# An automatically discovered chain-of-thought prompt generalizes to novel
models and datasets

Konstantin Hebenstreit1,2,∗, Robert Praas1,3,∗, Louis P Kiesewetter4, Matthias
Samwald1,§
1Institute of Artificial Intelligence, Medical University of Vienna, Austria
2Johannes Kepler University Linz, Austria
3 KTH Royal Institute of Technology, Stockholm, Sweden
4Humboldt University Berlin, Germany
*equal contribution, §corresponding author
matthias.samwald [@] meduniwien.ac.at

###### Abstract

Emergent chain-of-thought (CoT) reasoning capabilities promise to improve
performance and explainability of large language models (LLMs). However,
uncertainties remain about how reasoning strategies formulated for previous
model generations generalize to new model generations and different datasets.
In this small-scale study, we compare different reasoning strategies induced
by zero-shot prompting across six recently released LLMs (davinci-002,
davinci-003, GPT-3.5-turbo, GPT-4, Flan-T5-xxl and Cohere command-xlarge) on a
mixture of six question-answering datasets, including datasets from scientific
and medical domains. Our findings demonstrate that while some variations in
effectiveness occur, gains from CoT reasoning strategies remain robust across
different models and datasets. GPT-4 has the most benefit from current state-
of-the-art reasoning strategies and exhibits the best performance by applying
a prompt previously discovered through automated discovery.

##  1 Motivation

Emergent chain-of-thought (CoT) reasoning capabilities in large language
models (LLMs) promise to improve both predictive performance and
explainability of models when applied to complex tasks Wei et al. (2021).
While good performance can be reached by few-shot in-context prompting with
exemplars suitable to a specific task at hand, zero-shot prompting setups do
not require such task-dependent selection of exemplars Kojima et al. (2022).
The recent success of models optimized for dialog, such as GPT-3.5, further
increases the expectation that models reach robust performance with ad-hoc
reasoning strategies and are less influenced by minor variations. This study
empirically investigates how well previously discovered zero-shot CoT
prompting styles generalize to new model generations and datasets and how they
compare to newly developed reasoning strategies. We conduct our evaluations on
six question-answering datasets of varying levels of complexity, including
scientific and medical domains.

##  2 Methods

###  2.1 Datasets

For our study, we used the ThoughtSource framework Ott et al. (2023), which
provides a comprehensive meta-dataset and software library designed for
streamlined generation, evaluation, and annotation of chain-of-thought (CoT)
reasoning. We covered a sizable range of topics and complexity levels by
selecting subsamples of six datasets spanning common-sense Talmor et al.
(2019); Geva et al. (2021), scientific Xie et al. (2020); Mihaylov et al.
(2018), and medical domains Jin et al. (2021); Pal et al. (2022) (Table 1).
All of these question-answering datasets were multiple-choice, consisting of
two to five answer options with a single correct response.

Dataset |  Description
---|---
CommonsenseQA |  General domain crowd-sourced questions with high semantic complexity which command the use of prior knowledge.
StrategyQA |  General domain crowd-sourced questions which require implicit reasoning and multi-step answer strategies. Yes/No answers.
WorldTree v2 |  Elementary science questions for 3rd to 5th grade level, combining domain specific and world knowledge.
OpenBookQA |  Scientific and broad common knowledge questions, which require multi-step reasoning and rich text comprehension.
MedQA |  Questions from medical board exams. We used only examples from the US (USMLE subset).
MedMCQA |  Real-world medical entrance exam questions.
Table 1:  Descriptions of various datasets.

###  2.2 Prompts

We assembled a set of ten zero-shot reasoning strategies (Table 5 and Appendix
A) consisting of one baseline, two pre-existing, and seven novel designs:

  1. 1.

Direct prompting: No specific trigger, serving as a baseline for comparison.

  2. 2.

Kojima: A well-established CoT prompt, "Let’s think step by step." Kojima et
al. (2022)

  3. 3.

Zhou: An enhanced version created through automated prompt engineering, "Let’s
work this out in a step by step way to be sure we have the right answer." Zhou
et al. (2023)

  4. 4.

Seven original reasoning strategies designed by us, inspired by various public
resources OpenAI (2023a); Schulhoff (2022), and refined through iterative
adaptation based on analyzing outputs. One of these strategies employed a
self-critique strategy, requiring the model to provide an initial answer,
critique it, and then propose a revised response Madaan et al. (2023);
Saunders et al. (2022).

###  2.3 Models

We included six instruction-tuned models based on their reported capabilities
in CoT reasoning: davinci-002 Brown et al. (2020), davinci-003 Ouyang et al.
(2022), GPT-3.5-turbo OpenAI (2022), and GPT-4 OpenAI (2023b) from OpenAI,
Flan-T5-xxl from Google Chung et al. (2022), and command-xlarge-nightly from
Cohere Cohere.ai (2023). We used the LangChain framework Chase (2022) to
access several APIs. Between February and April 2023, we conducted 11,880
experiments, all with the model temperature set at 0 and a maximum token
length of 512.

###  2.4 Evaluation

We selected Krippendorff’s alpha as our evaluation metric Krippendorff (2011).
It allows for combining results from sub-datasets with different numbers of
answer choices by correcting for their corresponding base probability rates.
Krippendorff’s alpha measured inter-rater reliability on a scale from zero
(random chance) to one (complete agreement) and was used to compare model
predictions to gold standard answers Castro (2017).

To determine an appropriate sample size, we performed a power analysis with a
significance level set at 0.05, a medium Krippendorff’s alpha value of 0.8,
and a base correct probability of 0.2, considering the maximum of five answer
options in our sub-datasets (Appendix B). The analysis yielded a required
sample size of 164 items, which we increased to 198 items, divided into six
sub-datasets of 33 items each. We used bootstrapping (r = 1000) to compute
means and confidence intervals for the generated results. To guarantee
accurate Krippendorff scores, which depend on the number of options, we
bootstrapped each sub-dataset individually when needed and calculated
confidence intervals by pooling standard deviations.

##  3 Results

All scores within this paper are displayed with 95% confidence intervals (CI).

