On the statistical calculation of the density of the energy levels of the nuclei

Physica IV, no. 7 Juli l&37

ON THE STATISTICAL CALCULATION OF THE DENSITY OF THE ENERGY LEVELS

OF THE NUCLEI by C. VAN LIER and G. E. UHLENBECK

Natuurkundig Laboratorium der Rijks Unive;siteit, Utrecht.

Abstract The calculation of R e t h e concerning the density of energy levels

p(E) of a nucleus has been extended assuming for the density of the levels of the individual particles an arbitrary function 9(e). (5 2 Eq. 9 and 3 3 Eq. 10). The method of D a r w i n-F o w 1 e r, which seems especially adapted to the problem, has been used throughout. In $ 4 the influence of the incompleteness of the degeneration is investigated for the case of equidistant individual energy levels. The connection is shown with problems in the partitio numerorum. (see esp. Note III).

$ 1. Ivhodzcction. B e t h e 1) has attempted to compute the density of the energy levels of a heavy nucleus in an excited state by comparing it with a F e r m i-D i r a c gas of 2 protons and N neutrons in a spherical box and with a certain temperature corresponding to the excitation energy. For the density v(E)dE of the energy levels of the individzlal particles he used the expression:

cp(c)dc = xVEI’~ & (1)

which holds asymptotically for a Ijarticle freely moving in a certain volume v; x is a constant. Further B e t h e assumes I’ proportional to the total number of particles A = N + 2. It turns out that the temperature corresponding to an excitation energy Q of cu 1OMeV is so low that the gas is still almost completely degenerated. With this model B e t h e could explain the extremely rapid convergence of the distance between energy levels .of. the total nucleus. Ajso quantitatively he obtains results of the correct order of magnitude for medium heavy nuclei, However, as G o u d s m i t has pointed out, there is a difficulty for heavy nuclei, for which B e t h e finds

- 531 -

532 C. VAN LIER AND G. E. UHLENBECK,

such a small value for the distance between energy levels at Q CG lOMeT/, that one should expect for all heavy nuclei an extremely large absorption cross section for slow neutrons *). We have tried to see if one can remove this difficulty by gen,eralizing the calculation of B e t h e in two respects.

a. First we have taken instead of (1) a general distribution function ~J(E)&, where ‘p may Aso depejnd oh tlie number of particles. This is done in 5 2 for one kind of particles and in 5 3 for two kinds of particles in temperature equilibrium’, which we treat, in contrast to B e t h e, with the.same method.

b. In the’second place we have considered for a special choice oj r&) corresponding to equidistant energy levels what the result will be when the degeneration of the gas is not almost complete. This may happen when the excitation energy Q becomes comparable with the zero point energy of the particles which can be excited. In a recent note 2) & o u d s m i t has discussed this second point for the same choice of (P(E), and~has drawn attention to the connection with problems in the partitio numerorum. We will complete his discussion by treating more in ,detail the case of intermediate degeneration.

From the physical point of view the results, whi,ch will be suma- rized in $5, are more or less negative. They show that with the gas model for the nucleus it will be quite difficult to explain finer details of the scheme of energy levels. We shall use throughout the method of D a r w i n-F o w 1 e r 3) t) which seems to us especially adapted to this problem.

$2. Orte kind of fiarticles. Let Ed be the individual energy levels, and ni (= 0 or 1) the number of particles with energy E+ Write :

N=Efii i

E = I: qni i

for the total number of particles resp. total energy. The density p(E, N) of the energy levels of the whole gas around the energy E

*) To calculate the distance between energy levels in which a slow -neutron can be captured, one has to take only those levels with an angular mdmentum I, f +, when I,, is the spin of the original nucleus (camp. B e t h e l.c. 5 4). But even with this correction the distance turns out too small.

t) In contrast to B e t h e, who uses the method of G i b b s.

DEN~ITYOFTHEENERGYLEVIX~OFNUCLEI 533

will be given by the number of ways in which the state E can be realized *).

This will be given by the coefficient of x”y” in the development:

B $ . . . . xJJ% n,==o n,-0

y+i = l-l(l + xy’i). i

Therefore :

where the integrals are taken around the origins of the complex x and y planes. The integrand has along the real x and y axis sharp minima at x = a and y = b where a and b are given by:

Iv+1=x l i ,p+/% + 1

when a = - log a, p = - log b. By letting the paths of integration pass through these points one can apply the usual method of steepest descent for the evaluation of (2) (See Note I). The result is:

PW,E) = 1

27t~/xoxz - n: exp {aN + PE + IS log (1 + e,“-bC+}, (4)

i

where :

(5)

For the further calculation the summations must be replaced by integrals, whereby :

when cp(z) is the density of the individual energy levels. For the case

*) One sees this most easily for the case when the si are whole numbers when measured in a certain small unit of energy 6. Then also E is a whole number and the distance between the energy levels of the whole gas will also be 8. Each of these levels,wilJ be strongly degenerated and the degree of degeneration will be given by the number of realizations of E. This number divided by 6 will give the density of the energy levels. It is not surprising therefore that the final answer (4) has the correct dimension of a reciprocal energy!

