Source: "Nuclear Physics"byJohn Lilley

Source: "Introductory Nuclear Physics" by Kenneth S. Krane

Source: "Nuclear Physics" by Irving Kaplan

Source: "Concepts of Modern Physics" by Arthur Beiser

Source: "Modern Physics" by Randy Harris

Source: "Radiation Physics for Medical Physicists" by Ervin B. Podgorsak

Source: "Modern Physics" by Paul A. Tipler and Ralph A. Llewellyn

Nuclear Physics 25 (1961) 1 - -135 ; ~ ) North-HolMnd Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

S E M I E M P I R I C A L ATOMI C MASS LAW

P H I L I P A. S E E G E R

California Institute of Technology, Pasadena, Cali/ornia

and

Los AMmos Scientific Laboratory, University of California, Los AMmos, New Mexico *

Rece ived 30 J a n u a r y 1961

A b s t r a c t : New t e r m s are p roposed for t h e a t o m i c m a s s law wh ich incorpora te effects of nuc lear d e f o r m a t i o n a n d shell s t r u c t u r e inc lud ing in te r fe rence effects be tween unfi l led n e u t r o n and p ro ton shells. I n addi t ion , a s u r f a c e - s y m m e t r y t e r m is inc luded. Cou lomb t e r m s are f ixed b y r ecen t radi i d e t e r m i n a t i o n s in e lec t ron sca t t e r ing . The s t a n d a r d dev ia t i on of ca lcu la ted m a s s e s is r educed b y t h e add i t i on of t he new t e r m s as is t h e er ror in e x t r a p o l a t i n g f rom one side of t he va l l ey of b e t a s t ab i l i t y to t h e o ther . T he use fu lness of t h e law for ca lcu la t ing t he m a s s a n d b ind ing ene rgy of a t o m s w i t h large n e u t r o n excess is d iscussed. A table g iv ing m a s s excess, n e u t r o n , p r o t o n a n d a lpha-pa r t i c l e b ind ing energies, and be t a decay t r an s i t i on energies is appended .

1. Introduct ion

An improved expression for atomic masses as a function of position in the NZ-plane has been sought for the purpose of studying in detail the process of nucleosynthesis by rapid neutron capture 1, ~). For this purpose, it must be possible to extrapolate the mass law to atoms with high neutron excess, but there exists no means of directly testing the law in this region of the NZ- plane. Criteria for testing the law in the valley of beta stabili ty are listed in sect. 2.

The familiar Weizsiicker semiemperical mass law 3, 4) consists of terms whose forms follow from the statistical model of the nucleus, each multiplied by a coefficient to be determined empirically. The mass excess of the atom with Z protons and N neutrons was taken to be

AMw(Z, A) = NMn+ ZMH--A--ocA +~I2/A + v A i + (3e2/bRo)ZP"/A½, (1)

where Mn and MH are the neutron and hydrogen atom masses, A = N + Z, and I = N- -Z . Modifications of this original form will be discussed in sects. 3 and 4.

The free coefficients, such as ~, r, and y in eq. (1) and similar quantities appearing in modified forms of the mass law, are found from least squares analysis by fitting either to the mass excess M-A or to the packing fraction (M-A)/A. Fitting to the packing fraction is equivalent to using a weighting factor of 1/A ~ in the mass excess, and since A varies over a wide range, the results here reported were determined by fitt ing to the mass excess. In many

¢ W o r k suppo r t ed in p a r t b y t h e jo in t p r o g r a m of t h e Office of N a v a l Resea r ch and t he U. S. A t o m i c E n e r g y Commiss ion .

