The elemental metals

34; TETRAVALENT

4. The elemental metals

is 3.5 kcal. The difference between the 4f-5d multiplet coupling in the free atom and the solid is therefore 8.4 minus 3.5 kcal or about 5 kcal. The same coupling is also present with about the same magnitude in other trivalent free rare earth atoms. The cohesive e n e r g y - relative to the trivalent free a t o m s - should therefore be about 95 kcal/mol rather than 100 kcal/mol for metals where there is no f shell. Similarly, the measured cohesive energy of the trivalent actinide element curium is about 90 kcal/mol (Ward et al. 1975), which may be taken to be representative of the cohesive energy of the trivalent actinide metals - again relative to the corresponding trivalent (5f" 6d 7s 2) free-atom configuration.

COHESION IN RARE EARTHS AND ACTINIDES 181

400

200

0

-200

IJJ

-400

-600

Band energies

. ~ Cd

I i i i I ~ i

Ice INd ISm IGd IDy lEr I Lu La Pr Pm Tb Ho Tm

Fig. 13. An overview of the electronic structure of the trivalent lanthanides, evaluated at a common Wigner-Seitz radius of 3.75 au (after Skriver 1983b). The bottom, Bz, top, A» centre, C» and square-well pseudopotential, V~, are plotted, and V(S) is the potential and ex~ the exchange-correlation density at the sphere boundary. The Fermi energy is labelled by E v.

The 5s and other core states contract with increasing atomic number since the added 4f electron does not shield them from the increased nuclear charge completely. Ortho- gonalization to the core states increases the kinetic energy of the 6s states and tends to "exclude" them from the core region. Since the core region shrinks with incomplete screening of the increased nuclear charge, the kinetic energy of the 6s states decreases as the series is traversed. The same applies to the 6p states, which also fall rapidly in energy across the series. However, the bottom of the 6p band remains quite high throughout the series, never falling below the Fermi energy, and the partial 6p o c c u p a t i o n - which is due entirely to hybridization with states below the Fermi energy - increases less rapidly than the partial 6s occupation. To a lesser extent, the 5d states, which are themselves only partially shielded by the added 4f electron, also contract - the band mass, ~tSd, increases from two to three as the partial 5d density at the Wigner-Seitz sphere boundary decreases - and fall in energy across the series.

The partial 4f and 5d densities in Gd metal are shown in fig. 14. At an energy corres- ponding to the bottom of the 5d bands the bonding 5d density is very large, whereas

o

1L~f 5d-bond

1 2 3

RADIUS la.u.)

Fig. 14. The calculated 4f and 5d radial densities for Gd metal. The 5d density is shown for the bottom (bond) and top (anti-bond) of the 5d bands.

)

Z

g

0

La Fr Pm Eu Tb Ho Tm Lu

Fig. 15. The calculated s, p and d occupation numbers across the lanthanide series. The dashed lines corres- pond to divalent metals (Ba, Eu and Yb).

it is relatively small at an energy corresponding to the top of the 5d bands. The states at the bottom of the 5d bands are, therefore, better shielded from the increased nuclear charge and the energy of the bottom of the 5d bands falls less rapidly across the series than does the energy of the top of the bands. Since it is the bottom of the 5d bands that is occupied, the result is a net transfer of 5d to 6s electrons as the series is traversed. The partial s,p and d occupation numbers calculated for the di- and trivalent metals by Eriksson et al. (1990c), are plotted as a function of atomic number in fig. 15. The d occupation numbers decrease along the series for both values of the

0

-200

-~,00

-600

B Energy

c

i I I Ba Eu Yb

Bo.nd H o ~ 3

2 / / - ~ JJd

Y ~

0[I I I

Bo Eu Yb

Eo Occupati0n

ä

Bo Eu Yb

Fig. 16. An overview of the electronic structure of the divalent lanthanides in the left panel, evaluated at a common Wigner Seitz radius of 4.10au (after Skriver 1983b). The bottom, Bz, top, A» centre, C» and square-well pseudopotential, Vz, are plotted, and V(S) is the potential and exc the exchange-correlation density at the sphere boundary. The Fermi energy is labelled by E v. The centre panel displays the corresponding band masses and the right panel the occupation numbers.

C O H E S I O N IN RARE EARTHS A N D ACTINIDES 183

valency, and it is precisely the d occupation number that is correlated with the crystal structure sequence discussed in sects. 2.3 and 4.5.

