22.101 Applied Nuclear Physics (Fall 2006)

Lecture 1 (9/6/06) Basic Nuclear Concepts

References –

Table of Isotopes , C. M. Lederer and V. S. Shirle y, ed. (W iley & Sons, New York, 1978), 7 th ed.

Table of Nuclides <http : //atom .kaeri.re.kr/>

Physics Vad e Mecum , H. L. Anderson, ed. (Am e r i can Institute of Physics, New York, 1081).

P. Marm ier and E. Sheldon, Physics of Nuclei and Particles (Academ ic Press, New York, 1969), vol. 1.

General Re m a rks:

This subject treats foundational knowledge for all students in the Departm e nt of Nuc l ear Science and Engineering. Over the year s 22.101 has evolved in the hands of several instru cto r s, each of whom dealt with the subject contents in s o m e what different fashio ns. Two topics, in particular, were not given th e sam e treatm ent by the different faculty who have taught 22.101, quantum m echa n ics and the in teraction of radiation with m a tter.

There was also som e difference in ho w m u ch nuc lear structure and nuclear m odels we re taught from the perspective of a course on nuclear physics in a physics departm e nt.

In the present version of 22.101 we inte nd to em phasize the nuclear concepts, as opposed to traditional nuclear physics, essential for understand ing nuclear radiations and their interac t ions with m atte r. The ju stif ica tion f o r is th at we s ee our stud e n ts as nuc lea r engineers rather than nuclea r physicists. Nuclear engineer s work with all kinds of nuclear devices, from fission and fusion reacto rs to accelerato r s and detection system s. In all these com p lex system s nuclear radiations play a central role. In generating nuclear radiations and using them fo r beneficial purposes, scie ntists and engineers m ust understand the properties of th ese radiations and how th ey interact with their surroundings. It is through the control of radi ations interactions th at we can develop new devices or optim ize existing ones to m a ke the m more safe, powerful, durable, or econom ical. This is the sim ple reason why radiation interaction is the essence of 22.101.

Because nuclear physics is a very large subject and in view o f our focus on radia tion interac tions, w e will no t be covering so m e of the standard m a teria l on nucle ar structure and m odels that would be norm a lly treated in a physics course. Students interested in these t opics are encouraged to read up on your own. W e should also note that in 22.10 1 we will study the different types of reactions as single-collision phenom ena, described through variouscross sec tions, and leave the accumulated effects of m a ny collisions to late r subjects (2 2.105 and 22.106). Alth ough we will not teach quantum m e chanics by itself, it is used in se veral descriptions of nuclei, at a m o re introductory level than in 22.51 and 22.106.

Nomenclature :

z X A denotes a nuclide, a specific nucle us with Z number of protons (Z = atomic number ) and A num b er of nucleons (neutrons or protons). T h e sym bol of nucleus is X which is either a single letter as in uranium U, or two le tters as in copper C u or plutonium

Pu. There is a one-to-one correspondence be tw een Z and X, so that specifying both is

actually redundant (but helpful since not ev eryone rem e m b e r s the atom ic num ber of all the elem ents). The number of neutrons N of th is nucleus is A – Z. Of ten it is suf f i cient to specify only X and A, as in U 235 , if the nucleus is a f a m iliar one (uran i u m is well known to have Z=92). T h e sym bol A is called the mass number since knowing the num b er of nucleons one has an approxim ate idea of what is th e m a ss of the particular nucleus. Th ere ex ist s e v e ral u r anium nuclide s with different m a ss numbers, such as U 233 , U 235 , and U 23 8 . Nuclid es with the s a m e Z but different A are called iso t op es . By the sam e token, nuclides with the sa m e A but different Z are called isobars , and nuclides with sam e N but different Z are called isotones . Isomers are n u clide s with the sam e Z and A in different excited states. For a com p ilation of the nuclides that are known, see the Table o f Nuclides (Kaeri) for which a website is given in the References.

W e are, in p r inc i ple, inte rested in a ll the elem ents up to Z = 9 4 (pluton i u m ).

There are ab out 20 m o re elem ents which are kno wn, m o st with very sho r t lif etim es; th ese are of interest m o stly to nuc lear physicists and chem ists, but not so m u ch to nuclear engineers. While each elem ent can have se veral isotopes of significan t abundance, n o t

all the elem ents are of eq ual in tere st to us in this class. The number of nuclides we m i ght encounter in our studies is probably no m o re than 20.

