- Email: [email protected]
Electron-positron pair momentum densities in Mg and Cd
Solid State Communications, Vol. Printed in Great Britain.
ELECTRON-POSITRON
70, No. 6, pp. 593-597,
PAIR
MOMENTUM
1989.
DENSITIES
0038-1098/89$3.00+.00
Pergamon Press plc
IN
Mg
AND
Cd
G. Kontrym-Sznajd and J. Majsnerowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences SO-9SO Wroclaw, P. 0. Box 937, Poland
(Received
15 January 1988 by B. Mrlhlschlegel )
A reconstruction of 3D electron-positron densities p(p) from 20 ACPAR for Mg and Cd has been performed and compared with theoretical results. An agreement between theory and experiment as concerns the momentum dependence of p(p) inside the Fermi surface and the anisotropy of this surface is reasonable. As far as a quantitative comparison of HMC contribution is concerned, some disagreement between theory and experiment, especially for Cd, has been observed. data
The densities @.p~ on t.he plane Pl reconstmcted from the experimental data, are displayed in Fig.l. It IS seen that p(p) inside the Fermi w-face IF.9 1s isotropic for both metals. However. while for Mg it corresponds to the spherical FS on this plane. in the case of Cd such shape of p(p) can describe the anisotropic FS ~p&Oio]~pp[ll50~~ if (as it follows from onr theoretical calculations) the density is not constant on the FS and p(p&OrO]) < p(pp[~?z~]). Some small fluctuations of p(p) inside the FS are negligible the more so as we present pip) withont smoothing, i.e. when all 89-n/2 coefficients a:: have been applied in calculation of p(p) [9 1) . As concerns the momentum dependence of the density inside the FS , an increase in pip) for p *pF was obtained for Mg, while for Cd, a lack of such a dependence is observed. An application of the electron-gas enhancement factors E [7,8] in Eq.(Z) led to strong momentnm dependence of pfp) (for p ,: p,) in Cd. Because the similar effect was observed also in Zn [6], being in disagreement with the experimental results, it points out that the E obtained on the basis of the Kahana theory [7,8] for low r, are too strong momentum dependent. For this reason in our theoretical calculation the modified E were used. For rs = 1.5 and - 2 valnes of E were interpolated by biparabolic %muia. In this way their values for 0.8~~ i p i p were substantially decreased. For lower r, the const”. E (i.e. s(pj=z(O)) were used. In both cases i Mg and Cd) the positron annihilation with core electrons was taken into account in the theoretical calculations. Comparison of p(p) reconstructed from the experimental data with theoretical results Is presented in Flg.2. As one can see. the application of the theory [6] gives, qualitatively, a similar behaviour of p(p). Some differences are connected first of all with smearing of the experimental spectra due to the finite equipment resolution. In this connection that we do not dispose the theoretical spectra (so also the theoretical densities) convoluted with a 2D Gaussian representlng the experlmental angular resolution and a finite positron energy (proportional to the temperature of the experiment ). we are able to compare all results only for p ‘: 0.8~~ and for p > 1.2 pF. It may be seen that in Mg an agreement between theory and experiment for p c 0.8~~ is good. There is also a good accordance in a higher momentum region (above 7.5 mrad)
