22.101 Applied Nuclear Physics (Fall 2006)

Lecture 1 8 (11/20/06)

Neutron Interactions: Energy and An gular Distributions, Thermal Motions

References :

R. D. Evans, Atom ic Nucleus (M cGraw-Hill New York, 1955), Chap. 12.

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), Sec. 3.3. John R. Lam a rsh, Nuclear Reactor Theory (Addison-W e sley, Reading, 1966), Chap 2.

W e will use the expr essions rela ting ener gy and s cattering an gles der i ved in the previous ch apter to determ ine the energy a nd angular distributions of an elastically scattered neutron. The energy distribution, in par ticular, is widely used in the analysis of neutron energy m oderation in system s where ne utrons are p r o duced at h i g h energies (Mev) by nuclear reactions and slow down to th erm a l energies. This is the problem of neutron slowing down, where the assum p tion of the ta rget nu cleus be ing initially at r e st is justified. When the neutron energy appro aches the therm a l regi on (~ 0.025 ev), the stationary target assum p tion is no longer va lid. One can relax this assumption and derive a m o r e general distribution which hol ds for neutron elastic scattering at any energy. This then is the result that shoul d be used for the analysis of the spectrum (energy distribution) of therm a l neutrons, a problem known a s neutron therm a lization.

As part of th is discu ssion we will hav e an opportu nity to s t udy the energy dependence of the elastic scattering cross section.

We have seen from our study of cross section calculation using the m e thod of phase shif t that f o r lo w-energy scattering (kr o << 1, which is equivalent to neutron energies below about 10-100 kev) only s-wa ve contribution to th e cross section is im portant, and m o reover, the angular distributi on of the scattered neutron is spherically symmetric in CMCS. This is the r e s u lt th at we will m a ke use of in deriv i ng the en er gy distribution of the elastic ally scattered neutron.

Energy Distribution of Elas tica lly Scatter e d Neutrons

We define P (  c ) as the probab ility tha t th e scat tered neutron will be going in the direction of the unit vector  c (recall this is a un it vector in angular space). We should also understand that a more physical way of defining P is to say that

P (  c ) d  c = the probability that the neutron wi ll be s cattere d into an e l e m ent of solid angle d  c about  c

For s-wave scattering one has therefore

P (  c

) d  c

 d  c (16.1)

4 

Notice th at P (  c ) is a probability distribution in the two angular variables,  c and  c , and is properly norm a lized,

2  

 d  c  cos  c d  c P (  c )  1 (16.2)

0 0

Since there is a one-to - o n e relation b e tween  c and E 3 (cf. (15.13)), we can transform (16.1) to obtain a probability di stribution in the outgoing energy, E 3 . To do this we first need to reduce (16.1) from a distribution in tw o variables to a distri bution in the variable

 c . Let us define G (  c ) as th e p r obability of the sca tte rin g angle be in g  c . This qua ntity

can be obtained from (16.1) by sim p ly integra ting (16.1) over all valu es of the azim u thal angle  c ,

2 

G (  c ) d  c   d  P (  c ) sin  c d  c (16.3)

0

= 1 sin  d  (16.4)

2 c c

Now we can write down the transform a tion from G(  c ) to the e n ergy dis t r i b u tion in the outgoing energy. For the purpose of general discussion our notation system of labeling particles as 1 through 4 is not a good choice. It is m o re c onventional to label the energy

of the neutro n bef o re and af ter the collision as e ith er E and E’, or vice v e rsa . W e will

theref ore switch no tation at th is poin t and let E 1 = E and E 3 = E’, and denote the probability distr i bution f o r E’ as F ( E  E ' ) . The transform a tion between G(  c ) and F ( E  E ' ) is the sam e as that for any distribution function,

F ( E  E ' ) dE '  G (  c ) d  c (16.5) W ith G(  c ) given by (16.4) we obtain

F ( E  E ' )  G (  c

)

(16.6)

The Jacobian of transform a tion can be readily evaluated from (15.13) after relabeling E 1 and E 3 as E and E’. Thus,

F ( E  E ' )  1  E  E '  E

E ( 1   )

( 1 6 . 7 )

