Physics of atoms
and molecules
2nd edit ion
B.H. Bransden and
•
An Impr mt of
c.J. Joachain
Pearson Education
Harlow, England ' London · New York · Boston . San Franctsco . Torc nt o • Syd ney ' Smq apc re . Hong Kong
Tokyo ' Seo u! - Taipe i ' New Delhi ' Cape rown . Maond • Mell,ICO City . Am sterd am • Muruc h • Paru . Milan
374
Ma ny-e lectro n ato ms
8.3
A s we a lreudv showed in Section -1.2. the a llow~d co rnponern s of the wave LeCto r
k a re the n given b\ (see (-l.57))
2rr
J.. ,=/:",.
' rr
k =::"-11
l. .'
2rr
k.=-Il.
L
.
where II , . II and I L a re po-i tive or neg ativ e int eger s. or ze ro. Th e num ber o f spa.
tial o rb ita ls in th e ran ge dk = dk , dk , dk_ is ( L ,2rr)' dk , dk, dk , a nd th is number
m ust he multipl ied bv 1 to ta ke into accou nt the two possib le spin sta tes . A unit
vo lum e o f k-space will th er ef o re accomm od ate V (-Irr') el ectr on s (with V = L J).
Thus. th e indi vidua l e lect ro n sta tes havin g e ne rgies up to E = f1' k' /(2m ) will be
co nta ine d within a sphe re in k-s pace . o f radi us k . the number .\', of these sta tes
be ing given by
v = _\_' ':'rrk -'=
'.
-Irr 3
=~
(2I1l ,"J
:lrr' 11'
_ 1_Vk '
3rr'
VE "
(8.47)
in ag ree me nt wit h (K36 1.
We have see n above that in the grou nd sta te of th e Fermi e lec tro n za s the N
e lec tro ns fill a ll the sta tes up 10 the Fermi energy E,-. Thus in k -sp ace a ll"state s up
to a maximum va lue o f k eq ua l to k F are then filled. wh ile th e s ta tes for w hich
k > k, arc e mp ty. In o the r words a ll occupied spi n-o rbita ls o f a Fe rm i e lec tro n gas
a t 0 K fill a sp he re in k-space having radius k,. Thi s sp he re . whi ch is called the
Ferm i sphere. obvious ly con ta ins
_1_ Vk ' = V
31t ~
f
(8.48)
•
T he th eor y de ve loped indepen d e nt ly hv L.H . T hom a, and E . Fe rm i lor the
!!ro und sta te o f comple x atoms Io r ions hav ing a large numbe r of ~k~tron, IS
base d on 1"II1/;51;CII[ and '<'IIl ;'c!IISS;ca[ co nsiderat io ns. Th e .v ele ctrons o t the ,y 'te rn a re tre at ed as a Fermi el ect ron gas in th e gro und sta te . confined to a regio n
o f space by a ce n tra l potenti al V cr) whi ch vani sh e s at infini ty. It is ass ume d th at
this potential is slo wlv va rying o ver a dista nce which is la rge compa red wit h the
de Broglie wa velen gth s of the e lectro ns. so tha t eno ugh el ec trons a re pre sen t til a
vo lu me whe re VCr) i, nearly const ant , a nd therefore the sta tis tica l a pproac h used
in studvinz th e Fe rmi electr o n gas ca n be a pp lied. In ad d itio n. sinc e the n umbe r
o f electrons is lar ue . man y o f th em ha ve high prin cip al q ua nt um number s. so tha t
se mi-class ica l methods sho uld be usefu l.
