When is it appropriate to use 'admixture' rather than 'mixture'?

Solution 1:

Admixture is a minor ingredient, a smaller part of mixture, added to mixture

Mixture is irrelevant of ingredients proportions, and is one whole mixture.

In your quote, the author wants to stress that every single component that was added, plays its role in the final impression. To make such an emphasis, 'admixture' works better.

Technical example from Federal Highway Administration: (I am giving a technical context because in this case 'admixture' is the only acceptable term, and is absolutely different from 'mixture')

Chemical Admixtures

Chemical admixtures are added to concrete in very small amounts mainly for the entrainment of air, reduction of water or cement content, plasticization of fresh concrete mixtures, or control of setting time.

Seven types of chemical admixtures are specified in ASTM C 494, and AASHTO M 194, depending on their purpose or purposes in PCC. Air entraining admixtures are specified in ASTM C 260 and AASHTO M 154. General and physical requirements for each type of admixture are included in the specifications.

Examples

An article on Yourdictionary.com packed with lots of other context examples (follow the link).

Most dictionaries do not explain the difference, I found only two that mention that admixture as 'a minor ingredient':

The Oxford Pocket Dictionary of Current English (2009)

ad·mix·ture / adˈmikschər/ something mixed with something else,

• typically is a minor ingredient.

• the action of adding such an ingredient.

Collins dictionary article is good for giving synonyms and clearly splits admixture in two meanings, 'mixture' and 'component':

admixture

noun

• mixture, combination, compound, blend, fusion, alloy, medley, amalgamation, meld, intermixture
  • His heart beat with an admixture of aversion and thrill.
• ingredient, element, component, constituent

Example from the question: (mixture or admixture)

The 'Natura' section features a series of sixteen prose poems, redolent in many ways of Hill's Mercian Hymns in their admixtures of natural and urban landscape, dramatised autobiography and historical reference, fomal and colloquial styles.

In this case replacing 'admixtures' with 'mixtures' would not be a grammar mistake but it would shift the emphases. The author refers to different components of the Mercian Hymns, and enumerates them. But, if 'mixtures' were used, the emphasis would shift towards hymns themselves rather than components. The description would become less poetic.

Solution 2:

The point is about adding to mixture elements that are alien to some extent to the majority or to each other. As you can see, that collection of poems is a bit strange in their eclecticness:-). See the last example in the dictionary definition.

Let me give you the official separation first as per the mother of all synonym dictionaries:

Admixture adds to mixture the suggestion of the alien character of one or more of the constituent elements

Merriam-Webster's Dictionary of Synonyms: A Dictionary of Discriminated ...

Then, lemme give you the whole/real shebang of a definition :-)

ad·mix·ture

noun

1 a : the act of mixing

a favorable result will be obtained only by careful admixture of ingredients

b : the fact of being mixed

repeated sifting is necessary to secure complete admixture

2 a : an element or substance added by mixing

comic verses with an occasional admixture of mild bawdry — Alexander Cowie

b : a substance other than cement, aggregate, or water that is mixed with concrete

3 : a product of mixing:

a : a compound formed by mixing

by using wool, silk, cotton, and linen fibers in various admixtures — A. C. Morrison

b : something formed by combining two or more things

The presiding spirit of this place—an admixture of fear and what passes for hope among chronic gamblers—can be found in the basement beneath the grandstand … — David Samuels, Harper’s, February 1999

Merriam-Webster Unabridged Dictionary

Solution 3:

Affirming and expanding upon the good answer of javaNoobs:

1. Mixture is a more general term with broader application than admixture:

noun

1.0 A substance made by mixing other substances together:
form the mixture into a manageable dough

1.1 [MASS NOUN] The process of mixing or being mixed.

1.2 A combination of different things in which the component elements are individually distinct:
the old town is a mixture of narrow medieval streets and 18th-century architecture

1.3 Chemistry The product of the random distribution of one substance through another without any chemical reaction, as distinct from a compound.

1.4 The charge of gas or vapour mixed with air which is admitted to the cylinder of an internal-combustion engine, especially as regards the ratio of fuel to air:

2. Admixture shares some of the definitions of mixture, and in may cases is an acceptable synonym:

noun

1.0 A mixture:
he felt that his work was an admixture of aggression and creativity

[Entry 1.1 omitted for consideration below]

1.2 [MASS NOUN] The action of adding an ingredient to something else.

3. Entry 1.1 in the ODO is unique to admixture:

1.1 Something mixed with something else:
green with an admixture of black

The prefix ad carries admixture toward the definition of additive in some contexts:

noun

A substance added to something in small quantities to improve or preserve it:

Mixture is not generally used to describe an added ingredient in that way.

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