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Coherent neutron scattering from the rotational states of protonated samples
Solid State Communications, Vol. 23, PP. 765—768, 1977.
Pergamon Press.
Printed in Great Britaiu
COHERENT NEUTRON SCA11~ERINGFROM THE ROTATIONAL STATES OF PROTONATED SAMPLES A. Huller Institut fir Festkorperforschung der KFA, Jülich, 5170 Jtilich, West Germany (Received 26 April 1977 by PR. Dederichs) The librational ground state .multiplet of a tetrahedrally coordinated four proton group (CH4, NH~)consists ofseveral sublevels for the different spin isomers. The neutron scattering from these levels is shown to be coherent. Several neutron scatteringexperiments are proposed to test the predicted Q dependence of the scatteringintensity. NMR as well as neutron experiments can detect the exponential variation of the splitting with the potential. RECENT PROGRESS in NMR [1] and neutron scattering techniques [2, 3] has stimulated the study of the rotational excitations of molecules in crystals. The spin isomer splitting of the librational ground state (frequently referred to as tunnel splitting) is of special interest because it exhibits several unusual features:
dence of intermolecular forces. The pressure dependence of phonon frequencies (mode Grüneisen parameters) has been extensively used to obtain third derivativesof interatomic potentials. Contrary to the phonon problem where the frequency shift is a few percent for the technically attainable pressures one expects changes of 50%
(1) transition frequencies that vary over several orders of magnitude for different chemical environments of the reorienting molecule, (2) huge isotope effects, and (3) the linkage between rotational states and spin states due to particle exchange symmetry. In this letter extensive use will be made of the pocket state formalism [4J.The formalism is applicable to problems with shallow potentials, but it develops its full strength for strong rotational potentials [5] where an expansion into free rotor wave functions becomes tedious. The tunnel splitting of the librational ground state or reorienting molecules, molecular groups, and I atomic ions in the crystal depends exponentially on the potential barrier between equivalent orientations of the molecules. The splitting thus is very sensitive to the details of the intermolecular forces. Neutron scattering data have so far been compiled for the methane molecule [2], the methyl group [6] and for the ammonium ion [3, 7] in several environments. All these examples concern the motion of the hydrogen atoms contained in the molecules. The interest in the rotational tunneling therefore goes beyond physics of small molecular groups like cH4, CH3, and NIU. The.experimental results and their interpretation in terms of intermolecular forces are pertinent to the interaction of large molecules that contam hydrogen as e.g. polymers or biological molecules. Pressure measurements of tunneling frequencies provide very detailed information on the distance depen-
or more for tunneling experiments with comparable pressures [4]. Tunneling frequencies thus are an excellent probe of intermolecular forces. To calculate the neutron scatteringmatrix elements from a tetrahedrally coordinated four proton group in a rotational potential it is necessary to construct the corresponding wave functions. Consider the most general situation where a regular tetrahedron is in a potential with no additional symmetry elements. The eigensymmetry of the molecule then is the only symmetry of the problem. If a certain orientation of the molecule corresponds to a minimum value of the orientational potential V, then after a proper rotation which brings the molecule back onto itself, it is in an equivalent potential minimum. Such a rotation corresponds to an even permutation of the four protons. The group of proper rotations that transform a tetrahedron into itself is isomorphic with the group of even permutations of its four corners. Thus there are 12 equivalent minima in the orientational potential, also called the 12 pockets of the potential. A state where the tetrahedron performs zero point librations around one of the 12 minimum orientations in the potential, is called a pocket state In such a state each of the four protons of the nth molecule is assigned to one of the four lattice sited R~7(y•= 1, 2, 3,4) which are occupied by protons. The pocket states are no eigenstates of the Hamilton an H = 7’ + V of the molecule (T = rotational kinetic energy). In the subspace of the 12 pocket states only 8 of the 144 overlap matrix elements are independent: The diagonal element D, the elements for the four
—
765
~.