534 C. VAN LIER AND G. E. UHLENBECK

that the gas is almost completely degenerated (a Q 0) the integrals can be evaluated with the help of: “)

OQ au s 2!LEW gF(-a) + $F”(-a) ea+ + 1 da4

0

(6)

when F(0) = 0. Introducing the maximum energy co of the F e r m i distribution by :

N = 79(e) CEE (7) 0

and the excitation energy Q by:

Q = E -,i; (P(E) & (8)

one can express a and p in N an,d Q with the usual approximation method *). Substituting in (4) one obtains the final result:

p(N, E) = -L QdS

exp x 2Q U) 3x ’ (9)

where A = l/cp(~~) represents the spacing of the individual energy levels near the top of the F e r m i distribution. For the (P(E) given by (l), (9) reduces to B e t h e’s result (l.c. Eq. 21). It is remarkable that p depends only on N and r&) through A, which explains why G o u d s m i t with equidistant levels finds the same result as B e t h e, by taking their spacing equal to the value of A from the (P(E) chosen by B e t h e.

5 3. Two kinds. of fiarticles. For two kinds of particles the calculation is completely similar and we will therefore only indicate the results. Let Ebb, &ai be the energy levels, N,, N, the total numbers of the two kinds of particles and E again the total energy. The density p (N,, N,, E) of the energy levels of the whole gas will then be given by the coefficient of xyi $2 yE in:

[y + Xl Y'!i)l yv + 3 YWl*

Therefore one can again represent p by a threefold complex integral and, applying the method of steepest descent, one obtains:

p(Nl,N,,E)= &-& exp [BE +i!l(pjNi + X log (1 + e-?-&i)}] i

*) For the details, see Note II.

DENSITY OF THE ENERGY LEVELS OF NUCLEI 535

where ai and p are determined by the conditions:

E+1=L Cji j=l L eai+Pyi + 1

I x ! is the determinant:

x10 0 x11

IX I = 0 x20 x21

x11 x21 x12 + x22 where :

Let cpl(~) and ~~(4 be the densities of the indidual energy levels of the two kinds of particles. Then, by replacirg the sums by integrals one obtains in the same way .as before :

when both gases are highly degenerated. Here Q is again the excitation energy, Al .= 1 /cp,(~r~), A2 = I/c+J~(E~~), when clO, ~~~ are the maximum energies of the two Fe r m i distributions and l/A = l/A, + l/A,. For the case that A, = A2 = 2A one obtains:

PPL N21E) = 4 exp. x

The result given by B e t h e (l.c. Eq: 26) is a special case of this equation *) .

$4. Infitieme of the incom$etercess of the degerzeratiort. When the gas is not highly degenerated, the ,discussion of p(N,E) for a general cp(~) becomes quite complicated, because then the integrals for N, E, etc. have to be evaluated more exactly. We will therefore specialize 44 by taking ‘P( E cu tn where 12 is an integer. The physical sign& ) cance of this choice is for instance that this ,will represent the distribution of the energy levels for an isotropic harmonic oscillator

*) The proof which B e t h e has given seems to us less satisfactory, since he does not go back to the original picture of two gases in temperdture equilibrium.

536 C. VAN LIERAND G. E. UHLENBECK ,

in a space of PZ + 1 dimensions. Even for a general n the integrals can now be evaluated easily. Since however we are only interested in the order of magnitude and the trend of the influence of increasing the temperature we will only consider more in detail the case rt = 0, which corresponds to equidistant;energy levels. Let their spacing be A. This case has further the special interest that p(iV, E) becomes now equal to the number of partitions fiN(M) of M = Q/A in parts not exceeding N *).

For the case of high degeneration, which corresponds to N > M, &(M) will become independent from N. For large M Eq. (9’ gives then immediately :

(12)

which is identical with an asymptotic formula derived in another way by Hardy and Ramanuj an6). ‘For the case of intermediate. degeneration we must go back to Eq. (4), where again we will replace the sums by integrals. (Comp. also Note III). Calling:

one finds easily from (3), (7) and (8): M ~z. = 444 =

944 -4 log2 (1 + e-“) *

(13)

(14

Eq. (4) can then be written:

log.&(M) = f(a) 2/M - log M - g(a) where :

(15)

f(a)=-&lOg(l +I?“) +22/$lOg(l +e-“)

27q/2$e* - 1 g(a) =log +2/l +e-=log (1 + eWa)’

(16)

For the calculation of +, f and g we need therefore only to know cp(a). For positive values of a one has:

. . . . . . . .

