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The interaction of Ultra-Cold Neutrons (UCN) with liquid helium and a superthermal UCN source
Volume 62A, number 5
PHYSICS LETTERS
5 September 1977
THE INTERACTION OF ULTRA-COLD NEUTRONS (UCN) WITH LIQUID HELIUM AND A SUPERTHERMAL UCN SOURCE R. GOLUB and J.M. PENDLEBURY School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BNJ 9QH Sussex, UK Received 4 July 1977 4He and show that this interaction has We discuss the interaction of Ultra-Cold and Cold Neutrons with superfluid all the characteristics which are necessary for the achievement of extremely high densities of UCN.
The fact that Ultra-Cold Neutron (UCN) gas can be collected in a vessel at densities much higher than those that would be achieved at thermal equilibrium has been previously demonstrated [1]. The conditions for this are: (a) the vessel be filled with a medium which has a very small neutron absorption; (b) the medium has a critical energy for total reflection which is much smaller than that of the vessel’s walls and (c) the medium interacts with the UCN in such a way that it behaves as if there were only a single excited state with excitation energy E ~ T ~ E~where Tis the temperature of the medium and E~,the UCN energy is typically on the order of l07eV (10—3 K). At these energies UCN are totally reflected from many solid materials for all angles of incidence. It was suggested [1] that condition (c) would be fulfilled by a purely coherent scatterer with a Debye spectrum, but the nature of the excitation spectrum is not very We now attention the fact 4He important. is perhaps the only draw material which to satisfies that all the above conditions having no absorption (for pure pure coherent scattering, and a critical energy approximately ten times smaller than some common wall materials. In the present paper we show the relevant calculations of the interaction of UCN with liquid 4He and discuss a practical realization of a super-thermal UCN source. Our starting point is eq. 6 of ref. [1] which we rewrite using eq. (3) of ref. [1] for the case of a bottle with no exit hole, including the effect of the neutrons’ j3 decay: fdEOøO(EO)~(E0_~Eu)
=
~ + __
v~f~e~o/T ~
÷
,(l) (Eo+E~)dEo
where v = the average number of UCN collisions with the walls before “loss” (i.e. either absorption or upscattering), d’ = dl [1+d/R} for the cylinder of height d and radius R which we are now considering. Note that we have used the principle of detailed balance on the inelastic upscattering cross section in the denominator. We will approximate the dispersion curve for 4He [2] for the region in which we are interested (q ~ 0.7 A—1) by ~ = q and define *
flk
0=2mc.
(3)
This is the momentum of a neutron which can create a phonon and just come to rest (in the case of zero phonon width). We calculate the energy differential cross section from
Q do = 0b 1 dw 2
r2 s
~ d
4
where the energy transfer is ~ 0b is the bound atom cross section (1.1 barns for 4He) and k 0, k~are the initial and final neutron wave vectors. Q, the wave vector transfer varies between Q = k ÷k 5 1,2
0— u
Since ku ~ k0 we can write (4) as: ~~S(ko,w)k~. ~‘
(6)
0
For S(Q, w) we assume a Lorentzian form 337
Volume 62A, number 5
PHYSICS LETTERS
S(Q)FQ/lr
S(Q,o.)=
2+I’~
(7)
,
[w—w0(Q)]
with ~
a(T0)
5 September 1977
=
~../2mkI3T0= 5.7X l0~cm/sec for T0 = 20K,
as m a hydrogen or deuterium cold source in a reactor.
0(Q) given by (2). This form has been chosen to satisfy the zero-moment sum rule and is in the appropriate for thethat case kBT<<1TwO (1~w= E0 Eu). We form also assume ~ w 0(Q) and
The term in square brackets in (12) and (13) is equal to 4 sec/cm when we use r = i0~sec4He, and so thethe other parameters are as given above for liquid gain in density is iO~relative to the cold source at
E~~ F(k~)~ w0(k~), (8) in which case the factors multiplying the cross sections in the integrals in equation (1) can be set equal to their values atE~’rrlt2k*2/2m ck~1l. 0 For the numerical values of S(Q) we will use only the one-phonon contribution [21.Using (6), (7) and (8) we find
20K. Taking E~=l0~K (13) yields 4)X4X 1074X eq. l0~ n/cm3, PUCN= if we take
.
t= 10
(15)
n/cm /sec.
