22.101 Applied Nuclear Physics (F all 2006)

Lecture 6 (9/27/06)

The Neutron-Proton System: Bound State of the Deuteron

References –

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), App. A.

B. L. Cohen, Concepts o f Nuclear Ph ysics (McGraw-Hill, New. York, 1971), chap. 3.

“The investigation of this nuclear for ce has turned out to be a truly m onum ental task: Perhaps m o re m a n-hours of work have be en devoted to it than any other scientific question in the history of m a nkind” – B. L. Cohe n, Concepts of Nuclear P h ysics , p. 32.

A direct way to gain som e appreciation of th e essential nature of nuclear force for our purposes is to consider the sim p lest possi ble nu cleus, th e two - nucleon system of the deuteron H 2 nucleus which is com posed of a netu ron and a proton. W e will d i scuss two properties of this system using elem e n tary quantum m echanics, the bound state of the nucleus (this lectu r e) and the scattering of a neutron by a proto n (next lectu r e). The for m er problem is an application of our st udy of bound states in three-dim e nsions, while the la tte r is a new applic ation of solv ing the tim e-independent Schrödinger wave equation. F r om these tw o problem s a num ber of funda m e ntal features of the nuclear potential will em erge.

Bound State of the Deuteron

The deuteron is the only stable bound system of two nucleons – neither the di-neutron nor the di-proton are stable. Expe rim e ntally it is known that the deuteron exists in a bound state of energy 2.23 Mev. This energy is th e energy of a ga mm a ray given off in the reaction where a therm a l neutron is absorbed by a hydrogen nucleus,

n  H 1  H 2   (2.23 Mev)

The inverse reaction of us ing electrons of known en ergy to produce external bremsstrahlung for (  , n ) reaction on H 2 also has been studied. Besides the ground state no stable ex cited states of H 2 have been found; however, there is a virtual state at ~ 2.30 Mev.

Suppose we com b ine the inform ation on the bound-state energy of the deuteron with the bound-state solutions to the wave equation to see what we can learn about the potential well for neutron-prot on interac tion. A g ain we will assum e the inte raction between the two nucleon s is of the f o rm of a spherica l well,

V ( r )   V o r < r o

= 0 r > r o

We ask what is the energy level st ructure and what values should V o an d r o tak e in o r der to be consistent with a bound state at energy E B = 2.23 Mev.

There is good reason to believe that th e deuteron ground state is prim arily 1s (n=1, ℓ =0). First, the lowest energy state in p r ac tically all the m odel poten tia ls is an s- state. Secon d ly, the m a gnetic m o m e nt of H 2 is approxim a tely the sum of the proton and the neturon mom e nts, indicating that s n and s p are parallel and no orbital m o tion of the

proton relative to the neutron. This is also consistent with the tota l angular m o m e ntum of the ground state being I = 1.

W e therefore proceed by considering only the ℓ = 0 radial wave equation,

 ħ d 2 u ( r ) 

m d r 2

V ( r ) u ( r ) 

Eu ( r ) (6.1)

where E = - E B , and m is the neutron (or proton) m a ss. In view of our previous discussions of bound-state calc ulations we can readily wr ite down the interior and exter i or so lu tions,

u ( r )  A sin Kr K  [ m ( V  E )] 1 / 2 / ħ r < r o (6.2)

u ( r )  Be   r  

/ ħ r > r o (6.3)

Applying the boundary condition at the interface, we obtai n the relation between the potential well param e ters, depth a nd width, and the bound-state energy E B ,

 V  E

 1 / 2

K cot Kr o

   , or tan Kr o

   o B  (6.4)

E

 B 

To go further we can con s ider eith er num e rical or graphical solutions as discussed before, or approxim a tions based on som e special properties of the nu clear poten tial. Let us consider the latter op tion.

Suppose we take the potential well to be deep, that is, Vo >> E B , then the R H S (right hand side) of (6.4) is la rge and we get an approxim a te value for the argum ent of the tangent,

Kr o ~  / 2 ( 6 . 5 )

Then,

K ~

/ ħ ~  / 2 r o

   2 ħ 2

or V r 2 ~   ~ 1 Mev-barn (6.6)

o o  2  m

We see that a knowledge of E B allows us to determ ine the product of V r 2 , and not V o

o o

and r o sep a rately. F r om a study of neutron sc attering by a proton which we will dis c uss in the nex t c h apter, we will f i nd tha t r o ~ 2F. Pu tting this into (6.6) we g e t V o ~ 36 Mev,

which justif ies our tak i n g V o to be large com p ared to E B . A sketch of the nuclear potential deduced from our calcu lation is shown in Fig. 1.

Fig. 1. The nuclear potential for the neutron-prot on system in the form of a spherical well of depth V o and width r o . E B is the bound-state ener gy of the deuteron.

W e notice th at sin ce the interior wave function, sinKr, m u st m a tch th e exte rior wave function, exp(   r ) , at the interface, the quantity Kr o m u st be slightly greater than

 / 2 (a m o re accurate estim a te gives 116 o instead of 90 o ). If we write

K  2  /  ~  / 2 r o

then the ‘effectiv e wavelength’  is approxim a tely 4F, which suggests that much of the wave function is not in the interior region. The relaxation constant, or decay length, in the in terior region can be estim ated as

1  ħ



~ 4.3 F

This m eans that the two nucleons in H 2 spend a large fraction of their time at r > r o , th e region of negative kinetic energy that is class i cally forbidden. W e can calculate the ro ot- m ean-square radius of the deuteron wave function,

R 2 



d 3 rr 2 R 2 ( r )  r 2 drr 2 R 2 ( r )

 0

(6.7)

rms

 d 3 rR 2 ( r )



 r 2

0

drR 2

( r )

If we take R ( r ) ~ e   r f or all r (this should result in an overestim ate), we would obtain

R 2  ħ  3 F

This value can be com p ared with the es tim ate of nuclear radius based on the m a ss num b er,

(radius) 2 ~ (1.4x A 1/3 ) 2 ~ 3.1 F, or (1.2xA 1/3 ) 2 ~ 2.3 F