A great deal is known about the p r op erti es of nuclides. In ad dition to the Table of Nuclides , the student can consult the Table of Isotopes , cited in the References. It should be apprec iated that the g r eat inte res t in nuc lear structure and reactions is not just for scien tif ic kn owledge alo n e; the f a c t that the r e a r e two applic a tions th at af f ects the welf are of our society – nuclear power and nuclear w eapons – has everything to do with it.

We begin our studies with a review of the m o st basic physical attributes of nuclides to provide m o tivation and a basis to introduce what we want to accom p lish in this cou r se (see the Lecture Outline).

Basic Physical Attr ibutes of Nuclid es

Nuclea r M a ss

We adopt the unified scale where the m a ss of C 12 is exactly 12. On this scale, one m a ss unit 1 m u (C 12 = 12) = M(C 12 )/12 = 1.660420 x 10 -2 4 gm (= 931.478 Mev), where M(C 12 )

is ac tual m a ss of the nuc lide C 12 . S t udies of atom ic m a sse s by m a ss spectrograph shows that a nu clid e has a m a ss nearly equal to the m a ss num b er A tim e s the proton m a ss.

Three im portant rest m a ss values, in m a ss and energy units, to keep handy are: m u [M(C 12 ) = 12] Mev

electron	0.000548597	0.511006
proton	1.0072766	938.256
neutron	1.0086654	939.550

Reason we care abou t th e m a ss is tha t it is an indication of the stability of the nuclide. One sees this directly from E = Mc 2 , the higher the m a ss the higher the energy and the less s t able is the nuclide (think of nu clide b e ing in an excited state). W e will f i nd tha t if a nuclide can lower its en ergy by undergoing disi ntegration, it will do so – this is the sim p le explanation of radioactivity. Noti ce the proton is lighter than the neutron, suggesting the for m er is more stable than the latter. Indeed, if the neutron is not bound in

a nucleus (th a t is, it is a free neu t ron) it will decay into a p r oto n plus an electron (p lus an antineu tr ino ) with a ha lf -lif e of abou t 13 m i n.

Nuclear m a sses have been determ ined to quite high accuracy, precis ion of ~ 1 part in 10 8 , by the m e thods of m a ss spectrography and ener gy m easurem ents in nuclear reactions.

Using the mass data alon e we can get an idea of the stab ility of nuclid es. Define the m a ss defect as th e difference b e tween the actu al m a ss of a nuclide and its m a ss num b er,  = M – A; it is also ca lled the “m ass decrem ent”. If we plot  versus A, we get a curve sketched in Fig. 1. W h en  < 0, it m eans that taking the individual nucleons when

Fig. 1. Variation of m a s s decrem ent (M-A) show ing that nuclides with m a ss num bers in the range ~ (20-180) should be stable.

they are sep a rated far from each oth e r to m a ke the nucleus g i ves a produ ct whose m a ss is lighter than the sum of the com ponents. This can only happen if energy is given off during the form ation. In other words, to re ach a final state (the product nuclide) with sm aller m a ss than th e in itial s t ate (c ollec ti on of individual nucleons ) one must take away som e energy (m ass). It a l so f o llows that the f i na l state will b e more st able than the initial state, since energy m u st be put back in if one wants to reverse the process to go from the nuclide to the individual nucleons. We therefore expect that  < 0 m eans the nuclide is stable. Conversely,  > 0 m eans the nuclide is unstable. Our sketch shows that very light elem ents (A < 20) and heavy elem ents (A > 180) are not stable, and that m a xi m u m stability occurs around A ~ 50. W e will return to discuss this behavior f u rther when we consider the nuclear bi nding energy later.