The aim o f the Thom as-Fermi model is to provid e a method o f calc ula ting the
pot enti al VCr) a nd th e elec tro n de nsit y per). Th e se two quant ities ca n first be
rel at ed bv usin g the fo llowing a rgume nts. The to ta l e ne rgy o f a n e lec tro n I' wn tten as p' ;(2111) T V( r ). a nd th is e nergy ca nno t be po sit ive . oi herw isc th e e lec tro n
would esca pe to in finity. Sinc e th e max im um kineti c ene rgy o f an e lec tro n in a
Fermi e lect ron gas a t 0 K is the Fermi e ne rg y E r . we wri te for the tot al e ne rgy of
the most en erg;tic e lec tro ns o f th e sys te m th e cla ssical eq ua tio n
(1l.52)
It is cle ar tha t Em" m ust be inde pe nde nt o f r. becau se if thi s we re no t the case
e lectro ns would m igr at e to th at re gion o f space wh er e E"-,, is sma llest. Il1 o rder to
lowe r th e tot a l e ne-rg\ of th e syst~ m. Furth ermore. we mus t have Em" ,,;; O. We
not e from (8 .50) and'iS- 51) tha t the q ua ntity k, is now a funct io n o f r. Th at is.
k '(r) = 2m [E
EF -£
k'F
'1J11
fl ~
mJ\
- VC r)]
(8.53)
Usi ng (8 A1) a nd (\ 052) we the n have
k F = (3 rr'p) "
At th e surface o f th e Fermi sphe re. known
Ferm i e ne rgy
375
The Thom as- Fe rm i theory for multie lect ron ato ms an d ions
F
spin-orbita ls, so tha t
The Thomas-F ermi m od el of the ato m
(8.49)
,\>
(8.54)
the Fermi surface, the c ne rsv is the
,.
(8.50)
a nd we not e th at the result (K-I I) follo ws up on s ubs titutio n o f (K-I9) in (8.50). It
is a lso co nve nie nt to int roduce th e Fe rm i mom entum PI ' ve locity " F a nd te mpe ra t u re T, such th a t
a nd we se e tha t p va nishes w he n V = Em'" In the classica lly forb idd en reg io n
V > E we mu st se t p = O. since o t he rwise (8.5 1) wou ld yield a negati ve va lue
o f th e ;~ximum kin etic e nergy E,. Let us de not e by
1
e
6( r) = - -V (r )
(8.55)
th e ele ctros ta tic po te n tia l a nJ by Iil, = -Emje a non -nega tive co ns ta nt. Se tti ng
(8.5 1)
whe re k B is Bolt zm ann's co ns ta nt.
<I>(r ) = ¢( r) - ¢v
we see th at per) a nd <I>( r ) a re rel at ed by
(8.56)
8.3
Many-electro n ato ms
(8.57a )
(31l)' \
31t - fr
<I>
°
a nd
z (-X),"
=
r dr '
dx ' = x
~d', [r<l>(r)] = + ( 2~ )
,lrr E" Ii
On th e o the r hand , whe n <I>
1= 0,
dx'
..!:!-
, .-AI
-l1t £ ()
<1> "' 0
(8.59)
< 0 we see fro m (8.57b ) a nd (8.58) th a t
<I> < 0
(8.60 )
(8.61)
On th e other ha nd. since th e N electro ns o f the syste m a re ass ume d to be co nfined
to a sphe re o f rad ius r". we mu st have the ' no rma lisa tio n' co nd itio n
-+rr
j'" p(r) r ' dr = N
(8.62)
n
In or de r to simplify th e abo ve eq ua tions. it is co nve nie nt to int rod uce the new
dim en sio n less variable x a nd th e func tio n Xix) such that
r = bx.
r <l>(r ) =
Ze
-xix)
-+rrEo
_I'
·x " '
X< O
x (O) = 1
[e<l>(r )Y" .
(8.63)
(8.65b)
(8.66)
(8.67)
In ad ditio n. th e bound ary co nd ition a t r = 0, exp ress ed by (8.61 ), now reads
J/2
Fo r I' ~ 0 the leadi ng te rm of the electro sta tic po ten tial m ust be d ue to the
nucle us. so tha t the boun dar y co nditio n at r = 0 rea ds
lim r <l>(r ) =
x <o
d OX = O.
Ell
Th e eq ua tio ns (8 .57a ) and (8.58) a re t w o sim ulta ne o us equa tio ns fo r per ) a nd
<1>(1' ). E limi na ting per) fro m t he se eq ua tio ns. we find tha t for <I> '" 0
(8.65a )
.
