- Email: [email protected]
Interaction of ultra-cold neutrons with condensed matter
Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
Interaction of ultra-cold neutrons with condensed matter S.T. Belyaev*, A.L. Barabanov RRC **Kurchatov Institute++, 123182 Moscow, Russia
Abstract A general theory of neutron scattering (elastic and inelastic) is presented. It is applicable to the whole domain of slow neutrons and includes, as limiting cases, existing theories for thermal and cold neutrons and for the elastic scattering of UCN. A new expression for the inelastic scattering cross section for UCN is proposed. It di!ers from the one usually used by consistently taking into account re-scattering processes. Evidence for slight heating and cooling of UCN is given. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 25.40.Fq; 61.12.Bt Keywords: Ultra-cold neutrons; Inelastic scattering
1. Introduction Thermal and cold neutrons with wave length 0.03 nm4j41 nm are important tools for the investigation of condensed matter. The theory of their interaction with substances is well established (see, e.g., Refs. [1,2]). It is based on the use of a Fermi pseudopotential. For thermal and cold neutrons, re-scattering of secondary waves is unimportant and one may use the Born approximation which gives the following expression for the double di!erential cross section: d2p k@ " + bHb s (j,u). l l{ ll{ dX du 2pk ll{
(1)
Here j"k@!k, u"e!e@, where k and e are the momentum and energy of the incident neutron, k@
* Corresponding author.
and e@ are the same quantities for the scattered neutron, and b is a scattering amplitude on the lth l bound nucleus. The Fourier transform
P
`= s (j, t)e*ut dt (2) ll{ ~= of diagonal matrix elements of the operator of nuclear position correlation
s (j,u)" ll{
s (j, t)"SiDe*jRK l (t)e~*jRK l{ (0)DiT (3) ll{ between the initial eigenfunctions DiT of the target Hamiltonian is written here, that determines the target response on the scattered neutron wave. For ultra-cold neutrons (UCN), when j510 nm, re-scattering of the neutron wave in media is very important, and when k2(4pbn re-scattering becomes the dominant process and results in total re#ection from the surface of the target (for positive b). Thus, the Born approximation in general, and the cross section (1) in particular, cannot be used
0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 0 6 5 - 7
S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
for UCN. To describe the elastic scattering of UCN by matter one uses a multiple scattering wave approach for "xed (unmovable) nuclei (see, e.g., Refs. [3,4]). It gives an e!ective repulsive (optical) potential for a neutron inside the material, so a neutron wave with energy below the threshold decreases exponentially within the target. However, UCN escaping from vessels belongs to the inelastic scattering. In this paper we present the basic features of a general theory for elastic and inelastic scattering equally applicable for thermal and cold as well as UCNs, and which, as the neutron wave length decreases, smoothly transforms into the usual scattering theory giving the cross section (1).
2. General expressions A proper theory for UCN scattering should be based on the following postulates: (i) no Born approximation; (ii) no use of a Fermi potential; (iii) the target matter is a dynamical system. It is, of course, impossible to solve the many-body problem of neutron}target interaction without any approximations. In our problem there are two main small parameters: the short-range of the neutron}nuclei interaction (compared with interatomic distance and wave length), and the small neutron energy (compared with the depth of the interaction potential). The "rst condition allows us to consider only the s-wave part of the wave function of the neutron}nucleus center-of-mass motion, when their interaction is evaluated. The second condition allows us to neglect the energy of the relative neutron}nucleus motion in this evaluation, inside the interaction potential region. No speci"c model for the neutron}nucleus interaction potential is needed. The speci"c features described above (short range and large depth) allow us to use a scattering length approximation. From these considerations we obtain a general expression for the double di!erential cross section: d2p k@ " + /jH/j{ sjj{ (j,u#E !E ). i j l l{ ll{ dX du 2pk jj{ll{
