Neutron transmission probability troug a revolving slit for a continuous source and a divergent neutron beam J. Peters*, Han-Meitner-Institut Berlin, Glienicer Str. 00, D 409 Berlin Abstract: Here an analytical calculation for te transmission probability of neutrons travelling troug a revolving slit is being presented. For te first time, two effects ave been taen into account in te same approac, tat is on te one and te fact tat te neutron beam migt be divergent and, on te oter and, tat neutrons from a continuous source can arrive at different times at te copper. Furtermore, te neutron distribution at a given distance beind te copper as been calculated and tese teoretical results ave been compared wit simulated data obtained wit te VITESS simulation program. Te teoretical and te simulated curves are in good agreement. PACS: 6..Ld, 07.05.Fb Keywords: Fermi copper, neutron diffraction, transmission probability *Tel. number (030) 806 3068, email: peters@mi.de (J. Peters)
. Introduction Some decades ago, in te 940ies, E. Fermi, J. Marsall and L. Marsall [] proposed a new ind of velocity selector or rotating sutter to cop te continuous neutron beam into sort pulses, tus obtaining a well defined wavelengt-time correlation. Te first rotating sutter as been furter extended, improved and dribed by T. Brill and H.V. Lictenberger []. Tis ind of copper, composed of a sandwic pacage of neutron transmitting and neutron absorbing foils suc as aluminium and cadmium placed into a fast revolving cylinder, is now often referred to as Fermi copper. Tese early wors contain mostly mecanical and tecnical driptions of te copper and experimental results. In te 950ies, te sape of a neutron burst travelling troug a single revolving slit as been derived analytically for different cases: P.A. Egelstaff [3] as provided te formula for te transmission probability of a divergent beam of neutrons arriving all at equal time at te copper centre. He introduced is formalism from te perspective of te rotor s frame of reference, were te neutrons move on a parabolic curve. Marseguerra and Pauli [4] presented a teory for a curved parabolic Fermi copper wit a parallel beam of neutrons passing troug te centre of te copper at different times (tis applies, for instance, to a continuous source). Tey used te frame of reference of te neutrons. Later, Tol & Brugger and oyston [5] studied two consecutive revolving slits placed at a given distance from eac oter, wereas Becurts & Wirtz [6] summarised concisely some analytical results and applications. Wenever neutron instruments meet te requirement of ig resolution in time or in energy (for instance IN4 or IN6 at ILL/France, Paros at LANL/USA, HET at ISIS/UK), te Fermi copper can be very useful. Furtermore, te tecnique is now so powerful due to magnetic bearings as to allow very ig repetition rates per minute (up to 36.000 PM) and a very
precise pasing [7]. Wen a Fermi copper is placed in suc an instrument, it is used to produce time bursts of well defined time and wavelengt intervals. But in practice, te neutron beam is always divergent and te neutrons passing troug te slit are distributed in time. To dribe te two effects simultaneously, a unification of bot formalisms is necessary and tis can only be done witin one frame of reference. In tis paper, suc an analytical calculation of te transmission probability will be presented and confronted wit data obtained from te Monte Carlo simulation program VITESS written by te Han-Meitner-Institut (HMI) simulation group [8]. It can be sown tat tey are in good agreement. Te Fermi copper caracteristics will be suc as tey will be implemented at te new Extreme Environment Diffractometer (EXED), for wic tis study as been undertaen. Te analytical calculations compared to simulations of te complete instrument allow not only for a bencmaring of te simulation program, but enable us to evaluate te resolution of te instrument [9] and to optimise te total copper system, wic is composed of te Fermi copper and