*) This has been pointed out by G o u d s m i t *).,Wefound that an equivalent remark was already made by E u 1 e r (Introductio in Analysin Infinitesimalem, I, p. 2643, where one finds also a table ior $-N(M), with A’ < 11, M c 70. For a proof, see Note III.


For negative values of a, r&x) can be easily computed with the relation : *)

da> + d- 4 = d + $

Fig. 1. Dependence c If 4, f and g from the degeneration parameter a. (Comp. Eq. 15).

TABLE I

a = ra I + (4 1 f (4

0.01

i*; - 02

1 2

1: 100

100.3 0.6602 10.25 1.359 5.242 1.617 2.227 1.447 1.212 2.156 0.690 2.314 0.3561 2.444 0.2303 2.500 ,0.0746 2.557

Table I and Fig. 1 give.for different Glues of a : $(a), f(a) and g(a). From ( 16) follows that for large negative values (high degeneration) of a, /(,a) becomes x(2/3)‘/* = 2,565 and g(a) becomes 4 log 48

*) One has quite analogous ielations for integrals like (13) but with higher powers of u. They are alI pToved easily by differentiating after a.

538 C. VAN LIER AND-G. E. UHLENBECK

= 1,94, so that then (15) reduces to (12). .For large positive values of a (low degeneration), $(a) cu P, f(a) cu (a + 2) emal and. g,(a) cu log 2x. Eq. (15) may now be written.:

pN(M) 2 Mv-’ N! (Iv- I)! (17)

when one uses S t i r 1 i n g’s approximation. This is so to say the expression of the partition in B o 1 t z m a n n statistics. It can also be derived directly. One has in fact the recurrence relation:

fiN(M) = PN-1 W) + PNW - N> from which follows that for large M, &(M) is a polynomial of degree M - 1 in M, of which (17) is the first term. More precisely one finds:

MN-1

fid”) =N!(N- I)! . 1 1j-(N+1)N(N-l)-4@+..... . 1

$5. Discussiolz. Since Eqs. (9) and (11) have the same form as the results obtained by B e t h e, their consequences will be about the same. One may consider A as given approximately by the average energy of the y-rays between the first few excited states of the nucleus Experimentally A will then decrease with increasing A *). Conse- quently according to (9) and (1 l), p will increase rapidly and mo1zo- tovtically with A, so that one should expect that for heavy nuclei the distance of the nuclear levels near the~excitation energy Q will be very small. This monotonic behaviour may be weakened’somewhat since Q will decrease a little with increasing A. It is therefore vtot possible to change essentially the results of B e t h e by using an other distribution of the individual energy levels.

To see whether the degeneration is almost complete, one has to compute a g - ic, = - peO (see Note II). One finds:

a=-f3co=-GFA.

With Q = 8MeV, A = IO/A MeV, a becomes:

ElJd~ a g-- 7

when Ed is measured in MeV. With the v(c) of B e t h e E,, z 20tieV and this will always be the order of magnitude, so that even for light nuclei the degeneration will be very high (see Fig. 1). This conclusion .

*) One may take very roughly AN 1 O/A MeV. A will always be inversely proportional to A, when p(e) is directly Proportional to A.

DENSITY OF THE ENERGY LEVELS OF .NUCLEI 539

can only be changed if one assumes with G o u d s m i t a) that only a,few particles (say about five) can be excited. Taking equidistant. levels near the top of the F e r m i distribution one obtains then Ed = i(5 . 6) A = ISA and a = - 2O/dx, so that with increasing A the degeneration would also decrease. Even so, p will remain a monotonic function of Q/A; only instead of an exponential increase, p will be about proportional with (Q/A)*, as one sees from (17).

As an application of ( 15) we have still calculated fill(&) for a = exp (- a) = 2 . 5. According to (14) this gives M = 70,47, and from (15) one obtains then:

fi,,(70, 47) = 8,430 x 105.

From the tables of E u 1 e r one finds:

$,,(70) = 771474 $,,(71) = 852775.

Interpolating linearly, $,,(70,47) becomes 8,12 x 105, so that the error made by using (15) is about 4%.

NOTES

I. Consider in general:

I(N, E) - (2;ij2 dx dy 1 --

.$v+1 E+l efkY)*

Y (18)

The integrand will have minima for x = a and y = b where a and !j are given by:

aaf=N+l aa b$=E+l.

Take now for the paths of integration the circles x = aeig ard y = b&. The main contribution to the integral will then come from the neighborhood’ of 8 = 0, and cp = 0. Developing f(x, y) in powers. of 8 and ‘p, and using (19) one obtains approximately:

where :

af a2f xO=aaa+aZs,

a2f x1 = ab- af a2f aaab 1 x,=bab+bZw

540 C. VAN LIER AND G. E. UHLENBECK

By transforming the quadratic form in the exponent on principal axis, the integral can be’ easily evaluated, and the result is:

I(N, E) = ef(%b!