Nuclea r Size

According to Thom son’s “electr on” model of the nucleus (~ 1900), the size of a nucleus should be about an Angstrom , 10 -8 cm. W e now know this picture is wrong. The correct nuclear size was determ ined by Rutherford (~ 1911) in his atom ic nucleus hypothesis which put th e size at abo u t 10 -1 2 cm . Nuclear radius is not well defined, strictly speaking, because any result of a measurem ent de pends on the phenom e non involved (different experim e nts give different resu lts). On the other hand, all the results agr ee qualitatively and to som e extent also q u antitatively . R oughly speaking, we will take the nuclear rad i us to vary with the 1/3 pow er of the m a ss num ber, R = r o A 1/3 , with r o ~ 1.2 – 1.4 x 10 -1 3 cm . The lower v a lue of the coefficient r o com e s from electron sc attering which probes the charge distribution of the nucleus, while the higher value comes from nucl ear scattering which probes the range of nuclear force. Since nuclear radii have m a gnitudes of the order of 10 -1 3 cm , it is conventional to adopt a length unit called Ferm i (F), F  10 -1 3 cm. Because of particle-wav e duali ty we associate a wavelengt h with the m o m e ntum of a particle. The correspon ding wave is called the d e Broglie wave. Before discussing the connection between a w a ve property, the wavelength, and a particle property, the mom e ntum , let us first set down the relati vis tic k i nem a tic rela tions between m a ss, mom e ntum and energy of a particle with arb itr ar y velocity. C onsider a p a rticle with r e st m a ss m o moving with velocity v. There are tw o expressions we can write down for the tota l energy E of this par tic le. On e is the sum of it s kinetic energy E kin and its rest m a ss energy, E  m c 2 ,

E  E  E  m ( v ) c 2 ( 1 . 1 )

The second equality introduces the relativis tic m a ss m ( v) which depends on its velocity,

m ( v )   m ,   ( 1  v 2 / c 2 )  1 / 2 (1.2)

where is the Einstein factor. To understa nd (1.2) one should look into the Lorentz transform a tion and the special theory of relativi ty in any tex t . Eq.(1.1) is a first-o r der relation for the total energy. Another way to express the total en ergy is a second-order

rela tion

E 2  c 2 p 2  E 2 ( 1 . 3 )

where p = m(v)v is the m o m e ntum of the par tic le. Eqs. (1.1) – (1.3) are the genera l rela tions b e tween the to tal and kin e tic energ i es, m a ss, and mom e ntum . We now introduce the deBroglie wave by defining its wavelength  in term s of the m o m e ntum of the corresponding particle,

  h / p ( 1 . 4 )

where h is the Planck’s constant ( h / 2   ħ  1 . 055 x 10  27 erg sec). Two lim iting cas es are worth noting.

Non-rela tiv istic regim e :

E o >> E kin , p  ( 2 m E ) 1 / 2 ,   h /  h / m v (1.5)

o

Extrem e relativs i tic reg i m e :

E kin  E o , p  E kin / c ,   hc / E (1.6)

Eq.(1.6) applies as well to photons and ne utrinos which have zero rest m a ss. The kinem a tical relations just disc ussed are general. In practi ce w e can safely apply the no n - relativ istic expression s to neutrons, protons , and all nuclides, because their rest m a ss energies ar e always m u ch greater th a n any kinetic energ i es w e will en cou n ter. The s a m e is not true f o r ele c tron s, s i nce we will be inte res t e d in ele c tron s with energ i es in the M e v region. Thus, the two extrem e regimes do not apply to electrons, and one should use (1.3) for the energy-m o mentum relation. Si nce photons have zero rest m a ss, they are always in th e relativ istic regim e .

Nuclea r ch arge

The charge of a nuclide z X A is positive and equal to Ze, where e is the m a gnitude of the electron charge, e = 4.80298 x 10 -1 0 esu (= 1.602189 x 10 -1 9 Coulom b). We consider single atom s as exactly n e ut ral, the electron - pro t o n charge d i fference is < 5 x 10 -1 9 e, an d the charge of a neutron is < 2 x 10 -1 5 e.

We can look to high-energy electron scatte ring experim e nts to get an idea of how nuclear density and charge density are distri buted across the nucleus. Fig. 2 shows two typical nucleon density distribut ions obtained by high-energy electron scattering. One

Fig. 2 . Nucleon density distributions s howing nuclei having no sharp boundary.

can see two basic com ponents in each distributio n, a core region of constant density and a boundary region where the density decreases sm oothly to zero. Notice the m a gnitude of the nuclear density is 10 38 nucleons per cm 3 , compared to the atom ic density of solids and liquids wh ic h is in the range of 10 24 nuclei p e r c m 3 . What does this say about the packing of nucleons in a nucleus, or the average di stance between nucleons versus the separation between nuclei? Indeed the nucleons are pack ed together m u ch m o re closely than the nuclei in a s o lid. The sh ape of the distribu tio ns shown in Fig. 2 can be fitted to the expression, called the Saxon distribution,

