Electrical resistivity of zone-hardened Al Ag and Al Zn

(1)

HAL Id: jpa-00236687

https://hal.archives-ouvertes.fr/jpa-00236687

Submitted on 1 Jan 1962

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Electrical resistivity of zone-hardened Al Ag and Al Zn

R. Labusch

To cite this version:

R. Labusch. Electrical resistivity of zone-hardened Al Ag and Al Zn. J. Phys. Radium, 1962, 23 (10),

pp.823-823. �10.1051/jphysrad:019620023010082300�. �jpa-00236687�

(2)

823. ELECTRICAL RESISTIVITY OF ZONE-HARDENED Al Ag ^{AND Al Zn}

By ^R. LABUSCH,

Institüt fur Metallphysik und Institüt für Theoretische Physik ^der Universität, Göttingen.

Résumé.

²⁰¹⁴

On discute la résistivité d’alliages Âl Ag êt Âl ^{Zn due à} l’existence ^de ^zones de

Guinier-Preston, ^en ^tenant compte de la forme détaillée de la surface de Fermi dans l’aluminium.

Abstrat.

²⁰¹⁴

The resistivity ôf Al-Ag ând Âl-Zn alloys ^due ^to the existence of Guinier-Preston

zones is discussed, ^the ^exact form of the Fermi surface of aluminium being taken into account.

LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, ^OCTOBRE 1962,

Until now, attempts ^to explain ^the dependence

of the electrical resistivity p ôn the zone radius R in AI Ag ând ÂI ^Zn alloys ^have ^not ^been quite satisfactory. Most of them are based on the free electron model with three free electrons per âtom and can explain the fall-off in resistivity ^for large ^R only, ^not the maximum observed.

The magnitude ^{of the} effect, ^{which has} ^been

found experimentally ^to ^be up ^to 20-30 % ^{of the}

total resistivity, and the value of R at the resis-

tivity ^maximum (Rm

⁼

9-11 Á) ^cannot ^be explained by ^the ^free electron model at all.

To obtain a satisfactory theory, ^{the real} shape

of the Fermi-surface in Al must be taken into account. One such attempt ^has ^been ^made by Matyas [1], ^but unfortunately ^he ûsed ân încorrect

model for the conduction band and consequently

there is need for a new àttack on the problem.

The present investigation ^uses the band model of Harrison [2], which ^seems ^to be the best avail- able. In order to calculate the transport pro-

perties ôf â métal, ône ^has ^to solve the Bloch equa- tion. In general ^{this is} impossible because of the

complicated shape ^{of the} Fermi-surface, ^but by

means of suitable models one can make certain calculations on the properties ^{of the} ^solution ^{of this} equation. The result is the following. By ^a ^trans-

formation of the p(R) ^curve for free electrons using

an effective k, ^{where k} ^is ^the propagation ^{vector at}

the Fermi-surface, ^{one can} get ^the approximate shape ^{of the} P(R) ^curve for bound electrons.

’ This becomes quantitatively ^correct ^{for very} large R, ^but ^keff ^has ^a much smaller value than that for the case of three free électrons per ^atom.

Two electrons do not take part in transport pheno-

mena at all because, according to FJarrison’s ^model,

the first Brillouin zone is completely ^filled.

Only ^one ^electron ^remains free for the transport

of charge. ^In addition, the value of keti itself is affected by ^the ^curvature ^{of the} Fermisurface.

If the curvature is sharp ^at ^certain points, ^hit

becomes small. For smaller values of R, Umklapp-

processes play ^an important ^{role and} ^must ^{be taken}

into account in order to explain the value of R ^at the resistivity maximum. If one adopts ^certain parameters, ^which ^can ^be roughly ^estimated ^thèo- retically, ^thé present theory ^{is in} agreement ^with experiment. Moreover there is a term in the

resistivity equation, ^which depends ^on ^the ^concen-

tration of Ag ôr ^Zn âtoms withïn the zones. It is this term that explains â dependence ^{of the} ^resisti- vity ôn ^the înternal properties ^{of the} ^zones ^which

has also been found experimentally. ^A ^{more com-} plete ^account of this work will be published ^soon

elsewhere. The au thor wishes to thank the ^Deutsohe Forschungs Gemeinschaft for a travel grant.