766
ROTATIONAL STATES OF PROTONATED SAMPLES
Vol. 23, No. 10
different 120°rotations h1 h2, h~and h4 and for the The experimental situation in NH~ClO~ [3] is well three 180°rotations !I~,Hr~,and Hz. These rotations described by Hx = H~= Hz =0, hi = 0.038 peV, are defined for crystal fixed rotation axes of the tetrah2 = h3 = h4 = l.410j.teV. It is obvious that for hedron. narrow wave functions the 180°overlap should be much Here a different type of pocket states is defused: In smaller than the 120°overlap. The equality of h3 and h4 the state ~p3~z4)the quantum number y~.denotes is due to symmetry, the equality of h2 with h3 and h4 the spin state of the proton at the site R,~.The identiflleads to the observed accidental degeneracy of two cation of the proton at the site R~7 is not given it may T-levels. The appearance of one small and three large be any one of the four protons of the nth molecule. Spin 120°overlap matrix elements is in agreement with up is denoted by P~)= a, spin down by p.~.= j3. There observed librational amplitudes [8]. One obtains the are 16 pocket states [Pi ~.t2.t3~) corresponding to the 2 energy eigenvalue EA = D + 6/i2 + 2h1, E~ = E~ = possible values of~. By definition, the states D + h2 h1,Er1 = D 2(h2 —hi), and EE = D [P1I~2P3P4> are totally symmetric under even permu3h2 h~. tations of the protons (proper rotations of the tetraThe overlap matrix elements h1 and h2 depend hedron). exponentially on the potential. Thus the transition freSymmetry, under odd permutations also, does not quencies should show a strong pressure dependence. bother us. These operations are not considered in the N~UO~with h2 ~ h3 is of special interest. It is profollowing, posed to remove the accidental degeneracy of the T2 and The states [i.~1 ~~.L4) are classified according to the T3 levels by hydrostatic pressure. From the neutron z-component I~of the proton spins which may be 0, ±1, scatteringdata one concludes: ET2 ET3 <0.3 ,.zeV and ±2. M H does not involve the proton spin operators, ( 80 MHz). An NMR-experiment to detect a residual there are no matrix elements between pocket states with splitting is suggested. different I~.The overlap matrix between the six states In solid CH4 II [2] and (NH)2 SnCU [7] the site [aa/3j1), [~3aa>, [a~8a>, ~ [a~aj3), and [~3aj3a> with symmetry at the position of the tetrahedron is tetra= 0 is: hedral with the conseqent equality h1 = h2 = h3 = h4. ,
—
—
—
—
—
—
/
HxfHy
h1+h3
h2+h4
h1+h4
Hx+Hy
D+Hz
h2+h4
h1+h3
h2+h3
h~+h3
h2+h4
D+Hx
Hy+Hz
h1+h2
h3+h4
h1+h2
Hy+Hz
D+Hx
h3+h4
h1+h2
h2+h3
h1+h2
h3+h4
D+H~
Hx+Hz
h1+h4
h3+h4
h1+h2
Hx+Hz
D+Hy
H6=i
\h2+h3
There are the two groups of four states each [t3~aa), [al3aa), [aal3a), and [aaaj3>with I~= + I and ~ [j3a43~,[~j3aj3), and [~3~g3a> with I~= I which have an identical overlap matrix: —
7”D+2h1
H4=t
(
Hence: EA = D + 8Jz~,ET1 = ET2 = ET3 = D, and EE = D 4h1. Where h1 = 17.9 peV for CH.4 and h~= 0.37 peV for (NH~)2SnC1~.The rotational constants of CH4 and NH~are almost equal. The factor 50 in —
—
—
Hz+h3+h4
H~+h4+h2
Hx+hi+h~\
H~+h3+h~
D+2h2
Hx+hj+h4
H~+h1+h3
Hy+h4+h2
Hx+h1+h4
D+2h3
H~+h1+h3
Hz+hi+h2
\ Hz+hi+h2 D+2h4 / I
The diagonal matrix elements for the two states [aaaa) the tunnel splitting corresponds to a factor 3 in the with I~= + 2 and [j3j~j3j3> with I~= —2 are both equal to potential [41. D + (fix + Hy + Hz)+ 2(h1 + h2 + h3 + h4). These From the diagonalization of H6 and H4 one obtains two energy levels together with the eigenvalues of H6 the eigenstates Il/Ia> of H as linear combinations of the and H4 yield a five-fold ground state (with energy EA), pocket states [i.i1lz2Iz~,.L4>: three triplet states (ET1,ET2, and ET3) and a doublet = ~Sj~ [41/’12P31L4) (1) state (ER). The eigenvectors ofH4 and H6 yield the 1 2 #4 corresponding wave functions in terms of the pocket It is then easy to calculate the neutron scattering states [~j~z2~i~j.i4). matrix elements ‘
Vol. 23, No, 10