(4)
705
It contains the neutron amplitudes /j and the l Fourier transform of the non-diagonal matrix element of the correlation operator sjj{ (j, t)"S jDe*jRK l (t)e~*jRK l{ (0)D j@T (5) ll{ between the eigenfunctions D jT and D j@T of the target Hamiltonian. Note that E is the energy of the i initial target state DiT, and E corresponds to a state j D jT. A set of linear algebraic equations for neutron amplitudes /j is also found. l Neglecting in these equations terms that describe re-scattering we get for the amplitudes /j "d b (1!ia SiDJkK 2DiT) (6) l l ij l l where a and b are the scattering lengths on isol l lated and bound nuclei, respectively, and kK is an l operator of impact momentum in the center-ofmass system for the neutron and nucleus. Thus, for thermal and cold neutrons Eq. (4) really transforms into Eq. (1), and the usual relation between b and l b arises. l In condensed matter we have R "q #u , l l l where q is the equilibrium position of the lth l nucleus, and u is its shift from equilibrium. Thus, l the factors e~*kql and e*kql{ may be extracted in the matrix elements (5) and combined with the amplitudes /jH and /j{ in Eq. (4). Equations for new l{ l amplitudes tj(l)"(/j /b )e*kql are of the form l l (7) tj(l)"d e*kql ! + b Gjj{ tj{(l@). ij l{ ll{ j{l{ The coe$cients Gjj{ are expressed in terms of the ll{ matrix elements (5). Then we use an expansion over ku for the functions sjj{ (j,u), coe$cients Gjj{ , and amplitudes ll{ ll{ tj(l) (or /j ). The zero-order approximation (u"0) l corresponds to "xed nuclei and, therefore, results in only elastic scattering. Equations for the zero-order amplitudes t(0)j(l)"d tk (l) ij tk (l)"e*kql !+ b Gi(ll@)tk (l@), l{ l{ e*k@ql ~ql{ @ . (8) Gi(ll@)" Dq !q D l l{ coincide with the multiple scattering wave equations usually used to describe UCN elastic scattering.
SECTION 5b.
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S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
3. Inelastic scattering Inelastic scattering arises in the second-order approximation in ku. Analysis shows that there are four second-order terms in the inelastic cross section (4): + /jH/j{ sjj{ P/(0)iH/(0)is(2)ii#/(0)iH/(1)fs(1)if l{ ll{ l l{ ll{ l l l{ ll{ jj{ # /(1)fH/(0)is(1)fi l{ ll{ l (9) # /(1)fH/(1)fs(0)ff. l{ ll{ l The four terms on the right-hand side of Eq. (9) are illustrated in Fig. 1. To understand the physical meaning of these terms, it is instructive to compare our result with that based on an improvement of Eq. (1) by replacement of the Born amplitudes b by l the neutron amplitudes in an optical potential /(0) (see, e.g., Ref. [5]). In such an approach, an l expansion similar to Eq. (9) would evidently result in only the "rst term (Fig. 1a), where re-scattering is taken into account only for the incident neutron wave (already included in /(0)). l The other three terms in Eq. (9) describe the rescattering of out-going waves (through inelastic channels). They are directly and indirectly generated by the nondiagonal matrix elements sjj{ . The ll{ "rst-order term for the diagonal matrix element ( j"j@) is absent. The "nal expression for the second-order inelastic cross section is of the form
P
P
Note that Eq. (11) contains symmetrically the functions of the elastic and inelastic neutron channels. In the Born approximation, i.e., neglecting re-scattering both in the elastic and inelastic channels, we have from (8): tk (l)Pe*kql and t k (l)Pe~*k{ql . ~{ Thus, B(q)P!ij+ b e~*(q`j)ql l l
dp(2) 1 *% " d3k@ d(e@#u!e) du 2pmk d3q BH(q)B (q)X (q,u) ] b ab (2p)3 a
Fig. 1. Contributions to the second-order inelastic cross section: (a) scattering}scattering interference, (b) scattering}re-scattering interference, (c) re-scattering}re-scattering interference. Solid and dash lines represent neutrons in the elastic and inelastic channels, respectively. Open and crossed circles correspond to elastic and inelastic scattering, respectively.