six disc coppers. Tree of te disc coppers are frame overlap coppers and avoid contaminations by iger orders and terefore ave to be placed at well defined positions. Te only way to calculate tem is by using te exact nowledge of te lengt of te copper pulses.. Transmission probability troug a Fermi copper a) Straigt Fermi copper, non-divergent neutron beam, all neutrons pass troug te central plane of te copper (x = 0) at equal time t Let us first briefly recall te analytical approac [4] to calculate te transmission probability troug a straigt Fermi copper. In te following we will assume a copper wose slits all 3
ave te same lengt [see Fig. ] and we will suppose tat all neutrons are travelling troug te central plane of te copper (x = 0) at te time t = 0. Te copper is rotating wit te angular velocity. Figure Te slit lengt is assumed to be muc longer tan te slit eigt, so tat we can use te same approximations as in [4]. Te neutron transmission probability amplitude corresponds to te neutrons travelling troug te green surface in Fig.. Figure Te transmission probability per time unit for a neutron of velocity v to be transmitted troug te revolving system is tus given by te ratio (te neutron wavelengt λ being related to v troug λ[ǻ] = 0.3956 0 4 /v[m/s] P( v) v v m = = = = v v λ λ m, () were v m stands for te minimum velocity required to pass troug te Fermi copper, for wic P(v) vanises, and λ m is te corresponding wavelengt. For te maximum transmission probability, one evaluates te corresponding time of fligt t = /(). Te transmission probability as tus te form of te rigt alf plane of a triangle wit te maximum at te origin (see also Fig. 4). b) Straigt or curved parabolic Fermi copper, non-divergent neutron beam, te neutrons pass troug te central plane of te copper at different times t Following te derivation in [4], te formalism can be generalised to a parabolic copper, from were te straigt copper can be gained as a limiting case. Te curvature of te slit will now 4
follow te trajectory of a neutron wit a given velocity v 0 and will be passing troug te centre of te slit at t = 0. However, since te slit as a finite eigt, some neutrons wit a velocity v v 0 and passing troug x = 0 at t 0 will also be transmitted. Tus te transmission probability can be calculated as a function of wic depends on te deviation of v from te ideal value v 0 : =. () v v 0 Te case of a straigt Fermi copper can tus be obtained in te limit v. 0 Contrary to te case symmetric around x = 0, te transmission probability now explicitly depends on te time t, at wic te neutrons pass troug te plane x = 0 (see Fig. 3 for example). Figure 3 Te equations to dribe te transmission probability can be found in Table of [4]. Equation () can be obtained in te limiting case were P v) P(, t) ( lim =. t 0 v0 Te transmission probability is represented in Fig. 4 for a straigt and a curved Fermi copper; tey only differ in te fact tat one pacage as a curvature wit a radius of 4.4 mm, wic corresponds to an optimum transmitted wavelengt of.5 Ǻ at 36.000 PM. Te curved Fermi copper permits an exact definition of te wavelengt to be transmitted wit igest probability for a given angular velocity. Especially wen increases, te curved pacage can be used to limit te wavelengt band. Figure 4 5
c) Straigt or curved parabolic Fermi copper, divergent neutron beam, te neutrons pass troug te centre of te copper at different times t If te neutron beam is divergent, te neutrons can travel troug te copper following oblique pats. Tese pats, owever, are longer, so tat neutrons wit a velocity below a critical value cannot pass troug te copper any more. In consequence, te transmission probability is decreasing. Te base of te triangle representing te transmission probability P(,t) is proportional to te angle α troug wic a slit can turn wile neutrons are being transmitted (see Fig. 5). Figure 5 A maximum ape angle becomes possible if te gradient of te curves y* is less tan or equal to te gradient of te neutron pat at x = ± and