This is a formidable UCN density by current standards but its achievement would require the installation of several litres of liquid He, at a temperature below
f~(E 0~+Eu)dE0 = 2NOb(ku/k~)S(k~),
(9)
1K inside a 20K cold source in a reactor where the radiation heating may be 1 watt/gram or more an equally formidable technical task. However, the available gains are so large that one can place the system at the end of a neutron guide looking at a cold source. This will result in a reduction of UCN density in the vessel —
where N is the atomic density in the liquid helium, Using (9) we find that the condition for the last term in the denominator of (1) to be negligible in comparison with l/r~is: 2NubS(k~)e_E’0~T~
VuT/(~/Eu)
(10)
~,
whichyie1dsT
by a solid angle factor equal to eE~7>< l0~, (16) the critical angle of nickel for neutrons at where bA) is 1.7 X 10—2 radians. Multiplying (15) k~(X* 0c’
~ 0(E0) dE0 = r1(E0/T0) eE~T0dE0/T0,
(11)
i.e. a Maxwell distribution at temperature T0 and total flux 1, we obtain:
p~V~~) dEu C1 T =
—~-
~/
e~~T0 1 1rj~ ~ [~J~Jçj~ rS (k~)]dEu,
0 0 (12) where r is the storage time of UCN in the bottle. Integrating this with respect to Eu up to the critical energy of the bottle walls, Ec, we obtain the total steady-state UCN density in the bottle: PUCN
=
~
IE \ 3/2~/~ —~ e~’~T0 [2NabrS(k~)],
V
(13)
as compared to IE ~3/2 1T 1 ~ 0 ~ 0, ~ which would be obtained in the source at T PUCN
(14)
=
0.
by this factor yield~ / 3 PUCN~~AIUnjcm
which is a very substantial improvement over all existing 3, UCN which yield p ~ 1we n/cm In sources the above considerations neglected the wall losses, i.e. we assumed that r ~1(2vd/v W
u
)~r
This will be the case if it proves possible to obtain the theoretically expected values of s these are iO~for beryllium, boron free glass and solid oxygen. However, such low UCN absorption rates have not yet been achieved in practice [3,4]; the reasons for this remain controversial [4,5,6]. The present authors feel that inelastic by impurities in anomalous the walls is losses not yet ruled outscattering as a possible cause of the and that these losses can be reduced by a variety of tech—
niques. If we limit ourselves to values of v reported in the literature, i.e. v 5 X iO~[3] then the UCN density ~—
338
(17)
.
Volume 62A, number 5
PHYSICS LETTERS
5 September 1977
figures given above should be reduced by a factor of 10 for a cylindrical bottle with R = 10cm and d = 20 cm. This still leaves an expected density ‘~300n/cm3 for the bottle mounted on the end of a guide tube, What we have calculated is the steady state UCN density which would be achieved in the helium filled vessel after several storage times, so one would have to wait this long between periodic extractions of UCN from the source. However, this is not too great a limitation as many of the proposed applications of UCN such as the search for a neutron electric dipole moment and studies of the neutron ~3decay, which make use of the storage of UCN must be carried out with similar waiting times. In order for the above considerations to be valid the density of 3He must be low enough so that the neutron absorption by 3He does not significantly affect the storage time in the vessel. Thus we require
will not lead to any significant reduction in incident flux if the size of the storage vessel is small compared to the total mean free path of neutrons with a wavelength of 10 A in liquid helium [9] which is the case we are considering here. It may seem surprising that one can achieve UCN densities so much higher than those associated with thermal equilibrium but the point is that our system is in a steady-state far from thermal equilibrium, with energy flowing through the system from the downscattered neutrons, into phonons, and then into the thermal reservoir at a constant rate. Looked at in this way the system is analogous to a flame-driven refrigerator with the effective UCN temperature being much lower than the temperature of the liquid helium [10]. Note: Atkins and McClintock (Cryogenics 16 (1976) 733), describe a method which they estimate is capable
M3)u~(v)u’~l/r,
of reducing the 3He/4He ratio to 10—12.