2xaNbE 2/x0+ - XT *

In our case: /(x, y) = ZZ log (1 + xy’i) and by substituting one

obtains (4).

(21)

II. Write: zc = @, u. = PEG, cp(&) = Jr”(%) with 4(O) = 4’(O) = 0. Then the equations (3) for a and p can be written, neglecting the one :

mqJp4)z4dz4 P2E =J /y+u + 1

0

or, according to (6):

f3N = $‘(- a) + -2 +“‘(- a)

‘$

(22)

p2E = - a#‘(- a) - $(- a) + r {$“(- a) - a$“‘(- a)}. (23)

We will now first eliminate a. Since PN = $‘(zcg) according to (7), a will in first approximation be .A tie. By writing a = - 21, - ZJ for the next approximation one obtains:

+ 9”‘@0) v _

6.6”0’

Substituting in (23) and introducing Q according to (8) one gets :

P2Q = ;V@o) = ; (P(E~),

which gives p expressed in N and Q. The rest of the calculation is straightforward. One always uses the formula of S o m m e r f e 1 d (6). One substitutes then the value a = - zdo - v and develops according to :

F(- a) = F(zl,) f’v g.


One gets for instance:

/3 ~dq+) log ( 1 + e-“+) = +(.z+J - d (y - +,“I 0

P4(xox2 - XT) = n; {(J”(uo)}2.

By substituting in (4) one obtains (9).

III. The equivalence of p(N, E) and fi,(M) is illustrated by gig. 2. At the left we have an example of fi6( 16) ; at the right we have added columns 6, 5,4,3,2, 1, that is the zero point energy &N(iV + 1). By considering the r,ows one sees now that one has 0btained.a partition of E in N unequal parts, or in other words a distribution of a F e r- m i-D i r a c gas of N particles over equidistant energy levels, so that the total energy is E.

w WI . 5 l 5 . . .

4 . . . Q. . . . . .

3 . . . 3 . . . . . . . 2 . . . . .2 . . . . . . . . 1 . . . . . 1 . . . . . . . . . . .

t&16=5+4+4+2+1 ~&~=,6+21~6+5+5+4+4+4+5+2+2+1+1

Fig. 2.

,It may be of interest to remark that the determination of&(M) is also equivalent with two problems in B 0.s e statistics. This is illustrated in Fig. 3. One sees by considering either the columns or the rows that any partition of &( 16) corresponds either to a distribution of a B o s e gas with energy Q and with an indetermined number of particles over 6 equidistant energy levels, or to a distribution of a B o s e gas of N particles over an infinite number of equidistant energy levels so that again the total energy is Q. With the help of the generating functions in B o s e statistics one obtains therefore the following other expressions for &,,(M) : *)

(25)

*) Also these have been essentially given by E u I e r.

542 DENSITY OF THE ENERGY LEVELS OF NUCLEI

For the ordinary partition 9(M) where N > M the product in (25) can be taken till infinity and in this form (25) has been used by H a r d y and R am a n u j a n. When one applies the method of 5 2 and Note I to these integrals one obtains without trouble the main term in (12) and (15). To get the other terms one has to be careful with the replacement of the sums by integrals *). The expression forp(M)connected with(25) gives for instance only the correct

9 9’

N4 7 , 5 l N,6 6

4 . . . 5- ‘5 - J . . . 4- 4-

2 . . . . 3 3- I . . . . . 2- 2

M,16=5+4+4+2+1 1 - 1- Q-

Fig. i.

power of M in the denominator of ( 12)) when one uses the next term in the E u 1 e r-M a c L au r i n development. The constant term 2/48 one does not obtain at all. When one starts from (26) one gets the first two terms of (15) even when one uses throughout integrals instead of sums, but one does rtot get the same expression for g(g). The form (16) of g(a) is therefore rather doubtful. However since it gives the correct limiting values and since the variation is small, the true expressions for g(a) will probably not differ very much from ( 16).

Received May 20, 1937.

REFERENCES

1) B et h e, Phys. Rev. 50. 332 (1936). 2) G o u d s m i t, Phys. Rev. in press. 3) Comp. F o w 1 e r, Statistical Mechanics, Ch. II (Cambridge Univ. Press 1929). 4) Sommerfeld, 2. Phys. 47, 1 (1928). 5) Hardy and Ramanujan,~Proc.LondonMath.Soc.17,75(1918); Rama-

n u j a n, Collected Papers, Cambridge 1927.

*) This is connected with the difficulty regarding the so-called .,condensation pheno- menon” which according to E i n s t e i n, a B o se gas would show at very low tem- peratures. This prediction was not correct, again because for high degeneration it is not allowed to replace the sums by integrals.

Documents

On the statistical calculation of the density of the energy levels of the nuclei