T his is kn own as the Th oma s-Fermi equation . For negat ive X we see fro m (8 .60)
a nd (8.63) that
(8.58)
p(r )
x
O.
d 'X
T reat ing the cha rge d e nsity -ep( r) o f the e lec tro ns as co ntinuo us. we may use
Poisson 's eq uatio n o f electros ta tics to write
1 d'
e
= ----;[r <l>( r )J =-
(8.64)
a nd th e impo rta nt eq uatio n (8.59) ma v he writte n in d imen sion less fo rm as
the po int cha rge Z e o f th e nu cleu s. loca ted a t th e o rigin:
(ii) th e d istrib utio n of e lectricity d ue to the N ele ctrons .
( i)
d'
_'.\
= (-+rrE,,)f,' /(m e' ) is the first Boh r rad ius. The relati o n (8.5 7) then becomes
110
p= - -+ rrb'
°
---;- [r <l>( r)
dr -
_
(8.5Th )
<0
Th e equa tio n <I> =0 (tha t is. <D = iPo or F = Em.... ) may be th ough t o f as dete rmining
the 'boundary' r = r" of the atom (io n ) in this mod el. Now. for a neu tr al atom
( N = Z ) the elec tro sta tic po tenti a l ot r) va nishes at the boundary. so th at we sha ll
set I!J ) = in th at case. O n the o the r han d l!Ju > for an Io n. .
.
A second rela tio n bet ween p er ) a nd <I>(r ) ma y be obta ine d as follows. Th e
sources o f the e lectro sta tic potent ial cp(r) a re :
r dr -
_,
b = 2""" Q"Z ' = 0.88)3 11"Z .
u.
17 ' <I>( r)
377
whe re
(2m)" [e<l>(r )r. ' .
per) = -I, ---;-
=
The Thomas- Fermi model of the a to m
(8.68)
It is clear fro m (8.66) a nd (8.6 7) th at xix) has at mo st o ne ze ro in the int e rval
(0. +00). Let Xo be the positio n o f this zero. From o ur a bove d iscussio n we have
Xu = r" l b , whe re r o is the 'b oundary ' o f the syste m. We also not e th at X > 0 for x < X o
a nd X < 0 for x > x". Mo reover. the eq ua tio n (8 .67) has th e so lutio n X = C(x - x o).
wher e C is a nega tive co nsta nt, wh ich mu st be eq ua l to X'(xo). As a resul t, the
solu tio n xix) is e ntirely de te rmined if we kn o w it for X '" O. We also re mark tha t
for a n y finite Xo the q ua nt ity X' (xo) must be diffe re nt from zero . since othe rw ise
bo th X a nd X' wo uld va nis h at x = x o, a nd the equa tio n (8.66) wo uld yield the
una cceptabl e trivia l so lu tio n X = O.
Th e Th om as-Fermi eq ua tion (8.66) is a ' unive rsa l' eq ua tio n, which d oes not
de pend o n Z . nor o n physica l co nsta nts such as "' . m o r e which have been 'scaled
o ut' by per forming the change of vari able s (8.63). We also no te tha t it is a seco ndorde r , no n-linear diff er en tial eq uatio n. Since the bo u ndary co nd ition at the origin
(8.68) only specifies o ne co nstraint, the re e xist a who le family of so lutio ns xix)
sa tisfy ing the Th om as-Fermi eq ua tio n (8.66) an d the co nditio n (8.68) . which differ by their initial slope x '(O). It is also clea r fro m (8.66) that all th ese so lu tio ns
must be co nca ve upwards. As illustra ted in Fig. 8 .4, we ca n classify them in to th ree
cat ego ries:
1.
2.
3.
a so lutio n whic h is asy mp to tic to the x axis:
so lutio ns whi ch van ish fo r a finite value x = X o:
so lutio ns whic h never va nish and diver ge for lar ge x .