(18)
where the superscript refers to ~ and o~= 5.3 X i03 barns at v = 2200 rn/sec so that we require N(3) <1012/cm3 for the case when the storage time is determined by i3 decay. This corresponds to a relative concentration of 3He of 0.5 X 10—10. In the case when wall losses dominate this requirement can be increased by a factor of 10. Although this is somewhat lower than 3He concentrations which have been reported in the literature [7], existing techniques should be capable of achieving the required purity [8]. Multi-phonon processes which we have ignored until now, cannot contribute significantly to the loss of UCN from the bottle because the maximum momentum that can be transferred to the neutron is equal to the algebraic sum of the individual phonon momenta and this transfer is only possible if the final energy of the neutron is greater than E*. Thus multi-phonon upscattermg processes will also be proportional to exp [E * IT] and will be negligible at low enough temperature. For the down scattering into the UCN region we have only considered single phonon processes but it is clear that multi-phonon processes can only increase the production rate of UCN. The increased attenuation of the incident neutron flux due to multi-phonon processes
—
—
The authors are grateful for useful and encouraging conversations with A.D.B. Woods, D.O. Edwards, Douglas Brewer, W.S. Truscott, Arnold Dahm and Norman Ramsey.
References [1] R. Golub and J.M. Pendlebury, Phys. Lett. 53A (1975) 133. [2] R.A. Cowley and A.D.B. Woods, Can. J. Phys. 49 (1971) 177. [3] F.L. Shapiro, in: Proc. Conf. on Nuclear structure study with neutrons, Budapest, 1972. [4] V.1. Luschikov, in: Proc. of the Int. Conf. on the Interactions of neutrons with nucleii, Lowell, Mass., 1976, p. 118. [5] R. Golub and J.M. Pendlebury, Phys. Lett. 50A (1974) 177. [6] W.L. Lanford, K. Davis, P. Lamarche and R. Golub Bull. Amer. Phys. Soc. 22 (1977) 548; W.A. Lanford, R. Golub et al., to be published. [7] D.O. Edwards et al., Phys. Rev. Bil (1975) 4734. [8] D.O. Edwards, private communication. [9] H.S.Sornmers, J.G. Dash and L. Goldstein Phys. Rev. [101R. Golub, in: Proc. mt. Conf. on Statistical Physics Budapest, 1975.
339
PHYSICS LETTERS
5 September 1977
THE INTERACTION OF ULTRA-COLD NEUTRONS (UCN) WITH LIQUID HELIUM AND A SUPERTHERMAL UCN SOURCE R. GOLUB and J.M. PENDLEBURY School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BNJ 9QH Sussex, UK Received 4 July 1977 4He and show that this interaction has We discuss the interaction of Ultra-Cold and Cold Neutrons with superfluid all the characteristics which are necessary for the achievement of extremely high densities of UCN.
The fact that Ultra-Cold Neutron (UCN) gas can be collected in a vessel at densities much higher than those that would be achieved at thermal equilibrium has been previously demonstrated [1]. The conditions for this are: (a) the vessel be filled with a medium which has a very small neutron absorption; (b) the medium has a critical energy for total reflection which is much smaller than that of the vessel’s walls and (c) the medium interacts with the UCN in such a way that it behaves as if there were only a single excited state with excitation energy E ~ T ~ E~where Tis the temperature of the medium and E~,the UCN energy is typically on the order of l07eV (10—3 K). At these energies UCN are totally reflected from many solid materials for all angles of incidence. It was suggested [1] that condition (c) would be fulfilled by a purely coherent scatterer with a Debye spectrum, but the nature of the excitation spectrum is not very We now attention the fact 4He important. is perhaps the only draw material which to satisfies that all the above conditions having no absorption (for pure pure coherent scattering, and a critical energy approximately ten times smaller than some common wall materials. In the present paper we show the relevant calculations of the interaction of UCN with liquid 4He and discuss a practical realization of a super-thermal UCN source. Our starting point is eq. 6 of ref. [1] which we rewrite using eq. (3) of ref. [1] for the case of a bottle with no exit hole, including the effect of the neutrons’ j3 decay: fdEOøO(EO)~(E0_~Eu)
=
~ + __
v~f~e~o/T ~
÷
,(l) (Eo+E~)dEo
where v = the average number of UCN collisions with the walls before “loss” (i.e. either absorption or upscattering), d’ = dl [1+d/R} for the cylinder of height d and radius R which we are now considering. Note that we have used the principle of detailed balance on the inelastic upscattering cross section in the denominator. We will approximate the dispersion curve for 4He [2] for the region in which we are interested (q ~ 0.7 A—1) by ~ = q and define *
flk
0=2mc.