(10)
B(q)"+ b e~*qql + (t k (l)tk (l)) (11) l l ~{ l where X (q,u) is related to the Fourier transform ab of the correlation function by the equation
P
d3q du q q q SiDu( (t)u( (0)DiT" e* ( l ~ l{ )~*utX (q,u). la l{b ab (2p)4 (12)
(13)
and the usually used formula for the inelastic cross section arises (see, e.g., Ref. [4]). In Ref. [5] an attempt was made to improve this approach by replacing the plane wave e*kql in Eq. (13) by the damping function tk (l). This attempt is clearly inconsistent as such a replacement should be made in Eq. (11) before di!erentiation with respect to q . l 4. Results To illustrate the possibilities of our approach we studied slight heating and cooling of UCN in the
S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
simplest model. Let us consider UCN that fall normally onto a thick layer of uniform matter with an energy e below a threshold ;. Taking the correlation function in the phonon model we obtain the following expressions for the probabilities of inelastic scattering per bounce: dw(2) *% de@ dw(2) *% de@
K K
e{yU e{;U
707
by other collective excitations in condensed matter. In particular, when the propagation speed of the excitation is of the same order as the velocity of UCN, the in#uence of matter #uctuations on rescattering processes may be maximal. Our study of UCN interaction with di!usion and thermal wave modes is now in progress.
2kb ¹ v@ " p; Ms2 s 5. Summary v@/s kb ¹ " pe@ Ms2 1!(v@/2s)2
(14)
where ¹ is the target temperature, M is the mass of the target nuclei, s is the speed of sound, and v@"J2e@/m is the velocity of the scattered neutron. Note that the second formula is not valid in the small region just above the barrier, where oscillations governed by narrow resonances in transmission and re#ection are important. However, these oscillations are rapidly damped as e@ increases. Half of the neutrons with energy e@'; are re#ected from the target, while the other half is transmitted through it. The spectrum of inelastically scattered neutrons in the model considered has a maximum at e@K;. Indeed, it increases as &v@ below the barrier and falls o! as &1/v@ above the barrier. Thus, qualitatively it is just of the form needed to explain the slight heating and cooling of UCN in vessels. However, the magnitude of the e!ect in the phonon model is low, as kb&10~6 and v@/s&10~3. Nevertheless, it should be noted that an evaluation of the inelastic scattering probability in the Born approximation, i.e., with B(q) (13), gives
A general theory of neutron scattering (elastic and inelastic) is presented. It is applicable for the whole domain of slow neutrons and includes, as limiting cases, existing theories for thermal and cold neutrons and for the elastic scattering of UCN. The only small parameters used are those for the interaction potential which is assumed to be short range and relatively deep. This is equivalent to the scattering length approximation for the interaction. An expression for the inelastic cross section is given. It di!ers from that usually used by consistently taking into account re-scattering in the inelastic channel. It is shown that in the phonon model our approach qualitatively explains the low-energy transfer processes. Probably, the small heating [7] and cooling [6] observed recently belong to the inelastic process described. However, to provide the large observed probabilities of small heating and cooling of UCN in vessels, other collective excitations of condensed matter in the limit of small q and u should apparently be taken into account.
Acknowledgements dwB kb ¹ v@ *% & . de@ ms2 Ms2 s
(15)
This spectrum, "rst, has no maximum in the lowenergy region and, second, is additionally suppressed by the factor &;/ms2 as compared with Eq. (14). This results from the direct proportionality of B(q) (13) to j vanishing for low energy transfer from the neutron to the target or vice versa. One should expect that the low-energy transfer processes are governed not by phonons but rather
This work was supported by RFBR Grant 96-1596548.
References [1] V.F. Turchin, Slow Neutrons, Gosatomizdat, Moscow, 1963 (in Russian). [2] I.I. Gurevich, L.V. Tarasov, Physics of Low-Energy Neutrons, Nauka, Moscow 1965 (in Russian). [3] V.K. Ignatovich, Physics of Ultra-cold Neutrons, Nauka, Moscow 1986 (in Russian).
SECTION 5b.
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S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
[4] R. Golub, D. Richardson, S.K. Lamoreaux, Ultra-Cold Neutrons, Adam Hilger, Bristol, 1991. [5] D.I. Blokhintsev, N.M. Plakida, Phys. Stat. Sol. B 82 (1977) 627.
[6] L. Bondarenko, P. Geltenbort, E. Korobkina, V. Morozov, Yu. Panin, A. Steyerl, Pis'ma ZETP 68 (1998) 663. [7] V.V. Nesvizhevsky, A.V. Strelkov, P. Geltenbort, P.S. Yadjiev, ILL Annual Report, 1997, pp. 62}64.
