Nuclear resonance photon scattering from 15N in the form of BN and NH4Cl

Nuclear Phys/cs A339 (1980) 157-166; © NordIrEollarad Pr6tWiGtp Co., dterterdant

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NUCLEAR RESONANCE PHOTON SCATTERING FROM 'sN IN THE FORM OF BN AND NH + CI O. SHAHAL, R. MOREN fand M . PAZI

Nuclear Research Center-Neveu, Beer Sheoa, Israel and Ben-Gorton Unitxrsity o( the Neveu, Beer Sheoa, Israel Received 7 May 1979 (Revised 30 October 1979)

Ahtraet : The temperature dependen~x of We resonance scattering cross section o, of the 6.324 MeV photons from '°N in the form of BN and NH,CI was measured in the range 78K-640K. Using these data the effective temperatures T, of the N-atom in the two compounds as a function of the temperature T was deduced. At T = 293K we obtained T. = 561 í28K, and 521 í57K for BN

and NH 4CI respectively. The values of T, and the variation versus T were reproduced theoretically by considering the Doppler broadening of the nuclear level, caused by the motion of the N-atom, and described by the frequency distribution function of the lattice in the two compounds. The effort of the values of T, 'on We determination of the radiative width of nuclear levels is discussed .

E

NUCLEAR REACTION ' SN(y, y), E = 6.324 MeV ; measured temperature dependence of a . Deduced effective temperatures . Natural BN, enriched NH,CI targets.

1. Introdnctioto In measuring the widths of nuclear levels using the method of resonance fluorescence ' -'.~, it is usually neces.~ary to know the effective temperature T~ of the scattering nucleus' . The quantity Te appears in the expression of the Doppler width d = E(2k T~lMc 2)~ where E is the excitation energy of the nuclear level, and M is the nuclear mass. An accurate value of T~ is. very important for obtaining precise values of level widths. This statement is true not only for the resonance scattering measurements of the type discussed in the present work s ~ s) but it is even more so for the case of resonance fluorescence measurements using bremsstrahlung ~~ 9). In this latter case, it is sometimes possible to determine the ground-state radiation width I'o by a self-absorption measurement if Te is precisely known. Further, the use of incorrect values of Te may lead to deviations of ~ 25 ~ in the determination of l'a in light nuclei as shown recently by Moreh et al. 2~ in the case of ''B. f Present address : Physics Department, University of Illinois at Urbana-Champaign.

157

15 8

O . SHAHAL et al.

The notion of effective temperature was first introduced by Lamb 1 ~ to take into account the quantum mechanical behavior of the motion of an atom in a metallic solid. Thus; instead of the classical ~ = kT for the average energy per mode of vibration for a classical oscillator in a solid, he employed the quantum mechanical expression . Further, he used the Debye approximation which assumes a v2 dependence of the frequency distribution together with an upper cutoff frequency vD related to the characteristic Debye temperature Bp, thus yielding es-1 2 a From eq. (1) it may be seen that in general Te > T. The deviation of Ta from T is caused essentially by the contribution of the second term in the integrand which expresses the contribution of the zero-point energy of the quantum oscillators to the atomic motion and increases the effective velocity of the atom. For scattering from targets other than metals namely in the form of chemical compounds; diatomic gases or polyatomic gases, the Lamb treatment is no longer true because of the contribution of discrete frequencies arising from the vibrational normal modes of the molecule'" s). It was only recently that Moreh et al.') studied this problem and defined the effective temperature for more complicated chemical compounds. To do that it was necessary to consider the high molecular vibrational frequencies ofthe scattering atoms (taken from infrared spectroscopic data) to account for the "internal" degrees of freedom. The "external" degrees of freedom of the scattering atom was taken from specifc heat data . The contribution of the optical frequencies to the kineticenergy of the atom is usually very large and may consitute the dominant part of T~. Using this procedure, the T~ of the N-atom in ' SN2 and Li'sN03 was defined' " s) and an excellent agreement with measured values of the scattering cross section (dependent on Doppler broadening) was obtained. In the present work, the resonance scattering cross section of photons from the 6.324 MeV level e-s) of 1sN in the chemical forms NH4C1 and BN, was measured as a function oftemperature. Theexperimental behavior was reproduced theoretically by defining an effective temperature T~ using the corresponding phonon distribution frequencies of the two compounds. The method used here for defining Te is essentially more geIIeral than that used') in LiN0 3 and illustrates some new features ofthe problem not mentioned earlier. It can be seen (sect. 4) that the direct use of specific heat data for deducing Te can be misleading for targets in a form other than a pure metallic element, causing large errors in the determination of l'o. Bp

