of mass of

the initial state of the molecule along the direction of Q, as

well as our expectation that increasing coverage tends to re-

strict motion along the plane and localize molecules. The

finding that the Debye–Waller factor of Ib tends to be larger

than that of Im is consistent with an interpretation of Ib as

J1?, which would tend to be excited when Q has a large com-

ponent perpendicular to the adsorption plane. The narrow

adsorption potential means that molecules are strongly quan-

tized perpendicular to the plane, making them more delocal-

ized in this direction.

The results for HS;0B reveal that most of the difference in

its spectrum is contained within Im, which is larger, asymmet-

rically broadened, and peaks at a significantly lower energy. Ib

Table 2 – Debye–Waller factors in A2 of the mobile and bound ssample and coverage.

Coverage Im 60% Im 90% Im 12

MWCNT 0.085 0.081MSC-30 0.084 0.0793 K 0.086 0.079 0.07HS;0B 0.074 0.073 0.06

and the roto-vibrational peak do not move compared to the

other samples, although both do broaden. The Debye–Waller

factors are also significantly different from those of the other

samples. Previous theory predicts that the presence of a sig-

nificant number of sub-nanometer pores would result in

changes to all peak locations [23]. We briefly offer two possible

explanations for the results of this sample, which depend on

the details of peak identification.

The first possibility is that indeed, the sample does contain

significant numbers of sub-nanometer pores, but only one of

the rotational peaks is moved because the two peaks ob-

served do not represent J1? and J1jj, as previous theory has as-

sumed. While our preliminary theoretical calculations do

indeed find coupling between rotational and translational

motion, the resultant states are still primarily composed of

either J1? or J1jj. However, we have not included J > 1, or the

third translational dimension. It is quite possible that the

inclusion of either of these will significantly change the nat-

ure of the resultant rotational states, and thus change the

peak identification. Why, then, does the roto-vibrational peak

also not move? In our previous calculations we found that the

location of the roto-vibrational peak was largely increased in

energy for pores with widths under 7.5 A, it was slightly de-

creased for pores with widths between 7.5 and 9 A. Therefore,

a distribution of pores with widths �6–10 A could create a

roto-vibrational peak which is broadened but not significantly

moved, when compared to the roto-vibrational peak created

by pores with widths >10 A.

The possibility is that there are in fact no sub-nanometer

pores in the sample (and the peak pore size is closer to the

12 A calculated by NLDFT analysis of nitrogen adsorption iso-

therms rather than the 8 A calculated by QSDFT analysis), but

that some other mechanism, caused by a structural feature of

the sample other than pore width, moves one of the peaks.

TEM images of HS;0B were previously collected [26], but due

to low resolution of the instrument, the sub-nanometer pores

tates, calculated from the 30 meV spectra as a function of

0% Ib 60% Ib 90% Ib 120%

0.10 0.0980.11 0.090

7 0.10 0.085 0.0819 0.057 0.054 0.051

56 C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8

which make up the bulk of the pore size distribution (Fig. 3)

were not visible. Higher resolution images of this sample

are needed to understand if it has any other significant struc-

tural features which may affect its IISN spectra.

While we cannot yet determine precisely how the struc-

ture of HS;0B causes its unique spectrum, it is clear that the

standard model, which cannot explain asymmetrical broad-

ening of the rotational peak, is not yet complete. Full solution

of the five-dimensional problem as a function of pore width

(and perhaps other structural features) and further experi-

mental work is needed to understand the physical origin of

the two different rotational peaks (which are observed as dis-

tinct peaks at low coverage in Fig. 8 for HS;0B).

8. Summary

We have presented an extensive data set of Q-dependent IINS

measurements of molecular hydrogen adsorbed in four car-

bon samples. We have identified several common features

which are unexplained by previous theory, including a signif-

icantly reduced effective mass in the spectrum of recoil tran-

sitions and evidence of coupling between rotational and

translational motion. We find that the spectra of most sam-

ples, included a sample of multi-walled carbon nanotubes

and two activated carbons, are quite similar to one another.

This finding supports a model of nanoporous carbons which

are quite similar on the sub-nanometer scale. One sample,

manufactured by the pyrolosis of a polymer, shows asymmet-

rical broadening of the rotational peak on the low energy side.