ROTATIONAL STATES OF PROTONATED SAMPLES
767
Table 1. The angle dependent scattering cross-sections in a case where Hx = H~= Hz = 0, h2 = h3 = h.4 *0, and h1 = 0. The angular averages are compared with the experimental intensities in NH4 C104 [3/.The argument of f is2Qp/~/3 Levels concerned
Theoretical cross-section [4~~/288]
Angular average a~(l —f) 24
2 5l3G1—G2—G3—G41 2 + 10J2G 2 30 lG3 G4I2 + 2I2G 2 G3 —G412 1G3 —G41 2 —G3 46 12G 2+ 1 21G —G4J 2 2 G32 +G41 32 G4l 18IG 213G 12+ lG3—G41 1 —G21 2 1 + G2 2G3 2G4I
A~T1 A T 2, T3 T 2,÷~T3E T 2,T3~’E
—
~+
—
Intensity [arb. units] 49 104
10 2 4
—
—
Experiment [3] Energy [peV] 11.28 7.17
22
4.11
50
1.51
84
5.65
—
—
8
—
4
AZ’a’~,= (.z’i/i,,~’J~ A”7 e1~’~”flpi/’na>
(2) each other, giving rise to a coherent scatteringintensity.
y1
and to obtain the neutron scatteringcross-section d2u dlldw = ~PuPat5((.~) C~’a,a’)IA~’a’,ual2. (3)
EE
—
Il/Ina> is the state of the nth molecule,
Iii>
the neutron
spin state Q and hw are the transferred momentum and energy. Unprimed symbols relate to initial, primed symbols to final states. P,4 and Pa denote the probabilities of the initial states of neutron and molecule, A”” is the interaction operator of the neutron with the protonatR~1, A””
= aCOh+
2ainc 1)Smfl7 ~j(i+
(4)
S and I,,., denote the neutron and proton spin operators, respectively. In the following aCOh is set equal to zero (only the spin dependent scatteringamplitude is considered). The matrix elements ofA”” between the states [l.11p2p3p4> are calculated for the infmitely narrow proton wave functions in the pockets. Thus the results are valid only for neutrons with a wavelength larger than the extension of the proton wave function. The matrix elements between these pocket states then are linear combinations of Grn,, = exp (iQ R~) (5) where R~.,is one of the four equilibrium positions of the protons of the nth tetrahedron. From the unitary transformation (1) the matrix elements AZ’a’, ~ are calculated and then inserted into equation (3) to obtain the scattering cross-section. These cross-sections are listed in Table I for a situation with Ji~= H~= !i~= 0, h1 = 0, and h2 = h3 = h4 *0 which comes close to the NH~CIO~case. It is clearly seen that these cross-sections depend on Q. The spin dependent scattering amplitudes from the four protons of one molecule interfere with
The coherent nature of the scatteringis due to the close connection between the spin states and the rotational states of the molecule. For each transition matrix element AZ’a’,~the initial as well as the fmal state of the proton spins is defined. Therefore the scatteringis coherent. The scattering cross sections for CH 4 II and (NH)2 SnCl6 with ET3 = ET2 = ET1 are obtained from sums over the individual cross sections of Table 1. Polycrystalline intensities of the tunneling lines are compared with the averages of the cross sections over the angles of the scattering vector Q. From 2 ifX’y _________
G7G~+ = 2f(2Qp/\/3) ifX*-y (6) one sees that all inelastic lines have the same radial dependence. Here the bar denotes an angular average and p the distance of a proton from the center of the molecule. The function f(x)is defined by the series:
f(x)
~, ~ ______________
=
2,n
rn~o(2m+I)!!m!X
The angular averages are also listed in Table 1 and they are compared with the experimental intensities in NH~QO~.The agreement is very good. Two types of neutron scatteringexperiments are suggested. In single crystal experiments the angular dependence of the scattering cross section should be tested. These experiments may be used to determine the proton positions in the crystal. Deviations of the cross section from the predictions of Table 1 at high momentum transfer contain information on the librational amplitude of the molecule. The prediction that the radial dependence of the intensities of all tunneling lines is governed by the factor I —f(2Qp/’.,/3) can be tested in an experiment with a polycrystalline sample.