(3)
This is the momentum of a neutron which can create a phonon and just come to rest (in the case of zero phonon width). We calculate the energy differential cross section from
Q do = 0b 1 dw 2
r2 s
~ d
4
where the energy transfer is ~ 0b is the bound atom cross section (1.1 barns for 4He) and k 0, k~are the initial and final neutron wave vectors. Q, the wave vector transfer varies between Q = k ÷k 5 1,2
0— u
Since ku ~ k0 we can write (4) as: ~~S(ko,w)k~. ~‘
(6)
0
For S(Q, w) we assume a Lorentzian form 337
Volume 62A, number 5
PHYSICS LETTERS
S(Q)FQ/lr
S(Q,o.)=
2+I’~
(7)
,
[w—w0(Q)]
with ~
a(T0)
5 September 1977
=
~../2mkI3T0= 5.7X l0~cm/sec for T0 = 20K,
as m a hydrogen or deuterium cold source in a reactor.
0(Q) given by (2). This form has been chosen to satisfy the zero-moment sum rule and is in the appropriate for thethat case kBT<<1TwO (1~w= E0 Eu). We form also assume ~ w 0(Q) and
The term in square brackets in (12) and (13) is equal to 4 sec/cm when we use r = i0~sec4He, and so thethe other parameters are as given above for liquid gain in density is iO~relative to the cold source at
E~~ F(k~)~ w0(k~), (8) in which case the factors multiplying the cross sections in the integrals in equation (1) can be set equal to their values atE~’rrlt2k*2/2m ck~1l. 0 For the numerical values of S(Q) we will use only the one-phonon contribution [21.Using (6), (7) and (8) we find
20K. Taking E~=l0~K (13) yields 4)X4X 1074X eq. l0~ n/cm3, PUCN= if we take
.
t= 10
(15)
n/cm /sec.
This is a formidable UCN density by current standards but its achievement would require the installation of several litres of liquid He, at a temperature below
f~(E 0~+Eu)dE0 = 2NOb(ku/k~)S(k~),
(9)
1K inside a 20K cold source in a reactor where the radiation heating may be 1 watt/gram or more an equally formidable technical task. However, the available gains are so large that one can place the system at the end of a neutron guide looking at a cold source. This will result in a reduction of UCN density in the vessel —
where N is the atomic density in the liquid helium, Using (9) we find that the condition for the last term in the denominator of (1) to be negligible in comparison with l/r~is: 2NubS(k~)e_E’0~T~
VuT/(~/Eu)
(10)
~,
whichyie1dsT
by a solid angle factor equal to eE~7>< l0~, (16) the critical angle of nickel for neutrons at where bA) is 1.7 X 10—2 radians. Multiplying (15) k~(X* 0c’
~ 0(E0) dE0 = r1(E0/T0) eE~T0dE0/T0,
(11)
i.e. a Maxwell distribution at temperature T0 and total flux 1, we obtain:
p~V~~) dEu C1 T =
—~-
~/
e~~T0 1 1rj~ ~ [~J~Jçj~ rS (k~)]dEu,
0 0 (12) where r is the storage time of UCN in the bottle. Integrating this with respect to Eu up to the critical energy of the bottle walls, Ec, we obtain the total steady-state UCN density in the bottle: PUCN
=
~
IE \ 3/2~/~ —~ e~’~T0 [2NabrS(k~)],
V
(13)
as compared to IE ~3/2 1T 1 ~ 0 ~ 0, ~ which would be obtained in the source at T PUCN
(14)
=
0.
by this factor yield~ / 3 PUCN~~AIUnjcm
which is a very substantial improvement over all existing 3, UCN which yield p ~ 1we n/cm In sources the above considerations neglected the wall losses, i.e. we assumed that r ~1(2vd/v W
u
)~r
This will be the case if it proves possible to obtain the theoretically expected values of s these are iO~for beryllium, boron free glass and solid oxygen. However, such low UCN absorption rates have not yet been achieved in practice [3,4]; the reasons for this remain controversial [4,5,6]. The present authors feel that inelastic by impurities in anomalous the walls is losses not yet ruled outscattering as a possible cause of the and that these losses can be reduced by a variety of tech—
niques. If we limit ourselves to values of v reported in the literature, i.e. v 5 X iO~[3] then the UCN density ~—
338
(17)
.