2. ExperimenW procedure As mentioned above, we deal here with a resonance photon scattering process in which the 6.324 MeV level of l'N is photoexcited by a chance overlap to within - 30 eV with one of the y-lines of the Cr(n, y) reaction . The incident photon beam

1 sN~Y. Y)

159

was thus generated from the (n, y) reaction on three separated chromium discs, 7.5 cm in diameter and 1 .5 cm thick. The discs were placed along a horizontal tangential beam tube near the core of the IRR-2 reactor. The 6.324 MeV y-line arises from the s3Cr(n, y) reaction and it is one of the weak lines occurring in the y-spectrum of the Cr(n, y) reaction.ll). Its intensity in the incident photon beam is about 5 x 10`/cmz ~ s on the target position . The BN target consisted of polycrystalline powder of hexagonal structure resembling that of graphite, having 4.5 cm diem and 8 g/cmZ thick and natural composition (0.36 % 1 sN) . The powdered 1 sNH4C1 sampler enriched to 99 .5 ~ 1 sN, has 4 .5 cm diem and 0.54 g/cm 2 thick. The scattered radiation was detected using a 12.5 x 12.5 cm s NaI crystal . For measuring the non-resonant background a target of C(graphite) was used (for simulating the atomic effects of B and N) and a natural NH4Cl target for the other samples. The temperature variation of the scattering cross section of the sample was carried out by using a liquid nitrogen cryostat for cooling down to 78K and an electric heater for raising the temperature up to ~ 600K . Details of the experimental system may be found elsewhere a. i z), The scattering cross section of the 6.324 MeV y-line from the two targets at room temperature was obtained by measuring the cross section ratio relative to an enriched Li1sN03 (99.3 ~) target . The absolute cross section for the latter target was determined by measuring the intensity of the incident 6.324 MeV photons (in the direct beam) and the scattered photons from the enriched Li1 sN03 target under similar geometrical conditions 6" '). 3. T`eoredçal remarhs In the following we discuss the procedure used for extracting Te from scattering cross-section data and for calculating it using the phonon spectrum of both BN and NH4C1. In doing so we shall rely heavily on the discussion and the definitions given in a previous publication'). We will first write the relation between the measured scattering cross section and the effective temperature Te of the scatterer. We are dealing with the resonance scattering of a y-line arising from the Cr(n, y) reaction sect the 6:324 MeV level in 1 sN. The overlap between the broadened shapes of the incident y-line (corrected for recoil energy) and the resonance level occurs through the tails of the two lines whose peak energies E, sect Eo are separated by 8 = ~ Eo-,F.,I = 29 .5 eV [ace fig. 1 of ref.')]. The cross section for such a resonance scattering event from a very thin target is given by 2 " 3) (2) = ao ~o(x~ t). where Q o = 2><.t~ g(l'oll~ = 122.3 b is the peak cross section without the Doppler broadening; I'o = T = 2.9f0.2 eV is the radiative width') of the 6.324 MeV level Q.

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O . SHAHAL er al.

in 1 sN whose spin J = ~ and g = 2. The function ~o accounts for the effect of the thermal and the zero-point vibrational motion of the nuclei ofboth the scattererand the y-source, and its given by 1