37 8
M an y-ele ctron atom s
8.3
:<
Table 8.5
Values of the function
xl» ~
xl»~
1.0
N
Z
Z
CD
0
X'I
x,
.r
0.00
0.02
0.0 4
0.06
0 .08
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Th e Th omas-F ermi m od el of th e ato m
for neut ral atoms .
xl»~
1.000
0.972
0.947
0.924
0.902
0.8 82
0.793
0.721
0.660
0.607
0.5 6 1
0.5 21
0.4 85
0.9
10
1.2
1.4
16
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
379
0.45 3
0.425
0.375
0.33 3
0.29 8
0.26 8
0.242
0.220
0.20 1
0. 185
0. 171
0. 158
0. 146
xl»~
3.4
3.6
3.8
4.0
4 .5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
0.1 35
0.125
0.11 6
0.10 8
0.091 8
0.07 87
0.0 679
0.0592
0.0521
0.0461
0.0409
0.0365
0 .0327
xl»~
9.0
9.5
10
11
12
13
14
15
20
25
30
40
50
0.02 95
0.02 6 8
0.0244
0.0204
0.0172
0.0147
0.0126
0.0109
0.0058
0.00 35
0.002 3
0.00 1 1
0.00063
Figure 8.4 The th ree catego ries of solutions of the Thomas-F ermi equation:
.
(1) neutral atom solution ; (2) solution corresponding to a positive ion (N < l ); ( 3) solution corresponding to a neutral atom under pressure.
T he physica l me anin g of the sol utio ns bel on gin g to th e first two ca tegories
mav be ob ta ine d bv loo ki ng a t th e ' norma lisa tio n ' co nd itio n (8.62) . T akin g into
acc~u n t C8.63). (8.65) a nd (8.66) . thi s co nditio n re ad s
tV = Z
J'"x' 'x'" d.r
u
=
z J.:
x(x) = 1 - 1.588x + .
xX" dx
= Z[ ·\X' -
xl<: '
(8.69)
Us ing th e bo u nda ry co nd itio n (8.68) and the fac t th at X(x,,) = 0, we then have
N -Z
x"X'(x o) = - Z
- -
(8.71)
and is the refo re the (uniq ue ) so lutio n cla ssified above in the first cat eg ory . We
remark th a t since X(x) vanishes o nly a t infinity. there is no 'bo unda ry" to the
neu t ral ato m in the T homas-Fermi model.
(8.72)
Knowing t he universa l fu nctio n X(x) . we ca n ob tai n the fu nc tio n <I>(r ). a nd
hence th e e lec trosta tic pot ent ial <1)( r). th e pot ent ial ene rgy VCr) a nd th e densi ty
per). Us ing (8.55) . (8.56). (8.63). a nd rem ember ing th a t <Po = 0 for a neutral atom.
we see that in the Th om as-Fe rmi mod el the ce nt ra l p ot ential VCr) is give n for
ne utra l a to ms by
(8.70)
Le t us first co nside r neu tral atoms fo r which .V = Z . T he co nd itio n (8.70) then
req uires tha t X' (x o) = O. so tha t X' sho uld vanish at the sa me poi nt as X. Sinc e th is
co nd itio n ca nnot be satisfied for a finite val ue .r., by no n-trivial sol ut ions , th e poi nt
.r, mu st be at infinity. As a conse q ue nce . the so lu tio n X(x) co rres po nd ing to a
neutral ato m mu st be asy mpto tic to the x ax is, na me ly
xH =O
T he Th om as-Fermi equa tio n (8.66) an d the bo u nda ry co nd itio ns (8.68) a nd
(8.71) define a universal j unc tio n X(x ) for a ll neutral a to ms . V alu es o f th is
fu nct ion , obt a ined by num e rical inte gration. are give n in Table 8.5. We rem ark
from th is tab le th at X(x ) is mon ot onica lly d ecr e asin g . It ca n be show n that the
as ymp tot ic fo rm o f X(x) for lar ge .r is give n by th e fun ction 14-1x " . At .r = 0 o ne
has X' (O) = - 1.588 so that in the vici nity of th e or igin
Ze'
V(r ) = - -- x
(-Ire f o )r
(8.73)
As r -7 O. we have VC r) ..., - Ze'/(-IrefoT). More p recise ly. we ded uce from (8.63) .