= 0 otherwise

The distribution, which is sketched in Fig. 16.1, is so sim p le that one can understand com p letely all its f eatu r es. The dis t ribution is u n if orm in the inte rval ( E, E) because the sca tte rin g is spher i ca lly sym m etric (inde p e nd ence of scattering ang l e translates into independence of outgoing energy because of the one-to-one correspondence). The fact that the outgoing energy can only lie in a par ticu l ar in terva l f o llows f r om the rang e of scattering an gle (0, ). Since depends on the m a ss of th e targ et, being zero for hydrogen and approaching unity as M >> m , th e interval can vary from ( 0 , E) for hydrogen to a vanishing value as A >> 1. In other words, the neutron can lose all its energy in one collision w ith hydrogen, and loses pr actically no energy if it collides with a

Fig. 16.1. Scattering frequency giving the probabili ty tha t a neutron e l as tic s cattered at energy E will have an energy in dE’ about E’.

very heavy target nucleus. Although sim p le, the distribut ion is quite usef ul for the analysis of neutron energy m oderation in the slowing down regim e . It also represents a reference behavior for discussing conditions whe n it is no lon g er valid to assum e the scattering is spherically s y mm etric in CMCS, or to assum e the targ et nuc leus is a t res t . W e will co m e back to these two situations later .

Notice th at F is a dis t r i b u tion, so its dim e nsion is the reciprocal of its argum e nt, an energy. F is also p r o p erly norm a lized, its integral ov er th e range of the outgoing energy is n e cessarily un ity as required by particle conservatio n. Knowing the probability distribution F one can construct th e energy differential cro ss section

d  s  

d E '

s ( E ) F ( E  E ' ) (16.7)

such that

dE ' d  s

d E '

  s

( E ) (16.8)

which is th e ‘total’ (in th e sense th at it is th e integral of a differe ntial) scattering cross section. It is im portant to keep in m i nd that  s ( E ) is a f unction of the initia l (in c om ing)

neutron en ergy, whereas the in tegration in (16.8) is over the final (outgoing) neutron energy. The quantity F ( E  E ' ) is a distribu tio n in the va ri able E’ and also a function of E. W e can m u ltiply ( 16. 7) by the num b er density of the ta rget nuc lei N to obtain

N  s ( E ) F ( E  E ' )   s ( E  E ' ) (16.9)

which is som e ti m e s known as the scattering ke rnel. As its nam e suggests, this is the quantity that appears in the neutron balan ce equation for neutron slowing down in an absorbing m e dium ,

E / 

  s ( E )   a ( E )   ( E )   dE '  s ( E '  E )  ( E ' ) (16.10)

E

where  ( E )  vn ( E ) is the neutron flux and n(E) is the neutron num b er density. Eq.(16.10) is an exam ple of the usefulness of the energy different ial s catte ring cro ss section (16.7).

The scattering distribution F ( E  E ' ) can be u s ed to calculate various energy-

averaged qu antities p e rtaining to elastic s cat terin g . For example, the average loss for a collision a t e n ergy E is

E E

 dE ' ( E  E ' ) F ( E  E ' ) 

 E

( 1   ) (16.11)

2

For hydrogen the energy loss in a collision is one-half its en ergy before the collision, whereas for a heavy nucleus it is ~ 2 E /A.

Angular Distribution of El astica lly Scatter e d Neutrons

We have already m a de use of the fact that for s-wave scattering the angular distribution is spherically symme tric in CMCS. This m eans that the angu lar d i f f e rential scattering cross section in CMCS if of the for m

d  s   (  )   ( E ) 1

(16.12)

d  s c s 4 

One can ask what is the angular differential scattering cross s ection in LCS? The answer can be obtained by transfor m i ng the results (16. 12) from a di stribution in the unit vector

 c to a distribution in  . As before (cf. (16.5)) we write

 s (  ) d    s (  c ) d  c (16.13)

or 

(  )  

(  ) sin  c

d  c (16.14)

s s c

sin  d 

From the relation between cos  and cos  c , (15. 15), we can calcu late

d (cos  )  sin  c d  c

d (cos  c ) sin  d 

Thus

 (  ) 

 ( E )   2  2  cos 

 1  3 / 2

(16.15)

s 4  1   cos 

with = 1/A. Since (16.15) is a function of  , the factor cos  c on the right hand side should be expressed in term s of cos  in accordance with (15.15). The angular distribution in LCS, as given by (16.15), is so m e what too co mplicated to sketch s i m p ly. From the relation between LCS and CMCS indicat ed in F i g. 15.2, we can expect that if the distribution is isotropic in LCS, then th e distribution in C M CS should be peaked in the forward direction (simply b ecaus e the scattering angle in LCS is always less than the angle in CM CS). One way to dem o nstrate th at th is is indeed the case is to calcu late the average v a lu e of   cos  ,

1 1

d  cos 

(  )

 d  s (  )  d  c  (  c )  s (  c )

   s   1   1  2

(16.16)

 d   s (  ) 3 A

 d  s (  )  d  c  s (  c )

 1  1

The fact that  > 0 m eans that the ang u lar d i stribu tion is p eaked in the forward direction. This bias becom e s less pronounced the heavier the target m a ss; for A >> 1 the distinction between LCS and CMCS vanishes.