768
ROTATIONAL STATES OF PROTONATED SAMPLES A full account of this work will be published elsewhere.
Acknowledgement The author gratefully acknowledges many stimulating discussions with Dr. Werner Press and Dr. Michael Prager. —
REFERENCES 1.
PINTAR M.M., NMR Basic Princ. and Pmgr. 13, 125 (1976).
2. 3.
PRESS W. & KOLLMAR A., Solid State Commun. 17,405 (1975). PRAGER M., ALEFELD B. & HEIDEMANN A., Colloque Ampere. Heidelberg (1976).
4.
HULLER A. & KROLL D.M., J. Chem. Phys. 63,4495(1975).
5. 6.
RAICH J. & GILLIS N.S., 1. Chem. Phys. 65,2088 (1976). ALEFELD B., KOLLMAR A. & DASANNACHARYA B.A.,J. Chem. Phys. 63,4415(1975).
7.
PRAGER M., PRESS W., ALEFELD B. & HULLER A. (to be published).
8.
CHOIC. S., PRASK Hi. & PRINCE E., J. Chem. Phys. 61,3523 (1974).
Vol. 23, No. 10
Pergamon Press.
Printed in Great Britaiu
COHERENT NEUTRON SCA11~ERINGFROM THE ROTATIONAL STATES OF PROTONATED SAMPLES A. Huller Institut fir Festkorperforschung der KFA, Jülich, 5170 Jtilich, West Germany (Received 26 April 1977 by PR. Dederichs) The librational ground state .multiplet of a tetrahedrally coordinated four proton group (CH4, NH~)consists ofseveral sublevels for the different spin isomers. The neutron scattering from these levels is shown to be coherent. Several neutron scatteringexperiments are proposed to test the predicted Q dependence of the scatteringintensity. NMR as well as neutron experiments can detect the exponential variation of the splitting with the potential. RECENT PROGRESS in NMR [1] and neutron scattering techniques [2, 3] has stimulated the study of the rotational excitations of molecules in crystals. The spin isomer splitting of the librational ground state (frequently referred to as tunnel splitting) is of special interest because it exhibits several unusual features:
dence of intermolecular forces. The pressure dependence of phonon frequencies (mode Grüneisen parameters) has been extensively used to obtain third derivativesof interatomic potentials. Contrary to the phonon problem where the frequency shift is a few percent for the technically attainable pressures one expects changes of 50%
(1) transition frequencies that vary over several orders of magnitude for different chemical environments of the reorienting molecule, (2) huge isotope effects, and (3) the linkage between rotational states and spin states due to particle exchange symmetry. In this letter extensive use will be made of the pocket state formalism [4J.The formalism is applicable to problems with shallow potentials, but it develops its full strength for strong rotational potentials [5] where an expansion into free rotor wave functions becomes tedious. The tunnel splitting of the librational ground state or reorienting molecules, molecular groups, and I atomic ions in the crystal depends exponentially on the potential barrier between equivalent orientations of the molecules. The splitting thus is very sensitive to the details of the intermolecular forces. Neutron scattering data have so far been compiled for the methane molecule [2], the methyl group [6] and for the ammonium ion [3, 7] in several environments. All these examples concern the motion of the hydrogen atoms contained in the molecules. The interest in the rotational tunneling therefore goes beyond physics of small molecular groups like cH4, CH3, and NIU. The.experimental results and their interpretation in terms of intermolecular forces are pertinent to the interaction of large molecules that contam hydrogen as e.g. polymers or biological molecules. Pressure measurements of tunneling frequencies provide very detailed information on the distance depen-