Prompt |  GPT-4 $\alpha$ (CI) n per prompt = 198 |  Model avg. $\alpha$ (CI) n per prompt = 1188
---|---|---
Zhou | .83 (.77, .90) | .53 (.50, .57)
Kojima | .80 (.73, .87) | .51 (.47, .55)
Zhou-instr. | .79 (.72, .86) | .50 (.46, .54)
Articulate | .79 (.71, .86) | .52 (.48, .56)
Rephrase | .78 (.71, .85) | .54 (.51, .58)
Plan | .77 (.71, .84) | .50 (.46, .54)
Elaborate | .77 (.70, .84) | .51 (.47, .55)
Self-critique | .76 (.69, .84) | .49 (.45, .53)
Converse | .74 (.66, .81) | .47 (.43, .51)
Direct | .71 (.64, .79) | .49 (.45, .52)
Table 2: Krippendorff’s alpha ($\alpha$) performance of prompts averaged over
datasets. Average taken solely for GPT-4 and over all six models, n total =
11880.

Although the performance of many prompts averaged over all datasets is notably
similar, we see that applying reasoning strategies outperforms direct
prompting. A closer examination of the results obtained from the latest model,
GPT-4, highlights the advantage of employing specific prompts (Table 2). It
shows the retained performance of the automatically discovered prompt by Zhou
et al. (2023), which also has a notable result in the score averaged over
models. Interestingly, the self-critique prompt yielded relatively low scores.
It also resulted in the generation of multiple answers in various observed
instances, which were excluded from the scoring process. Creating an
instruction prompt by placing Zhou’s reasoning prompt before instead of after
the question did not yield better outcomes.

Better models are finding WorldTree v2 and CommonsenseQA increasingly easy
(Table 9), while StrategyQA suffers from peculiar items. This highlights the
necessity for developing more refined general-knowledge datasets or employing
domain-specific datasets, such as the two medical ones (Table 3).

Dataset |  $\alpha$ (CI) n per dataset = 1980
---|---
WorldTree v2 | .83 (.81, .85)
CommonsenseQA | .71 (.68, .73)
OpenBookQA | .65 (.63, .68)
StrategyQA | .31 (.27, .36)
MedMCQA | .31 (.28, .34)
MedQA | .21 (.19, .24)
Table 3: Krippendorff’s alpha ($\alpha$) performance on data-sets averaged
over models and prompts. N total = 11880.

FLAN-T5 shows good performance for its size, but its results are possibly
affected by data contamination: It was instruction-fine tuned on the sub-
datasets CommonsenseQA and StrategyQA. GPT-3.5-turbo and GPT-4 performed best
(Table 4) and were the only models that displayed decent performance on
medical datasets (Table 9).

Model |  $\alpha$ (CI) n per model = 1980
---|---
GPT-4 | .78 (.76, .81)
GPT-3.5-turbo | .62 (.59, .65)
Davinci-003 | .47 (.45, .50)
Flan-T5-XXL | .45 (.42, .47)
Davinci-002 | .41 (.38, .44)
Command-XL | .32 (.29, .35)
Table 4: Krippendorff’s alpha ($\alpha$) performance of models averaged over
datasets and prompts. N total = 11880.

Further detailed results, as well as results reported as accuracy values can
be found in the appendix.

##  4 Limitations

The presented work has several limitations. Our study aimed to test a wide
variety of combinations of prompts, datasets, and models under budgetary
constraints. We therefore chose to subsample datasets based on a statistical
power analysis. This limits the direct comparison of our results to
evaluations on full benchmark test sets. Upon inspecting results for some of
the academic benchmark datasets generated through crowdsourcing we found that
the quality of a sizable subset of examples was not optimal. One common
pattern we found was that questions and answer choices did not allow for
clearly picking a best answer. More advanced models tend to correctly point
out such problems in their reasoning response and refrain from selecting a
single answer choice. We did not use methods such as self-consistency Wang et
al. (2022) that maximize final accuracy at the expense of practical
interpretability, i.e., we targeted situations in which users expect a single,
high-quality and easily interpretable reasoning chain rather than a collection
of noisy reasoning chains. Results achieved when using prompts in conjunction
with ensemble methods might potentially differ. Our study included state-of-
the-art closed-source models which are undergoing constant change, making
replication and comparisons over time difficult. We partially address this
concern by making all data generated by models at the time of our experiment
openly available. The lack of documentation of closed models also leads to
concerns about contamination of training data with benchmark datasets. While
our comparison of different prompts is not severely impacted, we caution
against strongly interpreting results across different models for this reason.
We noted that Flan-T5 Longpre et al. (2023), which was instruction-finetuned
on the subsets of CommonsenseQA and StrategyQA, outperformed GPT-3.5-turbo on
CommonsenseQA.

##  5 Discussion

Related work. Several related studies evaluated zero-shot prompting
performance. As a notable example, Liévin Liévin et al. (2022) performed a
comparable zero-shot CoT evaluation focused on medical datasets. Earlier work
evaluating multiple models and datasets zero-shot includes commonsense data
Zhou et al. (2020) and the assessment of T0 performance on multiple-choice
tasks Orlanski (2022). HELM Liang et al. (2022) covers a wide range of model
comparisons. Our study added to current knowledge by focusing on finding
simple and versatile chain-of-thought prompting approaches that work across a
spectrum of question-answering datasets and models.

Future work. The current study can be extended by evaluating prompts and
datasets with additional models, particularly the multitude of openly
available LLMs like LLaMa, the Pythia suite, dialog-tuned models like Alpaca
Touvron et al. (2023); Biderman et al. (2023); Taori et al. (2023), StableLM
Stability AI (2023), and OpenAssistant LAION (2023). Finally, user evaluations
of the quality and explanatory utility of reasoning chains generated by
different prompts and models need to be conducted.

##  6 Acknowledgements

We thank the Cohere team for providing custom API access, enabling faster
inference and unrestricted analysis of medical question datasets that were
occasionally flagged by the standard API.

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##  Appendix A Model input templates

This Appendix provides an overview of the text templates and prompt structures
used in our research. The model input is structured as follows:

    {instruction}

    {question}
    {answer_choices}

    {cot_trigger}

Placeholder descriptions:

  * •

{question}: The multiple-choice question that the model is expected to answer.

  * •

{answer_choices}: The options provided for the multiple-choice question.

Two methods of prompting are employed (only one is used at a time):

  * •

{instruction}: Placed before the question and answer choices.

  * •

{cot_trigger}: Placed after the question.

Each prompt is accompanied by its type in parentheses in Table 5.

