Zeeman spectroscopy of vacancy-charge-compensated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Er</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> sit

We present polarization-resolved optical absorption measurements on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msup> <a:mrow> <a:mi>Er</a:mi> </a:mrow> <a:mrow> <a:mn>3</a:mn> <a:mo>+</a:mo> </a:mrow> </a:msup> </a:math> ions in <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msub> <b:mi>CaWO</b:mi> <b:mn>4</b:mn> </b:msub> </b:math> under vector magnetic fields, focusing on charge-compensated sites arising from local <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msup> <c:mrow> <c:mi>Ca</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> <c:mo>+</c:mo> </c:mrow> </c:msup> </c:math> vacancies. While the known axial <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msup> <d:mrow> <d:mi>Er</d:mi> </d:mrow> <d:mrow> <d:mn>3</d:mn> <d:mo>+</d:mo> </d:mrow> </d:msup> </d:math> site displays a single symmetric Zeeman-split transition pattern consistent with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi mathvariant="normal">S</e:mi> <e:mn>4</e:mn> </e:msub> </e:math> symmetry, two additional sites exhibit more complex spectral behavior, including sets of transitions that interchange under <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msup> <g:mn>90</g:mn> <g:mo>∘</g:mo> </g:msup> </g:math> crystal rotations-evidence of reduced, rhombiclike symmetry. From these polarization- and temperature-dependent spectra, we extract effective <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>g</h:mi> </h:math> factors. Our findings are corroborated by electron paramagnetic resonance measurements and support a model of multiple inequivalent <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:msup> <i:mrow> <i:mi>Ca</i:mi> </i:mrow> <i:mrow> <i:mn>2</i:mn> <i:mo>+</i:mo> </i:mrow> </i:msup> </i:math> vacancies around <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msup> <j:mrow> <j:mi>Er</j:mi> </j:mrow> <j:mrow> <j:mn>3</j:mn> <j:mo>+</j:mo> </j:mrow> </j:msup> </j:math> sites in the host lattice. This detailed characterization contributes to understanding defect-engineered rare-earth sites for quantum information applications.