is α = [3]. In general, tis / condition is dependent on velocity and time and requires a careful distinction between cases. T = 0 Having suggested first tat t = 0, te gradient of te neutron pat becomes / and te gradient of te curves y* becomes / at x =. Tus, one finds te maximum ape angle for velocities from down to v c (see [3]): 4 =. (3) v c If te gradient of y* is greater tan /, te ape angle is reduced. One calculates te gradient of y* to be r/ at every point r of te curve. If te neutron its te curve y u * at te point r (becomes tangent) ten it leaves te copper slit at 4r y = wit respect to y l * (see Fig. 6). Te equation of te tangent at any point r can tus be written: 6
rx r y ( x) =. (4) From eq. (4) one obtains y, were te tangent meets x =, r r y y( ) = (5) and te interval y (see Fig. 6): ( r) * y = yu ) y = ( Figure 6. (6) Te sum of 4r y =, from wic one derives an expression for r: r =. (7) Tus, te ape angle is found to be: 4r = 4 = ( ) α, (8) and as a maximum value at = v c. Te minimum neutron velocity v m to pass troug te copper is obtained for α = 0 and again results in (see eq. ()): vc m =. (9) v = 4 Te normalised transmission probability is proportional to te transmission amplitude and to te ape angle and can be expressed in terms of v m : vm P( ) = ( ) vm P( ) = 4( )( v m vm ) for for 4v v m m 4v m. (0) 7
T 0 For neutrons wic pass troug te central plane of te copper at a time different from zero, we ave to distinguis tree possible cases. As te gradient of te curves y u,l * is now / t, te gradient of te neutron pat / can become smaller tan tis even for neutron velocities greater tan v c, for times t greater tan a certain limiting value. (i) Let us first consider a gradient of te neutron pat greater tan te gradient of y u,l * (see Fig. 7). Te point r i can be determined as intersection point of te functions f(x) and g(x). F(x) and g(x) are bot linear functions, wic are written for positive times t: Figure 7 f ( x) = x t g( x) = ( t ) x. () Te point r i is ten obtained by setting f(r i ) = g(r i ): r t = i t /. () Te ape angle is calculated to be i α =. (3) r i i It can be easily proved tat te expressions of r i and te ape angle α are still valid for te case of negative times t, provided tat t is substituted by t in eqs. () and (3). For r i max i 0, one finds again tat α = α ( = 0) = / as it sould. From te condition r i t, it follows tat t for tis case. 8
() If now < t for positive times t, te ape angle α is again reduced. First we ave to determine te point r (see Fig. 8), were te neutron pat becomes tangent to te curve y u *, to find te gradient of te function (x). Ten we ave to determine r, to calculate te ape angle α. Figure 8 To determine r, one calculates te equation of te tangent y(x) at x = to get: y y( ) = r ( r ) t. (4) Wit te elp of tis expression and similar to eq. (6), one finds furtermore y = ( r ). (5) Te sum of y r 4 t =, from were one gets: ( t) =. (6) r In te limiting case t 0 one obtains correctly eq. (7). Te linear functions (x) and (x) can now be derived: r r ( x) = t x ( x) = ( t ) x and permit to determine te intersection point r : t (7) r = r t. (8) r 9
Similar to eq. (8), one can now calculate te ape angle: 4r t α =. (9) r It can be easily proved tat te expressions of r and te ape angle α are still valid for te case of negative times t, provided tat t is substituted by t in eqs. (6), (8) and (9). From te condition < t, it follows tat t > for tis case and tus te limiting case t 0 can never be reaced. Case (i) and case () can bot be realised for neutron velocities > v c, wic one applies depending on te time t, wen te neutrons pass troug te central plane of te copper. If < v c, te neutron pats cannot become tangent to y u * at x = anymore. Here we find a new situation: (i) For < v c, we find two linear functions l(x) and m(x) corresponding to te limiting neutron pats (see Fig. 9), wic become tangential to y u * at points r i and r i absolutely smaller tan. Figure 9 First one as to determine te points r i and r i as dribed above, ten te functions l(x) and m(x) can be obtained in a similar way as before to find finally te intersection point r i 3. One gets: r i = r (see eq. (6)), r i = -r (see eq. (7)), and r i i ( r r t ) = (0) i ( r r ) i 3 i 0