Interaction of ultra-cold neutrons with condensed matter S.T. Belyaev*, A.L. Barabanov RRC **Kurchatov Institute++, 123182 Moscow, Russia
Abstract A general theory of neutron scattering (elastic and inelastic) is presented. It is applicable to the whole domain of slow neutrons and includes, as limiting cases, existing theories for thermal and cold neutrons and for the elastic scattering of UCN. A new expression for the inelastic scattering cross section for UCN is proposed. It di!ers from the one usually used by consistently taking into account re-scattering processes. Evidence for slight heating and cooling of UCN is given. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 25.40.Fq; 61.12.Bt Keywords: Ultra-cold neutrons; Inelastic scattering
1. Introduction Thermal and cold neutrons with wave length 0.03 nm4j41 nm are important tools for the investigation of condensed matter. The theory of their interaction with substances is well established (see, e.g., Refs. [1,2]). It is based on the use of a Fermi pseudopotential. For thermal and cold neutrons, re-scattering of secondary waves is unimportant and one may use the Born approximation which gives the following expression for the double di!erential cross section: d2p k@ " + bHb s (j,u). l l{ ll{ dX du 2pk ll{
(1)
Here j"k@!k, u"e!e@, where k and e are the momentum and energy of the incident neutron, k@
* Corresponding author.
and e@ are the same quantities for the scattered neutron, and b is a scattering amplitude on the lth l bound nucleus. The Fourier transform
P
`= s (j, t)e*ut dt (2) ll{ ~= of diagonal matrix elements of the operator of nuclear position correlation
s (j,u)" ll{
s (j, t)"SiDe*jRK l (t)e~*jRK l{ (0)DiT (3) ll{ between the initial eigenfunctions DiT of the target Hamiltonian is written here, that determines the target response on the scattered neutron wave. For ultra-cold neutrons (UCN), when j510 nm, re-scattering of the neutron wave in media is very important, and when k2(4pbn re-scattering becomes the dominant process and results in total re#ection from the surface of the target (for positive b). Thus, the Born approximation in general, and the cross section (1) in particular, cannot be used
0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 0 6 5 - 7
S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
for UCN. To describe the elastic scattering of UCN by matter one uses a multiple scattering wave approach for "xed (unmovable) nuclei (see, e.g., Refs. [3,4]). It gives an e!ective repulsive (optical) potential for a neutron inside the material, so a neutron wave with energy below the threshold decreases exponentially within the target. However, UCN escaping from vessels belongs to the inelastic scattering. In this paper we present the basic features of a general theory for elastic and inelastic scattering equally applicable for thermal and cold as well as UCNs, and which, as the neutron wave length decreases, smoothly transforms into the usual scattering theory giving the cross section (1).
2. General expressions A proper theory for UCN scattering should be based on the following postulates: (i) no Born approximation; (ii) no use of a Fermi potential; (iii) the target matter is a dynamical system. It is, of course, impossible to solve the many-body problem of neutron}target interaction without any approximations. In our problem there are two main small parameters: the short-range of the neutron}nuclei interaction (compared with interatomic distance and wave length), and the small neutron energy (compared with the depth of the interaction potential). The "rst condition allows us to consider only the s-wave part of the wave function of the neutron}nucleus center-of-mass motion, when their interaction is evaluated. The second condition allows us to neglect the energy of the relative neutron}nucleus motion in this evaluation, inside the interaction potential region. No speci"c model for the neutron}nucleus interaction potential is needed. The speci"c features described above (short range and large depth) allow us to use a scattering length approximation. From these considerations we obtain a general expression for the double di!erential cross section: d2p k@ " + /jH/j{ sjj{ (j,u#E !E ). i j l l{ ll{ dX du 2pk jj{ll{