Assumption s in Derivin g F ( E  E ' )

In arriving at the scattering distribu tion (also som e tim es calle d the sca tte ring frequency), (16.7), we have m a de use of three assum p tions, nam e ly,

(i) elas tic s cattering

(ii) target nucleus at rest

(iii) scattering is isotrop i c in CMCS (s-wave)

These assu mptions im ply certa in restrictions pertaining to the energy of incom i ng neutron E an d the tem p erature of the scat tering medium . Ass u m p tion (i) is valid provided the neutron energy is not high enough to excite the nuclear levels of the com pound nucleus formed by the target nucleus plus the incom i ng neutron. On the other hand, if the neutron energy is high enough to excite the fi rst nuclear energy level above the ground state, then inelastic scattering becom e s energetical ly possible, Inelastic scattering is a thresho l d reaction (Q < 0), it can occur in heavy nuclei at E ~ 0.05 – 0.1 Mev, or in m e dium nuc lei at ~ 0.1 – 0.2 Mev. Typically the cross se ctio n f o r inelas tic scattering,  ( n , n ' ) , is of the order of 1 barn or less. In c o m p arison with ela s tic scattering, which is always presen t n o m a tter wh at other reactions can tak e place and is of order 5 – 10 barns except in the case of hydrogen where it is 20 barns as w e have previously discussed.

Assum p tion (ii) is valid when the neutron energy is large co mpared to th e kinetic energy of the target nucleus, typically taken to be k B T assum i ng the m e dium is in equilibrium at tem p erature T. This would be the case for neutron energies ~ 0.1 ev and above. W h e n the incident neutron energy is comparable to the energy of the target

nucleus, the assum p tion of stati onary target is clearly invali d. To take into account the therm a l m o t i ons of the target, one should know what is the state of the target since the nuclear (atom i c) m o tions in solids are differe nt from those in liquids , v i br ations in the f o rm er and dif f u sion in the la tte r. I f we assu m e the scatterin g m e dium can be treated as a gas at temperatu r e T, then the targ et nucleus m oves in a s t r a ight line with a speed th at is given by the Maxwellian distribution. In this case one can derive the scattering distribution which is an extensi on of (16.7) [see, for exam ple, G. I. Bell and S. Glasstone, Nuclear Rea c tor Theory (Van Nostrand reinhold, New Yo rk 1970), p. 336]. W e do not go into th e d e tails he re e x cept to sho w the qualitative b e havior in Fig. 16 .2. From the way the scattering distribution changes w ith incom i ng energy E one can get a good intuitive f e e l ing f o r how the m o re general F ( E  E ' ) evolv e s from a spread-ou t

distribution (the curves f o r E = k B T) to the m o re restricted form given by (16.7).

1.0

1.0

0.5

0.5

0 0

0 0.5 1.0 1.5 2.0 0

0.5

1.0 1.5 2.0

E'/E E'/E

Figure b y MIT OCW . Adapted f rom Bell and Glasstone.

Fig. 16.2. Energy distribution of elastically scat tered neutrons in a gas of nuclei with m ass A = M/m at tem p er ature T. (from Bell and Glasstone)

Notice th at for E ~ k B T there can b e appreciable upscattering which is no t possib l e when assum ption (ii) is invoked. As E becom e s larger com p ared to k B T, upscattering becom e s less im portant. The condition of stati onary nucleus also m eans that E >> k B T.

When thermal m otions have to be taken into account, the scattering cross section

 s ( E ) is also changed; it is no longer a constant, 4  a 2 , where a is th e sca t ter i ng leng th . This occurs in the energy region of neutron ther m alization; it cove rs the range (0, 0.1 -

0.5 ev). W e will now discuss the en ergy depend ence of  s ( E ) . For the case of the scatter i ng m e dium being a gas of ato m s with m a ss A and at te m p erature T , it is still relatively straightforward to work out the expression for  s ( E ) . W e will giv e the

essential steps to give th e studen t so m e feeli ng for the kind of analysis that one can carry out even for more com p licated s ituations such as neutron elastically in solids and liquids.

