Na-feldspar: temperature, pressure and the state of order

• Main Authors: H. Kroll, H. U. Bambauer, H. Pentinghaus
• Format: Article
• Language: English
• Published: Copernicus Publications 2020-07-01
• Series: European Journal of Mineralogy
• Online Access: https://ejm.copernicus.org/articles/32/427/2020/ejm-32-427-2020.pdf
• ISSN: 0935-1221; 1617-4011

<p>In feldspars, mean tetrahedral T–O bond lengths (T <span class="inline-formula">=</span> Al,Si) are the standard measure of the tetrahedral Al content. However, for a sophisticated assessment of the Al,Si distribution, factors have to be accounted for (1) that cause individual T–O bond lengths to deviate from their tetrahedral means and (2) that cause mean tetrahedral lengths to deviate from values specified by the Al content. We investigated low albite, <span class="inline-formula">Na[AlSi₃O₈]</span>, from six X-ray crystal structure refinements available in the literature. The Al,Si distribution of low albite is fully ordered so that Al,Si–O bond length variations result only from bond perturbing factors. For the <i>intra</i>-tetrahedral variation <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M3" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi mathvariant="normal">Δ</mi><mtext>T–O</mtext><mo>≡</mo><mtext>T–O</mtext><mo>-</mo><mo>〈</mo><mtext>T–O</mtext><mo>〉</mo></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="103pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="de4a18840c900a64a92c51c9eeb720fa"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-32-427-2020-ie00001.svg" width="103pt" height="13pt" src="ejm-32-427-2020-ie00001.png"/></svg:svg></span></span>, only two factors turned out to be effective: (1) the sum of bond critical point electron densities in the Na–O and T–O bonds neighbouring the T–O bond under consideration and (2) the fractional <span class="inline-formula"><i>s</i></span>-bond character of the bridging oxygen atom. This model resulted in a root mean square (rms) value for <span class="inline-formula">ΔT–O</span> of only 0.002 Å, comparable to the estimated standard deviations (esd's) routinely quoted in X-ray and neutron structure refinements. In the second step, the <i>inter</i>-tetrahedral differences <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M6" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi mathvariant="normal">Δ</mi><mo>〈</mo><mtext>T–O</mtext><mo>〉</mo><mo>≡</mo><mo>〈</mo><mtext>T–O</mtext><mo>〉</mo><mo>-</mo><mo>〈</mo><mo>〈</mo><mtext>T–O</mtext><mo>〉</mo><mo>〉</mo></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="124pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="6abfb304be1d8b97681671ce7a3bdf5f"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-32-427-2020-ie00002.svg" width="124pt" height="13pt" src="ejm-32-427-2020-ie00002.png"/></svg:svg></span></span> were considered. Here, apart from the tetrahedral Al content, the only size-perturbing factor is the difference between the tetrahedral and the grand mean fractional <span class="inline-formula"><i>s</i></span>-characters. The resulting rms value was as small as 0.0003&thinsp;Å.</p> <p>From this analysis, Al site occupancies, t, can be derived from observed mean tetrahedral distances, <span class="inline-formula">〈T–O〉_obs</span>, as </p><div class="disp-formula" content-type="" id="Ch1.Ex1"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M9" display="block" overflow="scroll" dspmath="mathml"><mrow><mstyle class="stylechange" displaystyle="true"/><mi mathvariant="normal">t</mi><mo>=</mo><mn mathvariant="normal">0.25</mn><mo>(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mi>n</mi><mrow class="chem"><mi mathvariant="normal">An</mi></mrow></msub><mo>)</mo><mo>+</mo><mfenced open="(" close=")"><mrow><mo>〈</mo><mtext>T–O</mtext><msub><mo>〉</mo><mrow class="chem"><mi mathvariant="normal">adj</mi></mrow></msub><mo>-</mo><mo>〈</mo><mo>〈</mo><mtext>T–O</mtext><mo>〉</mo><mo>〉</mo></mrow></mfenced><mo>/</mo><mn