REFERENCES

[1] ^MATYAS (Z.), ^Phil. Mag., 1949, 40, 324.

[2] ^HARRISON (W. A.), ^The ^Fermi Surface, ^John Wiley ^and Sons, Inc., 1960, p. 28.

[3] ^GEROLD (V.) ^{and MERZ} (W.), Z. Metall., 1962, 53, ^341.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010082300

Electrical resistivity of zone-hardened Al Ag and Al Zn

HAL Id: jpa-00236687

https://hal.archives-ouvertes.fr/jpa-00236687

Submitted on 1 Jan 1962

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Electrical resistivity of zone-hardened Al Ag and Al Zn

R. Labusch

To cite this version:

R. Labusch. Electrical resistivity of zone-hardened Al Ag and Al Zn. J. Phys. Radium, 1962, 23 (10),

pp.823-823. �10.1051/jphysrad:019620023010082300�. �jpa-00236687�

823.

ELECTRICAL RESISTIVITY OF ZONE-HARDENED Al Ag AND Al Zn

By R. LABUSCH,

Institüt fur Metallphysik und Institüt für Theoretische Physik der Universität, Göttingen.

Résumé.

On discute la résistivité d’alliages Al Ag et Al Zn due à l’existence de zones de

Guinier-Preston, en tenant compte de la forme détaillée de la surface de Fermi dans l’aluminium.

Abstrat.

The resistivity of Al-Ag and Al-Zn alloys due to the existence of Guinier-Preston

zones is discussed, the exact form of the Fermi surface of aluminium being taken into account.

LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, OCTOBRE 1962,

Until now, attempts to explain the dependence

of the electrical resistivity p on the zone radius R in AI Ag and AI Zn alloys have not been quite satisfactory. Most of them are based on the free electron model with three free electrons per atom and can explain the fall-off in resistivity for large R only, not the maximum observed.

The magnitude of the effect, which has been

found experimentally to be up to 20-30 % of the

total resistivity, and the value of R at the resis-

tivity maximum (Rm

9-11 Á) cannot be explained by the free electron model at all.

To obtain a satisfactory theory, the real shape

of the Fermi-surface in Al must be taken into account. One such attempt has been made by Matyas [1], but unfortunately he used an incorrect

model for the conduction band and consequently

there is need for a new àttack on the problem.

The present investigation uses the band model of Harrison [2], which seems to be the best avail- able. In order to calculate the transport pro-

perties of a métal, one has to solve the Bloch equa- tion. In general this is impossible because of the

complicated shape of the Fermi-surface, but by

means of suitable models one can make certain calculations on the properties of the solution of this equation. The result is the following. By a trans-

formation of the p(R) curve for free electrons using

an effective k, where k is the propagation vector at

the Fermi-surface, one can get the approximate shape of the P(R) curve for bound electrons.

’ This becomes quantitatively correct for very large R, but keff has a much smaller value than that for the case of three free électrons per atom.

Two electrons do not take part in transport pheno-

mena at all because, according to FJarrison’s model,

the first Brillouin zone is completely filled.

Only one electron remains free for the transport

of charge. In addition, the value of keti itself is affected by the curvature of the Fermisurface.

If the curvature is sharp at certain points, hit

becomes small. For smaller values of R, Umklapp-

processes play an important role and must be taken

into account in order to explain the value of R at the resistivity maximum. If one adopts certain parameters, which can be roughly estimated thèo- retically, thé present theory is in agreement with experiment. Moreover there is a term in the

resistivity equation, which depends on the concen-

tration of Ag or Zn atoms withïn the zones. It is this term that explains a dependence of the resisti- vity on the internal properties of the zones which

has also been found experimentally. A more com- plete account of this work will be published soon

elsewhere. The au thor wishes to thank the Deutsohe Forschungs Gemeinschaft for a travel grant.