Energy Dependence of Sc atter i ng Cross Sectio n  s ( E )

W h en the ta rget nuc leus is not a t re st , one can write down the expression for the elas tic s catte ring cro ss se ction m easured in the la boratory (we w ill c a ll it the m easured cross sectio n),

v  meas ( v )   d 3 V v  V  theo ( v  V ) P ( V , T ) (16.17)

where v is th e neutron spe e d in LCS, V is the targe t nucleus ve lo city in LCS,  theo is the scattering cross section we calculate theoretically, such as what we had p r evious ly studied us in g the phase - shif t m e thod and solving the wave eq uation f o r an ef f ective on e-

body proble m (notice th at the resu lt is a f u nc tion of the relative speed between neutron

and targ et n u cleus), and P is the th erm a l distri bution of the target nucleus velocity which depends on the tem p erature of the medium . Eq.( 16.17) is a general relation between what is calculated theoretica lly, in solving the effective on e-body problem, and what is m easured in the labo rato ry where one necessa rily has only an average ov er all possib l e targe t nucle u s veloc ities . W h at we call the sca t te ring cro ss se ction  s ( E ) we mean

 meas . It tu rns ou t tha t we can reduce (16. 17) further b y using for P the Maxwellian distribution and obtain the result

 ( v )   so    2  1  erf (  )  1  e   2  (16.18)

s  2  



 

where erf ( x ) is th e error f u nc tion integ r al

erf ( x )  2  dte  t 2

0

(16.19)

and  2  AE / k T , and E  mv 2 / 2 . Given that the erro r f unction ha s the lim iting behavior for sm all and large argum ents,

erf ( x ) ~

2 ( x  x

3

 ...) x << 1

( 1 6 . 2 0 )

e  x 2  1 

1   1  2 x 2  ...  x >> 1

 

we have the two lim iting behavior,

 s ( v ) ~  so / v    1 (16.21)

 s ( v ) ~  so   1 (16.22)

The physica l signif icanc e of this ca lc ulation is th at one sees the elas tic sc atte ring c r os s section has a 1/v behavior at low energy (or hi gh tem p erature) and a constant behavior at high energy. The expres sion (16.18) is ther efore a useful exp r ession g i ving the energ y varia tion of the sca tte rin g cross se ction over th e entire energ y range fro m therm a l to Mev, so far as elastic s catterin g is concern e d. Note tha t this resu lt has been obtained by assum i ng the targ et nuclei m o ve as in a gas of noninteracting atom s. This assum p tion is not rea lis tic when the sc atte ring m e dium is a solid or a liquid. For these s i tuations one can also work out the ex pression s for the cross s ection, bu t the results are m o re com p licated (and beyon d the scope of this cour s e ). W e will theref ore settle f o r a b r ie f , qualitative look at what new features can be seen in the en erg y dependence of the elas tic scattering cross sections of typi cal so lids (crys t als ) and liquids .

Fig. 16.3 shows the total and elas tic s cattering cross sections of graphite (C 12 ) over the entire ene r gy ra nge of intere st to th is class. At the v e ry low-en ergy end we see a num b er of features we have not discussed prev io usly. These all hav e to d o with the f a ct that the targ et nucleus (atom ) is bound to a crys tal lattice and theref ore the position s of the nuclei are fixed to well-defined latt ice s ites a n d the atom ic m o tions are sm all- am plitude vibrations abo u t thes e sites . Ther e is a sharp drop o f the cross s ection b e lo w an energy m a rked Bragg cutoff. Cutoff here refers to Bragg reflection wh ich occu rs when the co ndition for constructive interferen ce (reflection) is satisfied, a condition th at depends on the wavelength of the neutron (h en ce its energy ) and the spacing between the lattice p l ane s in the crys tal. W h en the wavelength is too lo ng (energ ies be low the cuto f f ) f o r the Brag g condition to be satisf ie d, the cros s section drops sharply. W h at is then lef t is the inte rac tion be twee n the neutro n and th e vibrational m o tions of the nuclei, this process invo lves th e tran sfer of energy from the vibra tions to the neutron which has m u ch lower ene r g y . Since the r e is m o re e x cita tion of the vibra tion a l m odes at higher tem p erature s , this is re as on why the cross se ctio n below the Bragg cutof f is very s e ns itiv e to tem p eratu r e, inc r eas in g with inc r e a sing T.

