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Interference effects in neutron scattering on protons in a double minimum potential
Volume 27A. number 9
PHYSICS
LETTERS
23 September
1968
We thank the Batelle Institute, Geneva, and in particular Dr. Schachner, for X-ray diffraction analyses of the Se targets, and also CERN for its hospitality. for ?I and p, respectively. The values are corrected for self-absorption. The errors are standard deviations. If there were no effects, all the numbers listed in the table would be unity. Effects are evident for all target pairs; these effects are usually stronger for muons than for pions. The most interesting case is that of Se, where the targets differ only in their physical form. Measurements with a Se target which consisted mostly of Se in monoclinic form give, for some transition ratios, even stronger effects than those obtained with amorphous Se. The results are not listed in table 1 because of uncertainties in the target composition. For a discussion of the condensed-state effects we refer to earlier papers [3,5].
References 1. V.G.Zinov. 2.
3. 4. 5.
A.D.Konin, A.I.MukhinandR.V.Polyakova. Yadern. Fiz. 5 (1967) 591; Soviet J. Nucl. Phys. 5 (1967) 420. D.Kessler, H. L.Anderson, M.S.Dixit. H. J.Evans. R. J. McKee, C.K. Hargrove, R.D.Barton, E. P. Hincks and J.D.McAndrew, Phrs. Rev. Letters 18 (1967) 1179. H.Daniel. H.Koch, G. Poelz, H.Schmitt. L.Tauschei G. Backenstoss and S. Charalambus. Phys. Letters 26B (1968) 281. H.Daniel. G. Poelz, H.Schmitt. G.Backenstoss. H. Koch and S.Charalambus. Z. Phys. 205 (1967) 472. H.Daniel. Naturwissenschaften 55 (1968) 314.
**+**
INTERFERENCE ON PROTONS
EFFECTS IN NEUTRON IN A DOUBLE MINIMUM
SCATTERING POTENTIAL
C. SCHENK and B. WECKERMANN Dizjisionof Experimental Neutron Physics I Department of Reactor Physics, Euratom? Joint Research Received
Center,
Ispra. Italy
12 August 1968
Based on the theoretical treatments by Stiller and Stamenkovich of the incoherent neutron scattering at a proton in a double minimum potential well experiments had been performed with KH2P04 single crystals to prove the predicted interference effect.
Stiller [l] and lately Stamenkovich [2] have shown that neutrons which are scattered truly elastically and incoherently by protons tunneling in a double minimum potential well, should show an intereference effect. KH2PO4 single crystals are most suitable for an experimental proof, as the hydrogen bonds are oriented along the x and y axis [3]. Therefore the ratio V of scattered intensity with x vector parallel to the x-axis to the one with 1~vector parallel to the z axis is a sensitive quantity to possible interference effects. (x stands for momentum transfer). In order to distinguish between an anisotropic single well potential and a double minimum potential we dis582
cuss the /x 1 dependence of the ratio V. Assuming a harmonic oscillator with the m 9jm squgre displacement $ along the bond and Ub and uc perpendicular to it, the ratio V is given by ‘harm
J
=z
exp =
t-x2utj +exp (-x2+
2 exp(-x2u2)c ’ In fig. 1 tiharm is plotted as full line for ub = uc and fitted to a measurem nt recently done by Stiller [4] at (X / = 5.5A- ‘i . For the case of the double minimum potential well, one obtains according to Stiller [l] for the scattering law:
Volume
27A. number
i
i
“0
PHYSICS
9
b H
y-‘]
lb
B
LETTERS
12
Fig. 1. Ratio V of scattered neutron intensities with X parallel and perpendicular to the H bond versus / XI .
se1 =
inc = exp (-X2”:)
{ 1 -&+&eD(k)[l
D(X) = exp [-
+cos(x.R)]},
&x2(“22 -u!)] ,
where u2 and U: are the isotropic mean square amplitu 2 es for the protons in the left or right well, E is the assymmetry parameter of the potential and R the distance between the equilibrium sites. Taking these parameters from Stiller’s experimental results [4], which seem to support the tunneling model, V interference gives the dashed line in fig. 1. Stamenkovich [2] gives the elastic cross section for a symmetric double minimum potential as
da++/dS2 = a2[1 + exp (-R2/8u2)lW2 x X
[cos~(X*I?)+ exp(-R2/8u2)12
and for the assymmetric
exp(-x2&
23 September
1968
Our measurements were carried out with a rotating crystal spectrometer [5]. The two intensities which are forming the ratio V were obtained by rotating the KH2PO4 crystal around its y axis, which does not change the scattering geometry in respect with the outer dimensions of the sample. The energy resolution of the spectrometer is given in fig. 1 for each measured point. At high x values the data have been corrected for for the inelastic contributions within the energy resolution of the instrument. This was only possible by using the results of earlier cold neutron measurements [6]. The computation was done according to the Sjorlander multi-phonon formalism [7] taking into account 4 phonon terms and using partly the Neutof program [8]. The corrected ratios are plotted as double bars in fig. 1. The results support the double minimum theory, if the assumption ub = uc for the harmonic oscillator is correct. As our experiments were partly done “i&h a beam of first and second order, neutrons uc could be calculated. As the ratio of scattered intensities is proportional exp (- 3U$ Grim&) a measurement at two different scattering angles provide the 4 necessary intensities. For vanadium we got u2 = 0.4 x 10m2f 25% which is in rather good agreement with the calculated value u2 = 0.61 X 10m2for a Debye temperature of 0 = 388OK. The deviation is mainly due to unresolved inelastic scattering contributions. For KH PO4 we obtain a value of u2 = = 0.03 f 20% A%. This is in good agreeme& with the diffraction measurements done by Bacon and Pease [9] who give U! = B6/8n2 = 0.033A2.