(4)
705
It contains the neutron amplitudes /j and the l Fourier transform of the non-diagonal matrix element of the correlation operator sjj{ (j, t)"S jDe*jRK l (t)e~*jRK l{ (0)D j@T (5) ll{ between the eigenfunctions D jT and D j@T of the target Hamiltonian. Note that E is the energy of the i initial target state DiT, and E corresponds to a state j D jT. A set of linear algebraic equations for neutron amplitudes /j is also found. l Neglecting in these equations terms that describe re-scattering we get for the amplitudes /j "d b (1!ia SiDJkK 2DiT) (6) l l ij l l where a and b are the scattering lengths on isol l lated and bound nuclei, respectively, and kK is an l operator of impact momentum in the center-ofmass system for the neutron and nucleus. Thus, for thermal and cold neutrons Eq. (4) really transforms into Eq. (1), and the usual relation between b and l b arises. l In condensed matter we have R "q #u , l l l where q is the equilibrium position of the lth l nucleus, and u is its shift from equilibrium. Thus, l the factors e~*kql and e*kql{ may be extracted in the matrix elements (5) and combined with the amplitudes /jH and /j{ in Eq. (4). Equations for new l{ l amplitudes tj(l)"(/j /b )e*kql are of the form l l (7) tj(l)"d e*kql ! + b Gjj{ tj{(l@). ij l{ ll{ j{l{ The coe$cients Gjj{ are expressed in terms of the ll{ matrix elements (5). Then we use an expansion over ku for the functions sjj{ (j,u), coe$cients Gjj{ , and amplitudes ll{ ll{ tj(l) (or /j ). The zero-order approximation (u"0) l corresponds to "xed nuclei and, therefore, results in only elastic scattering. Equations for the zero-order amplitudes t(0)j(l)"d tk (l) ij tk (l)"e*kql !+ b Gi(ll@)tk (l@), l{ l{ e*k@ql ~ql{ @ . (8) Gi(ll@)" Dq !q D l l{ coincide with the multiple scattering wave equations usually used to describe UCN elastic scattering.
SECTION 5b.
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S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
3. Inelastic scattering Inelastic scattering arises in the second-order approximation in ku. Analysis shows that there are four second-order terms in the inelastic cross section (4): + /jH/j{ sjj{ P/(0)iH/(0)is(2)ii#/(0)iH/(1)fs(1)if l{ ll{ l l{ ll{ l l l{ ll{ jj{ # /(1)fH/(0)is(1)fi l{ ll{ l (9) # /(1)fH/(1)fs(0)ff. l{ ll{ l The four terms on the right-hand side of Eq. (9) are illustrated in Fig. 1. To understand the physical meaning of these terms, it is instructive to compare our result with that based on an improvement of Eq. (1) by replacement of the Born amplitudes b by l the neutron amplitudes in an optical potential /(0) (see, e.g., Ref. [5]). In such an approach, an l expansion similar to Eq. (9) would evidently result in only the "rst term (Fig. 1a), where re-scattering is taken into account only for the incident neutron wave (already included in /(0)). l The other three terms in Eq. (9) describe the rescattering of out-going waves (through inelastic channels). They are directly and indirectly generated by the nondiagonal matrix elements sjj{ . The ll{ "rst-order term for the diagonal matrix element ( j"j@) is absent. The "nal expression for the second-order inelastic cross section is of the form
P
P
Note that Eq. (11) contains symmetrically the functions of the elastic and inelastic neutron channels. In the Born approximation, i.e., neglecting re-scattering both in the elastic and inelastic channels, we have from (8): tk (l)Pe*kql and t k (l)Pe~*k{ql . ~{ Thus, B(q)P!ij+ b e~*(q`j)ql l l
dp(2) 1 *% " d3k@ d(e@#u!e) du 2pmk d3q BH(q)B (q)X (q,u) ] b ab (2p)3 a
Fig. 1. Contributions to the second-order inelastic cross section: (a) scattering}scattering interference, (b) scattering}re-scattering interference, (c) re-scattering}re-scattering interference. Solid and dash lines represent neutrons in the elastic and inelastic channels, respectively. Open and crossed circles correspond to elastic and inelastic scattering, respectively.