1

May 1961

2 P . A . SEEGER

past t reatments of the problem, a l imited number of masses for the atoms nearest the line of beta stabil i ty have been used. In this study, however, all available odd mass atoms were used, all data being weighted inversely as the first power of the estimated error. The principal source was the table of Ever- ling et al. 5); masses omit ted from that table but estimated in the earlier work of Huizenga e) were corrected to the later value of Pb 2°8 by addition of 1.123 mMU and used with the quoted errors doubled. Thus a total of 488 odd-mass atoms with A --> 19 were used to determine a mass law for odd-mass atoms. The calculation of atomic masses for even mass nuclei from the law is discussed in sect. 5. All computations were made on an IBM 704 at Los Alamos Scientific Laboratory.

2. Criteria for Test ing of Mass Laws

Three criteria for testing the mass laws have been considered. First, the errors ~ of the masses calculated by the law compared to the input data were plotted, and the standard deviation calculated by

f ' (2)

where n is the number of Jinput masses, k the number of free coefficients, and wi the weight of the input datum. This is a measure of the abi l i ty of the law to fit known masses; it is to be made as small as possible. A standard deviation a ~ 1 MeV/c2 ~ 1 mMU is considered to be satisfactory.

As another test, called the " interpola t ion" test, the 488 odd-mass atoms were divided at random into two groups. This was done by ascertaining whether the last bit in the binary expression for the mass excess was even or odd. The first group, consisting of 232 masses, was used to determine the coefficients of a mass law of the same form as the law in question, and then that law was used to "pred ic t " the masses of the 256 atoms in the second group. The standard deviation al of the first group was calculated by eq. (2), and that of the second group by

a s = \ Xw, / " (3)

The ratio (o'Z/O'l)ln t is taken as a measure of the abil i ty of the particular form of mass law to interpolate; a 1 should be approximately the same as a, and the ratio (a2/al)tnt should be less than, say, 1.05.

For our purposes the most significant criterion was a third test called the "extrapolat ion" test. The 488 odd mass atoms were again divided into two groups: the first containing those atoms on the line of beta stabil i ty plus those on the neutron poor side of the valley of beta stability, the second containing atoms only from the neutron rich side of beta stability. The first group had 282

SEMIEMPIRICAL ATOMIC MASS LAW 3

a toms and the second had 206. The s t anda rd devia t ions a 1 and a , and the ra t io (az/al)ex t were calculated as in the in te rpola t ion test . Once again al should be similar to a, and the ra t io (az/al)ext should be close to uni ty . I t is diff icult to es t imate an acceptable value for this rat io, bu t we might hope to make

(0"2/0"1)2Xt ~<~ 2 o r (o'2/O'l)e2xt ~ 1.4.

3. Standard F o r m of the M a s s L a w

For our " s t a n d a r d " form of the mass law, the s y m m e t r y t e rm was modi f ied to include ]I] as well as I2 as suggested b y Wigner 7, s) and the coefficient fl was modif ied to include composi t ion dependence of the surface effects 9). The fac tor 2 in 21I I was suggested b y E. E. Salpeter lo). Following Mozer n) , the Coulomb t e rm was modif ied to t rapezoidal charge d is t r ibut ion and exchange effects. The coefficients 3e2/5Ro = 0.8076 MeV/c 2 and -~(t/Ro)* = 2.29 were de te rmined using R o ~ 1.07 fm, t ---- 2.8 fm f rom sca t te r ing exper iments wi th high energy electrons 12). Then in MeV/c 2 on the 016 scale the mass excess becomes

n AMstan(Z,A)=8.3674N+7.5848Z--o~A+ (f l - --~) ( ~ )

(4) Z 2 ( 0.7636 2.29~

+yALe-0 .8076 ~-~ 1 Z{ A-~ ] "

The coefficients as de te rmined b y least squares are

,¢~= 16.11 MeV/c 2, y = 20.21 MeV/c 2, ' (4a)

fl = 20.63 MeV/c 2, ~7 = 48.00 MeV/c 2.

The errors in the calculated mass excesses are shown in fig. 1, and the three evaluat ion tests yield

S tanda rd devia t ion: a = 2.61 MeV/c*; In te rpola t ion : a 1 ---- 2.87 MeV/c*, ( a 2 / a l ) l n t = 0 . 8 4 ,

Ext rapo la t i on : a 1 ---- 2.19 MeV/c 2, (a2/al)ex t ~ 1.75.