##  Appendix B Power Calculation Formula

$T\left(P_{c},\alpha_{\text{min}},p\right)=$

$=\\\
2z_{p}^{2}\left(\frac{\left(1+\alpha_{\text{min}}\right)\left(3-\alpha_{\text{min}}\right)}{4\left(1-\alpha_{\text{min}}\right)P_{c}\left(1-P_{c}\right)}-\alpha_{\text{min}}\right)$

Where:

$\displaystyle P_{c}$ $\displaystyle=\text{the probability of value }c$
$\displaystyle\alpha_{\text{min}}$ $\displaystyle=\text{the smallest
}\alpha\text{ for coding to be accepted }$ as reliable $\displaystyle p$
$\displaystyle=\text{level of significance}$ $\displaystyle z_{p}$
$\displaystyle=\text{the standardized }z\text{-statistics at }p$

##  Appendix C Tables

Some prompts work specifically well on certain datasets. The rephrasing prompt
seems to help with the ambiguous questions which we found to be prevalent in
the StrategyQA dataset.

ID |  Prompt Name |  Text
---|---|---
None |  Direct |  "Direct prompting. No specific prompt is used. Just the question and answer choices are the input to the model."
kojima-01 (cot_trigger) |  Kojima |  "Answer: Let’s think step by step."
zhou-01 (cot_trigger) |  Zhou |  "Answer: Let’s work this out in a step by step way to be sure we have the right answer."
zhou-01-ins |  |
(instruction) |  Zhou-instruction |  "Let’s work this out in a step by step way to be sure we have the right answer."
qa-10 (instruction) |  Plan |  "First think step by step - describe your plan for how to get to the right answer, written out in great detail. Then answer the question."
qa-12 (instruction) |  Articulate |  "Carefully read the question & work this out in a step by step way to be sure you have the right answer. Be certain to spell out your thoughts & reasoning so anyone can verify them. Spell out everything in painstaking detail & don’t skip any steps!"
qa-13 (instruction) |  Rephrase |  "Instruction: First let’s rephrase the question to be sure we understood it correctly. Second, let’s work this out step by step by spelling out our thoughts & reasoning so anyone can verify them. Third, make sure we have the right answer."
qa-16 (instruction) |  Elaborate |  "Answer the following question through careful, concise step-by-step reasoning. First, complement the question with helpful knowledge and important additional facts. Second, generate sub-questions that are required to answer the original question, answer them until you can answer the original question."
qa-17 (instruction) |  Converse |  "Create a dialog between a professor and a student. The student asks sub-questions to the question. The professor works them out in a step by step way and makes sure that the student understood how they got to the right answer."
refl-01 (instruction) |  Self-critique |  "Answer the question, then critique the answer. Based on the critique, reconsider the other answer options and give a single final answer."
Table 5:  Used prompts and their corresponding ID and text.  Prompt |  Model avg. accuracy (CI) n per prompt = 1188
---|---
Zhou | .68 (.65, .70)
Articulate | .67 (.64, .70)
Rephrase | .67 (.64, .69)
Elaborate | .66 (.63, .69)
Zhou-instruction | .65 (.63, .68)
Plan | .65 (.62, .68)
Kojima | .64 (.62, .67)
Direct | .64 (.61, .67)
Self-critique | .64 (.61, .67)
Converse | .64 (.61, .66)
Table 6: Accuracy of prompts averaged over datasets. In Table 5, text corresponding to the prompt names can be found. Average taken over all six models. N total = 11880. Dataset |  Accuracy (CI) n per dataset = 1980 | Base Rate
---|---|---
WorldTree v2 | .88 (.86, .89) | .25
CommonsenseQA | .77 (.75, .79) | .2
OpenBookQA | .74 (.72, .76) | .25
StrategyQA | .67 (.65, .69) | .5
MedMCQA | .49 (.46, .51) | .25
MedQA | .38 (.36, .40) | .2
Table 7: Accuracy on datasets averaged over models and prompts. Base rate for random chance, dependent on number of answer choices in datasets. N total = 11880 Model |  Accuracy (CI) n per model = 1980
---|---
GPT-4 | .85 (.83, .86)
GPT-3.5-turbo | .74 (.72, .76)
Davinci-003 | .63 (.61, .65)
Flan-T5-XXL | .61 (.59, .63)
Davinci-002 | .59 (.56, .61)
Command-XL | .52 (.50, .55)
Table 8: Accuracy of models averaged over datasets and prompts. N total =
11880.

model | Command-XL | Flan-T5-XXL | GPT-3.5-turbo | GPT-4 | Davinci-002 | Davinci-003
---|---|---|---|---|---|---
dataset |  |  |  |  |  |
CommonsenseQA | .57 (.50, .64) | .81 (.75, .85) | .70 (.64, .76) | .82 (.76, .87) | .68 (.62, .74) | .68 (.62, .74)
MedQA | .06 (.01, .13) | .02 (.00, .07) | .40 (.32, .47) | .55 (.47, .61) | .09 (.03, .15) | .17 (.11, .24)
MedMCQA | .08 (.01, .14) | .10 (.03, .17) | .51 (.44, .58) | .73 (.67, .79) | .20 (.13, .27) | .21 (.14, .28)
OpenBookQA | .43 (.36, .50) | .69 (.63, .76) | .77 (.71, .83) | .91 (.87, .95) | .45 (.37, .52) | .66 (.59, .72)
StrategyQA | .10 (.00, .21) | .23 (.12, .34) | .44 (.33, .55) | .69 (.61, .76) | .20 (.09, .32) | .22 (.12, .31)
WorldTree v2 | .67 (.61, .73) | .77 (.72, .83) | .89 (.85, .93) | .97 (.95, .99) | .84 (.79, .89) | .84 (.80, .89)

Table 9: Krippendorff’s alpha ($\alpha$) performance of models per dataset
averaged over prompts. The low score of Flan-T5-XXL on MedQA illustrates that
Krippendorff’s alpha ($\alpha$) corrects the accuracy in Table 10 for the base
rate in Table 7.

model | Command-XL | Flan-T5-XXL | GPT-3.5-turbo | GPT-4 | Davinci-002 | Davinci-003
---|---|---|---|---|---|---
dataset |  |  |  |  |  |
CommonsenseQA | .66 (.61, .71) | .85 (.81, .89) | .76 (.71, .81) | .85 (.81, .90) | .75 (.70, .79) | .75 (.70, .80)
MedQA | .27 (.22, .32) | .22 (.17, .26) | .53 (.47, .58) | .65 (.60, .70) | .28 (.23, .33) | .35 (.30, .40)
MedMCQA | .31 (.26, .36) | .35 (.30, .40) | .63 (.58, .69) | .80 (.76, .85) | .41 (.35, .46) | .41 (.36, .47)
OpenBookQA | .58 (.52, .63) | .78 (.73, .82) | .83 (.79, .88) | .93 (.91, .96) | .59 (.54, .65) | .75 (.70, .80)
StrategyQA | .57 (.51, .62) | .62 (.57, .68) | .73 (.68, .79) | .85 (.81, .89) | .63 (.57, .68) | .63 (.58, .69)
WorldTree v2 | .75 (.71, .80) | .83 (.79, .87) | .92 (.89, .95) | .98 (.96, .99) | .88 (.85, .92) | .88 (.85, .92)