(10)
B(q)"+ b e~*qql + (t k (l)tk (l)) (11) l l ~{ l where X (q,u) is related to the Fourier transform ab of the correlation function by the equation
P
d3q du q q q SiDu( (t)u( (0)DiT" e* ( l ~ l{ )~*utX (q,u). la l{b ab (2p)4 (12)
(13)
and the usually used formula for the inelastic cross section arises (see, e.g., Ref. [4]). In Ref. [5] an attempt was made to improve this approach by replacing the plane wave e*kql in Eq. (13) by the damping function tk (l). This attempt is clearly inconsistent as such a replacement should be made in Eq. (11) before di!erentiation with respect to q . l 4. Results To illustrate the possibilities of our approach we studied slight heating and cooling of UCN in the
S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
simplest model. Let us consider UCN that fall normally onto a thick layer of uniform matter with an energy e below a threshold ;. Taking the correlation function in the phonon model we obtain the following expressions for the probabilities of inelastic scattering per bounce: dw(2) *% de@ dw(2) *% de@
K K
e{yU e{;U
707
by other collective excitations in condensed matter. In particular, when the propagation speed of the excitation is of the same order as the velocity of UCN, the in#uence of matter #uctuations on rescattering processes may be maximal. Our study of UCN interaction with di!usion and thermal wave modes is now in progress.
2kb ¹ v@ " p; Ms2 s 5. Summary v@/s kb ¹ " pe@ Ms2 1!(v@/2s)2
(14)
where ¹ is the target temperature, M is the mass of the target nuclei, s is the speed of sound, and v@"J2e@/m is the velocity of the scattered neutron. Note that the second formula is not valid in the small region just above the barrier, where oscillations governed by narrow resonances in transmission and re#ection are important. However, these oscillations are rapidly damped as e@ increases. Half of the neutrons with energy e@'; are re#ected from the target, while the other half is transmitted through it. The spectrum of inelastically scattered neutrons in the model considered has a maximum at e@K;. Indeed, it increases as &v@ below the barrier and falls o! as &1/v@ above the barrier. Thus, qualitatively it is just of the form needed to explain the slight heating and cooling of UCN in vessels. However, the magnitude of the e!ect in the phonon model is low, as kb&10~6 and v@/s&10~3. Nevertheless, it should be noted that an evaluation of the inelastic scattering probability in the Born approximation, i.e., with B(q) (13), gives
A general theory of neutron scattering (elastic and inelastic) is presented. It is applicable for the whole domain of slow neutrons and includes, as limiting cases, existing theories for thermal and cold neutrons and for the elastic scattering of UCN. The only small parameters used are those for the interaction potential which is assumed to be short range and relatively deep. This is equivalent to the scattering length approximation for the interaction. An expression for the inelastic cross section is given. It di!ers from that usually used by consistently taking into account re-scattering in the inelastic channel. It is shown that in the phonon model our approach qualitatively explains the low-energy transfer processes. Probably, the small heating [7] and cooling [6] observed recently belong to the inelastic process described. However, to provide the large observed probabilities of small heating and cooling of UCN in vessels, other collective excitations of condensed matter in the limit of small q and u should apparently be taken into account.
Acknowledgements dwB kb ¹ v@ *% & . de@ ms2 Ms2 s
(15)
This spectrum, "rst, has no maximum in the lowenergy region and, second, is additionally suppressed by the factor &;/ms2 as compared with Eq. (14). This results from the direct proportionality of B(q) (13) to j vanishing for low energy transfer from the neutron to the target or vice versa. One should expect that the low-energy transfer processes are governed not by phonons but rather
This work was supported by RFBR Grant 96-1596548.
References [1] V.F. Turchin, Slow Neutrons, Gosatomizdat, Moscow, 1963 (in Russian). [2] I.I. Gurevich, L.V. Tarasov, Physics of Low-Energy Neutrons, Nauka, Moscow 1965 (in Russian). [3] V.K. Ignatovich, Physics of Ultra-cold Neutrons, Nauka, Moscow 1986 (in Russian).
SECTION 5b.
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S.T. Belyaev, A.L. Barabanov / Nuclear Instruments and Methods in Physics Research A 440 (2000) 704}708
[4] R. Golub, D. Richardson, S.K. Lamoreaux, Ultra-Cold Neutrons, Adam Hilger, Bristol, 1991. [5] D.I. Blokhintsev, N.M. Plakida, Phys. Stat. Sol. B 82 (1977) 627.
[6] L. Bondarenko, P. Geltenbort, E. Korobkina, V. Morozov, Yu. Panin, A. Steyerl, Pis'ma ZETP 68 (1998) 663. [7] V.V. Nesvizhevsky, A.V. Strelkov, P. Geltenbort, P.S. Yadjiev, ILL Annual Report, 1997, pp. 62}64.