Above the Bragg cutoff the cross s ection shows s o m e oscillations. These correspond to the onse t o f additiona l ref l ect ions b y planes which have sm alle r la ttice spacings. A t energ i es around k BT the cross sectio n approaches a constan t value up to ~

0.3 Mev. This is the reg i on where o u r previo u s calcu lation of cross section would apply. Between 0.3 and 1 Mev the scattering cross section decreases gradually, a behavior which we can still understand using sim p le theory (beyond what we had discussed). Above 1 Mev one sees s cattering res onances, wh ich we have not yet di scu ssed; also th ere is now a difference b e tw een total and scattering cross sections (which sho u ld be attributed to absorption).

10

5

1.0

0.5

0.0001

1020 o K

720 o K

478 o K

0.001

Bragg cutof f

0.01

Cross section Constant to 0.01 MeV

0.1 1

Neutr on Energy , eV

10

5

1.0

0.5

0.1

0.01

0.1

1 10 100

Neutr on Energy , MeV

F i g u r e b y M I T O C W .

Fig. 16.3. Total and elas tic s catte rin g cross sections of C 12 in the f o rm of graphite. (f r o m Lam arsh).

Fig. 16.4 shows the m easured to tal c r oss sec tion of H 2 O in the f o rm of water. The cross section is the sum of contributions fr om two hydrogen and an oxygen. Com p ared to Fig. 16.3 the low-energy behavior here is quite different. This is no t un expected sin ce a crystal and a liqu i d are really very d i fferent with regard to th eir a t om ic structur e and atom ic m o tions. In the case of the liquid the cro ss section ris es from a constant v alue at energies abo v e 1 ev in a m anner like the 1/v be havior given by (16.21 ). Notice th at the constant value of about 45 barns is just what we know from t he hydrogen cross section

 so of 20 barns per hydrogen and a cross sect ion of about 5 barns for oxygen.

200

150

100

50

0

0.001

0.01

0.1

1 10

Neutr on Energy , eV

F i g u r e b y M I T O C W .

Fig. 16.4. Total c r oss se ction of water. (from Lam a rsh)

The im portance of hydrogen (water) in neutron sc atte ring ha s led to ano t h er in terp reta tion of the rise of the cros s section with d ecrea sing neutron energy, one which focuses on the effect of chem ical bindin g . The idea is that at h i g h energies (relative to th erm al) the neutron doe s not see the water m o lecule. Ins t ea d it se es only the ind i vid u al nuclei as targets which are free-standing and essentially at rest. In th is energy rang e (1 ev and above) the interaction is the sam e as that between a neut ron and free protons and oxygen nuclei. Th is is why the c r oss sec tion is ju st the sum of the individual contributions.

W h en the cross section s t arts to rise as the en erg y decreases, this is an ind i cation tha t the chem ical binding of the protons and oxygen in a water m o lecule starts to have an effect. W h en the neutron energy is at k BT th e neutron no w sees the e n tir e water m o lecule r ather than the individual nuc le i. In th at ca se the scattering is effec tively between a neutron and a water m o lecule. W h at this m eans is th at as the neu t ro n energy decreases the target changes fro m individual nuclei with their in div i d u al m ass to a water m o lecule with m ass

18. Now one can show the scattering cross se ction is actually proportional to the square of the reduced m a ss of t he scatterer ,

   2   mM 

 A  2

 

(16.23)

s  m  M   A  1 

13

One can define a fre e -ato m cross section appropriate for the en ergy rang e where the cross section is a constan t , an d a bound-atom cross section for the energy rang e where the cross section is ris i ng, w ith th e rela tion

2

 free   bound  A  1  (16.24)

 

For hydrogen these two cross se ctions would have the values of 20 barns and 80 barns respec tive l y.

W e close this chapter with a brief con s ideration of assum p tion (iii) us ed in deriving (16.7). W h en t h e neutron energy is in the 10 Kev range and higher, the contributions from the higher angular m o m e ntum (p-wave and above) scattering m a y becom e significant. In that case we know th e angular distribution wi ll be m o re forward peaked. This m eans one should rep l ace (16.1 ) b y a different for m of P (  c ) . Without

going through any m o re details, we show in Fig. 16.5 the general behavior that one can expect in the scattering d i stri bution F when scattering in CMC S is no longer isotropic.

F(E E')

Backward scattering

Forward scattering

Isotropic sca ttering

E'

 E E

F i g u r e b y M I T O C W .

Fig. 16.5. Energy distribution of elastically scatte red neutrons by a stationary nucleus. (from Lam a rsh)

14