or more for tunneling experiments with comparable pressures [4]. Tunneling frequencies thus are an excellent probe of intermolecular forces. To calculate the neutron scatteringmatrix elements from a tetrahedrally coordinated four proton group in a rotational potential it is necessary to construct the corresponding wave functions. Consider the most general situation where a regular tetrahedron is in a potential with no additional symmetry elements. The eigensymmetry of the molecule then is the only symmetry of the problem. If a certain orientation of the molecule corresponds to a minimum value of the orientational potential V, then after a proper rotation which brings the molecule back onto itself, it is in an equivalent potential minimum. Such a rotation corresponds to an even permutation of the four protons. The group of proper rotations that transform a tetrahedron into itself is isomorphic with the group of even permutations of its four corners. Thus there are 12 equivalent minima in the orientational potential, also called the 12 pockets of the potential. A state where the tetrahedron performs zero point librations around one of the 12 minimum orientations in the potential, is called a pocket state In such a state each of the four protons of the nth molecule is assigned to one of the four lattice sited R~7(y•= 1, 2, 3,4) which are occupied by protons. The pocket states are no eigenstates of the Hamilton an H = 7’ + V of the molecule (T = rotational kinetic energy). In the subspace of the 12 pocket states only 8 of the 144 overlap matrix elements are independent: The diagonal element D, the elements for the four
—
765
~.
766
ROTATIONAL STATES OF PROTONATED SAMPLES
Vol. 23, No. 10
different 120°rotations h1 h2, h~and h4 and for the The experimental situation in NH~ClO~ [3] is well three 180°rotations !I~,Hr~,and Hz. These rotations described by Hx = H~= Hz =0, hi = 0.038 peV, are defined for crystal fixed rotation axes of the tetrah2 = h3 = h4 = l.410j.teV. It is obvious that for hedron. narrow wave functions the 180°overlap should be much Here a different type of pocket states is defused: In smaller than the 120°overlap. The equality of h3 and h4 the state ~p3~z4)the quantum number y~.denotes is due to symmetry, the equality of h2 with h3 and h4 the spin state of the proton at the site R,~.The identiflleads to the observed accidental degeneracy of two cation of the proton at the site R~7 is not given it may T-levels. The appearance of one small and three large be any one of the four protons of the nth molecule. Spin 120°overlap matrix elements is in agreement with up is denoted by P~)= a, spin down by p.~.