mathvariant="normal">0.12466</mn><mspace linebreak="nobreak" width="0.125em"/><mo>(</mo><mn mathvariant="normal">17</mn><mo>)</mo><mo>,</mo></mrow></math><div><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="256pt" height="16pt" class="svg-formula" dspmath="mathimg" md5hash="fe07b7a1b3c638e07d3983e3abcfb650"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-32-427-2020-ue00001.svg" width="256pt" height="16pt" src="ejm-32-427-2020-ue00001.png"/></svg:svg></div></div> <p>with the observed distance <span class="inline-formula">〈T–O〉_obs</span> adjusted for the influence of the fractional <span class="inline-formula"><i>s</i></span>-character, <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M12" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>〈</mo><mtext>T–O</mtext><msub><mo>〉</mo><mrow class="chem"><mi mathvariant="normal">adj</mi></mrow></msub><mo>=</mo><mo>〈</mo><mtext>T–O</mtext><msub><mo>〉</mo><mrow class="chem"><mi mathvariant="normal">obs</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">0.1907</mn><mo>(</mo><mn mathvariant="normal">51</mn><mo>)</mo><mspace width="0.125em" linebreak="nobreak"/><mo>[</mo><mo>〈</mo><msub><mi>f</mi><mi>s</mi></msub><mrow class="chem"><mo>(</mo><mi mathvariant="normal">O</mi><mo>)</mo></mrow><mo>〉</mo><mo>-</mo><mo>〈</mo><mo>〈</mo><msub><mi>f</mi><mi>s</mi></msub><mrow class="chem"><mo>(</mo><mi mathvariant="normal">O</mi><mo>)</mo></mrow><mo>〉</mo><mo>〉</mo><mo>]</mo></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="245pt" height="15pt" class="svg-formula" dspmath="mathimg" md5hash="76271aee603b405952b6274f3466ce59"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-32-427-2020-ie00003.svg" width="245pt" height="15pt" src="ejm-32-427-2020-ie00003.png"/></svg:svg></span></span>. This equation served to determine the site occupancies of 16 intermediate to high albites and one analbite from their mean tetrahedral distances. It was found that the individual site occupancies t<span class="inline-formula">₁</span>0, t<span class="inline-formula">₁</span>m and t<span class="inline-formula">₂0=</span> t<span class="inline-formula">₂</span>m all vary linearly with the difference <span class="inline-formula">Δt₁=</span> t<span class="inline-formula">₁0−</span> t<span class="inline-formula">₁</span>m. <span class="inline-formula">Δt₁</span>, in turn, varies linearly with the length difference, <span class="inline-formula">Δ</span>tr[110], between the unit cell repeat distances [<span class="inline-formula">1∕2<i>a</i></span>, <span class="inline-formula">1∕2<i>b</i></span>, 0] and [<span class="inline-formula">1∕2<i>a</i></span>, <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M25" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mi>b</mi></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="35pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="2fda8e0b28ae904e7551963328325caf"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-32-427-2020-ie00004.svg" width="35pt" height="14pt" src="ejm-32-427-2020-ie00004.png"/></svg:svg></span></span>, 0]. Then, from the <span class="inline-formula">Δ</span>tr[110] indicator, values of <span class="inline-formula">t</span> were obtained as </p><div class="disp-formula" content-type="" specific-use="align"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M28" display="block" overflow="scroll" dspmath="mathml"><mtable columnalign="left" displaystyle="true"><mtr><mtd><mrow><mstyle displaystyle="true" class="stylechange"/><msub><mi mathvariant="normal">t</mi><mn mathvariant="normal">1</mn></msub><mn mathvariant="normal">0</mn></mrow></mtd><mtd><mrow><mstyle class="stylechange" displaystyle="true"/><mo>=</mo><mo>(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">0</mn></msub><mo>)</mo><mo>+</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">0</mn></msub><mspace linebreak="nobreak" width="0.125em"/><mo>(</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">2</mn></msub><mspace