(8.72) a nd (8.73) tha t for sma ll r
Z" )
Z + 1.794 VCr) = - e' - ( - 4rrco
r
(8.74)
au
o r. using a to m ic units.
Z
VCr ) = - - + 1.794Z' , + . . .
r
(8.75)
380
Many-electron ato ms
8 .3 The Thomas-F ermi m od el of the a to m
T he rirst te rm is the nuclear a tt rac tio n while the second o ne . which is re pul ive,
ar ises fro m the co ntrib utio n of the e lectro ns. Whe n r --> ~. we see fro m (8.71) and
(8.73 ) that r Ver) --> O. so th at the T homas-Fe rm i pot e ntial (K73) falls o ff more
ra pid ly than I/r for la rge r. Th is be hav io ur is a t va ria nce with the res ult (8.&)
wh ich we ob ta ine d in o ur d iscussio n of the ce nt ral field a ppro xima tion. The
reason is that the pot enti al V d iscussed in Section 8.1 is the o ne felt by a n a to mic
electro n. wh ile the T ho mas-Fe rmi po te ntial ( .73) is tha t e xperie nce d by an
infini tesimal negati ve test cha rge. Th e d iffe ren ce betw ee n the two po te ntials is
du e to the sta tistical a nd se m i-classical approx ima tio ns mad e in the Th om asFe rmi mode l. the Th om as- Fe rmi result becom ing e xact in the limit whe n fl a nd e
te nd to ze ro . wh ile the numberV (= Z ) o f electr o ns becom es infinite.
T urni ng now to the e lectro n de nsity p ( r ). we see fro m ( .65a ) th at it is similar
fo r all a to ms . e xce pt fo r a d iffer ent len gth sca le. which is det erm ined by the
qu antit y b (see (8.6-+ )) a nd is proporti on al to Z - I.3 A s a result, the ra d ial sca le of
p( r) co ntracts accord ing to Z-11 whe n Z incr eases. We rem ar k that fo r fixed Z the
T homas-Fe rmi me thod is inaccurat e a t bo th small r (r < ao/Z ) a nd larg e r (r ~ a,,).
whe re it ov er estim at es the elec tro n density. Indeed. the Th om as-Fe rmi electro n
den sity (8.65a) d ive rges a t the o rigin (as r --'>O ) a nd falls o ff like r - o as r --> ~ . while
the co rrec t e lectro n den sity sho uld rem ain finite a t r = O. a nd d ec re ase ex po ne ntia lly for lar ge r. Thus th e a pplica tio n o f th e Th om as-Fe rmi met hod is limit ed
to 'i n te rmed iate' dist an ces r betw een a"IZ a nd a few times a". It is wo rt h no ting,
how ever , that in co mplex ato ms most of th e electron s a re to be found pr ecise ly
in thi s spatia l region . Thus we e xpect the Th om as-Fermi method to be useful in
calcu la ting q ua ntities which de pe nd o n the 'ave rage electro n'. suc h as th e tot al
e ne rgy o f the ato m. On th e o the r ha nd. q uan tities which rely o n the p rop ert ies
o f th e 'o ute r' electrons (such as the ion isati on pot en tial) a re po or ly give n in the
Th om as-Fe rmi model.
We have sho wn a bove that a ne utra l atom has no ' bo u nda ry' in the Th om asFermi model. Never the less, it is po ssib le to defin e in this cas e a n a to mic ' rad ius'
R{a ) as the radius o f a sp he re ce ntred a t the o rigin a nd conta ining a give n fracti on
(I - a) of the Z e lectro ns. We th en ha ve {see (8.62))
R, a l
-+11:
f
0
p{r)r' dr = (l - a )Z
(8.76)
Making th e change of va ria ble
R{a ) =b X{ a)
(8.77)
381
o the r hand. if we se t a = Z -' . the n R( 7. I ) = bX( Z ' ) is the rad ius o f a sp he re
cont aining all the atomic e lectro ns e xce pt o ne . Th e qua ntity R(Z 1) is fo und to be
a slo wly inc reas ing func tio n o f Z. suc h that -+lI" < R( Z -I ) < 611". T hus 1Il bo th cases
the a to mic radi us is nea rly ind e pende nt o f Z . Similar ly. the e nergy o f the 'out er'
electrons - a nd hen ce the ioni sa tion pot e ntial o f the a to m - is almost indep end ent
o f Z . A s a conseq ue nce . the T ho mas-Fe rmi mod el ca nno t acco u nt for th e peri od ic
pr o perties o f ato ms as a function o f Z . discussed in Sect ion 8.2.