Table 10: Accuracy of models per dataset averaged over prompts.

dataset | CommonsenseQA | MedQA | MedMCQA | OpenBookQA | StrategyQA | WorldTree v2
---|---|---|---|---|---|---
prompt |  |  |  |  |  |
Direct | .68 (.60, .76) | .21 (.12, .30) | .28 (.18, .37) | .65 (.56, .73) | .24 (.10, .38) | .84 (.77, .90)
Kojima | .69 (.61, .77) | .22 (.14, .31) | .25 (.16, .35) | .61 (.52, .70) | .46 (.32, .59) | .79 (.72, .86)
Zhou | .72 (.64, .79) | .23 (.14, .32) | .37 (.27, .46) | .74 (.66, .81) | .32 (.19, .44) | .83 (.77, .89)
Plan | .73 (.65, .80) | .19 (.11, .28) | .30 (.21, .40) | .65 (.56, .73) | .27 (.12, .42) | .82 (.75, .88)
Articulate | .72 (.64, .80) | .22 (.14, .31) | .35 (.25, .45) | .67 (.59, .75) | .27 (.13, .40) | .88 (.83, .93)
Rephrase | .75 (.68, .82) | .21 (.13, .29) | .31 (.22, .41) | .61 (.51, .70) | .42 (.30, .55) | .87 (.82, .92)
Elaborate | .68 (.60, .76) | .25 (.17, .34) | .36 (.25, .45) | .64 (.56, .72) | .33 (.20, .47) | .82 (.75, .88)
Converse | .63 (.55, .72) | .20 (.12, .29) | .32 (.23, .41) | .63 (.55, .72) | .30 (.16, .43) | .78 (.71, .85)
Self-critique | .73 (.65, .80) | .19 (.11, .27) | .25 (.16, .34) | .66 (.56, .74) | .23 (.09, .37) | .82 (.75, .88)
Zhou-instruction | .73 (.66, .81) | .19 (.11, .28) | .26 (.17, .36) | .65 (.57, .74) | .28 (.14, .42) | .86 (.80, .92)

Table 11: Krippendorff’s alpha ($\alpha$) performance of prompts per dataset
averaged over models.

dataset | CommonsenseQA | MedQA | MedMCQA | OpenBookQA | StrategyQA | WorldTree v2
---|---|---|---|---|---|---
prompt |  |  |  |  |  |
Direct | .74 (.68, .81) | .38 (.31, .45) | .46 (.39, .53) | .74 (.68, .80) | .64 (.57, .71) | .88 (.84, .93)
Kojima | .75 (.69, .81) | .39 (.32, .46) | .44 (.37, .51) | .71 (.65, .78) | .73 (.67, .80) | .85 (.79, .90)
Zhou | .78 (.72, .84) | .40 (.33, .47) | .54 (.47, .61) | .81 (.75, .87) | .67 (.60, .74) | .87 (.83, .92)
Plan | .78 (.73, .84) | .36 (.30, .43) | .48 (.41, .56) | .74 (.68, .80) | .66 (.59, .73) | .87 (.82, .91)
Articulate | .78 (.72, .84) | .39 (.32, .46) | .52 (.44, .59) | .76 (.70, .82) | .66 (.59, .73) | .91 (.87, .95)
Rephrase | .80 (.75, .86) | .38 (.31, .45) | .49 (.42, .56) | .71 (.64, .77) | .72 (.65, .78) | .91 (.87, .95)
Elaborate | .75 (.68, .81) | .41 (.34, .48) | .53 (.46, .60) | .74 (.67, .80) | .68 (.61, .75) | .87 (.82, .92)
Converse | .71 (.64, .78) | .37 (.30, .44) | .51 (.43, .58) | .73 (.67, .79) | .66 (.59, .73) | .84 (.79, .89)
Self-critique | .79 (.73, .84) | .37 (.30, .43) | .44 (.37, .51) | .75 (.69, .81) | .62 (.55, .69) | .87 (.82, .92)
Zhou-instruction | .79 (.73, .85) | .37 (.30, .44) | .45 (.38, .52) | .74 (.68, .81) | .66 (.59, .73) | .90 (.86, .94)

Table 12: Accuracy of prompts per dataset averaged over models.

model | Command-XL | Flan-T5-XXL | GPT-3.5-turbo | GPT-4 | Davinci-002 | Davinci-003
---|---|---|---|---|---|---
prompt |  |  |  |  |  |
Direct | .26 (.18, .33) | .49 (.41, .58) | .61 (.53, .69) | .71 (.64, .79) | .41 (.31, .50) | .44 (.35, .53)
Kojima | .25 (.16, .34) | .46 (.38, .55) | .66 (.57, .75) | .80 (.73, .87) | .42 (.33, .51) | .45 (.36, .54)
Zhou | .35 (.27, .43) | .44 (.37, .51) | .62 (.53, .71) | .83 (.77, .90) | .53 (.45, .62) | .50 (.41, .59)
Plan | .34 (.25, .42) | .45 (.37, .53) | .61 (.52, .70) | .77 (.71, .84) | .37 (.30, .45) | .46 (.37, .55)
Articulate | .33 (.26, .40) | .50 (.42, .58) | .59 (.49, .68) | .79 (.71, .86) | .44 (.35, .53) | .52 (.43, .60)
Rephrase | .42 (.33, .51) | .46 (.38, .54) | .61 (.52, .70) | .78 (.71, .85) | .44 (.35, .53) | .46 (.37, .55)
Elaborate | .34 (.26, .42) | .42 (.33, .51) | .61 (.51, .70) | .77 (.70, .84) | .51 (.42, .60) | .43 (.35, .51)
Converse | .31 (.22, .40) | .44 (.35, .52) | .58 (.49, .67) | .74 (.66, .81) | .35 (.26, .43) | .46 (.38, .54)
Self-critique | .32 (.26, .39) | .41 (.35, .47) | .58 (.49, .68) | .76 (.69, .84) | .38 (.30, .47) | .48 (.39, .57)
Zhou-instruction | .38 (.30, .46) | .43 (.35, .51) | .64 (.54, .73) | .79 (.72, .86) | .33 (.26, .40) | .49 (.41, .58)