The theory previously used to interpret these spectra predicts

pore-size-dependent changes to the energy of both J1jj and J1?,

which should result in more symmetric broadening.

We have also shown that even if rotational motion and

translational motion are independent of one another, transi-

tions in both of these types of motion have a mutual depen-

dence on the orientation of Q, resulting in very different

shapes for peaks representing different orientations of the ex-

cited rotational state. We presented an analysis technique

which considers both types of transitions together, and sepa-

rates spectra into states which either do or do not contain re-

coil broadening. Comparison of these peaks between samples

allows characterization of super-nanometer structural fea-

tures which affect spectra in more subtle ways than gross

movement of the main peaks, such as in differences in the

Q-dependence, width in DE, and the relative proportions of

the different peaks. Spectra of two activated carbons are

found to be consistent with models in which the substrate

is composed of graphene-like plaquettes with a small number

of non-hexagonal rings creating curvature along the surface.

This finding is also consistent with TEM images of activated

carbon, but with the advantage that IINS simultaneously

characterizes the entire sample rather than just the few sur-

face pores imaged. We find that the activated carbons studied

have few of the very small pores in which significantly deeper

binding potentials are created by the overlap of the binding

potential from each wall. Results suggest that simultaneous

treatment of translational and rotational degrees of freedom

and consideration of the potential corrugation are necessary

for improved theoretical understanding of INS spectra of ad-

sorbed hydrogen and development of INS as a sub-nanometer

characterization technique.

Acknowledgments

We would like to thank Enrique Robles for capable experi-

mental assistance. This research was supported by the

Department of Energy Office of Basic Energy Science (DOE-

BES) under contract DE-FG02-07ER46411. Research at Oak

Ridge National Laboratory’s Spallation Neutron Source was

sponsored by the Scientific User Facilities Division, Office of

Basic Energy Sciences, U.S. Department of Energy. H.T. was

supported by the National Science Foundation (NSF) under

contract number DMR-0705974 and DGE-1069091. R.J.O. was

also supported in part by the DOE Office of Energy Efficiency

and Renewable Energy (EERE) Postdoctoral Research Awards

under the EERE Fuel Cell Technologies Program, administered

by the Oak Ridge Institute for Science and Education (ORISE)

for the DOE. ORISE is managed by Oak Ridge Associated Uni-

versities (ORAU) under DOE contract number DEAC05-

06OR23100. All opinions expressed in this paper are the

authors’ and do not necessarily reflect the policies and views

of DOE, ORAU, or ORISE.

Appendix A. Analysis details

We begin our analysis with the assumption that the scattering

intensity IðQ;DEÞ around the main rotational peak is com-

posed of two types of excitations. The first, IbðQ;DEÞ, is a nar-

row peak which represents rotational excitations that are

translationally bound and exhibit no recoil broadening. The

second, ImðQ;DEÞ, is a broad peak extending into the high en-

ergy region that represents rotational excitations which are

mobile with respect to their translational motion parallel to

the substrate. In the high-energy region, within the roto-re-

coil tail, only mobile transitions are present. Therefore, to

separate mobile and bound states, we wish to perform a fit

to the roto-recoil tail, extend that fit into the main rotational

peak, and subtract it from the experimental data. Whatever

remains should represent the scattering from the bound

states.

Our analysis utilizes the full two-dimensional intensity

map of each sample. The intensities and fits for each step

of the analysis are shown in Fig. A1. To identify an effective

fitting function for the roto-recoil tail which uses the entire

two-dimensional data set, we observe that the Q-dependence

of the roto-recoil tail in Fig. 1 seems to remain quite similar in

shape as energy transfer increases, but shifts by the recoil

momentum QR ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MðE� EdÞ

p=�h. This behavior is illustrated

in Fig. A2, which shows the Q-dependence of the roto-recoil

tail at several values of DE, which were chosen to lie outside

of the main rotational and roto-vibrational peaks. The values

of M = 0.64 amu and DEd ¼ 14:0 meV are determined by a qua-

dratic fit of the Q values for the ridge of maximum intensity as

a function of DE of the roto-recoil tail in the 90 meV intensity

map over the ranges DE ¼ 17–22 meV and 32–63 meV (outside

of the main peaks), as shown in Fig. 1. We find that these val-

ues vary little from sample to sample.