x° =_

/'m

o -2b E,~ ___~

r

r

e

-(s -s)2/4t

dó+d ; to =_-r2

and d° _ (E°lcx2kT~/M°)}, d, _ (E,lcx2kT,lM,)~ are the Doppler widths of the incident y-line and the resonance level respectively ; M° is the mass of the isN scatterer ; M, is the mass of the emitting nucleus s4Cr in the y-source ; T, is the effective temperature of the Cr y-source, and T is the effective temperature of the N-atom in the scatterer. The Cr source was in a solid metallic form ; its temperature during reactor operation was T = 650K. Using a Debye temperature of Bp = 460K and employing eq . (1), one fords T, = 666K from which a value d, = 9.6 eV is obtained Our procedure was to measure rr, versus T for 1 sN from BN and NH4C1 then extract T, of the N-atom using the Q, versus T~ curve (fig . 4). These experimental values of T~ were then compared with calculated values . To calculate T, we note that it should reflect the average kinetic energy of the atom per degree of freedom (including the zero-point kinetic energy) As mentioned in sect . 1, the value of T~ for isNZ gas and LiN0 3 was calculated elsewhere') and the effect of the normal modes of vibrations of NZ and N03 molecules were taken into account. Generalizing this procedure i; may be expressed using the frequency distribution function g(v), namely the phonon spectrum, as follows

where g(v) is normalized so that ~~ g(v~v = 3nN, 0

(6)

where nN is the total number of atoms in the lattice containing N unit cells with n atoms per unit cell of which only a single atom is a nuclear scatterer. S(v) is the fraction of the kinetic energy shared by the N-atom in the molecule'). The frequency distribution g(v) is taken ifavailable from solid state litterature 1 `~ i s), It is extracted from data on inelasitc scattering of cold neutrons and infra-red spectroscopy using the dynamic equations of motion of the atoms in the lattice.

i

161

eN~Y, Y)

This matrix is then diagonalized ; the eigenvalues and the eigenvectors are then used for calculating S(v) and g(v). Details of this procedure may be found in the literature on lattice dynamics' 6-18 ). It should be remarked that for highly oriented compounds such as HN (see below) one should define III and as the average kinetic energy of the N-atom parallel and perpendicular to the hexagonal planes. In that case, ~ would be given by

is

e = s(El+2ell)

and the analogous quantities gll(v) and g l(v) would be normalized as follows : ~~ ~41(v)+2911(v)~dv = 3nN. 0

(6~)

3 .1 . EFFECTIVE TEMPERATURE OF N IN BN

In the litterature, no neutron scattering data is available on BN, and hence no calculation of the frequency distribution spectrum g(v) has been reported yet. This is due to the high n-absorption cross section of l °H in the thermal region . Instead, we made use of the distribution function g(v) reported by Nicklow et al. ' a) for graphite. This procedure is justified by noting that C in graphite form has the same crystallographic structure as the BN sample used here, namely hexagonal planar layers. The vibrational energies .in both BN and graphite are expected to be close to each other because the mass of the C-atom is medium between that of the N and B atoms. In addition, the lattice constants in both cases are again close to each other and so are the . spring constants between neighboring atoms. Perhaps the best evidence for the similarity .of the two spectral distributions may be found in ref. zl) where the ratio of the nth order experimental moments of g(v) for both pyrolytic graphite and pyrolytic BN were obtained and found to be nearly constant for all values of n between n = - 2 to n = + 6.

z a a_

m

m a â ó~ FRE~UNCY (10'Z CPs) Fig. 1 . The solid line shown the 6~equeacy~istribution function for~graphite C taren from ref. "), while the dashed curve shows ~ ""similar' " frequency distribution-function for BN .

162

O. SHAHAL et a1.

In order to account for the different atomic masses in BN and graphite, a scaling factor, to be determined empirically was applied to the function g(v) of graphite taken from ref. t4) . Thus, we assumed a frequency distribution spectrum g(av) for BN and the value of a was determined in such a way as to yield a best fit between the measured and calculated results of T~ versus T (see sect . 4). It was also assumed that the kinetic energy of B and N, in each normal mode, is distributed according to their mass ratio. Thus, the value of a in g(av) was found M be a = 0.82. Fig. 1 depicts the two functions g(v) for graphite taken from Nicklow et al. t4) and thé extracted function g(av) for BN respectively . The value of Te for each T may then be calculated by numerical integration using eq . (5) and fig. 1 . It must be remarked that the above procedure should be treated with some reserve as mentioned in more detail in sect. 4. 3.2 . EFFECTIVE TEMPERATURE OF N 1N NHaCI