REFERENCES

[1] MATYAS (Z.), Phil. Mag., 1949, 40, 324.

[2] HARRISON (W. A.), The Fermi Surface, John Wiley and Sons, Inc., 1960, p. 28.

[3] GEROLD (V.) and MERZ (W.), Z. Metall., 1962, 53, 341.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010082300

ELECTRICAL RESISTIVITY OF ZONE-HARDENED Al Ag ^{AND Al Zn}

By ^R. LABUSCH,

Institüt fur Metallphysik und Institüt für Theoretische Physik ^der Universität, Göttingen.

On discute la résistivité d’alliages Âl Ag êt Âl ^{Zn due à} l’existence ^de ^zones de

Guinier-Preston, ^en ^tenant compte de la forme détaillée de la surface de Fermi dans l’aluminium.

The resistivity ôf Al-Ag ând Âl-Zn alloys ^due ^to the existence of Guinier-Preston

zones is discussed, ^the ^exact form of the Fermi surface of aluminium being taken into account.

LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, ^OCTOBRE 1962,

Until now, attempts ^to explain ^the dependence

of the electrical resistivity p ôn the zone radius R in AI Ag ând ÂI ^Zn alloys ^have ^not ^been quite satisfactory. Most of them are based on the free electron model with three free electrons per âtom and can explain the fall-off in resistivity ^for large ^R only, ^not the maximum observed.

The magnitude ^{of the} effect, ^{which has} ^been

found experimentally ^to ^be up ^to 20-30 % ^{of the}

tivity ^maximum (Rm

9-11 Á) ^cannot ^be explained by ^the ^free electron model at all.

To obtain a satisfactory theory, ^{the real} shape

of the Fermi-surface in Al must be taken into account. One such attempt ^has ^been ^made by Matyas [1], ^but unfortunately ^he ûsed ân încorrect

The present investigation ^uses the band model of Harrison [2], which ^seems ^to be the best avail- able. In order to calculate the transport pro-

perties ôf â métal, ône ^has ^to solve the Bloch equa- tion. In general ^{this is} impossible because of the

complicated shape ^{of the} Fermi-surface, ^but by

means of suitable models one can make certain calculations on the properties ^{of the} ^solution ^{of this} equation. The result is the following. By ^a ^trans-

formation of the p(R) ^curve for free electrons using

an effective k, ^{where k} ^is ^the propagation ^{vector at}

the Fermi-surface, ^{one can} get ^the approximate shape ^{of the} P(R) ^curve for bound electrons.

’ This becomes quantitatively ^correct ^{for very} large R, ^but ^keff ^has ^a much smaller value than that for the case of three free électrons per ^atom.

mena at all because, according to FJarrison’s ^model,

the first Brillouin zone is completely ^filled.

Only ^one ^electron ^remains free for the transport

of charge. ^In addition, the value of keti itself is affected by ^the ^curvature ^{of the} Fermisurface.

If the curvature is sharp ^at ^certain points, ^hit

processes play ^an important ^{role and} ^must ^{be taken}

into account in order to explain the value of R ^at the resistivity maximum. If one adopts ^certain parameters, ^which ^can ^be roughly ^estimated ^thèo- retically, ^thé present theory ^{is in} agreement ^with experiment. Moreover there is a term in the

resistivity equation, ^which depends ^on ^the ^concen-

tration of Ag ôr ^Zn âtoms withïn the zones. It is this term that explains â dependence ^{of the} ^resisti- vity ôn ^the înternal properties ^{of the} ^zones ^which

has also been found experimentally. ^A ^{more com-} plete ^account of this work will be published ^soon

elsewhere. The au thor wishes to thank the ^Deutsohe Forschungs Gemeinschaft for a travel grant.

[1] ^MATYAS (Z.), ^Phil. Mag., 1949, 40, 324.

[2] ^HARRISON (W. A.), ^The ^Fermi Surface, ^John Wiley ^and Sons, Inc., 1960, p. 28.

[3] ^GEROLD (V.) ^{and MERZ} (W.), Z. Metall., 1962, 53, ^341.