Le t us now briefly discus s the two o the r catego ries o f so lutio ns (see Fig. 8 A)
ment ion ed in o ur discussion o f the Th om as-Fermi equat io n (8.66). Returnin g to
(8.69 )-{8.70) , e re ma rk tha t so lu tio ns X(x ) which va n ish a t a finite va lue x = x"
(tha t is. which bel o ng to the seco nd ca tego ry) are suc h th a t ,V ", Z. a nd hen ce co rrespo nd to ions o f radius r o = bx.; Moreover, since the slo pe o f X is ne ga tive a t X o
(see Fig. 8.4) the equ ation (8.70) imp lies that these ion s mu st be positive ions. such
tha t Z > N [7J. Se tt ing ~ = Z - N. so that ~e is the net c ha rge o f th e ion . we no te
fro m (8 .70) that the q ua nt ity ~ /Z is re ad ily o b ta ined fro m th e ta nge nt to the c u rve
X a t x = x ", as sho wn in Fig. 8A. Since X(xn) = O. the elec tro n de nsity p(r ) vanishes
a t r = r., =b X I1' as see n fro m (8.65). On the o the r hand , lo oki ng at (8.55) , (8.56 ) a nd
(K 63). an d rem embering th at 1>" > 0 for a n ion. we se e tha t th e pot en tial V (r)
rem a ins finite a t r = r".
Th e so lutio ns of the Th o mas-Fermi eq uat io n belon g ing to the th ird ca tegory
(tha t is. those which ha ve no ze ro a nd d ive rge for lar ge x ) are mor e d ifficu lt to
inte rp re t. Fir st o f all, the electro n de nsit y p( r) does not va n ish in thi s ca se. a nd o ne
ma y co nside r that these so lutio ns co rr espo nd to nega tive va lues o f ifI" . A s se e n
fro m Fig. 8.4. suc h so lutio ns lie above th e ' unive rsa l' c urve o f the neut ral a to m.
No w th e tot al cha rge inside a sp he re of ra d ius r = bx is ju st
Ze - -+11:e f ' p( r')r" dr' = Ze[x{x) - XX'(x )]
(8.79)
"
T h us, a t the po int r, = bx .. whe re
(8 .80)
th e to ta l cha rge insid e the sphe re r = r l va n ishes . a nd we no te tha t the ta nge nt to
X(x ) a t x = X l passes th ro ugh the o rigin (see Fig. , A). For x .,; .r, the curv e X(x )
th er efo re co rr espo nds to a neutral ato m hav ing a finite bou ndary a t r = r,. whe re
the de nsit y p(r) does not va nish. Thi s ma y be inte rpr e ted as a re present ati on o f a
ne utral atom unde r pre ssu re [8]. Further de velopme nts of the T ho mas-Fe rmi
met ho d ca n be fou nd in the mo nograph o f E ngle rt ( 1988).
a nd taking into acco unt (8.63). (8. 65) a nd (8 .66) , we find for X the equa tio n
X( X) - XX ' (X ) = a
(8.78)
which mu st be so lved nu me rically . If the sa me value o f a is ado pt ed fo r a ll a to ms.
(8.78) beco mes a ' unive rsa l' eq ua tio n a nd X is the sa me fo r all a to ms. Using (8.6-+)
a nd (8.77) we see tha t the atomic radiu s R( a ) is then pr op ortional to Z - 13 On the
PI
v cganvc ions cannot be handled bv the Thomas- Fermi the ory.
[Xl W e are not con side ring IOns under pre...su re. since in deali ng with an e nse mble o f suc h ions. diffi -
cultie s due to the presence o f the Co ulomb force s be twee n ions woul d arise .