= j3. There observed librational amplitudes [8]. One obtains the are 16 pocket states [Pi ~.t2.t3~) corresponding to the 2 energy eigenvalue EA = D + 6/i2 + 2h1, E~ = E~ = possible values of~. By definition, the states D + h2 h1,Er1 = D 2(h2 —hi), and EE = D [P1I~2P3P4> are totally symmetric under even permu3h2 h~. tations of the protons (proper rotations of the tetraThe overlap matrix elements h1 and h2 depend hedron). exponentially on the potential. Thus the transition freSymmetry, under odd permutations also, does not quencies should show a strong pressure dependence. bother us. These operations are not considered in the N~UO~with h2 ~ h3 is of special interest. It is profollowing, posed to remove the accidental degeneracy of the T2 and The states [i.~1 ~~.L4) are classified according to the T3 levels by hydrostatic pressure. From the neutron z-component I~of the proton spins which may be 0, ±1, scatteringdata one concludes: ET2 ET3 <0.3 ,.zeV and ±2. M H does not involve the proton spin operators, ( 80 MHz). An NMR-experiment to detect a residual there are no matrix elements between pocket states with splitting is suggested. different I~.The overlap matrix between the six states In solid CH4 II [2] and (NH)2 SnCU [7] the site [aa/3j1), [~3aa>, [a~8a>, ~ [a~aj3), and [~3aj3a> with symmetry at the position of the tetrahedron is tetra= 0 is: hedral with the conseqent equality h1 = h2 = h3 = h4. ,
—
—
—
—
—
—
/
HxfHy
h1+h3
h2+h4
h1+h4
Hx+Hy
D+Hz
h2+h4
h1+h3
h2+h3
h~+h3
h2+h4
D+Hx
Hy+Hz
h1+h2
h3+h4
h1+h2
Hy+Hz
D+Hx
h3+h4
h1+h2
h2+h3
h1+h2
h3+h4
D+H~
Hx+Hz
h1+h4
h3+h4
h1+h2
Hx+Hz
D+Hy
H6=i
\h2+h3
There are the two groups of four states each [t3~aa), [al3aa), [aal3a), and [aaaj3>with I~= + I and ~ [j3a43~,[~j3aj3), and [~3~g3a> with I~= I which have an identical overlap matrix: —
7”D+2h1
H4=t
(
Hence: EA = D + 8Jz~,ET1 = ET2 = ET3 = D, and EE = D 4h1. Where h1 = 17.9 peV for CH.4 and h~= 0.37 peV for (NH~)2SnC1~.The rotational constants of CH4 and NH~are almost equal. The factor 50 in —
—
—
Hz+h3+h4
H~+h4+h2
Hx+hi+h~\
H~+h3+h~
D+2h2
Hx+hj+h4
H~+h1+h3
Hy+h4+h2
Hx+h1+h4
D+2h3
H~+h1+h3
Hz+hi+h2
\ Hz+hi+h2 D+2h4 / I
The diagonal matrix elements for the two states [aaaa) the tunnel splitting corresponds to a factor 3 in the with I~= + 2 and [j3j~j3j3> with I~= —2 are both equal to potential [41. D + (fix + Hy + Hz)+ 2(h1 + h2 + h3 + h4). These From the diagonalization of H6 and H4 one obtains two energy levels together with the eigenvalues of H6 the eigenstates Il/Ia> of H as linear combinations of the and H4 yield a five-fold ground state (with energy EA), pocket states [i.i1lz2Iz~,.L4>: three triplet states (ET1,ET2, and ET3) and a doublet = ~Sj~ [41/’12P31L4) (1) state (ER). The eigenvectors ofH4 and H6 yield the 1 2 #4 corresponding wave functions in terms of the pocket It is then easy to calculate the neutron scattering states [~j~z2~i~j.i4). matrix elements ‘
Vol. 23, No, 10
ROTATIONAL STATES OF PROTONATED SAMPLES
767
Table 1. The angle dependent scattering cross-sections in a case where Hx = H~= Hz = 0, h2 = h3 = h.4 *0, and h1 = 0. The angular averages are compared with the experimental intensities in NH4 C104 [3/.The argument of f is2Qp/~/3 Levels concerned
Theoretical cross-section [4~~/288]
Angular average a~(l —f) 24
2 5l3G1—G2—G3—G41 2 + 10J2G 2 30 lG3 G4I2 + 2I2G 2 G3 —G412 1G3 —G41 2 —G3 46 12G 2+ 1 21G —G4J 2 2 G32 +G41 32 G4l 18IG 213G 12+ lG3—G41 1 —G21 2 1 + G2 2G3 2G4I
A~T1 A T 2, T3 T 2,÷~T3E T 2,T3~’E
—
~+
—
Intensity [arb. units] 49 104
10 2 4
—
—
Experiment [3] Energy [peV] 11.28 7.17
22
4.11
50
1.51
84
5.65
—
—
8
—
4
AZ’a’~,= (.z’i/i,,~’J~ A”7 e1~’~”flpi/’na>
(2) each other, giving rise to a coherent scatteringintensity.