 ( r ) 

 o

1  exp[( r  R ) / a ]

( 1 . 7 )

where  = 1.65 x 10 38 nucleons/cm 3 , R ~ 1.07 A 1/3 F, and a ~ 0.55 F. A sketch of this distribution, given in Fig. 3, shows clearly the core a nd boundary com ponents of the

Fig. 3. Schem a tic of the nuclear density dist ribution, with R being a m easure of the nuclear radius, and the width of th e boundary region being given by 4.4a.

distribution. Detailed studie s based on high-energy electron scattering have also revealed that even the proton and the neutron have rather com p licated s t ructures. T h is is illus t r a ted in Fig. 4. W e note tha t m e sons are uns table p a rtic les of m a ss betw een

Fig. 4. Charge density distribu tion s o f the proton and the neu t ron showing how each can be decom p osed into a core and two m e son cl ouds, inner (vector) and outer (scalar). T h e core has a p o sitiv e cha r ge of ~0.35e with probab l e rad i us 0.2 F. The vector cloud has a radius 0.85 F, with charge .5e and -.5e for the proton and the neutron respectively, whereas th e scalar clou id has radiu s 1 . 4 F and charge .15e for both proton and neutron[adopted from Marm ier and Sheldon, p. 18].

the electron and the proton: -m esons (pions) play an im portant role in nuclear forces

( m  ~ 270 m e ), -m esons(muons) are important in cosm ic-ray processes ( m  ~ 207 m e ).

Nuclea r Spi n an d Magneti c Mom ent

Nuclear angular m o m e ntum is often known as nuclear spin ħ I ; it is m a de up of tw o parts, the in trinsic spin o f each nucleon and th eir orbital angu lar m o m e nta. W e call I the spin of the n u cleus, w h ic h can take o n integ r al o r half -integra l value s . The f o llow i ng is

usually accepted as facts . Neutron and proton bo th have sp in 1/2 (in un it of ħ ). Nucle i

with even mass num ber A have integer or ze ro spin, while nuclei of odd A have half- integer spin. Angular mom e nta are quantized.

Associated with the sp in is a m a gnetic m o m e nt  , which can take on any value because

I

it is no t qua ntized. The unit of m a gnetic m o m e nt is the m a gneton

    B = 0.505 x 10 -2 3 ergs /gaus s (1.8)

2 m p c 1836 . 09

where  B is the Bohr m a gneton. The relation between the nuclear m a gnetic mom e nt and the nuclear s p in is

   ħ I ( 1 . 9 )

I

where here is the gyromagnetic ra tio (no relation to the Ein s tein f actor in specia l rela tiv ity). Experim e ntally, spin and m a gnetic mom e nt are measured by hyperfine struc t ure (sp litting of ato m ic lines due to interaction between atom ic and nuclear

m a gnetic mom e nts), deflations in m o lecu lar beam under a magnetic field (Stern-

Gerlach), and nuclear magnetic resonance (precession of nucl ear spin in c o m b ined D C and m i crowave field). W e will say more about nm r late r.

Electri c Quadrupl e Mom ent

The electric mom e nts of a nuc leus reflect the charge dist ribution (or shape) of the nucleus. This inform ation is im portant for developing nuclear m odels. We consider a classical calculation of the energy due to electric quadruple mom e nt. Suppose the nuclear charge has a cylindrical symm etry about an axis along the nuclear spin I , see Fig. 5.

Fig. 5. Geom etry for calculating the Coulom b potential energy at the field point S 1 due to a charge distribution  ( r ) on the spheroidal su rface as sketched. The sketch is for r 1 located alon g the z – ax is.

The Coulomb energy at the point S 1 is

V ( r ,  )  d 3 r  ( r ) ( 1 . 1 0 )

1 1  d

where  ( r ) is th e charge den s ity, and d  r 1  r . W e will exp a n d this in teg r al in a power series in 1 / r 1 by noting the expansion of 1/d in a Legendre polynom ial series,

1 1   r  n

d  r   r

 P n (cos  ) ( 1 . 1 1 )

1 n  0  1 

where P 0 (x) = 1, P 1 (x) = x, P 2 (x) = (3 x 2 – 1)/2, … T hen (1.10) can be written as

V ( r ,  )  1   a n

( 1 . 1 2 )