The authors want to thank Dr. Kley for his continuous interest and support of this work and their colleagues of the Experimental Neutron Physics Division, to whom they owe many stimulating discussions.
case
do++/dS2 = = ,2[cos2 $(X.R) + B sin2 t(x.Ip)] exp (-x2u2) where 0 stands for scattering length of the proton, ~2 for its mean square amplitude in one well and B for a simplified asymmetry parameter. All the other notations are the same as in the Stiller formula. For the symmetr$ case the ratio V was calculated with R = 0.4A, which leads to the dotted line in fig. 1. For the assymmetric case, V (Stamenkovich) coincides with V (Stiller), if B = 0.4. However, it should be pointed out that ,&amenkovich assumed in the second formula the overlap integral between left and right well, which he gives as exp (-lp2/8u2), to be zero.
Ber. Bunsenges. Phys. Chem. 72 (1968) 94. 1. H.Stiller, Fiz. Tver. Tela 10 (1968) 861. 2. S. S. Stamenkovich, of KH2P04 see e.g. F. 3. for extensive description Jona and G. Shirane, Ferroelectric crystals (Pergamon Press, 1962) Ch. III. 4. Th. Plesser and H. Stiller. Sol. State Corn., to be published. 5. H.Meister, R. Haas, F. May, C. Schenk and B. Weckermann, Kerntechnik 10 (1968) 145. 6. C. Schenk, E. Wiener, B. Weckermann and W. Kley, Phys. Rev., to be published. Arkiv Fysik 14, 315. 7. A.SjFrlander, 8. G. Verdan, Euratom Report EUR 2779 e (1966).
**** 583
PHYSICS
LETTERS
23 September
1968
We thank the Batelle Institute, Geneva, and in particular Dr. Schachner, for X-ray diffraction analyses of the Se targets, and also CERN for its hospitality. for ?I and p, respectively. The values are corrected for self-absorption. The errors are standard deviations. If there were no effects, all the numbers listed in the table would be unity. Effects are evident for all target pairs; these effects are usually stronger for muons than for pions. The most interesting case is that of Se, where the targets differ only in their physical form. Measurements with a Se target which consisted mostly of Se in monoclinic form give, for some transition ratios, even stronger effects than those obtained with amorphous Se. The results are not listed in table 1 because of uncertainties in the target composition. For a discussion of the condensed-state effects we refer to earlier papers [3,5].
References 1. V.G.Zinov. 2.
3. 4. 5.
A.D.Konin, A.I.MukhinandR.V.Polyakova. Yadern. Fiz. 5 (1967) 591; Soviet J. Nucl. Phys. 5 (1967) 420. D.Kessler, H. L.Anderson, M.S.Dixit. H. J.Evans. R. J. McKee, C.K. Hargrove, R.D.Barton, E. P. Hincks and J.D.McAndrew, Phrs. Rev. Letters 18 (1967) 1179. H.Daniel. H.Koch, G. Poelz, H.Schmitt. L.Tauschei G. Backenstoss and S. Charalambus. Phys. Letters 26B (1968) 281. H.Daniel. G. Poelz, H.Schmitt. G.Backenstoss. H. Koch and S.Charalambus. Z. Phys. 205 (1967) 472. H.Daniel. Naturwissenschaften 55 (1968) 314.
**+**
INTERFERENCE ON PROTONS
EFFECTS IN NEUTRON IN A DOUBLE MINIMUM
SCATTERING POTENTIAL
C. SCHENK and B. WECKERMANN Dizjisionof Experimental Neutron Physics I Department of Reactor Physics, Euratom? Joint Research Received
Center,
Ispra. Italy
12 August 1968
Based on the theoretical treatments by Stiller and Stamenkovich of the incoherent neutron scattering at a proton in a double minimum potential well experiments had been performed with KH2P04 single crystals to prove the predicted interference effect.
