Dynamics of a crystal lattice containing isotopes
By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the qua...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
|---|
| Format: |
E-Artikel |
|---|---|
| Sprache: |
Englisch |
| Erschienen: |
1958 |
|---|
| Reproduktion: |
Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 |
|---|---|
| Übergeordnetes Werk: |
in: Physica - Amsterdam : Elsevier, 24(1958), 1-5, Seite 73-92 |
| Übergeordnetes Werk: |
volume:24 ; year:1958 ; number:1-5 ; pages:73-92 |
| Links: |
|---|
| Katalog-ID: |
NLEJ18020517X |
|---|
| LEADER | 01000caa a22002652 4500 | ||
|---|---|---|---|
| 001 | NLEJ18020517X | ||
| 003 | DE-627 | ||
| 005 | 20210706122048.0 | ||
| 007 | cr uuu---uuuuu | ||
| 008 | 070505s1958 xx |||||o 00| ||eng c | ||
| 035 | |a (DE-627)NLEJ18020517X | ||
| 035 | |a (DE-599)GBVNLZ18020517X | ||
| 040 | |a DE-627 |b ger |c DE-627 |e rakwb | ||
| 041 | |a eng | ||
| 245 | 1 | 0 | |a Dynamics of a crystal lattice containing isotopes |
| 264 | 1 | |c 1958 | |
| 336 | |a nicht spezifiziert |b zzz |2 rdacontent | ||
| 337 | |a nicht spezifiziert |b z |2 rdamedia | ||
| 338 | |a nicht spezifiziert |b zu |2 rdacarrier | ||
| 520 | |a By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. | ||
| 533 | |f Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 | ||
| 700 | 1 | |a Pirenne, J. |4 oth | |
| 773 | 0 | 8 | |i in |t Physica |d Amsterdam : Elsevier |g 24(1958), 1-5, Seite 73-92 |w (DE-627)NLEJ177041110 |w (DE-600)2205782-1 |x 0031-8914 |7 nnns |
| 773 | 1 | 8 | |g volume:24 |g year:1958 |g number:1-5 |g pages:73-92 |
| 856 | 4 | 0 | |u http://dx.doi.org/10.1016/S0031-8914(58)93831-X |
| 912 | |a GBV_USEFLAG_H | ||
| 912 | |a ZDB-1-SDJ | ||
| 912 | |a GBV_NL_ARTICLE | ||
| 951 | |a AR | ||
| 952 | |d 24 |j 1958 |e 1-5 |h 73-92 | ||
| matchkey_str |
article:00318914:1958----::yaisfcytlatccna |
|---|---|
| hierarchy_sort_str |
1958 |
| publishDate |
1958 |
| allfields |
(DE-627)NLEJ18020517X (DE-599)GBVNLZ18020517X DE-627 ger DE-627 rakwb eng Dynamics of a crystal lattice containing isotopes 1958 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 Pirenne, J. oth in Physica Amsterdam : Elsevier 24(1958), 1-5, Seite 73-92 (DE-627)NLEJ177041110 (DE-600)2205782-1 0031-8914 nnns volume:24 year:1958 number:1-5 pages:73-92 http://dx.doi.org/10.1016/S0031-8914(58)93831-X GBV_USEFLAG_H ZDB-1-SDJ GBV_NL_ARTICLE AR 24 1958 1-5 73-92 |
| spelling |
(DE-627)NLEJ18020517X (DE-599)GBVNLZ18020517X DE-627 ger DE-627 rakwb eng Dynamics of a crystal lattice containing isotopes 1958 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 Pirenne, J. oth in Physica Amsterdam : Elsevier 24(1958), 1-5, Seite 73-92 (DE-627)NLEJ177041110 (DE-600)2205782-1 0031-8914 nnns volume:24 year:1958 number:1-5 pages:73-92 http://dx.doi.org/10.1016/S0031-8914(58)93831-X GBV_USEFLAG_H ZDB-1-SDJ GBV_NL_ARTICLE AR 24 1958 1-5 73-92 |
| allfields_unstemmed |
(DE-627)NLEJ18020517X (DE-599)GBVNLZ18020517X DE-627 ger DE-627 rakwb eng Dynamics of a crystal lattice containing isotopes 1958 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 Pirenne, J. oth in Physica Amsterdam : Elsevier 24(1958), 1-5, Seite 73-92 (DE-627)NLEJ177041110 (DE-600)2205782-1 0031-8914 nnns volume:24 year:1958 number:1-5 pages:73-92 http://dx.doi.org/10.1016/S0031-8914(58)93831-X GBV_USEFLAG_H ZDB-1-SDJ GBV_NL_ARTICLE AR 24 1958 1-5 73-92 |
| allfieldsGer |
(DE-627)NLEJ18020517X (DE-599)GBVNLZ18020517X DE-627 ger DE-627 rakwb eng Dynamics of a crystal lattice containing isotopes 1958 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 Pirenne, J. oth in Physica Amsterdam : Elsevier 24(1958), 1-5, Seite 73-92 (DE-627)NLEJ177041110 (DE-600)2205782-1 0031-8914 nnns volume:24 year:1958 number:1-5 pages:73-92 http://dx.doi.org/10.1016/S0031-8914(58)93831-X GBV_USEFLAG_H ZDB-1-SDJ GBV_NL_ARTICLE AR 24 1958 1-5 73-92 |
| allfieldsSound |
(DE-627)NLEJ18020517X (DE-599)GBVNLZ18020517X DE-627 ger DE-627 rakwb eng Dynamics of a crystal lattice containing isotopes 1958 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 Pirenne, J. oth in Physica Amsterdam : Elsevier 24(1958), 1-5, Seite 73-92 (DE-627)NLEJ177041110 (DE-600)2205782-1 0031-8914 nnns volume:24 year:1958 number:1-5 pages:73-92 http://dx.doi.org/10.1016/S0031-8914(58)93831-X GBV_USEFLAG_H ZDB-1-SDJ GBV_NL_ARTICLE AR 24 1958 1-5 73-92 |