Table 13: Krippendorff’s alpha ($\alpha$) performance of prompts per model
averaged over datasets.

model | Command-XL | Flan-T5-XXL | GPT-3.5-turbo | GPT-4 | Davinci-002 | Davinci-003
---|---|---|---|---|---|---
prompt |  |  |  |  |  |
Direct | .47 (.40, .54) | .62 (.55, .69) | .75 (.69, .81) | .81 (.76, .87) | .59 (.52, .66) | .61 (.54, .68)
Kojima | .45 (.38, .52) | .62 (.55, .69) | .76 (.70, .82) | .86 (.81, .91) | .56 (.49, .63) | .61 (.54, .68)
Zhou | .55 (.48, .62) | .58 (.51, .65) | .75 (.69, .81) | .89 (.84, .93) | .65 (.58, .71) | .66 (.59, .73)
Plan | .53 (.46, .60) | .62 (.55, .69) | .73 (.66, .79) | .84 (.79, .89) | .55 (.48, .62) | .63 (.56, .70)
Articulate | .53 (.46, .60) | .65 (.58, .71) | .74 (.67, .80) | .86 (.81, .91) | .61 (.54, .69) | .65 (.58, .72)
Rephrase | .58 (.51, .65) | .62 (.55, .69) | .73 (.66, .79) | .84 (.79, .89) | .61 (.54, .68) | .63 (.56, .70)
Elaborate | .55 (.48, .62) | .59 (.52, .66) | .73 (.66, .79) | .84 (.79, .89) | .65 (.58, .72) | .60 (.53, .67)
Converse | .52 (.45, .59) | .61 (.55, .68) | .71 (.65, .78) | .81 (.76, .87) | .53 (.46, .61) | .62 (.55, .69)
Self-critique | .51 (.43, .58) | .58 (.51, .65) | .72 (.65, .78) | .84 (.79, .89) | .57 (.50, .64) | .64 (.57, .71)
Zhou-instruction | .56 (.49, .63) | .59 (.52, .66) | .75 (.69, .82) | .85 (.80, .90) | .54 (.47, .61) | .65 (.58, .71)

Table 14: Accuracy of prompts per model averaged over datasets.


Influence of pressure on magnetic phase transitions in the ferromagnetic
superconductor UGe2 \- phenomenological approach

Diana V. Shopova

Institute of Solid State Physics, Bulgarian Academy of Sciences,

1784 Sofia, Bulgaria<EMAIL_ADDRESS>
###### Abstract

We propose a thermodynamic model of free energy expansion up to eighth order
of magnetisation to describe the complex magnetic phase transitions in
ferromagnetic superconductor UGe$2$. The model successfully describes
transitions between ordered phases which take place without changing of
magnetic structure but only of magnetisation which is the case of UGe2 where
two magnetic phases with same structure but with different magnitude occur
with decreasing of temperature. We consider the influence of pressure on
magnetic transitions in the simplest form of including it only in pressure
dependence of Curie temperature for the transition between the paramagnetic
and the magnetic phase with the lower magnetic moment FM1. Our results show
the for pressures lower than some limiting pressure the transition between
low-and high-magnetisation phase remains of crossover type. Above this
limiting pressure this transition changes to real thermodynamic transition of
first order.

Keywords: magnetic phase transition, ferromagnetic superconductor, Landau free
energy

PACS: 74.20.De, 74.20.Rp, 74.40-n, 74.78-w

##  1  Introduction

The uranium-based ferromagnetic superconductor UGe2, [1], [2] is the first
discovered one and there the superconductivity occurs only under pressure in
the pressure interval between $\sim$ 1GPa and critical pressure $P_{c}$ = 1.5
GPa where both superconductivity and magnetic order disappear. The magnetic
phases are two with the same crystal nd magnetic structure but different
magnitude of magnetisation. The high temperature magnetic phase denoted by FM1
occurs by second order phase transition and Curie temperature at $P=0$ is
52.2K. With decreasing of temperature second ferromagnetic phase appears with
larger magnetic moment at temperature of 30 K and at P =0. The magnetic moment
$M$ of FM1 phase experimentally found is approximately 0.9 $\mu_{B}$/U, while
the magnetic moment of FM2 phase is 1.41 $\mu_{B}$/U [3]. Recent summarised
phase diagram with experimental data can be found in [4]. The second order
phase transition to FM1 with the growing of pressure turns into first order at
tricritical point T${}_{CP}\simeq 22$ K,and $P_{CP}\approx 1.42$ GPa .
It is experimentally found that at ambient pressure the transition between FM1
and FM2 ferromagnetic phases is of crossover type and only with increasing
pressure there is a true first order phase transition close to the appearance
of superconductivity with a maximum transition temperature of $\approx 0.7$ K
at P${}_{x}=$ 1.2 GPa. In the pressure interval of superconductivity
existence, the magnetic moment is $1\mu_{B}/U$ . It is established that UGe2
has very strong uniaxial magnetic anisotropy [5], and at some earlier stages
of study, it has been considered that 3D Ising model can well describe its
magnetic properties - see, for example, [6], and for recent review [7]. It is
experimentally found and theoretically justified [8] that the critical
exponents for UGe2 do not belong to any known universality class including 3D
Ising model and anisotropic next nearest neighbor 3D Ising model. One of the
reasons for this lays in the dualism of 5f-electrons in UGe2 where two subsets
of itinerant and localised ones exist. This fact is experimentally established
by positive muon relaxation measurements where the itinerant magnetic moment
at ambient pressure is found to be $\approx$ 0.02$\mu_{B}$ [9]. Later this
dualism of 5f-electrons is confirmed by magnetic, transport and specific heat
experiments, see, for example, [10].
Understanding of magnetic phase transitions from microscopic point of view is
important for understanding the appearance of superconductivity - for review,
see for example [3], but there is now up to now consensus on this problem.
Recently, an estimate is made on the basis of Takahashi’s spin fluctuation
theory about the itinerancy of 5f-electron systems in actinides [11]. The
comparison of this approach with the experimental data shows that UGe2 within
the criterion used in the paper is an intermediate case between strongly
localised and itinerant 5f-electron system. In the paper [4], measurements of
magnetization are performed at high pressure for UGe2 with the aim to make
clear the correlation between the superconductivity and pressure-enhanced
ferromagnetic fluctuations. From the experimental data a conclusion is made
that at lower pressure the ferromagnetic state is more itinerant in FM1
compared to FM2.
Usually microscopic models are focused on the phase transition between FM1 and
FM2 around the critical end point, where superconductivity occurs under
pressure [12]. At pressures $0\leq P<P_{x}$ the nature of FM1 to FM2
transition is considered a crossover, and in this pressure interval there is
small number of systematic measurements. Different theoretical explanations of
this crossover transition exist. In earlier studies performed with Stoner
theory of itinerant electrons, it is proposed that the electronic density of
states has two very closely placed peaks near the low-dimensional anisotropic
Fermi surface, whose topology changes by the appearance of magnetisation [13].
We have proposed in previous papers [14] that phenomenologically the magnetic
transitions in UGe2 fall in the class of isostructural transition, for which
the transitions between ordered phases occur without the change of crystal or
magnetic structure, and only some physical quantity is changed as the
magnitude of magnetisation in UGe2. It was proposed by [15] and later on
generalised see, for example [16], [17] that the description of such
transitions phenomenologically can be done by expanding the free energy up to
eighth order in changing physical quantity. For the case of UGe2 we expand the
free energy up to eight order in magnetisation. The properties of such
expansion at zero pressure are explained and applied to UGe2 in [14], but here
we include in our considerations also the influence of pressure and give
qualitative results as well as some ideas for further generalisation of the
model.