Three dimensional electron-positron (e-p.) densities p(p) were determined for Mg and Cd from ZD ACPAR (two dimensional angular correlation of positron annihilation radiation) experimental spectra N(p ,p ) =/ p(p)dp, by applying the Cormack method rl.Zf $h e exoerimental soectra were obtained bv Waiters et al.[3] for two projections: with a direction of integration pz along [IO~O](N~& and [it%](Nrx). The measurements were performed at the temperature IZOK, with the equipment resolution of the order 0.7 mrad (1 mrad = 0.137 a.u.) and the total connts at the peak were of the order 65000. The density p(p) was calculated from the formula P(P,Q) = ;
Yo.;‘”
(n+m+l)aER$p)cos(n8),
(1)
on the succeeding planes pr= const. and for n=0,6. These planes, perpendicular to the [ooor]diredion, are distant to each other 0.367mrad which corresponds to the distance between the experimental points. p = (p,8) is expressed in a polar system, R::(p) are the Zemike polynomials and the coefficientsa: were obtained from the expansion of NO(p)=(Nr~+Nr. )/Z and Ne(p)=(Nr -N ~)/2 into the series of Chebysev polynomials 6, ZP No and N 4 represent the radial coefficients of the expansion N pX,pr) into the Fourier cosine series. For both ‘metals we will present results on the plane PI (first plane for p,.= 0.1835mrad) and on the planes where the anisotropy was the greatest : the plane P.5 for Mg (pz= 1.6515 mrad) and P7 for Cd (p,= 2.38% mrad ). For theoretical calculations we used the LMTO method [4,5] taking into account e-p interaction in a local density approximation [6], i.e. : p(p)
=Lzl/fi)
Ji+(r)+k,n(r)exp(ip
rjdrf
, (2)
where Q+(r) and ckk,,(r) are the wavefunctions of annihilated particles: thermalised positron and electron in the nth band, respectively. These wavefunctions have lxen obtained from band structure calculations. The electron gas enhancement factors E (describing a disturbance of p(p) connected with e-p interaction) (7,8] were applied as a function of a local electron density. E q E((T,,(r)/T,(r)),r,(r)}, where T’s denote a local kinetic energy of (k,n) state and local kinetic Fermi energy. r, (r) is the local density parameter. 593
594
MOMENTUM DENSITIES IN Mg AND Cd
Fig. I. The e - p den.sity for (a) Mg and {b) Cd recons t r u c t e d f r o m the experimental data [3] on tile plane perpendicular to the [ooot] direction d i s t a n t 0.1835 mrad front tile c e n t r e o f the Brillouin zone.
Vol. 70, No. 6
Vol •
70,
595
MOMENTUM DENSITIES IN Mg AND Cd
No. 6
/
,,4
// ///
T%(p) 0.8[
Q[ O.Z
0.6 0.; 0.4 0.2
0
1
1
2
2
3
3
4
4
g p[mrad]
5
6
7
8
9
p[mred]
Fig. 2. The average o f t h e e - p density on t h e p l a n e p e r p e n d i c u l a r t o t h e [oool] direction. Full line describes the i s o t r o p i c average of t~(p) o n t h e plane PI r e c o n s t r u c t e d f r o m t h e e x p e r i m e n t a l data [3]. Tlle average of t h e t h e o r e t i c a l densities on t h e FMK p l a n e c r o s s i n g t h e c e n t r e of t h e Brillouin zone is described by d a s h e d l i n e : a) Mg and b ) Cd.
Fig. 3. T h e s a m e as in Fig. J b~ b u t f o r
py= 2.3855
mrad.
596
MOMENTUM DENSITIES
IN Mg AND Cd
Vol. 70, No. 6
-TLo
Fig.4. The density p(p) for Mg on t h e plane p e r p e n dicular to" the [oool] direction d i s t a n t 1.6SIS mrad f r o m the centre o f the Brillouin zone.