Volume 62A, number 5
PHYSICS LETTERS
5 September 1977
figures given above should be reduced by a factor of 10 for a cylindrical bottle with R = 10cm and d = 20 cm. This still leaves an expected density ‘~300n/cm3 for the bottle mounted on the end of a guide tube, What we have calculated is the steady state UCN density which would be achieved in the helium filled vessel after several storage times, so one would have to wait this long between periodic extractions of UCN from the source. However, this is not too great a limitation as many of the proposed applications of UCN such as the search for a neutron electric dipole moment and studies of the neutron ~3decay, which make use of the storage of UCN must be carried out with similar waiting times. In order for the above considerations to be valid the density of 3He must be low enough so that the neutron absorption by 3He does not significantly affect the storage time in the vessel. Thus we require
will not lead to any significant reduction in incident flux if the size of the storage vessel is small compared to the total mean free path of neutrons with a wavelength of 10 A in liquid helium [9] which is the case we are considering here. It may seem surprising that one can achieve UCN densities so much higher than those associated with thermal equilibrium but the point is that our system is in a steady-state far from thermal equilibrium, with energy flowing through the system from the downscattered neutrons, into phonons, and then into the thermal reservoir at a constant rate. Looked at in this way the system is analogous to a flame-driven refrigerator with the effective UCN temperature being much lower than the temperature of the liquid helium [10]. Note: Atkins and McClintock (Cryogenics 16 (1976) 733), describe a method which they estimate is capable
M3)u~(v)u’~l/r,
of reducing the 3He/4He ratio to 10—12.
(18)
where the superscript refers to ~ and o~= 5.3 X i03 barns at v = 2200 rn/sec so that we require N(3) <1012/cm3 for the case when the storage time is determined by i3 decay. This corresponds to a relative concentration of 3He of 0.5 X 10—10. In the case when wall losses dominate this requirement can be increased by a factor of 10. Although this is somewhat lower than 3He concentrations which have been reported in the literature [7], existing techniques should be capable of achieving the required purity [8]. Multi-phonon processes which we have ignored until now, cannot contribute significantly to the loss of UCN from the bottle because the maximum momentum that can be transferred to the neutron is equal to the algebraic sum of the individual phonon momenta and this transfer is only possible if the final energy of the neutron is greater than E*. Thus multi-phonon upscattermg processes will also be proportional to exp [E * IT] and will be negligible at low enough temperature. For the down scattering into the UCN region we have only considered single phonon processes but it is clear that multi-phonon processes can only increase the production rate of UCN. The increased attenuation of the incident neutron flux due to multi-phonon processes
—
—
The authors are grateful for useful and encouraging conversations with A.D.B. Woods, D.O. Edwards, Douglas Brewer, W.S. Truscott, Arnold Dahm and Norman Ramsey.
References [1] R. Golub and J.M. Pendlebury, Phys. Lett. 53A (1975) 133. [2] R.A. Cowley and A.D.B. Woods, Can. J. Phys. 49 (1971) 177. [3] F.L. Shapiro, in: Proc. Conf. on Nuclear structure study with neutrons, Budapest, 1972. [4] V.1. Luschikov, in: Proc. of the Int. Conf. on the Interactions of neutrons with nucleii, Lowell, Mass., 1976, p. 118. [5] R. Golub and J.M. Pendlebury, Phys. Lett. 50A (1974) 177. [6] W.L. Lanford, K. Davis, P. Lamarche and R. Golub Bull. Amer. Phys. Soc. 22 (1977) 548; W.A. Lanford, R. Golub et al., to be published. [7] D.O. Edwards et al., Phys. Rev. Bil (1975) 4734. [8] D.O. Edwards, private communication. [9] H.S.Sornmers, J.G. Dash and L. Goldstein Phys. Rev. [101R. Golub, in: Proc. mt. Conf. on Statistical Physics Budapest, 1975.
339