##  2  Analysis of pressure dependent expansion of free energy

UGe2 has orthorhombic crystal structure of ZrGa2-type with space group Cmmm
and highly anisotropic magnetisation of Ising type. This is due to localized
5f-electrons, the contribution of itinerant f-electrons to bulk magnetisation
is small of order $\approx$ 0.02$\mu_{B}$ at ambient pressure as found
experimentally. We denote the highest magnetisation along the easy
magnetisation a-axis by $M_{a}$; it comes from both itinerant and localised
5f-electrons. The perpendicular components of magnetic moment ($M_{b}$,
$M_{c}$) of itinerant 5f-electrons along b- and c- crystal axes, should also
be considered but due to their smallness compared to $M_{a}$ , they will be
omitted at this stage of calculations.
The thermodynamic analysis is made by using the expansion of free energy in
magnetisation up to 8-th order. Such expansion allows the description of so-
called isostructural transitions between two ordered phases in which the
crystal or magnetic structure are not changed , and only some other physical
quantity , for example, magnetisation as in UGe2 undergoes a change. The
properties of the model are explained in detail in previous papers [14] at
ambient pressure. Here we will continue the analysis of free energy expansion,
taking into account the influence of pressure on the magnetic transitions and
thermodynamic quantities.
The free energy density $F\left[M(T,P)\right]/V=g$ with $V$ the volume, in
general form up to 8-th order of $\overrightarrow{M}=(0,0,M_{a})$, is given by
the expression; the subscript $a$ will be further omitted.

$g=aM^{2}+\frac{b}{2}M^{4}+\frac{c}{3}M^{6}+\frac{v}{4}M^{8}.$ (1)

In the above equation, $a=\alpha\left[T-T_{c}(P)\right]$, where $T_{c}(P)$ is
the pressure dependent Curie temperature of second order transition to FM1,
and $\alpha$ is a material parameter. We will also consider that the
coefficient $b=b(P)$ may depend on pressure, which dependence will be
explained later; for the other two coefficients $c,\;v$ at this stage of
analysis we assume that they do not depend on temperature and external
pressure, namely $;c=c(T_{c0},P=0),\;v=v(T_{c0},P=0)$. By $T_{c0}=T_{c}(P=0)$
we denote the Curie temperature at $P=0$ for the transition between
paramagnetic phase to FM1. The sign of $b(P),\;c(P=0)$ may be either positive
or negative, but $v(P=0)>0$ in order to ensure convergence of $g$.
In the expression (1) there are too many unknown parameters, which in
principle may be derived from experiment and/or respective microscopic
calculations in zero external magnetic field. We will not make such
quantitative comparison, rather we will give the general qualitative picture
of magnetic transitions in UGe2.
To reduce the number of unknown parameters, as well as to make free energy
density $g$ dimensionless we redefine the order parameter $M$ by introducing:

$m=v^{1/8}M.$ (2)

The free energy density - Eq. (1) in dimensionless form reads:

$f=rm^{2}+\frac{u}{4}m^{4}+\frac{w}{3}m^{6}+\frac{1}{4}m^{8};$ (3)

and the coefficients in Eq.(3) are related to the initial ones in the
following way:

$\displaystyle u$ $\displaystyle=$ $\displaystyle\frac{b}{v^{1/2}}$
$\displaystyle w$ $\displaystyle=$ $\displaystyle\frac{c}{v^{3/4}}$

The reduced temperature is $r=\beta\left[T/T_{c0}-T_{c}(P)/T_{c0}\right]$ with
$\beta=\alpha T_{c0}/v^{1/4}$ .
Here we make some assumptions on the dependence of Curie temperature on
pressure. In the general case such dependence may be written as
$(P-P_{0})^{1/n}$, see, [6] and the arguments given there. The choice of
$P_{c}$ is somewhat arbitrary as for UGe2 there are two special critical
pressures: $P_{x}\sim 1.20$ GPa where the transition from FM1 to FM2 is of
first order, and $P_{c}\sim 1.5$ GPa, at which ferromagnetic phase - FM1
disappears through first order phase transition. Here we choose $P_{0}=P_{c}$
as far as we work at not very high pressures for the description of magnetic
phase transitions only and especially the transition from FM1 to FM2 with the
increase of pressure. Then $r$ becomes:

$r=\gamma(t-1+p)$ (4)

where $t=T/T_{c0}$ and $p=P/P_{c}$; the parameter $\gamma$ will be determined
in the process of calculations. The equation of state $(df/dm)$ :

$2m\left[\gamma(t-1+p)+um^{2}+wm^{4}+m^{6}\right]=0$ (5)

has an obvious solution for disordered (paramagnetic) phase $m=0$.
The stability condition is given by inequality:

$\frac{d^{2}f}{dm^{2}}=2\left[\gamma(t-1+p)+3um^{2}+5wm^{4}+7m^{6}\right]\geq
0$ (6)

and there are several methods to resolve this condition in analytical form:
see for example [16]. This may be also done numerically by direct substitution
of solutions of Eq. (5) in the above equation and analysing its positiveness.
If more than one solution is stable also a comparison between free energies of
respective solutions of Eq. (5) should be made in order to find out which one
in what domain of reduced temperature is an absolute minimum.
The equation of state (5) becomes standard 3-rd order algebraic equation with
analytical solutions, if we make the substitution $x=m^{2}\geq 0$, [18]:

$\gamma(t-1+p)+ux+wx^{2}+x^{3}=0.$ (7)

The solving of Eq.(7) begins with the analyses of quantity $Q$ given below,
which determines the number of its real solutions and the boundaries of their
existence:

$Q=\frac{1}{4}\left[\gamma(t-1+p)\right]^{2}+\frac{2w}{3}\left(\frac{2w^{2}}{9}-u\right)\gamma(t-1+p)+\frac{u^{2}}{27}\left(u-\frac{w^{2}}{4}\right)$
(8)

$Q(t)$ is a quadratic equation with respect to the reduced temperature, and we
can find those values of reduced temperature $t$ as function of parameters
$u,w$ by which we can easily determine the phase boundaries and the order of
magnetic transitions as well as the region of real non-negative solutions for
magnetisation as function of parameters $u$ and $w$.
The solutions of (8)with respect to t, are :

$t_{1,2}=\frac{1}{\gamma}\left[-\gamma
p+\frac{uw}{3}+\gamma-\frac{2}{27}w^{3}\pm\frac{2}{27}\sqrt{(w^{2}-3u)^{3}}\right]$
(9)

It is obvious that $t_{1,2}$ are real for $w^{2}\geq 3u$ and $t_{0}$ for
$u=w^{2}/3u$ is a special point, at which

$t_{1}=t_{2}=\frac{w^{3}}{27\gamma}-p+1$

. The calculations show that such point appears only for $w<0$ and $u\geq 0$.
At this point all three solutions of Eq.(7) are equal and real. As shown
before [14], this can be interpreted as the crossover temperature $t_{0cr}$
for the transition between FM1 and FM2 at zero pressure, namely in new
notations $t_{0cr}=t_{1}(p=0)=t_{2}(p=0)=w^{3}/27\gamma+1$ . The expression
for reduced temperature $t_{0cr}$ of crossover transition at $p=0$ can be used
to find the relation between $\gamma$ and $c$ with the help of respective
experimental values of $T_{c}=$52 K and $T_{x}=$ 30 K at $P=0$. The numerical
result is $\gamma=-26/297w^{3}$. We illustrate the dependence of $t_{1,2}$ on
parameter $u$ for fixed $w<0$ in Fig. 1.

Figure 1: Solutions of (8) as function of parameter $u$ for $w$=-2.252

The grey lines give the variation of $t_{1,2}$ with parameter $u$ at $p=0$. We
are mainly interested in the interval of $u$-values between the two vertical
green lines, as there a second order phase transition from paramagnetic phase
is possible, given by horizontal thick dash line, denoting $t_{cc0}$ Along the
thick green line, drawn at $u=w^{2}/3$ the transition at $t_{c0}$ to FM1 is of
second order and at point B a crossover occurs to FM2 phase with a jump of
magnetisation.For $t_{2}<t<t_{1}$ , all roots of (7) are real and this is the
region of first order phase transitions. For $t>t_{1}$ there is only one real
solution and along thick green line above point B, this is the region of phase
FM1. Below point B, $t<t_{2}$ again only one root is real and bigger in
magnitude - this is the region of FM2. Point C ($u=w^{2}/4$) is tricritical
point, at which the transition from disordered phase changes from first to
second order. Further we will show how this picture changes with inclusion of
pressure.

##  3  Results and discussion

As experiment shows, see for example [4] and the papers cited therein, the
transition between FM1 and FM2 persists to be of crossover type up to high
pressures of order $P_{x}\sim$ 1.2 GPa, where a first order phase transition
occurs and FM2 disappears. On the other hand the positive muon spin rotation
measurements in paper [19] give that the transition between FM1 and FM2
becomes a real thermodynamic phase transition at lower pressures $\sim$ 1 GPa.
In recent paper [20], a two-fluid model for magnetic transitions in UGe2 is
proposed on the basis of Hall resistivity measurement, where the existence of
first order transition in FM2 is questioned.
As the conclusions from different experiments are somewhat controversial, here
we accept that the transition to FM2 is of crossover-type up to some limiting
pressure followed by real thermodynamic transition probably of first order
through which FM2 disappears at $\sim$ 1.2 GPa. The free energy expansion we
use is a general phenomenological approach to depict transitions between two
ordered phases without a change of their structure, and even in its simplified
version with respect to influence of pressure on such transitions, it gives
indications that the above picture may be realised.
For crossover value of $u$, namely, $u=w^{2}/3$, the equation of state Eq.(7)
has solutions for $x=M^{2}$ given by simple formula:

$\displaystyle x_{1}$ $\displaystyle=$
$\displaystyle\left[\frac{w^{3}}{27}-\gamma(t-1+p)\right]^{\frac{1}{3}}-\frac{c}{3}$
(10) $\displaystyle x_{2,3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\frac{w^{3}}{27}-\gamma(t-1+p)\right]^{\frac{1}{3}}-\frac{c}{3}\pm
i\frac{\sqrt{3}}{2}\left[\frac{w^{3}}{27}-\gamma(t-1+p)\right]$ (11)

The calculations give that the smallest root $x_{3}$ describes the
magnetisation of FM1 phase with a second order phase transition at
$t_{c}=1-p$; the jump of magnetisation at crossover temperature $t_{0cr}$ is
given by the solution $x_{1}$ and at the crossover temperature the
magnetisation $m^{2}=x_{1}=x_{2}=x_{3}=-c/3$. We should mention here that the
expression for crossover temperature is not changed, when dependence on
pressure is included only in $t_{c}(p)$, and the other coefficients in the
free energy remain independent of $p$.
In figure below we illustrate this behaviour of $m^{2}$ for pressure $p=0.1$,
which corresponds to $P$ = 0.15 GPa.

