Sacrifice as a propitiation or expiation

How can I find out the way that sacrifice is found in the New Testament text as a propitiation or expiation. I have tried doing a word study on sacrifice but that did not do what I wanted. Any advice? 

This is not done with a word study, since it depends on whether people think God needs to be propitiated (and what exactly that means). You can probably find articles or sections in monographs that discuss this (I remember, John Stott wrote about this in The Cross of Christ)

I remember, John Stott wrote about this in The Cross of Christ

dug up the text. I'll give you the beginning of the most relevant part - check the complete book for more, and the footnotes. 

...To “propitiate” somebody means to appease or pacify his anger. Does God then get angry? If so, can offerings or rituals assuage his anger? Does he accept bribes? Such concepts sound more pagan than Christian. It is understandable that primitive animists should consider it essential to placate the wrath of gods, spirits or ancestors, but are notions like these worthy of the Christian God? Should we not have grown out of them? In particular, are we really to believe that Jesus by his death propitiated the Father’s anger, inducing him to turn from it and to look upon us with favor instead?
Crude concepts of anger, sacrifice and propitiation are indeed to be rejected. They do not belong to the religion of the Old Testament, let alone of the New. This does not mean, however, that there is no biblical concept of these things at all. What is revealed to us in Scripture is a pure doctrine (from which all pagan vulgarities have been expunged) of God’s holy wrath, his loving self-sacrifice in Christ and his initiative to avert his own anger. It is obvious that “wrath” and “propitiation” (the placating of wrath) go together. It is when the wrath is purged of unworthy ideas that the propitiation is thereby purged. The opposite is also true. It is those who cannot come to terms with any concept of the wrath of God who repudiate any concept of propitiation. Here, for example, is Professor A. T. Hanson: “If you think of the wrath as an attitude of God, you cannot avoid some theory of propitiation. But the wrath in the New Testament is never spoken of as being propitiated, because it is not conceived of as being an attitude of God.”1
It is this discomfort with the doctrines of wrath and propitiation that has led some theologians to reexamine the biblical vocabulary. They have concentrated on a particular word-group which the Authorized Version translated in “propitiatory” terms, namely the noun hilasmos (1 Jn 2:2; 4:10), the adjective hilastērios (Rom 3:25, where it may be used as a noun) and the verb hilaskomai (Heb 2:17; also Lk 18:13 in the passive, which should perhaps be rendered “be propitiated—or propitious—to me, a sinner”). The crucial question is whether the object of the atoning action is God or humans. If the former, then the right word is propitiation (appeasing God); if the latter, the right word is expiation (dealing with sin and guilt).
The British theologian who led the way in this attempted reinterpretation was Professor C. H. Dodd.2 Here is his comment on Romans 3:25: “the meaning conveyed … is that of expiation, not that of propitiation. Most translators and commentators are wrong.”3 He expresses a similar opinion in relation to 1 John 2:2, namely that the translation “propitiation for our sins” is “illegitimate here as elsewhere.”4 Since Professor Dodd was director of the panels that produced the New English Bible (New Testament 1961), it is not surprising that his view was reflected in its rendering of the verses just referred to. Romans 3:25 is translated “God designed him to be the means of expiating sin by his sacrificial death,” while in 1 John 2:2 and 1 John 4:10 the key phrase is rendered “he is himself the remedy for the defilement of our sins.” The RSV, whose New Testament was published a few years earlier (1946), has “expiation” in all three verses.
Professor Dodd’s argument, developed with his customary erudition, was linguistic. He acknowledged that in pagan Greek (both classical and popular) the regular meaning of the verb hilaskomai was to “propitiate” or “placate” an offended person, especially a deity. But he denied that this was its meaning either in Hellenistic Judaism, as evidenced in the Septuagint (LXX), or, on that account, in the New Testament. He argued that in the LXX kipper (the Hebrew verb for “atone”) was sometimes translated by Greek words other than hilaskomai, which mean to “purify” or “cancel”; that hilaskomai in the LXX sometimes translates other Hebrew words than kipper, which mean to “cleanse” or “forgive”; and that when hilaskomai does translate kipper the meaning is expiation or the removal of defilement. This is how he sums up: “Hellenistic Judaism, as represented by the LXX, does not regard the cultus as a means of pacifying the displeasure of the Deity, but as a means of delivering man from sin.”5 Indeed, it was generally believed in antiquity that “the performance of prescribed rituals … had the value, so to speak, of a powerful disinfectant.”6 Therefore, he concludes, the New Testament occurrences of the hilaskomai word-group should be interpreted in the same way. By his cross Jesus Christ expiated sin; he did not propitiate God.
Professor Dodd’s reconstruction, although accepted by many of his contemporaries and successors, was subjected to a rigorous critique by others, in particular by Dr. Leon Morris7 and Dr. Roger Nicole.8

1 A. T. Hanson, The Wrath of the Lamb (London: SPCK, 1959), p. 192.

2 C. H. Dodd contributed an article on hilaskesthai to the Journal of Theological Studies (“Hilaskesthai: Its Cognates, Derivatives and Synonyms in the Septuagint,” Journal of Theological Studies 32 [1931]: 352–60), which was subsequently republished in his Bible and the Greeks (London: Hodder & Stoughton, 1935). The same attempt to reinterpret “propitiation” as “expiation” is also expressed in his two Moffatt New Testament Commentaries The Epistle of Paul to the Romans (London: Hodder & Stoughton, 1932), and The Johannine Epistles (London: Hodder & Stoughton, 1946).

3 Dodd, Bible and the Greeks, p. 94. See also his Epistle of Paul, pp. 54–55.

4 Dodd, Johannine Epistles, pp. 25–26.

5 Dodd, Bible and the Greeks, p. 93.

6 Dodd, Johannine Epistles, pp. 25–26.

7 Leon Morris wrote an article on hilaskesthai in The Expository Times (“The Use of Hilaskesthai etc. in Biblical Greek,” The Expository Times 62, no. 8 [1951]: 227–33), and then expanded his thesis in his—The Apostolic Preaching of the Cross (London: Tyndale Press, 1955). He has also produced a further development and simplification of this book in The Atonement: Its Meaning and Significance (Downers Grove, Ill.: InterVarsity Press, 1983).

8 Roger R. Nicole’s article titled “C. H. Dodd and the Doctrine of Propitiation” appeared in the Westminster Theological ]ournal 17, no. 2 (1955): 117–57. He acknowledges some indebtedness to Leon Morris, although it is an independent study.

Source: John R. W. Stott, The Cross of Christ (Downers Grove, IL: IVP Books, 2006), 167–169. https://ref.ly/logosres/crossofchrist?ref=Page.p+167-169&off=2405