i 4r t i α =. () i r 3 We can cec te limiting case t 0: r i, r i - r i and r i 3 0 as it sould. i 4 In consequence, one finds α ( ) for t 0. Tere is anoter point to mention ere, and it applies to all cases (i) (i): te triangles, i i troug wic one determines te ape angles α α, are not oblique. Consequently, one sould calculate te ape angles troug te cosine law. However, since it still olds tat ypotenuse and adjacent side are muc smaller tan te opposite side, one can use te following in very good approximation: opposite side tan α α. adjacent side Furtermore, we used in section (c) above, / = / v / v0, to define te deviation from te optimal velocity v 0, instead of te true neutron velocity v to obtain general formulas for a curved or a straigt Fermi copper. We ave already mentioned, tat te velocity v may range from zero to infinity, wereas can be positive or negative. All equations derived ere are also valid for negative, provided tat is replaced by. We will summarise te results in Table : Table
3. Calculated and simulated neutron burst sapes produced by a realistic Fermi copper Te analytical results will now be applied to te Fermi copper, wic will be realised at te new Extreme Environment Diffractometer (EXED) of te HMI. Wen a Fermi copper is used at a steady state source of neutrons, lie for instance te reactor BEII at te HMI, te neutrons are arriving continuously at te copper, if no oter velocity selector is placed before it. At a given moment te Fermi copper is oriented in a certain direction wit respect to te neutron beam. Te angle between tese two directions will be called te pase angle φ 0 (see Fig. 0) of te copper. Tis is one of te parameters of te Fermi copper module in VITESS. Figure 0 Bot te transmission probability P(,t) and te ape angle α (,t) are independent functions of te neutron velocity and te time t, wen te neutrons travel troug te centre of te copper. However, for a divergent beam, new transmission possibilities exist as te copper rotates independently from t, wic means tat te copper may be by cance oriented in a favourable direction wit respect to te divergent beam (see Fig. ). Altoug te transmission probability is reduced ere for a non-divergent neutron beam, te ape angle may acieve a maximum value, i.e. /, for a divergent beam. Figure To tae tis effect into account, one as to convolute bot functions, te convolution integral aving as limits te minimum and maximum time te copper needs to rotate from te minimum to te maximum divergence angle:
τ max φmax = F( v, t) = P(, t τ) α (, τ) dτ φmin τmin= (3) If one wants now to compare teory and simulated data, one as to be aware tat in te experiment one as only access to te distribution sown by te neutrons after a fligt distance d (see Fig. 0). Tis function will be called A(d,T) as in [4] giving te number of neutrons reacing te plane x = d at te time T: 0 d A ( d, T ) = F( v, t ) dv. (4) v A series of cases will be presented now corresponding to te projected Fermi copper aving two slit pacages at te instrument EXED of te HMI, sowing tat te analytical approac derived above is in agreement wit simulated calculations in various situations. Te real Fermi copper will be situated at.8 m beind te moderators, but for te calculations te Fermi copper will be placed immediately beind te source. Te instrument will be built at a multispectral beam extraction system and tus will ave a view on bot termal and cold moderator. In consequence wavelengts between 0.7 and 0 Ǻ can be acieved ere. At EXED, te neutron pulses will be produced alternatively by a ig speed double disc copper, wic permits to ave a muc iger flux at te cost of resolution, or by te Fermi copper, wic reduces te flux by a factor of about 0, but allows for a muc better resolution. Te Fermi copper will be equipped wit two slit pacages rotating on te same axis. It will be installed on a lift system wic enables to place one of te two slit pacages into te neutron beam or to replace te wole Fermi copper by a guide piece and to operate only te double disc copper. One slit pacage is straigt, te oter one as a curvature radius of 4.4 m, wic corresponds to an optimum wavelengt of.5 Ǻ at 36.000 PM. Te simulation of te instrument, including te HMI source, and bot moderators sows tat te 3