Figure 2: Function $M^{2}=x$ of reduced temperature $t$ for pressure $p=0.1$,
i.e., $P=$ 0.15 GPa. Open circle denotes the crossover transition between FM1
- lower curve to FM2 - upper curve. The meaning of green lines is explained in
the text

The green lines there show the values of $x=M^{2}$ which outline the stability
of solutions of equation of state. They are found using the stability
condition Eq.(6), together with the equation of state Eq.(5). The result is
simple quadratic equation in $x$:

$\gamma(t-1+p)+\frac{2}{3}w^{2}x+wx^{2}\leq 0.$

The roots of this stability condition are depicted in Fig.2 by green lines,
the solutions for $M^{2}=x$ are stable outside the green lines, note that
$w<0$.
In Fig. 3 the behaviour of magnetisation with temperature for different
pressures is presented with the aim to show and discuss the limits of approach
we apply. It is seen from figure that both temperature of second order
transition to FM1, and the temperature of crossover transition decrease with
pressure increasing. The region of FM1 existence is also increasing with
respect of FM2 phase. But taking into account that only Curie temperature
depends on pressure in linear form immediately leads to some discrepancies, as
far as the magnetisation at the crossover temperature remains the same in
magnitude as the magnetisation at zero pressure. This is so because we fix the
parameter $u$, see Eq.(5) to its value $u=w^{2}/3$ at $p=0$.

Figure 3: Function $M^{2}=x$ of reduced temperature $t$ for different values
of pressure: black line is at $p=0$; the grey line is at $p=0.1$, i.e., $P=$
0.15 GPa; the red line is for $p=0.3$, i.e.,$P=$ 0.45 GPa; and the green line
is ar $p=0.45$,i.e., $P=$ 0.675 GPa; Solid circles of respective color denote
the crossover transition.

We suppose that in order to give more realistic picture of FM1 to FM2
transitions and especially to properly describe the behaviour of magnetisation
with temperature and pressure, we should assume that $u=u(p)$. The problem
with particular form of such dependence is not trivial, as we should take in
consideration that the transition from FM1 to FM2 remains a crossover up to
pressures $\sim 0.95-1$ GPa, and only for bigger pressures a first order phase
transition occurs with disappearing of FM2.
Even in our simplified assumption there are indications that such
transformation is taking place. This is illustrated in Fig. 4. As we pointed
in Section 2, the number of real solutions for magnetisation of equation of
state Eq.(5) are determined by the quantity $Q$, Eq.(8). Solving it with
respect to reduced temperature shows also the limits of existence and gives a
picture of possible phase transitions for free energy Eq.(3).

Figure 4: The dependence of solutions $t_{1,2}$ of Eq.(8) on the parameter $u$
for different pressures. Blue lines are drawn for $p=0.3$,i.e.,$P=$ 0.45 GPa.
Pink lines are given at limiting pressure $p_{l}=0.57$, $P=$ 0.86 GPa, and
light blue lines are drawn at $p_{x}=0.8$, i.e.,$P_{x}=$ 1.2 GPa. The
horizontal black lines denote the reduced Curie temperature $t_{c}=1-p$ for
the respective values of p

In Fig.4 we have given solutions of $Q$ with respect to $t$ as function of
parameter $u$ for different pressures. The crossover temperature $t_{cr}$ is
determined by the equality $t_{1}=t_{2}$, see Eq.(9) and the expression below
it. It is seen from the figure that $t_{cr}$ is decreasing with pressure
increase and above some limiting pressure, which we denote by $p_{l}$, the
point, where $t_{1}=t_{2}$ becomes negative and not physical, as $t$ should be
$\geq 0$. This demonstrates that for pressures $p_{l}<p<p_{x}$, our
description is too simplified.
We denote the limiting pressure as $p_{l}$, given by

$p_{l}=1+\frac{w^{3}}{27\gamma}.$

In Fig 4, $t_{l}^{1,2}$ for $p=p_{l}$ as function of $u$ are described by
violet lines and the crossover temperature lies on u-axis; it is denoted by
solid black circle. The circle on the upper violet line marks the crossing of
$t_{1}$ with line for second order transition $1-p_{l}$. The grey vertical
lines have the same meaning as the green lines in Fig.1. It is obvious that at
$p=p_{l}$ the transition from FM1 to FM2 remains of crossover type.
For pressures $p_{l}<p<p_{x}$, the transition cannot be of crossover type any
more. This is shown by light blue lines, depicted at $p=p_{x}$. The crossover
temperature within our simplified approach is negative, shown by solid black
diamond in Fig.4. The values of $u$ for which second order phase transition to
FM1 are limited between the crossing of upper blue line and u-axis and the
solid grey line. Experimentally, the first order transition between
paramagnetic phase and FM1 occurs at pressure $p\sim 0.95$, corresponding to
$P\sim$ 1.42 GPa and parameter $u$ should be such as to give such opportunity.
Fig. 4 shows that including the dependence of $u$ on pressure is necessary for
the proper description of crossover transformation to real first order phase
transition. Preliminary calculations show that if $u(P)$ is expanded around
the value of $u_{0}=w^{2}/3$ at $p=0$ and only linear terms in pressure are
preserved, the crossover temperature has proper behaviour, namely, it slightly
decreases with increasing of pressure, but another problem appears related to
very fast decrease of Curie temperature with pressure and the discrepancy with
the respective experimental data, see [4], is of order $6-10$ K. This
discrepancy grows with the pressure increase. Moreover, such linear dependence
of $u(p)$ on pressure is adequate for the description of transition from
paramagnetic phase to FM1, followed by crossover transition to FM2 only for
pressures $p<p_{l}$.

##  4  Concluding remarks

In this paper we apply the phenomenological approach to UGe2 for the
description of magnetic phase transitions using the expansion of free energy
up to eight order in magnetisation and taking into account in explicit form
the dependence on pressure. At this stage we include such dependence only
through the linear dependence of Curie temperature on P. Even in such
simplified form, the results show that for pressures smaller that some
limiting pressure, defined in the previous section, the transition of FM1 to
FM2 remains of crossover type as experimentally found.
As pointed above in order to give more precise picture of magnetic transitions
in UGe2 for our model it will be necessary to consider the coefficient in
front of fourth order term $u$ in the free energy expansion, Eq. (3), as
pressure dependent. The concrete form of such pressure dependence should be
derived taking into account the available theoretical and experimental data.
As it is generally accepted that the spin fluctuations of itinerant
f-electrons are responsible for occurrence of superconductivity of p-type, it
will be of great interest also to consider the influence of itinerant
f-electrons on magnetic transitions within this phenomenological approach we
use. The above issues will be the subject of further studies.

## Acknowledgements

This work is supported by Grant $KP-06-N38/6$ of the Bulgarian National
Science Fund.

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