Although the 4f stares are not part of the band structure, it is the 4f electrons that are responsible for the change in electronic structure across the series. Since the Wigner-Seitz radius used in the calculations summarized in fig. 13 was kept constant, the effect of the screening of the increased nuclear charge by the 4f electrons was isolated, and is the only cause of any changes in the conduction electron band structure.

When the number of 4f electrons for a given nuclear charge is changed, as in the divalent rare earths, the effect on the conduction electron band structure is much greater. An overview of the electronic structure of the divalent lanthanide and barium is shown in fig. 16. The figure shows the bottom, centre and top of the d bands and the bottom of the s and p bands, evaluated for an atomic radius of 4.1 au and for the fcc structure (Skriver 1983b). Since there is now one extra 4f electron and one less 5d electron, the 5d band becomes almost depleted for Yb. Otherwise the trends in the energy band structure are similar to those of the trivalent rare earths.

4.2. Electronic structure of the actinide metals

An overview of the electronic structure of the actinide metals is shown in fig. 17.

The figure shows the bottom, centre and top of the 5f bands and the bottom of the s, p and d bands of the entire series, Fr-Lr, as a function of the Wigner-Seitz radius, as evaluated by Skriver and co-workers (Brooks et al. 1984). For the left of the series, Ra-Th, at the equilibrium radius, So, the bottom of the 7s band always lies below the Fermi energy, the bottom of the 6d band moves through the Fermi energy from above between Fr and Th, and the bottom of the 5f band is above the Fermi energy.

Hence the 7s and 6d bands are the only occupied bands and Th is a 6d-band transition metal. However, the unoccupied 5f band is so close to the Fermi energy that it

Fr R~

0.8 06 0.4 0.2 0.0 w-0.2 -0.4 -0.6

-°6 t ~o

5.5 /,.5

Ac Th

v l s ) ..

So L.O 3.5

S [ a . u ]

Pa U Np

"'~~t

:~, 4 & -1

Bf i

So So "Sc / LL~ I , l , , , , ]

3.s 3.s3.o 3s

Pu Am

'\.\

:, ... ~,o

3.0 3.5 3.0 3,5

Cm Bk Lr

Af EF

Bf B p \ _ Bd

N N ;

rest...'"" I ... "1 v s

ù." So -' .So [,i . . . I, ,I . . . . ~1,

3.0 3.5 3.0 3.5 3.5 S[a.u]

Fig. 17. Energy bands for Fr and the actinides, evaluated as function of the atomic radius, S. The potential,

V(S), at the sphere, the bottom, B I, and the top, A» of the relevant bands, together with the Fermi energy,

Er, are plotted. So is the measured equilibrium atomic radius.

0.6

"~O.L Od

0.2

0.0 3.9

\ 3.3 31

\ \ \ , o s:zgo.~.

• ~

I

Ac

I I q E I I I ~ I

5f bandwidth

3.5 6

2

[ i t ~ I ~ t r Ctf 0

Th Pa U Np Pu Am Cm Bk

Fig. 18. The f bandwidth, Wc, evaluated as a funcion of atomic volume and atomic radius, S. The solid circles indicate the 5f bandwidths calculated at the experimental equilibrium atomic radii, while the open circles indicate the bandwidths obtained at the calculated (non-spin-polarized) equilibrium atomic radii.

influences the occupied part of the band structure through hybridization, and this effect has proven to be important for the details of the Fermi surface (Gupta and Loucks 1969, 1971, Koelling and Freeman 1975, Skriver and Jan 1980).

The bands rise in energy under compression, as in the lanthanides, due to increasing kinetic energy, where now the 7s states are required to be orthonormal to the 6s core states. As a function of the atomic number, band filling occurs as follows. The 7s band in Fr takes the first electron, between Ra and Th the 6d band takes most of the next three electrons, the first real occupancy of itinerant 5f states occurs in Pa, and thereafter the 5f band fills up to Lr.

The gradual filling of the 5f bands is analogous to the filling of the d bands in transition metals, therefore the atomic volumes of the actinides should be parabolic as a function of the atomic number as first the bonding and then the anti-bonding orbitals become filled. Such a parabolic trend is indeed observed for the metals T h - P u , and the analogy with the d transition series is correct for the light actinides. However, the 5f electrons are localized in the heavy actinides, where the trend in atomic volume is similar to that for the lanthanides.

The 5f bandwidth is shown in fig. 18, and, for fixed atomic radius, it decreases with increasing 'atomic number. This occurs because at the beginning of the series the potential due to the added proton contracts the 5f orbitals and lowers the amplitude of the wave function, hence, increases the band mass, at the Wigner-Seitz sphere.