| language |
English |
| source |
in Physica 24(1958), 1-5, Seite 73-92 volume:24 year:1958 number:1-5 pages:73-92 |
| sourceStr |
in Physica 24(1958), 1-5, Seite 73-92 volume:24 year:1958 number:1-5 pages:73-92 |
| format_phy_str_mv |
Article |
| institution |
findex.gbv.de |
| isfreeaccess_bool |
false |
| container_title |
Physica |
| authorswithroles_txt_mv |
Pirenne, J. @@oth@@ |
| publishDateDaySort_date |
1958-01-01T00:00:00Z |
| hierarchy_top_id |
NLEJ177041110 |
| id |
NLEJ18020517X |
| language_de |
englisch |
| fullrecord |
<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">NLEJ18020517X</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20210706122048.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">070505s1958 xx |||||o 00| ||eng c</controlfield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)NLEJ18020517X</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)GBVNLZ18020517X</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Dynamics of a crystal lattice containing isotopes</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">1958</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">nicht spezifiziert</subfield><subfield code="b">zzz</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">nicht spezifiziert</subfield><subfield code="b">z</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">nicht spezifiziert</subfield><subfield code="b">zu</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial.</subfield></datafield><datafield tag="533" ind1=" " ind2=" "><subfield code="f">Elsevier Journal Backfiles on ScienceDirect 1907 - 2002</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Pirenne, J.</subfield><subfield code="4">oth</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">in</subfield><subfield code="t">Physica</subfield><subfield code="d">Amsterdam : Elsevier</subfield><subfield code="g">24(1958), 1-5, Seite 73-92</subfield><subfield code="w">(DE-627)NLEJ177041110</subfield><subfield code="w">(DE-600)2205782-1</subfield><subfield code="x">0031-8914</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:24</subfield><subfield code="g">year:1958</subfield><subfield code="g">number:1-5</subfield><subfield code="g">pages:73-92</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">http://dx.doi.org/10.1016/S0031-8914(58)93831-X</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_H</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-1-SDJ</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_NL_ARTICLE</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">24</subfield><subfield code="j">1958</subfield><subfield code="e">1-5</subfield><subfield code="h">73-92</subfield></datafield></record></collection>
|
| series2 |
Elsevier Journal Backfiles on ScienceDirect 1907 - 2002 |
| ppnlink_with_tag_str_mv |
@@773@@(DE-627)NLEJ177041110 |
| format |
electronic Article |
| delete_txt_mv |
keep |
| collection |
NL |
| remote_str |
true |
| illustrated |
Not Illustrated |
| issn |
0031-8914 |
| topic_title |
Dynamics of a crystal lattice containing isotopes |
| format_facet |
Elektronische Aufsätze Aufsätze Elektronische Ressource |
| format_main_str_mv |
Text Zeitschrift/Artikel |
| carriertype_str_mv |
zu |
| author2_variant |
j p jp |
| hierarchy_parent_title |
Physica |
| hierarchy_parent_id |
NLEJ177041110 |
| hierarchy_top_title |
Physica |
| isfreeaccess_txt |
false |
| familylinks_str_mv |
(DE-627)NLEJ177041110 (DE-600)2205782-1 |
| title |
Dynamics of a crystal lattice containing isotopes |
| spellingShingle |
Dynamics of a crystal lattice containing isotopes |
| ctrlnum |
(DE-627)NLEJ18020517X (DE-599)GBVNLZ18020517X |
| title_full |
Dynamics of a crystal lattice containing isotopes |
| journal |
Physica |
| journalStr |
Physica |
| lang_code |
eng |
| isOA_bool |
false |
| recordtype |
marc |
| publishDateSort |
1958 |
| contenttype_str_mv |
zzz |
| container_start_page |
73 |
| container_volume |
24 |
| format_se |
Elektronische Aufsätze |
| title_sort |
dynamics of a crystal lattice containing isotopes |
| title_auth |
Dynamics of a crystal lattice containing isotopes |
| abstract |
By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. |
| abstractGer |
By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. |