width="0.125em" linebreak="nobreak"/><mi mathvariant="normal">Δ</mi><mrow class="chem"><mi mathvariant="normal">tr</mi><mo>[</mo><mn mathvariant="normal">110</mn><mo>]</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mstyle displaystyle="true" class="stylechange"/><msub><mi mathvariant="normal">t</mi><mn mathvariant="normal">1</mn></msub><mrow class="chem"><mi mathvariant="normal">m</mi></mrow></mrow></mtd><mtd><mrow><mstyle class="stylechange" displaystyle="true"/><mo>=</mo><mo>(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">0</mn></msub><mo>)</mo><mo>-</mo><mo>(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">0</mn></msub><mo>)</mo><mspace linebreak="nobreak" width="0.125em"/><mo>(</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">2</mn></msub><mspace width="0.125em" linebreak="nobreak"/><mi mathvariant="normal">Δ</mi><mrow class="chem"><mi mathvariant="normal">tr</mi><mo>[</mo><mn mathvariant="normal">110</mn><mo>]</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mstyle displaystyle="true" class="stylechange"/><msub><mi mathvariant="normal">t</mi><mn mathvariant="normal">2</mn></msub><mn mathvariant="normal">0</mn></mrow></mtd><mtd><mrow><mstyle class="stylechange" displaystyle="true"/><mo>=</mo><msub><mi mathvariant="normal">t</mi><mn mathvariant="normal">2</mn></msub><mrow class="chem"><mi mathvariant="normal">m</mi></mrow><mo>=</mo><mo>(</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">0</mn></msub><mo>-</mo><mn mathvariant="normal">0.5</mn><mo>)</mo><mo>-</mo><mo>(</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">0</mn></msub><mo>-</mo><mn mathvariant="normal">0.5</mn><mo>)</mo><mspace linebreak="nobreak" width="0.125em"/><mo>(</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">b</mi><mn mathvariant="normal">2</mn></msub><mspace width="0.125em" linebreak="nobreak"/><mi mathvariant="normal">Δ</mi><mrow class="chem"><mi mathvariant="normal">tr</mi><mo>[</mo><mn mathvariant="normal">110</mn><mo>]</mo></mrow><mo>)</mo><mo>,</mo></mrow></mtd></mtr></mtable></math><div><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="248pt" height="46pt" class="svg-formula" dspmath="mathimg" md5hash="cfecd6185490c16477e2a53bb9fd506f"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-32-427-2020-ue00002.svg" width="248pt" height="46pt" src="ejm-32-427-2020-ue00002.png"/></svg:svg></div></div> <p>with b<span class="inline-formula">₀=0.7288(16)</span>, b<span class="inline-formula">₁=0.1103(59)</span> and b<span class="inline-formula">₂=3.234(32)</span>&thinsp;Å<span class="inline-formula">⁻¹</span>.</p> <p>Finally, from an expression that converts the <span class="inline-formula">Δ2<i>θ</i>(131)</span> measure of order into <span class="inline-formula">Δ</span>tr[110] and thus into site occupancies, it was possible to obtain from the unique suite of bracketed high-pressure experiments performed<span id="page428"/> on albites by Goldsmith and Jenkins (1985) the evolution with equilibrium temperature of the thermodynamic order parameter <span class="inline-formula"><i>Q</i>_od</span> and of the individual Al site occupancies t at a pressure of 1&thinsp;bar. For that purpose, since the Goldsmith and Jenkins experiments were performed at <span class="inline-formula">≈18</span>&thinsp;kbar, a procedure was devised that accounts for the effect of pressure on the state of order. At 1&thinsp;bar, low albite is stable up to 590&thinsp;<span class="inline-formula">^∘</span>C, where it begins to disorder, turning into high albite above 720&thinsp;<span class="inline-formula">^∘</span>C. The highly though not fully disordered monoclinic state (monalbite) is reached at 980&thinsp;<span class="inline-formula">^∘</span>C, 1&thinsp;bar, and 1055&thinsp;<span class="inline-formula">^∘</span>C, 18&thinsp;kbar, respectively. Eventually, when applying the determinative equations given above to low microcline, full order is predicted as in low albite.</p>