1 1 n

1 n  0 1

with a o   d 3 r  ( r ) = Z e ( 1 . 1 3 )

a 1 

 d 3 rz  ( r ) = electric d i pole (1.14)

a 2 

 d 3

1

r ( 3 z 2

2

 r 2

)  ( r )  1

2

eQ (1.15)

The coefficients in the expansion for the en ergy, (1.12), are recogni zed to be the total charge, the dipole (here it is equal to zero), the q u adruple, etc. In (1 .15) Q is defined to be the quadruole m o m e nt (in unit of 10 -2 4 cm 2 , or barns). Notice th at if th e charge distribution were sphe rically symm etric, <x 2 > = <y 2 > = <z 2 > = <r 2 >/3, then Q = 0. We see also, Q > 0, if 3<z 2 > > <r 2 > an d Q <0, if 3<z 2 > < <r 2 >. The corresponding shape of the nucleus in these two cases would be prolate or oblate spheroid, respectively (see Fig. 6).

Fig. 6. Prolate and ob late spheroid al shapes of nuclei as ind i c a ted by a po sitiv e or negative value of the electric quadruple m o m e nt Q.

Som e values of the spin and quadruple m o m e nts are:

Nucleus I Q [10 -2 4 cm 2 ]

n	1/2	0
p H 2	1/2 1	0 0.00274
He 4	0	0
Li 6	1	-0.002
U 233 U 235 Pu 241	5/2 7/2 5/2	3.4 4 4.9

For calcu lations and quick estim ates, the fo llowing table of physical constants and conversion f actors, taken from Meyerhof, Elements of Nuclear Physics (1967), appendix D, is useful.

Speed of light in vacuum c Elementary char ge e A vogadro's number N Mass unit Electron rest mass m 0 Proton rest mass M P Neutron rest mass M n Faraday constant Ne Planck constant _ h h = h/2  Char ge-to-mass ratio for electron e/m 0 R ydber g constant 2  2 m 0 e 4 / h 3 c _ Bohr radius h 2 /m 0 e 2 Compton wavelength of electron _ h/m 0 c h/m 0 c Compton wavelength of proton _ h/M p c h/M p c	2.997925(1)	x 10 8 m s -1	x 10 10 cm s -1
	1.60210(2)	10 -19 C	10 -20 emu
	4.80298(7)		10 -10 esu
	6.02252(9)	10 26 kmole -1	10 23 mole -1
	1.66043(2)	10 -27 kg	10 -24 g
	9.10908(13)	10 -31 kg	10 -28 g
	5.48597(3)	10 -4 u	10 -4 u
	1.67252(3)	10 -27 kg	10 -24 g
	1.00727663(8)	u	u
	1.67482(3)	10 -27 kg	10 -24 g
	1.0086654(4)	u	u
	9.64870(5)	10 4 C mole -1	10 3 emu
	2.89261(2)		10 14 esu
	6.62559(16)	10 -34 J s	10 -27 er g s
	1.054494(25)	10 -34 J s	10 -27 er g s
	1.758796(6)	10 1 1 C kg -1	10 7 emu
	5.27274(2)		10 17 esu
	1.0973731(1)	10 7 m -1	10 5 cm -1
	5.29167(2)	10 -1 1 m	1 0 -9 cm
	2.42621(2) 3.86144(3)	10 -12 m 10 -13 m	10 -10 cm 10 -1 1 cm
	1.321398(13) 2.10307(2)	10 -15 m 10 -16 m	10 -13 cm 10 -14 cm

G e n e r a l P h y s i c a l C o n s t a n t s . T h e n u m b e r s i n p a r e n t h e s e s a f t e r e a c h q u o t e d v a l u e r e p r e s e n t t h e s t a n d a r d d e v i a t i o n e r r o r i n t h e f i n a l d i g i t s o f t h e g i v e n v a l u e , a s c o m p u t e d o n t h e c r i t e r i o n o f i n t e r n a l c o n s i s t e n c y . T h e u n i f i e d s c a l e o f a t o m i c w e i g h t s ( 1 2 C = 1 2 ) i s u s e d t h r o u g h o u t .

1 electron volt	1.6021 x 10 -1 9 J
1.6021 x 10 -1 2 erg
1 erg	1 gm c m 2 /sec
1 ħ	1.054 x 10 -2 7 erg sec
1k B	1.3806 x 10 -1 6 erg o K -1
Proton m a ss	931.478	Mev
Neutron m a ss	939.550	Mev
Electron m a ss	0.511	Mev