The narrowing is most pronounced at the beginning of the series because later the extra electron added to the system enters the 5f shell and partially screens the nuclear charge seen by the other 5f electrons. A similar trend is found for the 6d and 7s states, which are located outside the 5f shell and, therefore, also see ~the nuclear charge partially screened. F o r fixed atomic number the bandwidth decreases with increasing atomic radius since the amplitude of the wave function at the Wigner-Seitz sphere decreases. The 5f bandwidths evaluated at the measured atomic volume (solid circles in fig. 18) remain, however, nearly constant between Th and Pu, since the volumes

COHESION IN RARE EARTHS AND ACTINIDES 185

200

T o

~6 150 r Y

m 100

W Z

50

S t a r e d e n s i t y o f c c - s t r u c t u r e z% b c c - s t r u c t u r e o c~-U

- - non - p o l a r i z e d _ - - - spin - p o l a r i z e d ... s p d - c o n t r i b u f i o n

ù e x p e r i m e n f a l e l e c t r o n i c h e a t y

Fr Ra

Fig. 19. The state density at the Fermi energy calculated at the experimental equilibrium radius, compared to the experimentally obtained electronic specific heat, 7. The dotted curve represents the spd contribution to the total state density.

decrease. The difference between theory and experiment for N p and Pu is at least partly caused by the relativistic volume effect, and the large j u m p in volume at Am, due to localization of the 5f states, is a magneto-volume effect.

The calculated state densities at the Fermi energy have been collected and c o m p a r e d to experimentally observed specific heat coefficients in fig. 19. One should bear in mind that most of the calculations assume an fcc structure, and therefore one cannot expect too detailed an agreement between theory and experiment. In the beginning of the series, i.e. for Fr Th, the 5f contribution is small, and N(EF) for Th is typical for a transition metal. In Pa the 5f contribution starts to dominate the state density, which by Am has increased by an order of magnitude. The measured electronic specific heat coefficients show a similar trend up to, and including, Pu. However, in Am it is down by one order of magnitude with respect to the value for Pu, and is in fact close to the spd contribution to the state density. Hence, in this respect Am behaves like a rare earth metal. The interpretation of the above-mentioned observations is that the 5f electrons in P a - P u are metallic, hence the high electronic specific heat, in the same sense that the 3d, 4d and 5d electrons in the ordinary transition series are metallic, a n d ' that this metallic 5f state turns into a localized,one at Am, hence the relatively low electronic specific heat. Am and the later actinides form a second rare earth series.

4.3. Atomic volume

We now consider the partial 1-resolved contributions to the electronic pressure (sect. 3.4, eq. (37)) as~functions o f b o t h atomic volume and atomic number. The partial pressures originally calculated by Skriver (1983b) for the trivalent rare earth metals

A ,P

C ~

Q_- ⁰

-0.2 -0A.

-0.6 -0.8

I B P a r t i a l p r e s s u r e s O.8

/

o.~ g

0.2

f

_ ~

I I I I I I I I I I I I I ] La Ce Pr N d ~ S m fid T b ~ H o Er Tm

o Fig. 20. The partial s,p,d and total electronic pressures of the trivalent lanthanides, evaluated [ at a common Wigner-Seitz radius of 3.75au Lu (after Skriver 1983b).

F.[- (V,C) o.~

j ; ~

/

~ o

m -0.2

Z

g -o.~

z -0.8

, I , I , I I I I I I L I I

La üd

t ~(%C) filnS

-1

. . . I . . . I

.3Ppn

o ...-

-0.4 -0.8

Lu La üd Lu

BIn~~yS 2 61nS

4F d - -

3~~

0 -I - ~ t

i i I I i i J l l ] ] l l l

Lo ad Lu

Fig. 21. Decomposition of the partial pressures into bandwidth and band- position terms (lower panel) for the trivalent lanthanides. Approximate pressure expressions, eqs. (46) and (47) (after Skriver 1983b).

at a fixed atomic radius of 3.75 au in the fcc structure are shown in fig. 20. The s and p partial pressures are positive (repulsive) and the d partial pressure is negative (bonding). In terms of eq. (37) and the discussion following eq. (48), only the 5d states are both bonding and resonant states. Alternatively, in terms of eqs. (46) and (47), the partial pressures may be further resolved into band-centre and bandwidth contri- butions. At the beginning of the series the square-well pseudopotentials of the s and p stares are greater than

exc(S)

