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A03.$ 'I 'S ' ,;7 ;\ : * REFLECTION SPECTRA OF NEUTRONS FROM CRYSTALS by M. Ginsburg M. L. Goldberger F. Seitz Argonne National Laboratory SUiqIV OF LLI SDOCJMENTS DEPT US DEPOSITORY This document consists of 9 pages. Date of Manuscript: February 8, 1945 Date Declassified: June 24, 1947 This document is issued for official use. Its issuance does not constitute authority to declassify copies or versions of the same or similar content and title and by the same authorss. Technical Information Division, Oak Ridge Directed Operations Oak Ridge, Tennessee . r. MDDC 1035 UNITED STATES ATOMIC ENERGY COMMISSION  ii REFLECTION SPECTRA OF NEUTRONS FROM CRYSTALS By M. Ginsburg, M. L. Goldberger, and F. Seitz STRODDUCTION Experiments on the distribution of Bragg reflected neutrons have been performed by Messrs. W:Sin and Sturm at Argonne and Messrs. Wollan and Borst at Clinton. In each case a collimated beam :'of neutrons was allowed to impinge upon a crystal and the reflected intensity was measured. In one !:ase to be described herein, the source of neutrons was a graphite thermal column; in the others, Radiation taken directly from the pile was used.We have attempted to explain the observed results aon the basis of the theory developed by us in reports CP2419 and CP2512. GENERAL CONSIDERATIONS The expression for the reflected intensity from a crystallographic plan characterized by the generalized Miller indices HHHL is [CP2419, equation 45] na k2 2 kM2mkeT k2 sin2 e 2mkoT where S/ar sJ iKH. r (2) aH = aH,HHs = 4nno 4 e k = 2z/A, A is the neutron wave length, KH = 2 7 (H,bi + H2,b + Hb3), bi is the reciprocal lattice trans lational vector, 6 is the Bragg angle, ko is Boltzmann's constant, T is the average neutron tempera ture in degrees Kelvin, m is the neutron mass, no is the number of unit cells per unit volume, F) is the position vector to jth atom in the unit cell. k I K, and 0 are related by the Bragg equation which can readily be shown to be KH= KH =2k sin (3) It is to be noted that the quantity aH is just 4 no times the conventional expression for the Xray structure factor in which the atomic scattering power fj has been replaced by + The question of the ambiguity of algebraic sign entering here will be taken up later. Equation 1 must be modified to take into account the sensitivity of the detector. BF3 counters are . generally used in neutron experiments. On the basis of arguments presented in CP2512, we shall as sume that the detector sensitivity for the entire energy spectrum of the neutron beam is proportional MDDC 1035 [ 1 l2b4pTaf9.p 2 MDDC 1035 Ak to 1/v where v is the neutron velocity. Using the wellknown relation v = and combining m unimportant constants, we may write equation I as C aH k' k2mkoT (4) RH k.k' sins e 1012 where aH = a4 Using equation 3, this may be written as 4noj141 I 2 o csc2 9 RH =C a4 1H csc3 e (5) 2 fK7 where I = 8mkT H 8mkoT It is important to note in connection with equations 1, 2 and 5 that a different value of aH and KH must be used for every plane with different indices (H,,H2,H,). Stated otherwise, different values must be used for every "order" of reflection. There is ambiguity about the meaning of the term "order". The source of the ambiguity is as follows: It is customary among crystallographers to choose a large cubic unit cell in preference to the true translational unit cell when treating facecentered cubic crystals. We shall designate planes in the large cell by (HI,H2,H3), those in the true cell by (K,,K,,K,). It is found that only half of the values of (H1,H2,H,) can be associated with values of (K1,K,,Ks). Con sequently, since reflections are permitted only for planes corresponding to the indices (Ki,K,,K), it will be found that certain reflections will be forbidden when the (H,,H,,H3) designation is employed. For example, in reflecting from planes parallel to a cleavage surface in NaCl the only permitted re flections correspond to the indices (200), (400), (600), when (HI,H2,H,) are used. The ambiguity can be resolved by using (H,,H2,H) indices consistently and recognizing that reflections will always be absent for certain planes. It is convenient, however, to write H, = nh, H, = nh, H, = nh, where (h,,h,h3) are the Miller indices for the reflecting plane and the integer n, which will be called the "order" of the (h,,h2,h,) reflection, take successive integer values. Bragg's equation then becomes nA = 2 d dh ,hsin ao where d =h, hh,,h3 hh h, in which ao is the length of an edge of the cubic cell. The integer n takes only even values for the (100) planes. It is readily seen that S2 n H dh 12 b4p2ba MDDC 1035 [3 We shall rewrite 5 to show explicitly the dependence of order: a csc2 0 R = C a nK, csc3 e (6) n J We shall fix the constant C by the demand that the maximum of the first order reflectivity be unity. This condition yields C 3 e 3 la K, ( )(/2 The positions of the maxima are given by n = sinan The final form for the reflectivity which was used in the numerical computations is nas es/h a 2 csc 92 Snai;A e IanI anS 6 R a 3 a csc I e DETERMINATION OF an It has been pointed out previously that there is an ambiguity of algebraic sign appearing in the definition of an. This sign cannot be predicted with any degree of accuracy theoretically because of inadequate knowledge of nuclear energy levels. The only methods at present for determining these signs are by total reflection_experiments and measurements of the intensities of reflection spectra. We shall use a combination of these methods in resolving the problems treated in this report. The spectra to be analyzed are from CaCO3, LiF, and NaCl. (i) CaCO,.From total reflection experiments performed at Argonne, it is known that graphite has an index of refraction for neutrons of less than unity. This implies that the sign preceding the carbon cross section is negative. The signs for calcium and oxygen alone must be determined. The I all's for the first five orders taking all possible sign permutations are given in the following tabulation: p 13IBpShn MDDC 1035 1st 2nd 3rd 4th 5th (1) 2.971 1.029 2.971 12.971 2.971 (2) 3.353 7.353 3.353 6.647 3.353 (3) 1.029 2.971 1.029 11.029 1.029 (4) 7.353 3.353 7.353 17.353 7.353 The numerical values used were a Ca = 10 b, aC = 4.8 b, aO = 4.0 b. The positional coordinates of the atoms in the unit cell may be found in Compton and Allison, XRays in Theory and Experiment, p. 382. The cleavage plane in the unit cell described there is the 211 plane; the interplanar distance dh is 3.02904 A. Other numerical values used in computing the spectrum from equation 9 are an2 = 0.08564 n2, for T = 300"K, an2 = 0.06761 nr for T = 380*K. Referring to the experimental curve shown in Figure 1, we see that rows (2) and (3) in the above table represent impossible sign permutations, since they predict that the second order reflection should be larger than the first order. We are left therefore with (1) and (4) which correspond respec tively to positive phases for Ca and O, and negative phases for these elements. The reflectivities for the first five orders for these two possibilities were computed and added. The resulting curves when compared to the experimental curve indicated that case (1), that of positive phase for Ca and O, is the more probable. This point could be settled by a total reflection experiment with calcite, since the index of refrac tion will be determined primarily by the phase of the oxygen atoms in CaCO3, and the phase of calcium must be the same as that of oxygen. This experiment was performed with the result that calcite is not totally reflecting; the conclusion is that the phase of the oxygen and hence of calcium is positive, as was deduced above. (ii) and (iii) LiF and NaCl.The determination of Iani for both of these crystals may be discussed simultaneously because they are both of facecentered cubic structure. We consider reflections from the 100 or cleavage plane for which lan = 2(1 + cos nn)(f, + f,) where fj = /sJ. We notice that an is zero when n is an odd integer and takes equal nonvanishing values when n is an even integer. Thus we do not need to know the scattering cross sections in the cases of LiF and NaCl and an may be absorbed in the constant C of equation 9. THE REFLECTION SPECTRA (i) CaCOsThermal column radiation.The reflectivity computed with positive phases for Ca and O has been plotted for two temperatures in Figure 1. There is some uncertainty about the neutron temperature, although it is reasonable to expect that neutrons at the end of the thermal column are at room temperature. The maxima of the theoretical curve can be made to coincide with the experimental maximum by choosing a temperature of 340 K; however, this does not explain