where tile contribution o f higher m o m e n t u m c o m p o n e n t s ( H M C ) is very small. The l a t t e r p o i n t s o u t t h a t t h e p o s i t r o n annihilation with ionic core e l e c t r o n s was described in a . p r o p e r way. In the case o f Cd a quantitative a g r e e m e n t b e t w e e n theory, and experim e n t is n o t s a t i s f a c t o r y . The theoretical r e s u l t s give a twice s t r o n g e r c o n t r i b u t i o n of d - b a n d s t h a n t h e experimental ones. It s u g g e s t s t h a t either d bands are s t r o n g e r localised o r we observe s o m e t e m p e r a t u r e e f f e c t (Deb¥~ t e m p e r a t u r e for Cd is 209K ). As far as t h e anisotropy o f t h e density in the higher m o m e n t a region is concenled, our t h e o r e t i c a l r e s u l t s give a marked anisotropy o f (~(p) f o r b o t h Mg and Cd : the H M C c o n t r i b u t i o n f r o m the third band in FM direction f o r Mg (the same was found by Wakoh [10]) and f r o m the s e c o n d band in FM for Cd. ~ p ) r e c o n s t r u c t e d f r o m t h e experimental data gives t h e same e f f e c t s but only qualitatively : in t h e case of Mg t h e anisotropy of H M C is I.S times s t r o n g e r than t h a t given by t h e theory. For Cd t h e H M C c o n tributlon Is much w e a k e r and c a n n o t be seen in t h e scale o f Pig. 1. R e c o n s t r u c t e d densities @(p) on t h e plane P7 for Cd and on PS for Mg are displayed in Figs. 3 and 4, respectively. For Cd w e observe a s t r o n g anisotropy o f the PS ( F S c l o s e t o t h e Brlllouin zone boundary). However, taking into a c c o u n t t h a t ta(p) f o r Cd is n o t c o n s t a n t on t h e FS , we m u s t r e s t r i c t ourselves to t h e conclusion t h a t on this plane t h e anlsotropy of PS,
I.e. p F [ n ~ o ] - p F [ l o i o ] is not less than 0.20 -+ 0.04 tared. In t h e case o f Mg t h e anisotropy o f t h e FS on the plane PS was also found. Because it is weakly visible in the scale o f Fig.l we p r o p o s e a n o t h e r picture. It s e e m s t h a t for such a simple metal as Mg, tao(p) being g r e a t e r t h a n 0.6 (if ~IP) for p= 0 is normalised to 1 and for p < pp has values not g r e a t e r than 1.2) would describe t h e s t a t e s inside the FS. So, taking a cross-section o f t~o(P) by the plane o f an equal density, i.e. Po(P) = 0.6 (if (so(p) > 0.61 and fSo(p~= PoiP) ( if tea(p) < 0.6) we draw ~(p) where tao(p) is replaced by ~ ( p ) - Pig.4. In such a picture t h e anisotroplc part o f the density" is n o t only very well visible but is also localised with r e s p e c t t o t h e PS . As It can be seen, in this case the anlsotropy for b o t h H M C and FS Is observed : t h e absence o f t h e l - s t band hole (an increase o f pl= in the FK direction) and a reduction of t h e 3-rd band s t a r s (an decrease o f PF in t h e FM direction) being in a g r e e m e n t with t h e L M T O band s t r u c t u r e calculations [11]. A c k n o w l e d g e m e n t s - We are very grateful to Dr. P. A. Waiters, Dr. J. Mayers and Prof. R. N. W e s t f o r providing their experimental resultes. The work b e n e f i t t e d from partial financial s u p p o r t from the Suisse Office Federal de l'Education e t de la Science.
Vol. 70, No. 6
MOMENTUM DENSITIES
IN Hg AND Gd
597
REFERENCES 1. A. M. C o r m a c k , J. Appl. Phys. 35, 2908 (1964). 2. G. K o n t r y m - S z n a j d , Positron Annihilation ( eds. P.G.Coleman et ai.) A m s t e r d a m : N o r t h - H o l l a n d 1982. p 346. 3. P. A. W a i t e r s , J. Mayers and R. N. W e s t , ibid. p 334. 4,. O. K. A n d e r s e n , Phys. Roy. B 12 , 3060 (1975). S. A. K. Slngh and T. J a r l b o r g , J. Phys. F : Metal Phys. 1S , 727 (1985). 6 . S. Dalduk, G. K o n t r y m - S z n a j d , A. Rubaszek, H. Stachowiak, J. Mayers, P. A. -Waiters and R. N. W e s t , J. Phys. Iv : Metal Phys. 17, 1365 (1988).
7 . A. Rubaszek and H. Stachowiak, phys. 5tat. aol.(b) 124, 159 (1982). 8. A. Rubaszek, H. Stachowiak, E. Boronskt, Z. Szotek, Phys. Rev. B 3 0 , 2490 (1984). 9 . G. K o n t r y r n - S z n a j d , to be published. 10. S. W a k o h , J. Phys. Soc. Japan 5 0 , 4 9 0 (1981). 11, S. Daniuk, G. Kontrym-Sznajd, J. Majsnerowski and T. J a r l b o r 8 , to be published.