(fig. 13) and the corresponding band-position contribu- tion is positive, as shown in fig. 21. As the relative core size shrinks with increasing atomic number, the square-well pseudopotentials of the s and p bands fall below

exc(S)

and the band-position contributions change sign. The bandwidth contribution

C O H E S I O N I N R A R E E A R T H S A N D A C T I N I D E S 187

4.5

3

,~d ~.0 u3

3.5

A t o m i c r a d i i

\ I

/

I I

Ba JCe JNd JSm IGd [Dy lEr Iyb ISc i La Pr Pm Eu Tb Ho Tm Lu Y

Fig. 22. The Wigner Seitz radii of the rare earth metals. The points represents measurements and the full line the results of L M T O calculations by Skriver (1983b).

is also influenced by the decrease in the core size across the series, which causes ( E ) t - Vt in eq. (47) to increase. The bandwidth contribution therefore increases, and the sum of the position and bandwidth contributions to the partial s and p pressures remains positive throughout the series, providing the repulsive pressure required to balance the

bonding

pressure of the d electrons.

We now consider the band-centre term in eq. (46) for the d states. Since the band mass, #~, is always positive, the sign of this contribution is determined by the position of the band centre relative to the exchange correlation energy,

%(S),

at the sphere, see eq. (46). The centre, C» of the d bands lies rar above % (fig. 13) and the position contribution (fig. 21) is positive, i.e. repulsive, throughout the series. It is the bandwidth term in eq. (46) that provides the bonding since ( E ) z - C l is negative, the d bands being less than half-rille& As the number of d electrons decreases across the series so does this contribution, due to the factor n e in eq. (46).

The atomic radii calculated by Skriver (1983b) from an evaluation of eq. (37), are shown in fig. 22. The variations in the lattice constant and the total pressure at fixed

/. S

2

> 0

02 - - - W

('°_2

-A - So

11 I I i

3.3 3.5

p .I d

~ c

- C

r ] o o ot

Jranium

I I I ,I ,

3.3 3.5 3.3 3.5

S [ a . u l

f

_ C

So

I Il I I I

3.3 3.5

Fig. 23. The partial pressures for uranium decomposed into the band-centre contribution (C) and the bandwidth contribution (W) as functions of the atomic radius, and the partial pressures for U decomposed according to the first-order pressure expression, eq. (46), (full lines). The letter t denotes the total pressure and the open circles are the results of the full calculations.

radius, fig. 20, are similar and, for an approximately constant bulk modulus, the trend in the radius is due to the trend in the electronic pressure. The origin of the rare earth contraction is, therefore, the shrinking core size and resulting fall in the band centres and square-well pseudopotentials (fig. 13).

The rare earths are early transition metals where the number of d electrons changes only slightly. In the actinides the 5f electrons play the premier role in the bonding, In fig. 23 we show the volume dependences of the s, p, d and f pressures for uranium, analyzed according to eqs. (37) and (46).

Since the band mass, #» is always positive, the sign of the band-centre contribution is determined by the position of the band centre relative to the exchange-correlation energy, exc(S ), at the sphere, (eq. (37)), as before. This relative position may be judged from the bands in fig. 23 by noting that C~ is approximately halfway between B~ and A» and that exc(S) ~- 0.75V(S). It follows that the centre term is positive, i.e. repulsive, for all values of the angular momentum. The magnitude of the centre term is governed by the occupation number, n» the centre position, C» and the band mass, #z. The band masses for the 7s and 7p states are nearly equal, while Cp - exo is approximately three times C s - ex¢. The partial occupation numbers of the actinide metals treated as di- and trivalent metals, as well as with 5f stares in the energy bands (Brooks et al.

1984, Eriksson et al. 1990c), are plotted in fig. 24 as a function of atomic number.

S i n c e rtp is between one half and one third of n s in uranium metal, the centre terms for all the s and p states have similar magnitudes. F o r the 6d states C a - ex¢ is intermediate

u~

rr

]E Z Z 0

t~

0

d ~

J

p -

. . . I D

B~lThl ü I#uldmldfl~ml~ioi

Ac Pa Np A m Bk Es Md Lr

80

1+

20 (b

Fig. 24. (a) The calculated s, p and d occupation numbers across the actinide series when the 5f states are part of the core. The full line is for trivalent actinides and the dashed line is for divalent actinides. (b) The calculated s , p , d and f occupation numbers across the actinide series when the 5f states are in the energy bands.