The treatment of NH4C1 is somewhat similar to LiN0 3 in the sense that the principal contribution to the kinetic energy of the N-atom arises from the NH4 molecular ion. The NH4 is strongly bound with its normal frequencies almost independent of the particular compound such as NH 4N03, NH 4C1, NH 4Br, etc. Hence, we distinguish here between the "internal" vibrational modes of NH4 and the "external" vibrational modes with respect to the Cl - ion and the other surrounding molecules. 3.2.1 . The "internal" vibrational modes . The kinetic energy of the N-atom due to the internal vibrational modes may be obtained by considering the structure ofthe NH4 ion which is known to be a tetrahedron with the N-atom at the center t a), The N-H distance is 1 .06 Á and the H-H distance is 1 .682 A. This ion has nine normal frequencies, most of which are degenerate due to the symmetry of the molecule . The experimental normal frequencies taken from the literature t3) are given in table 1 for t4NH4. These values were used for calculating the force constants of NH4 using standard methods of molecular spectroscopy t s) as described in some detail in a previous work'). The same force constants were used for calculating the normal frequencies for the t sNH4 molecule and are given in table 1 . The table TeatE 1 Normal frequenciea of' 4NH4 ions, the calculated frequencies of °NH4 (in units of cm'' and 10'= cps), and the fraction St of the kinetic energy of the N-atom Frequency (degeneracy) vi(1) v~(2) v3(3) va(3)

is NH4

~s NH4

3033 1685 3134 1397

(~ -')

(~ -`)

is~4

(10"~)

S~is Nl

3033 1685 3127 1390

90 .93 50 .52 97 .75 41 .67

0 0 0.064 0.147

16 3

~ sN(Y. Y)

lists also the calculated fraction S~ of the kinetic energy shared by the N-atom in t sNH+ a" The force constant characterising the vibrational motion of NH 4 are (i) KN and Ktar related to the stretching forces of the N-H and H-H bonds; (ü) Kd/!2 related to the bending force stabilizing the H-N-H angle, where l is the N-H distance and (iii) Ka/r 2 related to the bending force stabilizing the H-H-H angle, where r is the H-H distance . The values of the force constants obtained by the standard procedure are : Kmt = 0.184 N ~ cm-t, K,e = 4.714 N ~ cm - ', Kó/l2 = 0 .358 N ~ cm- ',

Ka/r~ = 0.229 N ~ cm-t .

3.2.2. The "external" vibrations! modes. The contribution of the external modes to

the kinetic energy of the N-atom in NH 4Cl was obtained using the phonon spectrum of ND4 C1 reported by Cowley ts) . We will not enter into the details of the model employed for this calculation which was briefly outlined earlier. It is enough to mention that Cowley has obtained 22 parameters out of which 17 parameters Z 0.9

(b1

w

(a)

n

Is NI~CI

p 0.6 0.4 a ;~ 0.2

Q

10

Q

5

F_

I~ 1

1 1 P

I

2

3

4

5

6

7

8

9

10 11

FREQUENCY v(10 1=CPS)

12

Fig. 2. (a) Frequency distribution function g(v) for'sNHaCI calculated using the parameters ofCowely' °) . (b) The calculated fraction S(v) of the Itinedc energy shared by '°N in'sNH4C1.

related to the force constants and the remaining five, related to the polarizability ofthe ions. We used these parameters for calculating the dynamics! matrix and only interchanged the isotopic mass of the D with that of H and themass of t `N with t sN. Our results of g(v) and S(v) are given in graphical form in fig. 2. The actual kinetic energy of the N-atom is to a good approximation the sum of the internal aad external contributions .

16 4

O. SHAHAL er a!. 4. Results and diac~sion

The relative scattering intensities from a BN and NH4 C1 scatterers as a function of temperature is given in table 2. The scattering yield is normalized to T = 293K. Teat~ 2

Measured relative scattering intensities /r~l~sa of the 6.324 MeV photons from "N at various temperatures NH,CI 7~K) 78 293 343 364 374 407 461 513

BN

h/hq~

T,(~ 10 ~)

0.754 1 .000 1.059 1 .096 1 .104 1 .140 1 .193 1 .226

433 571 605 627 632 654 687 708

~K) 78 293 348 460 558 663

IT//393

0.928 1 .000 1 .047 1 .123 1 .202 1 .295

Ta(~ 5 ~) 522 561 589 633 682 740

The experimental value of T~ was deduced using fig. 4. T(~K) 2 .5

700 600

2A

500

L5

é 400 F~

2 .5

700 600

S

é

2 .0

500 400

100

500 300 T(~K)

Fig. 3. Absolute scattering aoss sections a, of the 6.324 MeV photons from "N in the form of BN and NH 4CL The scale of T, is also given. The solid linen represent theoretical values of T, (and hence of a~ obtained u~ing the fimdion g(v) given in figs. 1 and 2 respectively .