y1
and to obtain the neutron scatteringcross-section d2u dlldw = ~PuPat5((.~) C~’a,a’)IA~’a’,ual2. (3)
EE
—
Il/Ina> is the state of the nth molecule,
Iii>
the neutron
spin state Q and hw are the transferred momentum and energy. Unprimed symbols relate to initial, primed symbols to final states. P,4 and Pa denote the probabilities of the initial states of neutron and molecule, A”” is the interaction operator of the neutron with the protonatR~1, A””
= aCOh+
2ainc 1)Smfl7 ~j(i+
(4)
S and I,,., denote the neutron and proton spin operators, respectively. In the following aCOh is set equal to zero (only the spin dependent scatteringamplitude is considered). The matrix elements ofA”” between the states [l.11p2p3p4> are calculated for the infmitely narrow proton wave functions in the pockets. Thus the results are valid only for neutrons with a wavelength larger than the extension of the proton wave function. The matrix elements between these pocket states then are linear combinations of Grn,, = exp (iQ R~) (5) where R~.,is one of the four equilibrium positions of the protons of the nth tetrahedron. From the unitary transformation (1) the matrix elements AZ’a’, ~ are calculated and then inserted into equation (3) to obtain the scattering cross-section. These cross-sections are listed in Table I for a situation with Ji~= H~= !i~= 0, h1 = 0, and h2 = h3 = h4 *0 which comes close to the NH~CIO~case. It is clearly seen that these cross-sections depend on Q. The spin dependent scattering amplitudes from the four protons of one molecule interfere with
The coherent nature of the scatteringis due to the close connection between the spin states and the rotational states of the molecule. For each transition matrix element AZ’a’,~the initial as well as the fmal state of the proton spins is defined. Therefore the scatteringis coherent. The scattering cross sections for CH 4 II and (NH)2 SnCl6 with ET3 = ET2 = ET1 are obtained from sums over the individual cross sections of Table 1. Polycrystalline intensities of the tunneling lines are compared with the averages of the cross sections over the angles of the scattering vector Q. From 2 ifX’y _________
G7G~+ = 2f(2Qp/\/3) ifX*-y (6) one sees that all inelastic lines have the same radial dependence. Here the bar denotes an angular average and p the distance of a proton from the center of the molecule. The function f(x)is defined by the series:
f(x)
~, ~ ______________
=
2,n
rn~o(2m+I)!!m!X
The angular averages are also listed in Table 1 and they are compared with the experimental intensities in NH~QO~.The agreement is very good. Two types of neutron scatteringexperiments are suggested. In single crystal experiments the angular dependence of the scattering cross section should be tested. These experiments may be used to determine the proton positions in the crystal. Deviations of the cross section from the predictions of Table 1 at high momentum transfer contain information on the librational amplitude of the molecule. The prediction that the radial dependence of the intensities of all tunneling lines is governed by the factor I —f(2Qp/’.,/3) can be tested in an experiment with a polycrystalline sample.
768
ROTATIONAL STATES OF PROTONATED SAMPLES A full account of this work will be published elsewhere.
Acknowledgement The author gratefully acknowledges many stimulating discussions with Dr. Werner Press and Dr. Michael Prager. —
REFERENCES 1.
PINTAR M.M., NMR Basic Princ. and Pmgr. 13, 125 (1976).
2. 3.
PRESS W. & KOLLMAR A., Solid State Commun. 17,405 (1975). PRAGER M., ALEFELD B. & HEIDEMANN A., Colloque Ampere. Heidelberg (1976).
4.
HULLER A. & KROLL D.M., J. Chem. Phys. 63,4495(1975).
5. 6.
RAICH J. & GILLIS N.S., 1. Chem. Phys. 65,2088 (1976). ALEFELD B., KOLLMAR A. & DASANNACHARYA B.A.,J. Chem. Phys. 63,4415(1975).
7.
PRAGER M., PRESS W., ALEFELD B. & HULLER A. (to be published).
8.
CHOIC. S., PRASK Hi. & PRINCE E., J. Chem. Phys. 61,3523 (1974).
Vol. 23, No. 10