the important points of disagreement which will be discussed later. CaCO3Side hole radiation (Figure 2).The agreement between theory and experiment is quite good in this case. The average temperature of the neutrons is 400*K, a value which checks quite well with velocity measurements made by pyrex plate transmission. The discrepancy at the extreme high energy side will be explained below. (ii) LiF Side hole radiation (Figure 3).This crystal has particular interest since the nuclei occurring in it possess a spin, in contrast with the case of calcite. The agreement here is again quite good at the high energy end (except at very high energies) whereas there seem to be fewer low energy neutrons in the beam distribution than would be predicted by a Maxwellian distribution. The good agreement in this case indicates that the spin dependence of nuclear forces which has not been taken *.2644b.u J MDDC 1035 40 50 GLANCING ANGLE DEGREESI  Figure 1. Reflection spectrum from calcite (thermal column radiation). a~~' [ [  THEORETICAL CURVE (T= 400'K) o 0 EXPERIMENTAL CURVE 0.8 o 0.5 ______ ___ 0.6 o, o 0  0.4 02 0 I IJ I I I I I I L 40 50 GLANCING ANGLE (DEGREES) Figure 2. Reflection spectrum from calcite (side hole radiation). "u"4p. 4 a I i I I 0 2 4 6 10 6 ] MDDC 1035 into account explicitly in deriving equation 1 does not cause appreciable incoherent scattering in these diffraction experiments. (iii) NaC1Side hole radiation (Figure 5).The NaCI spectrum was measured by Mr. Wollan at Clinton, whereas the previous work was done by Messrs. Zinn and Sturm at Argonne. In addition to the resultant curve we have_ included a breakdown into orders (Figure 4). From this curve we see that to angles of about 11" the reflected spectrum consists of only first order reflection. The agreement shown in Figure 5 is extremely good, again indicating that the nuclear spin does not cause appreciable inco herent scattering. The temperature at the maximum is 548"K. DISCUSSION OF RESULTS The greatest source of error in the theoretical calculations arises from the uncertainty in the energy distribution of the incident beam. The disagreement between the theoretical and experimental curves shown in Figure 1 is most pronounced at two places: extremely low energies and energies slightly less than the most probable as computed from a Maxwellian distribution. The most .reasonable explanation of these discrepancies is that the energy distribution is not exactly Maxwellian, but is distorted on the low energy side of the maximum. The experimental results could be explained if we assumed that there is an excess of extremely cold neutrons and a deficit of those having energies somewhat less than the most probable value. This would explain the minimum at 25" as well as the slight rise from 350 to 500. Such a distribution would not be entirely unreasonable since it is known that very slow neutrons diffuse through great lengths of graphite without suffering many collisions and hence do not come into thermal equilibrium with the graphite. This effect is sometimes called the crys tal effect. The dip in the proposed distribution could be the result of a preferred absorption of neutrons having an energy near koT. No attempt has been made at the present time to derive a corrected energy distribution. The principal discrepancies in the remaining spectra can be explained as follows: The neutron distribution at high energies is no longer Maxwellian but is dE/E. This distribution is discussed by S. M. Dancoff in CP2171. SUGGESTION FOR FUTURE EXPERIMENT The fact that Ianl depends upon f, + f, for reflections from the (100) planes of crystals having the NaCI structure, whereas an depends upon f, f, for the (111) reflections, indicates that measured values of the reflectivity from the two types of planes for a given wave length can be used to determine the ratio f, f,. It is important to note that the quantities required to determine this ratio are not the absolute reflectivities but the relative reflectivities for a given incident beam. Since the reflectivity of a crystal is a function of its perfection, it evidently is almost necessary that the same crystal be used to determine the relative reflectivity from the two types of crystallographic planes. If the elements in the crystals are monoisotopic, fL and f, are proportional to the square roots of the scattering cross sections