neutron flux as two local maxima in tis guide: one at.4 Ǻ and te oter one at 3.8 Ǻ. Te coice of te curvature radius corresponds to an average wavelengt value, for wic te flux is very ig. Te slit pacages ave a cuboids form wit te dimensions of 6 x 0 x cm³ (widt x eigt x dept) for te straigt pacage and of 6 x 0 x.5 cm³ for te curved pacage. Tese dimensions will also be used for te calculations. Te distance beind te copper considered ere is d = 7.5 cm. Te simulations ave been done wit te instrument simulation program VITESS [8] written by an HMI group. For eac curve 5 million trajectories (= statistical events) were started and some ten tousand of tem typically passed troug te copper. In te following, all transmission probability curves are normalised as to ave te maximum at unity. Fig. sows te case of te straigt Fermi copper rotating at 36.000 PM, neutrons arriving at any time wit a divergence of 0. wit a fixed wavelengt of λ 0 =. Ǻ. For suc a small divergence of te neutron beam and a fixed wavelengt, one finds te typical triangular form of te burst sape (for instance, see te rigt-and side of Fig. 4). Te full widt alf maximum (FWHM) was evaluated to 4 µs. Figure Fig. 3 differs from te previous one by corresponding to a wavelengt bandwidt of 0.7 Ǻ λ.7 Ǻ and as a FWHM of 9.6 µs. Already at a distance of 7.5 cm beind te copper, te curve is enlarged by te different fligt times corresponding to te various wavelengts. Figure 3 Fig. 4 and 5 sow te same Fermi copper and te same wavelengt bandwidt, but te divergence of te incoming neutron beam corresponds now to and 3, respectively. As te instrument EXED will be equipped wit a guide aving coatings up to m = 3 and te 4
multispectral beam extraction system allowing for wavelengts up to 0 Ǻ, divergences up to 3.8 are realistic. Figure 4 Te iger te divergence, te more te smooting effect of te convolution (see eq. (3)) becomes visible. Te curves ave FWHM of 0.6 µs and 8 µs and are also dribed by te teory introduced ere. Figure 5 Finally, two results corresponding to te curved slit pacage at 36.000 PM wit / = 0.04 are sown. Only for tis case, te difference between and v (see eq. ()) becomes important. To avoid unequal integration intervals, one as to integrate over instead of v in eq. (4) taing into account te corresponding transformation factor. Fig. 6 is similar to te case sown in Fig. wit a fixed wavelengt. Due to te dept of te copper of =.5 cm, te FWHM is reduced now to 5.8 µs. Te curved slit pacage will be used to produce sortest neutron bursts, wic mae igest resolution possible at te instrument [9]. Here again one finds te typical triangular form. Figure 6 Fig. 7 finally offers te neutron pulse for a wavelengt bandwidt typically used on EXED of.7 Ǻ λ.5 Ǻ and a divergence of, wic enlarges te curve at a FWHM of 4.4 µs. Figure 7 5
4. Conclusion In tis wor it as been sown ow to calculate analytically te neutron burst sape produced by a rotating Fermi copper under te condition of a divergent incoming neutron beam and a continuous flux in time. Te calculations are compared to Monte Carlo simulation results for various situations and are in good agreement for bot a straigt slit pacage and a curved slit pacage. Te comparison wit experimental data will be presented in a separate publication [0] for te data obtained at HET in ISIS. 6