ELECTRON-POSITRON
70, No. 6, pp. 593-597,
PAIR
MOMENTUM
1989.
DENSITIES
0038-1098/89$3.00+.00
Pergamon Press plc
IN
Mg
AND
Cd
G. Kontrym-Sznajd and J. Majsnerowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences SO-9SO Wroclaw, P. 0. Box 937, Poland
(Received
15 January 1988 by B. Mrlhlschlegel )
A reconstruction of 3D electron-positron densities p(p) from 20 ACPAR for Mg and Cd has been performed and compared with theoretical results. An agreement between theory and experiment as concerns the momentum dependence of p(p) inside the Fermi surface and the anisotropy of this surface is reasonable. As far as a quantitative comparison of HMC contribution is concerned, some disagreement between theory and experiment, especially for Cd, has been observed. data
The densities @.p~ on t.he plane Pl reconstmcted from the experimental data, are displayed in Fig.l. It IS seen that p(p) inside the Fermi w-face IF.9 1s isotropic for both metals. However. while for Mg it corresponds to the spherical FS on this plane. in the case of Cd such shape of p(p) can describe the anisotropic FS ~p&Oio]~pp[ll50~~ if (as it follows from onr theoretical calculations) the density is not constant on the FS and p(p&OrO]) < p(pp[~?z~]). Some small fluctuations of p(p) inside the FS are negligible the more so as we present pip) withont smoothing, i.e. when all 89-n/2 coefficients a:: have been applied in calculation of p(p) [9 1) . As concerns the momentum dependence of the density inside the FS , an increase in pip) for p *pF was obtained for Mg, while for Cd, a lack of such a dependence is observed. An application of the electron-gas enhancement factors E [7,8] in Eq.(Z) led to strong momentnm dependence of pfp) (for p ,: p,) in Cd. Because the similar effect was observed also in Zn [6], being in disagreement with the experimental results, it points out that the E obtained on the basis of the Kahana theory [7,8] for low r, are too strong momentum dependent. For this reason in our theoretical calculation the modified E were used. For rs = 1.5 and - 2 valnes of E were interpolated by biparabolic %muia. In this way their values for 0.8~~ i p i p were substantially decreased. For lower r, the const”. E (i.e. s(pj=z(O)) were used. In both cases i Mg and Cd) the positron annihilation with core electrons was taken into account in the theoretical calculations. Comparison of p(p) reconstructed from the experimental data with theoretical results Is presented in Flg.2. As one can see. the application of the theory [6] gives, qualitatively, a similar behaviour of p(p). Some differences are connected first of all with smearing of the experimental spectra due to the finite equipment resolution. In this connection that we do not dispose the theoretical spectra (so also the theoretical densities) convoluted with a 2D Gaussian representlng the experlmental angular resolution and a finite positron energy (proportional to the temperature of the experiment ). we are able to compare all results only for p ‘: 0.8~~ and for p > 1.2 pF. It may be seen that in Mg an agreement between theory and experiment for p c 0.8~~ is good. There is also a good accordance in a higher momentum region (above 7.5 mrad)