The data points of fig. 3 are the measured values of the absolute scattering cross section v, as a fundión of T for BN and NH 4C1 . This was obtained as explained in sect. 2 by measuring the ratio of the scattering intensity relative to that of a Li t sN03 target for which the absolute scattering cross section is well known 6 " '). For each value of Q the effective temperature of the tsN atom was then found

i

16 5

sN(Y. Y)

3 . E 0 ó 2 b~

zoo

goo

coo

soc soc Te

goo

eoo

soc

Fig. 4. Calculated scattering cross section o. of the 6.324 MeV photons from 1 sN as a function of T, as obtained using «l. (2) wiW the parameters : b ~ 29.5 eV, f ~ To = 2.9 eV, d, = 9.6 eV, taken from ref.').

using the calculated curve of .fig. 4. This is a plot of a, as a function of Te obtained using the resonance scattering parameters of the 6.324 MeV, Cr(n, y), line scattered from tsN. As mentioned earlier, the experim~tal values of Te (table 2) are very useful in resonance fluorescence measurements for deducing correct values of l'o. 4.1 . TIC HN TARGET

The solid curve of fig. 3 represents the calculated values of Te versus T for BN. It shows an excellent agreem~t with. the experimental points and justifies the model by which the effective temperature of N in BN was calculated. It is very important to note that the above calculation reproduced not only the correct absolute values of Te but also its variation with T over a wide çnergy range of dT ~ 600K . Nevertheless, it should be emphasized that the above ranarkable agreement does not necessarily imply that the details of the phonon spectra of HN and graphite are the same nor that they are really related by a multiplicative factor. This is because Q. is an integral over the entire phonon spectrum and therefore can't be sensitive to the details of the actual phonon spectrum . From fig. 3, it is possible to extract the zero-point kinetic energy of vibration of t sN by extrapolating the BN results to T = 0. This is identical to the effective temperature of t sN. in B' SN at T = OK: it was found to be : Te = 520 f 14K, at T = OK implying a Debye temperature of BD = (1387 f 27)K at T = OK, to be compared with BD = 598K obtained from specific heat data t ~. The large difference in Bp is due to the fact that the specific heat is sensitive mainly to the low frequencies appearing in g(v) because at the relatively low temperatures at which specific heats are measured the high frequencies are not excited and heave cannot contribute. This deviation between the two value of Bp could mean a deviation of N 25 ~ in the values of I'o as deduced from self-absorption measurements using resonance

166

O. SHAHAL et al .

fluorescence . This deviation in I'o illustrates the kind of discrepancies caused by following the "conventional" method for calculating T~. 4.2 . THE NH,CI TARGET

The solid curve of fig. 3 representing the theoretical values of Te versus T for NH4C1 shows a systematic deviation of about 6 % between the experimental and theoretical values . In comparing the experimental and theoretical results, it should also be borne in mind that the phonon spectrum for the case of NH4C1 made use of 22 parameters all of which were determined using methods of lattice dynamics involving data of an entirely different class of experiments on inelastic neutron scattering and infrared spectroscopy . Hence the above agreement is satisfactory in view of the experimental and theoretical uncertainties involved in the whole procedure. The zero-point kinetic energy of vibration of t sN in NH4C1 may be obtained by extrapolating the data of fig. 3 to T = OK in the same fashion as was done for BN . The result is : T~ = 390±30K

(at T = OK)

implying a Debye temperature of Bp = 1040K. It is of interest to note that an attempt was made to fit the experimental results of NH4C1(fig. 3) using specific heat data to account for the "external" vibrational modes. However, no agreement between the calculated and measured values could be obtained. The authors would like to thank Prof. G. Gilat for helpful suggestions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

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