| abstract_unstemmed |
By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial. |
| collection_details |
GBV_USEFLAG_H ZDB-1-SDJ GBV_NL_ARTICLE |
| container_issue |
1-5 |
| title_short |
Dynamics of a crystal lattice containing isotopes |
| url |
http://dx.doi.org/10.1016/S0031-8914(58)93831-X |
| remote_bool |
true |
| author2 |
Pirenne, J. |
| author2Str |
Pirenne, J. |
| ppnlink |
NLEJ177041110 |
| mediatype_str_mv |
z |
| isOA_txt |
false |
| hochschulschrift_bool |
false |
| author2_role |
oth |
| up_date |
2024-07-06T02:46:27.834Z |
| _version_ |
1803796074027024384 |
| fullrecord_marcxml |
<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">NLEJ18020517X</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20210706122048.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">070505s1958 xx |||||o 00| ||eng c</controlfield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)NLEJ18020517X</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)GBVNLZ18020517X</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Dynamics of a crystal lattice containing isotopes</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">1958</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">nicht spezifiziert</subfield><subfield code="b">zzz</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">nicht spezifiziert</subfield><subfield code="b">z</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">nicht spezifiziert</subfield><subfield code="b">zu</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">By means of a moment method, the vibration spectrum of a monoatomic crystal containing several isotopes, whose atoms are distributed at random throughout the lattice points, is obtained in terms of the solution of the conventional vibration problem of a single isotope crystal and in terms of the quantities Δμp>Aυ=iciΔμip=ici(μi-μ>Aυ)p,where c"i is the concentration of the i-th isotope and Δμ"i the deviation of its inverse atomic mass μ"i from the average value <μ> = Σ"i" "c"i"μ"i.The frequency distribution is represented by a series development following the increasing powers of a heterogeneity parameter to which all the Δμ"i are assumed to be proportional. The first term is the frequency distribution of a fictious crystal whose atoms would all have the mass <μ>^A^v"-"1. The next two terms are proportional to <Δμ^2>"A"v and <Δμ^3>Av; the fourth term brings contributions in <Δμ^4>Av and in <Δμ^2>^2, etc. These three additional terms have been written down explicitly, but the following ones could be easily obtained.Each term but the first one brings contributions of the form ∫ δ(λ - λ') K(λ') dλ', where λ is the square of the angular frequency. These functionals may not be simply put equal to K'(λ), as K(λ) has singularities which must be carefully considered. In thermodynamic applications, however, this difficulty is irrelevant for then the δ' function is simply replaced by the λ-derivative of the desired thermodynamic function of the harmonic oscillator and the remaining integral is trivial.</subfield></datafield><datafield tag="533" ind1=" " ind2=" "><subfield code="f">Elsevier Journal Backfiles on ScienceDirect 1907 - 2002</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Pirenne, J.</subfield><subfield code="4">oth</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">in</subfield><subfield code="t">Physica</subfield><subfield code="d">Amsterdam : Elsevier</subfield><subfield code="g">24(1958), 1-5, Seite 73-92</subfield><subfield code="w">(DE-627)NLEJ177041110</subfield><subfield code="w">(DE-600)2205782-1</subfield><subfield code="x">0031-8914</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:24</subfield><subfield code="g">year:1958</subfield><subfield code="g">number:1-5</subfield><subfield code="g">pages:73-92</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">http://dx.doi.org/10.1016/S0031-8914(58)93831-X</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_H</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-1-SDJ</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_NL_ARTICLE</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">24</subfield><subfield code="j">1958</subfield><subfield code="e">1-5</subfield><subfield code="h">73-92</subfield></datafield></record></collection>
|
| score |
7.4010286 |