I t will be no ted tha t the s t anda rd form fails in the two most crucial tests: the s t anda rd devia t ion a and the ex t rapo la t ion ra t io ( a 2 / a l ) e x t a r e too large.

4. Addi t ional T e r m s in the Mass Law

In the a t t e m p t to f ind a mass law which would ex t rapola te more., ,ccessfully than the s t anda rd form (4), a funct ion S(N, Z), represent ing the increase i~ nuclear b ind ing energy bo th at closed shells due to magic numbers of N or Z and be tween closed shells due to the deformat ion of the nucleus, was sub t rac ted from zJMstan to give AMo, the mass excess for odd-mass atoms. Thus AM o ~ AMstan --S(N, Z). We m a y now wri te

SCN, Z) = ksCo+kd(Cl~--C2~2), (5)

er" o cE r~ taJ

P . A . S E E G E R

where k s and k a are shell and deformation coefficients which may vary from shell to shell, Co(N , Z) is the functional representation of the excess binding energy due to shell effects, and the extra energy due to deformation is approxi-

14 ' r ' ' ~ ' ~ ' ' ' , [4

o

~ 2

_ 4 ¸

-E

- 8

-tO

-12

- 14

-16

-18

-20 0

+ + + + 4 + 4- + + F " + 12

o e

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+

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4- +

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6 4 + + + + + + + +. . + " . + " . -

~ . . + 4 + . + + +" + .4; "" + " " ~ 4 • I e •

.e • • * .e •

- ~ ee ~ - • * J r - } - + ~ - J~ • ~ J ~ ° • -{- - 7 " - } - . +

eee e eo • %o ~ ° • • • • %

• ee ..." ". ."i • "" =*f,.. ." •

' - ' " " ' ": " ', I I r " P" r' • i J ,~ • o~ i • % n • •

" . . . . . : -, ,.-. • . ;! • . . . • o* , o • ° ° • • •

• ° . • • • . % | • • •

+ 4 . . I + t ~ ' . . , . + " " ' ~ " " "+ 4" -'I- . " i + • + s e . . . . " ' : e " " " ° "

e l l • • • • I ! •

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+ +." + " + "-# .4 . •

+ 4- +" + +

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4 '. k + "'+ J °8°Q : 8, • ee o%*

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4 F • - °

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e=

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+ + 4 + 4 + + + 4- 4- "

MASS NUMBER A

+

+

-4

-6

-8

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-12

-14

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-20 280

F i g . 1. E r r o r s ~ o f 4 8 8 a t o m i c m a s s e x c e s s e s f o r o d d - m a s s a t o m s c a l c u l a t e d b y t h e s t a n d a r d m a s s

l a w (4) w i t h c o e f f i c i e n t s ( 4 a ) , p l o t t e d v e r s u s a t o m i c m a s s n u m b e r .4 . N o t e t h e l a r g e s y s t e m a t i c

s h e l l e f f e c t s .

SEMIEMPIRICAL ATOMIC MASS LAW 5

mated by C17--C~7 ~. In this last expression 7 = A R / R is the spheroidal eccentricity, C17 is the excess nuclear binding energy due to deformation and C272 is the change in Coulomb and surface energies in deforming a uniformly charged sphere into a spheroid. By the usual argument that the nucleus takes that deformation 70 such that the deformation energy is a maximum,

d--~ (C17--C~72) = C1--2C2% = 0, o

C 1 C1 ~ 7o = 2-C~' C17°-C~702-- 4C---2"

The coefficient C~ can be expressed as 13)

C 2 ~ 8A' (1--0.005 ~ ) MeV/c 2.

In this study, C~ has been considered constant over the range of A included within any one shell of N or Z.