Ener

gy T

rans

fer (

meV

)Experimental Intensity I

10

15

20

Fit to Roto−Recoil Tail Sm

Bound Intensity Ib

10

15

20

Fit to Bound Intensity Sb

Momentum Transfer (A−1)2 4 6

Momentum Transfer (A−1)

Mobile Intensity Im

2 4 610

15

20

(b)

(c)

(a)

(d)

(e)

Fig. A1 – (a) The INS intensity map, I, for the MWCNT

sample, with an incident energy of 30 meV and temperature

of 23 K. (b) The fit to the roto-recoil tail extended into the low

energy region, Sm, as described by Eq. A.2. (c) The scattered

intensity from bound states, Ib, calculated by Eq. A.3. (d) The

fit to the bound intensity, Sb, as described by Eq. A.4. (e) The

scattered intensity from mobile states, Im, calculated by Eq.

A.5. All plots are shown on the same intensity scale which

starts at zero and has been chosen to show the roto-recoil

tail well, saturating the rotational peak. Axis values are the

same for each panel.

2 4 6 8 10Momentum Transfer (A−1)

Nor

mal

ized

Inte

nsity

(arb

. uni

ts)

Δ E=20 meV,QR=1.30 A−1

Δ E=23 meV,QR=1.62 A−1

Δ E=26 meV,QR=1.87 A−1

Fig. A2 – Q-dependence of the roto-recoil tail at several

different values of DE for the MWCNT sample. Incident

neutron energy is 90 meV and the temperature is 23 K. The

value of the recoil momentum QR is given at each DE. They

are quite similar to each other in shape, but shifted in Q

relative to the recoil momentum.

C A R B O N 5 8 ( 2 0 1 3 ) 4 6 – 5 8 57

Therefore, we perform Q-dependent fits of the 30 meV

incident energy spectrum (shown in Fig. A1(a) for the

MWCNTs) for each value of energy transfer in the range

DE ¼ 17–22 meV using the following function,

SðQÞ ¼ AFjjðQ � QRÞe�BQ2; ðA:1Þ

where FjjðQÞ is an isotropic average of our previously calcu-

lated form factor [23] for rotational states with parallel orien-

tation. (The choice of this form factor is based on the

discussion around Fig. 2(a), which shows that J1jj tends to oc-

cur along with a larger amount of recoil.) The values of A and

B are both observed to drop slowly with increasing DE, and are

fit with AðDEÞ ¼ aþ bDE and BðDEÞ ¼ decDE. These parameters

are then used to describe the scattering to mobile states;

SmðQ;DEÞ ¼ ðaþ bDEÞFjjðQ � QRÞe�expðcDEÞdQ2

; ðA:2Þ

where the function Sm for the MWCNT sample is shown in

Fig. A1(b). Because Sm was chosen to describe the roto-recoil

tail, it clearly does not describe the scattering from mobile

states well on the low energy side of the main rotational peak.

Instead, the full scattering from the mobile states is extracted

below (using Eq. A.5). At the current level of theoretical

sophistication, Eq. A.2 is a reasonable ansatz that takes into

consideration the recoil of the adsorbed hydrogen molecules.

Next, Sm is extended well past the main rotational peak, to

DE=5 meV, and subtracted from the experimental intensity,

IbðQ;DEÞ ¼ IðQ;DEÞ � SmðQ;DEÞ: ðA:3Þ

The result represents the experimental intensity of scattering

from bound states (Ib Fig. A1(c)) and is well described by a

Gaussian in DE, consistent with scattering to a bound state.

This justifies a posteriori our initial assumption of the pres-

ence of bound states, as well as the fitting procedure we have

used thus far. Next Ib is fit over the range DE ¼ 13:5–16 meV

with

SbðQ;DEÞ ¼ feðDE�EbÞ2=2g2

F?ðQÞe�hQ2

; ðA:4Þ

where F?ðQÞ is an isotropic average of our previously calcu-

lated form factor [23] for rotational states with perpendicular

orientation. The parameters of this fit are relatively insensi-

tive to the range of DE used. The function Sb for the MWCNT

sample is shown in Fig. A1(d). Likewise,

ImðQ;DEÞ ¼ IðQ;DEÞ � SbðQ;DEÞ ðA:5Þ

gives the scattering to mobile states (Ib, Fig. A1(e)), without

the arbitrary assumptions contained in Eq. A.2.

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