Stiller [l] and lately Stamenkovich [2] have shown that neutrons which are scattered truly elastically and incoherently by protons tunneling in a double minimum potential well, should show an intereference effect. KH2PO4 single crystals are most suitable for an experimental proof, as the hydrogen bonds are oriented along the x and y axis [3]. Therefore the ratio V of scattered intensity with x vector parallel to the x-axis to the one with 1~vector parallel to the z axis is a sensitive quantity to possible interference effects. (x stands for momentum transfer). In order to distinguish between an anisotropic single well potential and a double minimum potential we dis582
cuss the /x 1 dependence of the ratio V. Assuming a harmonic oscillator with the m 9jm squgre displacement $ along the bond and Ub and uc perpendicular to it, the ratio V is given by ‘harm
J
=z
exp =
t-x2utj +exp (-x2+
2 exp(-x2u2)c ’ In fig. 1 tiharm is plotted as full line for ub = uc and fitted to a measurem nt recently done by Stiller [4] at (X / = 5.5A- ‘i . For the case of the double minimum potential well, one obtains according to Stiller [l] for the scattering law:
Volume
27A. number
i
i
“0
PHYSICS
9
b H
y-‘]
lb
B
LETTERS
12
Fig. 1. Ratio V of scattered neutron intensities with X parallel and perpendicular to the H bond versus / XI .
se1 =
inc = exp (-X2”:)
{ 1 -&+&eD(k)[l
D(X) = exp [-
+cos(x.R)]},
&x2(“22 -u!)] ,
where u2 and U: are the isotropic mean square amplitu 2 es for the protons in the left or right well, E is the assymmetry parameter of the potential and R the distance between the equilibrium sites. Taking these parameters from Stiller’s experimental results [4], which seem to support the tunneling model, V interference gives the dashed line in fig. 1. Stamenkovich [2] gives the elastic cross section for a symmetric double minimum potential as
da++/dS2 = a2[1 + exp (-R2/8u2)lW2 x X
[cos~(X*I?)+ exp(-R2/8u2)12
and for the assymmetric
exp(-x2&
23 September
1968
Our measurements were carried out with a rotating crystal spectrometer [5]. The two intensities which are forming the ratio V were obtained by rotating the KH2PO4 crystal around its y axis, which does not change the scattering geometry in respect with the outer dimensions of the sample. The energy resolution of the spectrometer is given in fig. 1 for each measured point. At high x values the data have been corrected for for the inelastic contributions within the energy resolution of the instrument. This was only possible by using the results of earlier cold neutron measurements [6]. The computation was done according to the Sjorlander multi-phonon formalism [7] taking into account 4 phonon terms and using partly the Neutof program [8]. The corrected ratios are plotted as double bars in fig. 1. The results support the double minimum theory, if the assumption ub = uc for the harmonic oscillator is correct. As our experiments were partly done “i&h a beam of first and second order, neutrons uc could be calculated. As the ratio of scattered intensities is proportional exp (- 3U$ Grim&) a measurement at two different scattering angles provide the 4 necessary intensities. For vanadium we got u2 = 0.4 x 10m2f 25% which is in rather good agreement with the calculated value u2 = 0.61 X 10m2for a Debye temperature of 0 = 388OK. The deviation is mainly due to unresolved inelastic scattering contributions. For KH PO4 we obtain a value of u2 = = 0.03 f 20% A%. This is in good agreeme& with the diffraction measurements done by Bacon and Pease [9] who give U! = B6/8n2 = 0.033A2.
The authors want to thank Dr. Kley for his continuous interest and support of this work and their colleagues of the Experimental Neutron Physics Division, to whom they owe many stimulating discussions.
case
do++/dS2 = = ,2[cos2 $(X.R) + B sin2 t(x.Ip)] exp (-x2u2) where 0 stands for scattering length of the proton, ~2 for its mean square amplitude in one well and B for a simplified asymmetry parameter. All the other notations are the same as in the Stiller formula. For the symmetr$ case the ratio V was calculated with R = 0.4A, which leads to the dotted line in fig. 1. For the assymmetric case, V (Stamenkovich) coincides with V (Stiller), if B = 0.4. However, it should be pointed out that ,&amenkovich assumed in the second formula the overlap integral between left and right well, which he gives as exp (-lp2/8u2), to be zero.
Ber. Bunsenges. Phys. Chem. 72 (1968) 94. 1. H.Stiller, Fiz. Tver. Tela 10 (1968) 861. 2. S. S. Stamenkovich, of KH2P04 see e.g. F. 3. for extensive description Jona and G. Shirane, Ferroelectric crystals (Pergamon Press, 1962) Ch. III. 4. Th. Plesser and H. Stiller. Sol. State Corn., to be published. 5. H.Meister, R. Haas, F. May, C. Schenk and B. Weckermann, Kerntechnik 10 (1968) 145. 6. C. Schenk, E. Wiener, B. Weckermann and W. Kley, Phys. Rev., to be published. Arkiv Fysik 14, 315. 7. A.SjFrlander, 8. G. Verdan, Euratom Report EUR 2779 e (1966).
**** 583