eferences [] E. Fermi, J. Marsall & L. Marsall, Pys. ev. 7 (947) 93 96. [] T. Brill & H.V. Lictenberger, Pys. ev. 7 (947) 585 590. [3] P.A. Egelstaff, AEE N/ 3 (953) 38. [4] M. Marseguerra & G. Pauli, Nucl. Instr. Met. 4 (959) 40 50. [5] N.H. Tol &.M. Brugger, Nucl. Instr. Met. 8 (960) 03 0,.J. oyston, Nucl. Instr. Met. 30 (964) 84 0. [6] K.H. Becurts & K. Wirtz, Neutron Pysics, Springer Verlag, 964. [7] K.T. Sundset et al., Proc. of te Eigt International Symposium on Magnetic Bearings (ISMB 8), 00, Mito, Ibarai-Pref., Japan. [8] VITESS group: G. Zsigmond, K. Lieutenant, F. Mezei, HMI Berlin, ttp://www.mi.de/projects/ess/vitess/ [9] J. Peters, K. Lieutenant, D. Clemens & F. Mezei, Proc. of te European Powder Diffraction Conference, 004, submitted for publication. [0] J. Peters, G. Zsigmond, H. N. Bordallo, J.D.M. Campion & F. Mezei, to be publised. 7
Figures and Tables Figure : Scematic design of a Fermi copper rotor Figure : Cross sections of te copping surfaces wit te plane (x,y). * y u and * y l indicate te position of te upper and lower surface of te revolving slit at te transit time of a neutron. A nondivergent neutron beam can pass troug te Fermi copper only in te green dased sector. y y u * n x y l * 8
Figure 3: Case of a non-divergent neutron beam and of neutrons wic pass troug te central plane x = 0 of te Fermi copper at a negative time t. (y u * ) min (y l * ) max Figure 4: Transmission probabilities as function of te neutron wavelengt (velocity) for t fixed (ls) at zero and as function of te time at a fixed wavelengt (rs). Te red line corresponds to a straigt pacage and te blue line to a curved Fermi copper pacage. Te copper caracteristics are as follows: = 0.005 m, = 0.0005 m, f = 600 Hz. For bot curves, te value of λ fix =. Ǻ in te rs figure. P(λ, t = 0) P(λfix, t) λ[ǻ] t[s] 9
Figure 5: Divergent neutron beam passing troug te central plane of te Fermi copper at t = 0 wit a maximum ape angle. α Figure 6: Divergent neutron beam passing troug te central plane of te Fermi copper at t = 0 wit a velocity < v c and a reduced ape angle. 0 y r x y 4r y Figure 7: Divergent neutron beam passing troug te central plane of te Fermi copper at t > 0. f(x) α ι α ι g(x) r i 0
Figure 8: Divergent neutron beam passing troug te centre of te Fermi copper at t 0 wit a velocity > v c and a reduced ape angle. α (x) (x) y 4r t y t r r Figure 9: Divergent neutron beam wit velocity < v c passing troug te central plane of te Fermi copper at t > 0. α i l(x) m(x) r i r i 3 r i
Figure 0: Passage of a neutron beam troug te Fermi copper, wic is rotated by te angle φ 0 wit respect to te beam. moderator d φ 0 Figure : Divergent neutron beam passing troug te Fermi copper. If te copper is parallel to te beam at t = 0 for non-divergent straigt neutrons, it cannot be simultaneously parallel to te divergent neutrons at t = 0. α =/
Figure : Calculated (straigt line) and simulated (scatter points) time distribution curve at a distance d after te passage troug a straigt Fermi copper. Here λ 0 =. Ǻ, te divergence = 0.. Transmission probability,0 0,8 0,6 0,4 0, 0,0-0 0 0 40 60 80 t [µs] Figure 3: Straigt Fermi copper rotating at 36.000 PM. Here 0.7 Ǻ λ.7 Ǻ, te divergence = 0.. Transmission probability,0 0,8 0,6 0,4 0, 0,0-0 0 0 40 60 80 t [µs] 3
Figure 4: Straigt Fermi copper rotating at 36.000 PM. Here 0.7 Ǻ λ.7 Ǻ, te divergence =.,0 Transmission probability 0,8 0,6 0,4 0, 0,0-40 -0 0 0 40 60 80 t [µs] Figure 5: Straigt Fermi copper rotating at 36.000 PM. Here 0.7 Ǻ λ.7 Ǻ, te divergence = 3.,0 Transmission probability 0,8 0,6 0,4 0, 0,0-40 -0 0 0 40 60 80 t [µs] 4
Figure 6: Curved Fermi copper rotating at 36.000 PM. Here λ 0 =. Ǻ, te divergence = 0..,0 Transmission probability 0,8 0,6 0,4 0, 0,0-40 -0 0 0 40 60 80 t [µs] Figure 7: Curved Fermi copper rotating at 36.000 PM. Here.7 Ǻ λ.5 Ǻ, te divergence =.,0 Transmission probability 0,8 0,6 0,4 0, 0,0-40 -0 0 0 40 60 80 t [µs] 5
6 Table : Equations () for P(,t) and α (,t) wit te corresponding and t ranges for a divergent neutron beam. P(,t) ), ( t α t v m 0 0 0 t m v m v 4 0 t / ) ( 4 α 0 r t r i i i = 3 4 α t t 4 0 4 v m 4 0 t t v m / ) ( 4 / α α 0 i i r = α r t r = 4 α t v m t v t m 0 v t m 0 m v m t v