so that this ratio is given at once. Thus if one of the elements possesses isotopes in compara ble abundance since then fi is proportional to + gBat where a is summed over all the isotopes a and ga is the fractional abundance of the ath isotope. The principles involved in the foregoing method of determining the relative scattering cross section of the elements in a diffracting crystal are obviously not restricted to crystals having an NaCI structure, but can be used whenever different crystallographic planes involve the fi in independent ways. If the flux of the incident beam, the dimensions of the beam, the dimensions of counter chambers, and the counter efficiency, and the magnitudes of the scattering cross sections for a given crystal were known, it should be possible to predict from equation 1 the actual number of counts recorded by the. counters for every sign permutation in lanl and hence one would theoretically be able to determine the signs from a single measurement. s*Ist. p u MDDC 1035 4.0  THEORETICAL CURVE(T=400"K) 0o0oEXPERIMENTAL CURVE 0.  "  o 0 0.4  0.  04 1  o 1 I 00 0 2 .6 8 0 12 14 16 0 22 24 P" R O2 3 40 4 46 48 50 s2 54 56 56 60 62 64 66 68 70 22 74 76 GLANCING ANGLE (DEGREES)  Figure 3. Reflection spectrum from LIF (side hole radiation). 0,9 I 1 I I/ \ I F1 / 2 0.0 0.7 0. 0A o.3 4 0.1 6 01 "zzj^* SI I . I 0 2 4 6 8 40 I 40 50 GLANCING ANGLE (DEGREES) 70 an Figure 4. Reflection spectrum from NaCI (resolution of orders). I.Isls1 8 I MDDC 1035 4.0 _ I .0  THEORETICAL CURVEIT54.S K Saoo EXPERIMENTAL CURVE 0.9 0.7 I 0.4 _,_ _. 03 0.5 0.4 ______________________ 0 10 20 30 40 0o 60 70 so80 GLANCING ANGLE (DEGREES)  Figure 5. Rerlection spectrum from NaCI (side hole radiation). 0 0.8 _ I 0.5 0. I \ 0.4 D' . 40 50 GLANCING ANGLE iIN DEGREES) Figure 1. Reflkrlion spectrum from calkte (resolution of orders; side hole radiation). MDDC 1035 SUMMARY (1) The recent experiments at Argonne and Clinton on the reflection spectra from crystals are discussed. A theoretical treatment based on reports CP2419 and CP2512 is given. (2) The reflected intensity from a crystallographic plane characterized by the Miller indices (hh,hs) for a given order, n, is given by a 2 csc2 0 Rn = C an nKI csc Oe n 62 where K, = a 2= 2 , dh n 8mkoT an i 2 Tn(h,xj + hyj + hzj) a = + (b) e where C is an arbitrary constant, dh is the interplanar spacing, A is Dirac's constant, m is the neutron mass, ko is Boltzman's constant, T is the average neutron temperature, asj (b) is the scattering cross section of the jth atom in barns, and (xj,yj, zj) are the coordinates of the jth atom in the unit cell. (3) The question of the ambiguity of algebraic sign appearing in the definition of an is discussed and it is shown how by combining the results of reflection spectra and total reflection data this ambiguity may be resolved. This problem is discussed in detail for CaCO,. (4) The effect of nuclear spin in producing incoherent scattering is considered. The good agree ment between the theoretical and experimental curves in the cases of LiF and NaCl indicates that this effect is negligible since the derivation of the expression for Rn neglects spin effects. (5) On the basis of the reflection spectrum from calcite obtained with radiation from a graphite thermal column an energy distribution for the neutron beam'has been proposed. It is believed that the energy distribution is not exactly Maxwellian, but is distorted on the low energy side of the maximum. The experimental results could be explained if we assumed that there is an excess of very cold neu trons and deficit in those having energies somewhat less than the most probable. The energy distribution in the neutron beam coming directly from the pile is Maxwellian over the entire range with the exception of a high energy tail. This is explained by the fact that the high energy distribution is given by dE/E where E is the neutron energy. (6) An experimental method for determining the scattering cross sections of certain elements by the use of reflection spectra data is proposed. This method is feasible whenever the an's for different planes involve the scattering cross sections in independent ways. I lB4pBnonUrita UNIVERSITY OF FLORIDA HII IHIHlllHi1111I111111ll11111 N IIIIl lMi IfE 3 1262 08909 7918 .:. . S . .. ". 1" Si ! .. L:. I ... ti , 4N1(t . 
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