Three dimensional electron-positron (e-p.) densities p(p) were determined for Mg and Cd from ZD ACPAR (two dimensional angular correlation of positron annihilation radiation) experimental spectra N(p ,p ) =/ p(p)dp, by applying the Cormack method rl.Zf $h e exoerimental soectra were obtained bv Waiters et al.[3] for two projections: with a direction of integration pz along [IO~O](N~& and [it%](Nrx). The measurements were performed at the temperature IZOK, with the equipment resolution of the order 0.7 mrad (1 mrad = 0.137 a.u.) and the total connts at the peak were of the order 65000. The density p(p) was calculated from the formula P(P,Q) = ;
Yo.;‘”
(n+m+l)aER$p)cos(n8),
(1)
on the succeeding planes pr= const. and for n=0,6. These planes, perpendicular to the [ooor]diredion, are distant to each other 0.367mrad which corresponds to the distance between the experimental points. p = (p,8) is expressed in a polar system, R::(p) are the Zemike polynomials and the coefficientsa: were obtained from the expansion of NO(p)=(Nr~+Nr. )/Z and Ne(p)=(Nr -N ~)/2 into the series of Chebysev polynomials 6, ZP No and N 4 represent the radial coefficients of the expansion N pX,pr) into the Fourier cosine series. For both ‘metals we will present results on the plane PI (first plane for p,.= 0.1835mrad) and on the planes where the anisotropy was the greatest : the plane P.5 for Mg (pz= 1.6515 mrad) and P7 for Cd (p,= 2.38% mrad ). For theoretical calculations we used the LMTO method [4,5] taking into account e-p interaction in a local density approximation [6], i.e. : p(p)
=Lzl/fi)
Ji+(r)+k,n(r)exp(ip
rjdrf
, (2)
where Q+(r) and ckk,,(r) are the wavefunctions of annihilated particles: thermalised positron and electron in the nth band, respectively. These wavefunctions have lxen obtained from band structure calculations. The electron gas enhancement factors E (describing a disturbance of p(p) connected with e-p interaction) (7,8] were applied as a function of a local electron density. E q E((T,,(r)/T,(r)),r,(r)}, where T’s denote a local kinetic energy of (k,n) state and local kinetic Fermi energy. r, (r) is the local density parameter. 593
594
MOMENTUM DENSITIES IN Mg AND Cd
Fig. I. The e - p den.sity for (a) Mg and {b) Cd recons t r u c t e d f r o m the experimental data [3] on tile plane perpendicular to the [ooot] direction d i s t a n t 0.1835 mrad front tile c e n t r e o f the Brillouin zone.
Vol. 70, No. 6
Vol •
70,
595
MOMENTUM DENSITIES IN Mg AND Cd
No. 6
/
,,4
// ///
T%(p) 0.8[
Q[ O.Z
0.6 0.; 0.4 0.2
0
1
1
2
2
3
3
4
4
g p[mrad]
5
6
7
8
9
p[mred]
Fig. 2. The average o f t h e e - p density on t h e p l a n e p e r p e n d i c u l a r t o t h e [oool] direction. Full line describes the i s o t r o p i c average of t~(p) o n t h e plane PI r e c o n s t r u c t e d f r o m t h e e x p e r i m e n t a l data [3]. Tlle average of t h e t h e o r e t i c a l densities on t h e FMK p l a n e c r o s s i n g t h e c e n t r e of t h e Brillouin zone is described by d a s h e d l i n e : a) Mg and b ) Cd.
Fig. 3. T h e s a m e as in Fig. J b~ b u t f o r
py= 2.3855
mrad.
596
MOMENTUM DENSITIES
IN Mg AND Cd
Vol. 70, No. 6
-TLo
Fig.4. The density p(p) for Mg on t h e plane p e r p e n dicular to" the [oool] direction d i s t a n t 1.6SIS mrad f r o m the centre o f the Brillouin zone.