The function C17 can be calculated as a sum over all nucleons outside a closed shell. Each such nucleon is in a spheroidal rather than spherical potential, and the excess binding energy can be calculated b y the methods of Rainwater 14) or Nilsson 15). It is to be expected that C1 is zero at a nucleus for which both Z and N are magic, and has a maximum near the middle of each shell of Z or N. This behaviour makes it possible to approximate C 1 with sine functions.

The term C o must be a maximum at closed shells and reach a minimum near the middle of the shell. I t has therefore been assumed that C o = K - - C 1 , where K is a constant.

With these assumptions, (5) becomes

S = ksK--k~Cl+kaC12/4C2, (6)

where ksK, ks and ka/4C2 are constants within any given shell. For the nucleus with N neutrons and Z protons, let N ' and Z' be the fractional occupations of the last shells,

N ' - - N- -Nj Z'-~ Z--Zk N ~ + I _ N , Z~+I-- Z~' , (7)

where the magic closed shell numbers are

Nj , Z~ ----- 8, 20, 50, 82, 126, 184,

and

N~ ~_ N < Nj+I, Z~ g Z < Z~+ 1.

Consider eq. (6) if C1 is expanded in a Fourier sine series,

C 1 = A 1 s i n N ' ~ + A 2 sin 2 N ' g + . . . -t-B 1 sin Z ' ~ + B 2 sin 2 Z ' ~ + . . . .

6 P . A . S E E G E R

Since the square of the sine has the same qualitative properties as the sine itself, that is, goes to zero at magic numbers and has its maximum halfwayin between, such terms in C1 ~ add nothing essentially new to our approximation for S which is not already included in the linear terms in C 1. The cross-products in C1 ~, such as (sin N'~t)(sin Z'~r), are unlike any terms in C1 itself, and must be included in S. Previous at tempts 11,1e-is) to include shell structure have used S = / (N)+g(Z) , which neglects cross product terms. We feel it is essential to include these terms in order to have some representation of the effects of the proton shell structure on the neutron shell structure and vice versa. Thus, the final approximate form adopted for S was

Si~(N', Z') = ~ sin N ' z t + ~ sin Z'zt+v~ sin 2N'~

+v~ sin 2Z 'z t+ (¢j+$~) (sin N'zt) (sin Z'z~)+Z, (8)

where indices/" and k refer to the shells of N and Z respectively, and the ~ , v,, ¢~, and Z are constants approximately related to the constants in eq. (6). Note that Sik is symmetric in N and Z, so that it is charge independent. Also note that the constant term Z does not vary from shell to shell, in order to make S¢~ a continuous function. Then the mass law for odd mass atoms is

3Mo(Z, A) = AMstan(Z, A)--Sj , (N' , Z'). (9)

The coefficients c¢, fl, 7 and ~/ as determined by least squares are, in MeV/c 2,

,¢ = 17.06, fl = 33.61, 7 = 25.00, ~ /= 59.54. (9a)

The coefficients ~, v, ¢ and Z as determined by least squares are given in table 1.

TABLE 1

T he coeff icients 6, v, ~ a n d ~ (in MeV/c 2)

~ Z o r N < ~ v ~b Z

8, 20 20, 50 50, 82 82, 126

126, 184

4.008 - - 0.508 - - 7.636 - -15 .63 - -27 .59

- -0 .428 2.331 0.496

- -2 .284 2.660

- - 1 , 3 8 9

- -0 .463 1.950

10.67 26.99

13.51 13.51 13.51 13.51 13.51

A representation of the function Sj, versus N and Z is given in fig. 2. Fig. 3 is a plot of the error of the calculated mass excess versus A. The three evaluation tests give

Standard deviation: a = 0.75 MeV/c2; Interpolation: a 1 : 0.83 MeV/c z, ( a J f f l ) i n t = 1.01; Extrapolation: a 1 = 0.96 MeV/c ~, (a2/az)ex t = 1.35.

Note that the standard deviation is reduced to less than a third of the previous value, and that the ratio (a2/al)ex t is substantially reduced from that for the