where tile contribution o f higher m o m e n t u m c o m p o n e n t s ( H M C ) is very small. The l a t t e r p o i n t s o u t t h a t t h e p o s i t r o n annihilation with ionic core e l e c t r o n s was described in a . p r o p e r way. In the case o f Cd a quantitative a g r e e m e n t b e t w e e n theory, and experim e n t is n o t s a t i s f a c t o r y . The theoretical r e s u l t s give a twice s t r o n g e r c o n t r i b u t i o n of d - b a n d s t h a n t h e experimental ones. It s u g g e s t s t h a t either d bands are s t r o n g e r localised o r we observe s o m e t e m p e r a t u r e e f f e c t (Deb¥~ t e m p e r a t u r e for Cd is 209K ). As far as t h e anisotropy o f t h e density in the higher m o m e n t a region is concenled, our t h e o r e t i c a l r e s u l t s give a marked anisotropy o f (~(p) f o r b o t h Mg and Cd : the H M C c o n t r i b u t i o n f r o m the third band in FM direction f o r Mg (the same was found by Wakoh [10]) and f r o m the s e c o n d band in FM for Cd. ~ p ) r e c o n s t r u c t e d f r o m t h e experimental data gives t h e same e f f e c t s but only qualitatively : in t h e case of Mg t h e anisotropy of H M C is I.S times s t r o n g e r than t h a t given by t h e theory. For Cd t h e H M C c o n tributlon Is much w e a k e r and c a n n o t be seen in t h e scale o f Pig. 1. R e c o n s t r u c t e d densities @(p) on t h e plane P7 for Cd and on PS for Mg are displayed in Figs. 3 and 4, respectively. For Cd w e observe a s t r o n g anisotropy o f the PS ( F S c l o s e t o t h e Brlllouin zone boundary). However, taking into a c c o u n t t h a t ta(p) f o r Cd is n o t c o n s t a n t on t h e FS , we m u s t r e s t r i c t ourselves to t h e conclusion t h a t on this plane t h e anlsotropy of PS,
I.e. p F [ n ~ o ] - p F [ l o i o ] is not less than 0.20 -+ 0.04 tared. In t h e case o f Mg t h e anisotropy o f t h e FS on the plane PS was also found. Because it is weakly visible in the scale o f Fig.l we p r o p o s e a n o t h e r picture. It s e e m s t h a t for such a simple metal as Mg, tao(p) being g r e a t e r t h a n 0.6 (if ~IP) for p= 0 is normalised to 1 and for p < pp has values not g r e a t e r than 1.2) would describe t h e s t a t e s inside the FS. So, taking a cross-section o f t~o(P) by the plane o f an equal density, i.e. Po(P) = 0.6 (if (so(p) > 0.61 and fSo(p~= PoiP) ( if tea(p) < 0.6) we draw ~(p) where tao(p) is replaced by ~ ( p ) - Pig.4. In such a picture t h e anisotroplc part o f the density" is n o t only very well visible but is also localised with r e s p e c t t o t h e PS . As It can be seen, in this case the anlsotropy for b o t h H M C and FS Is observed : t h e absence o f t h e l - s t band hole (an increase o f pl= in the FK direction) and a reduction of t h e 3-rd band s t a r s (an decrease o f PF in t h e FM direction) being in a g r e e m e n t with t h e L M T O band s t r u c t u r e calculations [11]. A c k n o w l e d g e m e n t s - We are very grateful to Dr. P. A. Waiters, Dr. J. Mayers and Prof. R. N. W e s t f o r providing their experimental resultes. The work b e n e f i t t e d from partial financial s u p p o r t from the Suisse Office Federal de l'Education e t de la Science.
Vol. 70, No. 6
MOMENTUM DENSITIES
IN Hg AND Gd
597
REFERENCES 1. A. M. C o r m a c k , J. Appl. Phys. 35, 2908 (1964). 2. G. K o n t r y m - S z n a j d , Positron Annihilation ( eds. P.G.Coleman et ai.) A m s t e r d a m : N o r t h - H o l l a n d 1982. p 346. 3. P. A. W a i t e r s , J. Mayers and R. N. W e s t , ibid. p 334. 4,. O. K. A n d e r s e n , Phys. Roy. B 12 , 3060 (1975). S. A. K. Slngh and T. J a r l b o r g , J. Phys. F : Metal Phys. 1S , 727 (1985). 6 . S. Dalduk, G. K o n t r y m - S z n a j d , A. Rubaszek, H. Stachowiak, J. Mayers, P. A. -Waiters and R. N. W e s t , J. Phys. Iv : Metal Phys. 17, 1365 (1988).
7 . A. Rubaszek and H. Stachowiak, phys. 5tat. aol.(b) 124, 159 (1982). 8. A. Rubaszek, H. Stachowiak, E. Boronskt, Z. Szotek, Phys. Rev. B 3 0 , 2490 (1984). 9 . G. K o n t r y r n - S z n a j d , to be published. 10. S. W a k o h , J. Phys. Soc. Japan 5 0 , 4 9 0 (1981). 11, S. Daniuk, G. Kontrym-Sznajd, J. Majsnerowski and T. J a r l b o r 8 , to be published.